Investigations of mixed-valence and open-shell transition ... · Investigations of Mixed-Valence...

Investigations of Mixed-Valence and Open-Shell Transition-Metal Complexes Employing Modern

Density Functional Methods

vorgelegt von

Diplom-Chemiker

Matthias Parthey

aus Bad Soden-Salmünster

Von der Fakultät II – Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

– Dr. rer. nat. –

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. rer. nat. Arne Thomas

Berichter: Prof. Dr. rer. nat. Martin Kaupp

Berichter: Prof. Dr. phil. nat. Wolfgang Kaim

Tag der wissenschaftlichen Aussprache: 16.04.2014

Berlin 2014

D83

“No one ever achieved greatness by playing it safe“

Harry J. Gray

Für Hami

“Wir werden selber reich, wenn wir Freude schenken“ Brigitte Theilen

ACKNOWLEDGEMENT

“It is personalities, not principles, that move the age.”

Oscar Wilde

First, I want to thank Prof. Martin Kaupp. Martin, words can barely express how grateful I am for the three years of my PhD and the ideas we created and developed together during that time. But not only your support in science but also all the socializing activities like football, skiing, and of course our TV appearance made my PhD an incredibly enjoyable time.

I thank Manuel Renz, for introducing me to quantum chemistry, always giving extremely helpful advice, and sticking with me (and my Mac) through more than two years in the office; Anja Greif, my light on gray office days with her dog Sari being my kryptonite, when it comes to working efficiently; Johannes Schraut, amazing football mate, both active and watching; Florian Meier, for being too nice for this world; Hilke Bahmann, coding and social organizing queen; Martin Enke, football and Facebook expert; Toni Maier for filling Manuel’s spot; Kolja Theilacker, Robert Müller, Vladimir Pelmentschikov, Sebastian Gohr, Sascha Klawohn, and the whole Kaupp group for always helping and supporting me.

“I left home, but there is one thing that I still know: it’s always summer in my heart and in my soul!”

Ryan Key of Yellowcard in “Always Summer”

From Durham University, now partly University of Western Australia at Perth, I want to thank the P. J. Low group. Most of all of course Prof. Paul J. Low for all the great work we did together and even more the fun we had while doing it. Sorry for drinking so much coffee and stealing your words for this acknowledgement, Paul! Thanks to my lab mates: Josef Gluyas, the synthetic grandmaster and my English proofreader; Sören Bock, best football and dart partner ever and worst German DJ; “real” thanks to Sam Eaves; Kevin Vincent for the fresh air breaks; Santiago Marquez-Gonzales for his positive attitude; Ross Davidson, the Doughnut provider; Campbell Mackenzie, Marie-Christine Oerthel, and the rest of the group. I also want to thank Prof. Dave J. Tozer and Dr. Mark A. Fox for the quantum-chemical discussions. In this context I also want to thank Prof. D. J. Tozer, Prof. Trygve Helgaker, Dr. Enrico Benassi, and Dr. Christoph Jacob for their excellent lectures, talks, and/or discussion about general concepts and special issues of DFT and quantum-chemistry.

Representatively for all people at the TU Berlin, who supported me, I want to thank: Heidi Grauel and Nadine Rechenberger for helping with the enormous amount of formal work,

especially the countless forms for business trips, and the organization of group activities and the homepage; Prof. Christian Limberg, my second supervisor; Dr. Jean-Philippe Lonjaret, the head and heart of the BIG-NSE graduate school; Dr. Norbert Paschedag and Sven Grottke for their IT support; the Schoen group, especially Michael Melle, Tillman Stieger, and Marco Mazza, for being valuable targets for scientific slamming. The same holds true for Jean-Christophe Tremblay from the FU Berlin.

“They say ‘You don’t grow up, you just grow old’, it seems to say I haven’t done both, I made mistakes, I know, I know, but here I am alive!”

Ryan Key of Yellowcard in “Here I Am Alive”

I also want to thank my family: My parents, Almuth and Roland, for their endless support, life advice, and that they never pushed me towards any direction; my sister Johanna, my role model when it comes to being persistent, strong, and faithful and thus succeeding in the end; my brother Christian for always helping me with physics, being even more Apple addicted, and for losing the clash of science by working for a chemical company; my grandma Erika “Hami” Goralewski, Godehard and Siegrid Goralewski, my uncle and aunt, and my “uncle” Heinz Löken for their support. I also thank Werner and Christa Parthey, Erika and Josef Jöckel, and Ulrike Selig-Parthey.

“That’s what friends are for. They help you to be more of who you are.”

Christopher Robin (Disney character)

There are a lot of friends I met during different stages of my life, who mainly made my free time enjoyable, every place feel like home, and prevented me from getting to caught up in research, but also gave advice in scientific matters, and thus I want to thank:

From Hessen: Augustin Danciu, for broing and awesoming all over the place; Patrick Noll, best wing man (= Goose) ever, proud to be your best man; Sebastian Dietz, oldest friend and like wine our friendship improves with age; Jana Noll, for always being a great observer and of course the famous carnival lasagna; Markus Auhl, hessian philosopher; Natascha Klumpp, forever Troy; Christopher Hämel, doubles partner for life; Andreas Möller, for all the travels and career advice; Theresa and Horst Schmidt, Elke Thesenvitz, Susan Simon, Nina Wallenta, Natalie Alt, Antonia Spielmann, Thomas Hummel, Philipp Roth, Steffen Pfeifer, Till Bergen, and Uta Krammenschneider.

From Würzburg: Frank Brunecker, Steffen Kalinna, Marius Silaghi, Klaus Dück, Johannes Landmann, Alexander Mertsch, and Martin Kess, for making even the hard times enjoyable during my studies in Würzburg. I hope, our Christmas market tradition will continue. A

special thanks to Jost Henkel for following or inviting me to all different spots like London, Durham, Berlin, Hannover, … (hopefully this list will continue)

From Berlin: My BIG-NSE (Berlin International Graduate School of Natural Science and Engineering) mates Fanni Daruny Sypaseuth, Laura Vieweg, Moritz Baar, Daniel Gallego, Patrick Littlewood, Genwen Tan, Hong Nhan Nong, Jiao Linyu, Setareh Sadjadi, Xunhua Zhao, Heiner Schwarz, Swantje Wiebalck, Xenia Erler, Subhamoy Bhattacharya, María Gracia Colmenares, Inéz Monte Pérez, Laura Pardo Pérez, and all the other nice people from the school. Jakub Stejskal, Igor Anatzki, Marcus Beyerlein, Andreas Podlasly, André Zacher, Benjamin Mietling, Sven Vangermain, Kristen Van Dernoot, and the rest from the Grün-Gold Pankow tennis team.

From Durham: Matt Perks, César Segura, Charlie Rozier, Ash Routen, James Koranyi, and the Staff football team. Thanks to the team of the New Inn.

I want to thank Reinhold Stein, my former chemistry teacher, for introducing me to chemistry and almost more importantly to Top Gun “It takes a lot more than just fancy flying”.

For financial support and the opportunity to work in an exceptional research environment I want to thank the “Unifying Concepts of Catalysis” (UniCat) excellence cluster and the German Academic Exchange Service (DAAD).

“I hope you live a life you are proud of; if you are not, I hope you find the strength to start all over again”

F. Scott Fitzgerald

ABSTRACT I

ABSTRACT

Mixed-valence multinuclear transition-metal complexes have been in the focus of research since the discovery of the Creutz-Taube ion and related systems in the 1960s. The search for functional (opto-) electronic materials on the molecular scale, such as molecular wires and transistors, has added to the momentum of the field. Successful applications of organic mixed-valence systems as charge carriers - for example in organic light-emitting diodes or dye-sensitized solar cells - are known. Mixed-valence transition-metal complexes are even more versatile in their electronic properties due to the availability of d-orbitals. Hence they are appealing targets for the design of functional materials and additionally of central importance in electron transfer processes in nature - e.g. in metalloenzymes - and in catalysis.

A central question in all of these fields is that of the localization of charge on a given redox center (end-cap or bridging ligand) versus delocalization over the molecular framework. In the important model case of two redox centers linked by a bridging ligand, the description of electronic structure is usually made within the Robin-Day scheme, which is based on the extent of electronic coupling. The investigation of the electronic structure is complicated by the lack of information directly extractable from experimental data. Especially challenging is the presence of overlapping absorption bands in UV-vis-NIR spectra and the speculated existence of different thermally accessible conformers. Hence a reliable investigation of the extent of electronic coupling is only possible via combined experimental and quantum-chemical studies. The failure of Hartree-Fock and Density Functional Theory - the methods feasible for the system-size of typical mixed-valence systems - in describing the charge localization/delocalization behavior has been overcome in the well balanced global hybrid functional BLYP35, which is employed in combination with solvent models.

In this thesis it is shown that the BLYP35/solvent model combination is furthermore capable of describing optoelectronic properties of mixed-valence transition-metal complexes. Even the challenging charge-transfer excitations are reliably reproduced. In addition the long-standing question to which extent the conformation determines the electronic and spectroscopic properties of mixed-valence systems is investigated. Calculations yield rotational motion as explanation for the optical properties of polyynediyl complexes. This prediction is experimentally proven by the Low group. For diethynylphenyl-bridged ruthenium complexes the computations demonstrate that conformational motion may even average to some extent localized and delocalized electronic and molecular structures. These findings lead beyond the traditionally more one-dimensional understanding of the Robin-Day scheme. Additionally the performance of local and range-separated hybrid functionals in describing isotropic transition metal hyperfine coupling constants is validated.

ZUSAMMENFASSUNG II

ZUSAMMENFASSUNG

Gemischtvalente, mehrkernige Übergangsmetallkomplexe sind fester Bestandteil naturwissenschaftlicher Forschung seit der Entdeckung des Creutz-Taube Ions und verwandter Systeme in den 60ern. Auf Grund des Potentials diese als molekulare (opto-) elektronische Bauteile - z.B. als molekulare Drähte oder Transistoren - einzusetzen, rückten diese Untersuchungen weiter in den Fokus. Organische gemischtvalente Systeme werden erfolgreich als Ladungstransportmaterialien - z.B. in organischen Leuchtdioden oder Farbstoffsolarzellen - eingesetzt. Durch die Verfügbarkeit von d-Orbitalen bieten gemischtvalente Übergangsmetallkomplexe noch flexiblere elektronische Eigenschaften. Daher stellen sie attraktive Ziele für die Entwicklung funktioneller Materialien dar. Zudem sind sie von zentraler Bedeutung in Elektrontransferreaktionen in der Natur - z.B. in Metallenzymen - und in der Katalyse.

Für alle diese Anwendungen ist vor allem entscheidend, ob die Ladung lokalisiert an einem Redoxzentrum oder delokalisiert vorliegt. Das typische Musterbeispiel eines gemischtvalenten Systems besteht aus zwei Redoxzentren, die über einen Brückenliganden verknüpft sind. In diesem Fall wird das System mit Hilfe des Robin-Day Schemas beschrieben, welches auf der Stärke der elektronischen Kopplung beruht. Die nötigen Informationen können nicht oder nur selten direkt aus experimentellen Daten gewonnen werden. Besonders überlagernde Absorptionsbanden in UV-vis-NIR Spektren sowie der (vermutete) Einfluss verschiedener populierter Konformere erschweren die Analyse. Daher können nur Studien, bei denen experimentelle und quantenchemische Methoden kombiniert werden, Aufschluss über das Ausmaß der elektronischen Kopplung geben. Die quantenchemische Beschreibung gemischtvalenter Systeme beruht auf einem kürzlich entwickelten Protokoll, das geschickt die beiden Methoden Hartree-Fock und Dichtefunktionaltheorie kombiniert. Das Protokoll basiert auf dem BLYP35 Funktional, das üblicherweise in Kombination mit Lösemittelmodellen angewendet wird.

In dieser Arbeit wird die Übertragbarkeit dieses Protokolls auf die optoelektronischen Eigenschaften gemischtvalenter Übergangsmetallkomplexe demonstriert. Sogar die anspruchsvolle Beschreibung von Ladungstransferanregungen ist möglich. Zudem kann der Einfluss verschiedener Konformere auf die elektronische Struktur und experimentelle Spektren untersucht werden. Die quantenchemische Vorhersage, dass die optischen Eigenschaften butdiinyl-verbrückter Komplexe auf Rotation zurückzuführen sind, konnte experimentell von der Arbeitsgruppe Low bewiesen werden. Rechnungen zeigen, dass es für diethinphenyl-verbrückte Rutheniumkomplexe von der Konformation abhängt, ob Ladungslokalisierung oder -delokalisierung vorliegt. Dieser Sachverhalt ist nicht mit dem klassischen eindimensionalen Robin-Day Bild zu vereinbaren. Lokale und range-separated Hybridfunktionale wurden zudem für die Beschreibung von Übergangsmetallhyperfein-kopplungskonstanten validiert.

LIST OF PUBLICATIONS III

LIST OF PUBLICATIONS

Journal Articles

[8] Matthias Parthey and Martin Kaupp, “Quantum-chemical insights into mixed-valence systems: within and beyond the Robin/Day scheme”, Chem. Soc. Rev. 2014, DOI: 10.1039/c3cs60481k.

[7] Kevin B. Vincent, Matthias Parthey, Dmitry S. Yufit, Martin Kaupp, and Paul J. Low, “Synthesis and redox properties of mono-, di- and tri-metallic platinum-ethynyl complexes based on the trans-Pt(C6H4N{C6H4OCH3-4}2)(C≡CR)(PPh3)2 motif ”, Polyhedron, 2014, DOI: 10.1016/j.poly.2014.04.035.

[6] Matthias Parthey, Josef B.G. Gluyas, Mark A. Fox, Paul J. Low, and Martin Kaupp, “Mixed-Valence Ruthenium Complexes Rotating Through a Conformational Robin-Day Continuum”, Chem. Eur. J. 2014, DOI: 10.1002/chem.201304947.

[5] Matthias Parthey, Kevin B. Vincent, Manuel Renz, Phil A. Schauer, Dmitry S. Yufit, Judith A.K. Howard, Martin Kaupp, and Paul J. Low, “A Combined Computational and Spectroelectrochemical Study of Platinum Bridged Bis-Triarylamine Systems”, Inorg. Chem. 2014, 53, 1544-1554.

[4] Kevin B. Vincent, Qiang Zeng, Matthias Parthey, Dmitry S. Yufit, Judith A.K. Howard, František Hartl, Martin Kaupp, and Paul J. Low, ”The Syntheses, Spectro-Electrochemical Studies, Molecular and Electronic Structures of Ferrocenyl Ene-diynes”, Organometallics 2013, 32, 6022-6032.

[3] Matthias Parthey, Josef B.G. Gluyas, Phil A. Schauer, Dmitry S. Yufit, Judith A.K. Howard, Martin Kaupp, and Paul J. Low, “Refining the Interpretation of NIR Band Shapes in a Polyynediyl Molecular Wire”, Chem. Eur. J. 2013, 19, 9780-9784.

[2] Sören Bock, Samantha G. Eaves, Matthias Parthey, Martin Kaupp, Boris Le Guennic, Jean-François Halet, Dmitry S. Yufit, Judith A.K. Howard, and Paul J. Low, “The preparation, characterisation and electronic structures of 2,4-pentadiynylnitrile (cyanobutadiynyl) complexes“ Dalton Trans. 2013, 42, 4240-4243.

[1] Martin Kaupp, Manuel Renz, Matthias Parthey, Matthias Stolte, Frank Würthner, Christoph Lambert, “Computational and spectroscopic studies of organic mixed-valence compounds: where is the charge?” Phys. Chem. Chem. Phys. 2011, 13, 16973-16986.

LIST OF PUBLICATIONS IV

Talks and Posters

[8] Talk: “Mixed-Valence Transition-metal complexes: Essential in Catalysis, Challenging for Quantum Chemistry”; BIG-NSE Evaluation Seminar, Berlin (DE), May 2013.

[7] Talk: “Rotamers of Mixed-Valence Complexes: Spinning Classes for a Better (Band) Shape”; Workshop Modern Methods in Quantum Chemistry in Mariapfarr (A), February 2013.

[6] Talk: “Mixed-Valence Complexes and Quantum Chemistry – Insights and Challenges”; Group Seminar P.J. Low Group in Durham (UK), October 2012.

[5] Talk: “Electron Transfer Processes: Crucial for Catalysis, Challenging for Quantum Chemistry”; 2. Berliner Chemie Symposium, Berlin (DE), April 2012.

[4] Talk: “Investigations of Catalytically Active Open-Shell Systems Using Modern Density Functional Methods”; BIG-NSE Workshop, Berlin (DE), January 2012.

[3] Poster: “Towards Improved Hyperfine Couplings for Transition-Metal Nuclei with Improved Density Functionals: The Core-Shell vs. Valence-Shell Spin-Polarization Dilemma”; Symposium for Theoretical Chemistry in Sursee (CH), August 2011.

[2] Talk: “Quantum Chemical Studies of Mixed-Valent Transition-Metal Complexes”; Joint Group Seminar with Schwarz Group, Berlin (DE), April 2011.

[1] Poster: “A Reliable Quantum-Chemical Protocol for Calculating Mixed-Valence Compounds”; Chem-SyStM in Würzburg (DE), December 2010.

CONTENTS ABSTRACT .......................................................................................................................................... I

ZUSAMMENFASSUNG ................................................................................................................... II

LIST OF PUBLICATIONS .............................................................................................................. III

LIST OF ABBREVIATIONS ........................................................................................................ VII

INTRODUCTION ............................................................................................................................... 1 Mixed-Valence Systems in Biocatalysis ...................................................................................... 2 Mixed-Valence Systems in Molecular Electronics ...................................................................... 3 Quantum-Chemical Investigations of Mixed-Valence Systems .................................................. 4

1 THEORETICAL BACKGROUND ................................................................................................ 5 1.1 Mixed-Valence Systems ............................................................................................................ 5

Robin-Day Classes ....................................................................................................................... 5 Marcus-Hush and Mulliken-Hush Theory ................................................................................... 6 Experimental Classification ....................................................................................................... 10

1.2 Density Functional Theory ..................................................................................................... 13 Principles .................................................................................................................................... 13 Exchange-Correlation Functionals ............................................................................................. 18 Time-Dependent Density Functional Theory ............................................................................. 24 Spin in Density Functional Theory ............................................................................................ 27

1.3 Quantum-Chemical Description of MV Systems ................................................................. 30 1.4 Computational Details ............................................................................................................ 34

2 THE COMPUTATIONAL STATE OF THE ART ................................................................... 35

3 APPLICATIONS TO MV TRANSITION-METAL COMPLEXES ........................................ 39 3.1 Platinum Bridged Bis-Triarylamine Complexes .................................................................. 39

Introduction ................................................................................................................................ 39 Results and Discussion ............................................................................................................... 44 Conclusions ................................................................................................................................ 50

3.2 Ferrocenyl Ene-diynes ............................................................................................................ 53 Introduction ................................................................................................................................ 53 Results and Discussion ............................................................................................................... 54 Conclusions ................................................................................................................................ 58

4 SPINNING CLASSES: THE IMPORTANCE OF ROTAMERS ............................................. 59 4.1 The Interpretation of NIR Band Shapes in Polyynediyl Molecular Wires ....................... 61

Introduction ................................................................................................................................ 61 The Class III Ruthenium Complex ............................................................................................. 63 The Class III Osmium Complex ................................................................................................. 70 The Class II Molybdenum Complex .......................................................................................... 72 The Class III Rhenium Complex ................................................................................................ 76

4.2 Complexes Rotating Through a Conformational Robin-Day Continuum ........................ 79

Introduction ................................................................................................................................. 79 Results and Discussion ................................................................................................................ 84 Conclusions ............................................................................................................................... 101

5 THE CORE-SHELL VS. VALENCE-SHELL SPIN POLARIZATION DILEMMA ........... 103 Introduction ............................................................................................................................... 103 Computational Details ............................................................................................................... 104 Results and Discussion .............................................................................................................. 106 Conclusions and Outlook .......................................................................................................... 109

6 GENERAL CONCLUSIONS AND OUTLOOK ....................................................................... 111

REFERENCES ................................................................................................................................ 115

APPENDIX ....................................................................................................................................... 128

LIST OF ABBREVIATIONS VII

LIST OF ABBREVIATIONS

AM1 Austin model 1

a.u. atomic units

CASSCF complete active space self-consistent field

CASPT2 complete active space perturbation theory to second order

CC coupled cluster (theory)

CDFT constrained density functional theory

CI (SD) configuration interaction (singles & doubles)

COSMO(-RS) conductor-like screening model for real solvents

Cp cyclopentadienyl ligand

(C-)PCM (conductor-like) polarizable continuum model

CT charge transfer

dmpe 1,2-bis(dimethylphosphine)ethane

dppe 1,2-bis(diphenylphosphine)ethane

DFT density functional theory

EPR electron paramagnetic resonance

ET electron transfer

Fc ferrocene

GGA generalized gradient approximation

HF Hartree-Fock

HOMO highest occupied molecular orbital

IC interconfigurational

INDO intermediate neglect of differential overlap

IR infrared (spectral region)

IVCT intervalence charge transfer

LC long-range corrected

LDA local density approximation

LMCT ligand-metal charge transfer

LMF local mixing function

LIST OF ABBREVIATIONS VIII

LUMO lowest unoccupied molecular orbital

MAE mean absolute error

MD molecular dynamics

MLCT metal-ligand charge transfer

MO molecular orbital

MP2 Møller-Plesset perturbation theory to second order

MR multi-reference

MV mixed-valence

NIR near-infrared spectral region

NMR nuclear magnetic resonance

OEP optimized effective potential

PES potential energy surface

QI quantum-interference

RISM-SCF reference interaction site model self-consistent field

SCF self-consistent field

SOMO singly occupied molecular orbital

STM scanning tunneling microscopy

SVP split valence polarization (basis sets)

TAA triarylamine

TDDFT time-dependent density functional theory

THF tetrahydrofuran

UV ultraviolet spectral region

vis visible

ABBREVIATIONS OF DENSITY FUNCTIONALS

B Becke

BMK Boese-Martin for kinetics

LYP Lee-Yang-Parr

PBE Perdew-Burke-Ernzerhof

S Slater-Dirac

TPSS Tao-Perdew-Staroverov-Scuseria

VWN Vosko-Wilk-Nusair

INTRODUCTION

INTRODUCTION

A wide range of molecular and solid-state systems is embodied in the mixed-valence (MV) concept, and various models have been devised to classify their electron transfer (ET) characteristics. Generally, one views a MV system as a molecule or solid in which a given redox center appears at least twice, in two different oxidation states, connected by a suitable bridging unit that typically provides for some electronic coupling between the redox centers. The thermal or optically induced ET between the (two or more) redox centers is at the heart of attention, but ET from or to the bridging unit, which is becoming increasingly more widely recognized, also plays a role in the overall ET mechanisms and spectroscopic profiles. The large variety of redox centers and bridge units and the different coupling between them accounts for the multitude of MV systems that can be envisioned or exist, either purposefully or accidentally constructed by synthetic chemists or present in nature. The “classical” MV systems are based on transition-metal redox centers connected by bridging ligands. Examples are Prussian Blue on the “serendipity” branch of the field or the Creutz-Taube ion and its derivatives[1,2] that feature prominently in the purposeful study of ET in transition-metal systems. However, many other molecules have been known that were classified as MV only much later. Examples are the radical anions of aryl compounds with two or more nitro substituents. These have been known and studied spectroscopically since the early 1960s, but they were recognized as MV systems only much later, where the nitro-substituted part of the aryl ring features as redox center, and the remaining aryl part plays the role of bridging unit.[3-

5] This served to introduce organic redox centers to the field, which have received growing attention due to their involvement in organic molecular electronics. Notable examples are

INTRODUCTION 2

those based on triaryl-amine redox-active units (see below). Organic MV systems are usually also based on organic bridges, but ”inverted” MV systems with organic redox centers and transition-metal-based bridges are also known as discussed below.

The MV concept may even be widened to such exotic species as the H2+ ion, which may be

viewed as the smallest conceivable MV system. In this thesis only MV systems are discussed where the formal redox states differ by one unit, which for two redox centers typically leads to open-shell compounds in the ground and excited states.[6] Of course, MV systems with larger electron-number differences are conceivable. Their closed-shell variants have been classified as donor-acceptor systems.[6] Systems with non-identical redox centers further extend the picture and may call for a looser MV definition.[7] In many cases redox-non-innocence of the bridge or of terminal ligands of transition-metal complexes additionally complicates the redox-center assignment.[8-11]

Despite the complications arising from the definition, MV systems have drawn and continue to draw the unabated attention of experimental and theoretical scientists from chemistry, biology, physics, and related natural sciences, which is reflected in a large variety of reviews and special issues.[6,7,12-22] This is due to the fundamental importance of MV systems in ET processes in nature (e.g. in metalloenzymes[23]), in catalysis, and in the design of functional materials.[24] Furthermore these often complex reactions or at least key steps of the reactions can be mimicked and investigated using more readily accessible MV compounds.[25] Additionally the detailed understanding of and the theory behind ET processes is promoted by the investigations of MV systems.[6] MV compounds are versatile in their application, as by varying different factors, like the redox centers, the bridging unit, and/or the enzymatic or solvent environment, the whole range from charge-localized to -delocalized systems can be tuned.

Mixed-Valence Systems in Biocatalysis

ET reactions are of tremendous importance in heterogeneous, homogeneous, and enzymatic catalysis. In fact it may be argued that ET is involved the majority of important catalytic processes. Considering biocatalysis in particular, metallo-proteins often contain multinuclear metal sites that feature MV states during catalysis.[23] It is in particular the conversion of small, stable molecules (e.g. H2O, CH4, N2, CO2) that requires a sequence of elementary steps to circumvent high activation barriers.[26] Multinuclear metal sites, sometimes in combination with redox-active ligands, are frequently involved here.

The oxygen-evolving complex of the photosystem II, which catalyzes the light-driven water oxidation, represents a typical example for an active site with predominant MV active states.[27,28] The Mn4Ca(µ-On) core features a tetranuclear transition metal cluster, where different (MV) oxidation states are reached as a consequence of light-induced electron removal (i.e. photo-oxidation).[29] Broken-symmetry DFT calculations have clearly

INTRODUCTION 3

demonstrated localized Class II character for the relevant states, and standard functionals like B3LYP have been found to provide a reasonable picture of the electronic structure of these MV clusters, if broken-symmetry approaches are employed.

Another type of multinuclear enzyme site is represented by the dinuclear CuA site in various copper enzymes, e.g. cytochrome c oxidase. Here the MV Cu+ICu+II state appears to be a delocalized Class III system, but perturbations due to mutation may alter the protein environment sufficiently to cause partial localization.[23,30-32] In contrast, partial delocalization characterizes many of the important biological iron-sulfur clusters, which renders these systems a challenge for both clear-cut experimental and theoretical descriptions. Charge distribution in FeS clusters depends crucially on coordination number of the metals and the extent of magnetic coupling between them (ferromagnetic coupling favors charge delocalization, whereas antiferromagnetic coupling appears to give rise to charge localization).[23]

Mixed-Valence Systems in Molecular Electronics

Class III MV systems are obvious candidates for use as “molecular wires” or “nanojunctions” in the field of molecular electronics.[24] Due to their electronically delocalized character and vanishing thermal ET barriers, fast ET over distances in the nm range is achievable.[14,15,33-36] Good energy matching between the orbitals of the bridge and the redox centers (“end caps”) is essential.[24,37]

On the other hand, switching functions or data storage require a certain degree of localization (“trapping”) of charge carriers, pointing to Class II situations with appreciable barriers. Here the currently envisioned targets include the controversially discussed “quantum-dot cellular automata” made up of MV complexes. These may be related to coding information in quantum computers[38-40] and to molecular transistors.[24] Driven by the technological and economic pursuit of “Moore’s Law”, the functional area of a transistor has been halving every eighteen to twenty-four months over the last decades, allowing the number of transistors per chip to double in each technology generation, and thus giving rise to smaller and more powerful electronic devices.[24,41] Although studies suggest another two decades of potential progress in silicon nanoelectronics,[42] a switch from the size-limited traditional materials towards molecular components will be required to continue this process of transistor scaling. Due to their versatile adjustable electronic properties MV systems represent promising targets towards molecule-based electronics. Both organic and inorganic MV compounds are used in photovoltaic devices, where dye-sensitized solar cells represent a highlight. Here MV systems may act as sensitizer, as well as redox shuttle, and remarkable efficiencies have already been achieved.[17,43,44]

Specific applications of the MV systems investigated in this thesis are discussed in a small introduction in the respective chapters.

INTRODUCTION 4

Quantum-Chemical Investigations of Mixed-Valence Systems

To achieve a proper quantum-chemical description of the ET transitions, it is important to account for effects from both dynamical and non-dynamical correlation, keeping in mind that there is no clear-cut separation between these extremes. With large configuration interaction (CI) or coupled cluster (CC) calculations within an ab initio framework this problem could be addressed in principle. Unfortunately, due to the size of typical MV systems those methods are computationally too demanding. Hartree-Fock (HF) calculations give too localized and standard density-functional calculations too delocalized descriptions. To make things worse, most experimental data are obtained in a polar solvent environment, which inevitably stabilizes a localized charge-separated situation. A recently developed quantum-chemical protocol employing the BLYP35 (global) hybrid functional with 35% exact-exchange, together with continuum solvent models, has been shown to give near optimum results for ground- and excited-state properties of bis-triarylamine radical cations containing two N,N-di(4-methoxyphenyl)-moieties as redox centers bridged by various organic units[45,46] and a wide variety of other organic MV systems.[47-49] The outstanding feature is that the prediction is made without any assumptions and/or constraints. In this work it is shown that this protocol is also capable of describing the charge localization/delocalization behavior of MV transition-metal complexes in the ground state. Furthermore an accurate quantum-chemical perspective on the excited-state properties is obtained. As a consequence the quantum-chemical approach facilitates the unique assignment of UV-vis-NIR bands of MV systems near the Class II/Class III borderline, which is very difficult based exclusively on experimental data.

Results and parts of this work have already been published in the journals Inorganic Chemistry (Section 3.1), Organometallics (Section 3.2), Chemistry – A European Journal (Chapter 4), and Chemical Society Reviews.

CHAPTER 1

THEORETICAL BACKGROUND

1.1 Mixed-Valence Systems

The principles involved in ET processes and models connecting spectroscopic data of MV compounds with their electronic structure have attracted appreciable attention since the discovery of the Creutz-Taube ion, which resulted in the fundamental Marcus-Hush[50,51] and (generalized) Mulliken-Hush theories.[20,52-56] Regardless of the constituents and composition of the system, electronic absorption data are crucial to the characterization of MV complexes.[57,58] Whilst there is a wealth of information contained in the (often overlooked) metal-ligand charge-transfer (MLCT)/ligand-metal charge-transfer (LMCT) transitions that usually fall in the visible region of the spectrum,[55,57,59,60] the lower energy intervalence charge-transfer (IVCT) band typically found in the NIR region is usually the primary source of information concerning the electronic character of a MV system.[12,57,58,61,62] Other important spectroscopic techniques for the investigation of the electronic character of MV complexes, which involve somewhat different energy and time scales, include vibrational spectroscopies (IR, Raman),[63-67] Stark spectroscopy,[68] Mössbauer spectroscopy,[69-71] and electron paramagnetic resonance (EPR) spectroscopy.[5,72,73]

Robin-Day Classes

The description of electronic structure of MV systems is usually made within the Robin-Day scheme, which was introduced in 1967.[74] Although in principal, it is applicable to larger and

CHAPTER 1: THEORETICAL BACKGROUND 6

more sophisticated systems, the scheme will be illustrated by the important model case of two redox centers linked by a bridging ligand. It is based on the extent of electronic coupling of two diabatic localized potential energy curves to give adiabatic ground and excited states of the system. The three primary Robin-Day classes are simply denoted Class I, II, and III. Class I corresponds to the situation in which there is no coupling between the diabatic potential energy curves (Figure 1, left). Class II corresponds to partial localization of charge and spin due to the electronic coupling, 2Hab, being smaller than the (internal plus external) Marcus reorganization energy, λ. This leads to a double-well adiabatic ground-state potential curve with an activation barrier for thermal ET. In contrast, in Class III charge and spin are delocalized over both redox centers, and the ground-state barrier has vanished as 2Hab ≥ λ. As optoelectronic properties of a MV system are crucially dependent on the localization/delocalization of charge, the distinction between Class II and III behavior has been investigated in detail, through application of an increasingly wide and sophisticated range of spectroscopic and computational techniques and theoretical treatments. At the borderline between Class II and III, small activation barriers and fast ET processes may give rise to contradictory findings with these different techniques, partly due to the different time scales of internal and solvent reorganization processes. This convolution of internal reorganization and solvent dynamics has led Meyer and coworkers to introduce an intermediate Class II/III.[12] Finally, a Class IV was proposed by Lear and Chisholm by taking the vibronic progression into consideration.[59] The characteristics of Class IV, which may be considered a subclass of Class III, include the absence of (or minimal) vibronic coupling, and the solvent-independence of not only the IVCT but also the MLCT band.

Figure 1. Potential energy curves of the three primary Robin-Day classes: Class I: diabatic states, no coupling, fully localized, no thermal ET (left); adiabatic states, weak coupling, partly localized, ET barrier ∆G* (middle), and Class III adiabatic states, strong coupling, fully delocalized, no ET barrier (right).

Marcus-Hush and Mulliken-Hush Theory

Theoretical models to describe MV systems are based on the groundbreaking work on ET reactions based on diabatic potential-energy curves by Marcus, which was awarded with the nobel prize in 1992.[75] The first experimental investigations were performed on “self-


exchange reactions”. These correspond to ET between two atoms of the same element in different oxidation states (e.g. FeIII/FeII as found in Prussian Blue) and no chemical bonds are formed or broken. In addition “cross reactions” between different elements in aqueous solution were investigated. In principle, intra-molecular ET processes can be either thermally or optically induced. In the former case, the ET activation barrier ∆G*, also called free activation energy, between two charge-localized states is the rate-determining quantity.[6] Experimentally determined values of ∆G* are available for only few systems, as variable-temperature spectroscopic measurements, typically EPR experiments, have to be performed.[73] This thesis will focus on the investigation of optically induced ET processes and on the factors influencing excited-state parameters. In the case of two diabatic states A and B (Class I), ET proceeds directly via the absorption of a photon. Thus both the ground and excited-state feature charge-localized structures, and the electron is transferred from one redox center to another directly by the excitation. The excitation energy corresponds exactly to the so-called Marcus reorganization energy λ.[6,57] The reorganization energy is normally divided into contribution from slow modes λ0, which account for solvent reorientation after the ET took place, and fast modes λi primarily describing intra-molecular structural changes, e.g bond elongations/contractions associated with oxidation-state changes. For MV complexes exhibiting an ET rate in the frequency range between the solvent (1011-1012 s–1), and the internal modes (1013-1014 s–1) Meyer et al. proposed the intermediate Class II/III.[12]

(1.1.1)

The classical Marcus theory is only applicable to MV systems, in which there is no electronic coupling between the redox centers (Class I case). The analysis of systems, which exhibit electronic interaction between the redox moieties, requires an adiabatic treatment, namely Marcus-Hush theory.[50,51,55,76] In this extension to Marcus theory the electronic coupling 2Hab leads to a double-minimum potential in the ground state (Class II, Figure 1), which exhibits an ET activation barrier ∆G*. In the case of a symmetric MV complex, identical redox centers in the neutral complex respectively, ∆G* is given as:

(1.1.2)

This equation links thermally and optically induced ET in Class II systems.[77] Remarkably Marcus-Hush theory allows for a detailed analysis of the IVCT transition for Class II systems. It is only valid if the temperature is sufficiently high for the ground-state vibrational modes to be populated following a Boltzmann distribution, and the ET is induced via a vertical excitation following the Franck-Condon principle. Taking the vibronic coupling into account via a harmonic diabatic approach the IVCT band is then nearly symmetrical and Gaussian-

Etrans = h⌫ = λ = λ0 +X

i

λi

G⇤ =( 2Hab)

2

4


shaped in the case of relatively small electronic coupling (Figure 2, top), and the full band width at half maximum ν1/2 is exactly given as:

(1.1.3)

Figure 2. NIR band shapes caused by vibronic coupling between ground and excited-state for a MV complex with relatively small electronic coupling Hab resulting in a nearly Gaussian shaped band, (top) and for a complex with relatively strong coupling Hab and small ET activation barrier ∆G* resulting in an asymmetric band (bottom).

⌫1/2 =p16ln2kB⌫max


Unfortunately the comparison of this theoretical value with the experimentally observed band width is often taken as main criterion for the Robin-Day classification: if the Gaussian function fitted to the experimental IVCT band is broader than the value obtained from Equation 1.1.3, the complex is assigned as Class II system. If it is narrower, Class III behavior is assumed. Despite the difficulties to uniquely assign an IVCT band and the problems arising from the Gaussian-deconvolution fitting, which are discussed in the next section, the assumption of a Gaussian-shaped band becomes more and more inaccurate with increasing electronic coupling. This is due to the increasing asymmetry of the IVCT band, which can be illustrated by a simple picture of the vibrational levels of ground and excited-state and the resulting Franck-Condon excitations (Figure 2, bottom). The overlap integral of the ground- and excited-state vibrational wave function, corresponding to the Franck-Condon transition probability and intensity, is largest for the areas near the borders of the electronic potential and near the ET barrier. If the barrier is sufficiently small, and vibrational states, which are energetically higher than the barrier, are appreciably populated, an asymmetric IVCT band is observed. In reality the cut-off is not as steep as indicated in the figure, but the band envelope is strongly and continuously decreasing, resulting in a large deviation from of a Gaussian-shaped band.[6,57,78]

To account for this band asymmetry, more advanced models like the two-mode model have to be applied.[20,79,80] But despite its more complicated form compared to the Marcus-Hush/Mulliken-Hush models, it is only practical for strongly coupled MV systems.

Within the framework of Mulliken-Hush theory the electronic coupling 2Hab can be directly extracted from spectroscopic parameters.[53,54,57,81] Unfortunately the necessary difference in the diabatic dipole moments of the ground and excited states is not directly accessible experimentally. The difference is then typically estimated by the difference of the adiabatic dipole moments, but again the adiabatic dipole moment of the excited states is only rarely determinable by Stark spectroscopy and has to be either calculated by quantum-chemical methods or approximated using the structural estimates such as the redox center distance.[6]

The models discussed so far focus on the analysis of the IVCT band. But additionally there is a wealth of information contained in the MLCT/LMCT transitions that usually fall in the visible region of the spectrum.[55,57,59,60] To account for those transitions one has to advance to three-state models and/or generalized Mulliken-Hush theory.[53,54,56] But as these models get more and more sophisticated and several required quantities are not accessible experimentally (sometimes by quantum-chemical calculations), they are not discussed in detail in this thesis, and the focus will be on a purely quantum-chemical description of MV complexes.


Experimental Classification

Information on the extent of charge localization/delocalization in an MV system can be obtained by various experimental techniques. Most commonly the assignment to a Robin-Day class is based on the analysis of the IVCT band, which typically occurs in the NIR region of the electromagnetic spectrum. The potential for multiple electronic transitions of similar energy but different electronic origin, together with the asymmetric IVCT band-shapes that characterize strongly coupled MV systems, renders derivation of the ET characteristics and electronic structure from NIR spectra alone very difficult in many MV transition-metal complexes, despite the popularity of such analyses. In organic MV systems the IVCT band usually appears as the lowest-energy band in the spectrum, facilitating the assignment.[6]

Other important spectroscopic techniques involve somewhat different energy and time scales, e.g. vibrational spectroscopies (IR, Raman), Stark spectroscopy, Mössbauer spectroscopy, and EPR spectroscopy. At the borderline between Class II and III, small activation barriers and fast ET processes may give rise to contradictory findings with different spectroscopic techniques, due to the different time scales of the spectroscopic methods, which can be comparable to the rates of ET, inner-sphere reorganization processes, and solvent dynamics.

In some early studies the presence of symmetry-broken crystal structures was used as classification criterion, but of course the solid state may differ significantly from the situation in solution. Despite the fact that crystals are normally grown at low temperatures, at which the ET process is slowed down, crystal structures may be distorted due to packing-, counter ion-, or other crystal-effects.[82] For example salen-type complexes often exhibit symmetry-broken structures in the crystal, but may feature delocalized charge in solution.[83-86] Additionally, for complexes with free coordination sites, direct coordination of Lewis basic solvents may occur.

IR and vibrational Raman spectroscopy represent very fast and powerful methods to obtain information on the electronic structure of the ground state of MV systems.[63] They are often used in combination with UV-vis-NIR spectroscopy as part of a spectroelectrochemical approach.[87] Indications of charge localization are easily derived from the IR spectra of complexes, which are symmetric in their non-mixed-valence forms. The splitting of IR bands due to the energy difference in modes, which are degenerate in the delocalized case (e.g. ν(C≡C) vibrations) or the appearance of new IR bands, which do not involve changes of the permanent dipole moment in the case of a symmetrically distributed electron density, point to symmetry-broken structures.[64-66] Satellite groups, such as local auxiliary ligands, may allow monitoring of modes, e.g. ν(NO) or ν(CO), which depend strongly on the oxidation state of the metal center and thus are sensitive to the charge distribution.[67] Analogously, information on electronic structure may be derived from Raman spectra.[63,88]


Mössbauer spectroscopy is sensitive to oxidation state and charge distribution and corresponds to short time scales. The technique is limited to the Mössbauer-active nuclei, where 57Fe is the most prominent example.[69-71]

The paramagnetic nature of most MV systems complicates the use of NMR spectroscopy due to paramagnetic line broadening.

Variable-temperature EPR spectroscopy is widely used to determine ET barriers in organic MV systems.[5,72,73] Applications to MV transition-metal complexes are more limited, due to difficulties in extracting hyperfine coupling constants.[64] In principle the isotropic hyperfine coupling constant of a MV system corresponds to the value obtained for an analogous monometallic complex, if the charge is localized. For a Class III system it is close to half the value of the monometallic analogue. In addition, line broadening effects in the EPR spectrum may reveal ET processes that are slower than or comparable to the EPR time scale (10–9 s).[89]

Shortcomings of the State-of-the-Art Experimental Classification

A common way to analyze NIR spectra is to fit the spectroscopic data with Gaussian functions. In principle, the population of both ground- and excited-state vibrational levels can to some extent be taken into account assuming that these follow a Boltzmann distribution. Based on the semi-empirical model proposed by Meyer et al., experimental spectra of 3d and 4d complexes are normally fitted with three Gaussian functions.[12] This localized model assumes two interconfigurational (IC) bands, which arise from excitations from orbitals localized at the hole-carrying (more highly oxidized) ruthenium center to orbitals at the same center, and three IVCT bands originating from separate electronic excitations from three dπ orbitals of the other metal center. The IC transitions are normally parity forbidden and only gain intensity, if the system exhibits noticeable spin-orbit coupling. As the latter tends to be significant only for 5d systems, 3d and 4d metal complexes are typically expected to exhibit only the three IVCT excitations.

Unfortunately this assumption is only valid for clear-cut Class II or Class III systems in the Robin-Day scheme, as pointed out nicely by Creutz et al. and Lambert and Heckmann.[6,57] For systems close to the Class II/Class III borderline, the NIR band gets more and more asymmetric, as there is a cut-off at the smallest energy possible for an electronic transition (Figure 2). Due to this asymmetry of the band envelope, Gaussian functions are no longer a very suitable approximation. In addition to the vibrational progression, an alternative explanation for the band asymmetry of the Creutz-Taube ion was suggested by Meyer et al.:[12] the asymmetry of the IVCT band of the Creutz-Taube ion is explained by the coupling of two of the proposed IVCT transitions at 6320 cm–1 and 7360 cm–1. The fact that the band shape does not change in low-temperature experiments was used as an argument to rule out vibronic effects. High-level quantum-chemical studies did not reproduce these transitions,[90] and thus caution with Gaussian deconvolutions is advocated, when the investigated system is not a clear-cut Class II or Class III complex. Taking into account the band cut-off, which


appears at the low-energy side, one can in principle perform Gaussian deconvolution of the experimental NIR band by fitting only the high-energy part of the curve, which should correspond to a Gaussian shape, if only one transition is present (Figure 3). But a quantum-chemical approach, which facilitates assignment of the unique transition and thus of the character of NIR bands, is a clearly preferable method for the explanation of band shapes.

Figure 3. NIR data with Gaussian deconvolutions fitted to only the high-energy half of the curve for a complex with two transitions (left) and for a complex with one transition in this spectral region (right).


1.2 Density Functional Theory

Already in 1929 Dirac stated that “the general theory of quantum mechanics is now almost complete ... The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these equations leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.”[91] Density functional theory (DFT) was developed in the 1960s by Kohn, Hohenberg, and Sham as such an applicable approximate practical method.[92,93] But it took until the early 1990s to establish this method widely in applied quantum-chemistry. Ever since then the importance and number of applications of DFT has been growing, and it has become the method of choice for electronic-structure scientists ranging from quantum chemists and solid-state physicists to biophysicists. To explain this success one has to look at the machinery behind the approach and the reasons for its outstanding status among quantum-chemical methods.[94-97] In this section the general quantum-chemical concepts and the specific principles behind DFT as well as it advantages and shortcomings are discussed.

Principles

General Quantum-Chemical Principles and Hartree-Fock Theory

In the non-relativistic case the energy of any system is given by the time-dependent Schrödinger equation:

(1.2.1)

is the time-dependent Hamiltonoperator, which contains all interactions within the system and is normally divided into a potential-energy part and kinetic-energy part . Ψ is the wave function of the system, which depends on the time t and the particle coordinates r, and ħ represents the reduced Planck constant. In the stationary case the time-dependence can be separated and treated as a phase factor, thus giving the time-independent Schrödinger equation:

(1.2.2)

In principle, by solving this Eigenwert equation one obtains the total energy E and all information about the ground-state of any given system can be extracted from the wave

i~@

@t (r, t) = H(r, t) (r, t)

H

V T

H(r) (r) = E (r)


function Ψ. Unfortunately, as Dirac stated, this equation (or even more so its relativistic analogues) is much too complicated to be solvable, and approximations have to be made. Typically the first approach is to assume fixed nuclei and to thus obtain the electronic Schrödinger equation. Therefore the kinetic energy of the nuclei is zero and the nuclei-nuclei repulsion is given by a constant. After this so-called Born-Oppenheimer approximation, the electronic Hamiltonian is given by the kinetic energy of the electrons and the two potential-energy terms, the electron-electron repulsion and the electron-nuclei attraction υ(r), the so-called external potential in DFT. In the case of an atom or molecule in the absence of an external field, υ (r) is simply determined by the nuclear configuration and corresponds to the attractive point-charge Coulomb potential between nuclei and electrons. Thus for a system with N electrons and M nuclei the electronic Hamiltonian is given (in atomic units) by

(1.2.3)

with

In the following the subscript “e” will be dropped for clarity, as only electronic energies Ee will be discussed. As the wave function Ψ of a system with N-electrons depends on 3N spatial and N spin coordinates, the solution to the electronic Schrödinger equation is still horrendously difficult. In fact an exact solution can be obtained only for one-electron systems. In HF theory Ψ is approximated by a single-determinant approach. This Slater determinant is built of one-electron functions, which are called spin orbitals, as they contain the product of a spatial and a spin part.

(1.2.4)

The determinant is antisymmetrized and thus fulfills the Pauli principle. Please note that for quantum-chemical calculations the spin orbitals are normally expressed in an atomic orbital basis. These basis sets contain a finite number Nbasis of basis functions and the energy is minimized by varying only the linear coefficients ciν.

(1.2.5)

He = Te + Vee + Vne =

NX

i=1

✓−1

2r2

i

◆+

NX

i<j

1

rij−

NX

i=1

(r)

(r) =MX

A=1

ZA

rAi

s(r1, r2, · · · rN ) =

'1(r1) '1(r2) · · · '1(rN )'2(r1) '2(r2) · · · '2(rN )

......

. . ....

'N (r1) 'N (r2) · · · 'N (rN )

'i(r) =

NbasisX

i

ci⌫χ⌫


The energy expression in HF theory is given as the sum of nucleus-electron attraction Vne[{φi}] and the classical kinetic energy T[{φi}] and Coulomb energy J[{φi}] terms.

(1.2.6)

The exchange term ExHF is a correction to the classical Coulomb term arising from the

antisymmetry requirement following the Pauli principle.[98]

(1.2.7)

In HF theory the energy of an electron is calculated in the field of nuclei and the average field of the other electrons. While the so-called Fermi correlation is included in HF theory, following Löwdin the Coulomb correlation energy is defined as the difference between the HF limit (lowest energy achievable with a single Slater determinant) and the exact total energy of the system.[99] Nevertheless the single-determinant approach of HF is not accurate enough for most systems and the error is due to the missing Coulomb correlation. More sophisticated methods such as full-CI or full-CC are able to describe this correlation. Other methods like MP2 (Møller-Plesset perturbation theory to second order) can at least partly recover it, but the wave function-based approaches tend to scale poorly with system size due to the complicated quantity Ψ.[94,95]

Density Functional Theory

So is it possible to replace the wave function by a simpler quantity allowing for a better-scaling method, which is able to describe Coulomb correlation? Already in the 1920s attempts were made to represent two correlated quantities, the electronic kinetic energy (by Thomas and Fermi) and the exchange energy (Dirac), as functionals of the electron density ρ(r). The possibility to find any electron of a system with N electrons in the volume element dr1 is given as the product of the number of electrons and the integration of |Ψ|2 over all spatial coordinates ri of the electrons 2, 3 … N and the spin coordinates si of all electrons.

(1.2.8)

The density ρ(r1) thus only depends on three spatial coordinates. The number of electrons N is trivialy determined by integration of ρ(r1) over space.

EHF [{'i}] = Vne[{'i}] + TS [{'i}] + J [{'i}] + EHFx [{'i}]

EHFx [{'i}] = 1

2

X

i,j

ZZ'i(r1)'j(r1)'i(r2)'j(r2)

r12dr1dr2

✓N

Z· · ·

Z| (r1, r2, . . . rN, s1, s2, . . . sN )|2dr2dr3. . . drNds1ds2. . . dsN

◆dr1

= ⇢(r1)dr1


(1.2.9)

The rigorous proof that the electronic ground-state energy E can be expressed as functional of ρ(r) was done by Hohenberg and Kohn.[92] They could show that each electron density ρ(r) is the ground-state density of at most one external potential υ (r), which is uniquely determined up to an additive constant. Conversely the external potential determines ρ(r) up to a degenerate set. Analogously it can be proven that different potentials also have different wave functions. Hence the external potential υ (r) and the number of electrons N of a system are uniquely determined by the ground-state electron density ρ(r). Consequently the Hamiltonian is determined as well and thus the wave function and all information about the ground state of a system can be obtained from the electron density ρ(r). In this section no distinction between α- (spin up) and β-electron density (spin down) is made (spin-restricted case) and spin labels will be omitted for clarity. The electronic energy can then be expressed as a functional of ρ(r).[94,95]

(1.2.10)

with

Vne[ρ] corresponds to the nuclear-electron attraction, T [ρ] is the electronic kinetic energy, and Vee [ρ] the electron-electron repulsion. The Hohenberg-Kohn functional F [ρ] does not depend on the external potential and thus is universally valid for any given molecule. DFT would be an exact theory, if the explicit form of F [ρ] was known. The quality of a DFT calculation thus crucially depends on the quality of the approximation of F [ρ]. In their second theorem Hohenberg and Kohn proved that each approximated spin density ρ(r) different from the exact spin density ρ(r) gives a higher energy than the ground-state energy. Hence the variational principle holds for DFT and the Levy constrained search formalism can be applied to minimize the energy, the constraint being that the integration of the electron density over space gives the number of electrons.[100]

The breakthrough of DFT was initiated by Kohn and Sham, who proved that by the introduction of orbitals a large part of the kinetic energy T [ρ] can be calculated exactly, and only a small positiv contributions to T [ρ] remains.[93] As T [ρ] corresponds to a very large part of the electronic energy and even small errors in this term can render a theory inadequate, Kohn-Sham theory is essential. Kohn and Sham introduced a non-interacting reference system of N electrons, which has exactly the same density as the real system.[98] Analogously to HF theory, it is described by a Slater determinant built of spin orbitals (Equation 1.2.4).

Z⇢(r)dr = N, ⇢(r) 0

H

E[⇢] = Vne[⇢] + T [⇢] + Vee[⇢] =

Z⇢(r)υ(r)dr+ F [⇢]

F [⇢] = T [⇢] + Vee[⇢] = h |T + V ee| i


The density is then simply defined as the sum of the orbital densities.

(1.2.11)

and the kinetic energy of the non-interacting system Ts[ρ] is exactly given by:

(1.2.12)

In addition the electron-electron interaction is approximated by a classical Coulomb repulsion of the density.

(1.2.13)

The unknown parts of the electronic energy, the non-classical term of the repulsion, and the pre-mentioned positive contribution to the kinetic energy term due to electron correlation, are gathered in the exchange-correlation functional Exc[ρ].

(1.2.14)

Hence large parts of the electronic Kohn-Sham energy

(1.2.15)

are described exactly and the Kohn-Sham theory is a formally exact theory. Unfortunately the exchange-correlation functional Exc[ρ] is not known and needs to be approximated.[94-98,100] Also a systematic improvement of Exc[ρ] towards an exact theory is not usually achieved. Equation 1.2.15 is very similar to the energy expression in HF theory (Equation 1.2.6). In HF the kinetic energy term corresponds to only the classical kinetic energy of a Slater determinant. The classical kinetic and Coulomb energy terms are identical to the Kohn-Sham expression, but evaluated with different orbitals.

The HF exchange term, which is exact, is used in hybrid functionals, which are discussed in the next section. In these it is evaluated using Kohn-Sham orbitals, which would give the exact density for the exact Exc[ρ], instead of HF orbitals, which give an approximate wave function, and thus is called exact-exchange Ex

exact to emphasize the difference in the resulting energy values.[101]

⇢(r) =

NX

i

|'i(r)|2

Ts[⇢] =

* s

��

NX

i=1

r2i

�� s

+= −1

2

NX

i=1

Z'i(r)r2'i(r)dr

J [⇢] =1

2

ZZ⇢(r1)⇢(r2)

r12dr1dr2

Exc[⇢] = T [⇢] Ts[⇢] + Vee[⇢] J [⇢]

EKS [⇢] = Vne[⇢] + Ts[⇢] + J [⇢] + Exc[⇢]


Exchange-Correlation Functionals

As mentioned before the quality of a DFT calculation crucially depends on the quality of the approximation used for the unknown Hohenberg-Kohn functional F [ρ]. The unknown parts of F [ρ] are gathered in the exchange-correlation functional Exc[ρ], hence the approximations of Exc[ρ] analogously determine the quality of a DFT calculation. Traditionally Exc[ρ] is split into the sum of an exchange Ex[ρ] and a correlation part Ec[ρ], which are approximated individually. Usually the energy contribution of Ex[ρ] is much larger than Ec[ρ]. But the exchange functional may also cover parts of the correlation and a clear-cut separation between exchange and correlation is often not possible. Unfortunately no simple systematic procedure for improving Exc[ρ] exists. Recently strategies for a systematic improvement of Exc[ρ] based on the interaction between the electron density ρ(r) and exchange-correlation hole ρxc (r1, r2), which consists of the exchange hole and a correlation factor, have been developed.[102] The advantage of the correlation factor ansatz is the possibility to express exact conditions.

Commonly at the simplest level the exchange-correlation energy is approximated assuming a uniform electron gas. In this so-called local density approximation (LDA), or local spin density approximation, an inhomogeneous system consists of small boxes, in which the electron density and the potential are constant. This results in a homogeneous system for each box, and after introducing infinitesimally small boxes, the LDA Dirac exchange functional can be obtained.[94,95]

(1.2.16)

Typically the Slater-Dirac exchange is used in combination with Vosko-Wilk-Nusair (VWN) correlation in the Slater-Vosko-Wilk-Nusair (SVWN) functional.[103,104] Despite some error cancelation between the exchange and correlation part, the LDA and the underlying assumption of a uniform electron gas proved inaccurate for most relevant molecules, as the electron density varies strongly in space. The LDA approach yields reliable results only for solid state materials, especially metals, and thus the breakthrough of DFT in applied quantum-chemistry was not initiated before the inclusion of the density gradient ∇ρ into exchange-correlation functionals, which are then called semi-local. Due to this so-called generalized-gradient approximation (GGA) the inhomogeneity in the electron density can be better described. The typical exchange functional associated with the GGA was introduced by Becke in 1988, which is normally expressed using the LDA exchange energy density εxLDA and a gradient correction ∆εxB88.[105]

(1.2.17)

EDiracx [⇢] = Cx

Z⇢4/3(r)dr Cx = 3

4

✓3

⇡

◆ 13

EB88x [⇢,r⇢] =

Z⇢(r)

�"LDAx +"B88

x

�


with

s (r) is called reduced density gradient.

(1.2.18)

In β the Becke exchange functional contains an empirical parameter, which was obtained by a least-squares fit to the HF exchange of noble-gas atoms (= 0.0042). Becke exchange Ex

B88[ρ, ∇ρ] is usually used in combination with the correlation functional developed by Lee, Yang, and Parr (LYP) in the same year (BLYP)[106] or with the correlation functional introduced by Perdew (BP86).[107] In 1996 another widely used GGA functional was introduced by Perdew, Burke and Ernzerhof introduced (PBE), which depends on the reduced density gradient, s(r), the spin polarization, and the Wigner-Seitz radius.[108]

To explain the success of DFT in applied quantum-chemistry, one has to have a closer look at electron correlation and its different forms. For easier explanation, electron correlation is divided into two parts, keeping in mind that there is no clear-cut separation between these extremes: the dynamical and the non-dynamical, also often called static, correlation.[109] The electron-electron repulsion influences the movement of the electrons. Namely the probability of presence is reduced as two electrons approach each other. This short-range effect is called dynamic correlation and the GGA correlation functionals solely model dynamic correlation. In contrast, non-dynamical correlation is a long-range effect, and it is partly recovered by the exchange part of GGA functionals. It is significant for systems, in which the ground state cannot satisfactorily be described by single-determinant wave function approaches. It can also be recovered by multi-determinant post-HF approaches like multi-configurational self-consistent field methods, e.g complete active space self-consistent field (CASSCF). Without non-dynamical correlation the dissociation of molecules cannot be described correctly and thus its nature is best explained by considering H2 at molecular dissociation and especially the nature of the exchange-correlation hole ρxc(r1, r2). Several conditions hold for this point: There is no dynamical correlation, as the two single-electron atoms do not interact. Similarly the kinetic contribution to Exc[ρ] is absent and thus the exchange energy exactly equals the non-dynamical correlation term. The exchange energy hole is given by ρ(r) /2 and hence delocalized over both atomic centers, which is the case in HF theory. The exact exchange-correlation hole ρxc(r1, r2) for an electron near one nucleus, which can be calculated via computationally demanding full CI calculations, is localized at that nucleus. Hence non-dynamical correlation converts the delocalized exchange hole to a localized ρxc(r1, r2).[94,109] The advantage of LDA DFT is that the exchange part recovers the localized exchange-correlation hole despite being a single-determinant approach. Analogously GGA exchange

"B88x = β⇢(r)1/3

s(r)2

1 + 6βs(r)sinh1s(r)

s(r) =|r⇢(r)|

2(3⇡2)1/3⇢4/3(r)


functionals accounts for non-dynamical correlation effectively, but also introduces an improved description of the exchange energies. Together with an appropriate correlation functional, e.g. LYP, which recovers the dynamic correlation, a comprehensive description of exchange, non-dynamical and dynamical correlation can be achieved.[94,109]

To climb a further rank on the “Jacob’s Ladder to the heaven of chemical accuracy” (Figure 4), which was introduced by Perdew and coworkers,[110-112] so-called meta-GGA functionals were developed. These depend on either the Laplacian of the density and/or the non-interacting local kinetic-energy density τ(r).

(1.2.19)

The most widely-used meta-GGA functional TPSS was developed by Tao, Perdew, Staroverov and Scuseria.[113]

Figure 4. “Jacob’s Ladder to the heaven of chemical accuracy” introduced by Perdew and coworkers. n is the physicists notation for the electron density ρ. Figure reproduced with permission from J. P. Perdew, A. Ruzsinszky, L. A. Constantin, J. Sun, G. b. I. Csonka, J. Chem. Theory Comput. 2009, 5, 902. Copyright 2009 American Chemical Society.

⌧(r) =1

2

NX

i

|r'i(r)|2


Global Hybrid Functionals

The predominant shortcoming of DFT are self-interaction errors. Naturally, the two-electron contributions to the electronic energy must be non-existent for one-electron systems.

(1.2.20)

But very few approximated Exc[ρ] fulfill this exact condition and the deviation to the electronic energy arising from each electron interacting with itself is called self-interaction error. This self-interaction error can be reduced by the incorporation of exact exchange Ex

exact from the HF theory into the exchange-correlation functional Exc[ρ]. Functionals, which contain Ex

exact, are called hybrid functionals. While due to the incorporation of Exexact the self-

interaction error is reduced, LDA or GGA exchange serve the purpose of simulating non-dynamical correlation. Thus the main challenge is to find the right amount of exact-exchange admixture a to be able to reduce the self-interaction error and still take advantage of the mimicking of non-dynamical left-right correlation by local (LDA) or semi-local (GGA) exchange functionals.

In many standard implementations the energy of hybrid functionals is then minimized with respect to the orbitals, which is in principle not Kohn-Sham theory. Normally the exchange-correlation potential υxc (r) is readily evaluable by expansion:

(1.2.21)

The potential is purely local and multiplicative for LDA or GGA functionals but becomes non-local due to exact-exchange part. This problem can be solved rigorously by the introduction of the (often approximated) optimized effective potential (OEP), but often so-called “generalized Kohn-Sham” approaches based on functional derivatives with respect to orbitals without a subsequent OEP step are used.[114]

The first global (= constant exact-exchange admixture) hybrid functionals, the Becke “Half-and-Half” (BHHLYP) functional, which contains 50% of Ex

exact and 50% LDA exchange, and the still most popular DFT functional B3LYP, were introduced in 1993.[115,116] B3LYP contains three semi-empirical coefficients and incorporates 20% of exact-exchange admixture.

(1.2.22)

The BLYP35 functional, which is mainly used in this thesis, is based on a simple one-parameter form for global hybrid functionals and incorporates 35% of exact-exchange (a = 0.35).[117]

Exc[⇢] = J [⇢]

υxc(r) =δExc

δ⇢(r)=

@"xc@⇢(r)

−r @"xc@r⇢(r)

+r2 @"xc@r2⇢(r)

+ · · ·

EB3LY Pxc = (1 a) · ELDA

x + a · Eexactx + b ·EB88

x + (1 c)EVWNc + c ·ELY P

c


(1.2.23)

Other commonly used global hybrid functionals are the PBE0 functional,[118] which is a modification to the PBE functional with 25% Ex

exact, the TPSSH functional,[113] a modification to the TPSS functional with 10% Ex

exact, and the Boese-Martin functional for kinetics (BMK) functional, which is a τ-dependent hybrid functional with 42% Ex

exact.

Local and Range-Separated Hybrid Functionals

The development of more flexible functionals further improved the performance of DFT. By introducing a dependence of exact-exchange admixture on either the position or the interelectronic distance, the self-interaction error can be reduced locally, while non-dynamical correlation is still mimicked at the “right” places. There are two different approaches to this class of exchange functionals: local hybrid functionals, in which Ex

exact is varied in real space by introducing a local mixing function (LMF) a(r), and range-separated hybrid functionals, in which a dependence of Ex

exact on the interelectronic distance r12 is introduced, typically by an error function.[119-121]

(1.2.24)

An approach to separate the Coulomb interaction and Exc into a short-range (first term on the right-hand side) and a long-range region (last term) was first suggested by Stoll and Savin.[122-

124] The principal idea is to evaluate the short-range part using the exchange-correlation functionals from DFT to mimic left-right correlation and the long-range part using exact exchange to reduce the self-interaction error and regain the right asymptotic decay. LC-BLYP represents a typical example for such long-range corrected (LC) DFT functionals.[125] But also range-separated hybrid functionals with increased exact-exchange admixtures in the short-range have been introduced.[126] Unfortunately, in practice the exchange-correlation functionals have to be adopted for this approach. In the most popular range-separated hybrid functional CAM-B3LYP the short-range is described using the slightly modified B3LYP functional and the exact-exchange admixture increases up to 60% at long range.[127] More recently range-separated hybrid functionals containing a middle-range part were introduced.[128]

While the exact-exchange admixture in range-separated hybrid functionals is normally evaluated for the short-range and the long-range region as explained above, in local hybrid functionals it is varied in real space due to the introduction of the local mixing function a(r).

(1.2.25)

EBLY Paxc = (1 a) ·

(ELDA

x +EB88x

)+ a · Eexact

x + ELY Pc

r12 =1 erf(µr12)

r12+

erf(µr12)

r12

ELocHybxc =

Za(r) · "exactx (r)dr+

Z[1 a(r)] · "DFT

x (r)dr+ EDFTc


This is an advantage over global hybrid functionals, as there are some areas within the molecule, for which the reduction of the self-interaction error or non-dynamical correlation is more important and where more exact-exchange admixture is needed. Probably the most obvious condition is that in one-electron regions a(r) should approach unity (100% Ex

exact) to remove self-interaction. While LDA and GGA give good results in bonding regions, spin polarization (see previous sections) of the core shells, which is fundamental for the description of magnetic properties like nuclear magnetic resonance (NMR) or EPR parameters, is only sufficiently described by the inclusion of exact exchange.[129] Hence and due to formal reasons, a(r) is designed to be larger in the core than in the valence-shell region. Additionally the LMF a(r) is required to be positive and equal to or smaller than one at all points in space. Jaramillo, Scuseria, and Ernzerhof introduced the so-called t-LMF, which is based on the ratio of the Weizsäcker kinetic energy density τW (r) and the non-interacting local kinetic energy density τ(r) (Equation 1.2.19).[130]

(1.2.26)

with

The t-LMF approaches one in one-electron regions and zero in bonding regions (Figure 5) and thus meets the criteria discussed above. Interestingly, for the first-generation local hybrid functionals the best thermochemical benchmark performance of the t-LMF, in conjunction with only LDA exchange and correlation, was achieved by adding a scaling factor of 0.48.[131]

Figure 5. Plot of the (constant) exact-exchange admixture (20%) of the global hybrid functional B3LYP (left), and of the (position-dependent) exact-exchange admixture of a local hybrid functional (t-LMF with scaling factor 0.48) (right) of MnO3.

t(r) =⌧W (r)

⌧(r)

⌧W (r) =|r⇢(r)|2

8⇢(r)


Analogously to CAM-B3LYP, in no region full (100%) exact exchange is recovered for the scaled t-LMF (Figure 5), and in principle neither the self-interaction error nor the wrong asymptotic behavior are fully corrected for. Another interesting observation is that the local hybrid functionals perform best, if LDA exchange is incorporated. There are two possible reasons for that: The two extrema, self-interaction error in LDA and missing non-dynamical correlation in exact-exchange, are more pronounced and thus the tunable range becomes larger. Another explanation is the gauge problem. In principle any function that integrates to zero can be added to the energy density without changing the energy. As in global hybrid functionals the weighting is applied after the integration, there is no gauge problem. But energy densities that are locally mixed via LMFs have to share the same gauge.

More flexible LMFs, which depend linearly or exponentially on the τW (r)/τ (r) ratio, the reduced density gradient s (r) (Equation 1.2.18), and/or the spin polarization ζ, were designed.[114,132-134] The spin polarization ζ simply represents the spin density, the difference of the α- and β-electron density, divided by the electron density, which corresponds to the sum of α- and β-electron density.

(1.2.27)

Time-Dependent Density Functional Theory

To compute excited states, as well as properties such as excited-state structures, excitation energies etc. within the framework of DFT the time-dependent Kohn-Sham theory has to be derived. Initially the time-dependent Schrödinger equation is considered. Analogously to the time-independent case, the time-dependent electronic Hamiltonian consists of the kinetic energy of the electrons and the two potential terms, the electron-electron repulsion, and a time-independent external potential υ (r) describing the nuclei-electron attraction. Additionally it contains a time-dependent external potential.

(1.2.28)

Excited states are accessible via computationally demanding wave function methods, CI, CASPT2, or CASSCF being the typical examples. Again a better scaling in combination with reasonable accuracy can be achieved by introducing the electron density and its linear response to a time-dependent electromagnetic field.[119] This approach is called time-dependent DFT (TDDFT) and it is derived very similarly to ground-state DFT. Analogously

⇣(r) =⇢↵(r) ⇢β(r)

⇢(r)

He(r, t) = Te(r) + Vee(r) + Vne(r) + V (r, t)

=NX

i=1

✓−1

2r2

i

◆+

NX

i<j

1

rij−

NX

i=1

(r) +

NX

i=1

(r, t)


to the ground-state description, time-dependent equivalents of the two Hohenberg-Kohn theorems have to be introduced. In other words it has to be proven that there exists a one-to-one mapping of the time-dependent electron density ρ(r,t)and the time-dependent external potential υ (r,t) and that the variatonal principle holds in this case. In accordance with the time-independent Kohn-Sham theory the electron density of the non-interacting system is expressed as a single Slater determinant of time-dependent spin orbitals.[135]

(1.2.29)

Again spin labels will be omitted for clarity. The time-dependent analog to the first Hohenberg-Kohn theorem was developed by Runge and Gross.[136] The Runge-Gross theorem rigorously proves that two densities, which evolve through excitation by two external potentials differing by more than a purely time-dependent function c(t), to be different at initial time. As a result of the Runge-Gross theorem the quantum-mechanical action integral A[ρ] is a functional of time-dependent electron density ρ(r,t) and analogously to the ground-state Kohn-Sham theory all system independent terms are gathered in an universal functional B[ρ].[119]

(1.2.30)

with

The exact electron density is then determined by the Euler equation and can be obtained variationally.

(1.2.31)

Assuming a non-interacting system with the same exact electron density as the real system allows the introduction of the exchange-correlation part of the action integral Axc[ρ].

(1.2.32)

⇢(r, t) = N

Z· · ·

Z| (r1, r2, . . . rN, s1, s2, . . . sN , t)|2dr2dr3. . . drNds1ds2. . . dsN

=NX

i

|'i(r, t)|2

A[⇢] = B[⇢]Z t1

t0

Z⇢(r, t)υ(r, t)dtdr

B[⇢] =

Z t1

t0

⌧ [⇢]

��i@

@t T Vee

�� [⇢]

�dt

δA[⇢]

δ⇢(r, t)= 0

Axc[⇢] = BS [⇢]1

2

Z t1

t0

Z Z⇢(r, t)⇢(r0, t)

|r r0| dtdrdr0 B[⇢]


with

All unknown parts are gathered in Axc[ρ] and in accordance with ground-state DFT, the time-dependent Kohn-Sham equation can be derived.

(1.2.33)

Within the framework of the linear response formalism excited-state parameters can be derived. After perturbation by a time-dependent external field the density is assumed to correspond to the sum of the unperturbed ground-state density matrix given by the time-independent Kohn-Sham equation and a first order term describing the time-dependent change.[119] Analogously the Kohn-Sham Hamiltonian is given as the sum of its time-independent form and the first order change in the density matrix formalism. Assuming an infinitesimally small perturbation and making use of known conditions, such as the diagonal nature of the ground-state Kohn-Sham density matrices and Hamiltonian and that both are expressed in the basis of canonical orbitals, the TDDFT equations can be derived.

(1.2.34)

The excitation energies are directly given by ω. Defining a and b as occupied and i and j as virtual orbitals and using chemists’ notation, the A and B matrices, also called orbital rotation Hessians,[137] are given as:

(1.2.35)

In this equation fxc is the time-dependent exchange-correlation kernel. The main part of diagonal elements of the A matrix corresponds to the difference of the Kohn-Sham orbitals 𝜖𝜖! − 𝜖𝜖! involved in the excitation and the unoccupied orbitals are evaluated for the N-electron system. By neglecting the matrix B, i. e. by considering only the occupied-virtual block, the non-Hermitian eigenvalue equation 1.2.34 can be approximated as

(1.2.36)

(Tamm-Dancoff approximation). Within the adiabatic local density approximation, which assumes only slow variations of the electron density with time, the time-dependent exchange-

BS [⇢] =

Z t1

t0

⌧ [⇢]

��i@

@t T

�� [⇢]

�dt

i~NX

i=1

@

@t'i(r, t) =

NX

i=1

✓1

2r2

i + υ(r, t) +

Z⇢(r0, t)

|r− r0|dr0 +

δAxc[⇢]

δ⇢(r, t)

◆'i(r, t)

A BA⇤ B⇤

� XY

�= !

1 00 1

� XY

�

Aia,jb = δi,jδa,b(✏a ✏i) + [ia|jb] [ia|fxc|jb]Bia,jb = [ia|bj] [ia|fxc|bj]

AX = !X


correlation kernel is replaced by a time-independent analogue, which is simply the second functional derivative of standard ground-state exchange-correlation functionals. Analogously to the time-independent case, a hybrid functional formalism of the TDDFT equations can be derived and the incorporation of some exact-exchange admixture into the exchange-correlation potential usually improves the TDDFT accuracy, as the right asymptotic decay 1/r is partly recovered. Problems arise from the fact that the virtual HF orbitals are evaluated for a system with N+1 electrons, while TDDFT assumes an N-electron system.[119]

Regarding the excitation energies of MV systems, another, related problem with standard functionals, e.g. of the GGA type, arises. It is well known that for typical CT transitions with small overlap between the relevant initial- and final-state MO densities, such functionals underestimate the corresponding excitation energies dramatically, and the reasons have been discussed extensively.[138-141] But as nicely pointed out by Peach et al.,[142] some types of CT transitions are less critical in this context. Even more importantly, the problem can be partly remedied by including a suitable amount of exact exchange into the functional. Indeed, the BLYP35 hybrid functional discussed below, with 35% Ex

exact admixture, underestimates Class II IVCT excitation energies only moderately (see e.g. Chapter 2 and Section 3.1).

Spin in Density Functional Theory

Open-shell systems, which contain at least one unpaired electron, are in the main focus of this thesis, and thus spin properties have to be accounted for. In the previous sections DFT and TDDFT were introduced without distinction between α-electron density ρα (r) (spin up, s = +1/2) and β-electron density ρβ(r) (spin down, s = –1/2).

(1.2.34)

(1.2.35)

Obviously the sum of ρα (r) and ρβ(r) equals the electron density ρ(r) . The spin density is simply defined as the difference between ρα (r) and ρβ(r).[143] In principle either spin-restricted or unrestricted open-shell calculations can be performed, but the spin-unrestricted formalism is used in the vast majority of applications. For closed-shell systems the two approaches are equivalent. In the spin-restricted case the spin density is given as the positive sum over all singly occupied orbitals for α-spin orbitals, and the negative sum for β-spin orbitals. The restricted approach fails to describe many magnetic properties, as it is unable to describe spin

⇢↵(r) = N

Z· · ·

Z �� ✓r1, r2, . . . rN,+

1

2, s2, . . . sN

◆��2

dr2dr3. . . drNds2ds3. . . dsN

⇢(r) = N

Z· · ·

Z �� ✓r1, r2, . . . rN,1

2, s2, . . . sN

◆��2

dr2dr3. . . drNds2ds3. . . dsN


polarization. For example isotropic EPR hyperfine coupling constants are, in the non-relativistic formalism, directly proportional to the spin density at the core. In the spin-restricted case only s-orbitals or orbitals with s-character contribute to the spin density at the core, as p-,d- and f-type orbitals exhibit radial nodes at the core.[144] Hence isotropic EPR hyperfine coupling constants are incorrectly calculated to be zero if the singly occupied orbitals have no s-character at this center. In the spin-unrestricted approach the corresponding α- and β-spin molecular orbitals (MOs) are allowed to have different spatial parts.

Spin Polarization and Spin Contamination

Within the spin-unrestricted framework spin polarization is enabled. Spin polarization arises from the different magnitude of interaction of an unpaired electron with electrons of α- or β-spin from doubly occupied orbitals and is given by Equation 1.2.27. As a consequence of this interaction, the corresponding α- and β-spin molecular orbital “pairs”, which are degenerate in the restricted case, differ in their energy and spatial distribution. Hence often an assignment of proper “doubly occupied” MOs, and thus of the proper SOMO is only possible after an overlap matching of the corresponding α- and β-MOs for many MV transition-metal complexes.[145] Spin polarization of the core shells is of fundamental importance for the description of magnetic properties.

In contrast, exaggerated valence-shell spin polarization, especially for orbitals of metal-ligand antibonding character, leads to significant spin contamination,[129,146,147] which leads to errors in calculated structures, orbital populations, and especially spin densities, and thus has proven to be a potential problem. Spin contamination occurs, if more than one spin state contributes to the Slater determinant of the ground state. The ground-state wave function is not an eigenfunction of the S2 operator anymore. Hence the amount of spin contamination corresponds to the difference of the calculated and “spin-pure” expectation value of the S 2 operator.

(1.2.36)

For complexes with a singlet ground state (S = 0.0) the “spin-pure” value is zero, for a doublet ground state 〈S 2 〉 = 0.75 (S = 0.5), for a triplet 〈S2 〉 = 2.00 (S = 1.0), and for a quartet 〈S 2 〉 = 3.75 (S = 1.5). Spin polarization and thus spin contamination typically increases with the amount of exact exchange in the functional.[129,146-149] While the importance of spin polarization for the description of magnetic properties is undisputable, the overall significance and consequences of spin contamination are still actively discussed.[110,147,149-151] The Kohn-Sham determinant in DFT describes the non-interacting reference system and not the real system. In principle the ground states of the reference and the interacting system can be even of different spin multiplicities.[143] Hence one can argue that the expectation value of the S 2 operator derived from an auxiliary quantity is not of physical significance.[110,150] Nevertheless

⌦S2

↵= S(S + 1)


increased spin contamination can notably affect results, for instance electronic structures, EPR parameters, and the oxidation state assignment.[129,146,147,149,151] These findings suggest that the amount of spin contamination to some extent does indeed provide an indication of the quality of a DFT calculation.[150,151] Similar findings were made for TDDFT and the amount of spin contamination was suggested as main criterion to eliminate unphysical excited-states.[152] As a consequence DFT methods, which control the spin-contamination by constraints, have been developed, but so far have not been often applied.[153]

Broken-Symmetry Density Functional Theory

Many catalytically active systems, especially active sites of enzymes, feature more than one paramagnetic transition metal or redox non-innocent ligand center. These are either ferromagnetically or antiferromagnetically coupled. Modeling of such complexes or clusters requires an accurate description of both high-spin and low-spin states, which is a challenge for DFT. The energy difference between the different spin states equals twice the negative of the magnetic coupling constant, Jab, between the centers. The unrestricted single Slater determinant approach in the usual open-shell DFT approaches prohibits the calculation of uncontaminated spin eigenfunctions.[154] The description of magnetically coupled centers thus requires the so-called broken-symmetry DFT approach, in which the high-spin state is calculated and then open-shell orbitals are localized on the different moieties to obtain states of lower total spin. To obtain the energy of the spin-pure states from the broken-symmetry DFT results, spin-projection techniques have to be employed, which were mainly developed by Noodleman et al.[155-158] and Yamaguchi and coworkers.[159-161] In this thesis Yamaguchi’s approach is used, where the Jab is given by the energy difference between the energies of the low-spin ELS and high-spin state EHS divided by the difference in spin expectation values of the respective Slater determinants.

(1.2.37)

Jab =ELS EHS

hS2iHS hS2iLS


1.3 Quantum-Chemical Description of MV Systems

Given the sometimes tough or even impossible Robin-Day classification of systems close to the Class II/Class III borderline based exclusively on experimental data (see Section 1.1), a quantum-chemical description of MV systems is clearly desirable. Additionally especially excited-state properties, e.g. the structure and the adiabatic dipole moment of the excited state, are not or hardly accessible by experimental measurements.[6,162] Despite this fundamental importance, until some years ago there had been a lack of a quantum-chemical protocol able to reliably describe and even more importantly predict properties of MV complexes. For the description of MV systems the contributions from both dynamical and non-dynamical correlation have to be accounted for. As discussed in Section 1.2 only sophisticated post-HF ab initio methods, which are computationally too demanding and scale disadvantageously with system size for most MV systems of practical interest and have rarely been applied,[90,163,164] are able to recover both. Attempts to apply CI methods within a semi-empirical framework have been demonstrated to sufficiently describe a number of organic MV systems.[165-168] But of course due to the semi-empirical parameterization the predictive power of such methods is rather limited.

As the more sophisticated methods are still limited in their application possibilities, a quantum-chemical protocol to universally describe MV systems may take advantage of the error compensation of HF and DFT in hybrid functionals. Due to the lack of Coulomb correlation HF tends to give localized charge distributions even for clear-cut Class III systems. In contrast, the self-interaction error of DFT leads to a too delocalized picture in most cases. The importance of this error is manifested in the term “delocalization error” in DFT introduced by Yang et al.[169,170] Van Voorhis advocated the use of constrained DFT (CDFT) in such situations, where charge localization is enforced by adding constraints to the Hamiltonian.[171-173] Of course the predictive power of such an approach is very limited. But it has been argued convincingly that it provides the ET parameters needed for the Marcus-Hush and generalized Mulliken-Hush treatments based on (enforced localized) diabatic states.[39,171-

176]

Environment Effects and Solvation Models

Most of the relevant MV systems are charged, and experimentally they are typically studied in a polar solvent environment. In fact, experimental gas-phase data are essentially absent in this field. It is thus obvious, that solvent and counter-ion effects are crucial ingredients for a reliable quantum-chemical protocol. Polar solvents will usually stabilize a localized, valence-trapped situation compared to a delocalized charge distribution. Gas-phase calculations, which were the standard approach until about 10 years ago, thus are clearly biased towards a too delocalized description. In fact, there are clear spectroscopic indications that even a moderate increase of solvent polarity may change a Class III to a Class II system, provided it


is close to the borderline.[45,46,177] While sometimes calculations included continuum solvent models in single-point calculations of, e.g., excitation spectra,[178] neglect of solvent effects in the ground-state structure optimization of a Class II system often provided (incorrect) symmetrical structures to start with, thereby invalidating the subsequent spectroscopic calculations.

There are no suitable continuum models to include counter-ion effects. This is why these have so far been largely neglected. It may be expected that counter ions will receive more scrutiny, as molecular dynamics (MD) studies will enter the field. The importance of counter-ion effects depends of course on the system. Bulky MV systems like the typical triarylamine radical cations are likely affected less than smaller MV systems, e.g. some organic radical anions. As the complexes in this thesis mostly contain redox centers with bulky ligand spheres, which sterically protect the charge, counter ions likely have only a minor influence on the charge distribution. Hence counter ions are not included in the calculations.

An explicit inclusion of the solvent into the quantum-chemical model, or possibly a QM/MM ansatz where the solvent is treated classically with a QM treatment of the solute, can in principle provide a realistic description. However, the dynamical nature of the solute-solvent interactions requires such simulations to include the molecular dynamics on a sufficient time scale. While this type of explicit ab initio MD or QM/MM based MD studies has become an important tool in other areas of research, they have not yet entered the stage in the study of MV systems so far, at least not to a notable extent. This is probably due to the complications described in the previous two sections regarding the electronic-structure treatment itself, but it is expected that over the next years dynamical studies will gain substantial importance in this field.

As a consequence, solvents are usually included more efficiently via so-called continuum solvation models. Examples are the conductor-like screening model (COSMO)[179] as implemented in TURBOMOLE[180] and the (conductor-like) polarizable continuum model ((C-)PCM)[181] as implemented in Gaussian.[182] Only electrostatic interactions between solvent and solvated molecule are described by those continuum models and to take into account more specific interactions, e.g. solvent coordination or hydrogen bonding, explicit solvent molecules have to be included in the calculation.

In continuum solvation models the solvent is given as a homogeneous polarizable medium with a cavity, which contains the solvated molecule. Despite the shape of the cavity being subject of debate, it is agreed on that it should resemble the molecular shape.[183] Thus it is normally constructed by taking atom-centered spheres with the van-der-Waals radii and building the cavity as a union of those. Analogously the size of the solvent is estimated and with the two sizes the solvent-accessible surface is obtained. The size of the solvent accessible-surface differs for different quantum-chemical programs. Despite the solvent sphere, the dielectric constant ε is the only solvent-specific parameter used in the continuum solvation models. It determines the polarizability of the solvent. In a self-consistent procedure


the charge distribution within the solvated molecule polarizes the dielectric continuum, which then polarizes the solute charge distribution. This interaction potential, also called solvent reaction field, is added to the Hamiltonian and thus to the iterative self-consistent field procedure.[184,185]

A special aspect of solvent interactions has to be considered for excitation and thus influences the excited-state parameters obtained by TDDFT calculations. For excited states there are two ways to describe the solvent-solute interaction: Equilibrium solvation, which analogously to the ground state corresponds to the solvent orientation at the energy minimum and thus should be included into excited-state optimizations, and non-equilibrium solvation.[186-188] Non-equilibrium solvation describes the situation after a Franck-Condon excitation. While the electronic charge distribution of the solvent molecules can readily react on the optically induced ET, reorientation of solvent molecules is a much slower process. Hence the solvent orientation still corresponds to the ground-state case, and the computed excitation energy is larger for non-equilibrium than for equilibrium solvation. That is, for absorption spectroscopy, the ground-state molecule and the solvent are in equilibrium, whereas for the excited state non-equilibrium solvation has to be considered. For the description of experimental emission spectra the situation is reversed.[162] Differences in the excitation energies obtained with the TURBOMOLE[180] or Gaussian[182] codes mainly arise from the different implementations of non-equilibrium solvation.

Please note that there are reports of remarkably good results from gas phase calculations for selected MV systems, which were obtained using hybrid functionals with an increased exact-exchange admixture[189] or range-separated hybrid functionals,[85] but for other systems especially TDDFT results differ appreciably from the experimental data.[190,191] And the question arises whether the “right” answer was obtained for the “right” reasons.[47]

A Reliable Quantum-Chemical Protocol

The above discussion makes clear that, to reliably describe the properties of MV systems close to the borderline between Class II and Class III, a suitable quantum-chemical protocol has to account in detail for dynamical and non-dynamical correlation without suffering from extensive self-interaction errors, and it has to cover at the same time the relevant environmental effects. Notably, these features should already be included accurately in the ground-state structure optimization, in contrast to many studies found in the literature. Otherwise, computations of excitation energies or other spectroscopic parameters may suffer from the inaccurate structures. For an even more accurate description of excitation spectra, vibronic coupling has to be considered. In the absence of specific solvation effects, such as hydrogen bonding or the dynamic exchange of solvent molecules within the coordination sphere of a metal center, it is expected that the bulk solvent effects may be covered adequately by dielectric continuum solvent models like COSMO[179] or other “polarizable continuum models”.[183,192]


Within a Kohn-Sham DFT setting, the judicial inclusion of exact exchange into the functional should allow a balance to be reached between the treatment of left-right correlation in bonds (improved by semi-local ingredients of the exchange functional) and a reduction of delocalization errors due to self-interaction (improved by more exact exchange). This leads naturally to the use of hybrid functionals, where one may distinguish between constant exact-exchange admixture in global hybrids, an admixture depending on the interelectronic distance in so-called range-separated hybrids,[119-125,127,128,193-195] or a real-space position-dependent admixture in so-called local hybrids.[114,130-132,134,196-199] While detailed validation of the latter for MV systems is still pending, the two former classes have been evaluated in some detail (see Chapter 2), initially for triarylamine (TAA) radical cations, subsequently for organic MV radical anions.

The initial validation of global hybrids for TAA radical cations[45,46] (see Chapter 2) in conjunction with continuum solvent models led to an optimum exact-exchange admixture of 35% in a customized one-parameter hybrid BLYP35, which was constructed analogous to the B1LYP model.[117] While this is an ad hoc construction, later studies on smaller and thus computationally more convenient organic MV radical anions[47,48] allowed an even larger set of functionals to be screened and showed, e.g., that the well-known BMK meta-GGA global hybrid functional[200] also provides a very reasonable compromise. The BMK functional incorporates 42% exact-exchange admixture, somewhat ameliorated by contributions from local kinetic energy density. Global hybrids with appreciably less exact exchange overestimate delocalization, functionals with larger exact-exchange admixtures give too localized electronic structure.[45-49] Indeed, it turned out that the exact-exchange admixture is the by far most important aspect of a global hybrid that has to be selected to achieve optimum performance for the treatment of MV systems.

Range-separated hybrids with 100% exact-exchange in the long range, such as ωB97X,[201] and LC-BLYP,[125] so far tended to provide a too localized description, whereas CAM-B3LYP[127] (which interpolates between 19% at short range and 65% at long range[127,142]) appeared to be just slightly too localized for organic MV radical anions. The currently popular double hybrid functionals exhibit too large exact-exchange admixture to be accurate for MV systems close to the borderline, and the MP2 contribution to the correlation functional tends to be problematic at the transition state of thermal ET for Class II systems.[45] These validation studies, which required the adequate inclusion of solvent effects, will be discussed in some more detail in the following chapter. Overall, the BLYP35/COSMO based protocol and its variations using either suitable alternative functionals or alternative solvent models have turned out to provide an unprecedentedly detailed picture of localization vs. delocalization in a wide variety of MV systems.

The successful extension of the protocol to MV transition-metal complexes and new insights into experimental findings gained from the quantum-chemical perspective are presented in this thesis.


1.4 Computational Detai ls

All calculations, including TDDFT,[202-204] have been performed using a version of the TURBOMOLE 6.4[180] or in rare cases TURBOMOLE 6.3[205] codes locally modified by the Kaupp group, if not mentioned otherwise. For selected cases, additional single-point TDDFT calculations were carried out using the Gaussian09 code,[182] which features a somewhat different treatment of the solvent model for excitations.[48] Following the previously mentioned computational protocol, the adjusted global hybrid functional BLYP35,[45] which contains 35% exact-exchange admixture (a = 0.35 in equation 1.2.23) was employed in conjunction with polarizable continuum solvent models. While not a thermochemically optimized functional, BLYP35 has been shown to provide a good balance between reduced self-interaction errors and a simulation of static correlation. Although other functionals like the BMK global hybrid[200] or the CAM-B3LYP range hybrid[127] have been found to provide a similar balance between localization and delocalization as BLYP35, these are not available in the TURBOMOLE code used in most studies.[47,48]

Solvents were considered by the conductor-like-screening solvent model (COSMO)[179] (and by the closely related C-PCM model[181,186] in the Gaussian09 TDDFT calculations). The solvents employed in experiments were considered without the respective supporting electrolyte. Most experiments were carried out in CH2Cl2 (permittivity ε = 8.93) and in MeCN (ε = 37.64). Notably, the TDDFT calculations took into account non-equilibrium solvation.[187,188]

For all calculations, split-valence basis sets def2-SVP on the lighter atoms and the associated Stuttgart effective-core potentials with a corresponding def2-SVP valence basis for transition metals were employed.[206-208] Calculated harmonic vibrational frequencies were scaled by an empirical factor of 0.95.[209,210]

Spin densities are plotted in form of isosurfaces (with values ± 0.002 a.u.) using light gray for positive and red for negative values. Similarly, molecular orbitals are presented as isosurface plots (± 0.03 a.u.) using light gray and blue colors for positive and negative signs, respectively. In the ball-and-stick plots, the atoms are also color-coded (iron, ruthenium, rhenium, osmium, and chlorine yellow, phosphorus green, carbon grey, molybdenum, nitrogen, and silicon blue, oxygen red, and platinum and hydrogen white). These plots were done with the Molekel program.[211]

Orbitals are designated with their spin (α spin-up, β spin-down) and relative to the number of the highest occupied α-spin-orbital labeled as α-SOMO and the second-highest α-spin-orbital labeled α-HOMO. The β-SOMO is thus the lowest unoccupied β-spin-orbital and the β-HOMO the highest occupied β-spin-orbital.

CHAPTER 2

THE COMPUTATIONAL STATE OF THE ART

Initially the BLYP35/COSMO protocol was validated for purely organic triarylamine (TAA) cations,[45] and later was transferred to a wider range of TAA systems, including neutral unsymmetrical neutral TAA-perchlorotriphenylmethyl radicals.[46] This work has been reviewed recently.[46] These systems are of interest due to a range of possible applications, and they have been studied by a wide variety of spectroscopic and computational methods, and some of the problems discussed in the previous sections were observed by a variety of authors.[6,45,46,49,78,212-223] Moreover, their character can be tuned very well by the choice of bridge unit, end cap, and even by the solvent, so that a wide variety of data for cases close to the border between Class II and Class III situations (from both sides) is available. Regarding the challenges for a computational treatment, the TAA-based systems have advantages and disadvantages: the relatively large size of the experimentally studied systems puts appreciable demands on computational efficiency, whereas the bulky aryl substituents at the amine redox centers tend to shield the positive charge well, causing specific solvation effects to be of minor importance in most cases. The latter aspect supports the convenient use of continuum solvent models.

The systematic variation of exact-exchange admixture in B1LYP-style global hybrids, and of the dielectric constant of the continuum solvent model appropriate to the condition of the experimental studies available for an appreciable variety of TAA-based MV radical cations led to the abovementioned preference for the BLYP35 functional combined with continuum solvent models as the basis for the initial version of the protocol discussed above.[45,46] Notably, TDDFT calculations using the same functional, solvent, and basis sets (in these

CHAPTER 2: COMPUTATIONAL STATE OF THE ART 36

initial studies SVP basis sets) provided surprisingly good agreement with the experimental excitation energies for the IVCT band (Figure 6). For Class II systems, the excitation energies were underestimated somewhat (red squares in Figure 6). This had to be expected, as in this case the IVCT band has distinct CT character, and DFT tends to underestimate such excitation energies, sometimes dramatically so. However, the maximum deviations were rather tolerable, about 1000-1500 cm-1, suggesting that 35% exact-exchange admixture turns out to be a reasonable compromise. It may be assumed that the deviations are largest for the most-weakly coupled systems. Indeed, for strongly coupled delocalized Class III cases, where the character of the IVCT band is more that of a delocalized single-chromophor π-π* excitation, the excitation energies were overestimated by up to ca. 1000 cm-1 (blue circles in Figure 6). Most importantly, the very systematic performance confirmed the accurate description of the underlying optimized ground-state structures, as the excitation energies crucially depend on whether a localized or delocalized structure is used.

Figure 6. Comparison of BLYP35/SVP/COSMO(CH2Cl2) IVCT excitation energies with experimental data for bis-triarylamine radical cations in CH2Cl2 solvent.[46] Adapted from M. Kaupp, M. Renz, M. Parthey, M. Stolte, F. Würthner and C. Lambert, Phys. Chem. Chem. Phys., 2011, 13, 16973. by permission of the PCCP Owner Societies.

These validation studies showed clearly that a) standard functionals like B3LYP exhibit too large delocalization errors to be reliable for Class II TAA systems, and b) the inclusion of solvent effects at least by a continuum solvent model is crucial for Class II cases. However, it appears that this knowledge is not yet sufficiently widespread, as even very recent studies have used inappropriate computational levels, resulting in rather inaccurate results.[224-228]


In line with the results shown in Figure 6, somewhat lower exact-exchange admixtures tend to lower IVCT excitation energies and thus actually improve agreement with experiment for Class III cases, as demonstrated by a recent study[229] at PBE0[117] level (25% exact-exchange; COSMO solvation), or by work on multibranched TAA systems using B3LYP and a PCM model.[230] In a TDDFT study of Class III triarylamine-substituted arylene bisimides,[178] the CAM-B3LYP range-separated hybrid functional (at B3LYP/gas phase optimized structures) yielded reasonable excitation energies. Use of a PCM model in the TDDFT single-points improved significantly the agreement between computed and experimental ionization potentials and electron affinities in CH2Cl2.

Figure 7. Left: Dinitro radical anions investigated in ref. [48]. Right: Computed excitation energies with different functionals for the Class III system DN1 (top) and for the Class II system DN2 (bottom) in two solvents (C-PCM) compared to experimental values in MeCN (11000 cm–1 for DN1 and 8320 cm–1 for DN2).[48] Reprinted with permission from M. Renz, M. Kess, M. Diedenhofen, A. Klamt and M. Kaupp, J. Chem. Theory Comput., 2012, 8, 4189. Copyright 2012 American Chemical Society.

A very recent DFT study on cross-conjugated TAA MV systems dealt with conformational effects on ET.[231] Notably, PBE0/6-31G(d) calculations with a PCM acetonitrile solvent model correctly reproduced localized ground-state minima for the relatively weakly coupled Class II systems involved. Not unexpectedly, the IVCT transition energies were, however, underestimated by a factor of about two, reflecting deficiencies of the TDDFT treatment with a functional containing only 25% Ex

exact admixture. Yet good agreement with experimentally determined electronic-coupling matrix elements was obtained, and these depended significantly on conformation. Interestingly, an evaluation of the range-separated hybrid

NO2

O2N

NO2O2N

NO2O2N

O2N NO2

NO2O2N

1,4-dinitrobenzene 1,3-dinitrobenzene

2,7-dinitronaphthalene 2,6-dinitronaphthalene

4,4'-dinitrotolane

2,2'-dimethyl-4,4'-dinitrobiphenyl

DN1

O2N

NO2

DN2

DN3


functionals CAM-B3LYP and LC-ωPBE indicated a drastic overestimate of the experimental IVCT excitation energies,[231] in particular for the latter functional.

This superior performance of the BLYP35/COSMO (or BLYP35/C-PCM) combination led us to investigate metal-bridged bis-TAA systems as starting point for the evaluation of the protocol for MV transition-metal complexes. But also for other organic MV systems, such as dinitro aryl[48] and diquinon radical anions[47] this approach performed best. Especially in the first case, a wide range of functionals including meta-GGA global hybrid functionals representing different exact-exchange admixtures (the Minnesota M05,[232] M06,[233] M05-2X,[234] M06-2X[233] and the BMK[200] functional), range-separated hybrid functionals (CAM-B3LYP,[127] ωB97X,[201] and LC-BLYP[125]), and double-hybrids (B2PLYP[235] and B2PLYPD[236]) was tested. A wide range of ground-state properties was considered, and TDDFT calculations were applied to the IVCT excitation (cf. Figure 7). Nevertheless, only the CAM-B3LYP and BMK functional were found to be competitive for these systems.

CHAPTER 3

APPLICATIONS TO MV TRANSITION-METAL COMPLEXES

3.1 Platinum Bridged Bis-Triarylamine Complexes

Introduction

Bis-biarylamines and bis-triarylamines, which incorporate a metal in the bridge unit,[237-241] exhibit behavior very similar to purely organic bis-triarylamine radical cations, as the bridge tends to be involved only marginally/indirectly in the ET process.[237,240-242] Yet, the spectroscopy of these systems may become richer in case of transition-metal systems, as in addition to the NàN+ IVCT transition also metalàN+ CT transitions are usually obtained in the UV-vis-NIR region. Hence they represent an interesting target to test the applicability of the protocol developed for purely organic bis-triarylamines to transition-metal complexes.

Previous Computational Approaches

So far it appears that mainly Class II systems have been reported.[237,240-242] Compounds in which thermal ET may proceed via the purely organic 2-2’-bipyridine part of the bridge (coordinating to different transition metals) may be a notable exception.[239] Here introduction of iridium leads to a Class II/Class III borderline complex, whereas ruthenium or rhenium bridges give Class II behavior. The BLYP35/C-PCM(CH2Cl2) protocol was applied to those

CHAPTER 3: APPLICATIONS TO MV TRANSITION-METAL COMPLEXES 40

systems. It gave good account of the influence of solvent effects, but the authors concluded that the distinction between purely organic and transition-metal-bridged systems was not completely faithful.[239]

For a weakly coupled Class II gallium-bridged TAA MV system, it was found that M06/def2-SV(P) calculations in an integral equation formalism variant of the PCM model (IEFPCM, e.g. reviewed in [183]) solvation model faithfully reproduced the Class II behavior. However, the corresponding TDDFT calculations underestimated the IVCT energy significantly (3764 cm–1 compared to 6390 cm–1).[242] This is not surprising in view of the results in Figures 6 and 7 and the fact that M06 exhibits only 27% exact-exchange admixture, i.e. less than the BLYP35-based protocol.

Figure 8. Platinum compounds studied in this section.

A previous report by Jones et al. has shown that the closely related platinum-bis(alkynyl) bridged bis-triarylamine compound trans-[Pt{C≡CC6H4N(C6H4OCH3-42)2}2(PEt3)2 also undergoes amine centered oxidations, giving rise to an unusual metal-bridged organic MV compound, the Pt center serving to limit extensive delocalization of the radical and engineering Class II MV character in the one-electron oxidized species.[237] In this earlier work, TDDFT calculations (B3LYP/LANL2DZ) were performed on the crystallographically determined structure of trans-[Pt{C≡CC6H4N(C6H4OCH3-42)2}2(PEt3)2 to support the spectroscopic observations made on the family of neutral, mono, and di-cationic complexes, the latter two being obtained by stoichiometric chemical oxidation. The key localized orbitals thought to be involved in the IVCT transition in the MV monocation were derived from a linear combination of the HOMO and HOMO–1 of the neutral compound following from a Koopmans’ Theorem based treatment of the electronic coupling. In order to further develop the quantum-chemical protocols and understanding of intra-molecular CT events, a combined experimental and computational study of the spectroscopic properties of the family of platinum-bis(alkynyl) bridged bis-triarylamine complexes trans-[Pt(C≡CC6H4NAr2)2(PR3)2]n+ (R = Et or Ph, Ar = C6H4CH3-4 or C6H4OCH3-4, Pt1 – Pt4; n = 0, 1, 2) and the model mono-amine complex trans-[Pt{C≡CC6H4N(C6H4Me-4)2}(C6H4CH3-4)(PPh3)2] (Pt5) was performed (Figure 8).

N

Me

Me

C C C C N

Me

Me

PtPR3

PR3

N

MeO

MeO

C C C C N

OMe

OMe

PtPR3

PR3

C C N

Me

Me

PtPPh3

PPh3

Me

Pt1: R=EtPt3: R=Ph

Pt2: R=EtPt4: R=Ph

Pt5


Experimental Situation

Figure 9. IR spectra of Pt1 (a) and Pt5 (b) showing the ν(C≡C) band in the various oxidation states generated by in situ electrochemical oxidation in CH2Cl2/0.1 M NBu4PF6 in an OTTLE cell collected by Kevin Vincent.

Spectroscopic data permit a more detailed assessment of the physical and electronic structures of the redox products than electrochemical measurements used in isolation. While the UV-vis-NIR spectrum of [Pt2]+ has been determined from samples prepared by chemical oxidation of Pt2,[237] Kevin Vincent conveniently obtained spectroscopic information for the series [Pt1 – Pt4]n+ by using spectroelectrochemical methods.[87] The IR spectra of the neutral complexes Pt1 – Pt4 are characterized by a single weak ν(C≡C) band (Table 1), consistent with the highly symmetric distribution of electron density over the molecular backbone. On oxidation, the IR band profile evolves through a unique ν(C≡C) band pattern associated with the comproportionated equilibrium mixture of the neutral, monocationic, and dicationic states, to a strong absorption feature characteristic of the dicationic state (Figure 9). The observation of multiple ν(C≡C) bands associated with the monocation, perhaps due to the effects of Fermi coupling,[243] in combination with the low comproportionation constant, which ensures solutions of the monocations also contain appreciable amounts of the neutral and dicationic forms, makes definitive assignment of the individual vibrational modes associated with the monocationic state difficult. The assignment is assisted, however as the unique features of the IR spectra associated with the monocationic states are coupled with a rise and fall in a unique low energy (NIR) electronic absorption band near 5000 cm–1 (the character of which is discussed in more detail below). The observation of IR spectral band profiles for the mono-oxidized complexes [Pt1 – Pt4]+ which resemble a superposition of the spectra of the associated neutral and dicationic forms suggests a description of these species in terms of a localized MV structure. This suggestion is also consistent with the IR spectra of the mono-amine compounds [Pt5]n+ (n = 0, 1) that shows a clear shift of the ν(C≡C) band on oxidation to the monocationic species from 2103 to 2013 cm–1 (Figure 9b).

Kevin Vincent collected the UV-vis-NIR spectra of [Pt1 – Pt4]n+ (n = 0, 1, 2) and [Pt5]n+ (n = 0, 1) using spectroelectrochemical methods from ca. 1 mM solutions in 0.1 M NBu4PF6/CH2Cl2 and data were corrected for the comproportionation equilibria.[244-246] Complexes Pt1 – Pt5 exhibit roughly the same overall electronic absorption pattern in the


UV-vis-NIR region in their different oxidation states (Figure 10). The neutral complexes each feature two pronounced bands near 33000 cm–1 and 25000 cm–1, arising from the two Nàπ* transitions commonly observed in triarylamine complexes of general form NArAr’2.[221,247,248] No transitions are evident below 22500 cm–1 in the electronic spectra of Pt1 – Pt5.

Table 1. Experimental and calculated C≡C stretching frequencies [cm–1] for complexes [Pt1 – Pt4]n+ (n = 0, 1, 2) and [Pt5]n+ (n = 0, 1).[a] Theoretical relative intensities [%] for the monocationic forms in parenthesis.

0 +1 +2 Pt1 exp. 2100 (s) 2018 (s)

2046 (sh) 2070 (sh) 2100 (s)

2018 (s) 2046 (sh) 2070 (sh)

BLYP35 2119 2031 (92) 2098 (100)

2043

B3LYP 2067 1994 2001 Pt2 exp. 2100 (s) 2025 (s)

2046 (sh) 2065 (sh) 2100 (s)

2025 (s) 2046 (sh) 2065 (sh) 2073 (s)

BLYP35 2119 2048 (100) 2110 (19)

2072

Pt3 exp. 2106 (s) 2024 (s)

2049 (sh) 2079 (sh) 2106 (s)

2024 (s) 2049 (sh) 2079 (sh)

BLYP35 2131 2029 (100) 2111 (55)

2043

Pt4 exp. 2106 (s) 2030 (s)

2050 (sh) 2070 (sh) 2106 (s)

2030 (s) 2050 (sh) 2070 (sh)

BLYP35 2130 2050 (100) 2126 (8)

2075

Pt5 exp. 2103 2013 (s)

2047 (sh)

BLYP35 2129 2021 [a]calculated IR frequencies were scaled by an empirical factor of 0.95.[ 209, 210]


Figure 10. UV-vis-NIR spectra of Pt1 (a), Pt2 (c), Pt3 (e), Pt4 (g) and Pt5 (i) in the various oxidation states generated by in situ electrochemical oxidation in CH2Cl2/0.1 M NBu4PF6 in an OTTLE cell collected by Kevin Vincent. Expansions of the NIR region are given in (b), (d), (f), (h) and (j) for [Pt1 – Pt5]+ respectively.

Upon oxidation of Pt1 – Pt4 to [Pt1 – Pt4]+, a broad, low intensity IVCT band near 6000 cm–1 and a distinct absorption band near 10000 cm–1 appears in each case (Figure 10). However, the visible spectra of the tolyl systems Pt1 (Figure 10a and b) and Pt3 (Figure 10e


and f) differ noticeably from those of the anisyl derivatives Pt2 (Figure 10c and d) and Pt4 (Figure 10g and h). Thus, while the spectra of Pt1 and Pt3 exhibit a series of unresolved transitions leading to a broad featureless absorption band envelope between 15000 and 20000 cm–1, for Pt2 and Pt4 a much better resolved band is apparent below 15000 cm–1 in each case (Figure 10), in addition to some unresolved bands at higher energy that are similar to those seen in the tolyl derivatives. For all complexes the two absorptions at 25000 cm–1 and 33000 cm–1 become less intense and new features arise between 20000 cm–1 and 25000 cm–1 upon oxidation. This change in shape for the localized amine transitions is consistent with previous studies of the electronic spectra of triarylamine radical cations.[221,247-250] Upon further oxidation to the dicationic forms, the low energy bands between 5000 cm–1 and 7000 cm–1 collapse, supporting the IVCT assignment made here and elsewhere for [Pt2]+.[237]

and all the other absorptions below 25000 cm-1 gain intensity (Figure 10). Oxidation of compound Pt5 (Figure 10i and j) gives rise to a spectrum that shows the changes of all the high-energy features of the spectra for [Pt1 – Pt4]+, with the absence of the lowest energy transitions.

Results and Discussion

Computational Results

Previously, the electronic and spectroscopic properties of [Pt2]+ were inferred from gas-phase calculations based on the crystallographic structure of Pt2, reflecting the available computational capacity at the time.[237] To better understand the underlying electronic structure of these prototypical metal-bridged organic MV systems, the complete series of complexes [Pt1 – Pt4]n+ (n = 0, 1, 2) and the mono-nuclear system [Pt5]n+ (n = 0, 1) was studied employing quantum-chemical methods, using the BLYP35 functional and the COSMO solvent model. This provides an alternative to the simpler computational models employed previously.[237] Unrestricted structure optimization of [Pt1´ – Pt4´]n+ (n = 0, 1, 2) and [Pt5´]n+ (n = 0, 1) gave stable minima evidenced by the absence of imaginary frequencies (the prime notation is used to distinguish the computational systems from the experimental complexes). For all neutral and dicationic complexes the singlet state gave the lowest energy. For the dications [Pt1´ – Pt4´]2+ the spin expectation value 〈S2 〉 of 1.1 indicates a broken-symmetry (BS) description for a system with one antiferromagnetically coupled pair of spins. Indeed the singlet-triplet separation, which is calculated using the Yamaguchi spin projection procedure,[159,161] is only 1.8 kJ/mol for [Pt1´]2+, 0.8 kJ/mol for [Pt2´]2+, 2.4 kJ/mol for [Pt3´]2+, and 0.8 kJ/mol for [Pt4´]2+. The IR frequencies (e.g. [Pt1´]2+: ν(C≡C) = 2046 cm–1 for the triplet vs. 2043 cm–1 for the singlet) and ground-state properties (e.g. [Pt1´]2+: d(N-C6) = 1.383 for the triplet vs. 1.380 for the BS singlet), which are discussed here, do not appreciably differ for the two spin states and thus only the singlet states are considered further. For the very important monocationic (MV) compounds, a doublet configuration is


obtained as would be expected, and calculations exhibit almost no spin contamination with the spin expectation values of 0.78 being only slightly larger than the theoretical value for doublet systems (= 0.75).

The neutral and dicationic structures [Pt1 – Pt4]n+ (n = 0, 2) are effectively symmetric, and some general trends can be observed across the series. In general the distances between the amine nitrogen atom and the first carbon atom of the bridge C6, d(N-C6), are the most influenced by the oxidation state: they contract upon oxidation (Table A1; for atom labeling, see Figure A1 in appendix). The d(N-C6) are slightly longer for the neutral tolyl systems Pt1´ and Pt3´ (1.411 Å and 1.413 Å) than for the anisyl complexes Pt2´ and Pt4´ (1.407 Å and 1.408 Å), and the d(N-C6) bond lengths are significantly shorter in the dicationic state for [Pt2´]2+ and [Pt4´]2+ by some 0.012 - 0.016 Å (1.395 Å and 1.392 Å) and shorter still for the tolyl derivatives [Pt1´]2+ and [Pt3´]2+ by 0.031 - 0.036 Å (1.380 Å and 1.377 Å). For Pt1´, Pt2´, Pt3´, [Pt1´]2+ and [Pt2´]2+ the two phenyl rings of the bridge and the P-Pt-P axis are in plane, as the averaged dihedral angle Ω (average of ∠(P1-Pt-C3-C4) and ∠(P2-Pt-C3-C8)) reaches a maximum value of 12.6 ° for [Pt2´]2+. In general this torsion angle is larger (> 23 °) for complexes with triphenylphosphine ligands Pt4´, [Pt3´]2+ and [Pt4´]2+, as expected due to the increased steric interactions.

Figure 11. Spin-density isosurface (top left, ±0.002 a.u.) and isosurface plots (±0.03 a.u.) of the α-SOMO (top right), α-HOMO (bottom left) and the β-SOMO (bottom right, virtual orbital) of [Pt1´]+. Small differences in the α-SOMO and β-SOMO arise from spin polarization.

For the mixed-valence monocationic structures [Pt1´ – Pt4´]+, minima with clearly localized electronic structures were obtained. In each case a significant asymmetry in the two halves of the molecule is observed, and the corresponding d(N-C6) distances differ by at least 0.16 Å (Table A2), clearly pointing towards a localization of the redox event in [Pt1´ – Pt4´]+ to one


amine, consistent with the large calculated total dipole moment (>25 D; Table A2) in each case, with the conclusions reached above on the basis of the IR data, and with the electronic spectroscopic data described by Jones et al.[237]

The localized behavior is further verified by the partial localization of the canonical orbitals and spin densities obtained for each of the compounds [Pt1´ – Pt4´]+ (cf. Figure 11). The spin density is predominantly centered at one of the triarylamine units with only a very small contribution from the platinum and supporting phosphine ligands. While calculations exhibit almost no spin contamination, the onset of spin polarization is observed: contrary to expectations a β-orbital (not shown in Figure 11) represents the occupied orbital with the highest energy ([Pt1´]+: –5.55 eV; [Pt2´]+: –5.39 eV; [Pt3´]+: –5.50 eV; [Pt4´]+: –5.34 eV). Taking the overlap as criterion this β-orbital can be matched with the energetically highest occupied α-orbital ([Pt1´]+: –5.60 eV; [Pt2´]+: –5.41 eV; [Pt3´]+: –5.54 eV; [Pt4´]+: –5.35 eV) to give the HOMO. The HOMO and SOMO are located on opposite halves of the complex (e.g. composition of the α-part Ar2C6H4C≡C/Pt(PEt3)2/C≡CC6H4N+Ar2 for [Pt1]+: HOMO: 0%/3%/91%; SOMO: 67%/9%/13 %) (Figure 11), and are similar in topology to the description by Jones et al.[237]

The subsequent vibrational frequency calculations show one ν(C≡C) band for the neutral and the dicationic complexes and two for the monocationic forms (Table 1). The frequencies obtained within a harmonic vibrational approach match the experimentally observed ones reasonably well after scaling by an empirical factor of 0.95,[209,210] although the calculations suggest a slightly larger intensity for the higher-energy band in [Pt1´]+ in contrast to the experimental data. For all other cationic complexes, computations give the correct relative intensities. Full optimization using the commonly employed hybrid functional B3LYP leads to a delocalized Class III charge distribution for [Pt1´]+ and thus to a single ν(C≡C) band at 1994 cm–1. However, it must be noted that the calculations do not reproduce the precise details of the splitting of the various ν(C≡C) bands, possibly due to neglect of Fermi coupling in the calculated results.

To gain more insight into the origin of the electronic transitions observed in the UV-vis-NIR spectroelectrochemical studies, and to both confirm the identity of the IVCT bands and test the accuracy of the computational methods, TDDFT calculations were performed for [Pt5´]+ and the mixed-valence monocations [Pt1´ – Pt4´]+ using the BLYP35 functional, the same basis sets, and solvent models (using non-equilibrium solvation as implemented in TURBOMOLE 6.4, Table 2). The results for [Pt1´]+ and [Pt2´]+ are illustrative for the entire series and will be discussed by way of example. The assignment of the broad low-energy NIR-band of [Pt1´]+ with its peak center around 5000 to 6000 cm–1 to an IVCT is straightforward on the basis of the quantum-chemical protocol. TDDFT calculations give the lowest excitation energy in [Pt1´]+ at 5572 cm–1 arising from the β-HOMO (231β) to β-SOMO (232β) transition (Figure 12). This transition has appreciable N→N+ character, with significant contribution from the ethynyl parts of the bridge (Figure 12, Table 2). As might be


expected from other studies reported elsewhere,[237,251] the mixing of platinum d-orbital character with the alkynyl π-system in the ground state is relatively limited, and there is only a small contribution from the Pt d-orbitals to the β-HOMO (2 %) and β-SOMO (6 %).

Table 2. Calculated excited-state parameters for [Pt1´]+ and [Pt2´]+: UV-vis-NIR transition energies Etrans [cm–1] and transition dipole moments µtrans [D], and character of the excitation at the BLYP35/COSMO(CH2Cl2) level.

[Pt1´]+ [Pt2´]+

# Etrans [cm–1] (µtrans [D])

character Etrans [cm–1] (µtrans [D])

character

1 5572 (15.3) N→N+, intervalence CT 5628 (13.0) N→N+, intervalence CT

2 11252 (6.6) π→N+ bridge to amine CT 11062 (8.2) π→N+ bridge to amine CT

3 12046 (4.0) Pt+→N+ metal to amine ligand CT

13588 (1.2) Pt+→N+ metal to amine ligand CT


16000 (1.6) π→N+ bridge to amine CT

5 16162 (1.3) π→N+ bridge to amine ligand CT

16529 (4.9) intra-ligand transition at triarylamine



7 19679 (1.7) intra-ligand transition at triarylamine


The TDDFT calculations also suggest that the intense asymmetric band envelope observed

in the experimental spectrum of [Pt1]+ at around 10000 cm–1 arises from two excitations, computed at 11252 cm–1 and 12046 cm–1 in [Pt1´]+. The transition at 11252 cm–1 is calculated to have substantial β-HOMO–1 (230β)→β-SOMO (69% contribution) and 228β→β-SOMO (12% contribution) character. The β-HOMO–1 and the 228β orbital are both unevenly distributed over the molecular backbone (Figure 12) and whilst the β-HOMO–1 exhibits noticeable Pt(d)/C≡C (13%/26%) character, there is substantially less d-orbital character (2%) in 228β. Thus the transition at 11252 cm–1 also exhibits CT character, and is best described as a πàN+ excitation, with considerably more bridge character in the donor orbital than for the IVCT transition at 5572 cm–1. The transition at 12046 cm–1 arises from transitions between 227β (41% contribution) and 229β (34% contribution) and the β-SOMO. Both orbitals are mainly localized in the metal coordination sphere (50 % and 24 %) and the ethynyl parts of the bridge (34% and 67%), giving this transition significant MLCT (Pt→N+) character. Four low-intensity excitations are computed for [Pt1´]+ between 15000 and 20000 cm–1, consistent


with the observation of multiple transitions in this range (Figure 10). The calculated transitions at 15790 cm–1 (µtrans = 1.0 D) and 17548 cm–1 (µtrans = 0.2 D) also have significant MLCT character, while the excitation at 16162 cm–1 (µtrans = 1.3 D) corresponds to a π→N+ transition. The highest energy calculated excitation at 19679 cm–1 (µtrans = 1.7 D) can be attributed to an intra-ligand transition associated with the oxidized triarylamine moiety.

Figure 12. Isosurface plots (±0.03 a.u.) of the orbitals involved in the first three UV-vis-NIR transitions for monocationic radical [Pt1´]+; transition energies and transition dipole moments are given in parentheses.

Based on the computed transition dipole moments, the calculations would appear to overestimate the relative intensity of the IVCT band significantly (in fact, the computed transition dipole moment of 15.3 D is larger than the sum of those for the second, µtrans = 6.6 D, and third transitions, µtrans = 4.0 D, while the experimentally observed IVCT band at 5000 to 6000 cm–1 exhibits much lower intensity than the band envelope near 10000 cm–1). The low intensity and the broadening of the IVCT band in Class II systems is well-known to be caused by vibronic coupling.[6,57] The computations do not take into account vibronic coupling and may thus not be expected to reproduce the relative intensities faithfully.

The first three transitions of [Pt2´]+ exhibit very similar character to that of the tolyl analogue [Pt1´]+. Interestingly, the calculated ‘IVCT’ transition in [Pt2´]+ at 5628 cm–1 (µtrans = 13.0 D) is blue-shifted by only 56 cm–1 relative to the analogous transition in [Pt1´]+


(Table 2). The quantum-chemical treatment thus suggests a less pronounced influence of the methoxy groups on the IVCT band than found experimentally (Figure 10). The BLYP35/COSMO combination has been found to moderately but systematically underestimate the IVCT excitation energies for related organic Class II MV radical cations.[45,46] It is conceivable that this underestimate will be more pronounced in excitations with greater ‘true’ CT character. In comparison to [Pt1´]+ the second (π→N+) transition of [Pt2´]+ is red-shifted by 190 cm–1, whilst the calculated third (MLCT) transition for [Pt2´]+ is some 1500 cm–1 higher in energy and of lower intensity than for [Pt1´]+. This compares well with the experimental observations (Figure 10) with [Pt1]+ featuring the band envelope at 9750 cm–1 whilst in [Pt2]+ comparable features are observed near 9950 cm–1

(Table 2).

Also consistent with experimental observation, the computed transitions between 15000 and 20000 cm–1 change appreciably when going from tolyl ([Pt1´]+) to anisyl ([Pt2´]+) substituents on the amine moieties (Table 2). The experimental data for [Pt2]+ in this region is dominated by an intense band centered at 14500 cm–1, which compares with a less intense broad band in [Pt1]+ between 14000 and 20000 cm–1. The TDDFT calculations for [Pt2´]+ features two excitations of similar energy at 16000 cm–1 and 16529 cm–1. While the first exhibits a relatively low transition dipole moment (µtrans = 1.6 D), the second is more intense (µtrans = 4.9 D), and thus is likely the major contributor to the experimentally observed absorption band. However, the character of these two transitions differs appreciably. The lower-energy, lower-intensity excitation at 16000 cm–1 originates from delocalized orbitals (244β and 246β) and goes to the β-SOMO (248β). It thus is best described in terms of a π→N+ transition (Table 2). The higher-energy, more intense excitation at 16529 cm–1 originates essentially from a single orbital, which is distributed over the same Ar2NC6H4C≡C moiety (94%) as the β-SOMO. The excitation thus exhibits no CT character and is better described as a localized transition arising from the amine radical cation (Table 2). The TDDFT calculations also predict three more low-intensity excitations at 17108, 19112, and 19562 cm–1 (µtrans ≤ 0.6 D), which are unlikely to have a significant role in determining the overall appearance of the absorption spectrum.

Turning attention briefly to the triphenylphosphine complex [Pt3]+ a small blue shift for the two lowest-energy absorption bands is observed in the experimental spectrum relative to those in the PEt3 analogue [Pt1]+ (Figure 10f). This trend is reproduced in the TDDFT calculations with the computational models [Pt3´]+ and [Pt1´]+ (Table 2, Table 3) and likely reflects the lower electron-donating ability of the PPh3 ligand which consequently lowers the energy of the various donor orbitals. The most obvious difference in the TDDFT results from [Pt3´]+ relative to [Pt1´]+ is the character of the calculated transition at 11295 cm–1 (5.5 D), which is of mixed π→N+ and MLCT character for [Pt3´]+. The IVCT excitation in [Pt3´]+ calculated at 5716 cm–1 is at almost the same energy as calculated for the PEt3 derivative [Pt1´]+, and the limited influence of the phosphine substituents on the IVCT band is a further indication of the limited role the metal center plays in this transition. The largest energy shift between [Pt1´]+ and [Pt3´]+ can be observed for the third transition (11960 cm-1), which is due to the coupling


of π→N+ type and MLCT transitions for the associated band. Similarly, the character of the transitions for [Pt4´]+ (e.g. 5626 cm–1, µtrans = 11.0 D) are nearly unchanged compared to [Pt2´]+.

Table 3. Calculated excited-state parameters for [Pt3´ – Pt5´]+. UV-vis-NIR transition energies Etrans [cm–1] and transition dipole moments µtrans [D], and character for the first three excitations at the BLYP35/COSMO(CH2Cl2) level.

[Pt3´]+ [Pt4´]+ [Pt5´]+

Etrans [cm–1] (µtrans [D])



character

5716 (13.8) N→N+ IVCT 5626 (11.0) N→N+ IVCT 10543 (7.0) π-tolyl→N+ CT

11295 (5.5) mixed π→N+ and Pt+→N+ MLCT


11532 (8.7) intraligand-transition at triarylamine



13892 (1.9) mixed π-tolyl→N+ and Pt+→N+ MLCT

As expected, TDDFT calculations from the monocation [Pt5´]+ does not exhibit any transitions below 10000 cm–1. The absorption pattern above 10000 cm–1 exhibits the same features as [Pt3´]+. The first excitation of [Pt5´]+ is calculated at 10543 cm–1, which is red-shifted by about 700 cm–1 compared to [Pt3´]+ (11295 cm–1). This can be explained by the nature of the involved orbital. For [Pt3´]+ this transition occurs from a delocalized orbital, which has significant contributions from the second triarylamine group (see SI of [252]). As this group is absent in [Pt5´]+, the orbital is destabilized (–6.41 eV compared to –6.60 eV for [Pt3´]+) in the ground state and is predominantly localized at the tolyl ligand (78 %) (see SI of [252]).

Conclusions

This combined spectroscopic and computational study demonstrates that the nature of the UV-vis-NIR transitions observed upon oxidation of the platinum-bridged bis-triarylamine compounds Pt1 – Pt4 are consistent across the series, and that the effect of the phosphine ligand on most of them is relatively insignificant. These transitions have been accurately modeled using a computational protocol based on the BLYP35 functional and a suitable solvent model, allowing a more detailed investigation than was possible a few years ago. As expected for localized MV systems, the lowest energy transition at around 5600 cm–1 corresponds to an IVCT excitation for all bis-triaryamine complexes [Pt1 – Pt4]+. Generally the agreement between computed and experimentally observed IVCT excitation energies is better for the tolyl complexes [Pt1]+ and [Pt3]+ than for the anisyl substituted compounds


[Pt2]+ and [Pt4]+. While the assignment of the second excitation to a CT transition from a delocalized π orbital and of the third to a CT from the platinum moiety to the cationic triarylamine unit is straightforward for [Pt1]+ and [Pt2]+, those excitations start to mix for the triphenylphosphine complexes [Pt3]+ and [Pt4]+. For complex [Pt5]+ no IVCT transition is observed due to the absence of the second triarylamine unit. Here the first excitation corresponds to a CT transition from a π orbital, which is mainly tolyl-centered, to the cationic triarylamine unit.


3.2 Ferrocenyl Ene-diynes

Introduction

So far only redox-active moieties linked by linear conjugated π-systems have been discussed. Additionally cross-conjugated structures are also attracting attention driven by similar ambitions.[253] The unique geometric properties of the prototypical diethynylethene fragment leads to wide range of cross-conjugated compounds, which comprise a series of materials with useful electronic and optical properties that complement those of the related linearly conjugated analogues.[254-268] In addition, cross-conjugated fragments are now being recognized as potential scaffolds through which to explore the influence of quantum-interference effects on the promotion and mediation of trans-molecule conductance in molecules and nascent molecular electronic devices.[269] In turn, this has led to a growing number of studies in which cross-conjugated carbon-rich ligands are being incorporated into metal complexes,[270-276] and attracting interest in the potential for intra-molecular ET between metal centers through the cross-conjugated bridge.[277,278]

Figure 13. A schematic of the Grozema molecular transistor. The arrows show the propagation of the components of the wave function directly between donor, D, and acceptor, A, and also the portion travelling via the gate, G. The moieties D and A may be distinct chemical groups or the source and drain electrodes of a device.[145]

Figure 14. Scheme of the ferrocene-containing cross-conjugated compounds addressed in this section.

Against this background of fundamental and applied research interests, it is unsurprising to note that perhaps the most ubiquitous of all redox probes, ferrocene, has been incorporated into a wide-range of cross-conjugated molecules.[260,279-283] The recent description of a chemically-gated quantum-interference based molecular transistor architecture based on a 1,1-diethynyl ethene (Figure 13) prompts renewed consideration of such structures.[269] In the model put forward by Grozema, a donor D (or source electrode) is connected to an acceptor, A (or drain electrode) via a cross-conjugated bridge. The gating component, G, is chosen to be capable of changing charge state through chemical or electrochemical means. The cross-conjugated structure limits charge flow from D to A as the components of the charge-carrier wave function propagating directly from D to A and that travelling via the channel to the gate G interact and form a interference pattern. This interference can be constructive or destructive

R R

HFe

Fc1: R = C6H4Me-4Fc2: R = C6H4NMe2-4 Fc3: R = C6H4NO2-4


depending on the chemical structure of the side chain; in the case of the structure shown in Figure 13 interference will be destructive and the transistor will be normally off. Chemical modification of the charge on the gating moiety (G) by (de)protonation or metal-ion binding to groups along the gate pathway changes the energy of the path, and under favorable conditions will prevent a fully destructive interference pattern from being formed, and hence increasing the flow of charge from D to A. Redox-active gate groups were also noted as providing a suitable means for switching these interference effects. With these points in mind, cross-coupling reactions of 1,1-dibromo-2-ferrocenyl ethene with terminal alkynes to generate the prototypical structures (Figure 14) in which the ferrocene moiety can serve as an electrochemically addressable means of introducing a point charge to gate the flow of charge were synthesized and spectroelectrochemically investigated by Kevin A. Vincent, Qiang Zeng, František Hartl, and Paul J. Low.[145] Crystallographic studies were performed by Dmitry S. Yufit, Judith A. K. Howard, Kevin A. Vincent, and Paul J. Low.


Here quantum-chemical calculations following the proposed protocol are used to explore the interactions between the ferrocenyl moiety and the cross-conjugated ancillary group and explain the obtained spectra from spectroelectrochemical studies. As representative examples complexes [Fc1]n, [Fc2]n, and [Fc3]n (n = 0, +1, Figure 14) were picked from the experimentally available compounds and investigated at the BLYP35/COSMO(CH2Cl2) level.

Figure 15. Isosurface plots of the spin density (± 0.002 a.u., left) and the SOMO 113α (± 0.03 a.u., right) of [Fc2]+.

The DFT calculations confirmed the experimental finding of an essentially ferrocenyl-centered oxidation in each case [Fc1]+, [Fc2]+, and [Fc3]+, with spin density almost exclusively localized on the ferrocenyl moiety (see SI of [145]). In the case of [Fc2]+ the calculated spin density on the ferrocenyl moiety (101 %) is matched with a small negative contribution form the two ethynyl units (-1 %) (Figure 15). Appreciable spin polarization is found at the chosen computational level, with appreciably negative (= β) spin density on the π-orbitals of the cyclopentadienyl rings (–0.27, –0.20 respectively) and mainly positive (= α) spin density on the iron atom (1.48). This spin polarization is accompanied by spin contamination (〈S 2 〉 = 0.88 compared to the nominal 0.75 for a doublet). This valence-shell


spin polarization and spin contamination at the hybrid DFT level is consistent with a significant Fe-Cp antibonding character of the SOMO, as has been analyzed in detail for other 3d transition-metal complexes in the context of computing hyperfine couplings.[129,146,147]

Table 4. Characteristic IR active vibrational modes [cm‒1] of [Fc1]n, [Fc2]n, and [Fc3]n (n = 0, 1) observed spectroelectrochemically in solution (CH2Cl2/10‒1 M Bu4NPF6) by Kevin Vincent and Qiang Zeng, and calculated at the BLYP35/COSMO(CH2Cl2) level (indicated by gray background).

ν(C≡C) ν(C=C) aryl

ν(C=C) vinyl ν(C−N) ν(NO2)

Fc1 2208, 2194 1609, 1510 1575

Fc1a 2233, 2220 1619, 1617 1499, 1495 1585

[Fc1]+ 2208, 2194 1605, 1510 1559

[Fc1]+[a] 2227, 2211 1615, 1556 1499 1561

Fc2 2188 1607, 1522 not observed

Fc2[a] 2223, 2210 1614, 1613 1514, 1511 –

[Fc2]+ 2167 1605, 1524 1545

[Fc2]+[a] 2203, 2178 1611, 1604 1518, 1510 1527

Fc3 2211, 2195 1594, 1491 1565 1520, 1343

Fc3[a] 2233, 2217 1601, 1566 1482, 1478

1568, 1567 1636, 1634

1395, 1393

[Fc3]+ 2215, 2201 1596, 1493 1565 1522, 1345

[Fc3]+[a] 2237, 2223 1608, 1605 1573, 1572 1483, 1463

1576 1641, 1639 1403, 1401

[a] calculated, with 0.95 correction factor applied.

The analysis of the spin density distributions in terms of MOs is complicated by the strong spin polarization, which leads to violation of the Aufbau principle. For example the “true” singly occupied MO in [Fc1]+ is not the α (i.e. spin-up) spin-orbital with the highest energy (which would be α-MO 115α), but rather is found at lower energy in the list of α-MOs (96α). An assignment of the proper “doubly occupied” MOs, and thus of the proper SOMO was only possible after an overlap matching of the corresponding α- and β-MOs. The same holds for the other complexes. In view of these aspects, a detailed MO-based analysis is not performed


but rather simply note that the α-SOMO of [Fc1]+ is ferrocenyl-based (90%), thus matching the spin-density distribution qualitatively. The localization of the α-SOMO (and thus of the spin density) on Fc+ is diminished in the presence of the donor substituents NMe2 in [Fc2]+ (113α, 78%) (Figure 15) but somewhat enhanced by the electron-withdrawing NO2 groups in [Fc3]+ (108α, 92%). In passing it is also noted that the SOMO involves σ-type orbitals on the cyclopentadienyl rings (Figure 15, right), contrast with the π-type negative spin density arising from spin polarization (Figure 15, left).

Harmonic vibrational frequency calculations were performed on each member of the series [Fc1]n, [Fc2]n, and [Fc3]n (n = 0, +1) (Table 4). In agreement with experimental spectroelectrochemical results, the ν(C≡C) frequencies of the neutral and the oxidized form differ little. For each oxidation state, two ν(C≡C) bands are computed. Taking [Fc2]n+ as an example, while the ν(C≡C) bands appear at 2223 cm–1 and 2210 cm–1 for the neutral species Fc2, upon oxidation to [Fc2]+ a small red shift to 2203 cm–1 and 2178 cm–1 occurs. While the ν(C≡C) frequencies of 2210 cm–1 (Fc2) and 2203 cm–1 ([Fc2]+) are in good agreement with principal features of the experimental band envelopes (Fc2 2188 cm–1; [Fc2]+ 2167 cm–1), the additional features at 2223 cm–1 (Fc2) and 2203 cm–1 ([Fc2]+) may explain in part the experimentally observed high-energy shoulder of the ν(C≡C) band. Overall, the calculations in each case overestimate the ν(C≡C) frequencies slightly (in spite of the customary scaling by an empirical factor of 0.95, cf. Computational Details).

Figure 16. Reversible UV-Vis-NIR (left) and IR (right) spectral changes resulting from the ferrocenyl-centered electrochemical oxidation of Fc2 in CH2Cl2/10‒1 M Bu4NPF6 within an OTTLE cell collected by Kevin Vincent and Qiang Zeng.[145]

To assign the spectroelectrochemically observed UV-vis spectra, TDDFT calculations were performed for [Fc1]+, [Fc2]+, and [Fc3]+ (Table 5). In each case the lowest-energy excitation of significant intensity is computed to be between 8200 - 8400 cm–1. It arises from a low-lying ferrocenyl-based MO and therefore has interconfigurational (or pseudo-dd) character. The first electronic transition above 10000 cm–1 is computed at 10123 cm–1 (µtrans = 2.0 D) for [Fc2]+, which is in excellent agreement with the broad observed absorption at 10480 cm–1 for this complex and similar bands in the others (c.f. Figure 16). Several orbitals contribute to the


transition. The predominant character is that of a CT from one ene-yne unit to the ferrocenyl cation, although the orbitals are more delocalized in the case of [Fc3]+ than the other systems.

Table 5. Calculated excited-state parameters in the range from 0 to 20000 cm‒1: UV-vis-NIR transition energies Etrans, transition dipole moments µtrans and character of [Fc1]+, [Fc2]+, and [Fc3]+ at the BLYP35/COSMO(CH2Cl2) level.[a]

Etrans [cm‒1] µtrans [D] character

[Fc1]+ 8242 0.7 interconfiguration

11807 1.1 cross-conjugated fragment to vinyl ferrocene charge transfer

19432 5.7 cross-conjugated fragment to vinyl ferrocene charge transfer

19838 2.6 cross-conjugated fragment to ferrocene charge transfer


10123 2.0 trans Me2NC6H4C≡C to vinyl ferrocene charge transfer

16108 2.9 cis Me2NC6H4C≡C to vinyl ferrocene charge transfer

16309 3.4 interconfiguration

16560 6.2 trans Me2NC6H4C≡C to vinylferrocene charge transfer

16678 3.8 mixed interconfiguration and charge transfer

19640 7.2 cis Me2NC6H4C≡C to vinyl ferrocene charge transfer


12168 0.7 delocalized π/π* to vinyl ferrocene charge transfer

[a] Transitions with a µtrans < 0.5 D are neglected.

Multiple transitions close in energy are computed around 16500 cm–1.They can be assigned to the broad feature observed experimentally near 15000 cm–1. In the case of [Fc2]+ the excitations at 16108 cm–1 (µtrans = 2.9 D) and at 16560 cm–1 (µtrans = 6.2 D) arise from transitions in which the hole is transferred from the vinyl ferrocenyl moiety to one branch of the cross-conjugated organic fragment. At 16309 cm–1 (µtrans = 3.4 D) TDDFT gives a purely ferrocenyl centered excitation, which does not involve CT. Precise assignment of the


character of the excitation at 16678 cm–1 (µtrans = 3.8 D) is not straightforward, as multiple transitions of both CT and IC character contribute to the absorption feature (Table 5).

The most intense excitation in [Fc2]+ (µtrans = 7.2 D), computed at 19640 cm–1, has again ene-yne to metal charge-transfer character. The excitation energy of 19640 cm–1 corresponds well with the band at 19900 cm–1 determined experimentally. This band exhibits a weak shoulder, which can be attributed to a computed low-intensity transition at 20127 cm–1 (µtrans = 1.5 D). Similar conclusions can also be drawn from the TDDFT results for [Fc1]+ and [Fc3]+.

Conclusions

The alkynyl substituents have a modest electronic influence on the ferrocenyl moiety, to which they are linearly coupled, as is evidenced by the sensitivity of the ferrocenyl oxidation potential to the electronic character of these remote groups found in experiment. Quantum-chemical calculations at the BLYP35/COSMO(CH2Cl2) level reproduce the charge localization at the ferrocenyl moiety. The modest shift in both experimental and computed ν(C≡C) and ν(C=C)vinyl bands in response to a change in the oxidation state of the ferrocene moiety together with the presence of a number of CT transitions in the spectra of [Fc1]+, [Fc2]+, and [Fc3]+ also points to a small interaction between the metallocene and cross-conjugated fragments. However, strong ground-state electronic coupling is not an essential criterion from the point of view of the Grozema molecular transistor designs and the limited structural rearrangement (evidenced by the small ν(C≡C) variations in response to charge state changes) may also help both preserving QI effects and aid integration into molecular electronic circuits where large structural changes during operation would be detrimental to long term device stability.

CHAPTER 4

SPINNING CLASSES: THE IMPORTANCE OF ROTAMERS

The importance of conformations for ET transfer rates has been acknowledged for a long time in various contexts, way beyond the scope of the present thesis, but of course including the field of MV systems.[284-294] One means to restrict conformational freedom is the introduction of steric constraints,[293,294] e.g. by tethering.[295-297] Linkers have been constructed specifically, e.g. to force certain dihedral angles within the bridge. In not too restricted situations, the results of spectroscopic experiments almost always correspond to an average over a sampling of conformational degrees of freedom.

Here quantum-chemical calculations may aid in the evaluation of the effects of such conformational motion, and a variety of studies along these lines has been performed.[298-302] Regarding certain optoelectronic properties, it is in fact important to describe correctly not only the ground-state conformational energy surface but also that in crucial excited states. Shortcomings of TDDFT in case of CT type excited states have been noted,[303,304] and more sophisticated methods[305] may have to be applied. In this chapter, the focus is on conformational effects in selected organometallic systems, but note that conformational effects are increasingly also in the focus of quantum-chemical studies on other MV systems. This includes recent work on organic TAA-based systems (see Chapter 2) with partly saturated bridges, where the predominant pathways for ET depend crucially on conformation.[231]

As a prerequisite for studies on binuclear MV systems, a number of computational studies focused on spin-density delocalization as a function of conformation in mononuclear Fe and

CHAPTER 4: SPINNING CLASSES: THE IMPORTANCE OF ROTAMERS

60

Ru complexes bearing a diethynylaromatic ligand that becomes the bridge in corresponding dinuclear species.[18,71,190,302,306,307] This provides insight into the coupling between metal d-orbitals and the ligand π-system as a function of rotation angle. A number of computational studies have evaluated the relative orientation of the two end caps in all-carbon-bridged MV complexes. As for the abovementioned mononuclear complexes, most of these studies used standard functionals and gas-phase conditions and thus are expected to provide an overall too delocalized description. BP86/gas-phase calculations on a truncated model of the C2-linked complex [{Ru(dppe)Cp}2(µ-C≡C)]+[301] provided a lowest-energy minimum structure with a Cp(midpoint)-Ru-Ru-Cp(midpoint) torsion angle of 55°, likely due to direct steric interactions between the end caps. A transoid minimum (torsion angle 180°) at somewhat higher energy was also found. In addition to a main excitation near 16200 cm–1, TDDFT calculations for this latter minimum provide low-intensity peaks at lower energies. However, due to the relatively large barriers the NIR spectra are likely dominated by the lowest-energy minimum in this case. Rotamers were also found in similar BP86/gas-phase computations on trimetallic ruthenium complexes.[300]

Experimental research on conformational effects in organometallic carbon-bridged species of the type discussed above has been reviewed by Low.[19] It is hard to access different rotational conformers with experimental methods such as tethering. In contrast investigations on a quantum-chemical basis are more or less straightforward and thus appealing, if a proper description of the charge-localization/delocalization behavior can be achieved. As the BLYP35/COSMO protocol yielded promising results for MV systems, in which the spectrum is dominated by the lowest-energy conformer, the influence of different rotameric forms on the UV-vis-NIR spectral profile was investigated in close collaboration with the Low group.

Figure 17. Complexes reviewed and/or investigated in this chapter.

[M][M]

[M] = FeCp*(dppe) Fe1[M] = RuCp(PPh3)2 Ru1[M] = Mo(η−C7H7)(dppe) Mo1[M] = ReCp*(NO)(PPh3) Re1[M] = OsCp*(dppe) Os1

[M] = FeCp*(dppe) Fe3[M] = RuCp*(dppe) Ru4

[M] [M]

[M] [M]

[M] = FeCp*(dppe) Fe2[M] = RuCp*(dppe) Ru2[M] = RuCl(dppe)2 Ru3

NNRu

H3N NH3

NH3H2N

H3N Ru

NH3H3N

H3N NH3

NH3

1

1 2 34

CTI


61

4.1 The Interpretation of NIR Band Shapes in Polyynediyl Molecular Wires

Introduction

Polyynediyl-bridged systems have attracted attention for more than two decades, with early synthetic efforts complemented more recently by experimental and computational investigations of electronic structure.[308] Polyynediyl complexes serve as models of the linear carbon allotrope carbyne,[309-314] and permit investigations of intra-molecular ET reactions and MV characteristics.[62,308,315,316] The highly delocalized electronic structures often associated with {LnM}{µ-(C≡C)n}{MLn} complexes has also led to consideration of their potential applications in molecular electronics.[24,34,35] Although impressively long polyynediyl complexes are known,[309-314] the greatest concentration of work has focused on the properties of diynediyl-bridged (µ-C≡CC≡C) d6/d6 transition-metal complexes, which have been synthesized, characterized, and investigated in their various electrochemically accessible redox states.[62,64,65,89,308-323] A variety of reviews and book sections address these all-carbon bridged MV systems.[34,62,308,315,316,318,324]

The degree of delocalization along the {LnM}(µ-C≡CC≡C){MLn} backbone, and hence the nature of the redox-derived products, is sensitive to the identity of the metal end-capping group. By varying the terminal cap, and thus the metal d-orbital/carbon π-system interactions, [{LnM}(µ-C≡CC≡C){MLn}]+ complexes ranging from weakly (Class II in the Robin-Day scheme; e.g. {MLn} = [Mo(dppe)(η-C7H7)], [Mo1]+)[89] to strongly coupled (Class III; {MLn} = [Fe(dppe)Cp*]) systems can be obtained (Figure 17).[34,318,325] Charge delocalization along the six-atom MC4M chain is increased in complexes of the heavier metals, {MLn} = [Ru(PPh3)2Cp] ([Ru1]+),[317] [Ru(dppe)Cp*],[322] [Os(dppe)Cp*] ([Os1]+),[321] and [Re(NO)(PPh3)Cp*] ([Re1]+).[64,65] The increased C4-bridge π-character in the frontier molecular orbitals of these complexes suggests that descriptions in terms of metal-stabilized carbon centered radicals might be more appropriate than MV classifications.[11]


Two metal atoms coupled by a “carbon wire” made such systems attractive targets for computational studies. The majority of this work dealt with strongly coupled Class III systems, where obviously the standard gas phase DFT treatments with functionals like BP86 or B3LYP correctly reproduced the delocalized, symmetrical ground-state character of the MV species. Examples are the Class III MV iron complexes [{Fe(CO)2Cp}2(µ-(C)n]+ (n = 4-8), models for the related Fe-C4-Fe and Re-C4-Re monocationic complexes [Fe1]+ and [Re1]+ (Figure 17) and their mixed Fe-Re complex,[65] a series of related diiron and dirhenium


62

molecular wires,[326] diynediyl-bridged manganese MV systems,[36,327,328] the ruthenium complex [Ru1]+,[317] as well as iron/ruthenium and iron/rhenium analogues of this system.[329]

Interestingly, a relatively early HF calculation on the dirhenium MV complex [Re1]+ indicated SCF convergence problems.[64] While the DFT calculations were generally consistent with the delocalized Class III situations derived experimentally for the homodinuclear complexes, some localization onto the iron center was computed (B3LYP, gas phase) on the mixed Fe-Re complex.[65] Yet, a too delocalized spin-density distribution compared to the experimentally established Class II character[66] may be discerned.

Often, the extent of spin density on the carbon bridge vs. the metal centers was of central interest. For the dirhenium complex [Re1]+ and analogues with longer bridges up to 20 carbon atoms, Reiher et al. investigated spin-state energies (also for the neutral and dicationic complexes) and spin densities at BP86 and partly B3LYP* level (15% Ex

exact admixture in the latter case).[330] In addition to the doublet-quartet splitting of the MV cationic form, the singlet-triplet splitting of the dication was investigated for the full systems. Not unexpectedly, spin-state splitting was found to decrease with increasing chain length. While the doublet ground state of the MV system had some tendency to localize the spin at the Re centers, the quartet exhibited more spin on the bridge, in particular for the longer bridges. The gas-phase nature of the calculations and the functionals used may exaggerate the delocalization onto the bridge somewhat.

Computationally studied counterexamples to the predominantly Class III molecular wires discussed above are provided by the weakly coupled Class II dimolybdenum complex [Mo1]+,[89] by the diruthenium complex [Ru4]+ (Figure 17),[189] and by some carborane-bridged molybdenum[331] and ruthenium[191] complexes. Neither B3LYP calculations on [Mo1]+ nor application of the MPW1K hybrid functional (Ex

exact admixture 42.8%) in the gas-phase calculations apparently reproduced the Class II character of this MV cation.[89] For the particularly weakly coupled carborane-bridged dimolybdenum and diruthenium systems, MPW1K calculations gave a localized structure and spin-density distribution, even though solvent effects were not considered.[191,331] Together with the clear-cut Class III examples discussed above, this shows that, as one moves away from the borderline between Class II and Class III, some of the shortcomings of the computational treatment become less serious.


63

The Class III Ruthenium Complex

The compound [{Ru(PPh3)2Cp}2(µ-C≡CC≡C)]+, [Ru1]+, is a well-known example of a highly delocalized butadiynyl-bridged bimetallic complex in which the carbon chain plays a significant role in stabilizing the unpaired electron.[317] However, reinvestigation of the NIR spectrum of [Ru1]+ revealed a high-energy shoulder that was not included in the original analysis (Figure 18).[317] The NIR band envelope in [Ru1]+ was found by Phil Schauer not to be solvatochromic (SI of [332]), consistent with a highly delocalized (or strongly coupled) system. However, the two-state Hush treatment of such strongly coupled systems predicts only a single asymmetric band,[6,12,57,58,61,333,334] no obvious vibrational progression fits the ca. 3000 cm-1 separation of the principal band and the shoulder, and spin-orbit splitting in Ru complexes is also too small (typically around 1000 cm-1)[12] to account for the observed separation of the component bands. We will show that quantum-chemical calculations allow an interpretation of this feature and show that the presence of rotamers is also a crucial variable that may influence the appearance of the IVCT band and the underlying electronic structure.

To provide a quantum-chemical perspective, calculations at density-functional theory (DFT) level were performed. Starting from a Ci-symmetric input the structure of [Ru1´]+ (the ´ notation being again employed to distinguish the in silico system from the experimental complex) was optimized without constraints using the global hybrid functional BLYP35 with a suitable continuum solvent model (CH2Cl2). The calculated spin density in the resulting fully optimized trans-[Ru1´]+ structure is evenly distributed over the RuC4Ru chain (see SI of [332]), whilst Mulliken population analysis of the α-SOMO, the α-HOMO and the β-HOMO show appreciable diynediyl bridge (45 %, 57 %, and 57 %, respectively) and metal character (34 %, 37 %, and 37 %) in line with earlier analyses.[317]

TDDFT calculations (BLYP35/CH2Cl2) with the Gaussian09 code revealed a single intense (µtrans = 9.5 D) transition at 11702 cm-1 arising from the excitation from the β-HOMO–1 to the β-SOMO. Given the distribution and nodal properties of these orbitals, this excitation is best described as a π-π* transition (Figure 18). As expected for transitions not involving charge transfer, the TDDFT result is in excellent agreement with the energy of the principal component of the experimentally determined spectrum at 11655 cm-1 (Figure 18). However, no significant transition in the TDDFT results could account for the higher-energy shoulder.

A number of rotamers of Ru1 have been observed crystallographically,[335-337] prompting consideration of the influence of different conformers on the electronic structure of the ground and excited states of [Ru1]+. To investigate the influence of different rotamers of [Ru1´]+ on the appearance of the NIR spectrum, constrained structure optimizations with a fixed P-Ru-Ru-P dihedral angle (Ω) were performed. Starting from the optimized nearly Ci-symmetric structure (Ω = 180 °) the dihedral angle was decreased to 0 ° in steps of 10 °. Interestingly, the energy of the resulting rotational energy potential (∆E) varies by only ca. 13 kJ/mol, with the energy maximum being at Ω = 0 ° (∆E = E0-E180 ° = 12.6 kJ/mol).


64

Figure 18. NIR spectra collected during spectroelectrochemical oxidation of [{Ru(PPh3)2Cp}2(µ-C≡CC≡C)] by Josef Gluyas and isosurface plots (± 0.03 a.u.) of the orbitals involved in the π-π* excitation of trans-[Ru1´]+ (right) and for the MLCT transition of perp-[Ru1´]+ (left) computed at the BLYP35/COSMO(CH2Cl2) level.


65

A plot of ∆E against Ω (Figure 19) revealed three minima at Ω = 30 ° (∆E = 1.0 kJ/mol); 110 ° (∆E = 6.9 kJ/mol) and 180 ° (reference, ∆E = 0). These three structures were subsequently used as starting points for three full unconstrained optimizations leading to three genuinely energy-minimized conformers: the previously noted trans-[Ru1´]+ (Ω ≈ 180 °, ∆E = 0), perp-[Ru1´]+ (Ω ≈ 112 °, ∆E = 7.5 kJ/mol) and cis-[Ru1´]+ (Ω ≈ 27 °, ∆E = 1.6 kJ/mol).

All three rotamers exhibit delocalized electronic structures with spin density distributed evenly over the six-atom RuC4Ru chain (c.f. Figure 20). Each structure gives a single ν(C≡C) vibration (trans-[Ru1´]+ ν(C≡C) = 1878 cm-1, perp-[Ru1´]+ ν(C≡C) = 1868 cm-1, cis-[Ru1´]+ ν(C≡C) = 1871 cm-1) which compares well with the broad band observed experimentally (ν(C≡C) = 1855 cm-1, see SI). Gaussian09 TDDFT calculations were performed for all nineteen values of Ω in the relaxed scan (SI) and for the three fully optimized structures using. The fully optimized structures all exhibit an intense β-HOMO–1 to β-SOMO transition near 11600 cm-1 [trans-[Ru1´]+, 11702 cm-1 (µtrans = 9.5 D); perp-[Ru1´]+ 11524 cm-1 (µtrans = 8.8 D); cis-[1´]+ 11572 cm-1 (µtrans = 9.8 D)] with π-π* character as described before. The transition dipole moment reveals trends in the diynediyl contribution to the β-HOMO–1 orbital (trans-[Ru1´]+: 19 %, perp-[Ru1´]+: 16 %, cis-[Ru1´]+: 19 %) and reflects the orbital overlap with the β-SOMO.

Figure 19. Energy ∆E relative to the most stable rotamer (Ω =180 °) of the structures of [Ru1´]+ versus the P-Ru-Ru-P dihedral angle Ω (0 ≤ Ω ≤ 180°).

For perp-[Ru1´]+ a second, less intense excitation at 13982 cm-1 (µtrans = 3.3 D) arising from a β-HOMO–3 to β-SOMO transition is also calculated. This transition has no calculated intensity for trans-[Ru1´]+ (µtrans = 0.0 D) and is only very weak in the case of cis-[Ru1´]+ (µtrans = 0.8 D). The β-HOMO–3 is largely metal in character (Ru/C≡CC≡C/Ru: trans-[Ru1´]+ 27%/3%/18%; cis-[Ru1´]+ 25%/4%/19 %; perp-[Ru1´]+ 18%/9%/27%), and the β-HOMO–3 to β-SOMO transition only gains appreciable intensity, when there is appreciable

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spatial overlap between these orbitals (i.e. in perp-[Ru1´]+, Figure 18). Very similar results have been obtained for two conformational isomers of the ethynediyl complex [{Ru(dppe)Cp}2(µ-C≡C)]+ although in this case the greater steric congestion around the C2-bridge gives rise to a greater energy barrier to rotation.[301] The higher-energy shoulder in [Ru1]+ is therefore assigned to an MLCT transition that gains intensity in rotameric forms with approximately orthogonal disposition of the Cp rings around the RuC4Ru axis.

Figure 20. Spin-density isosurface plots (± 0.002 a.u.) of trans-[Ru1´]+ (left), perp-[Ru1´]+ (right).

Thus, all of the nineteen computational structures of [Ru1]+ show a lower-energy transition in a narrow range around 11700 cm-1 and higher-energy transition around 14300 cm-1 (Table A3). For Ω = 160 °, the Gaussian09 calculations gave one negative-energy excitation, and thus the results are not considered in the discussion (Table A3). The β-HOMO–1 to β-SOMOtransition energies range from 11417 cm-1 (Ω = 90 °) to 11743 cm-1 (Ω = 40 °). The energy of the higher-energy excitation originating from the β-HOMO–3 varies from 13924 cm-1 (Ω = 110 °) to 14463 cm-1 (Ω = 90 °). The excitation near 11700 cm-1 exhibits significant intensity for all rotamers (µtrans ≥ 8.4 D, Figure 21). In contrast, the transition dipole moment of the higher-energy transition, which correlates with the shoulder in the experimental spectrum (Figure 18), steadily gains intensity from Ω = 0 ° (µtrans = 0.8 D) to Ω = 100 ° (µtrans = 3.8 D), and µtrans decreases gradually when advancing towards Ω = 180 ° (0.0 D).

To test the quantum-chemically predicted importance of rotamers for the shape of the NIR band of mixed-valence systems, a conformationally restricted diynediyl complex comparable to Ru1 was required. Gladysz and colleagues have used α,ω-bis(phosphines) in constructing ‘insulated’ molecular wire models based on polyynediyl complexes,[338,339] providing a conceptual basis for the work, which follows. Josef Gluyas prepared the pseudo-macrocyclic complex {RuCp}2(µ-C≡CC≡C)(µ-Ph2P(CH2)5PPh2)2 (Ru5, Figure 22) in 33% yield by phosphine exchange of Ru1 with 1,5-bis(diphenylphosphino)pentane. Dmitry Yufit and Judith Howard characterized Ru5 crystallographically. Full crystallographic, spectroscopic, and analytical data are given in [332] and the corresponding SI. The phosphine ligands constrain Ru5 to a narrow range of conformers with Ω being close to 0 ° and thus should exhibit properties similar to those of cis-[Ru1´]+.


67

Josef Gluyas obtained IR and UV-vis-NIR spectra of [Ru5]+ using the same spectroelectrochemical methods as employed for [Ru1]+ (Figure 22). The ν(C≡C) band in [Ru5]+ (1858 cm-1) is noticeably narrower than for [Ru1]+ (1855 cm-1), providing some evidence for restricted rotation in the tethered system. The NIR absorption bands in [Ru1]+ (Figure 18) and [Ru5]+ (Figure 22) are both clearly asymmetric, revealing the low-energy ‘cut-off’ associated with a strongly coupled system.[17,18] However, the high-energy shoulder evident in [Ru1]+ (Figure 18) is substantially reduced in intensity in the conformationally restricted compound [Ru5]+ (Figure 22). This observation is entirely in agreement with the TDDFT results for trans-[Ru1´]+, cis-[Ru1´]+and perp-[Ru1´]+.

Figure 21. Transition dipole moment µtrans for [Ru1´]+ as function of the P-Ru-Ru-P dihedral angle Ω (blue: β-HOMO–1 => β-SOMO transition, red: β-HOMO–3 => β-SOMO transition) calculated with the Gaussian09 code at the BLYP35/COSMO(CH2Cl2) level. For Ω = 160 °, the Gaussian09 calculations gave one negative-energy excitation.

Mixing of localized metal-based excitations, as featured in Meyer’s model,[19] is not applicable here (given the low spin-orbit coupling constant of Ru) and cannot explain the high-energy shoulder. In addition, whilst fitting of the band shape by deconvolution in a series of Gaussian-shaped sub-bands is obviously possible, such fitting-based solutions do not, per se, reveal the electronic origin of the underlying excitations. Moreover, interpretations based on the number of sub-bands necessary for an accurate fit to the experimental data can be misleading in systems close to the localized/delocalized borderline due to the inherently asymmetric shape of the band.

A full optimization of [Ru5´]+ starting from a Cs-symmetric structure led to a delocalized system with a P-Ru-Ru-P dihedral angle close to 0 °, and a calculated ν(C≡C) frequency of 1874 cm-1. TDDFT calculations give a single (π-π*) transition at 12012 cm-1 (µtrans = 10.1 D) occurring from the β-HOMO–1 to β-SOMO excitation. These orbitals are of very similar character for cis-[Ru1´]+ and [Ru5´]+ (Figure 23). The calculated band is slightly blue-shifted in comparison to cis-[Ru1´]+ (11572 cm-1) whilst the experimental band maximum is

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observed at 11120 cm-1. The β-HOMO–3 to β-SOMO transition has negligible intensity for the tethered system [Ru2´]+.

Figure 22. NIR spectra collected during spectroelectrochemical oxidation of [{RuCp}2(µ-C≡CC≡C)(µ-PPh2(CH2)5PPh2)2] [Ru5]+ to [Ru5]+ in CH2Cl2/0.1 M NBu4PF6 by Josef Gluyas. Note the loss of the shoulder on the high-energy side of the band.

In summary, the NIR band envelope observed for the simple diynediyl complex [Ru1]+ features two transition envelopes with distinct electronic character (π-π* and MLCT) arising from a distribution of conformers in solution. The observed NIR absorption profile of [Ru1]+ is in good agreement with the predictions based on a relaxed rotamer scan and well-matched with the results from the three fully optimized structures trans-[Ru1´]+, perp-[Ru1´]+ and cis-[Ru1´]+. An accurate interpretation of the NIR spectrum of [Ru1]+ must therefore allow for the different spectroscopic properties of the various rotameric forms. These conclusions likely apply to many other examples of ‘mixed-valence’ complexes with low axial symmetry and relatively free rotational elements. Detailed analyses of the NIR spectra of such systems should therefore consider not only the contributions to the band shape that can arise from vibronic coupling and transitions from lower-lying filled metal orbitals that gain intensity through low local coordination symmetry and/or spin orbit coupling but also the potential for rotamers with distinct spectroscopic profiles. The latter will not be adequately treated by interpretations drawn from a single lowest-energy conformer.


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Figure 23. Comparison of the isosurface plots (± 0.03 a.u.) of the orbitals involved in the π-π* excitation of cis-[Ru1´]+ and [Ru5´]+ at the BLYP35/COSMO(CH2Cl2) level.


70

The Class III Osmium Complex

Bruce et al.[321] reported the osmium complex [Os1]+ to exhibit Class III behavior very similar to its lighter homologue [Ru1]+. Again the NIR spectrum is dominated by a main band and a high-energy shoulder. In contrast to [Ru1]+ the main band and the shoulder are more distinguishable in [Os1]+, and in principle the band center of the shoulder can be extracted. Hence for [Os1]+ a similar scan at the BLYP35/COSMO(CH2Cl2) level was performed to test whether the presence of different rotamers explains the unusual band envelope. The relaxed scan yielded all nineteen structures within an energy range of 18 kJ/mol (Figure 24). Three minima are found at Ω = 50°, Ω = 110°, and Ω = 180°, which were reoptimized without constraints giving min-[Os1]+, perp-[Os1]+, and trans-[Os1]+. The lowest-energy structure min-[Os1]+ is favored by 5.9 kJ/mol compared to perp-[Os1]+, and by 6.5 kJ/mol compared to trans-[Os1]+. All three rotamers exhibit delocalized spin densities (see e.g Figure 25 left and Table A6) and the Class III assignment is further supported by the single ν(C≡C) frequency calculated for each of the conformers (min-[Os1]+: ν(C≡C) = 1877 cm–1, perp-[Os1]+: ν(C≡C) = 1870 cm–1, and trans-[Os1]+: ν(C≡C) = 1873 cm–1) reproducing the experimental IR band at 1860 cm–1 (Table A4).

Figure 24. Energy ∆E relative to the most stable rotamer (Ω = 50 °) of the structures of [Os1]+ versus the P-Os-Os-P dihedral angle Ω (0 ≤ Ω ≤ 180°) at the BLYP35/COSMO(CH2Cl2) level .

Similarly to [Ru1]+, for the transoid form trans-[Os1]+ an intense excitation at 12102 cm–1

(µtrans = 9.5 D) arising from the β-HOMO–1 to β-SOMO transition is computed using the Gaussian code (Table A5). In addition to this π-π* transition, the β-HOMO–2 to β-SOMO excitation is calculated to give rise to a low-intensity feature at 14537 cm–1 (µtrans = 1.2 D). The β-HOMO–2 has appreciably more metal (79%) than diynediyl bridge character (8%) and with the osmium centers (40%) contributing to a similar extent to the β-SOMO as the bridge (52%), the high-energy transition is of noticeable MLCT character. The main absorptions of

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min-[Os1]+ and perp-[Os1]+ are found to be of very similar energies at 11990 cm–1 (µtrans = 9.1 D), 11633 cm–1 (µtrans = 8.8 D) respectively, and are also of π-π* character. But in contrast to trans-[Os1]+, the β-HOMO–2 to β-SOMO transition exhibits only very little intensity (µtrans > 0.4 D) and instead the β-HOMO–3 to β-SOMO excitation has appreciable oscillator strength. It is computed at 15439 cm–1 (µtrans = 2.4 D) for min-[Os1]+ and at 15349 cm–1 (µtrans = 3.4 D) for perp-[Os1]+. But the orbital contributions (Table A4) reveal that this transition is of very similar character as the β-HOMO–2 to β-SOMO excitation in trans-[Os1]+ and hence is best described as MLCT absorption. Although the experimental excitation energy is overestimated by about 1500 cm–1 the π-π* transition calculated for all three rotamers can be assigned to the main experimental feature with a band center at around 10500 cm–1.[321] The broad high-energy shoulder between 11000 cm–1 and 19000 cm–1 of [Os1]+ peaking at around 12500 cm–1 can be attributed to the MLCT transition arising from different orbitals for different rotamers. While the transitions attributed to the respective main peak and its shoulder in the experimental spectrum are of very similar character, the shoulder of [Os1]+ is more pronounced than in its lighter analogue [Ru1]+ and exhibits a clearly distinguishable peak. This difference in the UV-vis-NIR band envelope can be explained by two considerations. In contrast to [Ru1]+ the MLCT excitation is present in the minimum-energy structure for [Os1]+. In addition the osmium complex [Os1]+ appears to be an even more clear-cut Class III system. Hence the band asymmetry is less pronounced leading to a decreased overlap of the main-absorption band and the shoulder.

Figure 25. Spin-density isosurface plot(left, ±0.002 a.u.) and isosurface plot (±0.03 a.u.) of the β-SOMO (right) of min-[Os1]+ calculated at the BLYP35/COSMO(CH2Cl2) level.


72

The Class II Molybdenum Complex

As discussed in the introduction by changing the redox centers from the group 8 metals ruthenium and osmium to molybdenum in [Mo1]+, and by changing the terminal-ligand environment to η-cycloheptatrienyl and dppe ligands, a transition from a charge-delocalized to a charge-localized system is achieved.[89,317] Experimental observations lead to the assessment of an ET rate of around 108-1010 s–1 in CH2Cl2. Can our computational protocol also reproduce the charge localization for [Mo1]+ encountered by Fitzgerald et al., and do different rotameric forms influence the UV-vis-NIR spectrum as for the ruthenium analogue?

Starting from a Ci-symmetric input the structure of [Mo1]+ was optimized at the BLYP35/COSMO(CH2Cl2) level. This yields the trans-[Mo1]+ conformer, which features a unsymmetrically distributed spin density to which both redox centers contribute (Figure 26). In addition the spin density exhibits spin polarization, which leads to an increased 〈S 2 〉 value of 0.81. A noticeable dipole moment of 8.9 D is computed and the corresponding bond lengths differ for the two halves (∆d(Mo-CBridge) = 0.05 Å) of the molecule indicating Class II behavior. As expected for a charge-localized system, two ν(C≡C) frequencies at 1930 cm–1 and 1868 cm–1 are observed in the experimental IR spectrum. Harmonic vibrational frequency calculations on trans-[Mo1]+ give 1930 cm–1 and 1897 cm–1, in good agreement with experiment (Table A4). The splitting of the ν(C≡C) band was attributed to the presence of different rotameric forms in the original analysis by Fitzgerald et al.[89]

Figure 26. Spin-density isosurface plots (± 0.002 a.u.) of trans-[Mo1´]+ (left), min-[Mo1´]+ (right) at the BLYP35/COSMO(CH2Cl2) level.

TDDFT calculations at the BLYP35/COSMO(CH2Cl2) level in TURBOMOLE gave two excitations with appreciable intensity below 15000 cm–1 (Table A7) The main transition at 7731 cm–1 arises from the β-HOMO to β-SOMO excitation (µtrans = 9.4 D). Both orbitals tend towards delocalization over the molecular Mo-C≡CC≡C-Mo backbone (Table A8). But as one molybdenum center bears the main contribution to the β-HOMO ([Mo]/C≡CC≡C/[Mo]: 14%/15%/59%) and the respective other mainly contributes to the β-SOMO ([Mo]/C≡CC≡C/[Mo]: 53%/27%/8%), significant CT is involved in the excitation and it can be assigned to an IVCT transition. The most intense experimental feature arises at around


73

8500 cm–1 (the given excitation energies were derived from Gaussian deconvolution). This is slightly underestimated by the quantum-chemical protocol, as expected for Class II systems (see e.g. Chapter 2). TDDFT gives a second, less intense excitation at 14047 cm–1 (µtrans = 1.7 D), which corresponds to a mixed β-HOMO–4 and β-HOMO–5 to β-SOMO transition.

With the findings of the importance of rotamers in the case of [Ru1]+ in mind, the influence of different conformers of [Mo1]+ has been investigated. Analogously to the previous section, a scan employing constrained structure optimizations with a fixed P-Mo-Mo-P dihedral angle (Ω) was performed (Figure 27). Starting from the optimized structure trans-[Mo1]+ (Ω ≈ 180 °), the dihedral angle was decreased to 0 ° in steps of 10 °. The first interesting finding is that trans-[Mo1]+ does not correspond to the most stable conformer and is no minimum. A maximum is found for the cisoid form. The lowest-energy structure is obtained for Ω = 60 °, but all structures are again within a range of 18.5 kJ/mol.

Figure 27. Energy ∆E relative to the most stable rotamer (Ω = 60 °) of the structures of [Mo1]+ versus the P-Mo-Mo-P dihedral angle Ω (0 ≤ Ω ≤ 180°) at the BLYP35/COSMO(CH2Cl2) level.

Unfortunately, the assignment of transitions to the different rotamers of [Mo1]+ is not as straightforward as for [Ru1]+, as most excitations arise from more than one orbital. Hence the spectral signature of fully-optimized minimum structure at Ω ≈ 60 °, min-[Mo1]+, which is favored by 8.7 kJ/mol compared to trans-[Mo1]+, will be in the focus of the discussion. The ground-state properties are very similar to those of trans-[Mo1]+. The structure of min-[Mo1]+ is symmetry-broken with a difference in the Mo-C bond length of 0.05 Å (Table A4), and only one molybdenum center contributes to the spin density (Figure 26). Frequency calculations give two ν(C≡C) at 1971 cm–1 and 1914 cm–1, which are in worse agreement with experiment than the ones obtained for trans-[Mo1]+.

Again the lowest-energy excitation of min-[Mo1]+ arises from the β-HOMO to β-SOMO transition. But it appears red-shifted by about 1650 cm–1 compared to trans-[Mo1]+ at 6073 cm–1 (µtrans = 2.4 D) and is best described in terms of an IVCT absorption. A second

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excitation is calculated at 10746 cm–1 (µtrans = 5.9 D), which corresponds to the β-HOMO–1 to β-SOMO transition. The β-HOMO ([Mo]/C≡CC≡C/[Mo]: 8%/25%/56%) and the β-HOMO–1 (8%/25%/58%) are located on the molybdenum moiety, which barely contributes to the β-SOMO (58%/24%/2%), and are of significant eg d-orbital character (Figure 28, Table A8). Thus IVCT (or metal-metal-CT) character can be assigned to both excitations.

Figure 28. Isosurface plots (±0.03 a.u.) of the β-SOMO (top), β-HOMO (bottom, left), and β-HOMO–1 (bottom, right) of min-[Mo1]+ calculated at the BLYP35/COSMO(CH2Cl2) level.

In the experimental spectrum a broad feature is observed with a peak center at about 4000 cm–1, in addition to the main peak at around 8500 cm–1 (Figure 29). Both computed excitation exhibit appreciable IVCT character the overestimation of about 2000 cm–1 appears rather puzzling. So far mostly underestimation for CT transitions energies has been reported, and the results for trans-[Mo1]+ are not in line with these conclusions. To probe the contribution of different rotamers to the UV-vis-NIR spectrum, the oscillator strength computed by TDDFT single-point calculation on each of the 19 structures were weighted by a Boltzmann factor exp(–∆E/kBT), where ∆E is the energy relative to the most stable conformer (Ω = 60 °), kB is the Boltzmann constant, and T was set to 298.14 K to correspond to the room-temperature experimental conditions. The resulting excitation energies and corresponding intensities were, for better comparison with the experimental spectra, convoluted with Gaussian broadening (σ = 500 cm–1, full width at half maximum FWHM = 1177 cm–1) using the Q-Spector program previously designed for IR spectra.[340] The resulting spectrum compared to the recorded UV-vis-NIR spectrum is shown in Figure 29. Qualitatively the computed band pattern matches the experimental one, but both bands are blue-shifted by about 2500 cm–1. In experiment the low-energy band is significantly broader than the high-energy feature. To reproduce this finding vibronic effects have to be considered.


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Due to the associated energy penalty the contribution of structures exhibiting a spectral signature very different to min-[Mo1]+ is negligible.

Figure 29. Computed envelope (red) compared to experimental spectrum (black) for [Mo1]+. The computed intensities were Boltzmann-weighted, the resulting stick spectrum was Gaussian broadened (σ = 500 cm–1, FWHM = 1177 cm–1). Experimental spectrum reproduced with permission from E. C. Fitzgerald, N. J. Brown, R. Edge, M. Helliwell, H. N. Roberts, F. Tuna, A. Beeby, D. Collison, P. J. Low, M. W. Whiteley, Organometallics 2012, 31, 157. Copyright 2012 American Chemical Society.


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The Class III Rhenium Complex

Brady et al. reported spectroelectrochemical investigations of the Class III complex [Re1]+ already in 1997.[64,65] Although Reiher et al. performed a comprehensive study on the spin densities and spin states of [Re1]+ and related polyynediyl complexes, the electronic excitations were not considered.[330]

Figure 30. Energy ∆E relative to the most stable rotamer (Ω = 170 °) of the structures of [Re1]+ versus the P-Re-Re-P dihedral angle Ω (0 ≤ Ω ≤ 180°).

In contrast to the other complexes investigated in this chapter, the coordination sphere of the rhenium centers contains a nitrosyl and only one phosphine ligand each. Interestingly, the introduction of these sterically less demanding ligands leads to an increased rotational barrier compared to [Ru1]+ (Figure 30). The energy profile of the relaxed scan at the BLYP35/COSMO(CH2Cl2) level reveals an almost Gaussian-shaped profile with two minima at Ω = 0° and Ω = 170° and a maximum at Ω = 90°. Indeed, structure optimization without constraints lead to trans-[Re1]+ (Ω ≈ 180°), when started from structures with Ω > 90°, and to cis-[Re1]+ (Ω ≈ 0°), when started from structures with Ω < 90°. The two structures are separated by only 2.4 kJ/mol and both exhibit delocalized spin densities (Tables A4 and A11). Interestingly, negative spin density is centered on the nitrosyl ligand indicating the onset of spin polarization, but the 〈S 2 〉 value of 0.78 is increased only slightly (Figure 31). The calculated ν(C≡C) frequencies at 1892 cm–1 for trans-[Re1]+ and 1889 cm–1 for cis-[Re1]+ are in good agreement with the 1872 cm–1 found in experiment.[64,65] Due to the nitrosyl group a characteristic, more intense ν(NO) absorption arises in the IR spectrum at 1665 cm–1. BLYP35/COSMO(CH2Cl2) calculations give vibrational frequencies of 1708 cm–1 for trans-[Re1]+ and 1710 cm–1 for cis-[Re1]+ with roughly five times the intensity of the ν(C≡C) mode. For cis-[Re1]+ a second nitrosyl vibrational frequency is computed at 1737 cm–1. This ν(NO) splitting is probably not observed experimentally due to the rather broad peak. While

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the computed ν(NO) is only in moderate agreement with experiment, the intensity profile is nicely reproduced.

Figure 31. Spin-density isosurface plot (left, ±0.002 a.u.) and isosurface plot (±0.03 a.u.) of the β-HOMO (right) of cis-[Re1]+ calculated at the BLYP35/COSMO(CH2Cl2) level.

The experimental NIR spectrum of [Re1]+ exhibits the asymmetric band typical for MV systems near the Class II/Class III borderline. In addition to the band peak at around 11000 cm–1, a distinct peak in the shoulder at around 10000 cm–1, and a flat band at about 8300 cm–1 are observed. As expected from the ground-state analysis, TDDFT yields very similar results for the two minima trans-[Re1]+ and cis-[Re1]+. The β-HOMO to β-SOMO transition at 10346 cm–1 (µtrans = 10.0 D), at 10287 cm–1 (µtrans = 10.0 D) respectively, is the most intense excitation in both cases, and is of π-π* character as expected for a Class III MV system (Tables A10 and A11). Additionally, a mixed excitation arising from the α-HOMO to α-LUMO and β-HOMO–1 to β-LUMO transitions is computed at 14237 cm–1 (µtrans = 3.0 D) and 14232 cm–1 (µtrans = 3.0 D). In the experimental spectrum a corresponding broad feature is found at around 13900 cm–1 (720 nm, scale in nm).

Hence the origin of the shoulder remains unsettled. For an explanation, all rotameric forms have to be considered. For some of these the β-HOMO–1 to β-SOMO transition gains appreciable intensity. This excitation is also present and appears at very similar energy as the main excitation for trans-[Re1]+ at 10238 cm–1 (µtrans = 0.8 D) and cis-[Re1]+ at 10269 cm–1 (µtrans = 0.2 D), but there it exhibits almost no intensity. Already for Ω = 10° and Ω = 170° the β-HOMO–1 to β-SOMO transition is significantly red-shifted to 9298 cm–1 (µtrans = 6.8 D) and 9315 cm–1 (µtrans = 6.7 D) respectively, and it gains appreciable intensity (Table A12). In contrast the β-HOMO to β-SOMO excitation appears blue-shifted at 11160 cm–1 (µtrans = 7.3 D) and 111325 cm–1 (µtrans = 7.4 D). Upon approaching Ω = 90° from either side the blue shift of the β-HOMO to β-SOMO transition and the red shift of the β-HOMO–1 to β-SOMO transition increase.

To consider the contribution to the experimental spectrum for each rotamer, single-point TDDFT calculations were performed. To take into account the thermal population, the calculated oscillator strengths were weighted by a Boltzmann factor (∆E relative to the most stable conformer at Ω = 170 °). Analogously to the previous section, the results were, for


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better comparison with the experimental spectra, convoluted with Gaussian broadening (σ = 500 cm–1, full width at half maximum FWHM = 1177 cm–1) using the Q-Spector program previously designed for IR spectra,[340] and afterwards transferred to the nm scale (Figure 32). The computed envelope matches the experimental band quite accurately, but is slightly red-shifted. In addition the shoulder appears not as pronounced as in the measured spectrum. Nevertheless the overall agreement is encouraging considering the fact that a Gaussian-shaped band and the neglect of vibronic effects are not fully adequate. Hence a Boltzmann-weighted averaging over the TDDFT results of different rotameric forms appears to be a proper analysis method and is applied in the following section.

Figure 32. Computed envelope (red) compared to experimental IVCT band (black) for [Re1]+. The computed intensities were Boltzmann-weighted, the resulting stick spectrum was Gaussian broadened (σ = 500 cm–1, FWHM = 1177 cm–1), and afterwards transferred to the nm scale. Experimental spectrum reproduced with permission from M. Brady, W. Q. Weng, Y. L. Zhou, J. W. Seyler, A. J. Amoroso, A. M. Arif, M. Bohme, G. Frenking, J. A. Gladysz, J. Am. Chem. Soc. 1997, 119, 775. Copyright 1997 American Chemical Society.


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4.2 Complexes Rotating Through a Conformational Robin-Day Continuum

Introduction

The organometallic complexes [{Ru(dppe)Cp*}2(µ-C≡CC6H4C≡C)]+, [Ru2]+, and [{trans-Ru(dppe)2Cl}2(µ-C≡CC6H4C≡C)]+, [Ru3]+, (Figure 17) both contain the 1,4-diethynyl-benzene (µ-C≡CC6H4C≡C) bridge, but differ in the composition of the supporting ligands, and in both cases, conflicting evidence exists regarding the MV classification.[190,341] Hence, in contrast to the previous section, which dealt with complexes containing linear bridges, the relative conformation of the bridge and of the redox centers has to be considered. For comparison purposes, the classical MV coordination complex, the Creutz-Taube ion,[1,2,342] [CTI]5+, for which the intermediate Class II/III was originally coined,[12] has also been studied using the same methods. Whilst a number of the principal spectroscopic features are reproduced by calculations based on the lowest-energy conformation, the optical and vibrational spectra of [Ru2]+ and [Ru3]+ (Figure 36) are better modeled by a series of structures that account for a distribution of relative conformations of bridge and redox centers in solution. As we will see, the thermal population of a conformational phase space encompassing both localized and delocalized charge distributions limits the usefulness of a description of such complexes in terms of a single, static Robin-Day class. A more accurate explanation of the spectroscopic properties and electronic characteristics requires consideration of the internal rotational dynamics of the molecule and description in terms of a continuum of Class II and Class III states rather than a specific single class.


Increasing the length of the all-carbon chains was found to deteriorate the chemical stability of the polyynediyl-bridged MV complexes. Introduction of aromatic spacers has therefore become a welcome means to increase the metal-metal distances while maintaining good stability and still appreciable, albeit possibly somewhat weaker, electronic coupling between the metal centers.[19,343] Substantial efforts have thus been invested into the study of organometallic complexes containing diethynylaromatic bridges (Figure 17), including computational work.

Interestingly, some of these dinuclear iron and polyyne-bridged ruthenium complexes (and some trinuclear iron species) proved to be sufficiently stable to be studied by scanning tunneling microscopy (STM) on a gold surface, including their MV states.[39,175,344,345] This allowed their single-molecule characterization in such an adsorbed environment, complementary to the usual spectroscopic studies in solution or in the solid state. These investigations were accompanied by DFT calculations to simulate the STM images. Figure 33


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shows examples of measured and simulated images of two diiron complexes, namely [Fe2]+ with a 1,4-diethynylbenzene bridge and [Fe3]+ with a 1,3-diethynylbenzene bridge (Figure 17). The stronger electronic coupling by the para-substituted linker in [Fe2]+ makes this a Class III/Class II borderline case in polar aprotic solution,[325,346] and Class III on a gold surface, as indicated by a relatively symmetrical image in the latter case (Figure 33).[39,175]

Figure 33. Comparison of simulated and experimental STM images under opposite biases for [Fe2]+ (a, left) and [Fe3]+ (b, right). CDFT (B3LYP) was employed to obtain charge localization.[175] Reprinted with permission from R. C. Quardokus, Y. Lu, N. A. Wasio, C. S. Lent, F. Justaud, C. Lapinte and S. A. Kandel, J. Am. Chem. Soc., 2012, 134, 1710. Copyright 2012 American Chemical Society.

In contrast, the meta-substitution in [Fe3]+ leads to a Class II situation, again both in polar solution[347] and on the gold surface[39,175] (Figure 33). Standard BP86 or B3LYP gas-phase calculations gave delocalized situations for both complexes[39,175,347,348] and thus failed to reproduce symmetry breaking for [Fe3]+ or for a corresponding trimetallic 1,3,5-triethynylbenzene-bridged iron complex.[347] Therefore, constrained DFT simulations were subsequently applied. Of course the predictive value of such an approach is limited, as the same constraints applied to [Fe2]+ will also give an asymmetric image.[39,175]

Simplified models of [Fe3]+ and of closely related diethynylpyridine-bridged complexes were also computed at B3LYP level by Costuas et al.[299] Some structural symmetry breaking could be observed in some of the optimizations. However, negligible energy differences (around 1 kJ/mol) between localized and delocalized structures indicated problems in the descriptions, which is not surprising at the given gas-phase DFT level. Subsequent CASSCF and MR-CI calculations provided localized electronic structures, in spite of missing environmental effects. This is unsurprising given that the CASSCF calculations do not include dynamical electron correlation, and the MR-CI calculations recovered very little of it, due to the very small basis sets used (STO-3G).[299]

Similarly to its iron analogue, the 1,3-diethynylbenzene-bridged ruthenium complex [Ru4]+ (Figure 17) exhibits Class II behavior in solution. Fox et al. modeled this system and a related complex with Ru(dppe)2Cl end caps, at the MPW1K gas phase level.[189] In spite of the neglected solvent effects, partially localized structures were obtained with this enhanced Ex

exact admixture, yet substantially more delocalization of spin density onto the bridge was


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computed compared to analogous iron complexes. This is confirmed when computing the spin density with a solvent model included, at the BLYP35/COSMO(CH2Cl2) level (Figure 34).[349] The same computational level provides a clearly delocalized Class III picture for the 1,4-diethynylbenzene-bridged isomer [Ru1]+, or for complexes with related para-substituted diethynylaromatic bridges, in agreement with experiment.[190] This type of system will be discussed in further detail below.

Figure 34. Spin-density isosurface plot(± 0.002 a.u.) of [Fe3]+ (left) and [Ru4]+ (right) calculated at the BLYP35/(def2-) SVP/COSMO(CH2Cl2) level.[349]

The Creutz-Taube Ion and Its Relatives

As the first synthetic MV transition-metal complex with predominantly delocalized character, and due to its importance in the early understanding of ET processes between two transition-metal centers, the famous Creutz-Taube ion [{Ru(NH3)5}2(µ-pz)]5+, [CTI]5+, has probably been studied more than any other MV transition-metal complex, both experimentally and computationally. In fact, a new Robin-Day Class II/III has been coined specifically for [CTI]5+ (see e.g. ref. [12] and references therein). To cover the extensive literature on the various different aspects of ET in the Creutz-Taube ion (vibronic effects, solvent effects, Class II/III), which have been reviewed many times, for example in ref. [12] and ref. [350], would exceed the dimension of this thesis. In particular, it is not attempted to evaluate the importance of vibronic effects, which have been studied in impressive detail as described in the references cited above. The focus will be on the electronic-structure methods and on the effects of environment. Yet even a comprehensive coverage of quantum-chemical studies, all the way back to the first extended-Hückel MO study,[69] does not seem too useful here. The wide range of methods applied and their advantages and disadvantages can probably be illustrated as follows:

The earliest studies at Xα and related levels did not attempt to optimize ground states but focused generally on electronic structure and electronic transitions. Given the local DFT and gas-phase character of these calculations, a clear bias towards Class III behavior is obvious.[351-353] Indeed, the first DFT-based structure optimizations used “pure” (LDA and/or GGA-type) functionals and thus inevitably provided equal Ru-N(pyr) bond lengths and overall symmetric structures.[354,355] Depending somewhat on the basis sets and functionals, the optimized Ru-N(pyr) distances were clearly larger than the experimental values, but the


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computed ΔSCF calculations of excitation energies gave reasonable results.[355] Hardesty et al.[356] obtained a similar overestimate of the Ru-N(pyr) distance at B3LYP hybrid level but much closer agreement with experiment at MP2 level (these calculations enforced C2v-symmetry and thus necessarily equal bond lengths). Subsequent DFT calculations that included solvent effects at the PCM level gave much shorter Ru-N(pyr) bonds and thus closer agreement with experiment.[357,358] Yet, these calculations used symmetry, and thus a possible symmetry breaking could not be evaluated. It appears likely that the much shorter bond lengths obtained at MP2 level were a compensation between sizeable basis-set superposition errors due to the too small basis sets used at that time[356] and the neglected environmental effects.

Symmetry breaking at the HF level and a tendency towards a more delocalized description upon including electron correlation for the Creutz-Taube ion had been mentioned earlier in a combined semi-empirical CNDO and ab initio CASSCF study.[359] Yet, at that time full optimizations at adequate levels could not yet be done. Nevertheless, these studies were the first that incorporated continuum solvent models in the quantum-chemical treatment. The first full optimizations probably pertain to an INDO-based study, used to demonstrate a new set of INDO parameters for Ru.[360] A somewhat asymmetric structure was obtained, consistent with the HF-like nature of the single-determinant INDO wave function. Consequently, subsequent INDO-CISD single-point calculations gave a more delocalized wave function.[360] The computed IVCT excitation energy was nevertheless significantly too low.

The probably most sophisticated ab initio computations on [CTI]5+ were Bolvin’s single-point CASSCF and CASPT2 calculations (done at a slightly idealized structure derived from crystallographic data).[90] In accordance with the findings of Broo and Larsson,[359] CASSCF yields rather poor agreement with experimental UV-vis-NIR data due to the missing dynamical correlation in the ground-state wave function. Subsequent MS-CASPT2 calculations (with PCM solvent) gave excellent agreement with measured excitation bands (e.g. 6400 cm–1 compared to the experimental IVCT band at 6370 cm–1). Interestingly, even transitions accessible only by magnetic circular dichroism spectroscopy were nicely reproduced, and EPR g-tensors in good agreement with experiment were obtained.[30]

The aforementioned first DFT calculations including solvent effects were part of a multi-step procedure aimed at evaluating the solvent contributions to the reorganization energy and thus to the IVCT energy.[357] While this involved conceptually diabatic localized states, the structure was taken to be delocalized. This relatively complicated procedure, which involved both TDDFT and CASSCF steps (with solvent effects included only in the latter), afforded overall also a good IVCT excitation energy (6170 cm–1).[357]

Even more sophisticated solvent treatments used classical MD for explicit water molecules, within a model approach.[361] MD for the solvent motion was also used at the PBE0 DFT level for the related but more localized cyano-bridged ruthenium complex [(H3N)5Ru-µ-NC-Ru(CN)5]–,[362] in the context of simulating the L3-edge X-ray absorption data of this Class II


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system. The comparison of experimental UV-vis-NIR data and TDDFT calculations at PBE0 and B3LYP level with COSMO(H2O) solvent model for this Class II system had been performed earlier.[363]

Computational Details

For [Ru2-Me]+, a two-dimensional relaxed scan of the potential-energy surface (PES) with a fixed P-Ru-Ru-P dihedral angle Ω (which defines a dihedral angle between the half-sandwich metal complex end groups) and a fixed P-Ru-C3-C4 dihedral angle Θreal (dihedral between bridge and a selected end group) was performed (Figure 35). Starting from a Ci-symmetric structure, both dihedral angles were varied in steps of 10°, from 180° to 0° for Ω and from +50° to –100° for Θreal, to cover a reasonable phase space of the relative conformation of the Ru fragments, and of the phenylene moiety in the bridging ligand.

Figure 35. Schematic relative orientation of the metal redox centers (black and grey) and the bridge phenyl plane (dashed blue) of [Ru2-Me]+ at various values of Ω, Θeff and Θreal (similar values hold for [Ru3-Me]+).

In the lowest-energy structures, which are found at Ω = 180° and Ω = 0° and correspond to transoid and cisoid forms of the complex, the plane of the phenylene moiety in the bridge bisects the P-Ru-P angle in each of the diphosphine chelate ligands. In these two structures, the P-Ru-C3-C4 dihedral angle Θreal of ca. 41° (Figure 35) can be translated into an effective X-Ru-C3-C4 dihedral angle Θeff of 0°, where X is the midpoint between the two phosphorus atoms of the chosen diphosphine ligand. As Θeff gives a somewhat more intuitive picture of the relative conformations (Figure 35), Θreal was transformed to Θeff for the entire relaxed scan, and we will discuss results predominantly based on conformations defined in terms of Θeff. A perpendicular arrangement of the phenylene plane relative to the chelate ligands thus translates from a Θreal near –50° to a Θeff of ca. +90° (Figure 35). Due to small variations in the remaining degrees of freedom throughout the scan and due to the reduced symmetry of the system along the scan profile, Θeff may deviate by some fraction of a degree from idealized values at the special points. For [Ru3-Me]+, a similar scan has been restricted to a somewhat


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smaller range of end-group and bridge dihedral angles (0° ≤ Ω ≤ 90° and –50° ≤ Θreal ≤ +50°, i.e. –8.3° ≤ Θeff ≤ +92.2 °), reflecting the higher local symmetry of the ligand environment (C2v) at the ruthenium centers.

Due to the even higher local symmetry of the coordination sphere in the Creutz-Taube ion [CTI]5+ (C4v), the phase space necessary to be covered explicitly was smaller (0° ≤ Ω ≤ 45° and 0° ≤ Θeff ≤ 45°) and was surveyed in 5° steps. Here Ω is the Neq-Ru-Ru-Neq dihedral angle and Θeff = 0° corresponds to the pyrazine plane bisecting the angle between two equatorial amine ligands. The resulting data were interpolated using the MATLAB griddata method 4 to construct the PES plots.[364] For crucial points on the obtained PES, selected minimum and transition state structures were subsequently reoptimized without constraints.

For each point on the PES created in this way, single-point TDDFT calculations were performed. The results are displayed graphically not only as stick spectra, but were, for better comparison with the experimental spectra, additionally convoluted with Gaussian broadening (σ = 300 cm–1, full width at half maximum FWHM = 706.4 cm–1) using the Q-Spector program previously designed for IR spectra.[340]

To take into account the thermal population of different parts of the considered conformational space, the computed data from all points were combined in one spectrum, based on weighting of the computed intensities by a Boltzmann factor as done in the previous section. As ∆E the energy relative to the most stable conformer (e.g. Ω = 180 ° and Θeff = 1.7 ° for [Ru2-Me]+) was taken, and T was set to 298.14 K to correspond to the room-temperature experimental conditions.


Analysis of [Ru2]+. Fox et al. have recently reported the appearance of the IR and NIR spectra of [Ru2]+ (Figure 36).[190] While asymmetry of the NIR bands at the low-energy side for Class II systems close to the Class II/Class III borderline is known to be caused by vibronic coupling effects,[6,57] these do not explain the high-energy features or multiple ν(C≡C) bands observed for [Ru2]+. Rather, the number and energy of ν(C≡C) vibrational modes together with the shape of the NIR absorption envelope was proposed to be due to thermal population of a range of conformers with distinct (localized/delocalized) electronic character.

These observations could not be corroborated by the B3LYP/3-21G* calculations performed on a single conformation using simplified molecular models.[190] However, whilst B3LYP is capable of modeling molecules with delocalized electronic structures quite well, in general density functionals with low exact-exchange admixture are less well suited to the description of Class II situations close to the border between Class II and Class III, due to extensive delocalization errors.[45,48] Moreover, in the experimental systems, charge distribution is likely biased by the solvent polarity and interactions with the counter-ion; in order to accurately


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model these situations computational models must also adequately address the nature of the medium.

Figure 36. The spectroelectrochemically generated NIR (upper) and IR (lower) spectra of [Ru2]+ (left)[190] and [Ru3]+ (right) in CH2Cl2/0.1 M NBu4BF4 as collected by Josef Gluyas, Mark Fox, and Paul Low.

To better model [Ru2]+, the previously introduced computational protocol should be employed. However, regardless of the computational methodology employed, any interpretation of quantum-chemical results based on a single, static, lowest-energy molecular structure will not accurately model systems in which molecular dynamics play an important role on the optoelectronic properties of a molecule. These points are illustrated further below. It is important to note that while the asymmetry of the NIR band at the low-energy side for Class II systems close to the Class II/Class III borderline is known to be caused by vibronic coupling effects, these do not explain the high-energy features observed for [1]+.[6,57]

Full BLYP35/def2-SVP/COSMO(CH2Cl2) structure optimization of [Ru2]+, starting from a Ci-symmetric input, gave a delocalized (Class III) structure with a trans-arrangement of the


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two redox centers, denoted trans-[Ru2]+ (c.f. Figure 35). The spin density in trans-[Ru2]+ (Figure 37) is evenly distributed over the molecular backbone with both the bridging ligand (C≡C/C6H4/C≡C 19 %/18 %/19 %) and the metal centers, Ru(dppe)Cp*, (21 %/21 %) contributing significantly (the large involvement of the bridge atoms in carrying the spin density supports redox non-innocent description of this ligand in such delocalized arrangements[8,11]). This solution corresponds to lowest-energy states calculated previously for [Ru2]+ and analogues.[190,365] The symmetrical Class III situation is further supported by a negligible dipole moment (µ = 0.0 D), and by the computed single harmonic ν(C≡C) vibrational frequency at 1978 cm–1 (scaled by 0.95, see Computational Details), which compares well with the very strong band at 1974 cm–1 in the experimental spectrum (Figure 36). TDDFT calculations with trans-[Ru2]+ at the same computational level gave a single, very intense (µtrans = 17.5 D) NIR transition at 6566 cm–1, in good agreement with the most intense peak at 5750 cm–1 in the experimentally determined spectrum of [Ru2]+ (even better agreement is obtained with the slightly different CPCM solvent implementation in the Gaussian 09 program, cf. SI of [366]). This excitation largely corresponds to a β-HOMO to β-SOMO transition which has substantial bridge π−π* character (Figure 37). However, the additional features at the high-energy side of the experimental NIR band cannot be explained from the TDDFT results.

Figure 37. Isosurface plots of the spin density (top, ± 0.002 a.u.) and the orbitals (± 0.03 a.u.) β-SOMO (bottom, left) and β-HOMO (bottom, right) of trans-[Ru2]+.

Several models for MV complexes have been described which can account for the appearance of multiple transitions of similar energy to the IVCT transition predicted from the Marcus-Hush two-state model. In the case of localized MV complexes towards the Class II/Class III boundary, Meyer and colleagues have shown that the combination of low symmetry, substantial metal-bridge orbital overlap and the use of heavy metals with high spin orbit coupling constants (e.g. Os(III), ξ ~ 3000 cm–1) can lead to the appearance of three IVCT and two dπàdπ transitions through the lifting of parity or LaPorte rules. In the case of lighter metals such as Ru(III) the lower spin orbit coupling constant (ξ ~ 1000 cm–1) not only


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serves to shift the dπàdπ transitions to lower energy (e.g. into the IR region), but also decreases the energy difference between the distinct IVCT transitions leading to a broad NIR absorption feature comprised of the overlapping bands.[12] Alternatively, extension of the two-state model by explicitly including both symmetric and asymmetric vibrational mode and addressing explicitly the bridge orbitals as a mediating state for ET (three-state model) and multiple d-electrons (four-state model) is useful in accounting for the observation of one or more MLCT/LMCT transitions in addition to the IVCT band in more weakly coupled MV systems and the pronounced asymmetry of these bands in strongly coupled (Class III) systems.[57] However, each of these frameworks assumes that the molecular system under investigation can be placed into one of the expanded Robin-Day Classes, and analysis or fitting of the spectroscopic data is used to extract the underlying electronic coupling information. Given the subtle distinctions that can arise from different treatments of the NIR spectra under localized (Class II or II/III) or delocalized (Class III or IV) regimes, the accurate interpretation of the electronic absorption data often requires supporting evidence for the time-scale of the ET event or delocalization. To this end, the observation or absence of IR active modes from both the supporting ligands on the metal centers and the bridging ligand itself is often critical in determination of the choice of method of analysis.

In the present cases of [Ru2]+ and [Ru3]+ the interpretation of the NIR spectra (Figure 36) in terms of a series of overlapping IVCT transitions and a formally Ru(II/III) d6/d5 MV system (at or near the Class II/III borderline) might account for the overlapping transitions that comprise the NIR band envelope, either in terms of multiple IVCT transitions or the presence of closely lying MLCT/LMCT transitions. Alternatively, the appearance of additional features on the NIR band envelope might arise from a vibronic progression due to coupling with the ν(C≡C) vibrational modes.[367] However, both the increasing intensity of the higher energy features in a closely related series of 1,4-naphthyl and 9,10-anthryl bridged complexes[190] and the IR spectra of [Ru2]+ and [Ru3]+ are difficult to reconcile with this interpretation. For example, in the case of [Ru2]+, whilst the ν(C≡C) bands at 2061 cm–1, 1915 cm–1 and the phenylene ring ν(C=C) band at 1564 cm–1 are consistent with a localized MV structure, the ν(C≡C) bands at 1997 cm–1 and 1974 cm–1 are not easily accounted for in terms of a localized model. Similar points apply to the spectra of [Ru3]+ (Figure 36). Although a more strongly coupled (delocalized) model might be more consistent with these latter ν(C≡C) IR bands, the three-state model predicts only an IVCT transition with an asymmetric band shape arising from the low energy ‘cut-off’ whilst the four-state model predicts a significant energy difference between the IVCT and the only MLCT transition with appreciable intensity. Indeed, the IR spectra are inconsistent with the various arguments that can be put forward based solely on the appearance of the NIR bands for assignment of [Ru2]+ and [Ru3]+ to any one of the conventional Robin-Day Classes II, II/III or III.

Given the importance that different molecular conformations play in the appearance of the UV-vis-NIR spectrum of the related complex [{Ru(PPh3)2Cp*}2(µ-C≡CC≡C)]+,[332] a full 2D


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relaxed scan (BLYP35/def2-SVP/COSMO(CH2Cl2) level) of metal fragment and bridge conformations for the truncated model [Ru2-Me]+ (see Computational Details) was performed. Two minima on this 2D PES were initially identified (Figure 38). These minima are almost isoenergetic, and correspond to the trans (Ω ≈ 180°, ∆E = 0.0 kJ/mol) and the analogous cis (Ω ≈ 0°, ∆E = 0.1 kJ/mol) orientations of the ruthenium fragments. In both of these minima, the bridge phenyl plane bisects the two P-Ru-P angles of the diphosphine ligands (i.e. Θeff ≈ 0 °) and hence are denoted trans(0)-[Ru2-Me]+ and cis(0)-[Ru2-Me]+. These two structures provide optimal overlap between the bridging ligand π-system and the metal d-orbitals of similar symmetry and hence the strongest electronic coupling of the two redox centers. Consequently the cis minimum cis(0)-[Ru2-Me]+ also features almost symmetrical structural parameters and an even distribution of the spin density over the molecular backbone (Figure 39), in a manner very similar to that described above for the trans structure. The apparent third minimum on the upper side (Ω ≈ 40°; Θeff ≈ 140 °) of Figure 6 is only part of the trough of a minimum equivalent to cis(0)-[Ru2-Me]+. However, the full optimizations without constraints furnished a third, very shallow minimum that is not apparent from data presented in Figure 38 for the model system [Ru2-Me]+, and which will be discussed further below.

Figure 38. Computed potential energy surface of [Ru2-Me]+ (BLYP35/def2-SVP/COSMO(CH2Cl2) level).

Figure 38 shows that, as expected, rotation of the phenylene moiety in the bridge relative tothe metal centers (i.e. Θeff) has a larger impact on the energy of the system than rotation of the metal end groups relative to each other (i.e. Ω). Maxima occur for Θeff ≈ 90° at Ω ≈ 180° and Ω ≈ 0°. Due to the perpendicular orientation of the phenylene moiety in these higher-energy model structures (with respect to the mirror plane bisecting the P-Ru-P angle in the Ru(PP)Cp


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moiety), electronic coupling is reduced, and the spin-density distribution exhibits partial symmetry breaking towards one of the metal centers (Figure 39). The symmetry breaking is also apparent from key structural parameters such as the difference in the Ru-C≡C bond lengths for the two halves ∆d(Ru-C1) of some 0.03 Å for both maxima. Indeed, most points with perpendicular bridge orientation (Θeff ≈ 90°) correspond to localized spin-density distributions (see Figure A2, Figure 39c, Figure 39d and Figure 41a). However, the highest-energy maximum occurs at only ca. 28.6 kJ/mol above the lowest-energy minimum, and most regions are at much lower energy. Figure A2 in the Appendix illustrates the progressive localization of the spin density as Θeff à 90° for Ω = 180° (i.e. rotation of the bridge) a process that is accompanied by a dramatic reduction of the bridge contributions to the spin density (within the bridging ligand only the C≡C unit close to the oxidized metal center always bears a significant share of the spin). That is, as the phenylene ligand rotates around the long molecular axis, [Ru2-Me]+ and hence by inference [Ru2]+ shifts from strongly coupled Class III situations with large bridge contributions for structures with Θeff = 0° towards more weakly coupled Class II situations with Θeff ≈ 90°. As the energy penalty associated with this rotation is so small, the entire conformational phase space is sampled at room temperature.

Figure 39. Spin-density isosurface plots (± 0.002 a.u.) of [Ru2-Me]+ for different points on the PES (BLYP35/def2-SVP/COSMO(CH2Cl2) level; cf. Figure 35 for the definition of dihedral angles and Figure 38 for the PES).

The relative orientation of the metal fragments, defined by dihedral angle Ω, also influences the distribution of spin density over the molecular framework, and hence the most appropriate Robin-Day classification at each point on the 2D PES: keeping Θeff ≈ 0° and rotating the end groups to a perpendicular orientation (Ω ≈ 90°) leads to a low-energy ridge (below 15 kJ/mol;


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Figure 38). The perpendicular orientation of the end groups also diminishes the electronic coupling to an extent that the charge may become localized (Figure 39d, Figure 41a).

It is worth noting at this point that one-dimensional cuts through the energy profile at either Ω ≈ 0° or 180°, and also through Θeff ≈ 0° superficially resemble the shape of the double-well ground-state potential in the two-state model of a Class II system. Despite the apparent similarity there is a fundamental difference: here the minima correspond to delocalized Class III situations, whereas the spin density becomes more and more localized upon approaching Ω = 90° or Θeff = 90° (Figure 38, Figure 39, Figure 41a).

The two conformational minima found in the relaxed scan for [Ru2-Me]+ (Figure 6) have been fully reoptimized (BLYP35/def2-SVP/COSMO(CH2Cl2) level), initially for the truncated complex [Ru2-Me]+ (with dmpe ligands) and subsequently for the full complex [Ru2]+ (with dppe ligands), with comparable results. The energies of the lowest-energy minima for the truncated system, trans-[Ru2-Me]+ and cis-[Ru2-Me]+ differ by only 0.1 kJ/mol. However, the spin densities in these fully-optimized structures are not completely symmetrical (SI of [366]) and whilst structural symmetry breaking is moderate the differences are sufficiently small that the electronic character is probably still in line with a Class III situation (differences in the Ru-C1 bond lengths are 0.014 Å for cis-[Ru2-Me]+ and 0.015 Å for trans-[Ru2-Me]+).

A third, very shallow minimum (indicated by the absence of imaginary frequencies) with perpendicular orientation of the end groups (Ω = 90°, Θeff = 0°), corresponding to the low-energy ridge in Figure 38 (10.4 kJ/mol above the lowest-energy minimum trans-[Ru2-Me]+) was also identified. Compared to this minimum energy structure, the energy goes slightly up when fixing Ω = 85° or Ω = 95°, and the structure remains a minimum when improving the integration grid (multiple grid m5[368]) and when using tighter structure optimization criteria. While this is certainly only a short-lived metastable structure, consideration of such extra minima will be useful for the interpretation of the IR features (see below). This extra minimum exhibits a localized spin density (SI of [366]) and the structural features of a Class II system (∆d(Ru-C1) = 0.046 Å), and it is denoted perp-[Ru2-Me]+.

Full optimization for the non-truncated complex [Ru2]+ afforded the same three minima, trans-Ru2]+ (already discussed above), cis-[Ru2]+, and perp-[Ru2]+, each of very similar energy (cis-[Ru2]+ and perp-[Ru2]+ are 0.1 kJ/mol and 7.5 kJ/mol, respectively, above trans-[Ru2]+). The spin-density distributions (SI of [366]) and structures of trans-[Ru2]+ (∆d(Ru-C1) = 0.002 Å) and cis-[Ru2]+ (∆d(Ru-C1) = 0.005 Å), are notably more symmetric than for the truncated complex, in agreement with clear Class III behavior, whereas the third, metastable minimum, perp-[Ru2]+, remains clearly localized (with a slightly larger bridge contribution than perp-[Ru2-Me]+; SI of [366]). Together, these results suggest a somewhat stronger electronic coupling between the redox centers for the full system (perhaps due to the relatively greater electron donating properties of dmpe ligands favoring more metal-based redox character and a greater energetic mismatch with the bridging ligand orbitals) but an


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overall very similar situation regarding the conformational profiles of [Ru2]+ and [Ru2-Me]+. This is a useful point to note for future studies of related systems with a view to managing computational effort.

The population of low-energy regions on the PES which feature molecular structures with delocalized to localized electronic characteristics is expected to heavily influence the appearance of the NIR spectrum of [Ru2]+ at ambient temperatures. This proposal has been tested by TDDFT calculations (BLYP35/def2-SVP/COSMO(CH2Cl2) level) for a large subset of points (a restriction to 0° ≤ Θeff ≤ 90° is justified due to approximate symmetry relations) on the 2D PES of [Ru2-Me]+ (cf. Figure 38). Figure 40 combines the Boltzmann-weightedsuperposition (cf. Computational Details) of the stick spectra (red) for all points sampled, with the experimental band profile for [Ru2]+. Additionally, the stick spectra have been convoluted with Gaussian functions (σ = 300 cm–1, full width at half maximum FWHM = 706.4 cm–1; grey shaded area). While being aware that Gaussian broadening and neglect of vibronic effects is not fully adequate, particularly for the low-energy side of the band.[6,57] agreement with the band shape at the high-energy side is encouraging (also given the use of a truncated model).

Figure 40. Computed Boltzmann-weighted TDDFT stick spectra (red) with Gaussian broadened envelope (σ = 300 cm–1, FWHM = 706.4 cm–1, grey) for [Ru2-Me]+ compared to experimental IVCT band[190] (black) of [Ru2]+.

The computed high-energy shoulder is less intense than in the experimental spectrum but at the correct position relative to the main band maximum (c.f. Figure 36, Figure 40). This may be due to the insufficient description of the band asymmetry of the main absorption, which


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would lead to a broader and “flatter” main feature, a more pronounced overlap between the bands of the CT and π−π* excitation, and thus to a higher shoulder.

The contributions from different parts of the conformational PES to the NIR spectral band shape are further analyzed in Figure 41. The differences in the Mulliken spin densities on the two metal fragments (Figure 41a) show that delocalized Class III behavior is concentrated around the cis(0)-[Ru2-Me]+ and trans(0)-[Ru2-Me]+ minima. Structures in these areas give rise to the transitions that dominate the main NIR absorption near 6000 cm–1 (Figure 41b).

Figure 41. Properties (BLYP35/def2-SVP/COSMO(CH2Cl2) level) as function of conformational phase space of [Ru2-Me]+. a) Color plot of Mulliken spin-density differences, ∆SD, between the two Cp*(dmpe)Ru-C≡C units. 0 % indicates fully delocalized and 100 % fully localized distributions (top). b) TDDFT transition dipole moment µtrans of the main π→π* excitation at around 6000 cm–1 (middle). c) TDDFT transition dipole moment of the IVCT excitation at higher energies 7350-9450 cm–1 (bottom). For Ω = 50°, Θeff = 91.7° and for Ω = 100°, Θeff = 101.6°, TDDFT did not converge, and the values were set to 0 (“holes”).


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The electronic coupling between the metal centers is weakest, and thus the localization most pronounced (Figure 41a) near the energy maxima at Ω = 180°, Θeff = 90° and Ω = 0°, Θeff = 90° (cf. Figure 38). These fully localized structures give rise to more intense excitations around 3500 cm–1 and above 12000 cm–1, and do not contribute to the band shoulder around 8000 cm–1 (Figure 41c, Figure 40). The higher-energy shoulder observed in the experimental spectrum arises from sections on the PES (yellow peaks in Figure 41c) that feature only partly localized spin densities (green areas in Figure 41a), but it may nevertheless be viewed as an IVCT band with considerable MMCT character. For structures in which the redox centers are nearly perpendicular and the bridge bisects one of the P-Ru-P angles, a third intense excitation arises (Figure A3), which corresponds to an IVCT transition and is very close in energy to the previously discussed IVCT excitation, and thus contributes to the shoulder as well.

Figure 42. Isosurface plots of the spin density (top left, ± 0.002 a.u.) and key orbitals (± 0.03 a.u.) involved in the NIR excitations of conformer perp-[Ru2-Me]+ (BLYP35/def2-SVP/COSMO(CH2Cl2).

The TDDFT results for the fully optimized minimum structures of the truncated system [Ru2-Me]+ may be used to illustrate these aspects further: starting with the truncated system, conformer trans-[Ru2-Me]+ contributes only one intense transition in the NIR region, at 6108 cm–1 (µtrans = 17.7 D) with considerable diethynyl benzene π-π* character. Similarly cis-[Ru2-Me]+ has only one intense transition at 6085 cm–1 (µtrans = 17.8 D). Clearly these “Class III” areas of the PES are responsible for the main feature in the NIR spectrum. In contrast, perp-[Ru2-Me]+ features one intense transition at 9849 cm–1 (µtrans = 9.0 D) and three lower-intensity transitions at 9334 cm–1 (µtrans = 4.2 D), 10196 cm–1 (µtrans = 5.9 D) and 14508 cm–1


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(µtrans = 2.5 D); the first three transitions are consistent with absorptions that give rise to the shoulder area.

The TDDFT results for the full system [Ru2]+ give very similar excitations for the trans and cis conformers, systematically blue-shifted by ca. 400 cm–1 relative to the similar conformations of [Ru2-Me]+ (see Table 6 below for a summary), again with delocalized π-π* character (β-HOMO→β-SOMO transition). The most pronounced differences occur for perp-[Ru2]+. The main transition appears at 9139 cm–1 (µtrans = 10.6 D) and arises mainly from the β-HOMO→β-SOMO excitation (75 %), with the β-HOMO–1→β-SOMO excitation also contributing (11 %). While the β-SOMO is mainly centered at one Cp*(dmpe)Ru-C≡C unit (64 %) and the aromatic part of the bridge (20 %), the β-HOMO and the β-HOMO–1 are localized on the opposite C≡C-Ru(dmpe)Cp* center (87 %, 78 % respectively), with little contribution from the phenyl ring (3 %, 7 % respectively) or the first ethynyl-metal part (0 %, 7 % respectively) (Figure 42). This transition thus exhibits significant charge-transfer character, but it involves the bridge somewhat more than for the truncated system.

Table 6. Comparison of computed IR and NIR parameters for three conformational minima of [Ru2]+ with experimental data.[a]

infrared (IR) near infrared (NIR)

conformer ν(C≡C)

[cm–1]

(Irel. [%])

ν(aryl)

[cm–1]

(Irel. [%])

νmax(π-π*)

[cm–1]

(µtrans [D])

νmax(MLCT)

[cm–1]

(µtrans [D])

trans-[Ru2]+ 1978 (100) / 6566 (17.5) /

cis-[Ru2]+ 1987 (100) / 6515 (17.6) /

perp-[Ru2]+ 2031 (100) 1566 (59) / 9138 (10.6)

1957 (58) 9412 (7.4)

10196 (5.9)

14248 (2.3)

exp.[190] [Ru2]+ 2061 (w) 1564 (m) 5600[b] 6600[b]

1997 (s) 8300[b]

1974 (vs)

1915 (w) [a]BLYP35/def2-SVP/COSMO(CH2Cl2) level. [b]Estimated from Gaussian deconvolution of the experimental NIR absorption band envelope (see text).

Overall symmetry breaking for this conformer is still notable but less pronounced than for the truncated complex (as noted above, the dmpe ligands appear to support somewhat more metal-localized redox character). Two further transitions between 9200 cm–1 and 20000 cm–1 are also computed. An excitation at 9412 cm–1 (µtrans = 7.4 D) contains contributions both


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from β-HOMO–1→β-SOMO (71 %) and β-HOMO→β-SOMO (16 %) excitations. These are the same orbitals that contribute to the main transition, although to a reversed extent, and hence the transition also has CT character. Finally, a mixed β-HOMO–2→β-SOMO (78 %) and β-HOMO–1→β-SOMO (13 %) transition, again with CT contributions, is found at 14248 cm–1 (µtrans = 2.3 D). Agreement of these excitations with the high-energy shoulder of the experimental IVCT band (Figure 36) is somewhat inferior compared to the truncated model. However, it should be noted that the determination of the experimental shoulder maximum has been based on a Gaussian fit that may well be inaccurate.[190] Moreover, the TDDFT excitation energies depend notably on the precise modeling of non-equilibrium solvation (as demonstrated by lower and thus more accurate excitation energies obtained with Gaussian 09, cf. Table A3).

In order to experimentally test our proposed assignments, the solvatochromic behavior of the NIR band envelope was also examined by Josef Gluyas, Mark Fox, and Paul Low. A sample of [Ru2]PF6 (prepared from Ru2 by treatment with AgPF6) was prepared, and the NIR spectrum was recorded in solutions of CH2Cl2 and CH2Cl2/acetone (1:6) (Figure 43). In the more polar mixture the high-energy shoulder (attributed by the computational study to the IVCT transition of the Class II component, Figure 39) shifts to higher energy and gains intensity while the main lower-energy feature (attributed to the β-HOMO to β-SOMO transition which has substantial bridge π−π* character in the Class III component, Figure 37) is essentially not solvatochromic.

Experimentally, the IR spectrum of [Ru2]+ features one very strong, one strong, and two weak bands in the ν(C≡C) stretching region, as well as a medium-strong band assigned to an aryl breathing mode at lower frequency (Figure 36, cf. Table 6).[190] As noted above, these observations are inconsistent with a pure sample of a symmetrically delocalized Class III complex, which should exhibit only one ν(C≡C) band, whereas the aryl breathing mode should be IR inactive for an essentially centrosymmetric system. The appearance of the IR spectrum was previously attributed to the population of structures with delocalized and localized electronic structures in solution, and suggested to be due to a distribution of conformers.[190] This proposal can now be refined through the computational work undertaken here, with the availability of three fully optimized conformational minima for [Ru2]+ allowing a detailed analysis of the experimental IR spectra by performing harmonic vibrational frequency analyses for all three structures (analogous results for the truncated model [Ru2-Me]+ are provided in the SI of [366]). Harmonic vibrational frequency analyses for trans-[Ru2]+ and cis-[Ru2]+ each provide one intense ν(C≡C) band at 1978 cm–1 and 1987 cm–1, respectively (scaled values, cf. Computational Details), which are consistent with the most intense features in the experimental spectrum (Table 6, Figure 36). Aryl breathing vibrations obtain negligible IR intensity for both of these minima. In contrast, the calculations for conformer perp-[Ru2]+ provide two ν(C≡C) frequencies of 2031 cm–1 and 1957 cm–1, as well as a ν(aryl) mode with significant intensity at 1566 cm–1. The splitting of the


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ν(C≡C) mode and the presence of the aryl breathing mode are fully consistent with the symmetry-broken Class II-nature and permanent dipole moment of this conformer.[189,191] Given that perp-[Ru2]+ is higher in energy than either trans-[Ru2]+ or cis-[Ru2]+ and thus will be less populated, the lower intensities of its features at 2061 cm–1, 1915 cm–1, and 1564 cm–1 in the overall experimental spectrum are easily understandable. The conformational PES is very shallow in the area around perp-[Ru2]+, but such minima may be sufficiently long-lived on the IR time scale to contribute to the spectrum. That is, the presence of rotamers with charge-localized character explains not only the shape and solvatochromic properties of the components of the NIR band envelope, but also the multiple features in the IR spectrum of [Ru2]+.

Figure 43. Overlay plots of the NIR spectra of [Ru2]+ (left) and [Ru3]+ (right) obtained by chemical oxidation in differing solvent mixtures by Josef Gluyas, Mark Fox, and Paul Low.

Extension to [Ru3]+. Can the simultaneous presence of valence-trapped and delocalized MV conformers for a single molecule in solution explain otherwise anomalous spectroscopic data in other systems? The complex [(trans-Ru(dppe)2Cl)2(µ-C≡CC6H4C≡C)]+,[341,369] [Ru3]+ offers a more symmetrical supporting ligand environment than the half-sandwich moieties in [Ru2]+. Klein et al.[341] have studied [Ru3]+ by UV-vis-NIR, IR, and EPR spectroscopies. Josef Gluyas and mark Fox collected the NIR spectrum in CH2Cl2/0.1 M NBu4BF4, whichis similar to that reported earlier in THF/0.1 M NBu4PF6,[341] and exhibits an intense peak with an apparent peak maximum at 6550 cm–1, a distinct high-energy shoulder near 8290 cm–1, giving a profile similar to that of [Ru2]+. In addition, a very weak low-energy shoulder near 4807 cm–1 also appears to be present in the NIR spectrum of [Ru3]+.[341] The IR spectrum of [Ru3]+ also exhibited multiple features that could not be reconciled with a simple Class III description (see below) and which were thought to indicate that the system was not fully delocalized on the IR time scale.[341]


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A similar 2D conformational relaxed scan as discussed above for [Ru2-Me]+ was performed for the truncated system [Ru3-Me]+, with dmpe replacing the dppe ligands in [Ru3]+. Given the more symmetrical metal-coordination environment, the end-group dihedral angle Ω was varied only from 0° to 90°, and the bridge dihedral angle Θeff from 0° to 90° (both in steps of 10°; CH2Cl2 solvent model was again used, see Computational Details). The conformational PES is shown in Figure A4 in Supporting Information. Interestingly, the surface is even much shallower than that obtained above for [Ru2-Me]+, and all points are within an energy of less than 8 kJ/mol. The lowest energy is obtained for Ω ≈ 0°, Θeff ≈ 0°, the highest (at 7.6 kJ/mol) for Ω ≈ 10°, Θeff ≈ 90°. A wide trough is found around Θeff = 0 °, but all features are much less pronounced than for [Ru2-Me]+. Obviously, the dependence of electronic coupling between the redox centers on conformation is reduced due to the more symmetric coordination sphere. Given the extremely flat conformational profile, the outcome of full structure optimizations depended markedly on starting structure, obviously reflecting small numerical inaccuracies (DFT integration grids, thresholds for optimization). Large low-energy motions throughout the entire conformational phase space should be expected, and less importance attached to the specific structures at true minima on the PES compared to the deeper minima observed for [Ru2-Me]+. Nevertheless, it can be noted that, for example, the lowest-energy structure obtained from a full optimization with Ω ≈ 0°, Θeff ≈ 0° (in the following termed deloc-[Ru3-Me]+) exhibits a fully delocalized and essentially symmetrical spin-density distribution (Cl(dmpe)2Ru/Ru(dmpe)2Cl: 24 %/22 %, Figure 44) and differences between d(Ru-C1) bond lengths in each half of the molecule of less than 0.003 Å. In contrast, a second minimum with Ω ≈ 45°, Θeff ≈ 45° (at 2.3 kJ/mol, in the following termed sb-[Ru3-Me]+) exhibits incipient symmetry breaking in both the spin density (Cl(dmpe)2Ru/Ru(dmpe)2Cl: 34 %/16 %; Figure 44) and in the d(Ru-C1) bond lengths differing by 0.025 Å. Still more pronounced charge localization is found for other points on the PES (Figure A5).

Figure 44. Spin-density isosurface plots (± 0.002 a.u.) of deloc-[Ru3-Me]+ (top) and sb-[Ru3-Me]+ (bottom) (BLYP35/def2-SVP/COSMO(CH2Cl2) level).

TDDFT results (BLYP35/def2-SVP/COSMO(CH2Cl2) level) for points across the entire 2D PES of [Ru3-Me]+ were obtained, and an applied Boltzmann weighting used to compare the computational results with the experimental NIR band (Figure 45). The intense peak and thehigh- and low-energy shoulders seen experimentally in [Ru3]+ are reproduced by the


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truncated model [Ru3-Me]+, but the high-energy shoulder is somewhat too close to the main peak.

Table 7. Comparison of computed IR and NIR parameters for two conformational minima of [Ru3-Me]+ with experimental data of [Ru3]+.[a]

infrared (IR) near infrared (NIR)

conformer ν(C≡C)

[cm–1]

(Irel. [%])

ν(aryl)

[cm–1]

(Irel. [%])

νmax(π-π*)

[cm–1]

(µtrans [D])

νmax(MLCT)

[cm–1]

(µtrans [D])

deloc-[Ru3-Me]+ 1982 (100) / 6301 (16.7)

7392 (2.8)

sb-[Ru3-Me]+ 1990 (31) 1584 (7) 6716 (15.6)

1974 (100) 5931 (1.7)

8040 (4.1)

exp. [Ru3]+ 2068 (w) 1571 (m) 6550[c]

2007 (sh) 4807[c]

1966 (vs) 1807 (vw)[b] 8290[c]

1916 (sh) [a]BLYP35/def2-SVP/COSMO(CH2Cl2) level. [b]Likely from contaminant. [c]Centers of the experimental NIR absorption band envelope.

The minima deloc-[Ru3-Me]+ and sb-[Ru3-Me]+ are used to illustrate how the shape of the NIR band is affected by conformational motion, but it should again be emphasized that structures from across the entire PES contribute to the observed spectroscopic profile. TDDFT calculations for deloc-[Ru3-Me]+ give one very intense (µtrans = 16.7 D) excitation at 6301 cm–1, which can be assigned to the main absorption feature in the experimental spectrum. This π-π* transition occurs from the β-HOMO to the β-SOMO (Figure A6) and corresponds to the IVCT or charge resonance band associated with a delocalized (or Class III) complex. A second, weaker excitation at 7392 cm–1 (µtrans = 2.8 D) is also calculated. This transition is of mixed character, as both the β-HOMO–2→β-SOMO (64 %) and β-HOMO–1→β-SOMO (29 %) excitations contribute significantly. The β-HOMO–2 and β-HOMO–1 are nearly degenerate (their energies differ by only 484 cm–1) and are located on opposite metal centers. Given the delocalized nature of the β-SOMO, this transition may be assigned MLCT character, and it corresponds to a bridge-to-metal hole transfer. However, this transition appears to be too close in energy to those responsible for the main absorption band at 6550 cm–1 to fully explain the observed high-energy shoulder at 8290 cm–1.

TDDFT calculations on sb-[Ru3-Me]+ reveal a main transition at 6716 cm–1 (µtrans = 15.6 D) originating from a β-HOMO→β-SOMO excitation with π-π* character, that is blue-shifted


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relative to that in deloc-[Ru3-Me]+. A second excitation at 8040 cm–1 (µtrans = 4.1 D) originates solely from the β-HOMO–1→β-SOMO transition with MLCT character (Figure A6) and appears to be better matched to the 8290 cm–1 absorption. Due to partial charge localization, mixing of this excitation with another MLCT transition is absent here (the near-degeneracy of β-HOMO–2 and β-HOMO–1 is lifted, their energies differ by 3647 cm–1), explaining the blue shift and the enhanced intensity of this excitation compared to deloc-[Ru3-Me]+. In addition, sb-[Ru3-Me]+ exhibits a third, low-intensity TDDFT transition at 5931 cm–1 (µtrans = 1.7 D) with more distinct β-HOMO–2→β-SOMO composition and also MLCT character (Figure S5). This transition may be connected to the experimentally observed low-energy shoulder at 4807 cm–1. Thus, in sb-[Ru3-Me]+ the MLCT transitions associated with the valence trapped forms occur at both higher and lower energy than the primary IVCT (or charge resonance) band associated with the delocalized (or Class III) forms. Overall, it is clear that conformational motion again is responsible for the weaker features of the NIR band of [Ru3]+, albeit in a somewhat different manner than for [Ru3]+. The weakly solvatochromic nature of the NIR spectrum of [Ru3]PF6 when measured in CH2Cl2 and CH2Cl2/acetone (1:6) (Figure 43) also supports these assignments drawn from the models based on various conformations of [Ru3-Me]+.

Figure 45. Computed Boltzmann-weighted TDDFT stick spectra (red) with Gaussian broadened envelope (σ = 300 cm–1, FWHM = 706.4 cm–1, grey) for [Ru3-Me]+ compared to experimental IVCT band (black) of [Ru3]+.


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Klein et al. report three experimental IR frequencies for solutions of [Ru3]+ at 2068 cm–1 (m), 1966 cm–1 (vs) and 1570 cm–1 (s) in CH2Cl2/0.1M NBu4PF6.[341] Closer inspection of the data from samples in CH2Cl2/0.1 M NBu4BF4 gave peaks at 2068 cm–1 (w), 2007 cm–1 (sh), 1966 cm–1 (vs), 1916 cm–1 (sh), 1807 cm–1 (vw), 1571 cm–1 (m) (Figure 36). The two minima deloc-[Ru3-Me]+ and sb-[Ru3-Me]+ allow a rationalization of the main IR bands. Only one ν(C≡C) frequency at 1985 cm–1 is computed for deloc-[Ru3-Me]+, and this likely contributes to the observed band at 1966 cm–1. In contrast, the slight localization and symmetry breaking for sb-[Ru3-Me]+ suffices to generate two ν(C≡C) stretching frequencies at 1990 cm–1 (rel. intensity 31 %) and 1974 cm–1 (100 %), which may correlate with the experimentally observed features at 2007 cm–1 and 1966 cm–1, and one aryl breathing mode at 1584 cm–1 (7 %). Due to the less pronounced localization the splitting of ν(C≡C) stretching frequencies is smaller than for [Ru2]+, although contributions from other areas of the PES, which may help explain the other smaller features at 2068 cm–1 and 1916 cm–1, cannot be excluded. The notion of a flatter PES and more, shallow minima calculated for [Ru3-Me]+ is consistent with the experimental spectrum of [Ru3]+, as the aryl breathing mode at 1571 cm–1 and the ν(C≡C) bands at 2007 and 1966 cm–1, which can be attributed to a Class II form of the complex, are less intense for [Ru3]+ than the analogous features in [Ru2]+ (Figure 36).

The Creutz-Taube Ion [CTI]5+. The prototypical MV complex, the Creutz-Taube ion, provides an obvious means through which to explore conformation effects in systems that ostensibly resemble [Ru2]+ and [Ru3]+. The main differences in [CTI]5+ compared to [Ru2]+ or [Ru3]+ are the Classical Werner-type coordination environment of the ruthenium centers (ammonia and pyrazine ligands coordinated to the metal center by their nitrogen atoms[1,2,61,342]) in the Creutz-Taube ion, the higher local symmetry at the metal centers (C4v) and the larger positive charge. No attempt has been made here to include counter ions in the structural models, but it can be assumed that some screening of charge is affected by the continuum-solvent treatment. Compound [CTI]5+ has probably been investigated more extensively than any other MV complex, through detailed experimental measurements in the solid state[82,370,371] and in solution,[68,88,342] and also using quantum-chemical methods. [90,355-

358,361] The UV-vis-NIR spectrum of [CTI]5+ exhibits a single asymmetric band envelope with no notable shoulders (preventing simple Gaussian fitting). The asymmetric band shape can be explained by vibronic coupling,[6,57] which is not considered here.

For comparison with the other two systems of this study, a 2D relaxed conformer scan was conducted, varying the end-group Neq-Ru-Ru-Neq dihedral Ω and the Neq-Ru-Nbridge-C1 bridge dihedral Θeff both from 0° to 45° in steps of 5°. In contrast to [Ru2]+ and [Ru3]+, the PES of [CTI]5+ (computed in acetonitrile, MeCN, for comparison with experimental conditions; cf. Figure A7) is dominated by a single minimum at Ω = 0°, Θeff = 0°. In this conformation, the plane of the bridging pyrazine ring bisects the angle between neighboring amine ligands. A single maximum is obtained at Ω = 0°, Θeff = 45°, indicating the eclipsing of the equatorial amine ligands by the bridge to dominate the energy profile. This maximum has a relative


101

energy of ca. 28 kJ/mol, giving the conformational PES an energy window comparable to that of [Ru2]+ (see above). Full structure optimization of [CTI]5+ confirms the Ω = 0°, Θeff = 0° as a true minimum. Interestingly, this structure exhibits slight symmetry breaking, as indicated by two different Ru-Nbridge bond lengths (2.027 Å vs 2.017 Å). Nevertheless, the spin density distribution is essentially delocalized (Figure 46). In contrast to [Ru2]+ and [Ru3]+, the BLYP35/def2-SVP/COSMO(MeCN) calculations exhibit appreciable spin contamination 〈S 2 〉 = 0.99 (compared to 0.75 for a pure doublet state). This is manifested in the appreciable (but unphysical) negative spin density on the bridging pyrazine (Figure 46, left;). Comparable spin contamination problems in open-shell transition-metal complexes in other calculations with hybrid density functionals have been noted previously.[129,147] In all cases examined so far, this has been connected with significant metal-ligand antibonding character of the singly occupied molecular orbital(s) (cf. Figure 46, middle).

Figure 46. Isosurface plots of the spin density (left, ± 0.002 a.u.) and β-SOMO (middle)/β-HOMO (right) (± 0.03 a.u.) of [CTI]5+ (BLYP35/def2-SVP/COSMO(MeCN) level).

In spite of the spin contamination, TDDFT calculations for the minimum energy structure give a single π-π* excitation at 7046 cm-1 (µtrans = 10.4 D), in good agreement with experimental[342] band maximum at 6250 cm–1 (Gaussian09 with its different solvent model provides 6210 cm–1, in even better agreement with experiment). A complete Boltzmann-weighted sum of TDDFT stick spectra across the entire conformational 2D PES provides, in addition to the most intense peak from the minimum energy structure, blue-shifted peaks arising from the higher-energy regions of the PES. For example, the maximum at Ω = 0°, Θeff = 45° exhibits a localized spin density, ((NH3)5Ru/pyrazine/Ru(NH3)5, 93 %/-7 %/9 %) and a single excitation at 8198 cm–1. But, due to the lower intensity (µtrans = 4.4 D) and the Boltzmann weighting, this excitation barely contributes to the observed NIR spectrum. Therefore, although the calculations suggest that dynamic conformational processes in [CTI]5+ can shift the electronic character from localized to delocalized, in contrast to [Ru2]+ or [Ru3]+, the most localized forms of the Creutz-Taube ion are almost NIR silent and so there is no appreciable distortion of the experimentally observed band envelope by transitions arising from variations in the relative orientation of the {Ru(NH3)5}n+ and the bridging ligand.

Conclusions

Delocalized Class III and valence-trapped Class II structures may be part of a conformational continuum for MV transition-metal complexes such as [Ru2]+, [Ru3]+, and


102

[CTI]5+, rendering an assignment to a single Robin-Day class inappropriate. The present study suggests that conformational dynamics should be considered when a) the electronic coupling depends significantly on the conformational degrees of freedom to allow switching between delocalized and valence-trapped structures, and b) the energy landscape associated with these changes is sufficiently shallow to allow thermal sampling of delocalized and localized areas at the given temperature. Based on a suitable quantum-chemical protocol, it has been shown that these conditions hold for the complexes [{Ru(dppe)Cp*}2(µ-C≡CC6H4C≡C)]+, [Ru2]+ and [{trans-RuCl(dppe)2}2(µ-C≡CC6H4C≡C)]+, [Ru3]+, to a varying extent: the conformational dependencies in [Ru2]+ and its truncated model [Ru2-Me]+ were found to be stronger than in [Ru3]+ (or [Ru3-Me]+), resulting in a particularly shallow conformational profile for the latter, where the significance of minima is only marginal.

Both in [Ru2]+ and [Ru3]+ the conformational sampling of delocalized and valence-trapped structures is necessary and sufficient to explain a) the non-trivial shape of the IVCT band in the NIR, and b) the simultaneous observation of vibrational frequencies in the IR consistent with Class II and III behavior. In turn, the ability to simulate these spectra with the chosen quantum-chemical approach lends strong support to its correct description of the delocalized and valence-trapped portions of the conformational continuum, consistent with previous successful applications of the protocol to various organic and transition-metal MV systems. The conformation analysis for the third system studied in this section, the classical Creutz-Taube ion [CTI]5+, also showed both delocalized and valence-trapped structures. However, its simulated NIR spectrum suggests that the bands from valence-trapped conformers do not affect the overall band shape significantly, unlike the observed and simulated NIR bands for [Ru2]+ and [Ru3]+.

The Robin-Day classification system will, undoubtedly, remain an important vehicle for the description of MV complexes. The present work serves to highlight that the asymmetries observed in NIR band shapes of MV complexes may be complicated by the presence of multiple conformers each with different electronic couplings. These effects are most significant in systems of low symmetry with relatively flat - but not too flat - potential energy surfaces. They reinforce the essential role that the concerted application of vibrational and electronic spectroscopic methods play in determining the best overall class, or across which range of classes, a complex may belong. Together with the assignment of MV systems to a continuum of MV classes, the availability of reliable quantum-chemical methods to describe localization/delocalization in mixed-valence systems also opens the door to a much more detailed combined analysis of spectroscopic data and electronic structure in MV systems than hitherto possible.

CHAPTER 5

THE CORE-SHELL VS. VALENCE-SHELL SPIN POLARIZATION DILEMMA

Introduction

The thermal ET barrier of MV systems is usually extracted from variable-temperature EPR spectroscopy. To investigate the influence of the more flexible Ex

exact admixture of modern density functionals on spectroscopic properties, a test set of relatively small open-shell transition-metal complexes was chosen and the respective Aiso values calculated. One of the weaknesses of approximate DFT is the often semi-empirical character of the applied exchange-correlation potentials. The functionals are usually adapted to a given problem by varying the amount of Ex

exact admixture, as most properties are more sensitive to the exchange than the correlation part of the functional. While calculations of EPR parameters require high Ex

exact admixtures, gradient-corrected density functionals, like BP86, or hybrid functionals with small Ex

exact admixtures often give superior accuracy describing bonding regions. Furthermore there are phenomena like the previously discussed ET in MV systems, where no conventional functional offers an adequate description and the specially customized functional BLYP35 is thermochemically and kinetically not competitive. These few examples clearly show the need for functionals, which are applicable to a wider range of problems.

One very promising approach is the use of local hybrid functionals (see Section 1.2). ). Here the Ex

exact admixture becomes position-dependent, governed by a LMF. Different local and range-separated hybrid functionals have been evaluated for a series of paramagnetic transition-metal complexes, which was used for validation of functionals before.[129,146] The

CHAPTER 5: HYPERFINE COUPLING CONSTANTS VALIDATION

104

superiority of local hybrids over, e.g., B3LYP for thermochemistry and kinetics has been demonstrated before.[114,131] Besides comparison with experimental isotropic hyperfine coupling constants the spin contamination is monitored. Significant spin contamination leads to errors in calculated structures, population analysis, and especially spin densities and thus has proven to be a potential problem, if EPR parameters for transition metals are calculated at DFT level.[129] This is due to exaggerated valence-shell spin polarization by hybrid functionals with large (constant) Ex

exact admixture, if the SOMO is of significant antibonding character.[129,146,147] At the same time, significant core-shell spin polarization is fundamental for, e.g., accurate hyperfine couplings at transition metal nuclei.[129] In these complexes the negative core-shell contributions of the spin polarization to the spin density at the metal nucleus ρN dominate.[129] The isotropic HFCC is directly proportional to ρN and thus Aiso can be used to probe the amount of spin polarization. For example Aiso decreases with increasing Ex

exact admixture in global hybrid functionals. But for these functionals the spin polarization needed for the proper description of Aiso is accompanied by the onset of significant spin contamination, if the SOMO is of significant antibonding character.[129,146,147] Thus in this chapter it is explored, if first-generation local or range-separated hybrid functionals can provide an improved balance between core- and valence-shells to escape this dilemma.

Computational Details

The structures of the 3d complexes investigated were taken from [129] without further optimization for better comparison. Those complexes included 2ScO, 2TiN, 3TiO, 3VN, 4VO, 6MnO, 6MnF2, 7MnF, 7MnH, 2TiF3, 2MnO3, 6[Mn(CN)4]2-, 6[Cr(CO)4]+, 2[Mn(CO)5], 2[Fe(CO)5]+, 2[Mn(CN)5NO]2-, 2[Mn(CN)4N]-, 2[Ni(CO)3H], 2[Co(CO)4], 2CuO, and 2[Cu(CO)3]. Single-point calculations with the global hybrid functionals B3LYP and BHHLYP,[115,116] local hybrid functionals, and the GGA functionals BLYP and BP86, were performed using a customized TURBOMOLE 5.10 code.[372] Decontracted QZVP basis sets without g-functions were employed for all elements.[373]

The local hybrid functionals included the so-called t-LMF based on the weighted ratio of the von Weizsäcker kinetic energy density τW (r) and the non-interacting local kinetic energy density τ (r) (Equation 1.2.26):[130]

and a so-called t-ρ-LMF:

a(r) = b · ⌧W (r)

⌧(r)

a(r) =

"1 e

⇣c· ⌧W (r)

⌧(r) d·⇢(r)5/3⌘#


105

While the weighting factor of the t-LMF was varied (the thermochemical and kinetic optimum was found to be b = 0.48[114,131]), the prefactors of the t-ρ-LMF were kept constant at c = 1.1275 and d = 0.0165. These functionals were tested in combination with different correlation functionals such as VWN[104] and LYP,[106] but only VWN results are reported here.

Calculations using range-separated hybrid functionals, CAM-B3LYP,[127] LC-ωPBE, ωB97XD,[201] LC-BP86, and LC-BLYP,[125] were performed with the Gaussian09 code,[182] employing contracted QZVP basis sets without g-functions[208] for all complexes except 3VO, for which TZVP basis sets[374] were used. The resulting Kohn-Sham orbitals were transferred from TURBOMOLE/Gaussian09 to the in-house program MAG-ReSpect, which was used to compute isotropic hyperfine coupling constants Aiso.[375] For better comparison, spin-orbit contributions were calculated at the BP86/QZVP level and subtracted from the experimental values given in [129].[376] Spin-density plots and plots of the position dependent Ex

exact admixture were obtained with the Molekel program.[211] Spin densities are plotted in form of isosurfaces (with values ± 0.001 a.u.) using light gray for positive and red for negative values. In the ball-and-stick plots, the atoms are also color-coded (manganese yellow and oxygen red). As experiments were mostly carried out in inert matrices, gas-phase calculations were performed.

Figure 47. Plot of the position-dependent Ex

exact admixture of the local hybrid functionals t-LMF with scaling factor b = 0.48 (right) and of the t-ρ-LMF 2MnO3. See Figure 5 for color scale and atomic positions.


106


Before the performance of the different functionals will be validated, the Exexact admixture

distribution of the local hybrid functionals is examined. As already discussed, high Exexact

admixture is needed in the core region and asymptotically. In contrast high (semi-)local exchange admixture is sufficient or even superior in bonding regions. Both the t-LMF and the t-ρ-LMF were specifically designed to fullfill these conditions.

For the t-LMF the Exexact admixture can be varied between zero and the weighting parameter

b. Hence for b < 1 no area of the molecule exhibits 100% Exexact. The factor b = 0.48 was

found to be optimal for thermochemistry, in spite of the fact that Exexact then never exceeds

48% (Figure 47). Interestingly the most popular range-separated hybrid functional CAM-B3LYP reaches its Ex

exact admixture maximum at 60%, despite the theoretical finding that 100% are needed to successfully recover the correct asymptotic behavior of the potential. Minimum Ex

exact admixture for the t-LMF is found in the bonding area. For the t-ρ-LMF, in which the Ex

exact admixture can be more flexibly tuned, the full 100% Exexact are reached in the

core region. In this region Exexact admixture also nicely resembles the shell structure indicating

the potential to describe negative core-shell contributions of the spin polarization to ρN.

Figure 48. Mean absolute error in isotropic hyperfine coupling constants Aiso for selected functionals. Values for all complexes and functionals are given in Tables A13 and A14.

To evaluate the different functional, the isotropic hyperfine coupling constants of the transition-metal complex series were calculated. The mean absolute errors (MAE) in Aiso compared to experimental results (after subtraction of spin-orbit contributions, see computational details) are shown in Figure 48. The largest errors are found for the t-ρ-LMF in combination with VWN correlation and for ωB97XD. In contrast the t-LMF with VWN correlation (b = 0.48) and the LC-BP86 functional give results (MAE = 52.63 MHz and MAE = 50.10 MHz) comparable to the best-performing functional B3LYP (MAE = 48.38 MHz). These functionals provide the right amount of Ex

exact admixture to

0

20

40

60

80

100

120

MA

E o

f Ais

o [M

Hz]


107

properly balance self-interaction error and non-dynamical correlation. Functionals with less (e.g. BP86) or more (e.g. BHHLYP) Ex

exact perform worse. It is noteworthy that CAM-B3LYP gives a competitive MAE of 60.24 MHz.

Hence the range-separated and local hybrid functionals are no improvement over the most widely used functional B3LYP, when calculated Aiso values are compared for this specific test set of Aiso values. But can the more flexible position-dependent Ex

exact admixture reduce the spin contamination, while still modeling the spin polarization needed for the description of HFCCs? To answer this question calculated spin expectation values 〈S 2 〉 were compared to the nominal value of S(S+1) for all complexes (Figure 49). Interestingly two of the three functionals, which gave the best agreement for Aiso, B3LYP and t-LMF-VWN exhibit almost identical mean spin contamination ∆〈S 2 〉 = 〈S 2 〉calc. – 〈S 2 〉nominal (∆〈S 2 〉 = 0.063 and ∆〈S 2 〉 = 0.064). The third functional LC-BP86, exhibits less spin contamination (∆〈S 2 〉 = 0.056). Another interesting finding is that indeed the more flexible local hybrid functional t-ρ-LMF-VWN exhibits less spin contamination than t-LMF-VWN. Hence employing a more flexible LMF is a promising route out of the core- vs. valence-shell spin polarization dilemma.

Figure 49. Mean spin contamination ∆〈S 2 〉 = 〈S 2 〉calc. – 〈S 2 〉nominal for selected functionals.

To have a closer look at the extent of spin polarization, the spin densities for MnO3 were plotted for different selected functionals (Figure 50). This complex is prone to exhibit spin contamination, and the Ex

exact dependence of computed Aiso is particularly pronounced.[129] The Aiso value decreases with increasing Ex

exact admixture and hence the performance of different functionals can be directly connected to it. For MnO3, the range-separated CAM-B3LYP and LC-BP86 functionals perform best giving Aiso = 1634 MHz and Aiso = 1600 MHz, respectively. While LC-BLYP (Aiso = 1636 MHz) and ωB97XD (Aiso = 1665 MHz) display similar agreement with experiment, for LC-ωPBE provides 1738 MHz. The 〈S 2 〉 values of 1.01 for CAM-B3LYP and of 1.05 for LC-BP86 indicate the

0.000

0.050

0.100

0.150

0.200

0.250

∆〈〈S2〉〉


108

onset of appreciable spin contamination. Both local hybrid functionals overestimate Aiso

significantly. This overestimation is more pronounced for the t-ρ-LMF in combination with VWN correlation. Together with the relatively low mean spin contamination (Figure 49) this result may indicate a too low average Ex

exact admixture of this functional.

BLYP

〈S 2 〉 = 0.76

Aiso = 2031 MHz

B3LYP

〈S 2 〉 = 0.87

Aiso = 1731 MHz

t-LMF-VWN

〈S 2 〉 = 0.87

Aiso =1774 MHz

CAM-B3LYP

〈S 2 〉 = 1.01

Aiso = 1634 MHz

BHHLYP

〈S 2 〉 = 1.86

Aiso = 1275 MHz

Figure 50. Spin-density isosurface plots of spin density (± 0.01 a.u.), spin operator expectation values 〈〈S 2 〉〉 , and isotropic hyperfine coupling constants of 2MnO3 for different selected functionals. Experiment provides 1617 MHz in Ne and Ar matrices.[377] The nominal 〈S 2 〉 value for a doublet is 0.75.

Interestingly the extent of spin contamination is found to be similar for the local hybrid functional t-LMF-VWN and for B3LYP, despite the overestimation of Aiso. Functionals with VWN correlation give lower Aiso values for MnO3 compared to those incorporating LYP correlation.

Figure 51. Mean absolute error (MAE) in isotropic hyperfine coupling constants Aiso for different prefactors b of the t-LMF in combination with VWN correlation.

To test whether raising the Exexact admixture of the t-LMF leads to an improved agreement

between experimental and computed Aiso, the prefactor b was increased in steps of 0.01 from the thermochemically optimized 0.48 to 0.60. Indeed the MAE decreases with increasing b and reaches its minimum of 48.51 MHz at b = 0.59. For b = 0.60 the agreement between experimental and computed Aiso values is already worse, with an MAE of 48.63 MHz (Figure 51). The spin contamination increases almost linearly with the prefactor and thus the Ex

exact

48 49 49 50 50 51 51 52 52 53 53

0.48 0.50 0.52 0.54 0.56 0.58 0.60

MA

E o

f Ais

o [M

Hz]

prefactor b


109

admixture (Figure 52). The MAE of b = 0.59 is comparable to that of B3LYP (MAE = 48.38 MHz), but again the spin contamination is found to be more pronounced (∆〈S 2 〉 = 0.084 compared to 0.063 for B3LYP).

Figure 52. Mean spin contamination ∆〈S 2 〉 for different prefactors b of the t-LMF in combination with VWN correlation.

Conclusions and Outlook

The t-LMF based local hybrid functional as well as two of the tested range-separated hybrid functionals - CAM-B3LYP and LC-BP86 - perform quite well for the isotropic HFCCs of the test set studied, but are so far no improvement over B3LYP. Interestingly two of the three best-performing functionals (t-LMF and CAM-B3LYP) exhibit no regions of the molecules, in which pure (100%) Ex

exact is employed. The t-ρ-LMF, LC-ωPBE, ωB97XD, and LC-BLYP are not competitive for the given HFCC test set. This finding is rather surprising for the t-ρ-LMF, which was especially designed to have larger Ex

exact admixture in the core. In contrast, range-separated hybrid functionals, which use 100% Ex

exact in the long-range, have previously - e.g. in the context of MV systems - been found to exhibit the typical weaknesses of hybrid functionals with too high Ex

exact admixtures. While LC-BP86 also uses 100% Exexact in the

long-range, in combination with its pure GGA exchange in the short-range the functional seems to provide a good balance between self-interaction error and non-dynamical correlation.

For t-LMF based local hybrid functional results can be improved by increasing the factor b to 0.58 and thus the overall Ex

exact admixture of the functional. The mean absolute error is then very similar to that of B3LYP, but of course this prefactor is not optimal for thermochemical und kinetic performance. Hence the tested local hybrid functionals so far do not offer a way out of the core-shell vs. valence-shell spin polarization dilemma.

0.060

0.065

0.070

0.075

0.080

0.085

0.090

0.48 0.50 0.52 0.54 0.56 0.58 0.60

∆〈〈S2 〉〉

prefactor b


110

Future-generation functionals should include improved local mixing functions (e.g. higher prefactor of t-LMF) in combination with more flexible correlation functionals to keep the superior performance for thermochemistry and kinetics over B3LYP. The validation of such second-generation LMFs is an ongoing project. The idea of local range separation appears particularly appealing in this context.

For a wider range of applications of local hybrid functionals the implementation of gradients with respect to nuclear displacement enabling structure optimizations as well as implementation into the ESCF module of TURBOMOLE enabling the calculation of excited-state properties is necessary. Development of a quantum-chemical protocol to describe MV systems based on local hybrid functionals in combination with solvent models, which is also competitive for thermochemistry and kinetics, represents an appealing future target in the present context of MV systems.

CHAPTER 6

GENERAL CONCLUSIONS AND OUTLOOK

The investigation of mixed-valence complexes close to the Class II/Class III borderline remains challenging for both experimental and computational studies. Analysis methods to extract information from UV-vis-NIR band envelopes as well as quantum-chemical approaches, which had proven to yield reliable results for clear-cut Class II or Class III systems have been shown to have difficulties close to the borderline. Nevertheless quantum-chemical calculations can significantly aid the assignment of mixed-valence complexes to the different Robin-Day classes and can reveal information about the character of the transition found in the measured spectra. Due to their favorable scaling with system size, density functional approaches have become the most widely used methods.

In this thesis it could be shown that the BLYP35 functional in combination with suitable solvent methods provides not only valid connections to experimental observations but even predictive quality for mixed-valence transition-metal complexes. Additionally it has been demonstrated that a more reliable assignment of the UV-vis-NIR spectra is facilitated by time-dependent density functional theory computations using the same functional and solvent models. This finding is even more remarkable in view of the well-known problems of time-dependent density functional theory with standard functionals in describing charge-transfer excitations. The standard analysis of experimental spectra introduced by Meyer et al. is based on a simple molecular orbital scheme and the assumption of a localized situation. However, Meyer’s suggested deconvolution of the experimental band envelope by three Gaussian-shaped bands has been shown to be invalid for the mixed-valence complexes investigated in this thesis. The DFT-based analyses demonstrated here provide considerably more insight.

CHAPTER 6: CONCLUSIONS AND OUTLOOK

112

The importance of conformational motion for the coupling between the redox centers has been explored and emphasized in this thesis. While conformational effects on electron transfer have been appreciated for a long time, a quantitative computational procedure to model them had been lacking. Adequate computational studies have now demonstrated how different thermally accessible conformations may alter the spectral characteristics of inorganic mixed-valence systems. The Low group was able to constrain a ruthenium complex to a narrow range of conformers by introducing specially designed ligands. In agreement with our quantum-chemical predictions, the shoulder assigned to perpendicular rotamers is reduced for the tethered complex. Additionally, the agreement between computational and experimental band envelopes is tremendously improved, if contributions of all accessible conformers are taken into account by a Boltzmann-weighting. In case of a series of organometallic complexes, the computations demonstrated that conformational motion may even average to some extent localized and delocalized electronic and molecular structures, leading beyond the traditionally more one-dimensional understanding of the Robin-Day classification scheme. These findings were confirmed by solvatochromatic measurements by the Low group exhibiting an increased intensity of the Class II features in more polar solvents. As a consequence the standard tools to analyze and extract information from experimental spectra appear even more inappropriate.

The importance of environmental effects for charge localization can hardly be overemphasized. The continuum solvent models employed in this thesis yield excellent agreement for the aprotic, relatively nonpolar solvent CH2Cl2 used in most cases. In contrast this approach fails, if hydrogen bonding or the coordination of solvent models to the mixed-valence complex become important. While the latter requires explicit solvent modeling, at best by augmenting the calculations by molecular dynamics or Monte-Carlo simulations, D-COSMO-RS has proven to yield improved results over standard continuum solvent models for organic mixed-valence systems, in which hydrogen bonding influences appreciably ground- and excited-state properties, and even leads to charge localization. This performance is remarkable given the only slightly increased computational cost of D-COSMO-RS over continuum solvent models such as COSMO and (C-)PCM. Due to the limited availability of experimental data in protic solvents for transition-metal complexes, except the Creutz-Taube ion, this approach has not been applied in this thesis, but it represents an interesting future target. An area requiring substantially more efforts in the future is how to appropriately include counter-ion effects into such quantum-chemical treatments of electron transfer in mixed-valence systems. Obviously, modeling of the enzymatic environment is of fundamental importance for biocatalytically active mixed-valence complexes, and the use of continuum models then is not a valid approach anymore.

The modeling of vibronic effects on the UV-vis-NIR spectra is likely to yield an improved agreement between experimental and computed band envelopes. Due to the aforementioned asymmetry of the measured charge-transfer bands in Class II/Class III borderline mixed-valence systems, the simple broadening of the computed stick-spectra by Gauss functions is


113

an invalid approach. To model the Franck-Condon factors, not only the standard harmonic vibrational models, but more sophisticated anharmonic approaches have to be employed. In this context the implementation of COSMO in the EGRAD module of TURBOMOLE to enable excited-state optimizations with the solvent model is highly desirable.

Post-Hartree-Fock approaches are an obvious further direction to pursue for enhanced accuracy. However, many of the applications of such methods so far suffered in particular from far too small single-particle basis sets. These lead to a dramatic underestimation of dynamical correlation effects. As the latter tend to favor more delocalized electronic structure, too small basis sets tend to bias the calculations towards a too localized description. Sometimes compensation with errors arising from neglect of environmental effects (which tends to favor too delocalized charge and spin) might then even provide the qualitatively right answer for the wrong reason. In many mixed-valence systems examined so far, the consideration of a multi-configurational character of the ground-state wave function was less important than expected and than the influence of the other factors discussed, in particular those due to the environment. It will thus be interesting to evaluate in more detail than done so far the usefulness and performance of predominantly single-reference approaches like coupled-cluster theory for mixed-valence systems.

Further improved functionals, either of the range-separated or local hybrid variety with position-dependent exact-exchange admixture, or even more sophisticated constructions, may well provide further enhanced accuracy. In this thesis the performance of these new generations of functionals in describing isotropic hyperfine coupling constants of transition-metal complexes has been evaluated. Additionally the extent of spin contamination has been examined. So far the tested range-separated and local hybrids functionals provide no improvement over the standard functional B3LYP for the tested magnetic properties. Similar findings have been made for mixed-valence systems. The range-separated hybrid functionals tested so far tend to overestimate localized situations. This is likely due to the too high amount of exact-exchange admixture caused by the long-range correction. Only range-separated hybrid functionals such as CAM-B3LYP, which do not recover the full 100% exact-exchange admixture in the long-range, appear to be appropriate in the context of mixed-valence systems. Nevertheless neither in the context of mixed-valence systems nor for the hyperfine coupling constants of transition-metal complexes investigated in this thesis, CAM-B3LYP offers an improvement compared to the global hybrid functionals BLYP35 and B3LYP. Due to a lack of implementation structure optimizations and calculation of excited-state properties have not been available for local hybrid functionals. Hence the development of a quantum-chemical protocol based on thermochemically and kinetically competitive local hybrid functionals seems especially appealing.

A (more) universally applicable functional, which does not require adaptation to the given problem, is of course the envisioned idea pursued by method developers. As long as more sophisticated post-Hartree-Fock approaches are computationally too demanding and/or not


114

applicable to large systems, improved local mixing functions and combined range-separated local hybrid functionals represent the most promising approach.

But already the state-of-the-art computational methods based on appropriately balanced hybrid functionals are able to successfully model, and hence predict, optoelectronic properties of even relatively large and complex molecular materials. This promises much not only for the interpretation of experimental data, but also for computational screening and molecular design optimization in advance of synthetic effort, expenditure of chemical resources, and waste generation. When used sensibly, the predictive power of density functional theory methods may now be seen as a viable tool alongside experimental molecular design and feedback driven optimization as a route to molecular materials with designed optoelectronic properties.

“There are three stages in scientific discovery. First, people deny that it is true, then they deny that it is important; finally they credit the wrong person.”

Bill Bryson

REFERENCES 115

REFERENCES

[1] C. Creutz, H. Taube, J. Am. Chem. Soc. 1969, 91, 3988. [2] C. Creutz, H. Taube, J. Am. Chem. Soc. 1973, 95, 1086. [3] S. F. Nelsen, M. N. Weaver, J. I. Zink, J. P. Telo, J. Am. Chem. Soc. 2005, 127,

10611. [4] J. P. Telo, G. Grampp, M. Shohoji, Phys. Chem. Chem. Phys. 1999, 1, 99. [5] J. P. Telo, A. S. Jalilov, S. F. Nelsen, J. Phys. Chem. A 2011, 115, 3016. [6] A. Heckmann, C. Lambert, Angew. Chem. Int. Ed. 2012, 51, 326. [7] P. Day, N. S. Hush, R. J. H. Clark, Philos. T. Roy. Soc. A 2008, 366, 5. [8] M. D. Ward, J. A. McCleverty, J. Chem. Soc., Dalton Trans. 2002, 275. [9] W. Kaim, Inorg. Chem. 2011, 50, 9752. [10] O. F. Koentjoro, R. Rousseau, P. J. Low, Organometallics 2001, 20, 4502. [11] P. A. Schauer, P. J. Low, Eur. J. Inorg. Chem. 2012, 390. [12] K. D. Demadis, C. M. Hartshorn, T. J. Meyer, Chem. Rev. 2001, 101, 2655. [13] M. D. Ward, Chem. Soc. Rev. 1995, 24, 121. [14] J.-P. Launay, Chem. Soc. Rev. 2001, 30, 386. [15] J.-P. Launay, Coord. Chem. Rev. 2013, 257, 1544. [16] W. Kaim, B. Sarkar, Coord. Chem. Rev. 2013, 257, 1650. [17] C. A. Bignozzi, R. Argazzi, R. Boaretto, E. Busatto, S. Carli, F. Ronconi, S. Caramori,

Coord. Chem. Rev. 2013, 257, 1472. [18] M. H. Chisholm, Coord. Chem. Rev. 2013, 257, 1576. [19] P. J. Low, Coord. Chem. Rev. 2013, 257, 1507. [20] N. S. Hush, in Mixed-Valence Compounds, Vol. 58 (Ed.: D. Brown), Springer

Netherlands, 1980, pp. 151. [21] C. P. Kubiak, Inorg. Chem. 2013, 52, 5663. [22] J. Hankache, O. S. Wenger, Chem. Rev. 2011, 111, 5138. [23] E. I. Solomon, X. Xie, A. Dey, Chem. Soc. Rev. 2008, 37, 613. [24] P. J. Low, Dalton Trans. 2005, 2821. [25] H. Dau, C. Limberg, T. Reier, M. Risch, S. Roggan, P. Strasser, Chemcatchem 2010,

2, 724. [26] V. K. K. Praneeth, M. R. Ringenberg, T. R. Ward, Angew. Chem. Int. Ed. 2012, 51,

10228. [27] S. Schinzel, J. Schraut, A. V. Arbuznikov, P. E. M. Siegbahn, M. Kaupp, Chemistry –

A European Journal 2010, 16, 10424. [28] P. E. M. Siegbahn, Acc. Chem. Res. 2009, 42, 1871. [29] B. Kok, B. Forbush, M. McGloin, Photochem. Photobiol. 1970, 11, 457. [30] W. E. Antholine, D. H. W. Kastrau, G. C. M. Steffens, G. Buse, W. G. Zumft, P. M.

H. Kroneck, Eur. J. Biochem. 1992, 209, 875. [31] P. M. H. Kroneck, W. E. Antholine, D. H. W. Kastrau, G. Buse, G. C. M. Steffens, W.

G. Zumft, FEBS Lett. 1990, 268, 274. [32] J. A. Farrar, W. G. Zumft, A. J. Thomson, Proc. Natl. Acad. Sci. USA 1998, 95, 9891. [33] F. Coat, C. Lapinte, Organometallics 1996, 15, 477. [34] F. Paul, C. Lapinte, Coord. Chem. Rev. 1998, 178, 431.

REFERENCES 116

[35] S. Ballmann, W. Hieringer, D. Secker, Q. L. Zheng, J. A. Gladysz, A. Gorling, H. B. Weber, ChemPhysChem 2010, 11, 2256.

[36] K. Venkatesan, O. Blacque, H. Berke, Dalton Trans. 2007, 1091. [37] W. B. Davis, W. A. Svec, M. A. Ratner, M. R. Wasielewski, Nature 1998, 396, 60. [38] M. Lieberman, S. Chellamma, B. Varughese, Y. L. Wang, C. Lent, G. H. Bernstein,

G. Snider, F. C. Peiris, in Molecular Electronics Ii, Vol. 960 (Eds.: A. Aviram, M. Ratner, V. Mujica), New York Acad Sciences, New York, 2002, pp. 225.

[39] Y. Lu, R. Quardokus, C. S. Lent, F. Justaud, C. Lapinte, S. A. Kandel, J. Am. Chem. Soc. 2010, 132, 13519.

[40] S. B. Braun-Sand, O. Wiest, J. Phys. Chem. 2003, 107, 285. [41] M. Lundstrom, Science 2003, 299, 210. [42] J. D. Meindl, Q. Chen, J. A. Davis, Science 2001, 293, 2044. [43] A. Orbelli Biroli, F. Tessore, M. Pizzotti, C. Biaggi, R. Ugo, S. Caramori, A.

Aliprandi, C. A. Bignozzi, F. De Angelis, G. Giorgi, E. Licandro, E. Longhi, J. Phys. Chem. C 2011, 115, 23170.

[44] A. Yella, H. W. Lee, H. N. Tsao, C. Yi, A. K. Chandiran, M. K. Nazeeruddin, E. W. G. Diau, C. Y. Yeh, S. M. Zakeeruddin, M. Gratzel, Science 2011, 334, 629.

[45] M. Renz, K. Theilacker, C. Lambert, M. Kaupp, J. Am. Chem. Soc. 2009, 131, 16292. [46] M. Kaupp, M. Renz, M. Parthey, M. Stolte, F. Würthner, C. Lambert, Phys. Chem.

Chem. Phys. 2011, 13, 16973. [47] M. Renz, M. Kaupp, J. Phys. Chem. A 2012, 116, 10629. [48] M. Renz, M. Kess, M. Diedenhofen, A. Klamt, M. Kaupp, J. Chem. Theory Comput.

2012, 8, 4189. [49] S. F. Völker, M. Renz, M. Kaupp, C. Lambert, Chem. Eur. J. 2011, 17, 14147. [50] N. S. Hush, Coord. Chem. Rev. 1985, 64, 135. [51] N. S. Hush, Prog. Inorg. Chem. 1967, 8, 391. [52] J. R. Reimers, N. S. Hush, Chem. Phys. 1989, 134, 323. [53] R. J. Cave, M. D. Newton, Chem. Phys. Lett. 1996, 249, 15. [54] R. J. Cave, M. D. Newton, J. Chem. Phys. 1997, 106, 9213. [55] C. Creutz, M. D. Newton, N. Sutin, J. Photochem. Photobiol. A-Chem. 1994, 82, 47. [56] M. D. Newton, in Electron Transfer-from Isolated Molecules to Biomolecules, Pt 1,

Vol. 106 (Eds.: J. Jortner, M. Bixon), John Wiley & Sons Inc, New York, 1999, pp. 303.

[57] B. S. Brunschwig, C. Creutz, N. Sutin, Chem. Soc. Rev. 2002, 31, 168. [58] D. M. D'Alessandro, F. R. Keene, Chem. Soc. Rev. 2006, 35, 424. [59] B. J. Lear, M. H. Chisholm, Inorg. Chem. 2009, 48, 10954. [60] D. M. Dattelbaum, C. M. Hartshorn, T. J. Meyer, J. Am. Chem. Soc. 2002, 124, 4938. [61] C. Creutz, Prog. Inorg. Chem. 1983, 30, 1. [62] P. Aguirre-Etcheverry, D. O'Hare, Chem. Rev. 2010, 110, 4839. [63] J. T. Hupp, R. D. Williams, Acc. Chem. Res. 2001, 34, 808. [64] M. Brady, W. Q. Weng, Y. L. Zhou, J. W. Seyler, A. J. Amoroso, A. M. Arif, M.

Bohme, G. Frenking, J. A. Gladysz, J. Am. Chem. Soc. 1997, 119, 775. [65] H. J. Jiao, K. Costuas, J. A. Gladysz, J. F. Halet, M. Guillemot, L. Toupet, F. Paul, C.

Lapinte, J. Am. Chem. Soc. 2003, 125, 9511. [66] F. Paul, W. E. Meyer, L. Toupet, H. J. Jiao, J. A. Gladysz, C. Lapinte, J. Am. Chem.

Soc. 2000, 122, 9405. [67] F. Coat, M. A. Guillevic, L. Toupet, F. Paul, C. Lapinte, Organometallics 1997, 16,

5988. [68] D. H. Oh, M. Sano, S. G. Boxer, J. Am. Chem. Soc. 1991, 113, 6880.

REFERENCES 117

[69] T. Y. Dong, X. Q. Lai, Z. W. Lin, K. J. Lin, Angew. Chem.-Int. Edit. Engl. 1997, 36, 2002.

[70] M. E. Pandelia, D. Bykov, R. Izsak, P. Infossi, M. T. Giudici-Orticoni, E. Bill, F. Neese, W. Lubitz, Proc. Natl. Acad. Sci. USA 2012, 110, 483.

[71] F. Paul, F. Malvolti, G. da Costa, S. Le Stang, F. Justaud, G. Argouarch, A. Bondon, S. Sinbandhit, K. Costuas, L. Toupet, C. Lapinte, Organometallics 2010, 29, 2491.

[72] S. F. Nelsen, D. A. Trieber, J. J. Wolff, D. R. Powell, S. Rogers-Crowley, J. Am. Chem. Soc. 1997, 119, 6873.

[73] K. Lancaster, S. A. Odom, S. C. Jones, S. Thayumanavan, S. R. Marder, J.-L. Bredas, V. Coropceanu, S. Barlow, J. Am. Chem. Soc. 2009, 131, 1717.

[74] M. B. Robin, P. Day, Adv. Inorg. Chem. Radiochem. 1967, 10, 247. [75] R. A. Marcus, Nobel Lectures, Chemistry 1991-1995, World Scientific Publishing Co.,

Singapore, 1997. [76] N. S. Hush, Electrochim. Acta 1968, 13, 1005. [77] N. Sutin, Prog. Inorg. Chem. 1983, 30, 441. [78] C. Lambert, G. Nöll, J. Am. Chem. Soc. 1999, 121, 8434. [79] S. B. Piepho, J. Am. Chem. Soc. 1988, 110, 6319. [80] S. B. Piepho, E. R. Krausz, P. N. Schatz, J. Am. Chem. Soc. 1978, 100, 2996. [81] B. S. Brunschwig, N. Sutin, Coord. Chem. Rev. 1999, 187, 233. [82] U. Furholz, S. Joss, H. B. Burgi, A. Ludi, Inorg. Chem. 1985, 24, 943. [83] T. Storr, P. Verma, R. C. Pratt, E. C. Wasinger, Y. Shimazaki, T. D. P. Stack, J. Am.

Chem. Soc. 2008, 130, 15448. [84] Y. Shimazaki, T. D. P. Stack, T. Storr, Inorg. Chem. 2009, 48, 8383. [85] L. Chiang, A. Kochem, O. Jarjayes, T. J. Dunn, H. Vezin, M. Sakaguchi, T. Ogura, M.

Orio, Y. Shimazaki, F. Thomas, T. Storr, Chem. Eur. J. 2012, 18, 14117. [86] Y. Shimazaki, T. Yajima, F. Tani, S. Karasawa, K. Fukui, Y. Naruta, O. Yamauchi, J.

Am. Chem. Soc. 2007, 129, 2559. [87] W. Kaim, A. Klein, Spectroelectrochemistry, RSC Publishing, Cambridge, 2008. [88] V. Petrov, J. T. Hupp, C. Mottley, L. C. Mann, J. Am. Chem. Soc. 1994, 116, 2171. [89] E. C. Fitzgerald, N. J. Brown, R. Edge, M. Helliwell, H. N. Roberts, F. Tuna, A.

Beeby, D. Collison, P. J. Low, M. W. Whiteley, Organometallics 2012, 31, 157. [90] H. Bolvin, Inorg. Chem. 2007, 46, 417. [91] P. A. M. Dirac, Proc. R. Soc. Lond. 1929, 123, 714. [92] P. Hohenberg, W. Kohn, Phys. Rev. B: Condens. Matter 1964, 136, 864. [93] W. Kohn, L. J. Sham, Phys. Rev. A: At. Mol. Opt. Phys. 1965, 140, 1133. [94] D. J. Tozer, Density Functional Theory Lecture 2013, University Durham. [95] T. Helgaker, Density-functional theory Lecture 2012, 12th Sostrup Summer School. [96] W. Koch, M. Holthausen, A Chemist's Guide to Density Functional Theory, 2nd ed.,

Wiley-VCH, Weinheim, 2002. [97] F. Jensen, Introduction to Computational Chemistry, 2nd ed., Wiley-VCH, Weinheim,

2006. [98] F. Neese, Coord. Chem. Rev. 2009, 253, 526. [99] P. O. Löwdin, Adv. Chem. Phys. 1959, 2, 207. [100] R. G. Parr, W. Yang, Density-functional theory of atoms and molecules, Vol. 47,

Oxford University Press, John Wiley & Sons, Inc., New York, Oxford, 1989. [101] H. Bahmann, Implementation, Development and Assessment of Local Hybrid Density

Functionals, PhD thesis, Julius-Maximilians-Universität Würzburg 2010. [102] H. Bahmann, M. Ernzerhof, J. Chem. Phys. 2008, 128. [103] J. C. Slater, The Self-Consistent Field for Molecular and Solids, Quantum Theory of

Molecular and Solids,, Vol. 4, McGraw-Hill, New York, 1974.

REFERENCES 118

[104] S. H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 1980, 58, 1200. [105] A. D. Becke, Phys. Rev. A: At. Mol. Opt. Phys. 1988, 38, 3098. [106] C. Lee, W. Yang, R. G. Parr, Phys. Rev. B: Condens. Matter 1988, 37, 785. [107] J. P. Perdew, Phys. Rev. B 1986, 33, 8822. [108] J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 1996, 77, 3865. [109] M. J. G. Peach, D. J. Tozer, N. C. Handy, Int. J. Quant. Chem. 2011, 111, 563. [110] J. P. Perdew, A. Ruzsinszky, L. A. Constantin, J. Sun, G. b. I. Csonka, J. Chem.

Theory Comput. 2009, 5, 902. [111] J. P. Perdew, A. Ruzsinszky, J. M. Tao, V. N. Staroverov, G. E. Scuseria, G. I.

Csonka, J. Chem. Phys. 2005, 123. [112] J. P. Perdew, K. Schmidt, in Density Functional Theory and Its Application to

Materials, Vol. 577 (Eds.: V. Van Doren, C. Van Alsenoy, P. Geerlings), AIP Conference Proceedings, Melville, New York, 2001, p. 1.

[113] J. Tao, J. P. Perdew, V. N. Staroverov, G. E. Scuseria, Phys. Rev. Lett. 2003, 91, 146401.

[114] A. V. Arbuznikov, M. Kaupp, Int. J. Quant. Chem. 2011, 111, 2625. [115] A. D. Becke, J. Chem. Phys. 1993, 98, 1372. [116] P. J. Stephens, J. F. Devlin, C. F. Chabalowski, M. J. Frisch, J. Phys. Chem. 1994, 98,

11623. [117] C. Adamo, V. Barone, Chem. Phys. Lett. 1997, 274, 242. [118] C. Adamo, M. Cossi, V. Barone, J. Mol. Struct.-Theochem. 1999, 493, 145. [119] A. Dreuw, M. Head-Gordon, Chem. Rev. 2005, 105, 4009. [120] T. M. Henderson, B. G. Janesko, G. E. Scuseria, J. Phys. Chem. A 2008, 112, 12530. [121] T. M. Henderson, B. G. Janesko, G. E. Scuseria, J. Chem. Phys. 2008, 128, 194105. [122] H. Stoll, A. Savin, in Density Functional Methods in Physics (Eds.: R. M. Dreizler, J.

da Providencia), Plenum, New York, 1985, p. 177. [123] A. Savin, in Recent Developments and Applications of Modern Density Functional

Theory (Ed.: J. M. Seminario), Elsevier, Amsterdam, 1996. [124] T. Leininger, H. Stoll, H. J. Werner, A. Savin, Chem. Phys. Lett. 1997, 275, 151. [125] H. Iikura, T. Tsuneda, T. Yanai, K. Hirao, J. Chem. Phys. 2001, 115, 3540. [126] J. Heyd, G. E. Scuseria, M. Ernzerhof, J. Chem. Phys. 2003, 118, 8207. [127] T. Yanai, D. P. Tew, N. C. Handy, Chem. Phys. Lett. 2004, 393, 51. [128] T. M. Henderson, A. F. Izmaylov, G. E. Scuseria, A. Savin, J. Chem. Phys. 2007, 127,

221103. [129] M. Munzarova, M. Kaupp, J. Phys. Chem. A 1999, 103, 9966. [130] J. Jaramillo, G. E. Scuseria, M. Ernzerhof, J. Chem. Phys. 2003, 118, 1068. [131] H. Bahmann, A. Rodenberg, A. V. Arbuznikov, M. Kaupp, J. Chem. Phys. 2007, 126,

011103. [132] A. V. Arbuznikov, H. Bahmann, M. Kaupp, J Phys Chem A 2009, 113, 11898. [133] A. V. Arbuznikov, M. Kaupp, Chem. Phys. Lett. 2007, 440, 160. [134] M. Kaupp, H. Bahmann, A. V. Arbuznikov, J. Chem. Phys. 2007, 127, 194102. [135] R. van Leeuwen, Phys. Rev. Lett. 1999, 82, 3863. [136] E. Runge, E. K. U. Gross, Phys. Rev. Lett. 1984, 52, 997. [137] F. Furche, R. Ahlrichs, J. Chem. Phys. 2002, 117, 7433. [138] A. Dreuw, J. L. Weisman, M. Head-Gordon, J. Chem. Phys. 2003, 119, 2943. [139] D. J. Tozer, J. Chem. Phys. 2003, 119, 12697. [140] O. Gritsenko, E. J. Baerends, J. Chem. Phys. 2004, 121, 655. [141] N. T. Maitra, J. Chem. Phys. 2005, 122, 234104. [142] M. J. G. Peach, P. Benfield, T. Helgaker, D. J. Tozer, J. Chem. Phys. 2008, 128,

044118.

REFERENCES 119

[143] C. R. Jacob, M. Reiher, Int. J. Quant. Chem. 2012, 112, 3661. [144] M. Kaupp, J. Comput. Chem. 2007, 28, 320. [145] K. B. Vincent, Q. Zeng, M. Parthey, D. S. Yufit, J. A. K. Howard, F. Hartl, M. Kaupp,

P. J. Low, Organometallics 2013, 32, 6022. [146] M. L. Munzarova, P. Kubacek, M. Kaupp, J. Am. Chem. Soc. 2000, 122, 11900. [147] C. Remenyi, M. Kaupp, J. Am. Chem. Soc. 2005, 127, 11399. [148] J. Baker, A. Scheiner, J. Andzelm, Chem. Phys. Lett. 1993, 216, 380. [149] C. Remenyi, R. Reviakine, M. Kaupp, J. Phys. Chem. B 2007, 111, 8290. [150] A. J. Cohen, D. J. Tozer, N. C. Handy, J. Chem. Phys. 2007, 126, 214104. [151] A. S. Menon, L. Radom, J. Phys. Chem. A 2008, 112, 13225. [152] A. Ipatov, F. Cordova, L. J. Doriol, M. E. Casida, J. Mol. Struct.-Theochem. 2009,

914, 60. [153] J. R. Schmidt, N. Shenvi, J. C. Tully, J. Chem. Phys. 2008, 129, 114110. [154] F. Illas, I. D. R. Moreira, C. de Graaf, V. Barone, Theor. Chem. Acc. 2000, 104, 265. [155] L. Noodleman, J. Chem. Phys. 1981, 74, 5737. [156] L. Noodleman, D. A. Case, Adv. Inorg. Chem. 1992, 38, 423. [157] L. Noodleman, E. R. Davidson, Chem. Phys. 1986, 109, 131. [158] L. Noodleman, J. G. Norman, J. Chem. Phys. 1979, 70, 4903. [159] M. Shoji, K. Koizumi, Y. Kitagawa, T. Kawakami, S. Yamanaka, M. Okumura, K.

Yamaguchi, Chem. Phys. Lett. 2006, 432, 343. [160] M. Shoji, K. Koizumi, Y. Kitagawa, S. Yamanaka, M. Okumura, K. Yamaguchi, Y.

Ohki, Y. Sunada, M. Honda, K. Tatsumi, Int. J. Quant. Chem. 2006, 106, 3288. [161] T. Soda, Y. Kitagawa, T. Onishi, Y. Takano, Y. Shigeta, H. Nagao, Y. Yoshioka, K.

Yamaguchi, Chem. Phys. Lett. 2000, 319, 223. [162] C. Adamo, D. Jacquemin, Chem. Soc. Rev. 2013, 42, 845. [163] P. Karafiloglou, J. P. Launay, J. Phys. Chem. A 1998, 102, 8004. [164] W. Helal, S. Evangelisti, T. Leininger, D. Maynau, J. Comput. Chem. 2009, 30, 83. [165] C. Lambert, S. Amthor, J. Schelter, J. Phys. Chem. 2004, 108, 6474. [166] S. Barlow, C. Risko, S.-J. Chung, N. M. Tucker, V. Coropceanu, S. C. Jones, Z. Levi,

J.-L. Bredas, S. R. Marder, J. Am. Chem. Soc. 2005, 127, 16900. [167] S. F. Nelsen, F. Blomgren, J. Org. Chem. 2001, 66, 6551. [168] S. F. Nelsen, M. N. Weaver, A. E. Konradsson, J. P. Telo, T. Clark, J. Am. Chem. Soc.

2004, 126, 15431. [169] E. R. Johnson, P. Mori-Sanchez, A. J. Cohen, W. T. Yang, J. Chem. Phys. 2008, 129,

204112. [170] T. Heaton-Burgess, W. T. Yang, J. Chem. Phys. 2010, 132. [171] B. Kaduk, T. Kowalczyk, T. Van Voorhis, Chem. Rev. 2012, 112, 321. [172] Q. Wu, T. Van Voorhis, J. Phys. Chem. A 2006, 110, 9212. [173] Q. Wu, T. Van Voorhis, J. Chem. Phys. 2006, 125, 164105. [174] T. Ogawa, M. Sumita, Y. Shimodo, K. Morihashi, Chem. Phys. Lett. 2011, 511, 219. [175] R. C. Quardokus, Y. Lu, N. A. Wasio, C. S. Lent, F. Justaud, C. Lapinte, S. A.

Kandel, J. Am. Chem. Soc. 2012, 134, 1710. [176] F. Ding, H. Wang, Q. Wu, T. Van Voorhis, S. Chen, J. P. Konopelski, J. Phys. Chem.

A 2010, 114, 6039. [177] S. Barlow, C. Risko, V. Coropceanu, N. M. Tucker, S. C. Jones, Z. Levi, V. N.

Khrustalev, M. Y. Antipin, T. L. Kinnibrugh, T. Timofeeva, S. R. Marder, J.-L. Bredas, Chem. Commun. 2005, 764.

[178] A. Pron, R. R. Reghu, R. Rybakiewicz, H. Cybulski, D. Djurado, J. V. Grazulevicius, M. Zagorska, I. Kulszewicz-Bajer, J.-M. Verilhac, J. Phys. Chem. C 2011, 115, 15008.

REFERENCES 120

[179] A. Klamt, G. Schüürmann, J. Chem. Soc., Perkin Trans. 2 1993, 5, 799. [180] TURBOMOLE V6.4 2012, a development of University of Karlsruhe and

Forschungszentrum Karlsruhe GmbH, 1989-2007, TURBOMOLE GmbH, since 2007. [181] V. Barone, M. Cossi, J. Phys. Chem. A 1998, 102, 1995. [182] Gaussian 09, Revision A.02, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E.

Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. Montgomery, J. A., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. M. Rega, J. M., M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, D. J. Fox, Gaussian, Inc., Wallingford, CT, 2009.

[183] J. Tomasi, B. Mennucci, R. Cammi, Chem. Rev. 2005, 105, 2999. [184] C. J. Cramer, D. G. Truhlar, Acc. Chem. Res. 2008, 41, 760. [185] C. J. Cramer, D. G. Truhlar, Acc. Chem. Res. 2009, 42, 493. [186] M. Cossi, N. Rega, G. Scalmani, V. Barone, J. Comput. Chem. 2003, 24, 669. [187] G. Scalmani, M. J. Frisch, B. Mennucci, J. Tomasi, R. Cammi, V. Barone, J. Chem.

Phys. 2006, 124. [188] A. Klamt, J. Phys. Chem. 1996, 100, 3349. [189] M. A. Fox, J. D. Farmer, R. L. Roberts, M. G. Humphrey, P. J. Low, Organometallics

2009, 28, 5266. [190] M. A. Fox, B. Le Guennic, R. L. Roberts, D. A. Brue, D. S. Yufit, J. A. K. Howard, G.

Manca, J.-F. Halet, F. Hartl, P. J. Low, J. Am. Chem. Soc. 2011, 133, 18433. [191] M. A. Fox, R. L. Roberts, T. E. Baines, B. Le Guennic, J.-F. Halet, F. Hartl, D. S.

Yufit, D. Albesa-Jové, J. A. K. Howard, P. J. Low, J. Am. Chem. Soc. 2008, 130, 3566.

[192] A. Pedone, M. Biczysko, V. Barone, ChemPhysChem 2010, 11, 1812. [193] I. C. Gerber, J. G. Angyan, Chem. Phys. Lett. 2005, 415, 100. [194] M. J. G. Peach, T. Helgaker, P. Salek, T. W. Keal, O. B. Lutnaes, D. J. Tozer, N. C.

Handy, Phys. Chem. Chem. Phys. 2006, 8, 558. [195] O. A. Vydrov, G. E. Scuseria, J. Chem. Phys. 2006, 125, 234109. [196] M. Kaupp, A. Arbuznikov, H. Bahmann, Z. Phys. Chemie 2010, 224, 545. [197] J. Perdew, V. Staroverov, J. Tao, G. Scuseria, Phys. Rev. A 2008, 78, 052513. [198] B. G. Janesko, G. E. Scuseria, J. Chem. Phys. 2007, 127, 164117. [199] B. G. Janesko, G. E. Scuseria, J. Chem. Phys. 2008, 128, 084111. [200] A. D. Boese, J. M. L. Martin, J. Chem. Phys. 2004, 121, 3405. [201] J. D. Chai, M. Head-Gordon, J. Chem. Phys. 2008, 128, 084106. [202] R. Bauernschmitt, R. Ahlrichs, Chem. Phys. Lett. 1996, 256, 454. [203] R. Bauernschmitt, M. Haser, O. Treutler, R. Ahlrichs, Chem. Phys. Lett. 1997, 264,

573. [204] F. Furche, D. Rappoport, Density functional theory for excited states: equilibrium

structure and electronic spectra, Vol. 16 of "Computational and Theoretical Chemistry", Elsevier, Amsterdam, 2005.

[205] TURBOMOLE V6.3 2011, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH, 1989-2007, TURBOMOLE GmbH, since 2007.

REFERENCES 121

[206] D. Andrae, U. Häussermann, M. Dolg, H. Stoll, H. Preuss, Theor. Chim. Acta 1990, 77, 123.

[207] A. Schäfer, H. Horn, R. Ahlrichs, J. Chem. Phys. 1992, 97, 2571. [208] F. Weigend, R. Ahlrichs, Phys. Chem. Chem. Phys. 2005, 7, 3297. [209] A. P. Scott, L. Radom, J. Phys. Chem. 1996, 100, 16502. [210] J. C. Roder, F. Meyer, I. Hyla-Kryspin, R. F. Winter, E. Kaifer, Chem. Eur. J. 2003, 9,

2636. [211] U. Varetto, MOLEKEL 5.4, Swiss National Supercomputing Centre: Manno,

Switzerland. [212] M. Malagoli, J. L. Bredas, Chem. Phys. Lett. 2000, 327, 13. [213] J. Bonvoisin, J. P. Launay, W. Verbouwe, M. Van der Auweraer, F. C. De Schryver, J.

Phys. Chem. 1996, 100, 17079. [214] V. Coropceanu, M. Malagoli, J. M. Andre, J. L. Bredas, J. Chem. Phys. 2001, 115,

10409. [215] V. Coropceanu, M. Malagoli, J. M. Andre, J. L. Bredas, J. Am. Chem. Soc. 2002, 124,

10519. [216] V. Coropceanu, C. Lambert, G. Nöll, J. L. Bredas, Chem. Phys. Lett. 2003, 373, 153. [217] V. Coropceanu, N. E. Gruhn, S. Barlow, C. Lambert, J. C. Durivage, T. G. Bill, G.

Noll, S. R. Marder, J. L. Bredas, J. Am. Chem. Soc. 2004, 126, 2727. [218] C. Lambert, S. Amthor, J. Schelter, J. Phys. Chem. A 2004, 108, 6474. [219] C. Lambert, G. Nöll, Angew. Chem.-Int. Edit. 1998, 37, 2107. [220] D. R. Kattnig, B. Mladenova, G. Grampp, C. Kaiser, A. Heckmann, C. Lambert, J.

Phys. Chem. 2009, 113, 2983. [221] P. J. Low, M. A. J. Paterson, H. Puschmann, A. E. Goeta, J. A. K. Howard, C.

Lambert, J. C. Cherryman, D. R. Tackley, S. Leeming, B. Brown, Chem. Eur. J. 2004, 10, 83.

[222] Y. Hirao, M. Urabe, A. Ito, K. Tanaka, Angew. Chem.-Int. Edit. 2007, 46, 3300. [223] N. Utz, T. Koslowski, Chem. Phys. 2002, 282, 389. [224] M. Matis, P. Rapta, V. Lukes, H. Hartmann, L. Dunsch, J. Phys. Chem. B 2010, 114,

4451. [225] W.-H. Chen, K.-L. Wang, W.-Y. Hung, J.-C. Jiang, D.-J. Liaw, K.-R. Lee, J.-Y. Lai,

C.-L. Chen, J. Polym. Sci., Part A: Polym. Chem. 2010, 48, 4654. [226] R. Rybakiewicz, D. Djurado, H. Cybulski, E. Dobrzynska, I. Kulszewicz-Bajer, D.

Boudinet, J.-M. Verilhac, M. Zagorska, A. Pron, Synth. Met. 2011, 161, 1600. [227] S. Barlow, C. Risko, S. A. Odom, S. Zheng, V. Coropceanu, L. Beverina, J.-L.

Brédas, S. R. Marder, J. Am. Chem. Soc. 2012, 134, 10146. [228] S. Mi, N. Lu, J. Mol. Struct.-Theochem. 2010, 940, 1. [229] B. C. De Simone, A. D. Quartarolo, S. Cospito, L. Veltri, G. Chidichimo, N. Russo,

Theor. Chem. Acc. 2012, 131. [230] C. Liu, K.-C. Tang, H. Zhang, H.-A. Pan, J. Hua, B. Li, P.-T. Chou, J. Phys. Chem. A

2012, 116, 12339. [231] E. Göransson, R. Emanuelsson, K. Jorner, T. F. Markle, L. Hammarström, H.

Ottosson, Chem. Sci. 2013, 4, 3522. [232] Y. Zhao, N. E. Schultz, D. G. Truhlar, J. Chem. Phys. 2005, 123, 161103. [233] Y. Zhao, D. G. Truhlar, Theor. Chem. Acc. 2008, 120, 215. [234] Y. Zhao, N. E. Schultz, D. G. Truhlar, J. Chem. Theory Comput. 2006, 2, 364. [235] S. Grimme, J. Chem. Phys. 2006, 124, 034108. [236] T. Schwabe, S. Grimme, Phys. Chem. Chem. Phys. 2007, 9, 3397. [237] S. C. Jones, V. Coropceanu, S. Barlow, T. Kinnibrugh, T. Timofeeva, J.-L. Bredas, S.

R. Marder, J. Am. Chem. Soc. 2004, 126, 11782.

REFERENCES 122

[238] R. J. Bushby, C. Kilner, N. Taylor, R. A. Williams, Polyhedron 2008, 27, 383. [239] H.-J. Nie, X. Chen, C.-J. Yao, Y.-W. Zhong, G. R. Hutchison, J. Yao, Chem. Eur. J.

2012, 18, 14497. [240] C. J. Yao, J. N. A. Yao, Y. W. Zhong, Inorg. Chem. 2011, 50, 6847. [241] Y.-M. Zhang, S.-H. Wu, C.-J. Yao, H.-J. Nie, Y.-W. Zhong, Inorg. Chem. 2012, 51,

11387. [242] B. J. Liddle, S. Wanniarachchi, J. S. Hewage, S. V. Lindeman, B. Bennett, J. R.

Gardinier, Inorg. Chem. 2012, 51, 12720. [243] R. Denis, L. Toupet, F. Paul, C. Lapinte, Organometallics 2000, 19, 4240. [244] D. E. Richardson, H. Taube, Coord. Chem. Rev. 1984, 60, 107. [245] D. P. Arnold, G. A. Heath, D. A. James, J. Porphyrins Phthalocyanines 1999, 03, 5. [246] F. Barrière, N. Camire, W. E. Geiger, U. T. Mueller-Westerhoff, R. Sanders, J. Am.

Chem. Soc. 2002, 124, 7262. [247] P. J. Low, M. A. J. Paterson, A. E. Goeta, D. S. Yufit, J. A. K. Howard, J. C.

Cherryman, D. R. Tackley, B. Brown, J. Mater. Chem. 2004, 14, 2516. [248] P. J. Low, M. A. J. Paterson, D. S. Yufit, J. A. K. Howard, J. C. Cherryman, D. R.

Tackley, R. Brook, B. Brown, J. Mater. Chem. 2005, 15, 2304. [249] S. Amthor, B. Noller, C. Lambert, Chem. Phys. 2005, 316, 141. [250] S. Barlow, C. Risko, S. J. Chung, N. M. Tucker, V. Coropceanu, S. C. Jones, Z. Levi,

J. L. Bredas, S. R. Marder, J. Am. Chem. Soc. 2005, 127, 16900. [251] M. Mayor, C. von Hanisch, H. B. Weber, J. Reichert, D. Beckmann, Angew. Chem.-

Int. Edit. 2002, 41, 1183. [252] M. Parthey, K. B. Vincent, M. Renz, P. A. Schauer, D. S. Yufit, J. A. K. Howard, M.

Kaupp, P. J. Low, Inorg. Chem. 2014, 53, 1544. [253] A. B. Ricks, G. C. Solomon, M. T. Colvin, A. M. Scott, K. Chen, M. A. Ratner, M. R.

Wasielewski, J. Am. Chem. Soc. 2010, 132, 15427. [254] J. K. Burdett, A. K. Mortara, Chem. Mater. 1997, 9, 812. [255] L. Gobbi, P. Seiler, F. Diederich, V. Gramlich, Helv. Chim. Acta 2000, 83, 1711. [256] L. Gobbi, N. Elmaci, H. P. Luthi, F. Diederich, ChemPhysChem 2001, 2, 423. [257] Y. M. Zhao, S. C. Ciulei, R. R. Tykwinski, Tetrahedron Lett. 2001, 42, 7721. [258] N. N. P. Moonen, C. Boudon, J. P. Gisselbrecht, P. Seiler, M. Gross, F. Diederich,

Angew. Chem.-Int. Edit. 2002, 41, 3044. [259] Y. M. Zhao, A. D. Slepkov, C. O. Akoto, R. McDonald, F. A. Hegmann, R. R.

Tykwinski, Chem. Eur. J. 2005, 11, 321. [260] Y. M. Zhao, N. Z. Zhou, A. D. Slepkov, S. C. Ciulei, R. McDonald, F. A. Hegmann,

R. R. Tykwinski, Helv. Chim. Acta 2007, 90, 909. [261] F. Bures, W. B. Schweizer, J. C. May, C. Boudon, J. P. Gisselbrecht, M. Gross, I.

Biaggio, F. Diederich, Chem. Eur. J. 2007, 13, 5378. [262] N. N. P. Moonen, R. Gist, C. Boudon, J. P. Gisselbrecht, P. Seiler, T. Kawai, A.

Kishioka, M. Gross, M. Irie, F. Diederich, Org. Biomol. Chem. 2003, 1, 2032. [263] F. Mitzel, C. Boudon, J. P. Gisselbrecht, P. Seiler, M. Gross, F. Diederich, Helv.

Chim. Acta 2004, 87, 1130. [264] S. Kato, M. Kivala, W. B. Schweizer, C. Boudon, J. P. Gisselbrecht, F. Diederich,

Chem. Eur. J. 2009, 15, 8687. [265] M. Hasegawa, Y. Takatsuka, Y. Kuwatani, Y. Mazaki, Tetrahedron Lett. 2012, 53,

5385. [266] P. A. Bouit, M. Marszalek, R. Humphry-Baker, R. Viruela, E. Orti, S. M.

Zakeeruddin, M. Gratzel, J. L. Delgado, N. Martin, Chem. Eur. J. 2012, 18, 11621. [267] K. Lincke, M. A. Christensen, F. Diederich, M. B. Nielsen, Helv. Chim. Acta 2011,

94, 1743.

REFERENCES 123

[268] A. S. Andersson, L. Kerndrup, A. O. Madsen, K. Kilsa, M. B. Nielsen, P. R. La Porta, I. Biaggio, J. Org. Chem. 2009, 74, 375.

[269] A. A. Kocherzhenko, L. D. A. Siebbeles, F. C. Grozema, J. Phys. Chem. Lett. 2011, 2, 1753.

[270] F. Diederich, R. Faust, V. Gramlich, P. Seiler, J. Chem. Soc.-Chem. Commun. 1994, 2045.

[271] R. Faust, F. Diederich, V. Gramlich, P. Seiler, Chem. Eur. J. 1995, 1, 111. [272] K. Campbell, R. McDonald, N. R. Branda, R. R. Tykwinski, Org. Lett. 2001, 3, 1045. [273] K. Campbell, R. McDonald, M. J. Ferguson, R. R. Tykwinski, Organometallics 2003,

22, 1353. [274] P. Siemsen, U. Gubler, C. Bosshard, P. Gunter, F. Diederich, Chem. Eur. J. 2001, 7,

1333. [275] M. I. Bruce, N. N. Zaitseva, P. J. Low, B. W. Skelton, A. H. White, J. Organomet.

Chem. 2006, 691, 4273. [276] M. Akita, Y. Tanaka, C. Naitoh, T. Ozawa, N. Hayashi, M. Takeshita, A. Inagaki, M.

C. Chung, Organometallics 2006, 25, 5261. [277] Z. Cao, T. Ren, Organometallics 2011, 30, 245. [278] B. Xi, I. P. C. Liu, G. L. Xu, M. M. R. Choudhuri, M. C. DeRosa, R. J. Crutchley, T.

Ren, J. Am. Chem. Soc. 2011, 133, 15094. [279] G. L. Xu, B. Xi, J. B. Updegraff, J. D. Protasiewicz, T. Ren, Organometallics 2006,

25, 5213. [280] Y. M. Zhao, T. Luu, G. M. Bernard, T. Taerum, R. McDonald, R. E. Wasylishen, R.

R. Tykwinski, Can. J. Chem. 2012, 90, 994. [281] T. Shoji, S. Ito, T. Okujima, N. Morita, Eur. J. Org. Chem. 2011, 5134. [282] W. P. Forrest, Z. Cao, H. R. Hambrick, B. M. Prentice, P. E. Fanwick, P. S.

Wagenknecht, T. Ren, Eur. J. Inorg. Chem. 2012, 5616. [283] W. P. Forrest, Z. Cao, K. M. Hassell, B. M. Prentice, P. E. Fanwick, T. Ren, Inorg.

Chem. 2012, 51, 3261. [284] B. S. Brunschwig, N. Sutin, J. Am. Chem. Soc. 1989, 111, 7454. [285] R. J. Geue, A. J. Hendry, A. M. Sargeson, J. Chem. Soc.-Chem. Commun. 1989, 1646. [286] S. Woitellier, J. P. Launay, C. W. Spangler, Inorg. Chem. 1989, 28, 758. [287] J. P. Launay, M. Tourrelpagis, J. F. Lipskier, V. Marvaud, C. Joachim, Inorg. Chem.

1991, 30, 1033. [288] A. Harriman, Mol. Cryst. Liq. Cryst. 1991, 194, 103. [289] A. C. Benniston, V. Goulle, A. Harriman, J.-M. Lehn, B. Marczinke, J. Phys. Chem.

1994, 98, 7798. [290] J. Wang, A. D. Becke, V. H. Smith, Jr., J. Chem. Phys. 1995, 102, 3477. [291] A. C. Benniston, A. Harriman, Chem. Soc. Rev. 2006, 35, 169. [292] S. V. Rosokha, D. L. Sun, J. K. Kochi, J. Phys. Chem. A 2002, 106, 2283. [293] D. Hanss, O. S. Wenger, Eur. J. Inorg. Chem. 2009, 3778. [294] D. Hanss, M. E. Walther, O. S. Wenger, Coord. Chem. Rev. 2010, 254, 2584. [295] D. M. D'Alessandro, F. R. Keene, Chem. Phys. 2006, 324, 8. [296] D. M. D'Alessandro, F. R. Keene, Pure Appl. Chem. 2008, 80, 1. [297] D. M. D'Alessandro, F. R. Keene, S. D. Bergman, M. Kol, Dalton Trans. 2005, 332. [298] P. P. Romanczyk, K. Noga, A. J. Wlodarczyk, W. Nitek, E. Broclawik, Inorg. Chem.

2010, 49, 7676. [299] K. Costuas, O. Cador, F. Justaud, S. Le Stang, F. Paul, A. Monari, S. Evangelisti, L.

Toupet, C. Lapinte, J.-F. Halet, Inorg. Chem. 2011, 50, 12601. [300] C. Olivier, K. Costuas, S. Choua, V. Maurel, P. Turek, J. Y. Saillard, D. Touchard, S.

Rigaut, J. Am. Chem. Soc. 2010, 132, 5638.

REFERENCES 124

[301] M. I. Bruce, K. Costuas, B. G. Ellis, J.-F. Halet, P. J. Low, B. Moubaraki, K. S. Murray, N. Ouddai, G. J. Perkins, B. W. Skelton, A. H. White, Organometallics 2007, 26, 3735.

[302] M. H. Chisholm, F. Feil, C. M. Hadad, N. J. Patmore, J. Am. Chem. Soc. 2005, 127, 18150.

[303] P. Wiggins, J. A. G. Williams, D. J. Tozer, J. Chem. Phys. 2009, 131, 091101. [304] D. Rappoport, F. Furche, J. Am. Chem. Soc. 2004, 126, 1277. [305] A. Köhn, C. Hattig, J. Am. Chem. Soc. 2004, 126, 7399. [306] M. A. Fox, R. L. Roberts, W. M. Khairul, F. Hartl, P. J. Low, J. Organomet. Chem.

2007, 692, 3277. [307] F. Gendron, A. Burgun, B. W. Skelton, A. H. White, T. Roisnel, M. I. Bruce, J.-F.

Halet, C. Lapinte, K. Costuas, Organometallics 2012, 31, 6796. [308] M. I. Bruce, P. J. Low, in Advances in Organometallic Chemistry, Vol. 50, Vol. 50

(Eds.: R. West, A. F. Hill), Elsevier Academic Press Inc, San Diego, 2004, pp. 179. [309] W. Mohr, J. Stahl, F. Hampel, J. A. Gladysz, Chem. Eur. J. 2003, 9, 3324. [310] Q. L. Zheng, J. C. Bohling, T. B. Peters, A. C. Frisch, F. Hampel, J. A. Gladysz,

Chem. Eur. J. 2006, 12, 6486. [311] M. I. Bruce, K. A. Kramarczuk, B. W. Skelton, A. H. White, J. Organomet. Chem.

2010, 695, 469. [312] M. I. Bruce, B. K. Nicholson, N. N. Zaitseva, C. R. Chimie 2009, 12, 1280. [313] A. B. Antonova, M. I. Bruce, B. G. Elis, M. Gaudio, P. A. Humphrey, M. Jevric, G.

Melino, B. K. Nicholson, G. J. Perkins, B. W. Skelton, B. Stapleton, A. H. White, N. N. Zaitseva, Chem. Commun. 2004, 960.

[314] W. A. Chalifoux, R. R. Tykwinski, Nature Chemistry 2010, 2, 967. [315] K. Costuas, S. Rigaut, Dalton Trans. 2011, 40, 5643. [316] A. Ceccon, S. Santi, L. Orian, A. Bisello, Coord. Chem. Rev. 2004, 248, 683. [317] M. I. Bruce, P. J. Low, K. Costuas, J.-F. Halet, S. P. Best, G. A. Heath, J. Am. Chem.

Soc. 2000, 122, 1949. [318] J.-F. Halet, C. Lapinte, Coord. Chem. Rev. 2013, 257, 1584. [319] M. I. Bruce, M. L. Cole, K. Costuas, B. G. Ellis, K. A. Kramarczuk, C. Lapinte, B. K.

Nicholson, G. J. Perkins, B. W. Skelton, A. H. White, N. N. Zaitseva, Z. Anorg. Allg. Chem. 2013, 639, 2216.

[320] N. Le Narvor, L. Toupet, C. Lapinte, J. Am. Chem. Soc. 1995, 117, 7129. [321] M. I. Bruce, K. Costuas, T. Davin, J.-F. Halet, K. A. Kramarczuk, P. J. Low, B. K.

Nicholson, G. J. Perkins, R. L. Roberts, B. W. Skelton, M. E. Smith, A. H. White, Dalton Trans. 2007, 5387.

[322] M. I. Bruce, B. G. Ellis, P. J. Low, B. W. Skelton, A. H. White, Organometallics 2003, 22, 3184.

[323] P. J. Low, S. Bock, Electrochim. Acta 2013, 110, 681. [324] P. J. Low, M. I. Bruce, in Adv. Organomet. Chem., Vol. Volume 48, Academic Press,

2001, pp. 71. [325] N. Le Narvor, C. Lapinte, Organometallics 1995, 14, 634. [326] P. Belanzoni, N. Re, A. Sgamellotti, C. Floriani, J. Chem. Soc.-Dalton Trans. 1998,

1825. [327] F. J. Fernandez, K. Venkatesan, O. Blacque, M. Alfonso, H. W. Schmalle, H. Berke,

Chem. Eur. J. 2003, 9, 6192. [328] S. Kheradmandan, K. Heinze, H. W. Schmalle, H. Berke, Angew. Chem.-Int. Edit.

1999, 38, 2270. [329] M. I. Bruce, K. Costuas, T. Davin, B. G. Ellis, J.-F. Halet, C. Lapinte, P. J. Low, M. E.

Smith, B. W. Skelton, L. Toupet, A. H. White, Organometallics 2005, 24, 3864.

REFERENCES 125

[330] C. Herrmann, J. Neugebauer, J. A. Gladysz, M. Reiher, Inorg. Chem. 2005, 44, 6174. [331] N. J. Brown, H. N. Lancashire, M. A. Fox, D. Collison, R. Edge, D. S. Yufit, J. A. K.

Howard, M. W. Whiteley, P. J. Low, Organometallics 2011, 30, 884. [332] M. Parthey, J. B. G. Gluyas, P. A. Schauer, D. S. Yufit, J. A. K. Howard, M. Kaupp,

P. J. Low, Chem. Eur. J. 2013, 19, 9780. [333] S. F. Nelsen, A. E. Konradsson, M. N. Weaver, J. P. Telo, J. Am. Chem. Soc. 2003,

125, 12493. [334] S. F. Nelsen, M. N. Weaver, J. P. Telo, J. Phys. Chem. A 2007, 111, 10993. [335] M. I. Bruce, B. G. Ellis, M. Gaudio, C. Lapinte, G. Melino, F. Paul, B. W. Skelton, M.

E. Smith, L. Toupet, A. H. White, Dalton Trans. 2004, 1601. [336] M. I. Bruce, B. C. Hall, B. D. Kelly, P. J. Low, B. W. Skelton, A. H. White, J. Chem.

Soc.-Dalton Trans. 1999, 3719. [337] M. I. Bruce, P. Hinterding, E. R. T. Tiekink, B. W. Skelton, A. H. White, J.

Organomet. Chem. 1993, 450, 209. [338] L. de Quadras, E. B. Bauer, W. Mohr, J. C. Bohling, T. B. Peters, J. M. Martin-

Alvarez, F. Hampel, J. A. Gladysz, J. Am. Chem. Soc. 2007, 129, 8296. [339] J. Stahl, W. Mohr, L. de Quadras, T. B. Peters, J. C. Bohling, J. M. Martin-Alvarez, G.

R. Owen, F. Hampel, J. A. Gladysz, J. Am. Chem. Soc. 2007, 129, 8282. [340] V. Pelmenschikov, Y. Guo, H. Wang, S. P. Cramer, D. A. Case, Faraday Discuss.

2011, 148, 409. [341] A. Klein, O. Lavastre, J. Fiedler, Organometallics 2006, 25, 635. [342] C. Creutz, M. H. Chou, Inorg. Chem. 1987, 26, 2995. [343] S. I. Ghazala, F. Paul, L. Toupet, T. Roisnel, P. Hapiot, C. Lapinte, J. Am. Chem. Soc.

2006, 128, 2463. [344] N. A. Wasio, R. C. Quardokus, R. P. Forrest, S. A. Corcelli, Y. Lu, C. S. Lent, F.

Justaud, C. Lapinte, S. A. Kandel, J. Phys. Chem. C 2012, 116, 25486. [345] S. Guo, S. A. Kandel, J. Phys. Chem. Lett. 2010, 1, 420. [346] S. Ibn Ghazala, F. Paul, L. Toupet, T. Roisnel, P. Hapiot, C. Lapinte, J. Am. Chem.

Soc. 2006, 128, 2463. [347] T. Weyland, K. Costuas, L. Toupet, J.-F. Halet, C. Lapinte, Organometallics 2000, 19,

4228. [348] F. De Montigny, G. Argouarch, K. Costuas, J.-F. Halet, T. Roisnel, L. Toupet, C.

Lapinte, Organometallics 2005, 24, 4558. [349] M. Parthey, M. Kaupp, P. J. Low, unpublished results. [350] W. Kaim, A. Klein, M. Glockle, Acc. Chem. Res. 2000, 33, 755. [351] M. J. Ondrechen, D. E. Ellis, M. A. Ratner, Chem. Phys. Lett. 1984, 109, 50. [352] M. J. Ondrechen, J. Ko, L. T. Zhang, J. Am. Chem. Soc. 1987, 109, 1672. [353] L. T. Zhang, J. Ko, M. J. Ondrechen, J. Am. Chem. Soc. 1987, 109, 1666. [354] Z. Chen, J. Bian, L. Zhang, S. Li, J. Chem. Phys. 1999, 111, 10926. [355] A. Bencini, I. Ciofini, C. A. Daul, A. Ferretti, J. Am. Chem. Soc. 1999, 121, 11418. [356] J. Hardesty, S. K. Goh, D. S. Marynick, J. Mol. Struct.-Theochem. 2002, 588, 223. [357] J. R. Reimers, Z. L. Cai, N. S. Hush, Chem. Phys. 2005, 319, 39. [358] T. Todorova, B. Delley, Inorg. Chem. 2008, 47, 11269. [359] A. Broo, S. Larsson, Chem. Phys. 1992, 161, 363. [360] A. Broo, P. Lincoln, Inorg. Chem. 1997, 36, 2544. [361] J. R. Reimers, B. B. Wallace, N. S. Hush, Philos. T. Roy. Soc. A 2008, 366, 15. [362] B. E. Van Kuiken, M. Valiev, S. L. Daifuku, C. Bannan, M. L. Strader, H. N. Cho, N.

Huse, R. W. Schoenlein, N. Govind, M. Khalil, J. Phys. Chem. A 2013, 117, 4444. [363] I. Kondov, V. Vallet, H. B. Wang, M. Thoss, J. Phys. Chem. A 2008, 112, 5467. [364] MatLab R2012b (8.0.0783), 2012, The MathWorks Inc., Natick, Massachusetts.

REFERENCES 126

[365] D. J. Armitt, M. I. Bruce, M. Gaudio, N. N. Zaitseva, B. W. Skelton, A. H. White, B. Le Guennic, J. F. Halet, M. A. Fox, R. L. Roberts, F. Hartl, P. J. Low, Dalton Trans. 2008, 6763.

[366] M. Parthey, J. B. G. Gluyas, M. A. Fox, P. J. Low, M. Kaupp, Chem. Eur. J. 2014, DOI: 10.1039/c3cs60481k.

[367] M. Y. Choi, M. C. W. Chan, S. B. Zhang, K. K. Cheung, C. M. Che, K. Y. Wong, Organometallics 1999, 18, 2074.

[368] K. Eichkorn, F. Weigend, O. Treutler, R. Ahlrichs, Theor. Chem. Acc. 1997, 97, 119. [369] O. Lavastre, J. Plass, P. Bachmann, S. Guesmi, C. Moinet, P. H. Dixneuf,

Organometallics 1997, 16, 184. [370] N. S. Hush, A. Edgar, J. K. Beattie, Chem. Phys. Lett. 1980, 69, 128. [371] A. Stebler, J. H. Ammeter, U. Fürholz, A. Ludi, Inorg. Chem. 1984, 23, 2764. [372] R. Ahlrichs, M. Bär, M. Häser, H. Horn, C. Köhmel, Chem. Phys. Lett. 1989, 162,

165. [373] F. Weigend, F. Furche, R. Ahlrichs, J. Chem. Phys. 2003, 119, 12753. [374] A. Schäfer, C. Huber, R. Ahlrichs, J. Chem. Phys. 1994, 100, 5829. [375] V. G. Malkin, O. L. Malkina, R. Reviakine, A. V. Arbuznikov, M. Kaupp, B.

Schimmelpfennig, I. Malkin, T. Helgaker, K. Ruud, MAG-ReSpect, Vers. 1.2, 2003. [376] C. Remenyi, R. Reviakine, A. V. Arbuznikov, J. Vaara, M. Kaupp, J. Phys. Chem. A

2004, 108, 5026. [377] R. F. Ferrante, J. L. Wilkerson, W. R. M. Graham, W. Weltner, J. Chem. Phys. 1977,

67, 5904.

REFERENCES 127

Copyright

The cover figure, the introduction chapter, Chapter 2, previous computational work parts of chapter introductions, and parts of Chapter 6 are adapted from M. Parthey, M. Kaupp, Chem. Soc. Rev. 2014, DOI: 10.1039/c3cs60481k, by permission of The Royal Society of Chemistry.

Section 3.1 is adapted with permission from M. Parthey, K. B. Vincent, M. Renz, P. A. Schauer, D. S. Yufit, J. A. K. Howard, M. Kaupp, P. J. Low, Inorg. Chem. 2014, 53, 1544-1554. Copyright 2014 American Chemical Society.

Section 3.2 is adapted with permission from K. B. Vincent, Q. Zeng, M. Parthey, D. S. Yufit, J. A. K. Howard, F. Hartl, M. Kaupp, P. J. Low, Organometallics 2013, 32, 6022-6032. Copyright 2014 American Chemical Society.

Section 4.1 is adapted with permission from M. Parthey, J. B. G. Gluyas, P. A. Schauer, D. S. Yufit, J. A. K. Howard, M. Kaupp, P. J. Low, Chem. Eur. J. 2013, 19, 9780-9784. Copyright 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Section 4.2 is adapted with permission from M. Parthey, J. B. G. Gluyas, M. A. Fox, P. J. Low, M. Kaupp, Chem. Eur. J. 2014, DOI: 10.1002/chem.201304947. Copyright 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

APPENDIX 128

APPENDIX

Table A1. Calculated ground-state properties for complexes Pt1´ – Pt4´ and [Pt1´ – Pt4´]2+. Total dipole moment µa, spin expectation value 〈S 2 〉 (theoretical value for singlet systems would be 0.0), distance between nitrogen and the neighboring carbon atom of the bridge d(N-C6), average distance between the two terminal phenyl rings and nitrogen d(N-C9/16), C≡C bond length d(C≡C) and platinum carbon distances d(Pt-C1), the average of dihedral angles ∠(P1-Pt-C3-C4) and ∠(P2-Pt-C3-C8) Ω, and the average ∠(C6-N-C9-C10) and ∠(C6-N-C16-C17) Θ. Experimental data from the crystallographically determined structures of Pt1 and Pt2[237] are included for comparison.

Pt1´ Pt1 Pt2´ Pt2[237] Pt3´ Pt4´ [Pt1´]2+ [Pt2´]2+ [Pt3´]2+ [Pt4´]2+

µa [D] 0.1 0.1 0.0 0.1 97.0 0.2 0.1 0.1

〈S 2 〉 0.0 0.0 0.0 0.0 1.1 1.1 1.1 1.1

d(N-C6) [Å] 1.411 1.423(7) 1.407 1.399(6) 1.413 1.408 1.380 1.395 1.377 1.392

1.411 1.406 1.413 1.408 1.379 1.395 1.377 1.392

d(N-C9/16) [Å]

1.418 1.423(7) 1.420 1.442(6) 1.418 1.420 1.421 1.410 1.422 1.412

1.419 1.409(8) 1.421 1.432(6) 1.418 1.420 1.421 1.410 1.422 1.412

d(C≡C) [Å] 1.227 1.194(8) 1.227 1.171(7) 1.225 1.225 1.233 1.230 1.232 1.229

1.227 1.227 1.225 1.225 1.233 1.230 1.232 1.229

d(Pt-C1) [Å] 2.028 1.999(6) 2.029 2.008(5) 2.029 2.029 2.010 2.016 2.011 2.015

2.028 2.028 2.029 2.029 2.009 2.016 2.011 2.016

Ω [°]a 6.4 6.2(1) 3.2 46.1(1) 11.5 24.8 6.5 12.6 -29.0 -29.3

-1.1 -8.3 -10.7 -23.4 -11.0 -12.3 28.9 30.6

Θ [°]a 44.6 50.2(3) 49.0 59.7(3) 44.4 -48.0 45.8 -42.0 -46.7 -42.7

-44.9 -49.4 -44.4 48.2 -45.9 41.9 46.7 42.6

a Magnitude of torsion is given by the absolute value, the opposite sign of the dihedral angles is due to the Ci-symmetric input structure of the optimization.

APPENDIX 129

Table A2. Calculated ground-state properties for monocationic complexes [Pt1´ – Pt4´]+. Total dipole moment µa, spin expectation value 〈S 2 〉 (theoretical value for doublet systems would be 0.75), distance between nitrogen and the neighboring carbon atom of the bridge d(N-C6), average distance between the two terminal phenyl rings and nitrogen d(N-CPh), C≡C bond length d(C≡C), the average of dihedral angles ∠(P1-Pt-C3-C4) and ∠(P2-Pt-C3-C8) Ω, and the average ∠(C6-N-C9-C10) and ∠(C6-N-C16-C17) Θ.

[Pt1´]+ [Pt2´]+ [Pt3´]+ [Pt4´]+

µa [D] 25.1 30.9 25.3 31.0

〈S 2 〉 0.78 0.78 0.78 0.78

d(N-C6) [Å] 1.403 1.401 1.405 1.404

1.373 1.385 1.371 1.383

d(N- C9/16) [Å] 1.422 1.422 1.421 1.422

1.426 1.416 1.427 1.417

d(C≡C) [Å] 1.228 1.227 1.226 1.225

1.238 1.234 1.237 1.233

d(Pt-C1) [Å] 2.017 2.021 2.017 2.021

1.998 2.009 2.002 2.011

Ω [°]a 5.7 4.6 12.7 32.5

-11.4 -7.0 -15.3 -29.5

Θ [°]a 48.8 51.7 47.8 50.7

-50.1 -45.1 -50.7 -45.3

a Magnitude of torsion is given by the absolute value, the opposite sign of the dihedral angles is due to the Ci-symmetric input structure of the optimization.

Figure A1. A plot of a molecule of [Pt1] showing the atom labeling scheme, with thermal ellipsoids plotted at 50 %. Hydrogen atoms have been omitted and only one component of a disordered ethyl chain is shown for clarity. Selected bond lengths (Å) and angles (°): Pt1-P1 2.2667(19); Pt1-C1 1.999(6); C1-C2 1.194(8); C2-C3 1.453(8); N1-C6 1.423(7); N1-C9 1.423(7); N1-C16 1.409(8); Pt1-C1-C2 175.9(6); C1-C2-C3 178.5(7); C1-Pt1-P1 87.82(18); C1’-Pt1-P1 92.17(18). Crystal structure obtained by Dmitry Yufit and Judith Howard.

APPENDIX 130

Table A3. Computed energy difference to the most stable rotamer (Ω =180 °), ∆E, from TURBOMOLE 6.4, excitation energies E1 (β-HOMO–1 => β-SOMO transition) and E2 (β-HOMO–3 => β-SOMO transition) and corresponding transition dipole moment µ1 and µ2 from Gaussian09 and TURBOMOLE 6.4 calculations, respectively, for [Ru1´]+ as a function of P-Ru-Ru-P dihedral angle Ω.

Gaussian09 TURBOMOLE 6.4 Ω

[°]

∆E

[kJ/mol]

E1

[cm–1]

µ1

[D]

E2

[cm–1]

µ2

[D]

E1

[cm–1]

µ1

[D]

E2

[cm–1]

µ2

[D] 0 12.6 11572 9.8 14307 0.8 12089 9.2 14736 0.8 10 5.9 11614 9.7 14285 1.2 12102 9.0 14657 1.2 20 2.2 11660 9.6 14154 1.2 12192 8.9 14568 1.2 30 1.0 11729 9.5 14055 1.4 12242 8.8 14438 1.3 40 3.3 11743 9.4 13987 1.6 12337 8.7 14467 1.5 50 6.4 11698 9.3 13974 2.2 12299 8.6 14459 2.0 60 9.8 11587 9.2 13991 2.7 12204 8.5 14497 2.5 70 10.6 11465 9.0 14089 3.1 12072 8.4 14547 3.0 80 11.7 11417 8.7 14137 3.6 12131 8.0 14681 3.4 90 12.2 11616 8.4 14463 3.6 12227 7.8 14882 3.5 100 9.8 11427 8.5 13928 3.8 12136 7.9 14470 3.6 110 6.9 11516 8.7 13924 3.5 12166 8.0 14446 3.2 120 7.3 11547 8.9 14024 3.1 12200 8.2 14564 2.8 130 9.6 11538 9.3 14095 2.4 12231 8.6 14692 2.2 140 6.3 11582 9.5 14120 1.7 12139 8.9 14585 1.6 150 4.1 11601 9.6 14082 1.2 12157 8.9 14551 1.1 160 2.4 one negative-energy excitation 12228 8.9 14554 0.7 170 0.9 11698 9.6 14057 0.5 12301 8.9 14558 0.4 180 0.0 11708 9.5 13961 0.0 12344 8.8 14543 0.0

APPENDIX 131

Table A4. Calculated ground-state properties for different minima of obtained from full structure optimizations of [Os1]+, [Mo1]+, and [Re1]+. Energy difference to the respective most stable conformer ∆E (min-[Os1]+, min-[Mo1]+, and min-[Re1]+), total dipole moment µa, spin expectation value 〈S 2 〉 (theoretical value for doublet systems would be 0.75), and calculated C≡C stretching frequencies ν(C≡C). [Os1]+ [Mo1]+ [Re1]+ min perp trans min trans trans cis ∆E [kJ/mol] 0.0 5.9 6.5 0.0 8.7 0.0 2.4 µa [D] 2.8 2.3 0.0 12.1 9.2 0.1 10.0 〈S 2 〉 0.76 0.76 0.76 0.81 0.80 0.78 0.78 ν(C≡C) 1877 1870 1873 1971 1930 1892 1889 [cm–1] 1913 1897

APPENDIX 132

Table A5. Calculated excited-state parameters: UV-vis-NIR transition energies Etrans, transition dipole moments µtrans, and contributing orbital transitions for the minima of [Os1]+ obtained from full structure optimizations at the BLYP35/COSMO(CH2Cl2) level.

TURBOMOLE 6.4

min-[Os1]+ perp-[Os1]+ trans-[Os1]+

# Etrans

[cm–1]

µtrans

[D]

contributions Etrans

[cm–1]

µtrans

[D]


[cm–1]

µtrans

[D]

contributions

1 2415

1.4 312 β -> 313 β 1484 1.9 312 β -> 313 β 3406 0.0 312 β -> 313 β

2 12652

8.4 311 β -> 313 β 12318 8.1 311 β -> 313 β 12875 8.8 311 β -> 313 β

3 14406

0.3 310 β -> 313 β 14029 0.3 310 β -> 313 β 15259 0.9 310 β -> 313 β

4 16077

2.2 309 β -> 313 β 15912 3.1 309 β -> 313 β 16436 0.0 309 β -> 313 β

5 18620

0.2 308 β -> 313 β,

307 β -> 313 β

17551 0.1 308 β -> 313 β,

307 β -> 313 β

19773 0.2 308 β -> 313 β

6 23254 0.4 307 β -> 313 β,

308 β -> 313 β

22970 0.1 307 β -> 313 β,

308 β -> 313 β

23730 0.0 307 β -> 313 β,

303 β -> 313 β

7 24071

0.4 305 β -> 313 β 23571 0.4 305 β -> 313 β 24226 0.6 312 β -> 332 β

Gaussian09


# Etrans

[cm–1]

µtrans

[D]


[cm–1]

µtrans

[D]


[cm–1]

µtrans

[D]

contributions

1 2371

1.6 312 β -> 313 β 1471 2.2 312 β -> 313 β 3373 0.0 312 β -> 313 β

2 11990

9.1 311 β -> 313 β 11633 8.8 311 β -> 313 β 12102 9.5 311 β -> 313 β

3 13788

0.4 310 β -> 313 β 13440 0.4 310 β -> 313 β 14537 1.2 310 β -> 313 β

4 15439

2.4 309 β -> 313 β 15349 3.4 309 β -> 313 β 15679 0.0 309 β -> 313 β

5 18183

0.3 308 β -> 313 β 17155 0.0 307 β -> 313 β,

308 β -> 313 β

19210 0.2 308 β -> 313 β

6 22841

0.4 307 β -> 313 β,

308 β -> 313B

22539 0.1 307 β -> 313 β 23352 0.0 307 β -> 313 β

7 23667

0.1 305 β -> 313 β 23114 0.3 305 β -> 313 β 24019 0.0 305 β -> 313 β

8 24226

0.5 mixed excitation 24002 0.4 312 β -> 332 β 24303 0.6 312 α -> 333 α

9 24908

0.2 312 β -> 314 β 24607 0.1 312 β -> 314 β 25242 0.1 312 α -> 314 α

10 25047

0.1 312 β -> 315 β 24741 0.2 312 β -> 315 β 25249 0.3 312 β -> 315 β

APPENDIX 133

Table A6. Orbital energies (Eorb) and composition from Mulliken population analysis for [Os1]+ ([Os] = Os(dppe)Cp*). Virtual orbitals are marked with a *.


Orbital EOrb contributions [%] EOrb contributions [%] EOrb contributions [%]

[eV] [Os] C≡C C≡C [Os] [eV] [Os] C≡C C≡C [Os] [eV] [Os] C≡C C≡C [Os] spin density / 24 25 25 26 / 25 25 26 25 / 24 25 25 24

314 β* -0.65 37 0 0 43 -0.65 0 0 0 83 -0.62 2 0 0 63

α* -0.67 46 0 0 38 -0.67 0 0 0 85 -0.64 25 0 0 40

313 β* -2.75 20 27 27 19 -2.80 18 27 27 21 -2.65 19 26 26 21

α -5.25 23 22 22 26 -5.23 23 25 24 18 -5.15 22 22 22 25

312 β -5.15 20 25 25 20 -5.08 23 24 24 20 -5.20 19 26 26 19

α -5.29 19 25 25 20 -5.31 23 22 22 24 -5.31 21 26 26 21

311 β -5.94 39 10 9 33 -5.93 37 9 8 34 -5.90 33 11 11 37

α -6.24 36 4 4 42 -6.25 67 5 2 11 -6.25 38 4 4 42

310 β -6.18 34 4 4 44 -6.19 40 3 4 39 -6.21 37 4 4 42

α -6.33 43 4 4 35 -6.29 10 1 6 68 -6.40 35 6 6 39

309 β -6.41 40 7 6 35 -6.46 33 8 8 38 -6.34 36 5 5 39

α -6.68 34 7 6 36 -6.75 35 7 7 35 -6.54 34 5 5 40

308 β -6.88 34 8 8 35 -6.78 36 8 9 34 -7.01 39 4 4 38

α -7.03 37 7 7 35 -6.93 34 8 8 37 -7.11 38 4 4 38

307 β -7.13 31 4 5 42 -7.17 44 4 4 30 -7.03 29 8 8 34

α -7.25 36 4 4 36 -7.28 40 3 4 32 -7.19 30 6 6 34

306 β -7.29 44 2 1 29 -7.27 29 2 3 45 -7.31 35 1 1 38

α -7.46 42 1 3 33 -7.43 34 3 2 43 -7.45 35 2 2 34

305 β -7.40 37 3 3 40 -7.42 38 3 3 40 -7.36 36 3 3 36

α -7.55 36 2 3 44 -7.56 42 3 2 39 -7.54 36 3 3 32

APPENDIX 134

Table A7. Calculated excited-state parameters: UV-vis-NIR transition energies Etrans, transition dipole moments µtrans, and contributing orbital transitions for the minima of [Mo1]+ obtained from full structure optimizations at the BLYP35/COSMO(CH2Cl2) level.

TURBOMOLE 6.4

min-[Mo1]+ trans-[Mo1]+

# Etrans

[cm–1]

µtrans

[D]


[cm–1]

µtrans

[D]

contributions

1 6073 2.4 284 β -> 285 β,

281 β -> 285 β

7301 0.7 283 β -> 285 β,

281 β -> 285 β

2 10746 5.9 283 β -> 285 β,

280 β -> 285 β

7731 9.4 284 β -> 285 β,

280 β -> 285 β

3 12710 0.9 281 β -> 285 β,

282 β -> 285 β

14047 1.7 280 β -> 285 β,

279 β -> 285 β

4 14701 2.4 280 β -> 285 β,

279 β -> 285 β

14589 0.2 281 β -> 285 β,

283 β -> 285 β

5 15133 0.9 284 β -> 286 β,

285 α -> 286 α

15118 0.8 283 β -> 286 β,

284 α -> 286 α

6 16276 0.4 285 α -> 291 α,

285 α -> 290 α

15526 0.2 284 β -> 286 β,

285 α -> 286 α

7 16582 0.2 285 α -> 292 α

16954 0.8 285 α -> 289 α

Gaussian09

min-[Mo1]+ perp-[Mo1]+

# Etrans

[cm–1]

µtrans

[D]


[cm–1]

µtrans

[D]

contributions

1 6466 2.4 284 β -> 285 β 7545 1.0 281 β ->285 β,

283 β ->285 β

2 10988 5.9 283 β -> 285 β 7959 9.6 284 β ->285 β

3 12702 1.3 281 β -> 285 β 13927 1.9 279 β ->285 β,

280 β ->285 β

4 14729 2.8 280 β -> 285 β 14758 0.2 281 β ->285 β

5 15245 0.9 285 α -> 286 α 15182 0.8 283 β ->286 β

6 16382 0.4 285 α -> 290 α 15540 0.2 284 β ->286 β

7 16682 0.2 mixed excitation 17013 0.8 285 α ->289 α

8 18078 1.7 mixed excitation 17547 0.3 285 α ->286 α,

284 β ->286 β

9 18578 1.8 282 β ->285 β 18181 0.7 282 β ->285 β

10 18973 2.4 282 β ->285 β

18844 0.5 282 β -> 285 β

APPENDIX 135

Table A8. Orbital energies (Eorb) and composition from Mulliken population analysis for [Mo1]+ ([Mo] = Mo(η-C7H7)(dppe)). Virtual orbitals are marked with a *.

min-[Mo]+ trans-[Mo]+

Orbital EOrb contributions [%] EOrb contributions [%]

[eV] [Mo] C≡C C≡C [Mo] [eV] [Mo] C≡C C≡C [Mo] spin density / 73 7 10 5 / 66 8 11 11

286 β* -1.24 67 10 6 3 -1.36 55 12 9 11

α* -1.39 75 9 5 2 -1.45 65 10 7 8

285 β* -2.75 58 14 10 2 -2.79 53 15 12 8

α -5.00 5 12 16 56 -5.19 4 12 18 56

284 β -4.98 8 11 14 56 -5.11 14 6 9 59

α -5.36 4 13 20 56 -5.37 10 15 19 50

283 β -5.50 8 10 15 58 -5.34 17 15 18 45

α -6.05 2 4 6 79 -6.17 2 3 4 83

282 β -6.03 11 6 6 68 -6.16 2 2 5 84

α -6.36 30 15 8 36 -6.45 41 12 6 33

281 β -6.24 32 9 5 47 -6.29 45 9 4 34

α -6.64 25 20 14 31 -6.61 26 18 11 32

280 β -6.67 55 13 9 15 -6.64 48 15 12 11

α -7.13 89 3 0 0 -7.12 91 2 0 0

279 β -7.00 44 14 17 10 -7.02 54 12 15 5

α -7.41 7 0 3 77 -7.46 6 2 5 75

APPENDIX 136

Table A9. Computed energy difference to the most stable rotamer (Ω =180 °), ∆E, two lowest excitation energies E1 and E2 and corresponding transition dipole moment µ1 and µ2 from TURBOMOLE 6.4 calculations, respectively, for [Mo1]+ as a function of P-Mo-Mo-P dihedral angle Ω.

Ω

[°]

∆E

[kJ/mol]

E1

[cm–1]

µ1

[D]

E2

[cm–1]

µ2

[D] 0 18.3 7349 1.1 7938 9.1 10 16.6 7033 4.2 8170 8.1 20 13.7 6557 4.9 8498 7.6 30 7.6 6291 4.4 9115 7.2 40 3.0 6089 3.7 9682 6.8 50 0.6 6035 2.9 10285 6.3 60 0.0 6056 2.3 10833 5.9 70 1.0 6056 1.5 11127 5.3 80 2.3 6060 0.7 11327 5.1 90 3.5 6123 0.1 11318 4.9 100 4.6 6033 0.6 11050 5.2 110 5.0 6109 1.3 10781 5.5 120 5.6 6161 2.1 10239 5.9 130 6.3 6277 3.0 9773 6.4 140 6.5 6411 3.6 9363 6.9 150 7.3 6505 4.6 8694 7.3 160 10.7 6709 4.6 8415 7.9 170 9.6 7034 4.6 7864 8.3 180 9.1 7260 1.6 7655 9.4

APPENDIX 137

Table A10. Calculated excited-state parameters: UV-vis-NIR transition energies Etrans, transition dipole moments µtrans, and contributing orbital transitions for the minima of [Re1]+ obtained from full structure optimizations at the BLYP35/COSMO(CH2Cl2) level.

TURBOMOLE 6.4

trans-[Re1]+ cis-[Re1]+

# Etrans

[cm–1]

µtrans

[D]


[cm–1]

µtrans

[D]

contributions

1 10238 0.8 253 β -> 255 β

10269 0.2 253 β -> 255 β 96.7

2 10346 10.0 254 β -> 255 β,

254 β -> 256 β

10287 10.0 254 β -> 255 β,

253 β -> 256 β

3 14237 3.0 254 α -> 256 α,

253 β -> 256 β

14232 3.0 254 α -> 256 α,

253 β -> 256 β

4 16722 0.0 254 α -> 263 α,

253 β -> 263 β

16686 0.1 254 α -> 263 α,

253 β -> 263 β

5 18423 1.0 255 α -> 256 α,

251 β -> 255 β

18464 1.0 255 α -> 256 α,

251 β -> 255

6 19731 0.0 252 β -> 255 β,

249 β -> 255 β

19817 0.3 252 β -> 255 β,

249 β -> 255 β

7 19894 0.4 251 β -> 255 β

19910 0.4 251 β -> 255 β

APPENDIX 138

Table A11. Orbital energies (Eorb) and composition from Mulliken population analysis for [Re1]+ ([Re] = ReCp(PPh3)(NO)). Virtual orbitals are marked with a *.

trans-[Re]+ cis-[Re]+

Orbital EOrb contributions [%] EOrb contributions [%]

[eV] [Re] C≡C C≡C [Re] [eV] [Re] C≡C C≡C [Re] spin density / 28 20 20 28 / 28 20 20 28

256 β* -1.65 32 15 15 32 -1.66 31 15 15 31

α* -1.59 32 15 15 32 -1.60 32 15 15 32

255

β* -3.27 24 23 23 24 -3.28 24 23 23 24

α -5.72 26 20 20 26 -5.73 26 20 20 26

254

β -6.17 34 10 10 34 -6.17 34 10 10 34

α -6.45 22 26 26 22 -6.46 22 26 26 22

253

β -6.42 22 26 26 22 -6.43 22 26 26 22

α -6.97 38 5 5 38 -6.98 38 6 6 38

252

β -7.08 37 5 5 37 -7.10 38 5 5 37

α -7.13 38 5 5 37 -7.15 39 4 4 37

251

β -7.11 39 2 2 39 -7.13 38 3 3 39

α -7.15 38 4 4 39 -7.17 38 4 4 39

250

β -7.68 39 2 2 39 -7.68 39 2 2 39

α -7.76 43 2 2 43 -7.77 42 2 2 42

249

β -7.71 30 9 9 30 -7.71 34 8 8 34

α -7.79 42 2 2 41 -7.79 43 2 2 43

248

β -7.81 44 0 0 42 -7.82 44 0 0 44

α -7.84 43 0 0 43 -7.86 43 0 0 44

APPENDIX 139

Table A12. Computed energy difference to the most stable rotamer (Ω =180 °), ∆E, three lowest excitation energies E1, E2, and E3 and corresponding transition dipole moment µ1, µ2, and µ3 from TURBOMOLE 6.4 calculations, respectively, for [Re1]+ as a function of P-Re-Re-P dihedral angle Ω.

Ω

[°]

∆E

[kJ/mol]

E1

[cm–1]

µ1

[D]

E2

[cm–1]

µ2

[D]

E3

[cm–1]

µ3

[D] 0 2.5 10267 0.2 10288 10.0 14230 3.0 10 2.5 9298 6.8 11160 7.3 14249 2.9 20 3.7 8244 6.7 11884 7.5 14293 2.5 30 5.2 7074 6.6 12394 7.6 14317 1.7 40 8.2 5775 6.5 12707 7.7 14358 0.7 50 11.7 4418 6.5 12868 7.6 14384 0.5 60 15.7 3072 6.8 12882 7.3 14181 1.4 70 19.9 3484 4.2 13185 5.9 13852 2.7 80 22.0 4077 1.7 13214 5.1 14380 2.9 90 23.1 4300 0.3 13236 5.0 14652 2.6 100 22.1 4220 1.5 13256 5.0 14436 3.0 110 19.9 3622 3.9 13200 5.9 13913 2.9 120 15.3 3115 6.7 12891 7.4 14245 1.6 130 10.2 4404 6.6 12851 7.7 14401 0.6 140 6.5 5765 6.5 12671 7.8 14340 0.7 150 3.7 7033 6.6 12310 7.7 14288 1.7 160 1.3 8205 6.7 11807 7.5 14244 2.4 170 0.0 9315 6.7 11132 7.4 14234 2.9 180 0.1 10250 1.1 10343 9.9 14240 3.0

APPENDIX 140

Figure A2. Cut through the PES of [Ru2-Me]+ for Ω = 180° and spin-density isosurface plots (± 0.002 a.u.) for all structures of the relaxed scan of Θeff (BLYP35/def2-SVP/COSMO(CH2Cl2) level).

APPENDIX 141

Figure A3. Properties (BLYP35/def2-SVP/COSMO(DCM) level) as function of conformational phase

space of [Ru2-Me]+. a) TDDFT transition dipole moment µtrans of the third excitation at around

10000 cm–1 (9500-11500 cm–1) (top), and b) TDDFT transition dipole moment µtrans of the fourth

excitation (bottom) of [Ru2-Me]+. For two structures (Ω = 50°, Θeff = 91.7° and Ω = 100°,

Θeff = 101.6°) on the PES the excited-state is unstable and thus the transition dipole moment is set to

zero. BLYP35/def2-SVP/COSMO(CH2Cl2) level.

APPENDIX 142

Figure A4. Potential energy surface of [Ru3-Me]+ (BLYP35/def2-SVP/COSMO(CH2Cl2) level) using different color scales.

APPENDIX 143

Figure A5. Surface of the difference, ∆SD, of the Mulliken spin density contributions of the two

Cl(dmpe)2Ru-C≡C units for [Ru3-Me]+ (near 0% for delocalized structures and close to 100% for fully

localized charge distributions). BLYP35/def2-SVP/COSMO(CH2Cl2) level.

β-SOMO β-HOMO

β-HOMO–1 β-HOMO–2

Figure A6. Isosurface plots (± 0.03 a.u.) of β-SOMO (top left), β-HOMO (top right), β-HOMO–1

(bottom, left) and β-HOMO–2 (bottom, right) of sb-[ Ru3-Me]+ (BLYP35/def2-

SVP/COSMO(CH2Cl2) level).

APPENDIX 144

Figure A7. Conformational potential-energy surface of [CTI]5+ (BLYP35/def2-SVP/COSMO(CH2Cl2) level).

Figure A8. Surface of the difference, ∆SD, of the Mulliken spin density contributions of the two

(NH3)5Ru units in [CTI]5+ (0% for delocalized structures, close to 100% for localized charge

distributions (BLYP35/def2-SVP/COSMO(CH2Cl2) level).

APPENDIX 145

Figure A9. Computed Boltzmann-weighted TDDFT stick spectra of [CTI]5+ (BLYP35/def2-

SVP/COSMO(CH2Cl2) level).

APPENDIX 146

Table A13. Isotropic hyperfine coupling constants Aiso, spin expectation value 〈S 2 〉, and difference between experimental and calculated isotropic hyperfine coupling constant values ∆Aiso for different functionals implemented in TURBOMOLE.

Complex BP86 BVWN BLYP B3LYP BHH LYP

t-LMF-VWN

t-ρ-LMF-VWN

Exp.

ScO Aiso [MHz] 1919 1979 1999 1981 1938 1955 1940 1948

〈S 2 〉 0.752 0.751 0.751 0.751 0.751 0.753 0.753 0.750

∆Aiso [MHz] -29 32 52 33 -9 7 -8

TiN Aiso [MHz] -558 -572 -580 -576 -562 -565 -562 -559

〈S 2 〉 0.755 0.753 0.752 0.753 0.756 0.757 0.757 0.750

∆Aiso [MHz] 1 -13 -21 -17 -3 -6 -3

TiO Aiso [MHz] -246 -256 -256 -250 -238 -253 -257 -241

〈S 2 〉 2.013 2.010 2.008 2.012 2.017 2.023 2.023 2.000

∆Aiso [MHz] -5 -15 -14 -9 3 -12 -15

VN Aiso [MHz] 1352 1407 1408 1352 1144 1343 1385 1314

〈S 2 〉 2.046 2.036 2.034 2.085 2.409 2.134 2.111 2.000

∆Aiso [MHz] 38 93 94 38 -170 29 70

VO Aiso [MHz] 804 843 841 818 779 835 872 780

〈S 2 〉 3.796 3.785 3.782 3.800 3.824 3.835 3.832 3.750

∆Aiso [MHz] 24 63 62 38 -1 55 93

MnO Aiso [MHz] 528 548 550 528 535 531 597 481

〈S 2 〉 8.791 8.783 8.782 8.829 9.036 8.842 8.838 8.750

∆Aiso [MHz] 47 67 69 47 54 50 116

MnF2 Aiso [MHz] 299 318 325 251 157 233 318 105

〈S 2 〉 8.760 8.759 8.758 8.760 8.760 8.761 8.762 8.750

∆Aiso [MHz] 194 213 219 146 52 128 212

MnF Aiso [MHz] 486 513 514 479 429 474 537 443

〈S 2 〉 12.003 12.002 12.002 12.003 12.003 12.003 12.003 12.000

∆Aiso [MHz] 44 71 72 36 -13 31 94

MnH Aiso [MHz] 386 415 397 343 284 362 425 280

〈S 2 〉 12.004 12.004 12.003 12.003 12.002 12.003 12.004 12.000

∆Aiso [MHz] 106 135 116 63 4 82 145

APPENDIX 147


t-LMF-VWN

t-ρ-LMF-VWN

Exp.

TiF3 Aiso [MHz] -218 -216 -222 -196 -161 -200 -213 -185

〈S 2 〉 0.753 0.752 0.752 0.752 0.752 0.753 0.754 0.750

∆Aiso [MHz] -33 -31 -37 -11 24 -15 -28

MnO3 Aiso [MHz] 1979 2020 2031 1731 1275 1774 1902 1617

〈S 2 〉 0.769 0.765 0.764 0.871 1.857 0.871 0.835 0.750

∆Aiso [MHz] 362 402 414 114 -342 157 285

CuO Aiso [MHz] -642 -571 -609 -715 -666 -806 -811 -490

〈S 2 〉 0.761 0.759 0.761 0.766 0.766 0.767 0.768 0.750

∆Aiso [MHz] -152 -81 -119 -224 -176 -315 -321

[Mn(CN)4]2- Aiso [MHz] -95 -78 -82 -110 -147 -121 -48 -198

〈S 2 〉 8.765 8.762 8.761 8.762 8.762 8.765 8.766 8.750

∆Aiso [MHz] 103 120 116 88 51 77 150

[Cr(CO)4]+ Aiso [MHz] 24 20 20 25 33 25 13 41

〈S 2 〉 8.762 8.759 8.757 8.760 8.761 8.764 8.765 8.750

∆Aiso [MHz] -18 -21 -21 -16 -8 -16 -29

[Cu(CO)3] Aiso [MHz] -8 -13 -20 4 39 29 22 71

〈S 2 〉 0.752 0.752 0.751 0.753 0.757 0.754 0.754 0.750

∆Aiso [MHz] -79 -83 -90 -66 -31 -42 -48

[Ni(CO)3H] Aiso [MHz] 14 12 12 29 53 35 17 14

〈S 2 〉 0.759 0.759 0.759 0.796 0.992 0.776 0.781 0.750

∆Aiso [MHz] 0 -2 -2 15 39 21 3

Co(CO)4 Aiso [MHz] -4 10 7 -48 -182 -75 34 -57

〈S 2 〉 0.762 0.761 0.761 0.787 1.012 0.775 0.773 0.750

∆Aiso [MHz] 53 67 64 9 -124 -18 91

Mn(CN)4N- Aiso [MHz] -169 -143 -152 -240 -480 -268 -140 -271

〈S 2 〉 0.775 0.771 0.772 0.873 1.723 0.871 0.840 0.750

∆Aiso [MHz] 101 127 119 31 -210 3 130

APPENDIX 148


t-LMF-VWN

t-ρ-LMF-VWN

Exp.

Mn(CN)5NO2- Aiso [MHz] -147 -116 -123 -212 -290 -254 -91 -218

〈S 2 〉 0.866 0.854 0.860 1.434 2.093 1.363 1.269 0.750

∆Aiso [MHz] 72 102 95 6 -72 -36 127

Mn(CO)5 Aiso [MHz] 5 16 12 4 3 0 50 -4

〈S 2 〉 0.754 0.753 0.753 0.757 0.766 0.756 0.755 0.750

∆Aiso [MHz] 9 20 16 8 7 4 54

Fe(CO)5+ Aiso [MHz] 0 3 2 -2 -8 -5 7 -3

〈S 2 〉 0.757 0.756 0.757 0.762 0.768 0.761 0.760 0.750

∆Aiso [MHz] 3 5 5 0 -5 -2 10

APPENDIX 149

Table A14. Isotropic hyperfine coupling constants Aiso, spin expectation value 〈S 2 〉, and difference between experimental and calculated isotropic hyperfine coupling constant values ∆Aiso for different functionals implemented in Gaussian.

Complex CAM-B3LYP

LC-ωPBE

ωB97XD

LC-BLYP

LC-BP86

Exp.

ScO Aiso [MHz] 2100 1931 2022 2205 2115 1948

〈S 2 〉 0.751 0.752 0.751 0.751 0.752 0.750

∆Aiso [MHz] 152 -16 75 257 168

TiN Aiso [MHz] -610 -568 -592 -644 -618 -559

〈S 2 〉 0.752 0.755 0.752 0.751 0.754 0.750

∆Aiso [MHz] -51 -10 -33 -86 -59

TiO Aiso [MHz] -264 -256 -259 -280 -271 -241

〈S 2 〉 2.013 2.016 2.014 2.012 2.016 2.000

∆Aiso [MHz] -22 -15 -17 -38 -29

VN Aiso [MHz] 1369 920 1355 1424 1363 1314

〈S 2 〉 2.127 2.144 2.105 2.176 2.198 2.000

∆Aiso [MHz] 55 -394 41 110 48

VO Aiso [MHz] 858 863 -222 915 883 780

〈S 2 〉 3.804 3.807 3.781 3.797 3.807 3.750

∆Aiso [MHz] 78 83 -1002 135 104

MnO Aiso [MHz] 535 541 523 533 517 481

〈S 2 〉 8.833 8.806 8.832 8.802 8.809 8.750

∆Aiso [MHz] 54 60 42 52 36

MnF2 Aiso [MHz] 230 245 216 228 207 105

〈S 2 〉 8.759 8.758 8.758 8.756 8.757 8.750

∆Aiso [MHz] 125 140 111 123 102

MnF Aiso [MHz] 490 498 471 516 489 443

〈S 2 〉 12.002 12.002 12.002 12.001 12.001 12.000

∆Aiso [MHz] 47 55 29 73 47

MnH Aiso [MHz] 341 381 470 360 356 280

〈S 2 〉 12.002 12.002 12.006 12.001 12.001 12.000

∆Aiso [MHz] 61 101 189 80 76

APPENDIX 150

Complex CAM-B3LYP

LC-ωPBE

ωB97XD

LC-BLYP

LC-BP86

Exp.

TiF3 Aiso [MHz] -183 -179 -186 -174 -172 -185

〈S 2 〉 0.752 0.752 0.752 0.752 0.752 0.750

∆Aiso [MHz] 2 6 0 12 13

MnO3 Aiso [MHz] 1634 1738 1665 1637 1600 1617

〈S 2 〉 1.007 0.900 0.927 1.052 1.055 0.750

∆Aiso [MHz] 17 121 48 20 -17

CuO Aiso [MHz] -747 -535 -728 -561 -527 -490

〈S 2 〉 0.766 0.760 0.765 0.760 0.759 0.750

∆Aiso [MHz] -257 -45 -238 -70 -37

[Mn(CN)4]2- Aiso [MHz] -114 -87 -127 -108 -122 -198

〈S 2 〉 8.760 8.762 8.759 8.758 8.761 8.750

∆Aiso [MHz] 84 111 71 90 76

[Cr(CO)4]+ Aiso [MHz] 25 21 26 23 27 41

〈S 2 〉 8.760 8.763 8.758 8.758 8.762 8.750

∆Aiso [MHz] -16 -20 -15 -18 -14

[Cu(CO)3] Aiso [MHz] 22 48 17 38 52 71

〈S 2 〉 0.756 0.762 0.755 0.760 0.761 0.750

∆Aiso [MHz] -49 -22 -53 -33 -19

[Ni(CO)3H] Aiso [MHz] -2 -16 12 -5 -6 14

〈S 2 〉 0.758 0.757 0.759 0.757 0.757 0.750

∆Aiso [MHz] -16 -30 -2 -19 -20

Co(CO)4 Aiso [MHz] -63 -16 -117 -43 -55 -57

〈S 2 〉 0.802 0.778 0.805 0.783 0.780 0.750

∆Aiso [MHz] -6 42 -60 14 2

Mn(CN)4N- Aiso [MHz] -284 -204 -299 -289 -307 -271

〈S 2 〉 1.031 0.969 0.984 1.136 1.112 0.750

∆Aiso [MHz] -13 67 -28 -18 -36

APPENDIX 151

Complex CAM-B3LYP

LC-ωPBE

ωB97XD

LC-BLYP

LC-BP86

Exp.

Mn(CN)5NO2- Aiso [MHz] -67 -51 -56 -73 -81 -218

〈S 2 〉 0.884 0.841 0.962 0.804 0.808 0.750

∆Aiso [MHz] 151 168 163 145 137

Mn(CO)5 Aiso [MHz] 3 26 -4 15 7 -4

〈S 2 〉 0.758 0.760 0.758 0.760 0.761 0.750

∆Aiso [MHz] 7 30 0 19 11

Fe(CO)5+ Aiso [MHz] -4 1 -8 -2 -4 -3

〈S 2 〉 0.761 0.757 0.762 0.757 0.757 0.750

∆Aiso [MHz] -1 3 -5 1 -1

Investigations of mixed-valence and open-shell transition ... · Investigations of Mixed-Valence...

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