University of Groningen

Tuning energy transfer between chromophores. Switchable molecular photonic systemsHurenkamp, Johannes Henricus

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Chapter 6 Application of Density Functional Theory to the Electronic Structure of Perylene Bisimides, Coumarins and Dithienylcyclopentenes In recent years molecular modelling and theoretical calculations have become an invaluable tool in helping to understand and explain experimental observations. In this chapter calculations, primarily using density functional theory, will be applied to find an additional theoretical basis and explanation for several of the observations (vide infra) made in Chapters 2 and 4 .

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6.1 Introduction to computational chemistry1 Computational chemistry uses theoretical models as a basis for calculating the physical properties of molecules. These properties can range from energy optimized structures to properties including vibrational structure (IR), absorption spectra (UV/Vis), charges, dipoles and reaction pathways. The validity of the results depends on a multitude of factors and is mostly dependent on the assumptions and simplifications that are used. The results should always be validated (when possible i.e. when the molecule exists) by comparison with experimental data. The limiting factor in obtaining results that are in agreement with experimental results lies, in the end, in the calculating power available, which translates as computational time, the more accurate the method, the more time that is required to complete the calculation. For small molecules, consisting of only a few atoms, the most accurate methods can be used. However, for larger molecules less accurate methods are needed to be able to enable completion of calculations within an acceptable time period. In the following sections several common methods and their (dis)advantages will be discussed.

6.2 Basic methods These methods use a combination of Newtonian mechanics and values obtained from experimental data to perform calculations, which are fast, but provide results that often deviate significantly from experimental data.

6.3 Molecular mechanics (MM) This method employs Newtonian mechanics to calculate molecular properties and is usually used to find the energy minimized structure of a molecule. Atoms within the molecule are treated as single particles of a defined radius (typically the van der Waals radius is used). These particles are connected by springs which have an equilibrium distance that was obtained from either experiment or more exhaustive calculations. For a certain conformation the potential energy of the system is calculated by separating the total energy in to covalent and non-covalent contributions (Eq. 6.1).

alentnonalent EEE covcov += Eq. 6.1

These contributions are then separated further into smaller individual interactions (Eq. 6.2 and Eq. 6.3). From the covalent interactions the bond and angle components can be treated as harmonic oscillators of a certain equilibrium value. For the dihedral component this is

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more difficult, since several minima exist, and other methods (i.e. functionals) should be used.

dihedralanglebondalent EEEE ++=cov Eq. 6.2

The noncovalent components are divided into a van der Waals and an electrostatic component. The van der Waals term takes into account repulsion and van der Waals attraction forces. These forces are usually treated in the context of a 6-12 Lennard-Jones potential,2 which means that attractive forces decrease with distance by r-6 and repulsive forces by r-12, with r being the inter atomic distance. Electrostatic interactions are more complicated and have considerable effect over longer ranges, which cannot be ignored and can be important in large molecules (e.g., proteins). The common method of modelling the electrostatic interactions is to use the Coulomb potential as the basis functional, which has distance dependence of r-1.

ticelectrostasvanderWaalalentnon EEE +=cov Eq. 6.3

All these parameters and terms, combined with constants and values from experiments and theoretical calculations are known as the force field. Molecular mechanics is used for fast structure optimizations (also for quick optimization prior to using more advanced methods) and many of the fundamentals behind this method are applied in molecular dynamics simulations. However, the assumptions made and the fact that it is based on Newtonian mechanics make it unsuitable for use for prediction of electronic properties.

6.3.1 Semi-empirical methods

The semi-empirical methods are built on Hartree-Fock theory (vide infra), however several approximations, constants and parameters obtained from empirical data are employed. These changes overall lead to considerable reduction in calculation time compared to ab initio based methods, which permits it to be used for large molecules. However, if empirical parameters are used which were obtained from molecules which are not similar enough to the target molecule, then the results obtained can be unreliable. Well known methods used for semi-empirical calculations are AM13 and PM34.

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6.4 Ab initio methods

6.4.1 Molecular orbital theory

Ab initio (“from the beginning”) methods are built on molecular orbital theory. These methods use quantum mechanics and quantum field theory to solve the time-independent molecular Schrödinger equation that is coupled with the molecular Hamiltonian (Eq. 6.4).5

Ψ=Ψ EH Eq. 6.4

Ψ is the wave function, which is a mathematical function used in quantum mechanics to describe the wave characteristics of a particle or a system of particles (i.e. an atom or molecule). Its solution has no direct physical meaning, however, Ψ2 gives the probability of finding a particle at a certain position at a certain time. The operator H, that gives the systems energy E, is called the Hamiltonian operator. The common form of this operator has five contributions to the total energy of the system (Eq. 6.5).

∑∑∑∑∑∑<<

++−∇−∇−=lk kl

lk

ji iji k ik

kk

k ki

i e rZZe

re

rZe

mmH

2222

22

2

22hh

Eq. 6.5

Where i and j are electrons, k and l are nuclei, me is the mass of an electron, mk the mass of nucleus k , ∇2 the Laplacian operator, e the charge of an electron, Z the atomic number and rik the distance between particles i and k. From left to right these five terms represent: the kinetic energy of the electrons, the kinetic energy of the nuclei, the attraction between electrons and nucleus, the interelectronic repulsion and the internuclear repulsion. Methods that do not use empirical or semi-empirical parameters in their equations are considered to be ab initio (i.e. only based on theory or results from theoretical calculations). The solution from these calculations, however, is still not an exact one; it is an approximation built up from first principles (current quantum theory). The total energy of the system under investigation is determined by solving the time-independent Schrödinger equation, using the Born-Oppenheimer approximation and typically used without relativistic terms.6 The BO approximation states that since the movement of electrons is much faster than nuclear motion nuclear positions can be treated as fixed and therefore the electrons “feel” an average field generated by the nuclei. This allows for significant simplification of the calculations and the Hamiltonian in Eq. 6.5 can be broken up into a nuclear and an electronic part. The last term, internuclear repulsion becomes a constant for a certain geometry (and is often omitted from the Hamiltonian) and as the nuclei do not

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move the nuclear kinetic energy term becomes zero, thereby yielding the so-called electronic Hamiltonian (Helec) (Eq. 6.6).

∑∑∑∑

<

+−∇−=ji iji k ik

ki

i eelec r

erZe

mH

222

2

2h

Eq. 6.6

This leads to the following electronic Schrödinger equation, combining the electronic Hamiltonian with the electronic wave function, which depends on the electron coordinates:

elecelecelecelec EH Ψ=Ψ Eq. 6.7

Finally the total energy of the system (Etot) is the sum of the Eelec and the constant nuclear repulsion term:

nucrepelectot EEE += Eq. 6.8

It is now feasible to use operators to describe properties of the system, however it is still not possible to construct a wave function that describes the system. Nevertheless, an educated “guess” can be made. Combining this “guess” with the variational principle, which states that when you have a system, which has a lower boundary, it is possible to find the best solution for a problem be systematically varying the guess until a minimum value has been reached. This minimum is the best possible solution since the boundary condition, in this case originating in quantum mechanics, makes sure a solution which is lower in energy then the exact description of the system can never be reached. This combination gives the inequality shown in Eq. 6.9 and allows for the fitting of an approximate wave function.

02E

dr

drH≥

Φ

ΦΦ

∫∫

Eq. 6.9

If Φ is the exact wave function describing the system, the energy of that system will be E0 , every other approximate wave function will be higher in energy. By systematically trying “guessed” wave functions in principle a minimum integral can be obtained, which represents the lowest energy wave function for that level of theory.

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6.4.2 Hartree Fock Theory

In addition to the approximations described in the previous section Hartree-Fock (HF) theory makes further approximations:

• The wave function Φ of the system (used in combination with the variational principle) is assumed to be a linear combination of the atomic wave functions (ψ) (which is also known as the linear combination of atomic orbitals (LCAO)) of the atoms present in the system.

• The interelectronic repulsion term from Eq. 6.6 is approximated by letting each electron feel an average repulsion effect of all electrons, instead of each separately, thereby neglecting electron correlation.

This last approximation has the most significant effects. Neglecting the electron correlation can cause large deviations from experimental results. It also prevents the calculated energy from ever reaching the exact energy (E0) of a system. The minimum energy that is approached by increasing the basis set size (i.e. a set of functions used to describe the molecular orbitals, see also paragraph 6.5.4) of a Hartree-Fock calculation is called the Hartree-Fock limit (Figure 6.1).

Figure 6.1 Schematic representation of the Hartree-Fock limit.

Several methods have been developed to compensate for this weakness. These methods are collectively called post-Hartree-Fock methods (e.g., Møller-Plesset perturbation theory) and use different approaches to include electron correlation into the multi-electron wave function.

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6.5 Density functional theory (DFT)

6.5.1 Early work, the Thomas-Fermi model.

The basis for this theory was developed independently by Thomas and Fermi in 1927. Named the Thomas-Fermi model, it uses the electron density (ρ) to calculate the potential energy of a many-electron system, instead of using the wave functions as in Hartree-Fock theory. As discussed above, the electronic Hamiltonian depends only on the position and atomic number of the nuclei and the total number of electrons. This dependency on the total number of electrons makes it possible to describe the energy of a system by using the electron density: integrating the electron density over all space gives the total number of electrons (N) (Eq. 6.10)

( )drrN ∫= ρ Eq. 6.10

The location of the nuclei will correspond to local minima in the electron density, leaving only the atomic number (Z) as an unknown. The atomic number (or nuclear charge) can be derived from the properties of the maximum in electron density at the position (rk) of atom k by:

( ) ( )kk

rk

k rZrr

k

ρρ

20

−=∂

∂

=

Eq. 6.11

where rk is the radial distance from atom k and ρ is the spherically averaged density. Since

all parameters required can be described using the electron density it is possible to construct the Hamiltonian and solve the Schrödinger equation. The energy (E) can be separated into its potential (V) and kinetic (T) components. The most straightforward way is to treat the system in a classical manner; this gives the following components for the potential energy for the attraction between nuclei and the electron density:

( )[ ] ( )drr

rrZ

rVNulcei

k k

kne ρρ ∑ ∫ −

= Eq. 6.12

and for the potential energy for the self-repulsion of a classical charge distribution:

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( )[ ] ( ) ( )

2121

21

21 drdr

rrrrrVee ∫∫ −

=ρρ

ρ Eq. 6.13

For the kinetic energy component Thomas and Fermi used fermion statistical mechanics to derive a formula describing the kinetic energy of a uniform electron gas (an infinite number of electrons moving in an infinite volume filled with a uniformly distributed positive charge):

( )[ ] ( ) ( )drrrT ∫= 3/53/223

103 ρπρ

Eq. 6.14

The previous terms (Eq. 6.12 - Eq. 6.14) for potential and kinetic energy are a function of the density, and the density itself is a function of the three-dimensional spatial coordinates. This means they are a function of a function, which is called a functional. Therefore, the expressions above are referred to as “density functionals”, and hence the name: density functional theory. This theory, however, was incomplete and only a rough approximation of the true kinetic energy. Since it did not describe the exchange and correlation effects, which are predicted by Hartree-Fock theory, in this form the theory is unsuitable to describe real molecules. Independent work by Bloch (1929), Dirac (1951) and later by Slater (1951) added a correction functional for the exchange energy, leading to the Thomas-Fermi-Dirac model. This however did not improve accuracy sufficiently to enable practical usage.

6.5.2 Practical density functional theory

The first steps in the direction of a practical application of DFT were made by Hohenberg and Kohn in 1964.7 They showed with the Hohenberg-Kohn Existence theorem that the non-degenerate ground-state density determines the external potential of a system (i.e. the nuclei), and thereby the Hamiltonian and from there the wave function. Proving in this way that by using only electron density it is possible to determine the energy of a system. They showed also that a variational principle could be applied to DFT (similar to Eq. 6.9), showing that any energy calculated should be greater then or equal to the true ground state energy (E0). This made a systematic approach to finding the best fitting result possible, since step by step minimisation of the energy of the system would lead to a result that is closest to the real ground state of the system under investigation (i.e. within the limits of the approximations used).

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Kohn and Sham (1965) introduced an important simplification to the construction of the Hamiltonian operator. By using a system of non-interacting electrons that have the same density as the system as if they were interacting, it is possible to represent the Hamiltonian as a sum of one electron operators. It also allows a significant part of the energy to be determined exactly and therefore only a small part has to be approximated. This allows the energy of the system to be obtained as follows from the electron density:

( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ]rVeerTrVrVrTrE eeneni ρρρρρρ ∆+∆+++= Eq. 6.15

where Tni is the kinetic energy of non-interacting electrons, Vne the nuclear-electron interaction, Vee the classical electron-electron interaction, ∆T the correction to the kinetic energy due to electron interaction and ∆Vee all the non-classical corrections due to electron-electron interaction. It is important to note that in this functional all terms are determined exactly except for correction terms ∆T and ∆Vee. These two terms are grouped usually as the exchange-correllation energy (EXC). This functional can then be used to construct the one-electron (Kohn-Sham) operators for the system:

( )δρ

δρ xc

i

nuclei

k ki

ki

KSi

Edr

rrr

rrZ

h +−

+−

−∇−= ∫∑ ''

'21 2

Eq. 6.16

of which the terms on the right hand side are kinetic energy, nuclear-electron interaction, classical electron-electron interaction and the exchange-correlation energy. These Kohn-Sham operators are used to construct the secular equation which is used to minimise the energy of the system and thereby obtain the ground state

6.5.3 Exchange-correlational functionals

As discussed before the exchange-correlation energy (EXC) is the only part that cannot be solved exactly, because it has not yet been possible to produce a functional that can determine this energy and that is based on only the electron density (i.e. ρ). Several different functionals have been developed to deal with the difference in kinetic energy between the real and non-interacting system and the non-classical electron-electron interactions, of which the most commonly used one, Becke’s three parameter exchange functional (B3) combined with the Lee-Yang-Parr correlational functional (LYP) (which makes B3LYP), will be discussed in more detail. This functional can best be described as a combination of the best methods from several theories (it is therefore called a hybrid functional), and is depicted as:

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( ) ( ) LYPC

LSDAC

Bx

HFx

LSDAx

LYPBxc cEEcEbaEEaE +++∆++−= 113 Eq. 6.17

The first two terms are a combination of exact Hartree-Fock exchange energy (EHFX),

corrected using the local spin density approach (LSDA) which uses uniform-electron gas to determine the value of the energy at any position using electron density. EB

X is a functional by Becke which corrects for the fact that the density is not only dependent on the local value but also on the gradient (i.e. the way the density is changing locally). ELSDA

C is the local term for electron corelation based on LDSA and ELYP

C the gradient correction of the electron correlation based on the functional by Lee, Yang and Par. This hybrid functional uses 3 constants (i.e. a, b and c), which were optimized by fitting to heats of formation of small molecules and are set to 0.20, 0.72 and 0.81 respectively. This functional is the most widely used exchange-correctional function and shows good results8 (i.e. it compares well with experimental results) for small and relatively large systems in combination with relatively low calculation cost.

6.5.4 Basis set

To solve the Schrödinger equation modern computer programs use a basis set to approximate orbitals. These basis sets use a combination of mathematical functions to describe the atomic orbitals. The larger this basis set, the better the description of a system, an infinitely large basis set would, in theory, give the best results. However, this would be impossible to use in practice since this would require unacceptable computational time. Therefore, a compromise has to be made between the accuracy required and time available. For calculations described in this chapter the 6-31G(d) (also known as 6-31G*) basis set was used.9 This implies the 6-31G valence double-ζ basis set, (10s4p/4s)[3s2p/2s], which was combined with an additional set of 6 d functions on C and N to allow for polarisation of the p orbitals.

6.5.5 Advantages and limitations of DFT

Currently, comparing results (i.e. best fit to empirical results) and computational cost (i.e. computer power required to calculate using a certain method in a reasonable amount of time), DFT (i.e. in combination with appropriate functionals and basis-set) gives the best results compared to methods requiring similar amounts of computational power (i.e. HF, MP2). In combination with hybrid functionals, such as the popular B3LYP, which are among the most useful and cost efficient at the moment, DFT can be a useful tool in understanding experimental results.

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One of the disadvantages of DFT is that it is not possible to improve the methods systematically (unlike post-HF methods such as Møller-Plesset or coupled cluster theory) to reach the final goal, the exact description of a system. It is therefore not possible to estimate errors without comparison of results to other methods or experimental data. For the calculation of vibrational frequencies DFT categorically shows better results then HF. The calculation of IR spectra using B3LYP/6-31G(d) gives results comparable or better then second order Møller-Plesset theory (MP2) at reduced computational cost.10,11 Infrared vibrational frequencies are on average overestimated by ~ 10% due to the approximation of the electron correlation and thereby an overestimation of bond strength. The agreement with experimental results can be increased by using a scaling factor.11

6.6 Experimental Hartree-Fock (HF) and Density functional theory (DFT) calculations were carried out with the GAUSSIAN 03W (rev. B.04) program package.12 Gaussian 03W was run on a computer system equipped with an Intel® Pentium® D 3.2 GHz CPU and two gigabytes of RAM with Windows XP SP2 as the operating system. GaussView 3.0 was used to prepare jobs and to analyse the results.13 GaussSum14 was used to monitor jobs, calculate (partial) density of states ((P)DOS) spectra , and to convolute IR spectra. All the DFT calculations were performed on systems in the gas phase using the Becke’s three-parameter hybrid functional (B3) 15 with the LYP correlation functional16 (B3LYP) and 6-31G(d) basis set. The geometry optimization was followed by the frequency calculation to prove that the energy minimum was found. Obtained frequencies were scaled with a factor of 0.9614, a value which was determined by Scott and Radom,11 and was obtained by comparing 1066 experimental vibrations of 122 molecules to their calculated frequencies. This scaling factor is required to compensate for the HF character of B3LYP and by using it the B3LYP/6.31G(d) method gives the lowest RMS error of all methods studied in ref. 11. Assignments of vibrations were made using the possibility to visualize vibrations using GaussView 3.0.13 FT-IR spectra were recorded (as intimate mixtures in KBr) in reflectance mode using a Nicolet Nexus FT-IR spectrometer.

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6.7 Calculations on three perylene bisimide models

6.7.1 Introduction

Three differently substituted perylene bisimides (i.e. 1a, 2a and 3a) have been synthesized and their electronic and redox properties been investigated (see Chapter 2). The experimentally observed absorption maxima (λmax) showed, that when going from more electron donating (i.e. butyl) to a more electron withdrawing group (i.e. isophthalic) there was a decrease in the HOMO – LUMO gap. DFT calculations were employed to predict the orbital nature of the frontier molecular orbitals of perylene bisimide 1b, 2b and 3b in order to investigate if the observed effect could be contributed to the imide substituents. The validity of the calculations was assessed by comparison of predicted physical data with FT-IR spectroscopy, cyclic voltammetry and UV/Vis data of synthesized perylene bisimides 1a, 2a and 3a (vide infra).

Figure 6.2 The investigated perylene bisimide, both synthetic compounds 1a, 2a and 3a and simplified models for calculations 1b, 2b and 3b.

Calculations were performed on suitable models for perylene bisimide 1a, 2a and 3a. These models have structures similar to the synthesized compounds, some alkyl extensions were removed where possible to decrease calculation time and to increase symmetry. For all calculations instead of 4-tert-butylphenol, 4-methylphenol bay substituents were used, which allows for an increase in symmetry for both molecules. Since the tert-butyl groups are spatially quite separated from the rest of the molecule no steric effects are expected

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from this and because the pKa values for 4-tert-butylphenol and p-cresol only differ slightly (10.16 vs. 10.26, respectively) there is not expected to be any significant electronic effect. For 1a the symmetry group C2 could be used, which almost halves the calculation time for 1b. For 2b and 3b a D2 symmetry could be used. For 3b the calculation time could be reduced further by replacing the piperidine groups by dimethylamine moieties, a change which not expected to influence the properties of the amide, but could, however, have some steric influence on the area around the carbonyls. Structures were first optimized using the Hartree-Fock method and a 6-31G(d) basis set. After this first optimization a more strict optimization was performed using DFT with the Becke’s three-parameter hybrid functional15 using the LYP correlation functional16 and the 6-31G(d) basis set, which has shown a good agreement with experimental data in similar structures.17 These energy optimized structures were then used to perform a frequency calculation (IR) using the same functional and basis set.

6.8 Partial density of state diagrams Partial density of state diagrams (PDOS) can be used to show the contribution of different parts with a molecule to the density of states and energy of the whole molecule. The partial density of state diagrams in Figure 6.3 show the influence of the imide substituent of the perylene bisimide on the frontier orbitals. Going from 1b to 3b, it is clear that this influence increases when going from more electron donating (i.e. butyl) to a more electron withdrawing group (i.e. isophthalic). However, the substituent shows no direct influence on either HOMO or LUMO in any of the molecules examined. In all cases the perylene bisimide dominates the LUMO orbitals. This is the case for the HOMO also, however, the influence of the substituents on the frontier orbitals becomes larger when going to the more electron withdrawing substituents and for the 3b the substituent has a clear contribution to the HOMO-1. There is no clear trend in the comparison of the HOMO and LUMO difference values (∆eV) obtained by DFT with the ∆V values for the separation between first oxidation and first reduction (Table 6.1). The values obtained from cyclic voltammetry show a decrease in the gap between oxidation and reduction (which is related to the HOMO – LUMO gap), with decreasing electron withdrawing character of this substituent, in agreement with UV/Vis spectral data (i.e. a decrease in λmax) (Chapter 2). This trend is not apparent for ∆eV obtained from calculations, where only very small differences are observed. DFT calculations performed by Pichierri on perylene bisimides without bay substitution,18 however, show the same trend in the calculations when going from an alkyl group to a

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phenyl group to a phenyl group with electron withdrawing substituents. Both sets of calculations were performed in the gas phase, which may be the origin of the differences in the values that were calculated, compared with the data from cyclic voltammetry and UV/Vis spectra, which originate from measurements which where performed in solution (CH2Cl2). Since the differences observed are quite small, it is possible that solvent effects (i.e. polarity, H-bond formation) are dominant.19

Table 6.1 First oxidation and reduction potentials of 1a, 2a and 3a combined with calculated eV values for HOMO and LUMO.

Comp. Ox. (V)a

Red. (V)a

∆Vb Model HOMO (eV) c

LUMO (eV) c

∆eVc

1a 1.25 -0.77 2.02 1b -5,195 -2,888 2,307

2a 1.28 -0.72 2.00 2b -5,194 -2,883 2,311

3a 1.31 -0.66 1.97 3b -5,474 -3,201 2.273 a Differential pulse voltammetry, b ∆V is the separation between E½ of the first oxidation and the first reduction. (V vs. SCE, in CH2Cl2 / 0.1 M TBAPF6), c Values obtained from PDOS diagrams.

6.8.1 Molecular orbitals diagrams

The orbital diagrams for the three systems (i.e. 1b, 2b and 3b) of LUMO + 1, LUMO, HOMO and HOMO - 1 (Figure 6.4) show no electron density on the imide substituents and therefore they are not likely to have an influence on the HOMO–LUMO gap. The only exception is the HOMO - 1 of 3b, which is also the molecule which shows the largest shift in the UV (i.e. 3a) compared to 1a. This indicates that the influence of the substituents on the HOMO - LUMO gap is unlikely to be exerted through bond. However, another observation can also be made from the orbital diagrams. In all three molecules the perylene bisimide carbonyls are involved in both HOMO and LUMO. Electron density on the carbonyl groups is sensitive to environmental effects (e.g., solvent, hydrogen bonding, and Lewis acids).19 Even if the imide substituents are not involved directly (i.e. through bond, by influencing HOMO and/or LUMO) in the HOMO – LUMO energy gap, they do influence the direct environment of the carbonyls, either sterically (e.g., by hindering solvent molecules around the carbonyls) or electrostatically (e.g., through interaction with the π system of the differently substituted phenyl groups).

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1b

0 2 4 6 8 10 12

Perylene Butyl

LUMO

HOMO

-14

-12

-10

-8

-6

-4

-2

0

eV

2b

0 2 4 6 8 10 12

Perylene Phenyl

LUMO

HOMO

-14

-12

-10

-8

-6

-4

-2

0

eV

3b

0 2 4 6 8 10 12

Perylene IsophthalDiMeAmine

-14

-12

-10

-8

-6

-4

-2

0

LUMO

HOMO

eV

Figure 6.3 Partial density of states (PDOS) diagrams of 1b, 2b and 3b with contribution of R1 groups to the DOS (grey filled areas, Figure 6.2).

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1b 2b 3b

LUMO + 1 LUMO + 1 LUMO + 1

LUMO LUMO LUMO

HOMO HOMO HOMO

HOMO - 1 HOMO - 1 HOMO - 1

Figure 6.4 Frontier molecular orbital diagrams of LUMO + 1 , LUMO, HOMO and HOMO - 1 of 1b, 2b and 3b (DFT B3LYP/6-31G(d), isosurface = 0.03).

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6.8.2 Frequency calculations

Frequency calculations on the optimized structures provided the calculated IR spectra for compounds 1b, 2b and 3b.20 Comparison of the FT-IR spectra of 1a , 2a and 3a with the calculated IR spectra of 1b, 2b and 3b shows good agreement (Figure 6.5 and Figure 6.6 respectively). This indicates the optimized model structures obtained from the calculations (1b, 2b and 3b) show good resemblance with the molecules synthesized (1a, 2a and 3a). Comparison of the FT-IR spectra of 1a and 2a with the calculated spectra of 1b and 2b, respectively, show that overall a good agreement is obtained. Differences between the calculated and experimental spectra are due to vibrations of the tert-butyl groups on the bay phenyl substituents (R2, Figure 6.2), which are not observed for the calculated spectra due to the replacement of the tert-butyl by a methyl substituent. Comparing just the experimental (1a, 2a and 3a) or the calculated spectra (1b, 2b and 3b) there are, of course, differences due to different IR absorption bands for the butyl, phenyl and isophthalic imide substituents. Bands for the core perylene and bay phenyl substituents remain at the same position for both groups of molecules, both calculated and experimental (Table 6.2). More important are the two characteristic imide bands (i.e. imide C=O, νs and νas ) in the FT-IR spectra of 1a, 2a and 3a show a shift towards higher energy going from 1a to 2a to 3a (Table 6.2). This indicates an influence of the imide substituents on the carbonyl bond strength frequency and since the carbonyls have a contribution to the frontier orbitals as shown in the molecular orbital diagrams (Figure 6.4), the substituents, therefore, can still via their influence on the imide vibrations have an effect on the HOMO-LUMO levels of the perylene core. However, when comparing the IR absorption bands of 1b, 2b and 3b, as with the values obtained from the PDOS diagrams, the expected trend breaks down. In the FT-IR spectra the trend from 1a to 2a to 3a seems to be towards higher energy for the imide carbonyls, however, this trend is not observed in the calculated spectra (Figure 6.6). From 1b to 2b the increase in energy is present in accordance with the experimental data, however for 3b the energy is lower in energy, as was also observed for the PDOS diagrams. As discussed for the PDOS diagrams, the calculations were performed with molecules in the gas phase, which might be the cause of the different trend between calculated and measured IR data; the latter being performed in the solid state.

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Table 6.2 Characteristic bands from the calculated and experimental infrared spectra of 1a,b; 2a,b and 3a,b.

Experimental (ν, cm-1) Calculated (ν, cm-1)a

Assignmentsb 1a, Bu 2a, Ph 3a, Iso 1b, Bu 2b, Ph 3b, Iso

Imide C=O νs 1701 1707 1709 1700 1712 1705

Imide C=O νas 1662 1672 1674 1668 1685 1676

Iso imide C=O νsνas - - 1637 - - 1671

Perylene νC-C, δC-H 1591 1587 1587 1568 1568 1565

Perylene and bay phenyls

δC-H 1506 1504 1504 1493 1492 1492

Imide methylene δC-H 1437 - - 1437 - -

Perylene νC-C, δC-H 1412 1406 1408 1397 1383 1384

a DFT B3LYP/6-31G(d) scaled by 0.9614. b Obtained by visualization with Gaussview13 and checked with known literature values.

6.8.3 Conclusions

Even though the bisimide substituents do not have a direct contribution to the HOMO and LUMO orbitals, as seen in the PDOS and molecular orbital diagrams, the agreement of experimental and calculated IR spectra seems to indicate that the substituents are able to influence the frontier orbitals indirectly by influencing the electron density on the imide carbonyls and thereby make a considerable contribution to the HOMO and LUMO levels of 1b, 2b and 3b. However, the trend observed experimentally was not in full agreement with the calculations, indicating that the model used was not sufficiently realistic, i.e. other factors are involved such as solvent environment, etc.

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Figure 6.6 Calculated infrared spectra of 1b (top), 2b (middle) and 3b (bottom). (DFT B3LYP/6-31G(d) scaled by 0.9614)

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6.9 Exploring the remarkable different behaviour of two diphenyldithienylcyclopentene switches coupled with differently substituted coumarin donor groups

6.9.1 Introduction

In Chapter 4 two modified diarylethene switches were described that show different behaviour with respect to their fluorescence properties. Both switches 4 and 5 show full quenching of the coumarin fluorescence in the ring-closed state, which was expected because of the overlap in the donor emission and acceptor absorption spectra. However, in the open state different behaviour was observed. The mono-methoxy coumarin functionalized switch 4 (MMCS) showed no coumarin fluorescence in the open state. In contrast, the dimethoxy coumarin functionalized switch 5 (DMCS) shows no quenching of the coumarin fluorescence (i.e. compared to the coumarin model 7a) in the ring-open state. In this paragraph DFT calculations will be used in combination with experimental data to rationalize this difference in behaviour in the ring-open forms of 4 and 5. Special attention will be paid to the position of the relative energy levels of the coumarin models (6b and 7b) compared to each other and compared to the energy levels of the ring open form of the diarylethene switch. Calculations were performed as described in the experimental § 6.6 and as described in § 6.7.1 using suitable models (Figure 6.8) and the results from the frequency calculations were compared to experimental FTIR spectra of the compounds under investigation to validate the calculation method used.

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Figure 6.7 Structure of the diarylethene switches MMCS 4 and DMCS 5.

OO O OO

O

O

6a 7a

R1

O

R2

O

6b 7b

NNHBoc

S S

O

MMC DMC

PhSPhCO

R3

R1 = NMe2 R2 = NMe2

R3 = NMe28b

8a

NNHBoc

NH

R1 = R2 =

R3 =

Figure 6.8 Structure of the compounds prepared and as described in Chapter 4 (6a, 7a and 8a) and models used in the DFT calculations (6b, 7b and 8b).

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6.9.2 Partial density of state diagrams

The PDOS diagrams for MMCS 4 and DMCS 5 show the energy levels of the diphenyl dithienylcyclopentene units and coumarin units within the whole molecule and also of the separate model components (Figure 6.9 and Figure 6.10). Three important observations can be made: First the differences in the energy levels of the diphenyl dithienylcyclopentene unit are negligible in both 4 and 5 (Table 6.3), as expected, since this part of the molecule is identical in both systems and is separated from the coumarin by the piperazine bridge. Secondly the HOMO-LUMO gap for both coumarins is found to be of equal size (both in the separate models 6b and 7b as in 4 and 5). This is unexpected since this would imply an identical lowest energy λmax of absorption for both coumarins, which is not the case (λmax = for 6a 323 nm, for 7a 346 nm, ∆E = 23 nm ≈ 0.24eV). Thirdly, compared to values obtained with cyclic voltammetry (CV) (Chapter 4), the energy of the HOMO levels (from the first oxidation, obtained from CV) for the coumarins are reversed compared to what has been calculated using DFT. For 6a the first oxidation is observed at 1.78 V [vs SCE] and for 7a at 1.48 V [vs SCE]. This a considerable difference, both in absolute and relative energy, compared to that obtained from the DFT calculations. However, from CV it is clear that the first oxidations of both 4 and 5 are localised on the diarylethene units in agreement with the PDOS diagrams. The differences can be rationalised by consideration of the environment of the molecules (i.e. solvent, solvent polarity, solutes), both coumarins have groups that can interact with the environment (carbonyls, methoxy groups) and DMC has one extra methoxy group, so one would expect more possibilities for interaction, however it is surprising that the DFT calculations indicate that the extra methoxy group has no influence on the HOMO-LUMO gap, a result which seems highly unlikely.

The LUMO levels of the diarylethene and coumarins (both for the separate models and for 4 and 5) are grouped closely together (Figure 6.9), with DMC having the lowest energy. MMC and the diarylethene switch are close in their LUMO energies, if change in e.g., environment could change the position of one of these levels it would be possible that for MMCS 4 the LUMO of the whole molecule would be localised on the diarylethene instead of on the coumarin unit, thereby making energy transfer from MMC to the open diarylethene a possibility. This possibility could explain the absence of fluorescence for the open form of MMCS 4.

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Table 6.3 Energy values for the calculated orbitals of MMCS 4, DMCS 5 and individual components (values obtained from PDOS diagrams).

Compound Part HOMO (eV)

LUMO (eV)

∆eVa HOMO-LUMO gap (eV)b

MMCS 4 MMC -5.888 -1.504 4.384 PhSPhCO -5.448 -1.488 3.960

3.944

DMCS 5 DMC -6.048 -1.664 4.384 PhSPhCO -5.488 -1.544 3.944

3.824

6b MMC -5.816 -1.424 4.392 3.976c 7b DMC -5.984 -1.592 4.392 3.808c 8b PhSPhCO -5.400 -1.288 4.112 - a Difference between HOMO and LUMO level of the component switch and coumarin subunits. b Difference between the HOMO and LUMO level within the molecule as a whole. c Compared to the HOMO level of 8b.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

LUMO

HOMO

PhSPhCO MMC DMC

-8

-6

-4

-2

0

eV

Figure 6.9 Partial density of states (PDOS) diagrams of 6b, 7b and 8b showing the location of the HOMO and LUMO levels for MMC, DMC and PhSPhCO.

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a

0.0 0.5 1.0 1.5 2.0 2.5 3.0

MMC Piperazine PhSPh

-8

-6

-4

-2

0

LUMO

HOMO

eV

b

0.0 0.5 1.0 1.5 2.0 2.5 3.0

DMC Piperazine PhSPh

-8

-6

-4

-2

0

LUMO

HOMO

eV

Figure 6.10 Partial density of states (PDOS) diagrams of a) MMCS 4 and b) DMCS 5. Also showing the contribution of the different coumarin sidegroups (MMC and DMC) to the frontier orbitals (DFT B3LYP/6-31G(d)).

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6.9.3 Molecular orbital diagrams

For both MMCS 4 and DMCS 5 the frontier molecular orbital diagrams show similar features (Figure 6.11). HOMO-1, HOMO and LUMO+1 are located on the diarylethene moiety for both molecules. The LUMO for both molecules is located on the coumarins. Considering the physical properties of the open state of MMCS 4 and DMCS 5 this is unexpected in the case of 4. The similarities of their orbital structure would suggest that they should exhibit similar properties, however, as expected from the PDOS diagrams, the calculated frontier orbitals do not seem to correspond to the experimental data in the case of MMCS 4 (Chapter 4). For this compound one would expect a LUMO which is also localised on the diarylethene moiety, thereby making the diarylethene HOMO-LUMO transition the lowest energy one and thereby making MMC fluorescence unfavourable. Considering the fact that Dexter energy transfer requires wave function overlap (which is not necessarily through bond in character) and since both LUMO (located on the coumarin) and LUMO+1 (located on the diaryethene) are proximate it should be possible with the correct orientation of the coumarin and diarylethene for the orbitals to overlap. If this is combined with favourable matching of the donor and acceptor energy levels, it is possible that the coumarin excited state will be quenched by the diarylethene. Since the LUMO’s of the MMC and the diarylethene are closest in energy, as observed from the PDOS diagrams, the probability of energy transfer in that case is expected to be greater.

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MMCS LUMO + 1 DMCS

MMCS LUMO DMCS

MMCS HOMO DMCS

MMCS HOMO - 1 DMCS

Figure 6.11 Frontier molecular orbital diagrams of LUMO + 1 , LUMO, HOMO and HOMO - 1 of MMCS 4 and DMCS 5 (DFT B3LYP/6-31G(d) isosurface = 0.03).

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6.9.4 Frequency calculations

Frequency calculations were performed on the optimized structures of 4, 5, 6b, 7b and 8b. The calculated spectra of the models show good agreement with compounds 4 and 5, i.e. important IR absorptions (e.g., from carbonyls) that are present in the spectra of the models can be seen in the spectra of switches 4 and 5 at identical locations. The spectra of 4 and 5 are almost a summation of the calculated IR spectra of their respective component unit models (i.e. 6b and 8b for 4 and 7b and 8b for 5, Table 6.4). This indicates that for the DFT calculations of the separate units (6b, 7b and 8b) the results are electronically identical to those performed on the full switches 4 and 5, supporting the conclusion made in Chapter 4 that the components are electronically isolated (Figure 6.12b and Figure 6.13b). The same holds for the experimental FT-IR spectra of 4 and 5 compared to those of models 6a, 7a and 8a. The data are in agreement except for the additional absorption of the Boc group in the coumarin models, which shows a strong absorption at ~ 1700 cm-1. However, features are much more difficult to resolve due to broadness of the peaks. Comparison of the FT-IR spectra with the calculated spectra is not trivial in this case, however, using the spectra of the model compounds (some peaks are clearly present only in the coumarin model compounds (6 and 7, respectively) and some only in that of the switch model 8) it is possible to assign several common absorption bands. The area between 1300 and 1500 cm-1 is crowded and therefore it is not possible to assign bands to specific vibrations with reasonable confidence. For MMCS 4 the agreement is quite good between the FT-IR and calculated spectra, the largest differences are observed for groups which are sensitive to their environment (i.e. carbonyls, amides), which is possible due to the difference between relating to the gas phase of the calculations and the FT-IR spectra which were obtained in the solid state. For DMCS 5 a similar comparison can be made as for MMCS 4. There is, however, one exception; the position of carbonyl absorption band of the calculated dimethoxy coumarin (both for 5 and 7b) is shifted considerably towards higher energy compared to that observed in the FT-IR spectrum. There is no clear direct explanation for this observed shift (DFT optimization of the structure shows no abnormalities and no imaginary frequencies were obtained for the calculation, which would indicate a saddle point). Considering the fact that in the PDOS diagrams the HOMO – LUMO gap for both coumarins is identical, it is likely that for the dimethoxycoumarin the calculations in this form (i.e. level of theory / basis-set) do not provide reliable results.21

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Table 6.4 Comparison between calculated spectra of 4, 5, 6b, 7b and 8.a

Vibrations (ν, cm-1) MMC 6b MMCS 4 8b DMCS 5 DMC 7b

Coumarin C=O νs 1739 1739 - 1768 1767 Coumarin Amide νs 1687 1683 - 1694 1697 Switch Amide νs - 1670 1668 1670 - Coumarin rings νC-C, δC-H 1618 1618 - 1616 1615 Coumarin rings νC-C, δC-H 1611 1611 - 1606 1606 Switch Phenyls νC-C, δC-H - 1600 1600 1599 - Coumarins rings νC-C, δC-H 1545 1545 - 1531 1531 Switch δC-H - 1499 1496 1498 - Coumarin δC-H 1475 1474 - 1488 1488 Switch and Coumarin δC-H - 1453 1453 1453 1449 Piperazine and switch γC-H - 1410 - 1409 - Switch Amide Phenyl νC-C, δC-H 1399 1397 1397 1398 1400 Piperazine/amide γC-H 1388 1383 - 1383 1386 Amide νC-N, δC-H 1371 - 1372 - - DMC νC-C, δC-H - - - 1358 1357 Coumarin νC-C, δC-H 1285 1285 - 1278 1278 a DFT B3LYP/6-31G(d) scaled by 0.9614

Table 6.5 Comparison between experimental and calculated spectra of 4 and 5.

MMCS 4 DMCS 5

Vibrations (ν, cm-1) Exp Calc a Exp Calc a

Coumarin C=O νs 1720 1739 1720 1768 Coumarin Amide νs 1652(sh)b 1683 1651(sh)b 1694 Switch Amide νs 1636 1670 1635 1670 Coumarin rings νC-C, δC-H 1616 1618 1617(sh)b 1616 Switch Phenyls νC-C, δC-H 1606 1600 1606 1599 Coumarins rings νC-C, δC-H 1545 1567 1531 1561 Switch δC-H 1507 1499 1517 1498 Coumarin νC-C, δC-H 1285 1285 1281 1278 a DFT B3LYP/6-31G(d) scaled by 0.9614 b Shoulder, but present in respective model coumarins 6a/7a

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a

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*

b

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4

Figure 6.12 a) FT-IR spectra of 4 (top), 6a (middle, * = Boc group ) and 8a (bottom). b) Calculated infrared spectra of 4 (top), 6b (middle) and 8b (bottom) (DFT B3LYP/6-31G(d) scaled by 0.9614).

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a

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b

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Figure 6.13 a) FT-IR spectra of 5 (top), 7a (middle, * = Boc group ) and 8a (bottom) b) Calculated infrared spectra of 5 (top), 7b (middle) and 8b (bottom) (DFT B3LYP/6-31G(d) scaled by 0.9614).

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6.9.5 Conclusions

The results obtained for both switches give some good indications as to why there is such a difference in behaviour between MMCS 4 and DMCS 5 in the open form which was described in chapter 4 (i.e. possibility for Dexter type energy transfer in MMCS 4). However, it should be borne in mind, that to compare the energy differences between the two different coumarins (i.e. DMC and MMC) and the switch (PhSPhCO), it would be better to not compare the coumarin LUMO with the switch HOMO, but to compare the coumarin first excited state with the ground state (HOMO) of the switch. Calculations on the coumarin first excited state, however, were troublesome and, unfortunately, were unsuccessful until now. Furthermore, there appears to be a fundamental disagreement between the vacuum DFT calculations and experiment (identical HOMO-LUMO gap for the coumarins and an unexpected shift of the coumarin carbonyl band in DMCS 5), which makes it difficult to draw conclusive comparative inferences from the results. It also shows that, even though calculations are an increasingly commonplace tool, the gas phase (or vacuum) conditions under which calculations are performed are very different from the solvated environment for which physical data are recorded.

6.10 References

1 Based on the following sources: (a) C.J. Cramer, Essentials of Computational Chemistry, Wiley N.Y., 2002. (b) W. Koch and M.C. Holthausen, A Chemist's Guide to Density Functional Theory, Wiley-VCH, Weinheim, sec. ed., 2002.

2 J. E. Lennard-Jones, Proc. Phys. Soc. 1931, 43, 461-482. 3 M.J.S. Dewar, E.G. Zoebisch, E.F. Healy and J.J.O. Stewart, J. Am. Chem. Soc. 1985,

107, 3902-3909. 4 J.J.P. Stewart, J. Comput. Chem. 1989, 10, 209-220. 5 (a) E. Schrödinger, Ann. Phys. 1926, 79, 489-527. (b) E. Schrödinger, Phys. Rev. 1926,

28, 1049-1070. 6 M. Born and R.Oppenheimer, Ann. Phys. 1927, 84, 457-484. 7 P. Hohenberg, W. Kohn, Phys. Rev. B 1964, 136, B864. 8 Recent studies have shown, however, that for some larger molecules considerable

deviations can occur: P.R. Schreiner, Angew. Chem. Int. Ed. 2007, 46, 4217-4219. 9 (a) R. Ditchfield, W. J. Hehre and J.A. Pople, J. Chem. Phys. 1971, 54, 724-728. (b) W.

J. Hehre, R. Ditchfield and J. A. Pople, J. Chem. Phys. 1972, 56, 2257-2261. (c) P. C. Hariharan and J. A. Pople, Mol. Phys. 1974, 27, 209-214. ((d) M. S. Gordon, Chem. Phys. Lett. 1980, 76, 163-168. (e) P. C. Hariharan and J. A. Pople, Theo. Chim. Acta

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1973, 28, 213-222. (f) J. P. Blaudeau, M. P. McGrath, L. A. Curtiss and L. Radom, J. Chem. Phys. 1997, 107, 5016-5021. (g) M. M. Francl, W. J. Pietro, W. J. Hehre, J. S. Binkley, D. J. DeFrees, J. A. Pople and M. S. Gordon, J. Chem. Phys. 1982, 77, 3654-3665. (h) R. C. Binning Jr. and L. A. Curtiss, J. Comp. Chem. 1990, 11, 1206-1216. (i) V. A. Rassolov, J. A. Pople, M. A. Ratner and T. L. Windus, J. Chem. Phys. 1998, 109, 1223–1229. (j) V. A. Rassolov, M. A. Ratner, J. A. Pople, P.C. Redfern and L.A. Curtiss, J. Comput. Chem. 2001, 22, 976-984.

10 X.F. Zhou, C.J.M. Wheeless and R.F. Liu, Vib. Spectrosc. 1996, 12, 53-63. 11 A.P. Scott, L. Radom, J. Phys. Chem. 1996, 100, 16502-16513. 12 M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A.

Montgomery, Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, 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, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, Gaussian 03, Revision B.04, Gaussian, Inc., Wallingford CT, 2004.

13 GaussView, Version 3.09, R. Dennington II, T. Keith, J. Millam, K. Eppinnett, W.L. Hovell, and R. Gilliland, Semichem, Inc., Shawnee Mission, KS, 2003.

14 N. M. O'Boyle, GaussSum 2.0, 2006. Available at http://gausssum.sf.net. 15 A. D. Becke, J. Chem. Phys. 1993, 98, 5648-5652. 16 C. Lee, W. Yang and R. G. Parr, Phys. Rev. B 1988, 37, 785-789. 17 (a) D. Zoon and A.M. Brouwer, ChemPhysChem 2005, 6 (8), 1574-1580. (b) N.U.

Zhanpeisov, S. Nishio and H. Fukumura, Int. J. Quantum Chem. 2005, 105 (4), 368-375

18 F. Pichierri, J. Mol. Struct. (THEOCHEM) 2004, 686, 57-63. 19 (a) K. Das, B. Jain and H.S. Patel, J. Phys. Chem. A 2006, 110, 1698-1704. (b) I.

Georgieva, N. Trendafilova, A. Aquino and H. Lischka, J. Phys. Chem. A 2005, 109 (51), 11860-11869.

20 Software limitations unfortunately did not allow calculation of the IR spectrum of 3b via the standard FREQ method and FREQ=NUMER was used. Results obtained using this method are in agreement with results obtained from the PDOS diagrams.

21 By using larger basis sets it may be possible to obtain results which are closer to empirical data. This however would take extensive calculating effort, also because this would have to be performed for all molecules, since otherwise comparisons cannot be made and time did not allow for this.