Chemical Applications of Charge DensityLouis J FarrugiaJyvskyl Summer School on Charge Density August 2007

Chemical applications of charge densityJyvskyl Summer School on Charge Density August 20071. VSEPR rules and the Laplacian2. Hydrogen bonding3. Metal-metal bonding

VSEPR rules and the LaplacianJyvskyl Summer School on Charge Density August 2007VSEPR (Valence Shell Electron Repulsion) rules are used to teach the elements ofstructure for main group compounds.

Simple theory based on a few elementary concepts :Electrons in the valence shell around central atom are paired (Lewis structures)Electrostatic repulsion between pairs leads to structures with the maximal distance between themRepulsions for single bonds follows order lone-pair lone pair > lone-pair bond pair > bond pair bond pairMultiple bonds have stronger repulsion than single bondsBond pair repulsion decreases with increasing electronegativity of ligand6. Explains why H2O is bent, NH3 is pyramidal but BF3 is planar , ClF3 is T-shaped .....R. J. Gillespie & P. L. A. Popelier Chemical Bonding and Molecular Geometry, OUP, 2001.

VSEPR rules and the LaplacianJyvskyl Summer School on Charge Density August 2007R. J. Gillespie & P. L. A. Popelier Chemical Bonding and Molecular Geometry, OUP, 2001.

VSEPR rules and the LaplacianJyvskyl Summer School on Charge Density August 2007R. J. Gillespie & P. L. A. Popelier Chemical Bonding and Molecular Geometry, OUP, 2001.Laplacian emphasiseslocal charge concentration

VSEPR rules and the LaplacianJyvskyl Summer School on Charge Density August 2007R. J. Gillespie & P. L. A. Popelier Chemical Bonding and Molecular Geometry, OUP, 2001.Laplacian recovers the shell structure of the atom (here a free S atom -3P). This is completely invisible in the density . Shows alternate shells of charge concentrations and charge depletions. Here we are concerned with the Valence Shell Charge Concentration (VSCC). Unitsof Laplacian in experimental CD studies usually given as e -5.

VSEPR rules and the LaplacianJyvskyl Summer School on Charge Density August 2007R. J. Gillespie & P. L. A. Popelier Chemical Bonding and Molecular Geometry, OUP, 2001.Plots of L -2 for the SCl2 moleculein molecular plane close-up of VSCC of S atomplot in the plane perpendicular to (a)

Critical points are marked by dots

VSEPR rules and the LaplacianJyvskyl Summer School on Charge Density August 2007R. J. Gillespie & P. L. A. Popelier Chemical Bonding and Molecular Geometry, OUP, 2001.Plots of L -2 for the T-shaped moleculeClF3 in molecular plane plot in the plane perpendicular to (a)

5 charge concentrations 3 bond pairs and2 lone pairsab

VSEPR rules and the LaplacianJyvskyl Summer School on Charge Density August 2007R. J. Gillespie, R. F. W. Bader et al (1998) J. Phys Chem. A. 102, 3407The Lennard-Jones Function: A Quantitative Description of the Spatial Correlation of Electrons As Determined by the Exclusion Principle - provides a physical basis for electron pairing through the operation of the Pauli exclusion principle the conditional probability function for same spin electron is called the LJ function in honour of Lennard-Jones.

Is BF3 ionic ?Jyvskyl Summer School on Charge Density August 2007R. J. Gillespie et al (1997) Inorg Chem. 36, 3022.Arguments :Bader atomic charges indicate high ionic character.F-F non-bonded distances are remarkably constant, in line with observed structure being dominated by ligand-ligand repulsions.

Counter arguments :While the B-F bond is undoubtedly highly polar, an ionic model can only account for ~ 60% of bond energyThe F atom is strongly polarised, indicating significant covalent characterIt is a gas.Bader charge onfluorideBader charge oncentral atomA. Haaland et al (2000) J. Chem. Ed. 77, 1076

Hydrogen bonds Jyvskyl Summer School on Charge Density August 2007Hydrogen bonds were first described (probably) by M. L. Huggins in 1919, but Linus Pauling (1931) was the first to coin the term hydrogen bond to describe the bonding in the [F-H...F]- anion.G. A. Jeffrey (1997) An Introduction to Hydrogen Bonding, OUP, New York.

Jyvskyl Summer School on Charge Density August 2007G. Gilli, P. Gilli (2000) Towards a Unified H-bond Theory J. Mol. Struct 552, 1Hydrogen bonds

Jyvskyl Summer School on Charge Density August 2007R.F.W. Bader & C. Gatti (1998) Chem. Phys. Lett 287, 233 (source function)E. Espinosa et al (1999) Acta Cryst B55, 563. (topology of H-bonds)E. Espinosa & E. Molins (2000) J. Chem. Phys. 113, 5686 (interaction energies)Hydrogen bonds It is generally agreed that weak H-bonds are primarily electrostatic in nature, while strong H-bonds have considerable covalent character.

How can we use charge density methods to shed light on this area? use topological properties to characterise bonds use charge densities to extract interaction energiesThe values of (r)bcp and 2(r)bcp provide some information, but recently other tools, such as the Source Function, have proved to provide interesting additional information.

Approximate potential and kinetic energy densities may be derived from (r) and 2(r) using DFT type functionals exact electrostatic energies may be computed

Jyvskyl Summer School on Charge Density August 2007U. Koch & P.L. A. Popelier (1995) J. Phys Chem 99, 9747. Topological Definition of Hydrogen Bonding Koch & Popelier have examined the characterisation of C-H...O bonds on the basis of charge density. The derived criteria have been used to induce the presence or absence of weak to moderate H-bonds in general.1.A bond path and bcp between H and acceptor atom.The value of (r)bcp in the range 0.05 0.2 e-3The value of the Laplacian is positive and reasonable 0.5 3.3 e-5Mutual penetration of hydrogen and acceptor, r = r0 rbcp is positive for both.r0 is "non-bonded" radius, experimentally taken as van der Waals radius. This is a necessary and sufficient condition.5.Loss of charge on H atom basin charge in H-bonded complex minus free monomer.Energetic destabilisation of H atom.Decrease of dipolar polarisation of H atom.Decrease in atomic basis volume.Criteria 5-8 difficult to apply in experimental studies, but first four criteria often used.

Jyvskyl Summer School on Charge Density August 2007E. Espinosa et al (1999) Acta Cryst B55, 563. Topology of Hydrogen bonds Espinosa et al have examined the topological properties of a large number of H-bonded compounds.

Two clear cut cases emerged in terms of (r)bcp and 2(r)bcpfor the D-H and the H...A interactions strong H-bonds - (r)bcp ~ 1-1.3 e-3 2(r)bcp is negative ~ -5-15 e-5 weak H-bonds - (r)bcp ~ 0.5, 0.02 e-3 2(r)bcp is positive ~ 0.5 - 5 e-5

Clear correlations between these topological and geometrical parameters were found, the clearest being with 3(curvature along bond path)

The Source Function R. F. W. Bader & C. Gatti, (1998) Chem. Phys. Lett. 287, 233J. Overgaard et al (2001) Chem. Eur. J. 7 3756C. Gatti et al (2003) J. Comput. Chem. 24, 422Potentially available from experimental densityDoes not require the presence of a bond critical pointJyvskyl Summer School on Charge Density August 2007

Jyvskyl Summer School on Charge Density August 2007C. Gatti et al (2003) J. Comput. Chem. 24, 422Topology of Hydrogen bonds Gatti has examined the nature of H-bonds, within the scheme of Gilli & Gilliusing the Source Function. Posed the question how do the balance of contributions to H-bonding (covalent vs electrostatic) change with the nature of the H-bond ?Calculations on water dimer at varying D-H...Adistancesatomic percentage contributions change dramatically along reaction profileat long D-H...A distances, source for H is very negative (i.e a sink)at long D-H...A distances all atoms make significant contributions indicative of the importance of the delocalised electrostatic componentat short D-H...A distances, the 3-centre nature of the H-bond is emphasised

Jyvskyl Summer School on Charge Density August 2007C. Gatti et al (2003) J. Comput. Chem. 24, 422Topology of Hydrogen bonds Gatti has examined the nature of H-bonds, within the scheme of Gilli & Gilliusing the Source Function. Posed the question how do the balance of contributions to H-bonding (covalent vs electrostatic) change with the nature of the H-bond ?1.H5O2+ +CAHB2.formic acid-formate complex CAHB3/4. malonaldehyde (Cs equi and C2v t.s.) RAHBwater trimer PAHBwater dimer IHB

Clear that these different H-bonds have very different source contributions

Jyvskyl Summer School on Charge Density August 2007E. Espinosa et al (1998) Chem. Phys. Lett. 285, 1790.Energetics of Hydrogen bonds Espinosa et al have also shown strong correlations between approximate G(rbcp) and V(rbcp) (derived from Abramov functionals) and D(H...O). HOWEVER, Spackman has shown that the promolecule density gives very similar results !

M. A. Spackman (1999) Chem. Phys. Lett. 301, 425

Jyvskyl Summer School on Charge Density August 2007Energetics of Hydrogen bonds The question then arises are the experimental data providing anything more than noise about a trendline determined by the promolecule electron distribution ?M. A. Spackman (1999) Chem. Phys. Lett. 301, 425Conclusion : experimental electron densities in weak interactions are indeed systematically different from those of a promolecule

Jyvskyl Summer School on Charge Density August 2007Atomic Energies in QTAIM There are two forms of the kinetic energy expression :Y. A. Abramov, (1997) Acta Cryst. A53, 264.G(r) = (3/10)(32)2/3(r)5/3 + (1/6)2(r) Abramov approximation for G(r) at bcp (ONLY valid for regions where 2 < 0, i.e. closed-shell interactions).

V(r) = (1/4) 2(r) 2G(r) - Virial relationshipThese two forms are identical if the Laplacian vanishes, as it does when integrating over an atomic basin. Then E() = -G() = 1/2 V()

Exact expressions for G(r) are NOT available directly from experimental data

Jyvskyl Summer School on Charge Density August 2007Weak interactions to covalent bonds Mallinson et al have reported several experimental charge density studies on proton sponges which exhibit a range of H-bonded interactions - from very strong N-H..N to very weak C..C interactions. They studies reveal a continuous transition from weak H-bonding to covalent bonding.P. R. Mallinson et al (1003) J. Am. Chem. Soc. 125, 4259

Jyvskyl Summer School on Charge Density August 2007Limitations on Determining Interaction Energies de Vries et al have studied the intermolecular interaction energies in urea crystals, using the multipole model refined against theoretical data. They conclude that it is not possible to extract the effect of intermolecular interactions from diffraction data with the current multipolar refinement techniquesR. Y. de Vries, D. Feil & V. G. Tsirelson (2000) Acta Cryst. B56, 118.same data but with different random noise added (contours at 0.002 au)

Jyvskyl Summer School on Charge Density August 2007J. Dunitz & A. Gavezzotti (2005) Molecular Recognition in Organic Crystals Directed Intermolecular Bonds or Nonlocalized Bonding ? Angew. Chem. Int. Ed., 44, 1766.

Review chapters on AIM concepts and H-bonding in books :"Hydrogen Bonding - New Insights" Ed A. J. Grabowski (2006) Springer, Dordrecht."The Quantum Theory of Atoms in Molecules" Ed. C.E. Matta & R. Boyd (2007), Wiley-VCH, Weinheim.A Note of Caution for HO interactions, critical point densities bcp and their distance dependence are closely similar to those calculated for the overlap of the unperturbed atomic charge densities (the promolecule electron density) so that great care in the interpretation of experimental and theoretical intermolecular charge densities is called for.

Topological studies on metal-metal bondsJyvskyl Summer School on Charge Density August 2007Metal-metal bonding is an important aspect of organometallic chemistry.

Undergraduate treatment 18 electron rule. Very useful in explaining short(ish) metal-metal distances, but ignores the effects of bridging ligands.

Numerous theoretical studies based on orbital interpretations conclude thereis only (at best) a weak direct M-M interaction in bridged bonds.J. Reinhold, A. Barthel, C. Mealli (2003) Coord. Chem. Rev. 238-239, 333.Fe2(CO)9 Fe-Fe = 2.523

Short M-M distance Diamagnetic Required by 18-e rule

But what does CD say ?

Metal-metal bonding without bridging ligands Mn2(CO)10100 K, Nonius KappaCCDI2/aSpherical atom refinementR(F) =0.021 wR(F2)=0.061GOF 1.11 0.62 -0.817052 data 101 parameters

XD refinementR(F) =0.014 wR(F2)=0.016GOF 2.16 0.27 -0.206532 data 296 parameters Mn-Mn bondMn1-Mn1a 2.9031(2) 2.9042(8)(rbcp) 0.144(3)e -3 0.190(4) 0.1922(rbcp) 0.720(3) e -5 0.815(8) 0.124L. J. Farrugia, P. R. Mallinson, B. Stewart (2003) Acta Cryst. B59, 234R. Bianchi, G. Gervasio, D. Marabello (2000) Inorg. Chem. 39, 2360.Co2(CO)6(AsPh3)2 P. Macchi, D. E. Proserpio, A. Sironi (1998) J. Am. Chem. Soc. 120, 13429.Jyvskyl Summer School on Charge Density August 2007

Mn2(CO)10100 K, Nonius KappaCCDI2/aSpherical atom refinementR(F) =0.021 wR(F2)=0.061GOF 1.11 0.62 -0.817052 data 101 parameters

XD refinementR(F) =0.014 wR(F2)=0.016GOF 2.16 0.27 -0.206532 data 296 parameters Mn-Mn bondMn1-Mn1a 2.9031(2) 2.9042(8)(rbcp) 0.144(3)e -3 0.190(4) 0.1922(rbcp) 0.720(3) e -5 0.815(8) 0.124(Mn-Mn) 0.29L. J. Farrugia, P. R. Mallinson, B. Stewart (2003) Acta Cryst. B59, 234R. Bianchi, G. Gervasio, D. Marabello (2000) Inorg. Chem. 39, 2360.Co2(CO)6(AsPh3)2 P. Macchi, D. E. Proserpio, A. Sironi (1998) J. Am. Chem. Soc. 120, 13429.Jyvskyl Summer School on Charge Density August 2007Metal-metal bonding without bridging ligands

Source function for Mn2(CO)10 C. Gatti, D. Lasi (2007) Faraday Discuss., 135, 55.Contributions to the source function at theMn-Mn bcp from (a) Mn atoms (b) COeq and (c) COax Blue curve = real system, red curveto two non-interacting Mn(CO)5 groups Jyvskyl Summer School on Charge Density August 2007

Metal-metal bonding with carbonyl bridges P. Macchi, L. Garlaschelli, A. Sironi (2002) J. Am. Chem. Soc. 124, 14173.P. Macchi, A. Sironi (2003), Coord Chem. Rev. 238-239, 383 Evolution of metal-metal bonding in [FeCo(CO)8]- anion

Conclusion metal-metal bonding disappears early on in the profile, with initial formation of ring structureSymmetrized correlation plot of M(-CO)M from CCD and experimental charge densities in Co4(CO11)(PPh3) and Co2(CO)6(AsPh3)3Jyvskyl Summer School on Charge Density August 2007

Source function for Co2(CO)8 C. Gatti, D. Lasi (2007) Faraday Discuss., 135, 55.Source function shows that the basic picture is similar for the two isomers, the all-terminal D3d which has a bcp, and the C2v isomer which does not.

In the unbridged case, the Co atoms act as a sink at the bcp.

In the bridged case, they act (very marginally) as a source.

Due to differing behaviour of the Laplacian near the reference point (an order of magnitude more positive in the C2v isomer).Jyvskyl Summer School on Charge Density August 2007

The bent Co-Co bond in Co2(CO)8J.Reinhold & M. Finger (2003), Inorg. Chem. 42, 8128J.Reinhold et al (2005), Inorg. Chem. 44, 6494Jyvskyl Summer School on Charge Density August 2007Reinhold et al interpret the minimum in the total energy density H as indicative of a bent Co-Co bond.

The presence of such an interaction is also indicated from an orbital study.Co2(CO)8

Metal-metal bonding with hydride bridges P. Macchi, D. Donghi, A. Sironi (2005) J. Am. Chem. Soc. 127, 16494.we note that the M-H-M systems have undisputedly many subtler features associated with even smaller energy gradients that therefore hamper an easier rationalization of the bonding effectsExperimental study on [Cr2(-H)(CO)10]-Jyvskyl Summer School on Charge Density August 2007

Metal-metal bonding with alkylidyne bridges L. J. Farrugia, C. Evans (2005) Comptes Rendu Chimie. 8, 1566.Co3(3-CX)(CO)9 (X=H,Cl)GeometryDist Co1-Co2 2.477Co1-Co3 2.487Co2-Co3 2.473Co1-C1 1.892Co2-C1 1.895 Co3-C1 1.895C1-H1 1.084Jyvskyl Summer School on Charge Density August 2007

Metal-metal bonding with alkylidyne bridges L. J. Farrugia, C. Evans (2005) Comptes Rendu Chimie. 8, 1566.Co3(3-CX)(CO)9 (X=H,Cl)GeometryDist Co1-Co2 2.477Co1-Co3 2.487Co2-Co3 2.473Co1-C1 1.892Co2-C1 1.895 Co3-C1 1.895C1-H1 1.084Delocalization indices (A,B)

Jyvskyl Summer School on Charge Density August 2007

Metal-metal bonding with alkylidyne bridges ababAtomic graph of alkylidyne carbon. Critical points inLaplacian L -2 (3,-3) charge concentrations(3,-1) saddle points(3,+1) charge depletionsCo3(3-CX)(CO)9 (a=H, b=Cl)Isosurface of the Laplacian L -2 at +10 e-5Jyvskyl Summer School on Charge Density August 2007

Structural variability in M3(-H)(-CX)(CO)10AX=COMe, M=Ru= 94.9oRu.C = 2.921BX=Ph, M=Os= 78.2oOs.C = 2.585CX=H, M=Os= 69.7oOs.C = 2.353DX=Ph, M=Os= 66.6oOs.C = 2.286A - M. R. Churchill et al (1983) Organometallics . 2, 1179.B, C, D Shapley et al (1983) J. Am. Chem. Soc. 105, 140 ; Organometallics (1985) 4, 767,1898Jyvskyl Summer School on Charge Density August 2007

Charge density in Fe3(-H)(-COMe)(CO)10Experimental data 100 K, Nonius KappaCCDP21/cSpherical atom refinementR(F) =0.026 wR(F2)=0.059GOF 1.04 0.72 -0.9017177 data 243 parameters

XD refinementR(F) =0.018 wR(F2)=0.016GOF 1.53 0.30 -0.3013970 data 701 parameters Fe1-Fe2 2.64603(16)Fe1-Fe3 2.67669(17)Fe2-Fe3 2.60037(15)Fe1...C1 2.6762(4)Fe2/Fe3-C1 1.86 = 90.1(1) oJyvskyl Summer School on Charge Density August 2007

Main features :Ring cp for Fe(-H)(-CR)FeBond cps for other two Fe-Fe interactionsRing cp between FeC(alk)Charge density in Fe3(-H)(-COMe)(CO)10 (r) (e-3) 2 (e-5) Fe-Fe (bcp) 0.24 0.30 0.40Fe-Fe (rcp) 0.35 2.09 -Fe...C (rcp) 0.22 1.31 -optimised B3LYP 6-311G (C/H/O) Wachters+f Fe Geometry Fe1-Fe2 2.739Fe1-Fe3 2.744Fe2-Fe3 2.646Fe1...C1 2.685Fe2/Fe3-C1 1.86 = 87.5 oJyvskyl Summer School on Charge Density August 2007

Main features :Ring cp for Fe(-H)(-CR)FeBond cps for other two Fe-Fe interactionsBond cp between Fe-C(alk)Charge density in Fe3(-H)(-COMe)(CO)10 (r) (e-3) 2 (e-5) Fe-Fe (bcp) 0.23 0.46 0.85Fe-Fe (rcp) 0.34 2.28 -Fe-C (bcp) 0.23 1.48 0.68Geometry Fe1-Fe2 2.741Fe1-Fe3 2.752Fe2-Fe3 2.650Fe1...C1 2.653Fe2/Fe3-C1 1.86 = 85.9 ooptimised B3LYP Ahlrichs TZV Jyvskyl Summer School on Charge Density August 2007

Main features :Ring cp for Fe(-H)(-CR)FeNO bond cps for other two Fe-Fe interactionsBond cp between FeCCharge density in Fe3(-H)(-COMe)(CO)10 (r) (e-3) 2 (e-5) Fe-Fe (rcp) 0.24 1.44 -Fe..C (bcp) 0.24 1.40 2.37Experimental multipole densityJyvskyl Summer School on Charge Density August 2007

Multipole refinements with synthetic dataGas phase wavefunction B3LYP/6-331Gs.f. from 30 pseudo cubic unit cell refine elaborate XD multipole model refine XD multipole model project DFT density into s.f. CRYSTAL03 periodic DFT wavefunctionFe...C(alkylidyne) bcpFe...C(alkylidyne) bcphighly curved Fe-Fe bcpJyvskyl Summer School on Charge Density August 2007

Isosurfaces of density in Fe3(-H)(-COMe)(CO)10 viewed normal to the Fe3 plane 0.25 e -30.15 e -3Charge density in Fe3(-H)(-COMe)(CO)10Jyvskyl Summer School on Charge Density August 2007

Fe3(-H)(-COMe)(CO)10Delocalisation indicesDist (A-B)Fe1-Fe20.40Fe1-Fe3 0.38Fe2-Fe3 0.21Fe1-C15 0.19Fe2-C15 0.91 Fe3-C15 0.93Fe2-H27 0.45Fe3-H27 0.44

Fe-C(O) ~1.0Fe2-C18 0.13Fe-O ~0.17Optimised structure B3LYP 6-311G* (C/H/O) Wachters+f FeC18Jyvskyl Summer School on Charge Density August 2007

Source % contributions at Fe-Fe bcp (theoretical) and Fe-Fe midpoint (expt) Source Function in Fe3(-H)(-COMe)(CO)10Jyvskyl Summer School on Charge Density August 2007

Source % contributions at FeCalkylidyne rcp (theoretical) and bcp (expt) Source Function in Fe3(-H)(-COMe)(CO)10Jyvskyl Summer School on Charge Density August 2007

Structural variability in M3(-H)(-CX)(CO)10AX=COMe, M=Ru= 94.9oRu.C = 2.921BX=Ph, M=Os= 78.2oOs.C = 2.585CX=H, M=Os= 69.7oOs.C = 2.353DX=Ph, M=Os= 66.6oOs.C = 2.286A - M. R. Churchill et al (1983) Organometallics . 2, 1179.B, C, D Shapley et al (1983) J. Am. Chem. Soc. 105, 140 ; Organometallics (1985) 4, 767,1898Jyvskyl Summer School on Charge Density August 2007

Fe3(-H)(-CH)(CO)10Optimised structure B3LYP 6-311G* (C/H/O) Wachters+f FeJ. W. Kolis, E. M. Holt, D. W Shriver (1983) J. Am. Chem. Soc. 105, 7307. (spectroscopic data)GeometryDist Fe1-Fe2 2.674Fe1-Fe3 2.674Fe2-Fe3 2.669Fe1-C1 2.146Fe2-C1 1.877 Fe3-C1 1.877Fe2-H2 1.686Fe3-H2 1.686C1-H1 1.090

Fe1-C2-O2 166.5oFe1-Fe2-Fe3-C1 65.8oJyvskyl Summer School on Charge Density August 2007

Fe3(-H)(-CH)(CO)10GeometryDist Fe1-Fe2 2.674Fe1-Fe3 2.674Fe2-Fe3 2.669Fe1-C1 2.146Fe2-C1 1.877 Fe3-C1 1.877Fe2-H2 1.686Fe3-H2 1.686C1-H1 1.090

Fe1-C2-O2 166.5oFe1-Fe2-Fe3-C1 65.8oOptimised structure B3LYP 6-311G* (C/H/O) Wachters+f FeMolecular graph and critical points (bpss in red, ring cp in yellow)Jyvskyl Summer School on Charge Density August 2007