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The Lewis theory revisited Bernard Silvi Laboratoire de Chimie Théorique Université Pierre et Marie Curie 4, place Jussieu 75252 -Paris Is there a theory of the chemical bond? The point of view of molecular physics A molecule is a collection of interacting particles (electrons and nuclei) which are ruled by quantum mechanics HY=EY • Expectation values of operators • Density functions (statistical interpretation) • Information is available for the whole system or for single points • The chemical bond is not an observable in the sense of quantum mechanics The quantum theory is a paradigm Is there a theory of the chemical bond? The point of view of (empirical) chemistry Molecules are made of atoms linked by bonds • A bond is formed by an electron pair (Lewis) • The (extended) octet rule should be satisfied • Chemical bonds are classified in: – – – – Covalent Dative Ionic Metallic • Molecular geometry can be predicted by VSEPR Rationalise stoichiometry and molecular structure Is there a theory of the chemical bond? The point of view of quantum chemistry Gives a physical meaning to the approximate wavefunction • Valence bond approach • Molecular orbital approach Relies on the atomic orbital expansion Successful for semi-quantitative predictions • Ex: the Woodward-Hoffmann rules There is no paradigm for the chemical bond, why? Quantum mechanics is a paradigm but tells nothing on the chemical bond Lewis theory and the VSEPR model have no real mathematical models behind them The quantum chemical approaches violate the postulates of quantum mechanics and do not work with exact wavefunctions Is it possible to design a mathematical model of the Lewis approach? Find a mathematical structure isomorphic with the chemistry we want to represent There is no need of physics as intermediate • Ex: equilibrium `[H+][OH-]=10-14 Chemical objects Mathematical objects Is it possible to design a mathematical model of the Lewis approach? XX X X XX regions of space From quantum mechanics we know that: The whole molecular space should be filled The model should be totally symmetrical The answer is yes Gradient dynamical system bound on R3 vector field X=V(r) V(r) potential function defined and differentiable for all r Analogy with a velocity field X=dr/dt enables to build trajectories in addition V(r) depends upon a set of parameters {ai} the control space: V(r;{ai}) More definitions.... Critical points index: positive eigenvalues of the hessian matrix hyperbolic: no zero eigenvalue stable manifold • basin: stable manifold of a critical point of index 0 • separatrice: stable manifold of a critical point of index>0 Poincaré-Hopf relation I p 1 (M ) Structural stability condition: all critical points are hyperbolic p That’s all with mathematics A meteorological example: V(r{ai})=-P basin 2 basin 1 Back to bonding theory We postulate that there exists a function whose gradient field yields basins corresponding to the pairs of the Lewis structure Such a function is called localization function h(r; ai) ELF (Becke and Edgecombe 1990) is a good approximation of the ideal localization function What is ELF? The statistical interpretation of Quantum Mechanics enables to define density functions (r ) Y * ( x, x2 ,....., xN )Y( x, x2 ,....., xN )dx2 ....dxN d a (r ) (r ) (r , r ' ) Y * ( x, x' , x2, ....., xN )Y Y * ( x, x' , x2, ....., xN )dx2 ...dxN dd ' aa (r , r ) a (r , r ' ) a (r , r ' ) (r , r ' ) it is possible to calculate the number of pairs in a given region i (r , r ' )drdr ' ) i aa N ii a N ii a N ii N ii What is ELF? Minimization of the Pauli repulsion: the Pauli repulsion increases with the number of pair region within a region it increases with the same spin pair population Fermi hole: aa a a aa (r , r ' ) (r ) (r' ) 1 h (r , r ' ) What is ELF? Curvature of the Fermi hole: 0 D (r ) 2r (r' )h (r , r ' ) TS ( (r )) TvW ( (r )) -1 r’ Homogeneous gas renormalization 1 h (r ) 1 [ D (r ) / cF 5 / 3 (r )]2 Classification of basins Core and valence Synaptic order V(O, H) monosynaptic disynaptic (protonated or not) higher polysynaptic V(C, O) V(C, H) C(C) V(O) C(O) Populations and delocalization Basin population pair populations N ii (r , r ' )drdr ' i i Ni (r)dr i N ij (r , r ' )drdr ' i j Example CH3OH N aa a C(C) N N 2.12 1.13 0.20 C(O) 2.22 1.24 0.31 V(C, H) 2.04 1.04 0.34 V(O, H) 1.66 0.69 0.25 V(O) 2.34 1.37 0.74 V(C, O) 1.22 0.37 0.16 Populations and delocalization variance (second moment of the charge distribution) 2 ( Ni ) Nii Ni ( Ni 1) (Ni N j Nij ) Bij j i antiaromatic j i aromatic 1.832 0.122 1.91 0.28 2.8 Population rules V(C) Z-Nv Increases with Z V(X) > 2.0 can merge V(X, Y) <2.0 can merge V(X, H) 1.5-2.5 cannot merge Subjects treated Connection with VSEPR Elementary chemical processes Protonation Unconventional bonding metallic bond hypervalent molecules tetracoordinated planar carbons Connection with VSEPR Visualization of electronic domains X-A-X AX3 AX2E Connection with VSEPR Visualization of electronic domains AX3Y AX3E AX2E3 AX4E AX4E2 AX5E Connection with VSEPR Size of the electronic domains 12.8 6.8 0.13 0.9 0.05 8.6 11.7 Elementary chemical processes Described by Catastrophe Theory the varied control space parameters are the nuclear coordinates RA The Poincaré-Hopf relationship is verified along the reaction path topological changes occur through bifurcation catastrophes the universal unfolding of the catastrophe yields the dimension of the active control space Elementary chemical processes Covalent vs. Dative bond Elementary chemical processes Covalent vs. Dative bond cusp catastrophe unfolding: (-1)0=1 (-1)0+(-1)1+(-1)0=1 x ux vx 4 - the active control space is of dimension 2 2 Elementary chemical processes Covalent vs. Dative bond Protonation Least topological change principle Where does the proton go? Covalent protonation 4.7 2.6 Where does the proton go? agostic protonation Where does the proton go? predissociative protonation Proton transfer mechanism Metallic bond Body centred cubic structures Metallic bond Face centred cubic structures Hypervalent molecules Total valence population of an atom A N v ( A) N (V ( A)) N (V ( A, X )) in hypervalent molecules the number of valence basin is that expected from Lewis structures conforming or not the octet rule In fact Nv(A) close to the number of valence electron of the free atom • P 4.99 0.6 • S 6.160.4 • Cl 6.850.45 Hypervalent molecules Hydrogenated series PF5-nHn 10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 PF5 PHF4 PH2F3 PH3F2 PH4F PH5 Tetracoordinated planar carbon H3C Cl2Zr CH3 ZrCl2 CH3 – D. Röttger, G. Erker, R. Fröhlich, M. Grehl, S. J. Silverio, I. Hyla-Kryspin and R. Gleiter, J. Am. Chem. Soc., 1995, 117, 10503 Tetracoordinated planar carbon CH2 CH2 C (OH)3Cr Cr(OH)3 – R. H. Clayton, S. T. Chacon and M. H. Chisholm, Angew. Chem., Int. Ed. Eng, 1989, 28, 1523 Tetracoordinated planar carbon HO Cl2 Zr OH H3C ZrCl2 – S. Buchwald, E. A. Lucas and W. M. Davis, J. Chem., Int. Soc, 1989, 111, 397 Tetracoordinated planar carbon H2 C Co B C BH2 Co Tetracoordinated planar carbon H C VCl2 Cl2Zr C H Conclusions The mathematical model replaces electron pairs by localization basins integer by reals It extends the Lewis picture to metallic bond multicentric bonds It enables to describe chemical reactions to generalize the VSEPR rules to make prediction on reactivity Acknowledgements Laboratoire de Chimie Théorique (Paris): H. Chevreau, F. Colonna, I. Fourré, F. Fuster, L. Joubert, X. Krokidis, S. Noury, A. Savin, A. Sevin. Laboratoire de Spectrochimie Moléculaire (Paris): E. A. Alikhani Departament de Ciencés Experimentals (Castelló): J. Andrés, A. Beltrán, R. Llusar University of Wroclaw: S. Berski, Z. Latajka Centro per lo studio delle relazioni tra struttura e reattività chimica CNR (Milano): C. Gatti