The Lewis theory revisited

1. The Lewis theory revisited Bernard Silvi Laboratoire de Chimie Théorique Université Pierre et Marie Curie 4, place Jussieu 75252 -Paris

2. 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 mechanicsHY=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

3. 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

4. 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

5. 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

6. Mathematical objects Chemicalobjects 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

7. X X X X regions of space X X Is it possible to design a mathematical model of the Lewis approach? • From quantum mechanics we know that: • The whole molecular space should be filled • The model should be totally symmetrical

8. The answer is yes • Gradient dynamical system bound on R3 • vector field X=ÑV(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})

9. 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 • Structural stability condition: all critical points are hyperbolic • That’s all with mathematics

10. basin2 basin 1 A meteorological example: V(r{ai})=-P

11. 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 functionh(r; ai) • ELF (Becke and Edgecombe 1990) is a good approximation of the ideal localization function

12. What is ELF? • The statistical interpretation of Quantum Mechanics enables to define density functions • it is possible to calculate the number of pairs in a given region i

13. 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:

14. What is ELF? 0 • Curvature of the Fermi hole: • Homogeneous gas renormalization -1 r’

15. V(O, H) V(O) V(C, O) C(O) C(C) V(C, H) Classification of basins • Core and valence • Synaptic order • monosynaptic • disynaptic (protonated or not) • higher polysynaptic

16. Populations and delocalization • Basin population • pair populations • Example CH3OH

17. 1.832 0.122 0.28 1.91 2.8 Populations and delocalization • variance (second moment of the charge distribution) aromatic antiaromatic

18. Population rules

19. Subjects treated • Connection with VSEPR • Elementary chemical processes • Protonation • Unconventional bonding • metallic bond • hypervalent molecules • tetracoordinated planar carbons

20. Connection with VSEPR • Visualization of electronic domains X-A-X AX3 AX2E

21. Connection with VSEPR • Visualization of electronic domains AX3Y AX3E AX2E3 AX4E AX4E2 AX5E

22. 12.8 6.8 0.13 0.9 0.05 8.6 11.7 Connection with VSEPR • Size of the electronic domains

23. 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

24. Elementary chemical processes • Covalent vs. Dative bond

25. Elementary chemical processes • Covalent vs. Dative bond • cusp catastrophe • unfolding:  (-1)0=1 (-1)0+(-1)1+(-1)0=1    - the active control space is of dimension 2

26. Elementary chemical processes • Covalent vs. Dative bond

27. Protonation • Least topological change principle

28. 4.7 2.6 Where does the proton go? • Covalent protonation

29. Where does the proton go? • agostic protonation

30. Where does the proton go? • predissociative protonation

31. Proton transfer mechanism

32. Metallic bond • Body centred cubic structures

33. Metallic bond • Face centred cubic structures

34. Hypervalent molecules • Total valence population of an atom A • 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 6.160.4 • Cl 6.850.45

35. Hypervalent molecules • Hydrogenated series PF5-nHn

36. Tetracoordinated planar carbon • D. Röttger, G. Erker, R. Fröhlich, M. Grehl, S. J. Silverio, I. Hyla-Kryspin and R. Gleiter, J. Am. Chem. Soc., 1995, 117, 10503

37. Tetracoordinated planar carbon • R. H. Clayton, S. T. Chacon and M. H. Chisholm, Angew. Chem., Int. Ed. Eng, 1989, 28, 1523 CH2 CH2 C (OH)3Cr Cr(OH)3

38. Tetracoordinated planar carbon • S. Buchwald, E. A. Lucas and W. M. Davis, J. Chem., Int. Soc, 1989, 111, 397

39. Tetracoordinated planar carbon

40. Tetracoordinated planar carbon

41. 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

42. 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