The synaptic order: a key concept to understand multicenter bonding

1. The synaptic order: a key concept to understand multicenter bonding Bernard Silvi Laboratoire de Chimie Théorique Université Pierre et Marie Curie 4, place Jussieu 75252 -Paris

2. Multicenter bond • An apparently odd concept for the chemist • bonds are represented by lines between symbols • Not accounted for by the valence concept • bonds are characterized by: • bond lengths • bond energies • Energy decomposition • atom-atom pair potential • Axilrod-Teller term is very weak • Not in Lewis’s theory • original cubic atoms • Not in Pauling’s textbook but in Coulson’s

3. Origins of the concept • The diborane problem or classical Hydrogen bridges (Dilthey 1921) in contradiction with boron valence and electron count

4. Answer given by spectroscopy • high barrier to internal rotation (Stitt 1940) • infrared spectrum (Price 1947) • Electronic structure proposals • K shell binding (Hellriegel 1930) • resonance structures (Nekrassov 1940, Syrkin and Dyatkina 1941, Seel 1945)

5. The protonated double bond (Pitzer 1945) • MO representation (Longuet-Higgins 1949)

6. Generalisation: 3c-2e bond • In B2H6

7. Generalisation: 3c-2e bond • B-B-B bonds

8. Generalisation: 3c-4e bonds • F-Xe-F

9. Strengths and weaknesses • Acknowledged model in boron chemistry • Lipscomb’s works • predictive tool • Interpretation of the numerical procedure • chemical meaning given to the wave function • depends of the expansion basis • non invariance of the MO’s

10. Bonds and related concepts • Bonds are not quantum mechanical observables • They belong to an other (chemical) level of understanding • Definitions often lack precision • Is a three-center bond a bond?

11. Chemical concepts related to bonding • Describe molecules and solids in terms of bonds, lone pairs, etc... • Bonds are links between atoms • According to Lewis a bond is made of an electron pair • The octet rule should be satisfied • According to chemistry bonds are classified as: • Covalent, polar, dative, metallic, ionic • The VSEPR model enables to predict molecular geometry • These concepts ratonalize the stoichiometry and the molecular structure • The approach of Chemistry has been and still is very successful

12. Il me faut cependant avouer que la chimie proprement dite ne m’a jamais beaucoup intéressé.Pourquoi?Peut être parce que des notions telles que celles de valence, de liaison chimique etc., m’ont toujours semblé peu claires du point de vue conceptuel. René Thom Paraboles et catastrophesHowever I have to confess that I have never been very interested in ChemistryWhy?Maybe because notions such as those of valence, chemical bond, etc., seem to me unclear from a conceptual point of view

13. What should be a theory of the chemical bond • Investigate the local properties of matter with a well suited mathematical theory • mathematics reveal relationships and behaviours which are the consequence of the intelligibility of the Nature • mapping chemical concepts with mathematical objects should improve their definitions and enable to introduce new concepts • isomorphism provides the mathematical model

14. The starting point • Statistical interpretation of Quantum Mechanics • epistemologically valid • provides a bridge between microscopic and classical worlds • position space representation • pioneered by Daudel with the Loge theory • hampered by numerical complexity • what to do after?

15. Topological theories of bonding • Purpose: provide rigorous qualitative information: • mathematical model of Lewis’s theory • Non ambiguous definitions of bonds • Mathematical background: dynamical system theory • Achievements: AIM (R. Bader)

16. Some definitions • Gradient dynamical system bound on 3 • vector field X=V(r) • V(r) potential function defined and differenciable for all r • the analogy with a velocity field X=dr/dt enables to build trajectories • moreover V(r) depends upon a set of parameters {ai} called the control space, i.e.: V(r;{ai})

17. More definitions.... • Critical points • index: number of positive eigenvalues of the second derivative matrix (hessian) • hyperbolic critical point: all eigenvalues are non zero • stable manifold • basin: stable manifold of critical point of index 0 • separatrix: stable manifold of a critical point of index >0 • Poincaré-Hopf relation • structural stability: • condition: all the critical points are hyperbolic

18. Basin 2 Basin 1 A meteorological example: V(r{ai})=-P

19. Domains • Definition • That’s all with maths a b

20. Atoms In Molecules theory • Bond path • unstable manifold of an index 1 critical point • bond critical point • only 2 centre bonds are possible

21. Back to bonding theory • Lewis theory is based on the electron pair concept, therefore the potential function should be related to pair densities and to probe the efficiency of the Pauli principle • localization function h(r; ai) • ELF (Becke and Edgecombe 1990) has been elected by our community cf: Workshop “Content and interpretation of ELF and related functions” Dresden, june2001

22. What is ELF? • Taylor expansion of the spherically averaged conditional pair probability: • Physical scaling by the homogeneous electron gas • Cosmetic scaling to confine ELF in the 0-1 range • Can be determined from experimental densities

23. V(O, H) V(O) V(C, O) C(O) C(C) V(C, H) Analysis: classification of basins Graphical representation: isosurfaces of the function • Core and valence basins • Nomenclature • C(A) core • V(A, ..) valence • color code

24. valence cores O S lonepairs bond Analysis of localization domains: • Bounded by the isosurface h(r)=f • reducible and irreducible domains • core-valence separation Localization domains (rcv) • hierarchy of basins: bifurcation diagram parent domain

25. Detailed diagram:

26. Hierarchyof localization domains:complex • Valence-valence separation(rvv) • Ex: FH CO2

27. Hierarchyof localization domains:ionic pair • Core-valence separation(rcv) • Ex: LiF

28. The two processes hollowed-filled: 1 chemical single chemical object molecule, ion, chemisorption filled-filled: 2 chemical single chemical objects (or more) complex, ionic pair, physisorption

29. The synaptic order The synaptic order  of a valence basin or of a group of valence basins (cwm) is the number of cores belonging to the same single chemical object with which it shares a boundary (separatrix) • proton counted 1 • complementary of the valence concept

30. Synaptic order: CH3F disynaptic Protonated disynaptic monosynaptic

31. Synaptic order and chemical reactions • Covalent bond breaking: C2H6

32. Synaptic order and chemical reactions • Dative bond breaking: BH3NH3

33. 3c-2e multicenter bond • Protonated bonds: B2H6 Protonated trisynaptic

34. 3c-2e multicenter bond • B-B-B bonds: B4H4

35. 3c-2e multicenter bond • Agostic hydrogen: RuClCH 3CH2(PH3)

36. 3c-2e multicenter bond • Agostic protonation 0.223 0.235

37. 3c-2e bonds: high coordination of C • Pentacoordinated sp3: Al2H4(CH3)2

38. Planar tetracoordinated carbons R. Choukroun, B. Donnadieu, J-S. Zhao, P. Cassoux, C. Lepetit et B. Silvi, Organometallics, 19, 1901-1911 (2000)

39. Planar pentacoordinated carbon P. v. R. Schleyer, private communication

40. Planar hexacoordinated carbon P. v. R. Schleyer, private communication

41. Metallic bond • Metal cluster:Li6

42. Metallic bond • Bcc structures

43. Metallic bond • Fcc structures

44. Metallic bond • Electron-phonon interaction?

45. 3c-4e multicenter bonds • Hypervalent molecules: XeF2

46. Conclusions • The topological analysis of ELF provides • unambiguous position space definition of multicenter bonding • 3c-2e bonds are true multicenter bonds • 3c-4e bonds are not multicenter bonds • The synaptic order of a basin is a good descriptor • It is complementary of the valence concept • However, the choice of the localization function remains an open problem

47. Nevertheless… Many forms of localization functions have been tried, and will be tried in this world of sin and woe. No one pretends that ELF is perfect or all-wise. Indeed, it has been the worst form of localization function except all those others that have been tried from time to time. Many forms of government have been tried, and will be tried in this world of sin and woe. No one pretends that democracy is perfect or all-wise. Indeed, it has been the worst form of government except all those others that have been tried from time to time. W. Churchill

48. Acknowledgements • Laboratoire de Chimie Théorique (Paris): H.Chevreau, F. Colonna, H. Demirdjian, I. Fourré, F. Fuster, H. Gérard, C. Giessner-Prettre, A. Hénoux, L. Joubert, X. Krokidis, S. Noury, J. Pilme, A. Savin, A. Sevin • Laboratoire de Spectrochimie Moléculaire (Paris): E. A. Alikhani • Departament de Ciències Experimentals (Castelló): J. Andrés, A. Beltrán, M. Calatayud, M. Feliz, R. Llusar • Université de Wroclaw: S. Berski, Z. Latajka • Centro per lo Studio delle Relazioni tra Struttura e Reattività Chimica CNR (Milan): C. Gatti • Laboratoire de Chimie de Coordination (Toulouse): C. Lepetit • Universitad de Oviedo: J. M. Recio, P. Mori Sanchez • University of Helsinki: J. Lundell, M. Sundberg • McMaster University: R. G. Gillespie

49. That's all folks

50. Is a 3-center bond a bond? • Rely on the definition of the bond • A bond is not an observable • It belongs to Chemistry • Lewis’s definition • Pauling’s definition • Daudel’s definition: a chemical bond can be considered to be a part of space in which the fluctuation of the number of electrons is small and the correlation between them is high