The document discusses the concept of multicenter bonding from the perspective of topological theories of bonding. It introduces the electron localization function (ELF) which can be used to analyze localization domains and classify bonds. ELF analysis provides definitions of multicenter bonds like 3c-2e bonds seen in B2H6 and 3c-4e bonds in hypervalent molecules like XeF2. The synaptic order of a localization basin, defined as the number of cores it shares a boundary with, is proposed as a descriptor of multicenter bonds that is complementary to the valence concept. While ELF provides a way to rigorously analyze bonding, the choice of localization function remains an open issue.

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 B 2 H 6

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 catastrophes However I have to confess that I have never been very interested in Chemistry Why? 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. 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 =d r /dt enables to build trajectories moreover V( r ) depends upon a set of parameters {  i } called the control space, i.e.: V( r ;{  i }) Some definitions

17. 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 More definitions....

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

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  ( r ;  i ) ELF (Becke and Edgecombe 1990) has been elected by our community cf: Workshop “ Content and interpretation of ELF and related functions ” Dresden, june 2001

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

24. Analysis of localization domains : Bounded by the isosurface  (r)=f reducible and irreducible domains core-valence separation Localization domains  (r cv ) hierarchy of basins: bifurcation diagram parent domain cores valence S  lone pairs bond

25. Detailed diagram: Parent domain C(O ) valence C(S) V(S) Valence O V(S,O ) V 1 (O)+V 2 (O) V 1 (O) V 2 (0)

26. Hierarchy of localization domains: complex Valence-valence separation  (r vv ) Ex: FH CO 2 Parent domain HF CO2 valence C(F) C(O) valence C(O) , V(C,O) V(O)

27. Hierarchy of localization domains: ionic pair Core-valence separation  (r cv ) Ex: LiF Parent domain C(Li) F C(F) V(F)

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

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: CH 3 F mon osynaptic disynaptic Protonated disynaptic

31. Synaptic order and chemical reactions Covalent bond breaking: C 2 H 6

32. Synaptic order and chemical reactions Dative bond breaking: BH 3 NH 3

33. 3c-2e multicenter bond Protonated bonds: B 2 H 6 Protonated trisynaptic

34. 3c-2e multicenter bond B-B-B bonds: B 4 H 4

35. 3c-2e multicenter bond Agostic hydrogen: RuClCH 3 CH 2 (PH 3 )

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

37. 3c-2e bonds: high coordination of C Pentacoordinated sp 3 : Al 2 H 4 (CH 3 ) 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:Li 6

42. Metallic bond Bcc structures

43. Metallic bond Fcc structures

44. Metallic bond Electron-phonon interaction?

45. 3c-4e multicenter bonds Hypervalent molecules: XeF 2

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

51. 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

52. Integrated conditional probability (Dobson and Savin) Six come three Limit for q  0