The synaptic order: a key concept to The synaptic order: a key concept to understand multicenter bondingunderstand multicenter bonding

Bernard SilviLaboratoire de Chimie Théorique

Université Pierre et Marie Curie4, place Jussieu 75252 -Paris

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

Origins of the concept

The diborane problem

B B B

H

B

H

or

classical Hydrogen bridges (Dilthey 1921)

in contradiction with boron valence and electron count

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)

The protonated double bond (Pitzer 1945)

MO representation (Longuet-Higgins 1949)

Generalisation: 3c-2e bond

In B2H6

Generalisation: 3c-2e bond

B-B-B bonds

Generalisation: 3c-4e bonds

F-Xe-F

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

Bonds and related concepts

Bonds are not quantum mechanical observables

They belong to an other (chemical) level of understanding

Definitions often lack precisionIs a three-center bond a bond?

Chemical concepts related to bondingDescribe 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

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 ThomParaboles et catastrophes

However 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

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

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?

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)

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

{i} called the control space, i.e.: V(r;{i})

Some 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

More definitions....

)(1 MpI

p

A meteorological example: V(r{i})=-P

Basin 2Basin 2Basin 1Basin 1

Domains

Definition

That’s all with maths

a

b

Atoms In Molecules theoryBond path

unstable manifold of an index 1 critical point

bond critical point only 2 centre bonds are possible

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

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

)())(())(( ),( v2 rrrrr DTTP WScondr '

23/5 )](/)([1

1)(

rcrDr

F

Analysis: classification of basins

Core and valence basins

Nomenclature C(A) core V(A, ..) valence color code

V(O)

C(C) C(O)V(C, H)

V(O, H)

V(C, O)

Graphical representation: isosurfaces of the function

Analysis of localization domains:Bounded by the isosurface (r)=f

reducible and irreducible domains core-valence separation Localization domains

(rcv) hierarchy of basins: bifurcation diagram

parent domain

cores valence

S

lone pairs bond

Parent domain

C(O) valence C(S)

V(S) Valence O

V(S,O) V1(O)+V2(O)

V1(O) V2(0)

Detailed diagram:

Hierarchy of localization domains:complex Valence-valence separation (rvv)

Ex: FH CO2

Parent domain

HF CO2

valence C(F) C(O) valence C(O)

, V(C,O) V(O)

Hierarchy of localization domains:ionic pair

Core-valence separation (rcv) Ex: LiF

Parent domain

C(Li) F

C(F) V(F)

The two processes

filled-filled: 2 chemical single chemical objects (or more)complex, ionic pair, physisorption

hollowed-filled: 1 chemical single chemical objectmolecule, ion, chemisorption

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

Synaptic order: CH3F

monosynaptic

disynaptic

Protonated disynaptic

Synaptic order and chemical reactions

Covalent bond breaking: C2H6

Synaptic order and chemical reactions

Dative bond breaking: BH3NH3

3c-2e multicenter bond

Protonated bonds: B2H6

Protonated trisynaptic

3c-2e multicenter bond

B-B-B bonds: B4H4

3c-2e multicenter bond

Agostic hydrogen: RuClCH 3CH2(PH3)

3c-2e multicenter bond

Agostic protonation

0.223

0.235

3c-2e bonds: high coordination of C

Pentacoordinated sp3: Al2H4(CH3)2

Cl2Zr

CH

CH

VCl2

Planar tetracoordinated carbons

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

Planar pentacoordinated carbon

CC

C

C C

C

B

B

B

B

BB

P. v. R. Schleyer, private communication

Planar hexacoordinated carbon

P. v. R. Schleyer, private communication

B

B

B

B

B

B

C

Metallic bond

Metal cluster:Li6

Metallic bondBcc structures

Metallic bondFcc structures

Metallic bondElectron-phonon interaction?

3c-4e multicenter bonds

Hypervalent molecules: XeF2

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

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

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

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

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

Integrated conditional probability (Dobson and Savin)

Six come three

Limit for q0

qV cond dP rrrr ),()(

31

0 )(

)(

r

r

qr

qVq

)(),(

),()(

35

35

2

0

42 0

rrr

rrr

rr

rr

qP

dssP

cond

r

cond

qdqV

rr)(