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
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
H
B
B
or
B
B
H
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 precision
 Is a three-center bond a bond?
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
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
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)
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})
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
  1
Ip
  (M )
p

structural stability:
• condition: all the critical points are hyperbolic
A meteorological example: V(r{ai})=-P
Basin 2
Basin 1
Domains
 Definition
a
b
 That’s all with maths
Atoms In Molecules theory
 Bond 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 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,
june 2001
What is ELF?
 Taylor expansion of the spherically averaged
conditional pair probability:

 2r Pcond
(r, r' )  TS (   (r ))  TvW (   (r ))  D  (r )
 Physical scaling by the homogeneous electron
gas
 Cosmetic scaling to confine ELF in the 0-1
range
1
h (r ) 

5/3
2
1  [ D (r ) / cF  (r )]
 Can be determined from experimental
densities
Analysis: classification of basins
Graphical representation: isosurfaces of the function
 Core and valence
basins
 Nomenclature



V(O, H)
C(A) core
V(A, ..) valence
color code
V(C, O)
V(C, H)
C(C)
V(O)
C(O)
Analysis of localization domains:
 Bounded by the isosurface h(r)=f



reducible and irreducible domains
core-valence separation Localization domains
h(rcv)
hierarchy of basins: bifurcation diagram
parent domain
valence
cores
O
lone pairs
S
bond
Detailed diagram:
Parent domain
C(O)
valence
V(S)
C(S)
Valence O
V(S,O)
V1(O)+V2(O)
V1(O)
V2(0)
Hierarchy of localization domains:
complex
 Valence-valence separation h(rvv)

Ex: FH CO2
Parent domain
HF
valence
CO2
C(F)
C(O)
,
valence
V(C,O)
C(O)
V(O)
Hierarchy of localization domains:
ionic pair
 Core-valence separation h(rcv)

Ex: LiF
Parent domain
C(Li)
F
C(F)
V(F)
The two processes
hollowed-filled: 1 chemical single chemical object
filled-filled: 2 chemical single chemical objects (or more)
molecule, ion, chemisorption
complex, ionic pair, physisorption
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
disynaptic
Protonated
disynaptic
monosynaptic
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
Planar tetracoordinated carbons
H
C
VCl2
Cl2 Zr
C
H
R. Choukroun, B. Donnadieu, J-S. Zhao, P. Cassoux, C. Lepetit et B. Silvi, Organometallics,
19, 1901-1911 (2000)
Planar pentacoordinated carbon
B
B
C
C B
C
B
C B
C
C
P. v. R. Schleyer, private communication
B
Planar hexacoordinated carbon
B
B
B
C
B
B
B
P. v. R. Schleyer, private communication
Metallic bond
 Metal cluster:Li6
Metallic bond
 Bcc structures
Metallic bond
 Fcc structures
Metallic bond
 Electron-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
forms of
of government
localization have
functions
been tried,
have and
beenwill
tried,
be
and will
tried
in this
be world
tried inofthis
sin world
and woe.
of sin
No and
one woe.
pretends
No one
that
pretends that
democracy
is ELF
perfect
is or
perfect
all-wise.
or all-wise.
Indeed, itIndeed,
has been
it has
the
been the
worst
form
worst
of government
form of localization
except allfunction
those others
exceptthat
all
thosebeen
have
others
tried
thatfrom
havetime
beentotried
time.from time to time.
W. Churchill
 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):
Lepetit
 Universitad de Oviedo: J. M. Recio, P. Mori Sanchez
 University of Helsinki: J. Lundell, M. Sundberg
 McMaster University: R. G. Gillespie
Acknowledgements
C.
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
forms of
of government
localization have
functions
been tried,
have and
beenwill
tried,
be
and will
tried
in this
be world
tried inofthis
sin world
and woe.
of sin
No and
one woe.
pretends
No one
that
pretends that
democracy
is ELF
perfect
is or
perfect
all-wise.
or all-wise.
Indeed, itIndeed,
has been
it has
the
been the
worst
form
worst
of government
form of localization
except allfunction
those others
exceptthat
all
thosebeen
have
others
tried
thatfrom
havetime
beentotried
time.from time to time.
W. Churchill
Integrated conditional probability
(Dobson and Savin)
 Six come three

h(r)   Pcond (r, r)dr

Vq
Vq
 (r)dr  q
 Limit for q0
Vq 
q
 (r )
 q 
r0  


(
r
)


r0
h (r )   Pcond (r, r) rr  s 4 ds
2

0
1
3

  Pcond (r , r)
2
rr

q

5
5
3
3
(r )