the lewis theory revisited bernard silvi laboratoire de chimie théorique université pierre et...
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The Lewis theory revisited
Bernard SilviLaboratoire de Chimie Théorique
Université Pierre et Marie Curie4, place Jussieu 75252 -Paris
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 mechanics
=E• 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
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
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
There is no paradigm for the chemical bond, why?Quantum mechanics is a paradigm but tells
nothing on the chemical bondLewis theory and the VSEPR model have
no real mathematical models behind themThe quantum chemical approaches violate
the postulates of quantum mechanics and do not work with exact wavefunctions
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
Chemical objects
Mathematical objects
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
X XX X
X X
X X regions of
space
The answer is yes
Gradient dynamical system bound on R3
vector field X=V(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 {i}
the control space: V(r;{i})
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
)(1 MpI
p
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 function (r;i)
ELF (Becke and Edgecombe 1990) is a good approximation of the ideal localization function
What is ELF?The statistical interpretation of Quantum
Mechanics enables to define density functions
)()(
....),.....,,(),.....,,(*)( 222
rr
r
ddxdxxxxxxx NNN
)',()',()',(),(
'...).....,,',(*).....,,',(*)',( 2,2,2
rrrrrrrr
rr
dddxdxxxxxxxxx NNN
iiiiiiii NNNNddi
)')',( rrrr
it is possible to calculate the number of pairs in a given region i
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: )',(1)()()',( rrr'rrr h
What is ELF?Curvature of the Fermi hole:
Homogeneous gas renormalization
r’
))(())((
)',()()( 2
r r
rrr'
vWS
r
TT
hrD
-1
0
))(())((
)',()()( 2
r r
rrr'
vWS
r
TT
hrD
23/5 )](/)([1
1)(
rcrDr
F
Classification of basins
Core and valence Synaptic order
monosynaptic disynaptic
(protonated or not)
higher polysynaptic
V(C, H)
V(O, H)
V(C, O)V(O)
C(C) C(O)
Populations and delocalizationBasin populationpair populations
Example CH3OH
i
dNi rr)(
')',( i j
ddNij rrrr
')',( i i
ddNii rrrr
N N
N
C(C) 2.12 1.13 0.20
C(O) 2.22 1.24 0.31
V(C, H) 2.04 1.04 0.34
V(O, H) 1.66 0.69 0.25
V(O) 2.34 1.37 0.74
V(C, O) 1.22 0.37 0.16
Populations and delocalization
antiaromatic aromatic
1.832
1.91 2.8
0.1220.28
variance (second moment of the charge distribution)
ij
ijijjiij
iiiii BNNNNNNN )( )1()(2
Population rules
V(C) Z-Nv Increases with Z
V(X) > 2.0 can merge
V(X, Y) <2.0 can merge
V(X, H) 1.5-2.5 cannot merge
Subjects treated
Connection with VSEPRElementary chemical processesProtonationUnconventional bonding
metallic bond hypervalent molecules tetracoordinated planar carbons
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
Elementary chemical processes
Covalent vs. Dative bond cusp catastrophe
unfolding:
(-1)0=1
(-1)0+(-1)1+(-1)0=1
vxuxx 24
- the active control space is of dimension 2
Hypervalent molecules
Total valence population of an atom A
)),(())(()( XAVNAVNANv 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 6.160.4
• Cl 6.850.45
Hypervalent molecules
Hydrogenated series PF5-nHn
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
PF5 PHF4 PH2F3 PH3F2 PH4F PH5
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
CH3CH3
Cl2Zr ZrCl2
CH3
Tetracoordinated planar carbon
– R. H. Clayton, S. T. Chacon and M. H. Chisholm, Angew. Chem., Int. Ed. Eng, 1989, 28, 1523
Cr(OH)3
C
CH2
(OH)3Cr
CH2
Tetracoordinated planar carbon
– S. Buchwald, E. A. Lucas and W. M. Davis, J. Chem., Int. Soc, 1989, 111, 397
OHOH
ZrCl2Cl2ZrCH3
ConclusionsThe 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
AcknowledgementsLaboratoire 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. LatajkaCentro per lo studio delle relazioni tra struttura e
reattività chimica CNR (Milano): C. Gatti