Transcript The Lewis Theory Revisited

The Lewis theory revisited
Bernard Silvi
Laboratoire de Chimie Théorique
Université Pierre et Marie Curie
4, 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
HY=EY
• 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 bond
 Lewis theory and the VSEPR model have
no real mathematical models behind them
 The 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?
XX
X
X
XX
regions of
space
 From quantum mechanics we know that:


The whole molecular space should be filled
The model should be totally symmetrical
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 {ai}
the control space: V(r;{ai})
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
I
p  1   ( M )
Structural stability condition: all critical points are
hyperbolic
p

 That’s all with mathematics
A meteorological example: V(r{ai})=-P
basin 2
basin 1
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 h(r; ai)
 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
 (r )   Y * ( x, x2 ,.....,xN )Y ( x, x2 ,.....,xN )dx2 ....dx N d
  a (r )    (r )
 (r , r ' )   Y * ( x, x' , x2, .....,xN )Y  Y * ( x, x' , x2, .....,xN )dx2 ...dxN dd '
  aa (r , r )   a (r , r ' )   a (r , r ' )    (r , r ' )

it is possible to calculate the number of pairs in a
given region i
  (r , r ' )drdr ' ) 
i
aa
Nii

a
Nii

a
Nii


Nii
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:

aa
a
a

aa
(r , r ' )   (r )  (r' ) 1  h (r , r ' )

What is ELF?
 Curvature of the Fermi hole:
0
D (r )   2r   (r' )h (r , r ' )
 TS (   (r ))  TvW (   (r ))
-1
r’
 Homogeneous gas renormalization
1
h (r ) 
1  [ D (r ) / cF  5 / 3 (r )]2
Classification of basins
 Core and valence
 Synaptic order



V(O, H)
monosynaptic
disynaptic
(protonated or
not)
higher
polysynaptic
V(C, O)
V(C, H)
C(C)
V(O)
C(O)
Populations and delocalization
 Basin population
 pair populations
Nii     (r , r ' )drdr '
i
i

Ni    (r)dr
i
N ij     (r , r ' )drdr '
i
j
Example CH3OH
N
aa
a
C(C)
N
N
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
 variance (second moment of the charge distribution)
 2 ( Ni )  Nii  Ni ( Ni  1)   (Ni N j  Nij )   Bij
j i
antiaromatic
j i
aromatic
1.832
0.122
1.91
0.28
2.8
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 VSEPR
Elementary chemical processes
Protonation
Unconventional bonding



metallic bond
hypervalent molecules
tetracoordinated planar carbons
Connection with VSEPR
 Visualization of electronic domains
X-A-X
AX3
AX2E
Connection with VSEPR
 Visualization of electronic domains
AX3Y
AX3E
AX2E3
AX4E
AX4E2
AX5E
Connection with VSEPR
 Size of the electronic domains
12.8
6.8
0.13
0.9
0.05
8.6
11.7
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
Elementary chemical processes
 Covalent vs. Dative bond

cusp catastrophe

  

unfolding:
(-1)0=1
(-1)0+(-1)1+(-1)0=1
x  ux  vx
4
- the active control space is of dimension 2
2
Elementary chemical processes
 Covalent vs. Dative bond
Protonation
 Least topological change principle
Where does the proton go?
 Covalent protonation
4.7
2.6
Where does the proton go?
 agostic protonation
Where does the proton go?
 predissociative protonation
Proton transfer mechanism
Metallic bond
 Body centred cubic structures
Metallic bond
 Face centred cubic structures
Hypervalent molecules
 Total valence population of an atom A
N v ( A)   N (V ( A))   N (V ( A, X ))


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 6.160.4
• Cl 6.850.45
Hypervalent molecules
 Hydrogenated series PF5-nHn
10
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
PF5
PHF4
PH2F3
PH3F2
PH4F
PH5
Tetracoordinated planar carbon
H3 C
Cl2Zr
CH3
ZrCl2
CH3
– D. Röttger, G. Erker, R. Fröhlich, M. Grehl, S. J. Silverio, I. Hyla-Kryspin
and R. Gleiter, J. Am. Chem. Soc., 1995, 117, 10503
Tetracoordinated planar carbon
CH2
CH2
C
(OH)3Cr
Cr(OH)3
– R. H. Clayton, S. T. Chacon and M. H. Chisholm, Angew. Chem., Int. Ed.
Eng, 1989, 28, 1523
Tetracoordinated planar carbon
HO
Cl 2Zr
OH
H3C
ZrCl2
– S. Buchwald, E. A. Lucas and W. M. Davis, J. Chem., Int. Soc, 1989, 111,
397
Tetracoordinated planar carbon
H2
C
Co
B
C
BH2
Co
Tetracoordinated planar carbon
H
C
VCl2
Cl2Zr
C
H
Conclusions
 The 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
Acknowledgements
 Laboratoire 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. Latajka
 Centro per lo studio delle relazioni tra struttura e
reattività chimica CNR (Milano): C. Gatti