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

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

basin 2basin 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 (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

Connection with VSEPRVisualization of electronic domains

X-A-X

AX3 AX2E

Connection with VSEPRVisualization of electronic domains

AX3Y AX2E3

AX4E AX4E2 AX5E

AX3E

Connection with VSEPRSize of the electronic domains

11.7

12.8

0.13

6.8

8.6

0.9 0.05

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

vxuxx 24

- the active control space is of dimension 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

)),(())(()( 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

Tetracoordinated planar carbon

Co

BH2

BCH2

Co C

Tetracoordinated planar carbon

Cl2Zr

CH

CH

VCl2

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