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1Valence Bond Theory
Why doing VB? Basic principles Ab initio
methods Qualitative VB theory
2The Valence Bond model
electron pairs are in local bonds
one bond an interaction between two singly
occupied atomic orbitals
Each bond is mainly covalent, but has some minor
ionic character
The Valence bond wave function is the Quantum
Mechanical translation of theLewis structure
3Systems that cannot be described by a single
Lewis structure
- Molecules displaying electron conjugation
The VB wave function Y(1?2) for the ground
state is a combination of two VB structures Y1
and Y2
Y(1?2) C1(Y1) C2(Y2)
Estimation of C1, C2, resonance energy
4Why Doing Valence Bond and not only MO?
- Some vital concepts
- - hybridization
- - resonance
- - VSEPR
- - Lewis structures
- - Arrow-pushing language
- - IR frequencies of C-H bonds
- sp3-H, sp2-H, sp-H
- Application to reactivity (the VBSCD model)
- Standard teaching
- VB has failed
- The only correct theory is MO
5 Exemple the covalent H2 bond
Writing VB functions
Working hypothesis the electrons remain in
atomic orbitals
At equilibrium distance, 2 possible
déterminants
?ja jb ?
?jb ja ?
?ja jb ?
?jb ja ?
Correct wave function (for the covalent bond)
YHL
6Writing VB functions beyond the two-electron/two
center case
Exemple the p-system of butadiene
Ycov
Ycov
7Dissociation curve of H2
?ja jb ?
-20
-40
-60
-80
-100
8Dissociation curve of H2
?ja jb ?
?jb ja ?
YHL ?
-20
YHL
-40
-60
-80
-100
9Dissociation curve of H2
?ja jb ?
?jb ja ?
YHL ?
-20
Physical origin of the bond spin exchange
between AOs
YHL
-40
-60
-80
-100
10Comparaison with MO description (Hartree-Fock)
YHF ??g ?g ?
??a ?b ??? ??b ?a ??????? ??????a ?a ??? ??b
?b ??
HH
HH HH
??g ?g ?? 50 covalent 50 ionic
YHL 100 covalent
11Comparaison with MO description (Hartree-Fock)
YHF ??g ?g ?
??a ?b ??? ??b ?a ??????? ??????a ?a ??? ??b
?b ??
HH
HH HH
??g ?g ?? 50 covalent 50 ionic
YHL 100 covalent
72-79 covalent 21-28 ionic
12Exact description
Yexact ????a ?b ??? ??b ?a ???? ?????a ?a ???
??b ?b ???
HH
HH HH
l, ? and the orbitals are optimized
simultaneously VBSCF method (Balint-Kurti, van
Lenthe)
13Exact description
Yexact ????a ?b ??? ??b ?a ???? ?????a ?a ???
??b ?b ???
HH
HH HH
??g ?g ?? ??a ?b ??? ??b ?a ??????? ??????a ?a
??? ??b ?b ??
??u ?u ?? ??a ?b ??? ??b ?a ??????- ??????a ?a
??? ??b ?b ??
C1 ??g ?g ???? C2 ??u ?u ?? Yexact
14The Generalized Valence Bond Method (GVB)
  Â
15The Generalized Valence Bond Method (GVB)
  Â
YGVB ?????????a ?b ??? ??b ?a ???? ??????a ?a
??? ??b ?b ???
HH
HH HH
YGVB is formally covalent, but physically
covalent-ionic optimized
16The Generalized Valence Bond Method (GVB)
 GVB pair Overlapping distorted AOs
17The Generalized Valence Bond Method (GVB)
 GVB pair Overlapping distorted AOs
Generalization
 Four GVB pairsÂ
18Hartree-Fock, GVB, CASSCF and correlation energy
SCF
Non-dynamical correlation energy
80
GVB
CASSCF (1764 conf)
19Ab initio methods
Test case the bonding energy of FF
Nature of the wave function for F2
- Hartree-Fock too much ionic
-
FF FF
FF
Optimized covalent vs ionic coefficients
20Accuracy of the various methods
?E
Test case the dissociation of F2
FF F F
Calculation of ?E for F-F1.43Ã…, 6-31G(d) basis
- Hartree-Fock - 37 kcal/mol (repulsive!)
- Reason too much ionic
-
- Full configuration interaction (6-31G(d) basis)
30-33 kcal/mol
Only 15.7 kcal/mol
Reason we miss dynamic correlation.
What does this physically mean?
21What is wrong with GVB and VBSCF?
GVB/VBSCF a closer examination
FF FF
FF
- The coefficients and orbitals are optimized, but
- The same set of AOs is used for all VB
structures - optimized for a mean neutral situation
22What is wrong with GVB and VBSCF?
GVB/VBSCF a closer examination
FF FF
FF
- The coefficients and orbitals are optimized, but
- The same set of AOs is used for all VB
structures - optimized for a mean neutral situation
A better wave function
23The  Breathing-Orbital VB method (BOVB)
- Provides optimized covalent-ionic coefficients
(like GVB)
FF
FF FF
- Different orbitals for different VB structures
- Orbitals for FF will be the same as VBSCF
- Orbitals for ionic structures will be much
improved
- One expects
- A better description of ionic structures
- A better bonding energy
24Test case the dissociation of F2
?E
FF F F
Calculation of ?E for F-F1.43Ã…, 6-31G(d) basis
Iteration De(kcal) FF FF ?
FF Classical VB -4.6 0.813 0.187 GVB,VBSCF 15.
7 0.768 0.232 BOVB 1 24.6 0.731 0.269 2 27.9
0.712 0.288 3 28.4 0.709 0.291 4 28.5 0.710
0.290 5 28.6 0.707 0.293 Full CI 30-33
25Improvements of the BOVB method
- Improvement of the ionic VB structures
- - basic level
- improved level ( split-level  or S)
The  active orbital is split. This brings
radial electron correlation
26- Improvement of the interactions between
spectator orbitals
- Spectator orbitals can be
- local atomic orbitals
- bonding and antibonding combinations
Slightly better ( Delocalized level or D)
27The three levels of the BOVB method
All orbitals are localized, ionics are
closed-shell
All orbitals are localized, but active orbitals
in ionics are split
- Active orbitals are split in ionics
- Spectator orbitals are delocalized in all
structures
28Performances of the various BOVB levels
?E
Test case the dissociation of F2
FF F F
Method Req (Ã…) De(kcal/mol) 6-31G
basis set GVB 1.506 14.0
CASSCF 1.495 16.4 L-BOVB 1.485 27.9
SL-BOVB 1.473 31.4 SD-BOVB 1.449 33.9 Dunni
ng-Huzinaga DZP basis set SD-BOVB 1.443 31.6 E
stimated full CI 1.440.005
28-31 Experimental 1.412 38.3
29Electron correlation in BOVB
- Non-dynamic correlation (GVB, CASSCF)
-
- Non dynamic correlation gives the correct
ionic/covalent ratio
30Electron correlation in BOVB
- Non-dynamic correlation (GVB, CASSCF)
-
- Non dynamic correlation gives the correct
ionic/covalent ratio
- All the rest. This is what is missing in GVB.
- BOVB brings that part of dynamic correlation
that varies in the reaction
31What is an accurate description of two-electron
bonding?
- Spin exchange between two atomic orbitals
-
- Electrons are on different atoms and they
exchange their positions
32What is an accurate description of two-electron
bonding?
- Spin exchange between two atomic orbitals
-
- Electrons are on different atoms and they
exchange their positions
- Sometimes both electrons are on the same atom.
- There is some charge fluctuation. All orbitals
instantaneously rearrange in size and shape to
follow the charge fluctuation (orbitals
 breathe ). - This is differential dynamic correlation
33A Qualitative valence bond theory
Effective hamiltonian Heff (h(1) h(2) h(3)
.)
Parameters b, S, e
Similar model in the MO framework
34A Qualitative valence bond theory
Overlap between determinants
Generate permutations - between identical
spins - only one side
35A Qualitative valence bond theory
Hamiltonian matrix elements
Choice of an origin of energies
36A Qualitative valence bond theory
The two-electron bond
?a b ?
?b a ?
YVB
a,b pure AO gt purely covalent bond
a,b GVB pair gt fully correlated bond
2ßS
Reminder In MO theory De 2b/(1S)
De(2-e)
E(YVB )
1 S2
37A Qualitative valence bond theory
The triplet Pauli repulsion
?a b ?
?b a ?
YT
Reminder MO theory
-2ßS
E(YT )
1- S2
The same!
38Triplet Pauli repulsionwhy YMO ?YVB ?
? a b ?
YMO ??g ?u ?? ?(ab)(a-b) ? ?aa ?- ?ab ?
?ba ? - ?bb ? ? YVB ?
Whenever the MO and VB wave functions of an
electronic state are equivalent, the VB energy
can be estimated using qualitative MO theory
39Other cases where YMO ?YVB ?
The three-electron bond
Example the helium cation dimer He2
??a ?a ?b ?
??a ?b ?b ?
YMO ??g ?g ?u ??? ??a ?a ?b ??? ??a ?b ?b
????YVB
ß(1-3S)
Interaction energy De E(He\He) - E(He) -
E(He)
1- S2
40Elementary interaction energies in qualitative VB
vs MO theories
VB
MO
ß
ß
One-electron bond (A? B)
1 S
1 S
2ßS
2ß
Two-electron bond (A-B)
1 S2
1 S
ß(1-3S)
ß(1-3S)
Three-electron bond (A\B)
1- S2
1- S2
-4ßS
-4ßS
4-e repulsion (A?? ??????B)
1- S2
1- S2
-2ßS
-2ßS
Triplet repulsion (A? ?B)
1- S2
1- S2
41Thumb rule VB energy of a single determinant
-2nßS
Energy of a determinant with n (neighboring? ?????
1- S2
-2ßS/(1-S2)
triplet repulsion
(VB and MO)
(VB and MO)
-4ßS/(1-S2)
4-e repulsion
0
(VB only)
spin-alternated determinant
-2ßS/(1-S2)
(VB only)
3-e repulsion
bond single electron
-ßS/(1-S2)
(VB only)
bond bond
-ßS/(1-S2)
(VB only)
42The  failures of valence bond theory
1) Dioxygen molecule OO or ?OO? ?
OO, singlet state
?OO? , diradical displaying two 3-e
bonds, triplet state
43The  failures of valence bond theory
1) Dioxygen molecule OO or ?OO? ?
E(S)
E(T)
E(S) - E(T)
Qualitative VB predicts O2 to be a triplet
diradical Pauling, 1931 (!)
44The  failures of valence bond theory
2) The two ionization potentials of CH4
Two valence MO energies gt two IPs
Four equivalent local bonds gt only one IP ?
45The  failures of valence bond theory
2) The two ionization potentials of CH4
-E -b -b -b -b -E -b -b -b -b -E -b -b -b -b -E
0
Qualitative VB predicts CH4 to have two IPs
46Conclusions
- VB and MO two complementary theories.
- lower level MO too much ionic, VB not enough
- elaborate level MO ??VB
- There are nothing such as VB failures
- VB specific concepts
- Lewis structures, arrow pushing language,
- transferable local bonds, hybridization,
resonance energy - Applications to chemical reactivity
- Valence bond state crossing diagrams (next
lecture)
47Some reading
Valence Bond Theory. Its History, Fundamentals,
and Applications. A Primer. S. Shaik and P.C.
Hiberty, Reviews in Computational Chemistry 20,
1-100 (2004) A
conversation on VB vs MO Theory A Never-Ending
Rivalry? R. Hoffmann, S. Shaik and P.C. Hiberty,
Acc. Chem. Res. 36, 750-756 (2003)