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1
Breathing Orbital Valence Bond (BOVB)
A Valence Bond Method incorporating static an
dynamic clectron correlation effects
2
The Molecular Orbital model

• Example CH4

? 1s 1s j? j? j? j? j? j? j? j? ?
Y??
Electron pairs are in delocalized orbitals
3
The Valence Bond model
electron pairs are in local bonds
one bond an interaction between two singly
occupied atomic orbitals
The C-H bond is mainly covalent, but also has
some minor ionic character
The Valence bond wave function is the Quantum
Mechanical translation of the Lewis structure
4
Systems 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)
One can calculate the energy of a single VB
structure 1 or 2 (diabatic state). Application
Calculation of resonance stabilizations in
chemistry
RE E(1) - E(1?2)
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Systems that cannot be described by A single
Lewis structure

• Transition states of reactions

Example
A BC ???ABC? ????AB C
ABC? is a resonating mixture of two VB
structures
A // BC ? AB // C
Image of the reactants
Image of products

• The VB method can
• calculate the energy of each of these
• VB structures for any given geometry,
• - calculate the resonance energy

6
Application VB curve-crossing diagrams
The higher the G value, the higher the reaction
barrier
G is a property of the reactants
7
Covalent structures, ionic structures

• ExampleSN2 transition state

X H3CY ???XCH3Y ????XCH3 Y
XCH3Y

• Ionic vs covalent structures are clearly defined
• only if electrons are in atomic orbitals

8
What kind of VB method do we need?

• A compact wave function
• Not more than one VB function per Lewis
  structure

• A wave function which is clearly interpretable
• The VB functions must use pure atomic
  orbitals,
• so that they clearly correspond to one given
• Lewis structure

• Accuracy of the calculated energetics
• The VB methos must be able to accurately
• describe the elementary process of a reaction
• bond breaking or bond making
• Necessity of well taking electron correlation
• into account

9
Electron correlation in VB Theory

• Exemple the H2 molecule (Heitler-London, 1927)

Working hypothesis the electrons remain in
atomic orbitals
At equilibrium distance, 2 possible déterminants

?ja jb ?
?jb ja ?
Correct wave function (for the covalent bond)
?ja jb ?
?jb ja ?

YHL
10
Calculated dissociation energy curve
?ja jb ?
?jb ja ?

YHL ?
-20
-40
-60
-80
-100
Physical origin of the bond spin exchange
between AOs
11
Comparaison with MO description (Hartree-Fock)
YHF ??g ?g ?
??g ?g ?? ??a ?b ??? ??b ?a ??????? ??????a ?a
??? ??b ?b ??
HH
HH HH

• Simple MO description

??g ?g ?? 50 covalent 50 ionic

• Simple VB description

YHL 100 covalent

• Exact description

72-79 covalent 21-28 ionic
12
Dissociation curve OM (YHF) vs VB (YHL)
E

(kcal)
50 covalent 50 ionic
R
HH
-20
-40
Y
H
F
-60
100 covalent
Y
HL
-80
-100
Y
exact
Hartree-Fock does not dissociate well
13
Exact description

• In the VB framework

Yexact ????a ?b ??? ??b ?a ???? ?????a ?a ???
??b ?b ???
HH
HH HH

• In the MO framework

??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
14
The Generalized Valence Bond Method (GVB)
           

• At bonding distance

YGVB ?????????a ?b ??? ??b ?a ???? ??????a ?a
??? ??b ?b ???
HH
HH HH
YGVB is formally covalent, but physically
covalent-ionic optimized
15
Features of the various methods
Test case the description of F2
FF
Nature of the wave function for F2

• Hartree-Fock too much ionic

• VBSCF

FF FF
FF
Optimized covalent vs ionic coefficients

• GVB equivalent to VBSCF

16
Accuracy of the various methods
Test case the dissociation of F2
?E
FF F F

Calculation of ?E for F-F1.43Å, 6-31G(d) basis

• Hartree-Fock - 37 kcal/mol (repulsive!)
• Reason too much ionic

• Full configuration interaction (6-31-G(d) basis)
• 30-33 kcal/mol

• GVB, VBSCF

Only 15.7 kcal/mol
Reason ??
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What is wrong with GVB and VBSCF?

• They provide optimized covalent-ionic
• coefficients

• They greatly underestimate bond strength

GVB/VBSCF a closer examination
FF FF
FF

• The 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
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The  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

19
The  Breathing-Orbital  VB method (BOVB)
Test case the dissociation of F2
?E
FF F F

Calculation of ?E for F-F1.43Å, 6-31G(d) basis
Iteration De(kcal) FF FF ?
FF Classical VB -4.6 0.813 0.187 1 24.6 0.73
1 0.269 2 27.9 0.712 0.288 3 28.4 0.709 0.2
91 4 28.5 0.710 0.290 5 28.6 0.707 0.293
GVB 15.7 0.768 0.232
20
Improvements 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
21
Improvements of the BOVB method

• Improvement of the interactions between
• spectator orbitals

• Spectator orbitals can be

- local atomic orbitals
- bonding and antibonding combinations
Slightly better ( Delocalized  level or D)
22
The three levels of the BOVB method

• Basic L-BOVB

- All orbitals are localized, ionics are
closed-shell

• SL-BOVB

• All orbitals are localized, but active orbitals
  in
• ionics are split

• SD-BOVB

• Active orbitals are split in ionics
• Spectator orbitals are delocalized in all
  structures

23
Performances of the various BOVB levels
Test case the dissociation of F2
?E
FF F F

Method Req (Å) 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 Estim
ated full CI __ 30-33 Dunning-Huzinaga
DZP basis set SD-BOVB 1.443 31.6 Estimated
full CI 1.440.005 28-31 Experimental 1.41
2 38.3
24
Performances of the various BOVB levels
A polar molecule the dissociation of FH
?E
FH F H

Method Req (Å) De(kcal/mol) 6-31G
basis set GVB 0.920 113.4
L-BOVB 0.918 121.4 SL-BOVB 0.911 133.5
SD-BOVB 0.906 136.3 Bauschlicher Taylor
DZP basis set SD-BOVB 0.906 136.5 Full
CI 0.921 136.3 Experimental 0.917 14
1.1 Full CI and SD-BOVB dissociation curves are
indistinguishable within less than one kcal/mol
25
Electron correlation in BOVB

• Non-dynamic correlation

• Non dynamic correlation gives the correct
• ionic/covalent ratio for the bonds.

GVB (or valence CASSCF) has all the non
dynamic correlation

• Dynamic correlation

• All the rest. This is what is missing in GVB.
• BOVB brings that part of dynamic correlation
• that varies in the reaction

26
What is an accurate description of two-electron
bonding?

• Spin exchange between two atomic orbitals

• Electrons are on different atoms and they
• exchange their positions

• Charge fluctuation

• 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

27
Three-electron bonds (or (3e,2c) bonds)

• Example the cation dimer of NH3

H3NNH3
H3NNH3
Noted H3N?NH3

• Other examples
• Any molecule that has a lone pair can dimerize

, etc

• Very frequent in biochemistry the
  three-electron
• sulfur-sulfur linkage R2S?SR2

28
Electron correlation in (3e,2c) bonds
Example the helium cation dimer He2

• The MO picture

YMO ??g ?g ?u ??

• The VB picture

??a ?a ?b ?
??a ?b ?b ?

• Equivalence of qualitative MO and VB pictures

??g ?g ?u ??? ??a ?a ?b ??? ??a ?b ?b ????YVB
There is no left-right correlation in (3e,2c)
bonding Electron correlation is all dynamic
correlation
29
Importance of electron correlation in (3e,2c)
bonds
Test case the dissociation of 3e-bonded cations
?E
X?X X X
?E(kcal/mol)
Hartree-Fock MP2 MP4 He?He 34 39 43
H3N?NH3 24 40 39 H2O?OH2 23 46 44
HF?FH 20 48 45 Ne?Ne 9 39 37
H3P?PH3 19 27 27 H2S?SH2 20 30 30
HCl?ClH 17 30 29 Ar?Ar 11 24 24
Huge dynamic correlation!
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The BOVB description of three-electron bonds
Example F2 (F?F)

• Hartree-Fock or VBSCF

F F
F F
Same orbitals for both VB structures

• BOVB

All the bond is made of charge fluctuation A
large stabilization is expected
Question does the breathing orbital effect
bring all the dynamic correlation associated to
the bond?
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Performances of BOVB for 3-electron bonds
Test case the dissociation of the F2 anion

• ROHF or VBSCF

De - 4 kcal/mol (repulsive!)

• Breathing active orbitals

De 13.3 kcal/mol

• Fully Breathing (active and spectator orbitals)

De 28.0 kcal/mol (experiment 30.2)
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Some features of the BOVB method

• Compactness
• Only a few VB structures one per Lewis
  structure
• necessary to represent the electronic state

• Accuracy
• Yield reasonably accurate dissociation curves
• as compared to full CI levels in the same
  basis set

• Interpretability
• VB structures have a clear physical meaning,
• owing to the use of pure AOs

• Electron correlation
• gives non-dynamic correlation,
• and differential dynamic correlation, which
  closely
• corresponds to the  breathing orbital effect 

• Suitable for calculating diabatic states
• May calculate the energy of a single VB
  structure,
• or the energy curve of a VB structure along a
  reaction

33
An application of VB curve-crossing diagrams
Breaking of alkoxy radicals in the low atmosphere

• Importance for atmospheric pollution (cities)
• Endothermic reactions. DH depends on Ri
• The barrier DE? also depends an Ri
• DH and DE? are related in a strange way

34
Degradation of alkoxy radicals

• Prediction of the Hammond postulate

DE? should be a linear function of DH

• Experimentally, two different laws

• If one stabilizes the products by substituting
  R1
• DE? 1.20 DH cte (law 1 )

• If one stabilizes the products by substituting
  CO
• DE? 0.4 DH cte (law 2 )

35
VB curve-crossing diagram
36
If one substitutes R1

• DH decreases

• DE? decreases about as much as DH
• (law 1)

37
If one substitutes the carbonyl
DH P?P
H2CO ???? 97.6 MeHCO 13.7
107.8 Me2CO 10.4 114.9
DH decreases but the gap P ? P increases. DE?
does not decrease much law 2
38
Why does the gap P ? P increase upon carbonyl
substitution?
1
2

• Effect of R on the triplet state 1
• The doubly occupied p MO of R conjugates with
• the carbonyls p electrons

Repulsion is proportional to the number of
neighboring parallel spins. Conjugation does not
increase repulsion

• Effect of R on the non-bonding state 2

Conjugation increases repulsion
P is destabilized, P ? P increase
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Generalization the two relations DE? f(DH) for
alkoxy decomposition

• If one subtitutes R1 law 1

DE?
n
2
n
0
n
1
H
H
H
DH

• If one subtitutes the carbonyl law 2

m
e
t
h
y
l
R


1
p
r
o
p
y
l
R


1
e
t
h
y
l
R


1
l
)
DE?
i
-
p
r
o
p
y
l
R


1
t
-
b
u
t
y
l
R


1
DH
40
Acknowledgements
Prof. Wei Wu Kunming Dong Lingchun Song
Sason Shaik David Danovitch Avital Shurki