Valence Bond Theory

Beijing Normal University Summer School of Theoreticaland Computational Chemistry

Wei WuAugust 1, 2010

1. Introduction2. Ab initio Valence Bond Methods3. Applications4. Some available VB softwares

Quantum Chemistry

Molecular OrbitalTheory

Valence Bond Theory

Delocalized MO based Localized AO based

Roots of Valence Bond Theory

G. N. Lewis, 化学键的概念J. Am. Chem. Soc. 38, 762 (1916).The Atom and the Molecule

The paper predated the new quantum mechanics by 11 years, constitutes the first formulation of bonding in terms of the covalent-ionic classification.

G. N. Lewis

Zeits. für Physik. 44, 455 (1927).

Interaction Between Neutral Atoms and Homopolar

氢分子H2的量子力学处理

F. LondonW. Heitler

The Nature of the Chemical Bond, Cornell University Press, Ithaca New York,1939 (3rd Edition, 1960).

L. Pauling

History of Valence Bond Theory

A B A B A B

A B

1929 Slater 行列式方法

Phys. Rev. 34, 1293 (1929). The Theory of Complex Spectra.

1931 Slater 推广到n电子体系

Phys. Rev. 38, 1109 (1931). Molecular Energy Levels and Valence Bonds.

1932 Rumer 独立价键结构规则

Pauling和Slater 的多原子分子的量子化学理论

1931年 Pauling和Slater 杂化，共价－离子叠加，共振

Pauling 建立了量子力学与Lewis理论的关系

L. Pauling, The Nature of the Chemical Bond,Cornell University Press, Ithaca New York,

1939 (3rd Edition, 1960).

应用共价键－离子键讨论了任何分子体系的任何化学键－－共振论

价键理论是Lewis理论的量子理论形式

引言部分只引用Lewis的1916年的文章

Origins of MO Theory

1928年 Mulliken, Hund分子中电子的量子数与光谱

1929年 Lennard-Jones 分子轨道波函数（氧分子）指出价键理论处理氧分子的困难

1930年 Hückel-分离，C4H4, C8H8, 4n＋2规则,

Aromaticity and Antiaromaticity

MO

a b

H H• • • • • • • • • • • •

1S 1S

H H

1

2

11MO

VB

abba VB

2. Ab initio Valence Bond Methods

2.1 Evaluation of Hamiltonian Matrix

A many-electron wave function is expressed in terms of VB functions.

K KK

C

corresponds to a given VB structure

is given by solving secular equation

K

KC

( ) 0KL KL KK

H ES C | | , |KL K L KL K LH H S

H – H H- H+ H+ H-H2:

H2: H – H H- H+ H+ H-

baba

abba

21

21

)]2()1()2()1([2

1)]2()1()2()1([2

1cov

Heitler-London-Slater-Pauling (HLSP) Function

aaaaion

21)]2()1()2()1([

21)2()1(2

bbbbion

21)]2()1()2()1([

21)2()1(2

KK 0A

)()2()1( 210 NN

)()()]()()()([2

)]()()()([2

344321

122121

Np

K

kkkkkk

kkkk

In eq 4, the scheme of spin pairing (k1,k2), (k3,k4), etc, corresponds to the bond pairs that describe the structure K. Linearly independent pairing schemes may be selected by using the Rumer diagrams. A VB function with a Rumer spin function is called a Heitler-London-Slater-Pauling (HLSP)function.

Number of Independent VB Structures for Singly Occupied Configuration (Covalent Structures):

)!2/()!12/(!)12(

SNSNNSf

]1,2[][ 22/ SSN

Dimension of irreducible representation of symmetric group

Rumer Rule:

S = 0;S >0.

Young Tableaux of Symmetric Group:

The Total Number of Structures Including Ionic:

)!2/()!12/()!12/()!2/()!1()!1(

112

SNmSNSNmSNmm

mSf

Weyl Tableaux of Unitary Group

Dd KK

By expanding spin function in terms of elementary spin products, attaching the spatial factor, and antisymmetrizing, a VB function is expressed in terms of 2m determinants,

abc

def

1 | | | | | | | |

| | | | | | | |

abcdef abcdef abcdef abcdef

abcdef abcdef abcdef abcdef

abce

d

f2 (| | | | | | | |

| | | | | | | |)

adbcef adbcef adbcef adbcef

adbcef adbcef adbcef adbcef

C6H6

A VB function for a system with m covalent bonds is expressed in terms of 2m determinants.

For matrix element: 22m determinants

C2H6, N = 14, n = 7, 128 determinants for a VB function16384 determinants for a matrix element

General Cases

)())(( 22221111 mmmm babababababa

HC = EMC

KK

KC

LKKL HH

LKKLM

LKLL

KK CMCW

where Hamiltonian and overlap matrices are defined as follows:

VB structural weights

K KK

C K

KK DC

utsr

tursKLKL

rssr

KLrsLK SDggSDfDHD

utrsturs,,,

,,

)()()(||,,

Time scaling for a matrix element of determinants: N4

for one point: MN4+Nm4

Matrix elements in VB method

Löwdin, Phys. Rev. 97(1955) 1474.

Orbitals in Valence Bond Theory

OEO (overlap-enhanced orbital): delocalized freely. BDO (bond-distored orbital): delocalize over the two bonded centers. HAO (hybrid atomic orbital): strictly localized on a single center or fragment.

2.2 Valence Bond Self-consistent Field (VBSCF)

Valence Bond Self-consistent Field (VBSCF)

F F•••••

••

••••

••• F F••

••

••

••••

••F F••

••

••••

••C1 + C2 + C2•• ••••

F•—•F F– F+ F+ F–

0SCFVBSCFK

KKC

μii T

Numerical Algorithm: ( in XMVB program, 2006)

N2 matrix elements are required,Cost: N6+m4N

0EEcE ii cc

i

New Algorithm for Energy Gradient

KK

KC VB

KK

K DC

)...(ˆ21

KN

KKKD A

Ki

Ki c

A many-electron wave function

DK is built upon nonorthogonal localized orbitals

(J. Comput. Chem. 2009)

KK χT

Overlap matrix between the two orbital sets

LKKL STTV ~

KLKLM V

The overlap matrix element between the determinants

Defining a transition density matrix

LKLKKL TSTTTP ~)~( 1

))(21(

,,,,,

,

ggPPhPMH KLKLKL

KLKL

Hamiltonian matrix element

The first- and second-order Density Matrices

LK

LKL

KLK CPMCP,

LK

LKLKL

KLK CPPMC,

,

,,,

,,,

)(21 ggPPhPH KLKLKLN

KL

NKLKLKL HMH

Normalized Hamiltonian matrix element

KLKL GhF

,

,, )()( ggPPG KLKL

)(21 KLKLKLN

KL trtrH FPhP

Fock matrix

LKKLLK

LK

KLKLKLLK

LKKLLK

LKKLLK

MCC

trtrMCC

MCC

HCCE

,

,

,

,

2

)( FPhP

Variation and Gradients

]1[~)~(~)~(]1[ 11 KLLKLKLKLKKLKL SPTVTTVTSPP

iii '

The density matrix changes by

]~[]~[ LLKLKL

KKKLKLKL trMtrMM TYTY 1)( KLLK

KL VSTY 1)~( KLKLKL VSTY

]~)~()1[(]~)(]~)~1[( 11 LKLKKLKLKKLLKLKLNKL trtrH TVTFSPTVTFPS

The change in the ‘normalized’ Hamiltonian matrix element

The corresponding change in the overlap matrix element is

)~()~( LLKL

KKKLKL trtrH TZTZ

The change in Hamiltonian matrix element

1)(]~)~1([ KLLKLKLKLKL

KKL MSH VTFPSZ

1)~(])1([ KLKKLKLKLKL

LKL MSH VTFSPZ

KKLZ L

KLZ KLHKT LT

where and are the derivatives of with respect to the

andorbital coefficient matrices

The change in energy with respect to the variation of coefficients

LK

KLLK MCCM,

LK

KLLK HCCH,

}]~)[(]~)[({1,,

2 LK

LLKL

LKLLK

LK

KKKL

KKLLK HMTrCCHMTrCC

ME TYZTYZ

An efficient algorithm for energy gradients in VB theory is presented. the scaling for the evaluation of the first derivative of Hamiltonian matrix element is m4.

The new algorithm is especially efficient for the BOVB method.

Integral transformation is not required in the new algorithm.

Basically, the VBSCF method is quasi-equivalent to the CASSCF method, for a given dimension of the orbital space and if all the VB structures are considered.

VBSCF vs CASSCF

VBSCF Non-orthogonal localized AOs A few structures

CASSCF Orthogonal delocalized MOs Full configuration space

VBSCF provides qualitative correct description for bond breaking/forming, but its accuracy is still wanting.

VBSCF takes care of the static correlation, but lacks dynamic correlation.

2.3 Breathing Orbital Valence Bond (BOVB)

Breathing Orbital Valence Bond (BOVB)

F F•••••

••

••••

••• F F••

••

••

•• ••••F F••••

••••

••C1 + C2 + C2•• ••••

• Different orbital sets for different VB structures

F•—•F F– F+ F+ F–

Levels: L-BOVB; D-BOVB; SL-BOVB; SD-BOVB.

Hiberty, et al. Chem. Phys. Lett. 1992, 189, 259.

Levels of BOVB method:

L-BOVB: Localized AOs;

D-BOVB: Delocalized AOs for inactive electrons;

SL-BOVB: Splitting doubly active orbitals + L-BOVB;

SD-BOVB: Splitting doubly active orbitals + D-BOVB.

2.4 Valence Bond Configuration Interaction (VBCI) Method

In MO theory,

post Hartree-Fock methods, such as CI, MP2, and CCSD,are efficient tools for computing dynamic correlation.

Can we have post-VBSCF method?

Is it possible to use CI technique in VB theory?

Wave function in VB method should

• correspond to the concept of VB structure(strictly localized orbitals)

• be compact (only a few structures)

How to define localized VB orbitals?

A: atom or fragment

1 2 1 2 1 2{ } { , , , ; , , , ; , , ; }A B C

A A A B B B C C Cm m m

Localized occupied VB orbitals

A A Ai ic

Occupied VB orbitals are obtained by VBSCF calculations

Virtual orbitals may be defined by a projector:

1( )A A A A AP C M C S

CA: vector of occupied orbital coefficientsMA: overlap matrix of occupied VB orbitalsSA: overlap matrix of basis functions

It can be shown that

The eigenvalues of the projector are 1 and 0;

Eigenvectors associated with eigenvalue 1 is the occupied VB orbitals;

Eigenvectors associated with eigenvalue 0 may be used as the virtual VB orbitals.

Two important features:

Strictly localized;

Orthogonal to the occupized VB orbitals.

By diagonalizing the projectors for all blocks, we have allvirtual orbitals.

An excited VB structure iK is built from by

replacing occupied Aj with virtual orbital .A

iK and describe the same classical VB structure.

iK

iKi

CIK C '

By collecting all , we have a wave function corresponding to VB structure K.

iK

Corresponding to a VB structure.

Excited VB structures

K

SCFK

SCFK

VBSCF C

SCFK

SCFK

VBCI CI CIK K

Ki

Ki KK i

C

C

LK ji

jL

iKLjKi

LK ji

jL

iKLjKi

VBCI

CC

HCCE

, ,

, ,

K

SCFK

SCFK

VBSCF C

jL

iKLj

jiKi

CIKL HCCH

,

jL

iKLj

jiKi

CIKL CCM

,

i

KiK WW

,

i jKi Ki K L Lj

L j

W C C

All formulas are similar to those of VBSCF, and compact.

Levels of VBCI Method

VBCI(A,I), A= D, S; I = D, S

A = Active electrons that are involved in a chemical process I = Inactive electrons that are NOT involved in …

VBCI(D,D) = VBCISDVBCI(S,S) = VBCIS

VBCI(D,S)The “inactive” electrons play an indirect role in a chemical process, and the dynamic correlation of inactive electrons is quasi constant during the process.

VBCI Method with Perturbation Theory

iK

L L

iKL

SCFL

iKL

SCFL

iK EE

MCEHCE

0

20

00

The energy contribution of an excited structure is estimated by

iK

E < critical value iK is discarded in CI procedure

E > critical value iK is involved in CI procedure

VBCIPT Method

Davidson Correction of VBCISD

Size inconsistency problem is one of the most undesirable drawbacks in truncated CI methods.

DK

KQ EWE )1(

to estimate the contribution of quadruple excitations that are product of double excitations

58.040.2(522)

42.1(1089)

38.9(81)

35.626.241.614.5Cl2

38.030.9(507)

33.9(1089)

40.4(81)

31.510.928.3-33.1F2

101.297.9(189)

98.0(274)

92.0(40)

89.985.899.177.6HCl

137.2126.1(206)

126.0(274)

125.0(40)

115.9105.1127.294.9HF

56.649.0(49)

49.6(171)

42.8(27)

43.042.449.532.5LiH

104.2105.9(27)

105.9(55)

96.0(11)

96.095.8105.984.6H2

Exp.cVBCIPTVBCISDVBCISBOVBVBSCFCCSDHFMol.

Table 1 Bond energies with various methods (kcal/mol)

2.5 Valence Bond Second-Order Perturbation Theory (VBPT2) Method

Though the VBCI space is much smaller than those of MO-based methods. VBCI method is computational demanding.

Can we have a VB method that is accurate and cheap?

Valence Bond Second Order Perturbation Theory

SCFSCFSCFSCFSCF E CMCH

Orbitals:

Inactive. doubly occupied in VBSCF

Active. Occupied in VBSCF

Virtual.

K

SCFK

SCFK

VBSCF C

Chen; Song; Hiberty; Sason; Wu, J. Phys. Chem. A, accepted.

iii

SCF )( 1' STM

Inactive and virual orbitals are orthogonal,

Active orbitals are nonorthogonal mutually, but are orthogonal to the inactive and virtual ones.

Such definition of orbitals keeps VBSCF energy unchanged.

Excited VB structures:Excited structures are generated from the VBSCF

structures by replacing occupied orbitals with virtual orbitals.

The zeroth-order Hamilton:

SDSDKK PFPPFPPFPH ˆˆˆˆˆˆˆˆˆˆ000

i

ifF )(ˆˆ

nm

nmnmSCFmn iKiJDihif

,))(ˆ)(ˆ()(ˆ)(ˆ

nm

SCFmnpqpq qnpmmnpqDhf

, 21

The first-order wave function:)1()0(

KK

SCFK

SCF C)0(

SDVR

RRC )1()1(

0)1()0(

)0(10111)0(110

)1( )( CHMHC E

The second-order energy:

)0(10111)0(110

01)0()2( )( CHMHHC EE

1)0(

The most time-consuming part:

111)0(110 )( MH E

which is block diagonalized, due to the block-orthogonalitybetween different orbital blocks.

If the occupation numbers of inactive or virtual orbitalsare different in the two excited structures, the corresponding matrix element is zero.

VBPT2 is much cheaper than VBCI.

The structure weights in VBPT2 method:

)( R

RRKKKPTK XN

K

PTK

PTK

VBPT C

KKPTK NCC /)0( 2/1

,

11 )1( SR

SKRSRKK XMXN

L

PTL

PTKL

PTK

PTK CMCW

10

,

111)0(110

01)2( )( SLSR

RSKRKL HEHE MH

)2(KL

SCFKL

PTKL EHH

Example 1. The Spectroscopic Constants of H2

a. J. Phys. Chem., 1990, 94, 5483., where ANO(4s3p2d) basis set was used and orbitals 1σg and 1σu are taken as active orbitals.

Method re (a.u.) ωe (cm-1) De (eV)

FCI 1.405 4421 4.707

VBSCF 1.429 4193 4.121

VBPT2 1.408 4376 4.609

VBCISD 1.405 4421 4.707

CASSCFa 1.427 4255 4.14

CASPT2Na 1.410 4407 4.57

Example 2. The Spectroscopic Constants of O2 Ground State

a. J. Chem. Phys. 86, 5595 (1987).b. 12 fundamental VB structures are used.c. 105 fundamental structures are used, butthe orbitals are optimized

use 17 VB structures.d. Three orbital block according to σ, πx, πy are used.

e J. Compt. Chem. 28, 185 (2007),

where cc-pVTZ basis set are used.f. J. Chem. Phys. 96, 1218 (1992).

Method re (a.u.) ωe (cm-1) De (eV)

FCIa 2.318 1608 4.637

VBSCF(12)b 2.368 1580 2.999

VBPT2(12)b 2.333 1560 4.327

VBSCF(105)c 2.368 1581 3.045

VBPT2(105)c 2.324 1601 4.661

VBCIS(12)b ,d 2.321 1578 4.582

VBCISD(12)b, d 2.333 1594 4.154

VBCISDe 2.336 1545 4.77

CASSCFf 2.322 1566 3.678

CASPT2Nf 2.317 1607 4.658

Example 3. The Spectroscopic Constants of N2 Ground State

a. J. Chem. Phys. 86, 5595

(1987).

b. 17 fundamental VB structures

are used.

c. 175 fundamental VB structures

are used, but

the orbitals are optimized use 17

VB structures.

d. J. Chem. Phys. 96, 1218

(1992).

Method re (a.u.) ωe (cm-1) De (eV)

FCIa 2.123 2341 8.748

VBSCF(17)b 2.109 2388 8.086

VBPT2(17)b 2.115 2373 8.421

VBSCF(175)c 2.114 2364 8.190

VBPT2(175)c 2.120 2344 8.573

VBCIS(17)b 2.116 2348 8.287

VBCISD(17)b 2.121 2330 8.651

CASSCFd 2.119 2337 8.333

CASPT2Nd 2.122 2342 8.621

Example 4. The Barrier of Hydrogen Exchange Reaction

Method E(H3) (a.u.) E(H2+H) (a.u.) Barrier (kcal/mol)

VBSCF -1.61804 -1.65081 20.6

VBPT2 -1.65175 -1.66885 10.7

L-BOVB 1.63485 -1.65115 10.2

VBCISD -1.65655 -1.67246 10.0

CCSD(T) -1.65689 -1.67246 9.8

Example 5. Size Consistency Error

a. RNN=50a0, ROO=100a0, RFF=100Å.

Moleculea E(A2) 2E(A) ∆E(size) (mh)

2N -108.828718 -108.828718 <0.1*10-2

2O -149.697247 -159.697240 0.7*10-2

2F -199.199010 -199.199003 0.7*10-2

VBPT2-calculated energies (hartrees) of some supersystems of two

distant atoms as compared with the summed energies of the separate

atoms.

Molecule a E(A2) 2E(A) ∆E(size)

2N -108.828718 -108.828718 <0.1*10-5

2O -149.697247 -149.697240 0.7*10-5

2F -199.199010 -199.199003 0.7*10-5

a. RNN=50a0, ROO=100a0, RFF=100Å.

Size consistency

Summary

VBPT2 method provides a cheap VB tool, which is able to cover dynamical correlation. Test calculation shows thatVBPT2 is in good agreement with CASPT2.

Add-Ons: VB Methods for Solution Phase

VB wave function/density + Solvation model

2.9 Generlaized Valence Bond (GVB) Method

Goddard, te al

Strong Orthogonality (SO):

Perfect Paring (PP):

OEOs are used.

Advantages of GVB Method:Computer time-saving. MCSCF.

2.10 Spin-couple Valence Bond (SCVB) Method

Gerratt and coworkers

K

KKC 0SCVB

K

KKC 0SCVB

)()2()1( 210 NN

)()()]()()()([2

)]()()()([2

344321

122121

Np

K

kkkkkk

kkkk

Non-orthogonal OEOs are used in SCVB.

3. Applications

3.1 Accuracy of Modern VB Methods

Table 1. Bond dissociation energies calculated with valence bond methods, from Ref. 83.

De (kcal/mol) Bond Basis set BOVB VBCISDa CCSD(T) Expt F-F 6-31G* 36.2 32.3 32.8 cc-pVTZ 37.9 36.1 34.8 38.3 Cl-Cl 6-31G* 40.0 41.6 40.5 cc-pVTZ 50.0 56.1 52.1 58.0 Br-Br 6-31G* 41.3 44.1 41.2 cc-pVTZ 44.0 50.0 48.0 45.9 F-Cl 6-31G* 47.9 49.3 50.2 cc-pVTZ 53.6 58.8 55.0 60.2 H-H 6-31G** 105.4 105.4 105.9 109.6 Li-Li 6-31G* 20.9 21.2 21.1 24.4 H3C-H 6-31G** 105.7 113.6 109.9 112.3 H3C-CH3 6-31G* 94.7 90.0 95.6 96.7 HO-OH 6-31G* 50.8 49.8 48.1 53.9 H2N-NH2 6-31G* 68.5 70.5 66.5 75.4 ± 3 H3Si-H 6-31G** 93.6 90.2 91.8 97.6 ± 3 H3Si-F 6-31G* 140.4 b 151.1 142.6 160 ± 7 H3Si-Cl 6-31G* 102.1 101.2 98.1 113.7 ± 4

Table 2. Barriers for the hydrogen exchange reactions, X• + HX XH + X’• (X = CH3,

SiH3, GeH3, SnH3, PbH3 and H). Energies in kcal/mol.

Molecule HF CCSD VBSCF BOVB VBCISD VBCIPT

CH3a 35.1 26.5 33.0 23.1 25.8 25.5

SiH3a 25.2 19.3 25.5 19.1 19.7 19.0

GeH3a 22.0 16.6 25.5 18.0 18.1 17.0

SnH3a 18.5 13.5 20.5 14.9 15.3 14.1

PbH3a 15.2 13.0 17.3 12.3 12.5 11.5

Hb 9.8c 20.6 10.2 10.0 a 6-31G* basis set. Ref. 20 for columns 1-4, Ref. 19 for columns 5 and 6. b Aug-cc-pVTZ basis set. Ref. 44. c CCSD(T) calculation.

3.2 Chemical Reactivity

Two fundamental questions that any model of chemical reactivity would have to answer:

What are the origins of the barriers?

What are the factors that determine reaction mechanisms?

GrGp

B

Reaction Coordinate

R

R* P*

P

r p

E rp

E

Figure 1: VBSCD for a general reaction R P. R and P are ground states of reactants and products, R* and P* are promoted excited states.

BfGE r

Suppose that the diabatic profileIs a parabola, as

BGEEGGGfE 0

2rprp0p00 5.02

E f0G0 0.5Erp B

GrGp

B

Reaction Coordinate

R

R* P*

P

r p

E rp

E

neglecting the quadratic term and taking Gp/2Ga as ~1/2,

3.2.1. Hydrogen Abstraction Reactions

X• + HX’ -> XH + X’• (X = X’ = CH3, SiH3, GeH3, SnH3, PbH3)

Identity Reaction (X=Y)

VB structures (3 electrons/3 orbitals system)

(X = X’ = CH3, SiH3, GeH3, SnH3, PbH3)

The trend of weights follows the electronegativity.

Eqs. 21 and 22 capture the key factors controlling barrier.

HX2r DG

HX5.0 DB

HX5.02 DfE

Nonidentity Hydrogen Abstraction ReactionsJ. Phys. Chem. 2002, 106, 8226.

Modeling is more difficult, asNo symmetry. ( two sets of VB parameters);Driving force. (energy difference of reactant and product)

Valence Bond Structures

Avoided Crossing State (ACS)

Computed VBSCD for X=C, X’=Si.

ACS is a reasonably good approximation to the TS.So we may use the ACS as an approximation to the TS.

It seems that the resonance energy of the TS is determined by the weak bond.

The B (eq 18) values are lower somewhat than the B(VB) values,by 2.1 kcal/mol. The B (eq 12) values are closer to theB(VB) values. Because eq 12 is much simpler, it should be thechoice expression whenever eq 18 cannot be applied.

(12)

Eq. 29 contains a balance between an intrinsic term, faGa, and the reaction “driving force” term, Erp.

A simple expression derived from VBSCD model reproduces the VB barriers quite well.

A qualitative model can be coupled to a complex computational scheme to reproduce all the trends and show their dependence on fundamental properties of reactant and products.

Are the expressions valid for other hydrogen transfer reactions?

J. Am. Chem. Soc. 2004, 126, 13539.

H-H + X• H• + H-X (X = F, Cl, Br, I)

B = 0.5DW

3.3 New Concepts in Chemical Bonding: Charge-Shift Bonding

Application

The classical paradigm of the A—X bond

A-X = C1(A•–•X) + C2(A+ :X–) + C3(A:– X+)

A•–•X A+ :X– A:– X+

covalent ionic ionic

A X A X A X

-1.200

-1.150

-1.100

-1.050

-1.000

-0.950

0.5 1.0 1.5 2.0 2.5 3.0

Exact

Covalent

E(au)

R(Å)-0.780

-0.760

-0.740

-0.720

-0.700

-0.680

-0.660

-0.640

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

E(au)

R(Å)Exact

Covalent

It follows that the F-F bond owes its existence to the covalent-ionicfluctuation of the electron-pair even though its static charge is zero.

F-F is a “covalent bond” of a special type: a “charge-shift bond”

RECS

Dissociation energy curves

H2 → H· + H· F2 → F· + F·

Valence Bond TheoryCS Bond: Resonance energy dominates the bonding

energy.

RE/BDE > 50%

Digging into the literature…

X X

Separate atoms

F–F, Cl–Cl, …Deficit of densityin the bondingregion

H–H, H3C–CH3 …Density build-upin the bondingregion

Real phenomenon or VB artifact?

Other signs (not VB) that charge-shift bonds are special

H-H

H3C-CH3

H2N-NH2

HO-OH

F-F

Cl-Cl

Na-Cl

-1.39

-0.62

-0.54

-0.02

+0.58

+0.01

+0.18

2

0.27

0.25

0.29

0.26

0.25

0.14

0.03

Covalent

bonds

Charge-shift

bonds

Ionic bond

9.2

27.7

43.8

56.9

62.2

48.7

8.1

RE (kcal/mol)

and at the bond critical point in AIM theory2

Zhang; Ying; Wu; Hiberty; Shaik, Chem. Eur. J. 2009,15,2979.

Valence Bond TheoryCS Bond: Resonance energy dominates the bonding

energy.

RE/BDE > 50%

AIM TheoryCS Bond: Laplacian is positive or close to zero;

density is large.

The “inverted” bond in [1,1,1]propellane:a charge-shift bond

Wu; Gu; Song; Shaik; Hiberty, Angew. Chem. Int. Ed. 2009, 48, 1407.

H2C CH2

H2C

The problem of “inverted bonds” in propellanes

H2C CH2

H2C

H

H

H2C CH2

H2C

-2 H•

[1.1.1] propellane

∆E(S-T) = 109 kcal/mol

=> not a diradical

H2C CH2

H2C

2 = +10.3

2 = -13.0

2

- Very weak electron densitybetween the carbons

- Positive at bond critical point

- extra stability of 65 kcal/mol

The three features characterize charge-shift bonding

What kind of bond is it?

577211

1.60Å 1.8Å

E(kcal/mol)

RC-C(Å)ground state

covalent

Valence bond calculations (BOVB)

C

CH2

C

H2C

CH2

C

H2C

C

H2C

CH2

The covalent curve is repulsiveThe resonance energy is huge

C

C

C

C

C

A typical charge-shiftbond

Table 1. Computed Valence Bond Features for C-C Bonds of 1-13

Entrya Moleculeb Din-situc cov

d REcov-ionc RE/Din-situ

1 C2H6 131.1 0.694 28.5 0.217

2 C3H6 138.8 0.686 40.7 0.293

3 C4H6 140.4 0.674 50.0 0.356

4 [1.1.1]-(CH2)3 123.1 0.672 70.2 0.570

5 [1.1.1]-(NH)3 122.6 0.668 81.0 0.661

6 [1.1.1]-O3 105.7 0.684 89.6 0.848

7 [1.1.1]-(BH)3 65.2 0.821 43.4 0.666

8 [1.1.1]-(CO)3 82.8 0.769 55.1 0.665

9 [1.1.1]-(CF2)3 103.1 0.714 66.7 0.647

10 [1.1.1]-(OH)33+ 58.4 0.870 52.2 0.894

11 [2.1.1] 106.3 0.704 65.3 0.614

12 [2.2.1] 130.8 0.689 57.8 0.442

13 [2.2.2] 137.1 0.693 43.8 0.319

C CH3C C HHC CH2 CC

C

C

CH2

H2

H3 H2

4 (X=CH2)1 2 3

CN

C

N

NH

H

5 (X=NH)

H CO

C

O

O

6 (X=O)

CB

C

B

BH

H

7 (X=BH)

H CCC

C

C

8 (X=CO)

H2

O

OO

CC

CC

CF2

F2

9 (X=CF2)

F2CO

CO

OH

H

10 (X=OH+)

H +

+

+C C CH2

C

C

C

H2

11 (1.1.2)

CC

CH2

12 (1.2.2)

13 (2.2.2)

CC

14 (2.2.3)

CC

15 (2.3.3)

CC

16 (3.3.3)

L=-0.557G=0.056H=-0.250

L=-0.435G=0.088H=-0.284

L=-0.246G=0.130H=-0.320

L=0.068G=0.154H=-0.137

L=0.004G=0.167H=-0.166

L=-0.042G=0.176H=-0.186

L=0.261G=0.106H=-0.041

L=0.314G=0.126H=-0.047

L=0.191G=0.139H=-0.091

L=0.250G=0.147H=-0.085

L=0.013G=0.123H=-0.120

L=-0.301G=0.099H=-0.175

L=-0.649G=0.069H=-0.232

L=-0.567G=0.063H=-0.205

L=-0.513G=0.059H=-0.187

L=-0.472G=0.055H=-0.173

Shaik; Chen; Wu; Stanger; Danovich; Hiberty, ChemPhysChem, 2009, 10, 2058.

3.4 Direct Estimate of Hyperconjugation Energies

The Origin of Rotation Barrier in Ethane

))/12(/9.2 molkJmolkcalE

Origin of Barrier?

Steric Repulsion Model

L. Pauling, The Natrue of Chemical Bond, 3rd, 1960

M. Karplus, J. Chem. Phys. 1968,49, 2592.

Hyerconjugation Model

F. Weinhild, J. Am. Chem. Soc. 1979, 101, 1700; Angew. Chem. Int. Ed. 2003, 42, 4188.

L. Goodman, Nature, 2001,411, 565.E. J. Baerends, Angew. Chem.

Int. Ed., 2003, 42, 4183.

MO Method：

Delocalized MOs

Localized MOs

Minimum of Energy

Lower energy？

Possibility：Overestimate delocalization energy ?

VB method：Localized AOs

Orbtial transformation

optimize

Minimum of Energy

optimize

1

2

1

2

1'

hc del locE E E

eclipsed staggeredhc hc hcE E E

barrier hc sE E E

eclipsed staggereds loc locE E E

Ab initio VB: 14e/7 bonds/6-311G**

Energy analyses with the ab initio VB method and 6-31G(d)

0.912.711.80 (kcal/mol)

-10.30-79.33379-79.31737Eclipsed

-11.21-79.33811-79.32024Staggered

Ehc (kcal/mol)Edel (a.u.)Eloc (a.u.)

The hyperconjugation effect does favor the staggered structure but accounts for only around one-third of the rotation barrier, most of which comes from the sterichindrance.

The hyperconjugation effect does favor the staggered structure but accounts for only around one-third of the rotation barrier, most of which comes from the steric hindrance.

Mo, Wu, et al. Angew. Chem. Int. Ed. 43, 1986 (2004).

Figure 1.Comparison of energy profiles (energy E versus dihedral angle φ) for the ethane rotation.

3.5 VBSCF Applications to Aromaticity

Cyclopropane, Theoretical Study of σ-aromaticity

C C

CH

H

H

H

H

H

C C

C

aromaticity in C3H6 aromaticity in C3H6

πdel

σlocC

σloc A

πloc

σdelC

πloc A

σloc

σloc

σRE

σloc

πloc

πRE

ARE(Ref)CMARECMECRE

Table 8. Extra cyclic resonance energies (ECRE, in kcal/mol) for cyclopropane (C3H6), and cyclobutane (C4H8) and trisilacycloproane(Si3H6) with the basis sets of 6-31G(d) and cc-pVDZ .

X = C X = Si

Species 6-31G* cc-pVDZ 6-31G* cc-pVDZ

ECRE1σ (kcal/mol) 4.8 3.5 8.0 6.3

ECRE1π (kcal/mol) 1.8 1.8 -0.1 0.4

ECRE1σ+π (kcal/mol) 6.5 5.4 7.9 6.8

ECRE2σ (kcal/mol) 1.1 -0.7 6.2 4.2

ECRE2π (kcal/mol) -1.8 -2.7 -0.7 -0.3

X3H6

ECRE2σ+π (kcal/mol) -0.6 -3.2 5.5 3.9

ECRE1σ (kcal/mol) 2.0 1.6

ECRE1π (kcal/mol) -0.1 -0.2

ECRE1σ+π (kcal/mol) 2.0 1.6

ECRE2σ (kcal/mol) -1.6 -2.5

ECRE2π (kcal/mol) -3.6 -4.6

X4H8

ECRE2σ+π (kcal/mol) -5.2 -6.9

Basis set: 6-311+G** ECRE2σ(C3H6)= -1.5 kcal/mol

The extra s-stabilization energy (at most 3.5 kcal/mol) is far too small to explain the small difference in strain energy between cyclopropane (27.5 kcal/mol) and cyclobutane (26.5 kcal/mol) by σ-aromaticity. Thus, there is no need to invoke σ-aromaticity for cyclopropaneenergetically.

4. Some available VB softwares

The XMVB Program The TURTLE Software The VB2000 Software The CRUNCH Software

XMVB: An ab initio NonorthogonalValence Bond Program

Version 1.0

Lingchun Song, Yirong Mo, Qianer Zhang, Wei Wu*

Center for Theoretical Chemistry, State Key Laboratory for Physical Chemistry of Solid Surfaces, and Department of Chemistry,

Xiamen University, Xiamen, Fujian 361005, CHINAweiwu@xmu.edu.cn

Song; Mo; Zhang; Wu, J. Comp. Chem. 2005, 26, 514.

XMVB-G03

XMVB-GMS

XMVB

VB methods implemented in XMVB

• Hartree-Fock Method• VBSCF, BOVB, VBCI, LVB, VBPT2• VBPCM with GAMESS• VBSM with SM6-8• Total Energy, Energy for Individual Structure, Dipole Moments, Weights.Plot VB orbitals

The TURTLE Software

TURTLE is also designed to perform multistructure VB calculations and can execute calculations of the VBSCF, SCVB, BLW or BOVB types. Currently, TURTLE involves analytical gradients to optimize the energies of individual VB structures or multistructure electronic states with respect to the nuclear coordinates. TURTLE is now implemented in the GAMESS-UK program.

Verbeek, J.; Langenberg, J. H.; Byrman, C. P.; Dijkstra, F.; van Lenthe, J. H. TURTLE-A gradient VB/VBSCF program (1998-2004); Theoretical Chemistry Group, Utrecht University, Utrecht.

The VB2000 Software

VB2000178 is an ab initio VB package that can be used for performing non-orthogonal CI, multi-structure VB with optimized orbitals, as well as SCVB, GVB, VBSCF and

BOVB. VB2000 can be used as a plug-in module for GAMESS(US) and Gaussian98/03 so that some of the

functionalities of GAMESS and Gaussian can be used for calculating VB wave functions. GAMESS also provides

interface (option) for the access of VB2000 module.

Li, J.; Duke, B.; McWeeny, R.; VB2000,Version 1.8; SciNet Technologies: San Diego, CA, 2005.

The CRUNCH Software

The CRUNCH (Computational Resource for Understanding Chemistry) has been written originally in Fortran by Gallup, and recently translated into C. This program can perform multiconfiguration VB calculations with fixed orbitals, plus a number of MO-based calculations like RHF, ROHF, UHF (followed by MP2), Orthogonal CI and MCSCF.

Gallup, G. A. Valence Bond Methods; Cambridge University Press: Cambridge, 2002.

"A Chemist's Guide to Valence Bond Theory", by S. Shaik and P.C. Hiberty,Wiley, 2007.

Thanks !