Valence Bond Theory

Beijing Normal University Summer School of Theoretical and Computational Chemistry

Valence Bond Theory Wei Wu August 1, 2010

1. 2. 3. 4.

Introduction Ab initio Valence Bond Methods Applications Some available VB softwares

Quantum Chemistry

Molecular Orbital Theory

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

氢分子H2的量子力学处理

F. London

W. Heitler Zeits. für Physik. 44, 455 (1927).

Interaction Between Neutral Atoms and Homopolar

History of Valence Bond Theory A

B

A

B

A

B

A

B

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

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). 引言部分只引用Lewis的1916年的文章

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

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

2

H

VB H

H

• •• • • •• • • •• •

H

1S

1S

1 MO  11

a

b

VB   ab  b a

2. Ab initio Valence Bond Methods

2.1 Evaluation of Hamiltonian Matrix

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

 

C

K

K

K

 K corresponds to a given VB structure H2:

H–H

H- H+

H+ H-

CK is given by solving secular equation

 (H

KL

 ES KL )CK  0

K

H KL   K | H |  L ,

S KL   K |  L 

Heitler-London-Slater-Pauling (HLSP) Function H2:

H–H

H- H+

H+ H-

1 1 [ a (1)b(2)  b(1)a ( 2)] [ (1) (2)   (1) (2)] 2 2 1 1  ab  a b 2 2

 cov 

2

ion

 a (1)a (2)

1 1 [ (1) (2)   (1) (2)]  aa 2 2

2

ion

 b(1)b(2)

1 1 [ (1) (2)   (1) (2)]  bb 2 2

ˆ K  A 0 K  0  1 (1) 2 (2)  N ( N )  K  2 1 2 [ (k1 )  (k 2 )   (k 2 ) (k1 )]  2 1 2 [ (k3 )  (k 4 )   (k 4 ) (k3 )] (k p )  (k N ) 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 S  1) N ! f  ( N / 2  S  1)!( N / 2  S )! Dimension of irreducible representation of symmetric group

[ ]  [2 N / 2 S ,12 S ]

Rumer Rule: S = 0; S >0.

Young Tableaux of Symmetric Group:

The Total Number of Structures Including Ionic: f   

2S  1 (m  1)!(m  1)! m  1 ( N / 2  S )!(m  N / 2  S  1)!( N / 2  S  1)!(m  N / 2  S )!

Weyl Tableaux of Unitary Group

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,

 K   dK D 

C6 H6 a f

b

e

c d

1 | abcdef |  | abcdef |  | abcdef |  | abcdef |  | abcdef |  | abcdef |  | abcdef |  | abcdef |

a b c

f e d

 2  (| adbcef |  | adbcef |  | adbcef |  | adbcef |  | adbcef |  | adbcef |  | adbcef |  | adbcef |)

General Cases

  ( a1b1  a1b1 )( a 2 b2  a 2 b2 )( a m bm  a m bm ) 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 function 16384 determinants for a matrix element

   CK  K K

HC = EMC where Hamiltonian and overlap matrices are defined as follows: H KL   K H  L

M KL   K  L

VB structural weights WK   C K M KL C L L

Matrix elements in VB method  

C

K

K

K

   C K DK K

DK | H | DL   f rsKL D( S rs )  r ,s

KL KL ( g  g  rs ,tu rs ,ut ) D(Srs,tu )

r , s , t ,u

Löwdin, Phys. Rev. 97(1955) 1474.

Time scaling for a matrix element of determinants: N4 for one point: MN4+Nm4

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)  VBSCF   C KSCF  0K K

C1

••

••

F

••

•• F

•• •• F•—•F

••

+ C2

••

••

F

••

••

•• F

••

F– F+

i   Tμi   

••

+ C2

••

••

F

••

••

•• F

•• F+ F–

••

Numerical Algorithm: ( in XMVB program, 2006)

Eci ci    E0 E  ci 

N2 matrix elements are required, Cost: N6+m4N

New Algorithm for Energy Gradient (J. Comput. Chem. 2009) A many-electron wave function

   C KVB  K K

   C K DK K

DK is built upon nonorthogonal localized orbitals

ˆ ( K  K ... K ) DK  A 1 2 N

iK   cKi   

 K  χT K Overlap matrix between the two orbital sets

~ V KL  T K ST L The overlap matrix element between the determinants

M KL  V KL Defining a transition density matrix

P

KL

~ L K 1 ~ L  T ( T ST ) T K

Hamiltonian matrix element H KL  M KL (  PKL h   ,

1 PKL PKL ( g ,  g , ))  2  , , ,

The first- and second-order Density Matrices

P   C K M KL PKLC L

  ,   C K M KL PKL PKLC L K ,L

K ,L

Normalized Hamiltonian matrix element N H KL   PKL h 

 ,

1 KL KL P   P ( g ,  g , ) 2  , , ,

N H KL  M KL H KL

Fock matrix KL G ( P )   PKL ( g  ,  g  , )

F KL  h  G KL N H KL

C E C

K

C L H KL

K ,L

K

K ,L

 ,

1  (trP KLh  trP KL F KL ) 2

C L M KL



C K ,L

K

C L M KL (trP KL h  trP KL F) 2 C K C L M KL K ,L

Variation and Gradients i '  i  i The density matrix changes by

~

~

~

~

P KL  [1  P KLS]T K ( V KL ) 1 T L  T K ( V KL ) 1T L [1  SP KL ] The corresponding change in the overlap matrix element is

~

~

M KL  M KLtr[ YKLK T K ]  M KLtr[ YKLL T L ] YKLK  ST L ( V KL ) 1

~ YKLL  ST K ( V KL ) 1

The change in the ‘normalized’ Hamiltonian matrix element ~

~

~

~

~

H KLN  tr[(1  SP KL )F KL ]T L ( V KL ) 1T K ]  tr[(1  SP KL )F KL T K ( V KL ) 1T L ]

The change in Hamiltonian matrix element ~

~

L H KL  tr ( Z KKLT K )  tr ( Z KL T L )

L where Z KKL and Z KL are the derivatives of H KL

orbital coefficient matrices

with respect to the

TK and TL

~ ~ Z KKL  [ H KL S  M KL (1  SP KL )F KL ]T L ( V KL ) 1

~ L Z KL  [ H KL S  M KL (1  SP KL )F KL ]T K ( V KL ) 1 The change in energy with respect to the variation of coefficients

E 

1 ~K ~L K K L L { C C Tr [( M Z  H Y )  T ]  C C Tr [( M Z  H Y )  T ]}  K L KL KL K L KL KL 2  M K ,L K ,L M   C K C L M KL K ,L

H   C K C L H KL K ,L

 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.  Integral transformation is not required in the new algorithm.  The new algorithm is especially efficient for the BOVB method.

VBSCF vs CASSCF 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  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)

• Different orbital sets for different VB structures •• •• •• •• •• •• C1 • • F • • F •• + C2 •• F •• F •• + C2 •• F •• F •• •• •• •• •• •• •• 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? { }  {1A , 2A , , mA A ; 1B , 2B , , mB B ; 1C , C2  , CmC ;}

A: atom or fragment Localized occupied VB orbitals

iA   cAi  A 

Occupied VB orbitals are obtained by VBSCF calculations

Virtual orbitals may be defined by a projector:

PA  C A ( M A ) 1 C A  S A

CA: vector of occupied orbital coefficients MA: overlap matrix of occupied VB orbitals SA: 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 all virtual orbitals.

Excited VB structures

 VBSCF   C KSCF  SCF K K

An excited VB structure  iK is built from  SCF K

A  replacing occupied  with virtual orbital  .

A j

describe the same classical VB structure.  iK and  SCF K i  By collecting all K , we have a wave function

corresponding to VB structure K. ' i  CI  C   Ki K K i

Corresponding to a VB structure.

by

VBCI   CKCI  CI K K

  CKi  iK K

i

 VBSCF   C KSCF  SCF K K

E VBCI 

i j C C  H    Ki Lj K L K ,L i, j

i j C C     Ki Lj K L K ,L i , j

CI H KL   C Ki C Lj  iK H  Lj i, j

CI M KL   C Ki C Lj  iK  Lj i, j

WK   WKi i

WKi   CKi  iK  Lj CLj L, j

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) = VBCISD VBCI(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 The energy contribution of an excited structure  iK is estimated by  SCF i0 SCF i0    C L H KL  E 0  C L M KL  L  E Ki   L E 0  E Ki

2

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.

E Q  (1   WK )E D K

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

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

HF

CCSD

VBSCF

BOVB

VBCIS

VBCISD

VBCIPT

Exp.c

H2

84.6

105.9

95.8

96.0

96.0(1 1)

105.9 (55)

105.9 (27)

104.2

LiH

32.5

49.5

42.4

43.0

42.8(2 7)

49.6 (171)

49.0 (49)

56.6

HF

94.9

127.2

105.1

115.9

125.0( 40)

126.0 (274)

126.1 (206)

137.2

HCl

77.6

99.1

85.8

89.9

92.0(4 0)

98.0 (274)

97.9 (189)

101.2

F2

-33.1

28.3

10.9

31.5

40.4(8 1)

33.9 (1089)

30.9 (507)

38.0

Cl2

14.5

41.6

26.2

35.6

38.9(8 1)

42.1 (1089)

40.2 (522)

58.0

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

Though the VBCI space is much smaller than those of MObased methods. VBCI method is computational demanding. Can we have a VB method that is accurate and cheap?

Valence Bond Second Order Perturbation Theory

 VBSCF   C KSCF  SCF K K

H SCF C SCF  E SCF M SCF C SCF Orbitals: Inactive. doubly occupied in VBSCF Active. Occupied in VBSCF Virtual. Chen; Song; Hiberty; Sason; Wu, J. Phys. Chem. A, accepted.

       (STM '

SCF 1

) i i

i

 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:

Hˆ 0  Pˆ0 FˆPˆ0  PˆK FˆPˆK  PˆSD FˆPˆSD   Fˆ   fˆ (i ) i

SCF ˆ fˆ (i )  hˆ(i )   Dmn ( J nm (i )  Kˆ nm (i )) m ,n

1  SCF      f pq  h pq   Dmn pq mn pm qn    2 m ,n

The first-order wave function:  ( 0 )   SCF   C KSCF  K

   ( 0 )   (1)

K

 (1 ) 

(1 ) C  R R

RVSD

 ( 0 )  (1)  0

 (0)   1

C (1)  (H 110  E ( 0 ) M 11 ) 1 H10 C ( 0 ) The second-order energy:

E

( 2)

C

(0) 

H 01 (H110  E ( 0 ) M11 ) 1 H10C( 0 )

The most time-consuming part: (0) 11 1 (H11  E M ) 0

which is block diagonalized, due to the block-orthogonality between different orbital blocks. If the occupation numbers of inactive or virtual orbitals are different in the two excited structures, the corresponding matrix element is zero. VBPT2 is much cheaper than VBCI.

The structure weights in VBPT2 method:  KPT  N K ( K   X RK  R ) R

 VBPT   C KPT  PT K K

11 N K  (1   X RK M RS X SK ) 1 / 2

CKPT  CK( 0 ) / N K

R,S

PT WKPT   CKPT M KL CLPT L

(2) 01 10 EKL   H KR (H110  E ( 0 ) M11 ) RS1 H SL R ,S

PT SCF (2) H KL  H KL  E KL

Example 1. The Spectroscopic Constants of H2 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

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

Example 2. The Spectroscopic Constants of O2 Ground State 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

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

Example 3. The Spectroscopic Constants of N2 Ground State Method

re (a.u.) ωe (cm-1) De (eV)

FCIa

2.123

2341

8.748

a. J. Chem. Phys. 86, 5595

VBSCF(17)b

2.109

2388

8.086

b. 17 fundamental VB structures

b

VBPT2(17)

2.115

2373

8.421

VBSCF(175)c

2.114

2364

8.190

c

2.120

2344

8.573

VBPT2(175)

b

VBCIS(17)

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

(1987).

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).

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

Moleculea

E(A2)

2E(A)

∆E(size) (mh)

2N

-108.828718 -108.828718

50%

Real phenomenon or VB artifact? Other signs (not VB) that charge-shift bonds are special Digging into the literature…

X

X

Separate atoms

H–H, H3C–CH3 … Density build-up in the bonding region

F–F, Cl–Cl, … Deficit of density in the bonding region

 and  2 at the bond critical point in AIM theory 

2  

RE (kcal/mol)

Covalent

H-H

0.27

-1.39

9.2

bonds

H3C-CH3

0.25

-0.62

27.7

H2N-NH2

0.29

-0.54

43.8

Charge-shift

HO-OH

0.26

-0.02

56.9

bonds

F-F

0.25

+0.58

62.2

Cl-Cl

0.14

+0.01

48.7

Na-Cl

0.03

+0.18

8.1

Ionic bond

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

Valence Bond Theory CS Bond: Resonance energy dominates the bonding energy. RE/BDE > 50% AIM Theory CS Bond: Laplacian is positive or close to zero; density is large.

H2 C

H2C

CH2

The “inverted” bond in [1,1,1]propellane: a charge-shift bond Wu; Gu; Song; Shaik; Hiberty, Angew. Chem. Int. Ed. 2009, 48, 1407.

The problem of “inverted bonds” in propellanes

H2 C

H2 C H

H2C

H

-2 H• CH2

H2C

CH2

[1.1.1] propellane ∆E(S-T) = 109 kcal/mol

=> not a diradical

What kind of bond is it? 2

  = -13.0

H2 C

- Very weak electron density between the carbons - Positive

2

 at bond critical point

- extra stability of 65 kcal/mol

H2C

CH2

 2 = +10.3

The three features characterize charge-shift bonding

Valence bond calculations (BOVB)

E(kcal/mol)

covalent 11

72

57

RC-C(Å)

ground state 1.8Å

1.60Å

H2 C

H2 C

CH2

H2 C C

C

C

C H2

C

C C H2

C

C C

The covalent curve is repulsive The resonance energy is huge

C

A typical charge-shift bond

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

Moleculeb

Din-situc

covd

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

CH2 CH3 H3C L=-0.557 G=0.056 H=-0.250 1

CH2 H2 C L=-0.435 G=0.088 H=-0.284 2

NH

O

C HN

C L=0.004 NH G=0.167 H=-0.166 5 (X=NH) C F2 C

F2C C CF L=0.191 2 G=0.139 H=-0.091 9 (X=CF2)

C C L=-0.649 G=0.069 H=-0.232 13 (2.2.2)

C O

C L=-0.042 O G=0.176 H=-0.186 6 (X=O)

C C H H L=-0.246 G=0.130 H=-0.320 3 BH C C HB L=0.261 BH G=0.106 H=-0.041 7 (X=BH)

+ OH C + C + HO OH L=0.250 G=0.147 H=-0.085 10 (X=OH+)

C C H2C L=0.068 CH2 G=0.154 H=-0.137 4 (X=CH2) O C C C C O L=0.314C O G=0.126 H=-0.047 8 (X=CO) CH2

C

C

C H2C C CH2 L=-0.301 L=0.013 G=0.099 G=0.123 H=-0.175 H=-0.120 11 (1.1.2)

12 (1.2.2)

C

C

C C

C

C L=-0.567 G=0.063 H=-0.205

L=-0.513 G=0.059 H=-0.187

L=-0.472 G=0.055 H=-0.173

14 (2.2.3)

15 (2.3.3)

16 (3.3.3)

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

3.4 Direct Estimate of Hyperconjugation Energies

The Origin of Rotation Barrier in Ethane

E  2.9kcal / mol (12kJ / mol ))

Origin of Barrier? Steric Repulsion Model L. Pauling, The Natrue of Chemical Bond, 3rd, 1960

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

M. Karplus, J. Chem. Phys. 1968, 49, 2592. E. J. Baerends, Angew. Chem. Int. Ed., 2003, 42, 4183.

L. Goodman, Nature, 2001, 411, 565.

2 2 1' 1

MO Method：

1

optimize

Minimum of Energy

Delocalized MOs Orbtial transformation

Localized MOs

Lower energy？ optimize

Possibility：Overestimate delocalization energy ? VB method： Localized AOs

Minimum of Energy

Ebarrier  Ehc  Es Ehc  Edel  Eloc Ehc  Ehceclipsed  Ehcstaggered eclipsed staggered Es  Eloc  Eloc

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

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

Eloc (a.u.)

Edel (a.u.)

Ehc (kcal/mol)

Staggered

-79.32024

-79.33811

-11.21

Eclipsed

-79.31737

-79.33379

-10.30

1.80

2.71

0.91

 (kcal/mol)

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.

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

H C

H C

H

C

H C

C

C

H

H  aromaticity in C3H6

 aromaticity in C3H6



σ loc



π loc









 π  A  C  σloc del  π  A  C  σdel loc

RE σ   σloc   σloc π RE π   loc   σloc

ECRECM   ARECM  ARE(Ref)

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 Species

X3H 6

X4H 8

X = Si

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

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

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 cyclopropane energetically.

4. Some available VB softwares  The XMVB Program  The TURTLE Software  The VB2000 Software  The CRUNCH Software

XMVB: An ab initio Nonorthogonal Valence 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, CHINA [email protected]

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 !