Beijing Normal University Summer School of Theoretical and Computational Chemistry

Valence Bond Theory

Wei Wu August 1, 2010 1. Introduction 2. Ab initio Valence Bond Methods 3. Applications 4. Some available VB softwares Quantum Chemistry

Molecular Orbital Valence Bond Theory 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

W. Heitler F. London

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 VB  2 HH•••••••••••• H H

1S 1S a b 1     MO  11 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 KK K

K corresponds to a given VB structure

- + + - H2: H – H H H H H

CK is given by solving secular equation

()0HESCKL KL K  K

HHSKL K||  L  , KL  K |  L  Heitler-London-Slater-Pauling (HLSP) Function

H : H – H H- H+ H+ H- 2   1 1  cov  [a(1)b(2)  b(1)a(2)] [ (1) (2)  (1)(2)] 2 2 1 1  ab  ab 2 2   ion 1  1   a(1)a(2) [ (1) (2)  (1)(2)]  aa 2 2 2   ion 1  1   b(1)b(2) [ (1) (2)  (1)(2)]  bb 2 2 2  ˆ  K  A 0 K     0  1 (1)2 (2)N (N)     21 2[ (k ) (k )  (k ) (k )] K 1 2 2 1  1 2  2 [ (k3 ) (k4 )  (k4 ) (k3 )] (k p )(kN )

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):

(2S 1)N! f   (N / 2  S 1)!(N / 2  S)!

Dimension of irreducible representation of symmetric group []  [2N / 2S ,12S ] Rumer Rule:

S = 0; S >0.

Young Tableaux of Symmetric Group: The Total Number of Structures Including Ionic:

2S 1 (m 1)!(m 1)! f   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  K  d D     C6H6

a f b 1 ||||||||abcdef abcdef abcdef abcdef e c d ||||||||abcdef abcdef abcdef abcdef

a f b 2 (|adbcef | | adbcef | | adbcef | | adbcef | e c d ||||||||)adbcef adbcef adbcef adbcef General Cases

  (a1b1  a1b1 )(a2b2  a2b2 )(ambm  ambm ) 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  CK M KLCL L Matrix elements in VB method

 C K K   CK DK K K D | H | D  f KL D(S )  (g KL  g KL )D(S ) K L  rs rs  rs,tu rs,ut rs,tu r,s 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 SCF 0   CK  K K

• • • • • • C1 • FF• • • + C2 • FF• • + C2 • FF• • • • • • • • F•—•F F– F+ F+ F– 

i  Tμi    Numerical Algorithm: ( in XMVB program, 2006)

E Ec c   E0  i i ci 

N2 matrix elements are required, Cost: N6+m4N New Algorithm for Energy Gradient (J. Comput. Chem. 2009)

A many-electron wave function

VB   CK  K K

  CK DK K

DK is built upon nonorthogonal localized orbitals   D  Aˆ ( K K ... K ) K 1 2 N K K i  c i      K  χTK

Overlap matrix between the two orbital sets ~ V KL  T K ST L

The overlap matrix element between the determinants

KL M KL  V

Defining a transition density matrix ~ ~ PKL  T K (T LSTK )1 T L

Hamiltonian matrix element     KL 1 KL KL H KL  M KL ( P h  P P (g ,  g , ))     , 2 , , ,        The first- and second-order Density Matrices

KL KL KL P  CK M KL P CL  ,  CK M KL P P CL K ,L K ,L

Normalized Hamiltonian matrix element     N KL 1 KL KL H KL  P h  P P (g ,  g , )     , 2 , , ,    N   H KL  M KL H KL Fock matrix  KL KL G KL (P)  P KL (g  g ) F  h  G   , ,  ,   N 1 KL KL KL H  (trP h  trP F )  KL 2 

KL KL CK CL H KL CK CL M KL (trP h  trP F) E  K ,L  K ,L CK CL M KL 2CK CL M KL K ,L K ,L Variation and Gradients

i ' i i The density matrix changes by

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

K ~ K L ~ L M KL  M KLtr[YKLT ]  M KLtr[YKLT ]

K L KL 1 L K ~ KL 1 YKL  ST (V ) YKL  ST (V ) The change in the ‘normalized’ Hamiltonian matrix element

N ~ KL ~ KL L KL 1 ~ K KL KL K ~ KL 1 ~ L H KL  tr[(1 SP )F ]T (V ) T ]  tr[(1 SP )F T (V ) T ] The change in Hamiltonian matrix element

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

K L where Z KL and Z KL are the derivatives of H KL with respect to the K L orbital coefficient matrices T and T

K ~ KL ~ KL L KL 1 ZKL  [H KLS  M KL (1 SP )F ]T (V )

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

1 K K ~ K L L ~ L E  { C C Tr[(MZ  HY ) T ]  C C Tr[(MZ  HY )T ]} 2  K L KL KL  K L KL KL M K ,L K ,L

H  C C H M  CKCL M KL  K L KL K ,L 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 CASSCF  Non-orthogonal  Orthogonal localized AOs delocalized MOs  A few structures  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 • FF• • + 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?

{}{,,,AA  A ;,,,  BB  B ;,  CC ,  C ;}  12mmmABC 12 12

A: atom or fragment Localized occupied VB orbitals

A AA ii  c  

Occupied VB orbitals are obtained by VBSCF calculations Virtual orbitals may be defined by a projector:

1 PA  CMAA() CS AA

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 SCF SCF   CK  K K

An excited VB structure i is built from SCF by K  K A  A. replacing occupied j with virtual orbital  i andSCF describe the same classical VB structure. K  K

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

CI ' i  K  CKi  K i Corresponding to a VB structure. VBCI CI CI CKK  K i  CKiK Ki

VBSCF SCF SCF   CK  K K

i j CKiCLj  K H  L EVBCI  Kj,,L i i j CKiCLj  K  L Kj,,L i CI i j H KL  CKiCLj  K H  L i, j

CI i j M KL  CKiCLj  K  L i, j

WK  WKi i

ij WCKiKiKLLj C Lj,

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

i The energy contribution of an excited structure  K is estimated by

2  SCF i0 SCF i0  CL H KL  E0 CL M KL  i  LL EK  i E0  EK

i E < critical value  K is discarded in CI procedure

i E > critical value  K 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.

EQ  (1 WK )ED 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(1 105.9 105.9 104.2 1) (55) (27) LiH 32.5 49.5 42.4 43.0 42.8(2 49.6 49.0 56.6 7) (171) (49) HF 94.9 127.2 105.1 115.9 125.0( 126.0 126.1 137.2 40) (274) (206) HCl 77.6 99.1 85.8 89.9 92.0(4 98.0 97.9 101.2 0) (274) (189)

F2 -33.1 28.3 10.9 31.5 40.4(8 33.9 30.9 38.0 1) (1089) (507)

Cl2 14.5 41.6 26.2 35.6 38.9(8 42.1 40.2 58.0 1) (1089) (522) 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

VBSCF SCF SCF   CK  K K H SCF CSCF  E SCF M SCF CSCF

Orbitals: Inactive. doubly occupied in VBSCF Active. Occupied in VBSCF Virtual.

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

' SCF 1   (STM )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:

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ H 0  P0 FP0  PK FPK  PSD FPSD  Fˆ   fˆ(i) i

ˆ ˆ SCF ˆ ˆ f (i)  h(i)  Dmn (J nm (i)  Knm (i)) m,n

SCF  1  f  h  D pqmn  pm qn pq pq  mn   m,n  2  The first-order wave function:

(0) (1) (0) SCF SCF          CK  K K

(1) (1)   CR  R RVSD  (0)  (1)  0  (0)   1

(1) 11 (0) 11 1 10 (0) C  (H 0  E M ) H C

The second-order energy:

(2) (0)  01 11 (0) 11 1 10 (0) E  C H (H 0  E M ) H C The most time-consuming part:

11 (0) 11 1 (H0  E M ) 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:

PT  K  N K ( K   X RK  R ) R

VBPT PT PT   CK  K K

PT (0) 11 1/ 2 CK  CK / N K N K  (1  X RK M RS X SK ) R,S

PT PT PT PT WK  CK M KL CL L

(2) 01 11 (0) 11 1 10 EKL   H KR (H0  E M )RS H SL R,S PT SCF (2) H KL  H KL  EKL Example 1. The Spectroscopic Constants of H2

-1 Method re (a.u.) ωe (cm ) 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

b a. J. Chem. Phys. 86, VBSCF(12) 2.368 1580 2.999 5595 (1987). b. 12 fundamental VB VBPT2(12)b 2.333 1560 4.327 structures are used. c. 105 fundamental c structures are used, but VBSCF(105) 2.368 1581 3.045 the orbitals are optimized use 17 VB structures. c VBPT2(105) 2.324 1601 4.661 d. Three orbital block according to b ,d σ, πx, πyare used. VBCIS(12) 2.321 1578 4.582 e J. Compt. Chem. 28, b, d 185 (2007), VBCISD(12) 2.333 1594 4.154 where cc-pVTZ basis set are used. VBCISDe 2.336 1545 4.77 f. J. Chem. Phys. 96, 1218 (1992). CASSCFf 2.322 1566 3.678 CASPT2Nf 2.317 1607 4.658

Example 3. The Spectroscopic Constants of N2 Ground State

Method re (a.u.) ωe (cm-1) De (eV) a. J. Chem. Phys. 86, 5595 FCIa 2.123 2341 8.748 (1987). VBSCF(17)b 2.109 2388 8.086 b. 17 fundamental VB structures are used. b VBPT2(17) 2.115 2373 8.421 c. 175 fundamental VB structures VBSCF(175)c 2.114 2364 8.190 are used, but the orbitals are optimized use 17 c VBPT2(175) 2.120 2344 8.573 VB structures. d. J. Chem. Phys. 96, 1218 b VBCIS(17) 2.116 2348 8.287 (1992). 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

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

a. RNN=50a0, ROO=100a0, RFF=100Å. Size consistency

VBPT2-calculated energies (hartrees) of some supersystems of two distant atoms as compared with the summed energies of the separate atoms.

a Molecule 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. Summary

 VBPT2 method provides a cheap VB tool, which is able to cover dynamical correlation. Test calculation shows that VBPT2 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 Perfect Paring (PP):

Strong Orthogonality (SO):

OEOs are used. Advantages of GVB Method: Computer time-saving. MCSCF. 2.10 Spin-couple Valence Bond (SCVB) Method

Gerratt and coworkers

SCVB   CK 0K K SCVB    CK 0K  K     0  1 (1)2 (2)N (N)     21 2[ (k ) (k )  (k ) (k )] K 1 2 2 1  1 2  2 [ (k3 ) (k4 )  (k4 ) (k3 )] (k p )(kN )

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 b H3Si-F 6-31G* 140.4 151.1 142.6 160 ± 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 a CH3 35.1 26.5 33.0 23.1 25.8 25.5 a SiH3 25.2 19.3 25.5 19.1 19.7 19.0 a GeH3 22.0 16.6 25.5 18.0 18.1 17.0 a SnH3 18.5 13.5 20.5 14.9 15.3 14.1 a PbH3 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? R* P*

Gr Gp

r B p E R Erp P

Reaction Coordinate

 E  fGr  B

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. Suppose that the diabatic profile Is a parabola, as R* P*

Gr Gp

r B p E R Erp P

Reaction Coordinate

 2 E  f0G0  Gp 2G0 Erp  0.5Erp G0  B

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

 E  f0G0 0.5Erp  B 3.2.1. Hydrogen Abstraction Reactions Identity Reaction (X=Y)

X• + HX’ -> XH + X’• (X = X’ = CH3, SiH3, GeH3, SnH3, PbH3) VB structures (3 electrons/3 orbitals system)

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

The trend of weights follows the electronegativity.

Gr  2DX  H B  0.5DX  H E   2 f  0.5D X  H 

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

Nonidentity Hydrogen Abstraction Reactions J. Phys. Chem. 2002, 106, 8226.

Modeling is more difficult, as No 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. (12)

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 the B(VB) values. Because eq 12 is much simpler, it should be the choice expression whenever eq 18 cannot be applied. 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? H-H + X•  H• + H-X (X = F, Cl, Br, I)

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

B = 0.5DW

Application

3.3 New Concepts in Chemical Bonding: Charge-Shift Bonding The classical paradigm of the A—X bond

A X A X A X A•–•X A+ :X– A:– X+

covalent ionic ionic

+ – – + A-X = C1(A•–•X) + C2(A :X ) + C3(A: X ) Dissociation energy curves

H2 → H· + H· F2 → F· + F· E(au) E(au) -0.950 -0.640

-0.660 -1.000

-0.680

-1.050 Covalent -0.700 Covalent -0.720 -1.100 RECS -0.740 Exact -1.150 -0.760 Exact R(Å) R(Å) -1.200 -0.780 0.5 1.0 1.5 2.0 2.5 3.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

It follows that the F-F bond owes its existence to the covalent-ionic fluctuation 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” Valence Bond Theory CS Bond: Resonance energy dominates the bonding energy. RE/BDE > 50% Real phenomenon or VB artifact?

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

Digging into the literature…

X X H–H, H3C–CH3 … F–F, Cl–Cl, … Separate Density build-up Deficit of density atoms in the bonding in the bonding region 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

Ionic bond Na-Cl 0.03 +0.18 8.1

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 H2 C C H -2 H•

H C CH H C CH 2 H 2 2 2

[1.1.1] propellane

∆E(S-T) = 109 kcal/mol => not a diradical What kind of bond is it?

H 2 2   = -13.0 C

- Very weak electron density between the carbons 2 H2C CH2 - Positive  at bond critical point

2 - extra stability of 65 kcal/mol  = +10.3

The three features characterize charge-shift bonding Valence bond calculations (BOVB) E(kcal/mol)

covalent

11 72 57

ground state RC-C(Å)

1.60Å 1.8Å H2 C H2 H CH2 2 C C C C C C C A typical C H C C 2 C charge-shift H2 C C bond The covalent curve is repulsive The resonance energy is huge

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

a b c d c Entry Molecule Din-situ cov REcov-ion 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

3+ 10 [1.1.1]-(OH)3 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 C

CC C H3C CH3 H2C CH2 H H H C L= 0.246 2 CH2 L=-0.557 L=-0.435 - L=0.068 G=0.056 G=0.088 G=0.130 G=0.154 H=-0.250 H=-0.284 H=-0.320 H=-0.137 2 3 4 (X=CH ) 1 2 O BH NH O C C C C C C C HN C C HB BH C C NH O L=0.261 O L=0.314 O L=0.004 L=-0.042O G=0.106 G=0.126 G=0.167 G=0.176 H=-0.041 H=-0.047 H=-0.166 H=-0.186 5 (X=NH) 6 (X=O) 7 (X=BH) 8 (X=CO)

+ CH CF2 OH 2 C C C C + F C C HO C + H C C C 2 CF2 OH 2 CH2 L=0.191 L=0.250 L=0.013 L=-0.301 G=0.139 G=0.147 G=0.123 G=0.099 H=-0.091 H=-0.085 H=-0.120 H=-0.175 + 9 (X=CF2) 10 (X=OH ) 11 (1.1.2) 12 (1.2.2)

C C C C C C C C L=-0.649 L=-0.567 L=-0.513 L=-0.472

G=0.069 G=0.063 G=0.059 G=0.055 H=-0.173 H=-0.232 H=-0.205 H=-0.187 13 (2.2.2) 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 Hyerconjugation Model

L. Pauling, The Natrue of F. Weinhild, J. Am. Chem. Soc. Chemical Bond, 3rd, 1960 1979, 101, 1700; Angew. Chem. Int. Ed. 2003, 42, 4188. M. Karplus, J. Chem. Phys. 1968, 49, 2592. L. Goodman, Nature, 2001, 411, 565. E. J. Baerends, Angew. Chem. Int. Ed., 2003, 42, 4183. 2

2

1' 1 MO Method： 1 optimize Delocalized MOs Minimum of Energy Orbtial transformation Localized MOs Lower energy？ optimize Possibility：Overestimate delocalization energy ? VB method： Localized AOs Minimum of Energy EbarrierEE hc  s

EEhc del E loc

eclipsed staggered EEhc hc  E hc

eclipsed staggered EEslocloc  E

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

 (kcal/mol) 1.80 2.71 0.91

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 C H H H CC CC H H

 aromaticity in C3H6  aromaticity in C3H6 σ  σ π loc  AClocdel 

π  σ π loc  ACdelloc 

σ σ σ RE  loc  loc

π π σ RE  loc  loc

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 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 X3H6 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 X4H8 ECRE2π (kcal/mol) -3.6 -4.6 ECRE2σ+π (kcal/mol) -5.2 -6.9

σ Basis set: 6-311+G** ECRE2 (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 !