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Valence Bond Theory 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 11 VB ab ba 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 - + + - H2: H – H H H H H 1 1 cov [a(1)b(2) b(1)a(2)] [(1) (2) (1)(2)] 2 2 1 1 ab ab 2 2 1 1 ion a(1)a(2) [(1) (2) (1)(2)] aa 2 2 2 1 1 ion b(1)b(2) [(1)(2) (1)(2)] bb 2 2 2 ˆ K A 0 K 0 1 (1)2 (2)N (N) 1 2 K 2 [(k1) (k2 ) (k2 )(k1)] 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 / 2S ,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 ci 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 ˆ K K K DK A(1 2 ...N ) K K i ci 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 KL KL F h G G (P) P (g , g , ) , 1 H N (trPKLh trPKLFKL ) 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 2CK 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[YKLT ] M KLtr[YKLT ] 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 KLT ) tr(Z KLT ) K L where Z KL and Z KL are the derivatives of H KL with respect to the orbital coefficient matrices TK and TL 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 ;} 12mmmABC 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; .
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