Chemical Reviews REVIEW where summation includes the fundamental structure.There- where the VBSCF wave function is taken as the zeroth-order fore,a many-electron VBCI wave function is written as a linear wave function combination of VBCI functions: IΨ，=IΨsCF)=∑CCF1Φ》 (21) ΨBa-∑cgΦg=∑∑Ca CK=CCK Since higher order excitations do not contribute to the first- (15) order interacting space,the first-order wave function is written as where the coefficients CK,are determined by solving the secular a linear combination of the singly and doubly excited structures, equation.The total energy of the system is 更R时 ∑∑CC④KIHp) p=∑CIΦ) (22) K,L ij R∈VsD EVBCI (16) ∑∑CKCL(ΦI) The excited VB structures may be generated by replacing K,L ii occupied orbitals with virtual ones,as in the VBCI method However,to enhance the efficiency of VBPT2,the VB orbitals are The compact forms of the Hamiltonian and overlap matrices defined in a different way and are partitioned into three groups: may be respectively given by inactive,active,and virtual orbitals.The inactive orbitals are always H是=∑CKC④KH) (17) doubly occupied in the VBSCF wave function,the active orbitals are occupied orbitals,with variable occupancies,and the virtual orbitals are always unoccupied in the VBSCF reference.These and three groups maintain the following orthogonality properties: (i)The inactive and virtual orbitals are orthogonal within their M是=∑CC,④kI) (18 own groups.(ii)The active orbitals are kept in the VB spirit as mutually nonorthogonal,but are orthogonal to the inactive and A compact definition of the structure weights is virtual orbitals by a Schmidt orthogonalization,which is done in the following order: WK=∑W&=∑CKiMKL.C Mk=(I型) (1)The Lowdin orthogonalization is performed for the in- active orbitals. (19) (2)The Schmidt orthogonalization procedure is carried out between groups (inactive and active orbital groups), In this manner,the extensive VBCI wave function is con rather than each basis,where the order is inactive orbitals densed to a minimal set of fundamental structures,thus first and then active orbitals. ensuring that the VBCI method keeps the VB advantage of (3)The virtual orbitals are obtained by a two-step procedure: compactness. (3.1)group (Schmidt)orthogonalization between occu- The CI space can be truncated following the usual CI pied orbitals and basis functions (occupied orbitals first methodology.The levels of CI are fashioned as in the corre- and then basis functions);(3.2)removal of linearly inde- sponding MO-CI approach.Thus,VBCIS involves only single pendent vectors in the basis functions after step 3.1. excitations,while VBCISD involves singles and doubles,and so Such a definition of the orbitals keeps the VBSCF energy on.VBCI applications45 show that VBCIS gives results that are invariant,while the orthogonalization between orbital groups en- on par with those of D-BOVB,while the VBCISD method is sures the efficiency of the VBPT2 method. somewhat better,and its results match those of the MO-based In a fashion similar to that of MO-based multireference CCSD method.Furthermore,the VBCI method applies pertur- perturbation theory,a one-electron Fock operator is defined as bation theory to truncate less important excited structures and estimates their contribution by an approximated perturbation 间=间+∑D9 j-求间 (23) formula,resulting in a VBCIPT level. , 2.2.4.VBPT2 Method.The accuracy of VBCI applications is always satisfactory,454s5 but the method is computation- where and K are Coulomb and exchange operators, ally demanding.The stumbling blocks in a VBCI calculation are respectively,D is the VBSCF density matrix element,and (i)the construction of the Hamiltonian matrix with nonortho- m and n denote the valence bond orbitals.Using the Fock operator gonal AOs and(ii)the solution of the general secular equation, defined in eq 23,the zeroth-order Hamiltonian is defined as where the overlap matrix is nonunity.Perturbation theory is i。=Pp。+pxpx+Ps知前s知+… (24) known to be an economical assessment of electronic correlation and is widely applied not only in MO-based methods,but also in where F=Ef(i),Po=0)0 is a projector onto the VBSCF space, the VB framework,such as in the GVB11,156-159 and SCVB Pk is a projector onto the space complementary to that of the methods.152 VBSCF wave function,and Psp is a projector associated with The VBPT2 method1s uses perturbation theory to incorpo- singly and doubly excited structures from the reference wave rate dynamic correlation for the VB method,much like CASPT2 function. (e.g,VBPT2 suffers the same defect as CASPT2,e.g.,intruder On the basis of the Rayleigh-Schrodinger perturbation states).In the VBPT2 method,the wave function is written as the theory,the expansion coefficients of the first-order wave function sum of the zeroth-and the first-order wave functions and the second-order energy are written respectively as ΨBT)=Ψo,+平Ψ) (20) C()=(HI-E(0)M)-H10C(0) (25) dx.dol.org/10.1021/cr100228r |Chem.Rev.XXXX,XXX,000-000G dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW where summation includes the fundamental structure. There￾fore, a many-electron VBCI wave function is written as a linear combination of VBCI functions: ΨVBCI ¼ ∑ K CCI K ΦCI K ¼ ∑ K ∑ i CKiΦi K CKi ¼ CCI K Ci K ð15Þ where the coefficients CKi are determined by solving the secular equation. The total energy of the system is EVBCI ¼ ∑ K, L ∑ i, j CKiCLjÆΦi KjHjΦj Læ ∑ K, L ∑ i, j CKiCLjÆΦi KjΦj Læ ð16Þ The compact forms of the Hamiltonian and overlap matrices may be respectively given by HCI KL ¼ ∑ i, j CKiCLjÆΦi KjHjΦj Læ ð17Þ and MCI KL ¼ ∑ i, j CKiCLjÆΦi KjΦj Læ ð18Þ A compact definition of the structure weights is WK ¼ ∑ i WKi ¼ ∑ L, i, j CKiMij KLCLj Mij KL ¼ ÆΦi KjΦj Læ ð19Þ In this manner, the extensive VBCI wave function is con￾densed to a minimal set of fundamental structures, thus ensuring that the VBCI method keeps the VB advantage of compactness. The CI space can be truncated following the usual CI methodology. The levels of CI are fashioned as in the corre￾sponding MO
CI approach. Thus, VBCIS involves only single excitations, while VBCISD involves singles and doubles, and so on. VBCI applications145 show that VBCIS gives results that are on par with those of D-BOVB, while the VBCISD method is somewhat better, and its results match those of the MO-based CCSD method. Furthermore, the VBCI method applies pertur￾bation theory to truncate less important excited structures and estimates their contribution by an approximated perturbation formula, resulting in a VBCIPT level. 2.2.4. VBPT2 Method. The accuracy of VBCI applications is always satisfactory,143,145,154,155 but the method is computation￾ally demanding. The stumbling blocks in a VBCI calculation are (i) the construction of the Hamiltonian matrix with nonortho￾gonal AOs and (ii) the solution of the general secular equation, where the overlap matrix is nonunity. Perturbation theory is known to be an economical assessment of electronic correlation and is widely applied not only in MO-based methods, but also in the VB framework, such as in the GVB11,156
159 and SCVB methods.152 The VBPT2 method118 uses perturbation theory to incorpo￾rate dynamic correlation for the VB method, much like CASPT2 (e.g., VBPT2 suffers the same defect as CASPT2, e.g., intruder states). In the VBPT2 method, the wave function is written as the sum of the zeroth- and the first-order wave functions jΨVBPT2æ ¼ jΨð0Þ æ þ jΨð1Þ æ ð20Þ where the VBSCF wave function is taken as the zeroth-order wave function jΨð0Þ æ ¼ jΨSCFæ ¼ ∑ K CSCF K jΦ0 Kæ ð21Þ Since higher order excitations do not contribute to the first￾order interacting space, the first-order wave function is written as a linear combination of the singly and doubly excited structures, ΦR: Ψð1Þ ¼ ∑ R ∈ VSD Cð1Þ R jΦRæ ð22Þ The excited VB structures may be generated by replacing occupied orbitals with virtual ones, as in the VBCI method. However, to enhance the efficiency of VBPT2, the VB orbitals are defined in a different way and are partitioned into three groups: inactive, active, and virtual orbitals. The inactive orbitals are always doubly occupied in the VBSCF wave function, the active orbitals are occupied orbitals, with variable occupancies, and the virtual orbitals are always unoccupied in the VBSCF reference. These three groups maintain the following orthogonality properties: (i) The inactive and virtual orbitals are orthogonal within their own groups. (ii) The active orbitals are kept in the VB spirit as mutually nonorthogonal, but are orthogonal to the inactive and virtual orbitals by a Schmidt orthogonalization, which is done in the following order: (1) The L€owdin orthogonalization is performed for the in￾active orbitals. (2) The Schmidt orthogonalization procedure is carried out between groups (inactive and active orbital groups), rather than each basis, where the order is inactive orbitals first and then active orbitals. (3) The virtual orbitals are obtained by a two-step procedure: (3.1) group (Schmidt) orthogonalization between occu￾pied orbitals and basis functions (occupied orbitals first and then basis functions); (3.2) removal of linearly inde￾pendent vectors in the basis functions after step 3.1. Such a definition of the orbitals keeps the VBSCF energy invariant, while the orthogonalization between orbital groups en￾sures the efficiency of the VBPT2 method. In a fashion similar to that of MO-based multireference perturbation theory, a one-electron Fock operator is defined as ^fðiÞ ¼ ^hðiÞ þ ∑ m, n DSCF mn ^JmnðiÞ
1 2 K^mnðiÞ
ð23Þ where ^Jmn and K^mn are Coulomb and exchange operators, respectively, Dmn SCF is the VBSCF density matrix element, and mand n denote the valence bond orbitals. Using the Fock operator defined in eq 23, the zeroth-order Hamiltonian is defined as H^ 0 ¼ P^0F^P^0 þ P^KF^P^K þ P^SDF^P^SD þ ::: ð24Þ where F^ = ∑ i ^f (i), P^0 = |0æÆ0| is a projector onto the VBSCF space, P^K is a projector onto the space complementary to that of the VBSCF wave function, and P^SD is a projector associated with singly and doubly excited structures from the reference wave function. On the basis of the Rayleigh
Schr€odinger perturbation theory, the expansion coefficients of the first-order wave function and the second-order energy are written respectively as Cð1Þ ¼ ðH11 0
Eð0Þ M11Þ
1 H10Cð0Þ ð25Þ