Chemical Reviews REVIEW and the important VB structures was given in the recent study of the Mk红=（④xΦ） (7) various states of The VBSCF method permits complete flexibility in the VB structural weights can be evaluated by the Coulson- definition of the orbitals used for constructing VB structures. Chirgwin formula, 37 which is an equivalent of the Mulliken The orbitals can be allowed to delocalize freely during the population analysis: orbital optimization (resulting in OEOs),and then it will resemble the GVB and SC methods.The orbitals can be defined Wk=∑CkMkLCL also by pairs that are allowed to delocalize over the two bonded centers (bond-distorted orbitals,BDOs4),or they can be Apart from the Coulson-Chirgwin formula,other definitions for defined as strictly localized on a single center or fragment structural weights have also been proposed,such as Lowdin's (resulting in HAOs). symmetrical weights and Gallup'serse weightV 2.2.2.BOVB Method.The BOVB method!13-115 was de. structural weights are typically used to compare the relative vised with the aim of computing diabatic or adiabatic states with importance of individual VB structures and can be helpful in the wave functions that combine the properties of compactness understanding of the correlation between molecular structure unambiguous interpretability in terms of structural formulas,and and reactivity. accuracy of the calculated energies.The following features have to be fulfilled to retain interpretability and achieve reasonably 2.2.VB Methods of the HLSP Type good accuracy for the BOVB method:(i)the VB structures are 2.2.1.VBSCF Method.In the old classical VB method,VB constructed with HAOs,which means that covalent and ionic functions were built upon AOs,taken from the atom calculations, forms are explicitly considered;(ii)all the VB structures that are and the coefficients of structures were optimized to minimize the relevant to the electronic system being computed are generated; total energy of the system.Obviously,the computational results (iii)the coefficients and orbitals of the VB structures are were extremely poor due to the use of frozen atomic orbitals. optimized simultaneously.An important specificity of the BOVB The VBSCF method was the first modern VB approach that method is that the orbitals are variationally optimized with the also optimized orbitals.It was devised by Balint-Kurti and van freedom to be different for different VB structures.Thus,the Lenthe49s and was further modified and efficiently implemen- ted by van Lenthe and Verbeek.4014In the VBSCF method,the different VB structures are not optimized separately but in the presence of each other,so that the orbital optimization not only wave function is expressed in terms of VB functions as lowers the energies of each individual VB structure but also ΨBsCF=∑CCFΦg (9 maximizes the resonance energy resulting from their mixing. Since the BOVB wave function takes a classical VB form,its implementation is less practical for large electronic systems, where both of the structure coefficients(CCF)and VB functions because a large number of VB structures would have to be ()are simultaneously optimized to minimize the total energy. generated in such a case.As such,the usual way of using BOVB is The VB functions are optimized through their occupied orbitals, to apply it only on those orbitals and electrons that undergo which are usually expanded as linear combinations of basis significant changes during the process,such as bond breaking functions: and/or formation;the remaining orbitals are treated as doubly 中：=∑Taie occupied MOs.However,even though the"spectator electrons' (10 reside in doubly occupied MOs,these orbitals too are allowed to optimize freely,but are otherwise left uncorrelated. Basically,whenever VBSCF takes all independent VB struc- The difference between the BOVB and VBSCF wave functions tures and uses delocalized orbitals,e.g.,overlap-enhanced orbitals can be illustrated on the simple example of the description of the (OEOs),it will be equivalent to CASSCF with the same active A-B bond,where A and B are two polyelectronic fragments. electrons and orbitals.However,usually the VBSCF method Including the two HAOs that are involved in the bond in the employs only a few structures that are essential to describe the active space,and the adjacent orbitals and electrons in the system of interest,whereas CASSCF uses the complete set of spectator space,the VBSCF wave function reads configurations within the active-space window.One of the advantages of VBSCF,associated with purely localized HAOs, ΨscF=C(yp中年，|-|l4p中lD is having a compact wave function with a limited number of VB +C2pp中φl+C3ppφ中 (11) structures.Indeed,using pure HAOs to define the VB structures makes the neutral covalent structures largely predominant,as is where and o are the active orbitals,common to all the well-known in the two-electron two-orbital case (Scheme la). structures,and is a generic term that represents the product of Following this principle,the selection of VB structures can be spectator orbitals,also common to all structures.On the other done by chemical background in the polyatomic case.Thus,in hand,the BOVB wave function takes the following form: using VBSCF,it is usually advisable to remove the multi-ionic structures,which are generally of very high energy compared ΨovB=B1(p中，再l-|p师) with covalent and monoionic structures.Furthermore,symmetry (12 considerations are often helpful for removing additional structures +By'4,p1+B"”中"年"T which have no symmetry match to mix with the low-lying Physically,one expects the and o"orbitals to be more covalent and monoionic structures.For example,in the study of diffuse than and since the former are doubly occupied C2,using 92 VB structures in the VBSCF gives almost the same while the latter are only singly occupied.Similarly,the spectator result as the full set of 1764 structures,both numerically and orbitals in the different structures should have different sizes qualitatively.42 A discussion of the strategy of selecting only and shapes depending on whether they reside on cationic, dx.dol.org/10.1021/cr100228r Chem.Rev.XXXX,XXX,000-000E dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW and MKL ¼ ÆΦKjΦLæ ð7Þ VB structural weights can be evaluated by the Coulson
Chirgwin formula,137 which is an equivalent of the Mulliken population analysis: WK ¼ ∑ L CKMKLCL ð8Þ Apart from the Coulson-Chirgwin formula, other definitions for structural weights have also been proposed, such as L€owdin’s symmetrical weights138 and Gallup’s inverse weights.139 VB structural weights are typically used to compare the relative importance of individual VB structures and can be helpful in the understanding of the correlation between molecular structure and reactivity. 2.2. VB Methods of the HLSP Type 2.2.1. VBSCF Method. In the old classical VB method, VB functions were built upon AOs, taken from the atom calculations, and the coefficients of structures were optimized to minimize the total energy of the system. Obviously, the computational results were extremely poor due to the use of frozen atomic orbitals. The VBSCF method was the first modern VB approach that also optimized orbitals. It was devised by Balint-Kurti and van Lenthe94,95 and was further modified and efficiently implemen￾ted by van Lenthe and Verbeek.140,141 In the VBSCF method, the wave function is expressed in terms of VB functions as ΨVBSCF ¼ ∑ K CSCF K Φ0 K ð9Þ where both of the structure coefficients (CK SCF) and VB functions (ΦK 0 ) are simultaneously optimized to minimize the total energy. The VB functions are optimized through their occupied orbitals, which are usually expanded as linear combinations of basis functions: ϕi ¼ ∑ μ Tμiχμ ð10Þ Basically, whenever VBSCF takes all independent VB struc￾tures and uses delocalized orbitals, e.g., overlap-enhanced orbitals (OEOs), it will be equivalent to CASSCF with the same active electrons and orbitals. However, usually the VBSCF method employs only a few structures that are essential to describe the system of interest, whereas CASSCF uses the complete set of configurations within the active-space window. One of the advantages of VBSCF, associated with purely localized HAOs, is having a compact wave function with a limited number of VB structures. Indeed, using pure HAOs to define the VB structures makes the neutral covalent structures largely predominant, as is well-known in the two-electron two-orbital case (Scheme 1a). Following this principle, the selection of VB structures can be done by chemical background in the polyatomic case. Thus, in using VBSCF, it is usually advisable to remove the multi-ionic structures, which are generally of very high energy compared with covalent and monoionic structures. Furthermore, symmetry considerations are often helpful for removing additional structures which have no symmetry match to mix with the low-lying covalent and monoionic structures. For example, in the study of C2, using 92 VB structures in the VBSCF gives almost the same result as the full set of 1764 structures, both numerically and qualitatively.142 A discussion of the strategy of selecting only the important VB structures was given in the recent study of the various states of O2. 143 The VBSCF method permits complete flexibility in the definition of the orbitals used for constructing VB structures. The orbitals can be allowed to delocalize freely during the orbital optimization (resulting in OEOs), and then it will resemble the GVB and SC methods. The orbitals can be defined also by pairs that are allowed to delocalize over the two bonded centers (bond-distorted orbitals, BDOs144), or they can be defined as strictly localized on a single center or fragment (resulting in HAOs). 2.2.2. BOVB Method. The BOVB method113
115 was de￾vised with the aim of computing diabatic or adiabatic states with wave functions that combine the properties of compactness, unambiguous interpretability in terms of structural formulas, and accuracy of the calculated energies. The following features have to be fulfilled to retain interpretability and achieve reasonably good accuracy for the BOVB method: (i) the VB structures are constructed with HAOs, which means that covalent and ionic forms are explicitly considered; (ii) all the VB structures that are relevant to the electronic system being computed are generated; (iii) the coefficients and orbitals of the VB structures are optimized simultaneously. An important specificity of the BOVB method is that the orbitals are variationally optimized with the freedom to be different for different VB structures. Thus, the different VB structures are not optimized separately but in the presence of each other, so that the orbital optimization not only lowers the energies of each individual VB structure but also maximizes the resonance energy resulting from their mixing. Since the BOVB wave function takes a classical VB form, its implementation is less practical for large electronic systems, because a large number of VB structures would have to be generated in such a case. As such, the usual way of using BOVB is to apply it only on those orbitals and electrons that undergo significant changes during the process, such as bond breaking and/or formation; the remaining orbitals are treated as doubly occupied MOs. However, even though the “spectator electrons” reside in doubly occupied MOs, these orbitals too are allowed to optimize freely, but are otherwise left uncorrelated. The difference between the BOVB and VBSCF wave functions can be illustrated on the simple example of the description of the A
B bond, where A and B are two polyelectronic fragments. Including the two HAOs that are involved in the bond in the active space, and the adjacent orbitals and electrons in the spectator space, the VBSCF wave function reads ΨVBSCF ¼ C1ðjψψ̅ϕaϕ̅bj
jψψ̅ϕ̅aϕbjÞ þ C2jψψ̅ϕaϕ̅aj þ C3jψψ̅ϕbϕ̅bj ð11Þ where ϕa and ϕb are the active orbitals, common to all the structures, and ψ is a generic term that represents the product of spectator orbitals, also common to all structures. On the other hand, the BOVB wave function takes the following form: ΨBOVB ¼ B1ðjψψ̅ϕaϕ̅bj
jψψ̅ϕ̅aϕbjÞ þ B2jψ0 ψ̅0 ϕa 0 ϕ̅a 0 j þ B3jψ00ψ̅00ϕb 00ϕ̅b 00j ð12Þ Physically, one expects the ϕa 0 and ϕb 00 orbitals to be more diffuse than ϕa and ϕb since the former are doubly occupied while the latter are only singly occupied. Similarly, the spectator orbitals in the different structures should have different sizes and shapes depending on whether they reside on cationic