Chemical Reviews REVIEW included in the VBCI or VBPT2 method,the corresponding Scheme 4.Five Rumer Structures for the Benzene Molecule wave function of the system still retains a compact form by condensing the extensive VBCI/VBPT2 wave function into a minimal set of the fundamental structures that are used in the VBSCF calculation(e.g.,in Scheme 2).As such,all VB proper- ties such as weight and resonance energy are still clearly defined in both high-level methods in the same manner as in the VBSCF 2.AB INITIO VALENCE BOND METHODS method.We note that both the BOVB and VBCI come in various internal levels of sophistication,which will be described 2.1.Theoretical Background in the corresponding sections in some detail. In VB theory,a many-electron wave function is expressed in In addition to the above VB methods,there are add-ons that terms of VB functions: enable one to carry out calculations in solution using the polar- izable continuum model (PCM)11120 or SMx(x=1-8)models,21 Ψ=∑CKΦx (1) hence valence bond polarizable continuum model (VBPCM) and valence bond solvation model(VBSM),or by incorpora- where the VB function dy corresponds to a classical VB struc- tion of molecular mechanics (MM),hence VB/MM,to carry ture.In quantum chemistry,any state function Pg should be a reactions inside protein cavities.22 As such,with this arsenal of spin eigenfunction that is antisymmetric with respect to permu- methods,VB theory is coming of age and starting to be useful tations of electron indices.In general,a VB function is of the for the treatment of some real chemical problems,as this review form will show. In addition to the strict VB methods displayed in Scheme 3, ΦK=A2oOx (2) the review will also describe MO-based methods that generate where A is an antisymmetrizer,o is a direct product of orbitals VB-type information which can be used for some specific VB applications.The block-localized wave function(BLW) {中}as method23-127 is a type ofVB method that utilizes an HF wave 20=中(1)中2(2)…中x(N) (3) function with block-localized orbitals.By partitioning the mo- lecular orbitals to the subgroups of a molecule,a BLW can and Ok is a spin-paired spin eigenfunction,3 defined as describe a specific VB structure at the HF level.Thus,the BLW is capable of computing delocalization/resonance energies Ox =2-1[a(k)B(k2)-B(k)a(k2)] and charge transfer effects among molecules.The BLW approach x2-2[a(k3)B(k4)-k)α(k】.a(k)…a(kw) is related to the early Kollmar method,2 wherein the subgroup (4) orbitals were input and the energy for this so-generated"localized" reference was computed at zero iteration,without optimization of In eq 4,the scheme of spin pairing (ki,k2),(k3,k),etc., the orbitals.Both methods belong to a general class of MO-and corresponds to the bond pairs that describe the structure K. density functional theory (DFT)-based energy decomposition Linearly independent electron pairing schemes may be selected analysis(EDA)approaches1 which use as a reference by using the Rumer diagrams.35 In a Rumer diagram,we set either a fragment-localized wave function or a density and down the electron indices,1,2,..,N,in a ring,representing each thereby estimate the various interactions between the frag- factor 2-1/2[a(k)B(k)-B(k)a(k)]in eq4by an arrow from i ments,thus providing VB-related information from MO or to j.On the basis of the Rumer rule,the independent Rumer DET calculations.For space economy,we shall limit our structure set is obtained by drawing all possible Rumer diagrams coverage to the BLW method since this method performs the in which there are no crossed arrows.Scheme 4 shows the five energy decomposition closer to the VB spirit compared to Rumer structures for the benzene molecule,where,as is well- other EDA methods. known,the first two are Kekule structures and the last three are The molecular orbital valence bond method(MOVB)125 is an Dewar structures. extension of the BLW method,which uses a multireference wave Rumer's rule is applicable for singlet states with spin quantum function,thus allowing calculation of the electronic coupling number S=0.To extend to the general spin S,extended Rumer energy resulting from the mixing of two or more block-localized diagrams'36 should be applied,where a pole is added in the structures. diagram.A VB function with a Rumer spin function is called a The structure of the review follows the above ordering of VB Heitler-London-Slater-Pauling (HLSP)function. methods,which are detailed in section 2.This methodology An alternative way of writing the wave function is by use of a section,which will certainly interest the computation-oriented Slater determinant form,which will be used in this review.For reader,is followed by applications which demonstrate the cap- example,the (ki,k2)bond-paired wave function will be given by ability of VB theory to lead to lucid physical insight into a variety (5) of problems,now approaching "real size".Then section 4 Φx=(中p2-,中) describes algorithms and techniques which make modern VB where the bar over the orbital denotes a B spin while lack of it theory faster and more efficient.Lastly,section 5 illustrates the denotes spin a.In turn,dk will be written as a product of the current capabilities of modern VB methods by displaying ab initio Slater determinant forms for all the bond pairs. VB calculations for a sizable molecular system,(CO)Fe(C,H). The coefficients Ck ineq 1 can conveniently be determined by The review is written in such a way that the application-oriented solving the secular equation HC=EMC,where Hamiltonian and reader who is less interested in the methodological details can overlap matrices are defined as follows: skip parts of section 2 and then proceed to the applications in sections 3 and 5. H=〈ΦxHΦ)） (6) dx.dol.org/10.1021/cr100228rChem.Rev.XXXX,XXX,000-000D dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW included in the VBCI or VBPT2 method, the corresponding wave function of the system still retains a compact form by condensing the extensive VBCI/VBPT2 wave function into a minimal set of the fundamental structures that are used in the VBSCF calculation (e.g., in Scheme 2). As such, all VB proper￾ties such as weight and resonance energy are still clearly defined in both high-level methods in the same manner as in the VBSCF method. We note that both the BOVB and VBCI come in various internal levels of sophistication, which will be described in the corresponding sections in some detail. In addition to the above VB methods, there are add-ons that enable one to carry out calculations in solution using the polar￾izable continuum model (PCM)119,120 or SMx (x = 1
8) models,121 hence valence bond polarizable continuum model (VBPCM) and valence bond solvation model (VBSM), or by incorpora￾tion of molecular mechanics (MM), hence VB/MM, to carry reactions inside protein cavities.122 As such, with this arsenal of methods, VB theory is coming of age and starting to be useful for the treatment of some real chemical problems, as this review will show. In addition to the strict VB methods displayed in Scheme 3, the review will also describe MO-based methods that generate VB-type information which can be used for some specific VB applications. The block-localized wave function (BLW) method123
127 is a type of VB method that utilizes an HF wave function with block-localized orbitals. By partitioning the mo￾lecular orbitals to the subgroups of a molecule, a BLW can describe a specific VB structure at the HF level. Thus, the BLW is capable of computing delocalization/resonance energies and charge transfer effects among molecules. The BLW approach is related to the early Kollmar method,128 wherein the subgroup orbitals were input and the energy for this so-generated “localized” reference was computed at zero iteration, without optimization of the orbitals. Both methods belong to a general class of MO- and density functional theory (DFT)-based energy decomposition analysis (EDA) approaches129
133 which use as a reference either a fragment-localized wave function or a density and thereby estimate the various interactions between the frag￾ments, thus providing VB-related information from MO or DFT calculations. For space economy, we shall limit our coverage to the BLW method since this method performs the energy decomposition closer to the VB spirit compared to other EDA methods. The molecular orbital valence bond method (MOVB)125 is an extension of the BLW method, which uses a multireference wave function, thus allowing calculation of the electronic coupling energy resulting from the mixing of two or more block-localized structures. The structure of the review follows the above ordering of VB methods, which are detailed in section 2. This methodology section, which will certainly interest the computation-oriented reader, is followed by applications which demonstrate the cap￾ability of VB theory to lead to lucid physical insight into a variety of problems, now approaching “real size”. Then section 4 describes algorithms and techniques which make modern VB theory faster and more efficient. Lastly, section 5 illustrates the current capabilities of modern VB methods by displaying ab initio VB calculations for a sizable molecular system, (CO)4Fe(C2H4). The review is written in such a way that the application-oriented reader who is less interested in the methodological details can skip parts of section 2 and then proceed to the applications in sections 3 and 5. 2. AB INITIO VALENCE BOND METHODS 2.1. Theoretical Background In VB theory, a many-electron wave function is expressed in terms of VB functions: Ψ ¼ ∑ K CKΦK ð1Þ where the VB function ΦK corresponds to a classical VB struc￾ture. In quantum chemistry, any state function ΦK should be a spin eigenfunction that is antisymmetric with respect to permu￾tations of electron indices. In general, a VB function is of the form ΦK ¼ A^Ω0ΘK ð2Þ where A^ is an antisymmetrizer, Ω0 is a direct product of orbitals {ϕi } as Ω0 ¼ ϕ1ð1Þ ϕ2ð2Þ ::: ϕNðNÞ ð3Þ and ΘK is a spin-paired spin eigenfunction,134 defined as ΘK ¼ 2
1=2 ½Rðk1Þ βðk2Þ
βðk1Þ Rðk2Þ 2
1=2 ½Rðk3Þ βðk4Þ
βðk3Þ Rðk4Þ ::: RðkpÞ ::: RðkNÞ ð4Þ 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 electron pairing schemes may be selected by using the Rumer diagrams.135 In a Rumer diagram, we set down the electron indices, 1, 2, ..., N, in a ring, representing each factor 2
1/2[R(ki ) β(kj )
β(ki ) R(kj )] in eq 4 by an arrow from i to j. On the basis of the Rumer rule, the independent Rumer structure set is obtained by drawing all possible Rumer diagrams in which there are no crossed arrows. Scheme 4 shows the five Rumer structures for the benzene molecule, where, as is well￾known, the first two are Kekul
e structures and the last three are Dewar structures. Rumer’s rule is applicable for singlet states with spin quantum number S = 0. To extend to the general spin S, extended Rumer diagrams136 should be applied, where a pole is added in the diagram. A VB function with a Rumer spin function is called a Heitler
London
Slater
Pauling (HLSP) function. An alternative way of writing the wave function is by use of a Slater determinant form, which will be used in this review. For example, the (k1, k2) bond-paired wave function will be given by ΦK ¼ j:::ðϕk1 ϕ̅k2
ϕ̅k1 ϕk2 Þ:::j ð5Þ where the bar over the orbital denotes a β spin while lack of it denotes spin R. In turn, ΦK will be written as a product of the Slater determinant forms for all the bond pairs. The coefficients CK in eq 1 can conveniently be determined by solving the secular equation HC = EMC, where Hamiltonian and overlap matrices are defined as follows: HKL ¼ ÆΦKjHjΦLæ ð6Þ Scheme 4. Five Rumer Structures for the Benzene Molecule