Chemcal Reviews：Classical Valence Bond Approach by Modern Methods

CHEMICAL REVIEW REVIEWS pubs.acs.org/CR Classical Valence Bond Approach by Modern Methods Wei Wu,+Peifeng Su,Sason Shaik,and Philippe C.Hiberty" The State Key Laboratory of Physical Chemistry of Solid Surfaces,Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry,and College of Chemistry and Chemical Engineering,Xiamen University,Xiamen,Fujian 361005,China Institute of Chemistry and The Lise Meitner-Minerva Center for Computational Quantum Chemistry,The Hebrew University, Jerusalem 91904,Israel SLaboratoire de Chimie Physique,Groupe de Chimie Theorique,CNRS UMR 8000,Universite de Paris-Sud,91405 Orsay Cedex, France CONTENTS 3.5.1.Magnitude of Hyperconjugation in 1.Introduction A Ethane T 1.1.Family of Classical Valence Bond Methods C 3.5.2.Physical Origin of the Saytzeff Rule U 2.Ab Initio Valence Bond Methods D 3.5.3.Tetrahedranyltetrahedrane V 2.1.Theoretical Background D 3.6.VBSCF and BLW Applications to Aromaticity 2.2.VB Methods of the HLSP Type E 3.6.1.Energetic Measure of Aromaticity and W 2.2.1.VBSCF Method E Antiaromaticity 3.6.2.Cyclopropane:Theoretical Study of 2.2.2.BOVB Method E o-Aromaticity W 2.2.3.VBCI Method F 3.7.Electronic Structure of Conjugated Molecules 十 2.2.4.VBPT2 Method G 3.7.1.Ground States of S2N2 and S2+ + 2.3.Add-Ons:VB Methods for the Solution Phase H 3.7.2.o-and -Aromatic Dianion Al2- 2.3.1.VBPCM Method H 4.Algorithm Advances in ab Initio VB Methods Z 2.3.2.VBSM Method 4.1.VB Wave Function and Hamiltonian Matrix Z 2.3.3.Combined VB/MM Method 4.2.Orbital Optimization in the VBSCF Procedure 2.4.Molecular Orbital Methods That Provide 4.3.Spin-Free Form of VB Theory AA Valence-Bond-Type Information 4.4.Paired-Permanent-Determinant Approach AA 2.4.1.BLW Method 4.5.Direct VBSCF/BOVB Algorithm AB 2.4.2.MOVB Method 5.Current Capabilities of ab Initio VB Methods A 3.Applications 6.Concluding Remarks AD 3.1.Accuracy of Modern VB Methods Appendix:Some Available VB Software AD 3.2.Chemical Reactivity K XMVB Program AD 3.2.1.Hydrogen Abstraction Reactions L TURTLE Software AD 3.2.2.SN2 Reactions in the Gas Phase M VB2000 Software AE 3.2.3.SN2 Reactions in the Aqueous Phase from CRUNCH Software AE an Implicit Solvent Model SCVB Software AE 3.2.4.SN2 Reactions in the Aqueous Phase from Author Information Molecular Dynamics with Explicit Solvent Biographies Molecules Q Acknowledgment 3.3.Excited States of Polyenes and Polyenyl Radicals R List of Abbreviations 3.4.Quantitative Evaluation of Common Chemical References G Paradigms R 3.4.1.BLW Examination of the Role of Conju- gation in the Rotational Barrier of Amides R 1.INTRODUCTION 3.4.2.VBSCF Application to Through-Bond ver- Quantum mechanics has provided chemistry with two general sus Through-Space Coupling in 1,3- theories of bonding:valence bond(VB)theory and molecular Dehydrobenzene orbital (MO)theory.The two theories were developed at about 3.5.Direct Estimate of Hyperconjugation Energies Received:July 20,2010 by VBSCF and BLW Methods ACS PublicationsAmercan chemkcal soety dx.dol.org/10.1021/cr100228r Chem.Rev.XXXX,XXX,000-000

rXXXX American Chemical Society A dx.doi.org/10.1021/cr100228r | Chem. Rev. XXXX, XXX, 000–000 REVIEW pubs.acs.org/CR Classical Valence Bond Approach by Modern Methods Wei Wu,*,† Peifeng Su,† Sason Shaik,*,‡ and Philippe C. Hiberty*,§ † The State Key Laboratory of Physical Chemistry of Solid Surfaces, Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry, and College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, Fujian 361005, China ‡ Institute of Chemistry and The Lise Meitner-Minerva Center for Computational Quantum Chemistry, The Hebrew University, Jerusalem 91904, Israel § Laboratoire de Chimie Physique, Groupe de Chimie Th
eorique, CNRS UMR 8000, Universit
e de Paris-Sud, 91405 Orsay C
edex, France CONTENTS 1. Introduction A 1.1. Family of Classical Valence Bond Methods C 2. Ab Initio Valence Bond Methods D 2.1. Theoretical Background D 2.2. VB Methods of the HLSP Type E 2.2.1. VBSCF Method E 2.2.2. BOVB Method E 2.2.3. VBCI Method F 2.2.4. VBPT2 Method G 2.3. Add-Ons: VB Methods for the Solution Phase H 2.3.1. VBPCM Method H 2.3.2. VBSM Method I 2.3.3. Combined VB/MM Method I 2.4. Molecular Orbital Methods That Provide Valence-Bond-Type Information J 2.4.1. BLW Method J 2.4.2. MOVB Method J 3. Applications K 3.1. Accuracy of Modern VB Methods K 3.2. Chemical Reactivity K 3.2.1. Hydrogen Abstraction Reactions L 3.2.2. SN2 Reactions in the Gas Phase M 3.2.3. SN2 Reactions in the Aqueous Phase from an Implicit Solvent Model O 3.2.4. SN2 Reactions in the Aqueous Phase from Molecular Dynamics with Explicit Solvent Molecules Q 3.3. Excited States of Polyenes and Polyenyl Radicals R 3.4. Quantitative Evaluation of Common Chemical Paradigms R 3.4.1. BLW Examination of the Role of Conju￾gation in the Rotational Barrier of Amides R 3.4.2. VBSCF Application to Through-Bond ver￾sus Through-Space Coupling in 1,3- Dehydrobenzene S 3.5. Direct Estimate of Hyperconjugation Energies by VBSCF and BLW Methods T 3.5.1. Magnitude of Hyperconjugation in Ethane T 3.5.2. Physical Origin of the Saytzeff Rule U 3.5.3. Tetrahedranyltetrahedrane V 3.6. VBSCF and BLW Applications to Aromaticity V 3.6.1. Energetic Measure of Aromaticity and Antiaromaticity W 3.6.2. Cyclopropane: Theoretical Study of σ-Aromaticity W 3.7. Electronic Structure of Conjugated Molecules X 3.7.1. Ground States of S2N2 and S4 2+ X 3.7.2. σ- and π-Aromatic Dianion Al4 2
Y 4. Algorithm Advances in ab Initio VB Methods Z 4.1. VB Wave Function and Hamiltonian Matrix Z 4.2. Orbital Optimization in the VBSCF Procedure Z 4.3. Spin-Free Form of VB Theory AA 4.4. Paired-Permanent-Determinant Approach AA 4.5. Direct VBSCF/BOVB Algorithm AB 5. Current Capabilities of ab Initio VB Methods AB 6. Concluding Remarks AD Appendix: Some Available VB Software AD XMVB Program AD TURTLE Software AD VB2000 Software AE CRUNCH Software AE SCVB Software AE Author Information AE Biographies AE Acknowledgment AF List of Abbreviations AF References AG 1. INTRODUCTION Quantum mechanics has provided chemistry with two general theories of bonding: valence bond (VB) theory and molecular orbital (MO) theory. The two theories were developed at about Received: July 20, 2010

Chemical Reviews REVIEW Scheme 1.VB Wave Functions for a Two-Electron Bond In a philosophy similar to that of GVB,Gerratt,Raimondi,and between Atoms A and B Cooper developed their VB method known as the spin-coupled (SC)theor 23-31 and its CI-augmented version,the so-called (a)Classical VB with pure HAOs SCVB.26,32-35 Like GVB,SC/SCVB theory relies on semiloca- A -B Xo=A Xh= lized orbitals and includes formally covalent configurations only 平B=Cx元-zmod+C2 xaXol+C3xbz The difference between SC and GVB methods is that the former releases the orthogonality and perfect-pairing restrictions,which (b)GVB/SC VB with semi-localized orbitals are usually used in GVB applications.Thus,in SC all orbitals are 中=A○○B allowed to be nonorthogonal,and all possible spin couplings GVBSC=N(φ-pa) between the singly occupied orbitals are included in the wave function.The SC and SCVB methods were applied to aromatic and antiaromatic molecules the alyl radica Diels-Alder and the same time,but quickly diverged into two schools that have retro-Diels-Alder reactions,sigmatropic rearrangements,-47 competed on charting the mental map of chemistry.Until the 1,3-dipolar cycloadditions-so and so on. mid-1950s,chemistry was dominated by classical VB theory, Another VB method that was developed also starting in which expresses the molecular wave function as a combination of the 1980s is a semiempirical method based on the Heisenberg explicit covalent and ionic structures based on pure atomic Hamiltonian (HH)and AO determinants rather than spin- orbitals (AOs)or hybrid atomic orbitals(HAOs),as illustrated adapted VB structures.Initially,the method used semiempirical in Scheme la.However,the computational effort required to per- parameters and a zero-differential overlap approximation and was form ab initio calculations in the classical VB framework proved applied to the ground and excited states of hydrocarbonssi-s7 to be overly demanding,and as such,the theory was employed in and metal clusters.ss A nonempirical geometry-dependent ver- an oversimplied manner,neglecting ionic structures and using sion was subsequently derived in which the parameters were nonoptimized orbitals.At the same time when this early ab initio extracted from accurate ab initio calculations on simple mole- VB theory was lacking accuracy and did not progress,MO theory cules.960 The latter calculations use orthogonalized AOs,which was enjoying efficient implementations,which have provided consequently possess significant delocalization tails,but result in the chemical community with computational software of ever- computer-time savings.The method has been applied to ground increasing speeds and capabilities.VB theory was unable to come and excited states of conjugated hydrocarbons,s9 -61 heteroatomic up with equally popular and useful software,and as such it has conjugated systems,polyynes,6 and so on.Eventually this ab gradually fallen into disrepute and was almost completely aban- initio-parametrized method led to the molecular mechanics/ doned.Thus,MO theory took over. valence bond (MM/VB)method of Robb and Bearpark,65-73 However,from the 1980s onward,VB theory started making a which was extensively used for demonstrating conical intersec- strong comeback and has since enjoyed a renaissance,including tions in photochemical reactions.7071,74-76 the ab initio method development of the theory.A common Since VB theory is well-known for its deep chemical insight feature of all modern VB methods is the simultaneous optimiza- many methods have sprung to extract VB information from MO- tion of the orbitals and the coefficients of the VB structures, based methods.Some of these methods involve mapping of MO- which thereby lead to an improved accuracy.However,the and Cl-augmented wave functions into valence bond structures various modern VB methods differ in the manners by which the and can be dated to the pioneering studies of Slater and van Vleck VB orbitals are defined. and later to Moffitt in his treatment of electronic spectra for large The modern era began when one of the pioneers of ab initio molecules.77 The first practical implementation of Hartree- VB theory,Goddard,and his co-workers developed the general- Fock(HF)and post-HF wave functions was made by Hiberty ized VB (GVB)method,-s which employed semilocalized and Leforestier,who created such a "VB transcriptor"in 1978 atomic orbitals (having small delocalization tails as in Scheme 1b) and treated many molecules by showing the VB content of used originally by Coulson and Fischer for the H2 molecule.The their MO and MO-CI wave functions.Since then,the problem GVB theory does not incorporate covalent and ionic structures has been explored by others,for example,by Karafiloglou,?9 explicitly,but instead uses formally covalent structures based on Bachler,so,s1 Malrieu,and so on.Some important develop- semilocalized orbitals,which implicitly incorporate the contribu- ments along these lines were made by Cooper et al.ss-87 who tions of ionic structures to bonding (see Scheme 1b).This computed a wave function of the SC type by projecting CASSCF enables a drastic reduction of the number of VB structures;for wave functions onto VB structures using maximum overlap criteria example,the t-system of benzene requires a total number of 175 There are also various methods of VB readings of CASSCF wave covalent and ionic VB structures based on pure AOs compared functions through orbital localization techniques with only five formally covalent Kekule and Dewar structures and through wave function transformation using nonorthogonal based on semilocalized orbitals.It is noted that the GVB method orbitals 89,93 as implemented by Goddard is completely equivalent to a Concurrently to the developments of all the above methods strongly orthogonal geminal anzatz with two orbitals per pair. the progress that has occurred in computer technology and in Further progress was made after the initial development of the computational methodologies has enabled the re-emergence of method,when the GVB wave functions were used as starting points modern forms of classical VB in which both orbitals and the struc- for further configuration interaction (CI)or perturbative treat- tural coefficients are simultaneously optimized.The advantage of ments of electron correlation.-12 The method was applied, these modern classical VB methods over other brands of VB among others,to the electronic structure of 1,3-dipoles, 13-19 theory is two-fold:(i)owing to the strictly local characters of the resonance in the allyl radicalor cyclobutadiene, 17 dissocia- employed orbitals (either purely atomic or purely localized on tion energhalogen exchange reactionss organometallic fragments as in Scheme la above),the VB structures are very complexes, and so on. clearly interpreted and as close as possible to the intuitive Lewis dx.dol.org/10.1021/cr100228rChem.Rev.XXXX,XXX,000-000

B dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW the same time, but quickly diverged into two schools that have competed on charting the mental map of chemistry. Until the mid-1950s, chemistry was dominated by classical VB theory, which expresses the molecular wave function as a combination of explicit covalent and ionic structures based on pure atomic orbitals (AOs) or hybrid atomic orbitals (HAOs), as illustrated in Scheme 1a. However, the computational effort required to per￾form ab initio calculations in the classical VB framework proved to be overly demanding, and as such, the theory was employed in an oversimplied manner, neglecting ionic structures and using nonoptimized orbitals. At the same time when this early ab initio VB theory was lacking accuracy and did not progress, MO theory was enjoying efficient implementations, which have provided the chemical community with computational software of ever￾increasing speeds and capabilities. VB theory was unable to come up with equally popular and useful software, and as such it has gradually fallen into disrepute and was almost completely aban￾doned. Thus, MO theory took over. However, from the 1980s onward, VB theory started making a strong comeback and has since enjoyed a renaissance, including the ab initio method development of the theory. A common feature of all modern VB methods is the simultaneous optimiza￾tion of the orbitals and the coefficients of the VB structures, which thereby lead to an improved accuracy. However, the various modern VB methods differ in the manners by which the VB orbitals are defined. The modern era began when one of the pioneers of ab initio VB theory, Goddard, and his co-workers developed the general￾ized VB (GVB) method,1
5 which employed semilocalized atomic orbitals (having small delocalization tails as in Scheme 1b) used originally by Coulson and Fischer for the H2 molecule.6The GVB theory does not incorporate covalent and ionic structures explicitly, but instead uses formally covalent structures based on semilocalized orbitals, which implicitly incorporate the contribu￾tions of ionic structures to bonding (see Scheme 1b). This enables a drastic reduction of the number of VB structures; for example, the π-system of benzene requires a total number of 175 covalent and ionic VB structures based on pure AOs compared with only five formally covalent Kekul
e and Dewar structures based on semilocalized orbitals. It is noted that the GVB method as implemented by Goddard is completely equivalent to a strongly orthogonal geminal anzatz with two orbitals per pair. Further progress was made after the initial development of the method, whenthe GVB wave functions were used as starting points for further configuration interaction (CI)7,8 or perturbative treat￾ments of electron correlation.9
12 The method was applied, among others, to the electronic structure of 1,3-dipoles,13
15 resonance in the allyl radical16 or cyclobutadiene,17 dissocia￾tion energies,7 halogen exchange reactions,18 organometallic complexes,19
22 and so on. In a philosophy similar to that of GVB, Gerratt, Raimondi, and Cooper developed their VB method known as the spin-coupled (SC) theory23
31 and its CI-augmented version, the so-called SCVB.26,32
35 Like GVB, SC/SCVB theory relies on semiloca￾lized orbitals and includes formally covalent configurations only. The difference between SC and GVB methods is that the former releases the orthogonality and perfect-pairing restrictions, which are usually used in GVB applications. Thus, in SC all orbitals are allowed to be nonorthogonal, and all possible spin couplings between the singly occupied orbitals are included in the wave function. The SC and SCVB methods were applied to aromatic and antiaromatic molecules,35
41 the allyl radical,42 Diels
Alder and retro-Diels
Alder reactions,43,44 sigmatropic rearrangements,45
47 1,3-dipolar cycloadditions,48
50 and so on. Another VB method that was developed also starting in the 1980s is a semiempirical method based on the Heisenberg Hamiltonian (HH) and AO determinants rather than spin￾adapted VB structures. Initially, the method used semiempirical parameters and a zero-differential overlap approximation and was applied to the ground and excited states of hydrocarbons51
57 and metal clusters.58 A nonempirical geometry-dependent ver￾sion was subsequently derived in which the parameters were extracted from accurate ab initio calculations on simple mole￾cules.59,60 The latter calculations use orthogonalized AOs, which consequently possess significant delocalization tails, but result in computer-time savings. The method has been applied to ground and excited states of conjugated hydrocarbons,59
61 heteroatomic conjugated systems,62 polyynes,63,64 and so on. Eventually this ab initio-parametrized method led to the molecular mechanics/ valence bond (MM/VB) method of Robb and Bearpark,65
73 which was extensively used for demonstrating conical intersec￾tions in photochemical reactions.70,71,74
76 Since VB theory is well-known for its deep chemical insight, many methods have sprung to extract VB information from MO￾based methods. Some of these methods involve mapping of MO￾and CI-augmented wave functions into valence bond structures and can be dated to the pioneering studies of Slater and van Vleck and later to Moffitt in his treatment of electronic spectra for large molecules.77 The first practical implementation of Hartree
Fock (HF) and post-HF wave functions was made by Hiberty and Leforestier,78 who created such a “VB transcriptor” in 1978 and treated many molecules by showing the VB content of their MO and MO
CI wave functions. Since then, the problem has been explored by others, for example, by Karafiloglou,79 Bachler,80,81 Malrieu,82
84 and so on. Some important develop￾ments along these lines were made by Cooper et al.,85
87 who computed a wave function of the SC type by projecting CASSCF wave functions onto VB structures using maximum overlap criteria. There are also various methods of VB readings of CASSCF wave functions through orbital localization techniques80,81,83,84,88
92 and through wave function transformation using nonorthogonal orbitals.89,93 Concurrently to the developments of all the above methods, the progress that has occurred in computer technology and in computational methodologies has enabled the re-emergence of modern forms of classical VB in which both orbitals and the struc￾tural coefficients are simultaneously optimized. The advantage of these modern classical VB methods over other brands of VB theory is two-fold: (i) owing to the strictly local characters of the employed orbitals (either purely atomic or purely localized on fragments as in Scheme 1a above), the VB structures are very clearly interpreted and as close as possible to the intuitive Lewis Scheme 1. VB Wave Functions for a Two-Electron Bond between Atoms A and B

Chemical Reviews REVIEW Scheme 2.Complete Set of VB Structures for a Hydrogen Scheme 3.Tree of Modern VB Methods That Are Based on Abstraction Process Classical VB Theory X· 日一X X·H产X X·HX VB structures set: Covalent and ionic structures for the active electrons XHX' X:HX X H X 5 6 VBCI +CI HAOS Breathing VBSCF Orbitals X'H.:X X:H·X PT2 VBPT2 BOVB structures that constitute the language of chemists;(ii)because the Add-ons for Condensed Phases covalent and ionic structures are explicitly considered,it is possible to +PCM +SM +MM meaningfully calculate their weights or their quasi-variational en- ergies.As such,classical VB allows the VB structures to be dearly and accurately defined,which,as will be seen,is important for calculation VBPCM VBSM VB/MM of resonance energies,for diagrams in chemical reactivity,for in- depth study of the nature of chemical bonds,and so on. The resurgence of modern classical VB theory involves the development of several methodological advances which allowed covalent and six ionic structures which distribute the three new and more accurate applications of the theoryThus, "active electrons"in the three HAOs of the X---H---X'system. several significant advances have been achieved in overcoming the The wave function is a linear combination of such a structure set notorious"N!problem",associated with the nonorthogonality in and is optimized with respect to both the structural coefficients the VB method,and direct algorithms have considerably increased and the HAOs.The methods differ from each other by the levels the speed of VB computations,which are nowadays much faster in which the dynamic correlation energy is incorporated into than they used to be in the past.During this effort to speed up VB the calculations.The relationships among the various methods calculations,there have emerged VB methods that also enable follow a philosophy similar to the one used in ab initio MO- quantitative accuracy to be achieved.Thus,dynamic correlation based theory. has been incorporated into VB calculations,so that,at present, The tree ofthese VB methods is shown in Scheme 3.The basic sophisticated VB methods are able to achieve the accuracy of high- method,which was devised by Balint-Kurti and van Lenthe 94s level post-HF methods.Very recently,classical VB theory was is called the valence bond self-consistent field(VBSCF)method. extended to handle species and reactions in solution and is also The method optimizes VB orbitals and structural coefficients capable of treating transition-metal complexes. simultaneously and uses the same set of HAOs for all the struc- Indeed the VB culture is broad and involves a variety of tures.This is analogous to the MO-based CASSCF method,and techniques and approaches,and a review of all the methods both methods should be numerically quasi-identical if all struc- would be vast and too diffuse compared with a more focused tures for a given dimension of the active space are included.How- review that may benefit the general reader.Thus,since ever,usually VB methods employ only a few structures that are methods that use semilocalized orbitals such as GVB,SCVB essential for describing the system of interest and use the strictly HH,and MM/VB have already been amply reviewed localized orbitals.Consequently,the VBSCF results are often before,1-4,23-31,53-56,69-71 we shall not review these meth- less accurate than those of CASSCF.Nevertheless,both of the ods again,and the reader is advised to consult the existing methods include some degree of static electron correlation,but authoritative sources.The present review will focus on those lack dynamic correlation. modern VB methods that are based on classical VB theory, The VBSCF method branches into two sets of methods which,we recall,deals with purely localized orbitals and The one to the right is the breathing-orbital VB (BOVB) explicit consideration of covalent and ionic structures.The method,1 where one uses the same VBSCF wave func- combination of the lucid insight of VB into chemistry and the tion,but with an additional degree of freedom that allows the new computational methods is discussed in this review, HAOs to be different for the different structures.Thus,the hopefully establishing a case for the return of VB theory to orbitals adapt themselves to the instantaneous field of each the classroom and to the laboratory bench in the service of structure,which has the effect of introducing the dynamic cor- experimental chemists. relation that is necessary to provide accurate energies.The two branches to the left in Scheme 3 are two alternative ways of 1.1.Family of Classical Valence Bond Methods improving VBSCF by introducing dynamic correlation.This is In all the methods of the classical VB type to be described done by means of post-SCF treatments that are analogous to herein,one uses an active shell,which involves the electrons MRCI and MRPT2 in the MO theory.In the valence bond that participate in the electronic reorganization in a process, configuration interaction method (VBCI),7 the VBSCE which can be bond making/breaking or a reaction such as SN2, energy and wave function are improved by CI.On the other Diels-Alder,and so on.These electrons are then distributed in hand,the valence bond second-order perturbation method the valence atomic orbitals or HAOs to generate all the possible (VBPT2)118 uses perturbation theory,taking the VBSCF wave covalent and ionic structures.Scheme 2 shows this set of function as the zeroth-order reference.It is worthwhile to structures for a hydrogen abstraction process;there are two emphasize that,despite the excited VB structures that are dx.dol.org/10.1021/cr100228r Chem.Rev.XXXX,XXX,000-000

C dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW structures that constitute the language of chemists; (ii) because the covalent and ionic structures are explicitly considered, it is possible to meaningfully calculate their weights or their quasi-variational en￾ergies. As such, classical VB allows the VB structures to be clearly and accurately defined, which, as will be seen, is important for calculation of resonance energies, for diagrams in chemical reactivity, for in￾depth study of the nature of chemical bonds, and so on. The resurgence of modern classical VB theory involves the development of several methodological advances which allowed new and more accurate applications of the theory.94
112 Thus, several significant advances have been achieved in overcoming the notorious “N! problem”, associated with the nonorthogonality in the VB method, and direct algorithms have considerably increased the speed of VB computations, which are nowadays much faster than they used to be in the past. During this effort to speed up VB calculations, there have emerged VB methods that also enable quantitative accuracy to be achieved. Thus, dynamic correlation has been incorporated into VB calculations, so that, at present, sophisticated VB methods are able to achieve the accuracy of high￾level post-HF methods. Very recently, classical VB theory was extended to handle species and reactions in solution and is also capable of treating transition-metal complexes. Indeed the VB culture is broad and involves a variety of techniques and approaches, and a review of all the methods would be vast and too diffuse compared with a more focused review that may benefit the general reader. Thus, since methods that use semilocalized orbitals such as GVB, SCVB, HH, and MM/VB have already been amply reviewed before,1
4,23
31,53
56,69
71 we shall not review these meth￾ods again, and the reader is advised to consult the existing authoritative sources. The present review will focus on those modern VB methods that are based on classical VB theory, which, we recall, deals with purely localized orbitals and explicit consideration of covalent and ionic structures. The combination of the lucid insight of VB into chemistry and the new computational methods is discussed in this review, hopefully establishing a case for the return of VB theory to the classroom and to the laboratory bench in the service of experimental chemists. 1.1. Family of Classical Valence Bond Methods In all the methods of the classical VB type to be described herein, one uses an active shell, which involves the electrons that participate in the electronic reorganization in a process, which can be bond making/breaking or a reaction such as SN2, Diels
Alder, and so on. These electrons are then distributed in the valence atomic orbitals or HAOs to generate all the possible covalent and ionic structures. Scheme 2 shows this set of structures for a hydrogen abstraction process; there are two covalent and six ionic structures which distribute the three “active electrons” in the three HAOs of the X---H---X0 system. The wave function is a linear combination of such a structure set and is optimized with respect to both the structural coefficients and the HAOs. The methods differ from each other by the levels in which the dynamic correlation energy is incorporated into the calculations. The relationships among the various methods follow a philosophy similar to the one used in ab initio MO￾based theory. The tree of these VB methods is shown in Scheme 3. The basic method, which was devised by Balint-Kurti and van Lenthe,94,95 is called the valence bond self-consistent field (VBSCF) method. The method optimizes VB orbitals and structural coefficients simultaneously and uses the same set of HAOs for all the struc￾tures. This is analogous to the MO-based CASSCF method, and both methods should be numerically quasi-identical if all struc￾tures for a given dimension of the active space are included. How￾ever, usually VB methods employ only a few structures that are essential for describing the system of interest and use the strictly localized orbitals. Consequently, the VBSCF results are often less accurate than those of CASSCF. Nevertheless, both of the methods include some degree of static electron correlation, but lack dynamic correlation. The VBSCF method branches into two sets of methods. The one to the right is the breathing-orbital VB (BOVB) method,113
115 where one uses the same VBSCF wave func￾tion, but with an additional degree of freedom that allows the HAOs to be different for the different structures. Thus, the orbitals adapt themselves to the instantaneous field of each structure, which has the effect of introducing the dynamic cor￾relation that is necessary to provide accurate energies. The two branches to the left in Scheme 3 are two alternative ways of improving VBSCF by introducing dynamic correlation. This is done by means of post-SCF treatments that are analogous to MRCI and MRPT2 in the MO theory. In the valence bond configuration interaction method (VBCI),116,117 the VBSCF energy and wave function are improved by CI. On the other hand, the valence bond second-order perturbation method (VBPT2)118 uses perturbation theory, taking the VBSCF wave function as the zeroth-order reference. It is worthwhile to emphasize that, despite the excited VB structures that are Scheme 2. Complete Set of VB Structures for a Hydrogen Abstraction Process Scheme 3. Tree of Modern VB Methods That Are Based on Classical VB Theory

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,000-000

D 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

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,000-000

E 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

Chemical Reviews REVIEW Figure 1.L-BOVB-computed orbitals of F2.Note the size difference in the left-hand vs right-hand side orbitals of the ionic forms.The two F Figure 2.SL-BOVB split p=orbital of the F fragment in the FF atoms lie on the horizontal z-axis. ionic structure in F2.The p orbital is split into two singly occupied orbitals,one diffuse (faint color)and one more compact (strong color) Red and yellow correspond to different signs of the lobes. neutral,or ionic fragments.These differences are significant enough to be apparent with the naked eye,as shown in Figure 1, correlation via single-reference methods such as CISD and multi which displays some optimized z-orbitals for the three VB structures of difluorine,obtained by a BOVB calculation using reference-based methods such as MRCI.Similar ideas have also the 6-31+G**basis set. been applied to VB theory.25,116117,147-153 The CI technique usually requires a huge number of excited configurations;however, Thus,both the active and spectator orbitals can be viewed as instantaneously following the charge fluctuation by rearranging the aim of the VBCI method is to retain the conceptual clarity of in size and shape,hence the name "breathing orbital".The the VBSCF method while improving the energetic aspect by introducing further electron correlation. physical meaning of this"breathing-orbital effect"can be grasped by remembering that the CASSCF and VBSCF levels only bring The VBCI method is a post-VBSCF approach,where the initially calculated VBSCF wave function is used as a reference. nondynamic electron correlation and that the missing dynamic The VBCI wave function augments the VBSCF wave function correlation is obtained by further Cl involving single,double,etc. with excited VB structures,which are generated from the refer- excitations to outer valence orbitals.Now,as CI involving single excitations is physically equivalent to an orbital optimization ence wave function by replacing occupied (optimized VBSCF) orbitals with virtual orbitals.Different from MO-based methods, (to first order),it becomes clear that BOVB brings dynamic where virtual orbitals can be obtained from an SCF procedure correlation and is comparable to VBSCF singles-CI,with the further advantage that it keeps the wave function as compact as the virtual orbitals in VB theory are not available in the VBSCF calculation and should be defined for the VBCI method.To the VBSCF wave function.More specifically,BOVB confers only that part of the dynamic correlation that varies along a reaction generate physically meaningful excited structures,the virtual orbitals should also be strictly localized,like the occupied VB coordinate or throughout a potential surface.Therefore,it would be more exact to say that BOVB brings differential dynamic orbitals.In the original VBCI method,the virtual orbitals were correlation.As such,BOVB brings about better accuracy relative defined by using a projector: to the VBSCF,GVB,SC,and CASSCF levels,as shown in PA TA(TA+MATA)TA+SA (13) ek出hu th where Ta is the vector of orbital coefficients and the Ma and SA breathing orbitals and dynamic correlation is particularly well are,respectively,the overlap matrices of the occupied VB orbitals illustrated in three-electron bonds,where all the electron correla- and the basis functions,respectively,while the index A indicates tion is of dynamic nature. that all matrices are associated with fragment A.It can be shown The BOVB method has several lxevels of accuracy.At the most that the eigenvalues of the projector PA are 1 and 0.The basic level,referred to as L-BOVB,all orbitals are strictly localized eigenvectors associated with eigenvalue 1 are the occupied VB on their respective fragments.One way of improving the orbitals,while the eigenvectors associated with eigenvalue 0 are energetics is to increase the number of degrees of freedom by used as the virtual VB orbitals of fragment A.By diagonalizing the permitting the inactive orbitals to be delocalized.This option, projectors for all blocks,we can have all the virtual VB orbitals.A which does not alter the interpretability of the wave function, simpler way,which was implemented in the current versions of accounts better for the nonbonding interactions between the the VBCI and VBPT2 methods,uses Schmidt orthogonalization fragments and is referred to as D-BOVB.Another improvement that is imposed on each fragment. can be achieved by incorporating radial electron correlation in With localized occupied and virtual orbitals,one can generate the active orbitals of the ionic structures by allowing the doubly excited VB structures by replacing occupied orbitals with virtual occupied orbitals to split into two singly occupied orbitals that orbitals.To create chemically meaningful excited structures. are spin-paired.This option carries the label"S"(for split), the excitation should involve the replacement of an occupied leading to the SL-BOVB and SD-BOVB levels of calculation,the VB orbital only by those virtual orbitals that belong to the latter being the most accurate one.In this manner,the two same fragment as the occupied orbital.As such,the excited VB electrons are relocated into different regions of the space,as structure retains the same electronic pairing pattern and clearly seen in Figure 2,which shows the two split and spin- charge distribution asΦg.In other words,bothΦkandΦg paired p:orbitals of F in difluorine describe the same"classical"VB structure.A VBCI function 2.2.3.VBCI Method.An alternative way of introducing dy- is defined by adding all excited VB structures k to the namic correlation into the VB calculation is the VBCI method117 fundamental structure: which uses the configuration interaction technique to incorporate the correlation.In MO-based theory,configuration interaction Φg=∑CxΦk (14) provides a conceptually simple tool for describing dynamic F dx.dol.org/10.1021/cr100228r Chem.Rev.XXXX,XXX,000-000

F dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW neutral, or ionic fragments. These differences are significant enough to be apparent with the naked eye, as shown in Figure 1, which displays some optimized π-orbitals for the three VB structures of difluorine, obtained by a BOVB calculation using the 6-31+G** basis set. Thus, both the active and spectator orbitals can be viewed as instantaneously following the charge fluctuation by rearranging in size and shape, hence the name “breathing orbital”. The physical meaning of this “breathing-orbital effect”can be grasped by remembering that the CASSCF and VBSCF levels only bring nondynamic electron correlation and that the missing dynamic correlation is obtained by further CI involving single, double, etc., excitations to outer valence orbitals. Now, as CI involving single excitations is physically equivalent to an orbital optimization (to first order), it becomes clear that BOVB brings dynamic correlation and is comparable to VBSCF + singles-CI, with the further advantage that it keeps the wave function as compact as the VBSCF wave function. More specifically, BOVB confers only that part of the dynamic correlation that varies along a reaction coordinate or throughout a potential surface. Therefore, it would be more exact to say that BOVB brings differential dynamic correlation. As such, BOVB brings about better accuracy relative to the VBSCF, GVB, SC, and CASSCF levels, as shown in benchmark calculations of bond dissociation energies and reac￾tion barriers.113
115,145 The relationship between the effect of breathing orbitals and dynamic correlation is particularly well illustrated in three-electron bonds, where all the electron correla￾tion is of dynamic nature.146 The BOVB method has several lxevels of accuracy. At the most basic level, referred to as L-BOVB, all orbitals are strictly localized on their respective fragments. One way of improving the energetics is to increase the number of degrees of freedom by permitting the inactive orbitals to be delocalized. This option, which does not alter the interpretability of the wave function, accounts better for the nonbonding interactions between the fragments and is referred to as D-BOVB. Another improvement can be achieved by incorporating radial electron correlation in the active orbitals of the ionic structures by allowing the doubly occupied orbitals to split into two singly occupied orbitals that are spin-paired. This option carries the label “S” (for split), leading to the SL-BOVB and SD-BOVB levels of calculation, the latter being the most accurate one. In this manner, the two electrons are relocated into different regions of the space, as clearly seen in Figure 2, which shows the two split and spin￾paired pz orbitals of F
in difluorine. 2.2.3. VBCI Method. An alternative way of introducing dy￾namic correlation into the VB calculation is the VBCI method116,117, which uses the configuration interaction technique to incorporate the correlation. In MO-based theory, configuration interaction provides a conceptually simple tool for describing dynamic correlation via single-reference methods such as CISD and multi￾reference-based methods such as MRCI. Similar ideas have also been applied to VB theory.7,25,116,117,147
153 The CI technique usually requires a huge number of excited configurations; however, the aim of the VBCI method is to retain the conceptual clarity of the VBSCF method while improving the energetic aspect by introducing further electron correlation. The VBCI method is a post-VBSCF approach, where the initially calculated VBSCF wave function is used as a reference. The VBCI wave function augments the VBSCF wave function with excited VB structures, which are generated from the refer￾ence wave function by replacing occupied (optimized VBSCF) orbitals with virtual orbitals. Different from MO-based methods, where virtual orbitals can be obtained from an SCF procedure, the virtual orbitals in VB theory are not available in the VBSCF calculation and should be defined for the VBCI method. To generate physically meaningful excited structures, the virtual orbitals should also be strictly localized, like the occupied VB orbitals. In the original VBCI method, the virtual orbitals were defined by using a projector: PA ¼ TAðTA þMATAÞ
1 TA þSA ð13Þ where TA is the vector of orbital coefficients and the MA and SA are, respectively, the overlap matrices of the occupied VB orbitals and the basis functions, respectively, while the index A indicates that all matrices are associated with fragment A. It can be shown that the eigenvalues of the projector PA are 1 and 0. The eigenvectors associated with eigenvalue 1 are the occupied VB orbitals, while the eigenvectors associated with eigenvalue 0 are used as the virtual VB orbitals of fragment A. By diagonalizing the projectors for all blocks, we can have all the virtual VB orbitals. A simpler way, which was implemented in the current versions of the VBCI and VBPT2 methods, uses Schmidt orthogonalization that is imposed on each fragment.118 With localized occupied and virtual orbitals, one can generate excited VB structures by replacing occupied orbitals with virtual orbitals. To create chemically meaningful excited structures, the excitation should involve the replacement of an occupied VB orbital only by those virtual orbitals that belong to the same fragment as the occupied orbital. As such, the excited VB structure ΦK i retains the same electronic pairing pattern and charge distribution as ΦK 0 . In other words, both ΦK i and ΦK 0 describe the same “classical” VB structure. A VBCI function ΦK CI is defined by adding all excited VB structures ΦK i to the fundamental structure ΦK 0 : ΦCI K ¼ ∑ i Ci KΦi K ð14Þ Figure 1. L-BOVB-computed πy orbitals of F2. Note the size difference in the left-hand vs right-hand side orbitals of the ionic forms. The two F atoms lie on the horizontal z-axis. Figure 2. SL-BOVB split p1z orbital of the F
fragment in the F
F+ ionic structure in F2. The p1z orbital is split into two singly occupied orbitals, one diffuse (faint color) and one more compact (strong color). Red and yellow correspond to different signs of the lobes

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,000-000

G 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

Chemical Reviews REVIEW E()=C()+HOI(H-E(0)M1)-HIOC(0) (26) 0.12 2 52 where C()and C(1)are the coefficient column matrices of the 0.38 VBSCF,eq 21,and the first-order wave functions,eq 22, ■CASPT respectively,is the zeroth-order energy,and matricesH 0.88 H,H,and M are respectively defined as VBPT2 -1.38 ■BCIS (H)Rs=(ΦRHolΦs〉 (27) (H)kR=(④xHR》 (H)Rx=(Φx) Figure 3.Computational errors of bond dissociation energies relative to the FCI method.Aug-cc-pVTZ basis set for H2,DZP for N2 and O2, 28) and cc-pVTZ for F2 (M)s=(④RΦs】 (29) the ab initio VB level.VBPCM uses the IEF version of the PCM model,165-167 which is widely implemented in standard quantum In the above equations,index K and indices R and S are chemical programs.To incorporate solvent effects into a VB respectively for the fundamental and the excited structures.H is scheme,the state wave function is expressed in the usual terms as the total system Hamiltonian. a linear combination of VB structures,but now these VB It is obvious that the largest matrix is Ho,which is block- structures interact with one another in the presence of the diagonal,owing to the orthogonality constraints that are applied polarizing field of the solvent.The Schrodinger equation for to the different orbital sets.Therefore,the VBPT2 method is the VBPCM can be expressed as computationally efficient,compared to VBCI.No Hamiltonian matrix elements with nonorthogonal orbitals are required past (HO+VR)ΨVBPCM=EΨVBPCM (30) the VBSCF step.Owing to the orthogonality between different orbital groups,all matrix elements involved in the perturbation where Ho is the gas-phase Hamiltonian and the interaction correction procedure of VBPT2 may be easily computed by using potential Vg for the ith iteration is given as a function of the the Condon-Slater rules. electronic density of the (i-1)th iteration and is expressed in Though the VBPT2 wave function involves a large number of the form of one-electron matrix elements that are computed by a excited structures,the wave function is ultimately expressed in standard PCM procedure.The detailed procedures are as terms of a minimal number of fundamental structures,as in the follows: VBSCF,by partitioning the first-order wave function into the (1)A VBSCF procedure in a vacuum is performed,and the fundamental structures. electron density is computed. VBPT2 applications show that the method gives computa- (2)Given the electron density from step 1,effective one- tional results that are on par with those of the VBCISD method electron integrals are obtained by a standard PCM and match those of the MO-based MRCI and CASPT2 methods subroutine. (at the same basis sets).The total VBPT2 energies match those (3)A standard VBSCF calculation is carried out with the of CASPT2 if one uses a properly designed VB wave function as a effective one-electron integrals obtained from step 2.The reference.Figure 3 shows the computational errors of bond electron density is computed with the new optimized VB dissociation energies relactive to FCI with various methods, wave function. where 3 structures,1 covalent and 2 ionic,are involved for H2 (4)Repeat steps 2 and 3 until the energy difference between and F2,and 12 and 17 structures are used for O2 and N2. the two iterations reaches a given threshold. 2.3.Add-Ons:VB Methods for the Solution Phase Having the optimized wave function,the final energy of a 2.3.1.VBPCM Method.Solvation plays an important role in system in solution is evaluated by the molecular energy,structure,and properties.Theoretical treatments of solute-solvent interactions have been the subject E-Va) (31) of many studies in computational chemistry.In this sense,the continuum solvation model is one of the most economical tools By performing the above procedure,the solvent effect is taken for describing the solvation problem.A typical and commonly into account at the VBSCF level,whereby the orbitals and used continuum solvation model developed by Tomasiet a structural coefficients are optimized until self-consistency is is the PCM,wherein the solvent is represented as a homogeneous achieved.The VBPCM method enables one to study the energy medium,characterized by a dielectric constant and polarized by curve of the full VB state as well as that ofindividual VB structures the charge distribution of the solute.The interaction between the throughout the path of a chemical reaction and then reveal the solute charges and the polarized electric field of the solvent is solvent effect on the different VB structures as well as on the total taken into account through an interaction potential that is VB wave function.The method has been applied to the studies of embedded in the Hamiltonian and determined by a self-consistent SN2 reactions in aqueous solution, 168169 which will be reviewed reaction field(SCRF)procedure. in section 3,and to the heterolytic bond dissociation of C HoCl With its lucid insight into the understanding of chemical and CHSiCl in aqueous solution.Figure shows the reaction,VB theory is very well suited for elucidating solvent potential energy curves of C.HCl in the gas phase and in the effects in solution-phase reactions.Coupling the VB method with solvated phase,which illustrate intuitively different dissociation behaviors in the two different media,dissociating to radicals in was developed for exploring the solute-solvent interactions at the gas phase and to free ions in water. dx.dol.org/10.1021/cr100228r |Chem.Rev.XXXX,XXX,000-000

H dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW Eð2Þ ¼ Cð0ÞþH01ðH11 0
Eð0Þ M11Þ
1 H10Cð0Þ ð26Þ where C(0) and C(1) are the coefficient column matrices of the VBSCF, eq 21, and the first-order wave functions, eq 22, respectively, E(0) is the zeroth-order energy, and matrices H0 11, H01, H10, and M11 are respectively defined as ðH11 0 ÞRS ¼ ÆΦRjH^ 0jΦSæ ð27Þ ðH01ÞKR ¼ ÆΦKjH^ jΦRæ ðH10ÞRK ¼ ÆΦRjH^ jΦKæ ð28Þ ðM11ÞRS ¼ ÆΦRjΦSæ ð29Þ In the above equations, index K and indices R and S are respectively for the fundamental and the excited structures. H^ is the total system Hamiltonian. It is obvious that the largest matrix is H0 11, which is block￾diagonal, owing to the orthogonality constraints that are applied to the different orbital sets. Therefore, the VBPT2 method is computationally efficient, compared to VBCI. No Hamiltonian matrix elements with nonorthogonal orbitals are required past the VBSCF step. Owing to the orthogonality between different orbital groups, all matrix elements involved in the perturbation correction procedure of VBPT2 may be easily computed by using the Condon
Slater rules. Though the VBPT2 wave function involves a large number of excited structures, the wave function is ultimately expressed in terms of a minimal number of fundamental structures, as in the VBSCF, by partitioning the first-order wave function into the fundamental structures. VBPT2 applications show that the method gives computa￾tional results that are on par with those of the VBCISD method and match those of the MO-based MRCI and CASPT2 methods (at the same basis sets). The total VBPT2 energies match those of CASPT2 if one uses a properly designed VB wave function as a reference. Figure 3 shows the computational errors of bond dissociation energies relactive to FCI with various methods, where 3 structures, 1 covalent and 2 ionic, are involved for H2 and F2, and 12 and 17 structures are used for O2 and N2. 2.3. Add-Ons: VB Methods for the Solution Phase 2.3.1. VBPCM Method. Solvation plays an important role in the molecular energy, structure, and properties. Theoretical treatments of solute
solvent interactions have been the subject of many studies in computational chemistry. In this sense, the continuum solvation model is one of the most economical tools for describing the solvation problem. A typical and commonly used continuum solvation model developed by Tomasi et al.119,120 is the PCM, wherein the solvent is represented as a homogeneous medium, characterized by a dielectric constant and polarized by the charge distribution of the solute. The interaction between the solute charges and the polarized electric field of the solvent is taken into account through an interaction potential that is embedded in the Hamiltonian and determined by a self-consistent reaction field (SCRF) procedure. With its lucid insight into the understanding of chemical reaction, VB theory is very well suited for elucidating solvent effects in solution-phase reactions. Coupling the VB method with the PCM119,120,160
167 generates the VBPCM method,168 which was developed for exploring the solute
solvent interactions at the ab initio VB level. VBPCM uses the IEF version of the PCM model,165
167which is widely implemented in standard quantum chemical programs. To incorporate solvent effects into a VB scheme, the state wave function is expressed in the usual terms as a linear combination of VB structures, but now these VB structures interact with one another in the presence of the polarizing field of the solvent. The Schr€odinger equation for the VBPCM can be expressed as ðH0 þ VRÞΨVBPCM ¼ EΨVBPCM ð30Þ where H0 is the gas-phase Hamiltonian and the interaction potential VR for the ith iteration is given as a function of the electronic density of the (i
1)th iteration and is expressed in the form of one-electron matrix elements that are computed by a standard PCM procedure. The detailed procedures are as follows: (1) A VBSCF procedure in a vacuum is performed, and the electron density is computed. (2) Given the electron density from step 1, effective one￾electron integrals are obtained by a standard PCM subroutine. (3) A standard VBSCF calculation is carried out with the effective one-electron integrals obtained from step 2. The electron density is computed with the new optimized VB wave function. (4) Repeat steps 2 and 3 until the energy difference between the two iterations reaches a given threshold. Having the optimized wave function, the final energy of a system in solution is evaluated by E ¼ ÆΨVBPCMjH0 þ 1 2 VRjΨVBPCMæ ð31Þ By performing the above procedure, the solvent effect is taken into account at the VBSCF level, whereby the orbitals and structural coefficients are optimized until self-consistency is achieved. The VBPCM method enables one to study the energy curve of the full VB state as well as that of individual VB structures throughout the path of a chemical reaction and then reveal the solvent effect on the different VB structures as well as on the total VB wave function. The method has been applied to the studies of SN2 reactions in aqueous solution,168,169 which will be reviewed in section 3, and to the heterolytic bond dissociation of C4H9Cl and C3H9SiCl in aqueous solution.170 Figure 4 shows the potential energy curves of C4H9Cl in the gas phase and in the solvated phase, which illustrate intuitively different dissociation behaviors in the two different media, dissociating to radicals in the gas phase and to free ions in water. Figure 3. Computational errors of bond dissociation energies relative to the FCI method. Aug-cc-pVTZ basis set for H2, DZP for N2 and O2, and cc-pVTZ for F2

Chemical Reviews REVIEW 200 2007 180 180 160 1e0 140 140 R+:X 120 120 100 100 R·X 80- ◆ 60 RX R*:X 40 20 0 Bond length(A) Bond length(A) (a) (b) Figure 4.Dissociation curves of RX(R=(CH3)3C)(a)in the gas phase and(b)in the solvated phase. 2.3.2.VBSM Method.The VBPCM method incorporates qi(R)=Zk- (Sp.p-s/P)aa (34) PCM into the VB method,while the VBSM method applies the a∈k SMx(x=1-8)2L,171,172 solvation model.SMx (x=1-8), developed by Truhlar and co-workers,treats the electrostatics respectively,where Zk is the nuclear charge of atom k, due to bulk solvent by the generalized Born approximation with a is a basis function on atom k,and P and S are the one- self-consistent partial atomic charges.21 In the SMx(=1-8) electron density and overlap matrices,respectively.Then modelthe electrostatic ree energy of soation isaug the generalized Born polarization energy 21,171,172is mented by terms proportional to the solvent-accessible surface calculated by areas of the solute's atoms times empirical geometry-dependent 1-司9R)9R)R) 1 Gp = atomic surface tensions,accounting for cavitation,dispersion, and solvent structure effects,where the solvent structure effects (35) include short-range deviations of the electrostatics from the bulk electrostatic model.The solvation free energy of SMx(x=1-8) where E is the bulk dielectric constant of the liquid can be expressed as solvent.Then AGep is obtained. △Gs=△EE+Gn+GCDs (32) (4)Set AGEp into a new VBSCF procedure and solve a new where Gp is the negative electric polarization term,AEE is the secular equation including SM6 terms.After the iterations positive electronic energy term,and Gcps is the term accounting have converged,the total solvation energy at the ab initio for cavitation,dispersion,and solvent structure.The sum of Gp VB level is obtained. and△Ee is called△Gep Test calculations for a few systems show that the liquid-phase In the VBSM method,173 electronic density is computed by partial atomic charges obtained by VBSM are in good agreement using a VB wave function,which is used for computing the partial with liquid-phase charges obtained by the charge mode CM atomic charges,while all other parameters are taken from the Free energies of solvation are calculated for two prototype test normal SM6 solvation model.71 As such,the VBSM method is cases,namely,for the degenerate Sv2 reaction of Cl with able to produce all VB properties and thus provide intuitive CH3Cl in water and for a Menshutkin reaction in water.These insights into studied chemical problems in solution as the calculations show that the VBSM method provides a practical VBSCF method does in the gas phase. alternative to single-configuration self-consistent field theory for To perform a VBSM calculation with orbital optimization,the solvent effects in molecules and chemical reactions following steps are involved in the current version of XMVB: 2.3.3.Combined VB/MM Method.The VB/MM method, (1)Calculate (i)the Coulomb integrals yu(R)that enter the introduced by Shurki et al,presents a new kind of combined generalized Born calculation,where k and k label atoms, QM/MM method that combines the ab initio valence bond (ii)the solvent-accessible surface areas Ak(R),and method with MM by importing the ideas from the empirical (iii)the geometry-dependent factors that enter the Gcps valence bond (EVB)approach.74175 It utilizes the ab initio calculation.R denotes the current geometry of the solute VB approach for the reactive fragments and MM for the environ- molecule. ment and thus extends VB applications to large biological (2)Perform a standard VBSCF calculation and then obtain systems.In the VB/MM method,the Hamiltonian of the whole the gas-phase VB wave function. system is expressed as (3)Using the current VB density,we calculate the current partial atomic charges qu(R)by Mulliken (M)or Lowdin HVB/MIM =H'(VB)+HO(MM)+H(VB/MM) (L)population analysis,eqs 33 and 34 (36) q(R)=Zk-∑(PS)aa (33) where I and O stand for the inner (quantum)and the outer a∈k (classical)regions.H(VB)is the VB Hamiltonian of an isolated dx.dol.org/10.1021/cr100228r Chem.Rev.XXXX,XXX,000-000

I dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW 2.3.2. VBSM Method. The VBPCM method incorporates PCM into the VB method, while the VBSM method applies the SMx (x = 1
8)121,171,172 solvation model. SMx (x = 1
8), developed by Truhlar and co-workers, treats the electrostatics due to bulk solvent by the generalized Born approximation with self-consistent partial atomic charges.121 In the SMx (x = 1
8) model,121,171,172 the electrostatic free energy of solvation is aug￾mented by terms proportional to the solvent-accessible surface areas of the solute’s atoms times empirical geometry-dependent atomic surface tensions, accounting for cavitation, dispersion, and solvent structure effects, where the solvent structure effects include short-range deviations of the electrostatics from the bulk electrostatic model. The solvation free energy of SMx (x = 1
8) can be expressed as ΔGS ¼ ΔEE þ GP þ GCDS ð32Þ where GP is the negative electric polarization term, ΔEE is the positive electronic energy term, and GCDS is the term accounting for cavitation, dispersion, and solvent structure. The sum of GP and ΔEE is called ΔGEP. In the VBSM method,173 electronic density is computed by using a VB wave function, which is used for computing the partial atomic charges, while all other parameters are taken from the normal SM6 solvation model.171 As such, the VBSM method is able to produce all VB properties and thus provide intuitive insights into studied chemical problems in solution as the VBSCF method does in the gas phase. To perform a VBSM calculation with orbital optimization, the following steps are involved in the current version of XMVB: (1) Calculate (i) the Coulomb integrals γkk0(R) that enter the generalized Born calculation, where k and k0 label atoms, (ii) the solvent-accessible surface areas Ak(R), and (iii) the geometry-dependent factors that enter the GCDS calculation. R denotes the current geometry of the solute molecule. (2) Perform a standard VBSCF calculation and then obtain the gas-phase VB wave function. (3) Using the current VB density, we calculate the current partial atomic charges qk(R) by Mulliken (M) or L€owdin (L) population analysis, eqs 33 and 34 qM k ðRÞ ¼ Zk
∑ R ∈ k ðP 3 SÞRR ð33Þ qL k ðRÞ ¼ Zk
∑ R ∈ k ðS1=2 3 P 3 S1=2 ÞRR ð34Þ respectively, where Zk is the nuclear charge of atom k, R is a basis function on atom k, and P and S are the one￾electron density and overlap matrices, respectively. Then the generalized Born polarization energy121,171,172 is calculated by GP ¼
1 2 1
1 ε
∑ k ∑ k0 qkðRÞ qk0ðRÞ γkk0ðRÞ ð35Þ where ε is the bulk dielectric constant of the liquid solvent. Then ΔGEP is obtained. (4) Set ΔGEP into a new VBSCF procedure and solve a new secular equation including SM6 terms. After the iterations have converged, the total solvation energy at the ab initio VB level is obtained. Test calculations for a few systems show that the liquid-phase partial atomic charges obtained by VBSM are in good agreement with liquid-phase charges obtained by the charge model CM4.171 Free energies of solvation are calculated for two prototype test cases, namely, for the degenerate SN2 reaction of Cl
with CH3Cl in water and for a Menshutkin reaction in water. These calculations show that the VBSM method provides a practical alternative to single-configuration self-consistent field theory for solvent effects in molecules and chemical reactions. 2.3.3. Combined VB/MM Method. The VB/MM method, introduced by Shurki et al.,122 presents a new kind of combined QM/MM method that combines the ab initio valence bond method with MM by importing the ideas from the empirical valence bond (EVB) approach.174,175 It utilizes the ab initio VB approach for the reactive fragments and MM for the environ￾ment and thus extends VB applications to large biological systems. In the VB/MM method, the Hamiltonian of the whole system is expressed as HVB=MM ¼ HI ðVBÞ þ HOðMMÞ þ HI, OðVB=MMÞ ð36Þ where I and O stand for the inner (quantum) and the outer (classical) regions. HI (VB) is the VB Hamiltonian of an isolated Figure 4. Dissociation curves of RX (R = (CH3)3C) (a) in the gas phase and (b) in the solvated phase.

Chemical Reviews REVIEW quantum region(the gas-phase Hamiltonian)of all the atoms in of resonance theory.As a resonance structure is composed, the inner region,H(MM)is the energy of all the atoms in the by definition,of local bonds plus core and lone pairs,a bond outer region determined by use of an empirical force field,and between atoms A and B will be represented as a bonding MO HO(VB/MM)is the Hamiltonian that accounts for all the inter- strictly localized on the A and B centers,a lone pair will be an actions between the quantum and the classical atoms. atomic orbital localized on a single center,etc.With these The diagonal matrix elements of the Hamiltonian,which restrictions on orbital extension,the self-consistent field solution include all the nonbonded interactions,are calculated by can be decomposed to coupled Roothaan-like equation sets,each corresponding to a block.The final block-localized wave func- H欧=EK称=Ex(VB)十E (37) tion is optimized at the constrained Hartree-Fock level and is where Ekk(VB)is the Kth diagonal matrix element of the VB expressed by a Slater determinant.Consequently,the energy structure energy in n isolated quantum region and E describes difference between the Hartree-Fock wave function,where all the interactions of the ab initio VB active subsystem with the MM electrons are free to delocalize in the whole system,and the part involving all the nonbonded interactions,such as electro- block-localized wave function,where electrons are confined to static,van der Waals,bulk,etc. specific zones of the system,is defined as the electron delocaliza- Following the idea underlying the EVB methodology,7s tion energy.Recently,the BLW method has been extended to the off-diagonal matrix elements of the Hamiltonian can be DFT178 by replacing the Hartree-Fock exchange potential by a approximated as DFT exchange-correlation(XC)potential in the Roothaan SCF procedure.This improved BLW method,referred to as BLW- HKL HKL(VB)+(WkEKK WLEL)MKL (38) DFT,has the advantages over the original method to bring whereHL(VB)and MRL are the off-diagonal elements of the electron correlation to both the individual structure and the final adiabatic states. Hamiltonian and overlap matrices between the Kth and Lth 2.4.2.MOVB Method.The MOVB125 method is an exten- diabatic states(VB structures)in a vacuum,respectively.Wk and sion of BLW,which allows calculation of the electronic coupling WL are the respective weights of VB structures K and L given by energy resulting from the mixing of two or more diabatic states, eq 8. i.e., The VB/MM method combines molecular mechanics for the states corresponding to single-resonance structures.The diabatic states are first calculated by the BLW method,and then diagonal elements with ab initio VB calculations for the off- a nonorthogonal configuration interaction Hamiltonian is con- diagonal elements,hence avoiding parametrizations as well as interpolations.The method maintains the advantages of the EVB structed using these diabatic states as the basis functions.Thus MOVB is a mixed molecular orbital and valence bond method methodology and provides an ab initio VB wave function. Recently,a new version of VB/MM,called density-embedding since it makes use of a Hartree-Fock or DFT description for the VB/MM(DE-VB/MM),was developed.176 The improvement covalent bonds,while being able to calculate diabatic states of the DE-VB/MM method is that the electrostatic interaction Importantly,solvent effects can be incorporated into the MOVB method.Thus,the MOVB method has been used to model between the VB active subsystem and the MM environment is involved during the optimization of the VB wave function for the the proton transfer between ammonium ion and ammonia in QM fragment,which is implemented by adding effective one- water,25 as well as a solvated S2 reaction,7 using Monte Carlo simulations. electron integrals to the ab initio VB Hamiltonian,hence taking One concern of the MOVB method is that the adiabatic state into account the wave function polarization of the QM fragment due to the environment.A somewhat related method is the wave function is simply a mixture of two diabatic states without MOVB/MM approach of Mo and Gao.177 further reoptimization of the orbitals.To remedy this defect,an alternative BLW-based two-state model has been devised that 2.4.Molecular Orbital Methods That Provide Valence-Bond- is applicable to cases where a ground state o can be approxi- Type Information mately described in terms of a resonance between two diabatic 2.4.1.BLW Method.The BLW method is a simple bridge structures,ΦAandΦB: between MO and VB methods which provides VB-type infor- mation (e.g.,resonance energies).As such,it is considered ΨO=CAaΦA+CBΦB (39) herein as a variant of the ab initio valence bond method which Differently than in MOVB,here both the ground state and meewgacnri 127 diabatic states are independently optimized at the same Hartree- The basic principle consists of partitioning the full basis set of Fock or DFT level,yielding the ground-state energy o and the orbitals into subsets each centered on a given fragment.The diabatic energies EA and EB.The method assumes that,on one molecular orbitals are then optimized in a Hartree-Fock way, hand,the diabatic states are orthogonal to each other with the restriction that each orbital is expanded only on its own SAB=(ΦAΦB〉=0 (40】 fragment.The MOs of a given fragment are orthogonal among themselves,but the orbitals of different fragments have finite but,on the other hand,the nonorthogonality effects are absorbed overlaps. by the effective off-diagonal Hamiltonian (A HB),whose The applications of the BLW method are designed primarily to value is not directly computed but determined via a "reverse evaluate the electronic delocalization and charge transfer effects configuration interaction"procedure from the known values of between fragments/molecules.Thus,the block-localized wave Eo EA.and EB function represents a reference for evaluating delocalization The orthogonality constraint further allows the construction energies relative to the fully delocalized wave function. of the first excited state as A typical application of the BLW method is the energy calculation of a specific resonance structure in the context Ψ1=CB④A-CAΦE (41) dx.dol.org/10.1021/cr100228rChem.Rev.XXXX,XXX,000-000

J dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW quantum region (the gas-phase Hamiltonian) of all the atoms in the inner region, HO(MM) is the energy of all the atoms in the outer region determined by use of an empirical force field, and HI,O(VB/MM) is the Hamiltonian that accounts for all the inter￾actions between the quantum and the classical atoms. The diagonal matrix elements of the Hamiltonian, which include all the nonbonded interactions, are calculated by HKK ¼ EKK ¼ E0 KKðVBÞ þ Eint KK ð37Þ where EKK 0 (VB) is the Kth diagonal matrix element of the VB structure energy in an isolated quantum region and EKK int describes the interactions of the ab initio VB active subsystem with the MM part involving all the nonbonded interactions, such as electro￾static, van der Waals, bulk, etc. Following the idea underlying the EVB methodology,174,175 the off-diagonal matrix elements of the Hamiltonian can be approximated as HKL ¼ H0 KLðVBÞþðWKEint KK þ WLEint LLÞM0 KL ð38Þ where HKL 0 (VB) and MKL 0 are the off-diagonal elements of the Hamiltonian and overlap matrices between the Kth and Lth diabatic states (VB structures) in a vacuum, respectively. WK and WL are the respective weights of VB structures K and L given by eq 8. The VB/MM method combines molecular mechanics for the diagonal elements with ab initio VB calculations for the off- diagonal elements, hence avoiding parametrizations as well as interpolations. The method maintains the advantages of the EVB methodology and provides an ab initio VB wave function. Recently, a new version of VB/MM, called density-embedding VB/MM (DE-VB/MM), was developed.176 The improvement of the DE-VB/MM method is that the electrostatic interaction between the VB active subsystem and the MM environment is involved during the optimization of the VB wave function for the QM fragment, which is implemented by adding effective one￾electron integrals to the ab initio VB Hamiltonian, hence taking into account the wave function polarization of the QM fragment due to the environment. A somewhat related method is the MOVB/MM approach of Mo and Gao.177 2.4. Molecular Orbital Methods That Provide Valence-Bond￾Type Information 2.4.1. BLW Method. The BLW method is a simple bridge between MO and VB methods which provides VB-type infor￾mation (e.g., resonance energies). As such, it is considered herein as a variant of the ab initio valence bond method which can be used for some specific applications, with the advantage of retaining the efficiency of molecular orbital methods.123
127 The basic principle consists of partitioning the full basis set of orbitals into subsets each centered on a given fragment. The molecular orbitals are then optimized in a Hartree
Fock way, with the restriction that each orbital is expanded only on its own fragment. The MOs of a given fragment are orthogonal among themselves, but the orbitals of different fragments have finite overlaps. The applications of the BLW method are designed primarily to evaluate the electronic delocalization and charge transfer effects between fragments/molecules. Thus, the block-localized wave function represents a reference for evaluating delocalization energies relative to the fully delocalized wave function. A typical application of the BLW method is the energy calculation of a specific resonance structure in the context of resonance theory. As a resonance structure is composed, by definition, of local bonds plus core and lone pairs, a bond between atoms A and B will be represented as a bonding MO strictly localized on the A and B centers, a lone pair will be an atomic orbital localized on a single center, etc. With these restrictions on orbital extension, the self-consistent field solution can be decomposed to coupled Roothaan-like equation sets, each corresponding to a block. The final block-localized wave func￾tion is optimized at the constrained Hartree
Fock level and is expressed by a Slater determinant. Consequently, the energy difference between the Hartree
Fock wave function, where all electrons are free to delocalize in the whole system, and the block-localized wave function, where electrons are confined to specific zones of the system, is defined as the electron delocaliza￾tion energy. Recently, the BLW method has been extended to DFT178 by replacing the Hartree
Fock exchange potential by a DFT exchange-correlation (XC) potential in the Roothaan SCF procedure. This improved BLW method, referred to as BLW￾DFT, has the advantages over the original method to bring electron correlation to both the individual structure and the final adiabatic states. 2.4.2. MOVB Method. The MOVB125 method is an exten￾sion of BLW, which allows calculation of the electronic coupling energy resulting from the mixing of two or more diabatic states, i.e., states corresponding to single-resonance structures. The diabatic states are first calculated by the BLW method, and then a nonorthogonal configuration interaction Hamiltonian is con￾structed using these diabatic states as the basis functions. Thus, MOVB is a mixed molecular orbital and valence bond method, since it makes use of a Hartree
Fock or DFT description for the covalent bonds, while being able to calculate diabatic states. Importantly, solvent effects can be incorporated into the MOVB method. Thus, the MOVB method has been used to model the proton transfer between ammonium ion and ammonia in water,125 as well as a solvated SN2 reaction,177 using Monte Carlo simulations. One concern of the MOVB method is that the adiabatic state wave function is simply a mixture of two diabatic states without further reoptimization of the orbitals. To remedy this defect, an alternative BLW-based two-state model has been devised179 that is applicable to cases where a ground state Ψ0 can be approxi￾mately described in terms of a resonance between two diabatic structures, ΦA and ΦB: Ψ0 ¼ CAΦA þ CBΦB ð39Þ Differently than in MOVB, here both the ground state and diabatic states are independently optimized at the same Hartree
Fock or DFT level, yielding the ground-state energy ε0 and the diabatic energies εA and εB. The method assumes that, on one hand, the diabatic states are orthogonal to each other SAB ¼ ÆΦAjΦBæ ¼ 0 ð40Þ but, on the other hand, the nonorthogonality effects are absorbed by the effective off-diagonal Hamiltonian ÆΦA|H|ΦBæ, whose value is not directly computed but determined via a “reverse configuration interaction” procedure from the known values of ε0, εA, and εB. The orthogonality constraint further allows the construction of the first excited state as Ψ1 ¼ CBΦA
CAΦB ð41Þ