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,000-000B 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