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,000-000H 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