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,000-000F 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