The CASVB Method

It is known that, in the MO framework, the nondynamical electron correlation is accounted for by means of a so-called CASSCF calculation, which is nothing else than a full Cl in a given space of orbitals and electrons, in which the orbitals and the coefficients of the configurations are optimized simultaneously. If the active space includes all the valence orbitals and electrons, then the totality of the nondynamical correlation of the valence electrons is accounted for. In the VB framework, an equivalent VB calculation, defined with pure AOs or purely localized hybrid atomic orbitals (HAOs), would involve all the covalent and ionic structures that may possibly be generated for the molecule at hand. Note that the resulting covalent—ionic VB wave function would have the same dimension as the valence—CASSCF one (e.g., 1764 VB structures for methane, and 1764 MO SCF configurations in the CASSCF framework). 

The GVB and SC methods provide wave functions that are, of course, much more compact than the corresponding valence—CASSCF one (e.g., only 14 spin-coupling modes for methane with the SC method, and a single one with the GVB method). Owing to this difference in size, the GVB and SC methods cannot be expected to include the totality of the nondynamical correlation, even if these two methods treat well, by definition, the left—right correlation for each bond of the molecule. Physically, this is because the various local ionic 

Despite these restrictions, the GVB and SC methods generally provide energies that are much closer in quality to CASSCF than to Hartree Fock (19), and wave functions that are close to the CASSCF wave function having the same number of electrons and orbitals in the active space. This property has been used to devise a fast method to get approximate SC wave functions. 

Since a CASSCF calculation is faster than a direct SC calculation, owing to the advantages associated with orbital orthogonality in CASSCF, it is practical to extract an approximate SC wave function (or another type of VB function, e.g., a multiconfigurational one) from a CASSCF wave function. The conversion from one wave function to the other relies on the fact that a CASSCF wave function is invariant under linear transformations of the active orbitals. Based on this invariance principle, two different procedures were developed and both share the same name CASVB . Thus, CASVB is not a straightforward VB method, but rather a projection method that bridges between CASSCF and VB wave functions. 

In the CASVB method of Thorsteinsson et al. (22,23), one transforms the canonical CASSCF orbitals so that the wave function (which we recall, is kept unchanged in this process) involves a dominant component of a VB-type wave function, TVb, which is chosen in advance and may be single- or multiconfiguration, as in Equation 9.5  

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The CASVB method developed by Hirao et al. (24) differs from the previous one in the requirement that the final CASVB wave function, obtained after the... 

The CASVB method of Thorsteinsson et al. (22) is incorporated in the MOLPRO (67) and in the MOLCAS (68) packages. In addition to the features of the original CASVB method, the CASVB code also permits fully variational VB calculations, which can be single- or multiconfigura-tional in nature. 

MOPLRO A general-purpose package of programs. It contains the CASVB method. 

The complete active space valence bond (CASVB) method is an approach for interpreting complete active space self-consistent field (CASSCF) wave functions by means of valence bond resonance structures built on atom-like localized orbitals. The transformation from CASSCF to CASVB wave functions does not change the variational space, and thus it is done without loss of information on the total energy and wave function. In the present article, some applications of the CASVB method to chemical reactions are reviewed following a brief introduction to this method unimolecular dissociation reaction of formaldehyde, H2CO — H2+CO, and hydrogen exchange reactions, H2+X — H+HX (X=F, Cl, Br, and I). 

The CASVB functions [50,51] can be obtained by transforming the canonical CASSCF functions without loss of energy. First we transform the CASSCF delocalized MO to localized MO using the arbitrariness in the definition of the active orbitals. Then we perform a full Cl again in the active space. The CASVB method provides an alternative tool for describing the correlated wave functions. 

The reaction for fluorine (Rl) is highly exothermic, while the reactions for chlorine (R2), bromine (R3), and iodine (R4) are endothermic. The heats of these reactions are 30.8, — 1.2, — 16.7, and — 32.7 kcal/mol for reactions (Rl), (R2), (R3), and (R4), respectively. According to Hammond s postulate, reaction (Rl) should have an early TS, and reactions (R2) and (R3) should have late TSs. What the electronic states are during these reactions, and how the CASVB method describes the electronic structure, are our interests in this section. 

If we re-take the TS of chemical bonds as the origin, these facts well explain the shift of TS (from the early TS side to the late TS side) that Hammond s postulate predicts, indicating that the CASVB method is a powerful tool for describing the electronic structure and chemical bond during chemical reactions. 

Valence-bond methods have increased its applicability recently. One example is the CASVB (complete active space valence bond) method. A CASVB wave function can be obtained simply by transforming a canonical CASSCF function and readily interpreted in terms of the well-known classical VB resonance structures. The total CASVB wave function is identical to the canonical CASSCF wave function. In other words, the MO description and the VB description are equivalent, at least at the level of CASSCF. The CASVB method provides an alternative tool for describing the correlated wave functions. 

For cyclobutadiene, keeping the K0 term as necessary to describe the background exerted by a skeleton, the average of the two Kekule structures is K0-Ja-Jb. Correspondingly, for benzene, confined also to Kekule structures, the non-resonant part can be conventionally defined as K0-3Ja/2-3Jb/2. In this way we estimate a resonance stabilization for benzene of — 8.56 kcal/mol, comparable with the CASVB calculation [22] giving — 7.4 kcal/mol. Here one should note that with respect to the adopted definition and the method of estimation, the resonance energy is disputed in a very large interval (— 5 to — 95 kcal/mol) [23]. The representation in... 

H. Nakano, K. Sorakubo, K. Nakayama, K. Hirao, in Valence Bond Theory, D. L. Cooper, Ed. Elsevier, Amsterdam, The Netherlands, 2002, pp. 55-77. Complete Active Space Valence Bond (CASVB) Method and its Application in Chemical Reactions. 

Here TyB is the orthogonal complement of TVb to the CASSCF wave function, and SVb is the overlap between Tcas and Tv is- To ensure that the so-obtained VB function is as close as possible to the starting CASSCF wave function, the procedure transforms the orbitals in a manner that maximizes the overlap Svb- An alternative procedure is one that minimizes the energy of the VB function TVs- This latter procedure is, however, more expensive than the first one. As the two methods generally yield similar sets of orbitals, the method of Svb maximization is generally preferred (23). The 5Vb based CASVB method is implemented in the MOLPRO and MOLCAS packages. 

The complete active space valence bond (CASVB) method [1,2] is a solution to this problem. Classical valence bond (VB) theory is very successful in providing a qualitative explanation for many aspects. Chemists are familiar with the localized molecular orbitals (LMO) and the classical VB resonance concepts. 

In this article, we present applications of CASVB to chemical reactions the unimolecular dissociation reaction of formaldehyde, H2CO — H2+CO [5], and a series of hydrogen exchange reactions, H2+X — H+HX (X-F, Cl, Br, and I). The method in this article is based on the occupation numbers of VB structures that are defined by the weights of the spin-paired functions in the CASVB functions, so that we could obtain a quantitative description of the nature of electronic structures and chemical bonds even during reactions. 

We have proposed two types of CASVB method. The first one is a method where the valence bond structures are constructed from orthogonal localized molecular orbitals (LMOs) [1], and the second is one from nonorthogonal localized molecular orbitals [2]. 

One may re-define the active orbitals utilizing the invariance of the active orbital space. In the orthogonal CASVB method, the LMOs constructed by Boys localization procedure are used that is, active orbitals are transformed so as to have the minimum sum of expectation values. If the active orbitals are defined appropriately, the LMOs obtained nearly always turn out to be localized on a single atomic center with small localization tails on to neighboring atoms. In the non-orthogonal CASVB case, the atomic-like orbitals are constructed by Ruedenberg s projected localization procedure. 

VB2000 (63) is an ab initio VB package that can be used for performing nonorthogonal Cl, multistructure VB with optimized orbitals, as well as SCVB, GVB, and CASVB calculations in the spirit of Hirao s method (24) (see Section 9.2.3). 

CASVB A method that converts a CASSCF wave function to the closest possible VB wave function. 

With this method, we clarified the electronic structures of the ground and excited states of benzene, butadiene, methane, and hydrogen molecules [1,2]. We also applied the method to valence excited states of polyenes [3] and then-cations [4]. In previous studies, we put our focus on the formalism of CASVB and its applicability to molecules in their equilibrium structures. 

The various approximations implicit in the SCVB approach make the scaling of computational effort with numbers of electrons and of basis functions somewhat less severe than is the case for the multiconfiguration spin-coupled method of Penotti [20], making it possible to perform calculations on larger systems, and SCVB may compare favourable with CASVB, which is described in a later section. 

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