Complete active space wave functions

Although the primary emphasis of this chapter is on NBO analysis of existing wave functions, it is important to realize that these orbitals may also be useful in constructing new wave functions for photochemical phenomena. An important example is the CASNBO method, which uses NBOs as starting orbitals in multiconfigurational CAS (complete active space) wave functions. The CASNBO method allows the active space to be defined in the most specific and physically relevant way for a given chemical application (such as, e.g., the two-dimensional a B and space for a localized A-B bond-break-... 

For the MFP approach, most calculations have used SCF wave functions. However, Bak et aL have implemented the MFP theoretical approach at the correlated multiconfigura-tional self-consistent-field-level by using the complete active space wave functions (CASSCF) expressed over conventional and gauge invariant basis sets. In addition, the MFP approach has been implemented using correlated wave functions at the non-local DFT level and B3LYP (a hybrid HF/DFT) level. ... 

The wave function for the elechonic structure can in principle be any of the constructions employed in electronic structure theoiy. The prefened choice in this context is a wave funchons that can be classified as single and multi-configurational, and for the latter type only complete active space (CAS) wave... 

Some details of END using a multiconfigurational electronic wave function with a complete active space (CASMC) have been introduced in terms of an orthonormal basis and for a fixed nuclear framework [25], and were recently [26] discussed in some detail for a nonoithogonal basis with electron translation factors. 

Density functional theory, direct molecular dynamics, complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 404-411 Density operator, direct molecular dynamics, adiabatic systems, 375-377 Derivative couplings conical intersections, 569-570 direct molecular dynamics, vibronic coupling, conical intersections, 386-389 Determinantal wave function, electron nuclear dynamics (END), molecular systems, final-state analysis, 342-349 Diabatic representation ... 

When the HF wave function gives a very poor description of the system, i.e. when nondynamical electron correlation is important, the multiconfigurational SCF (MCSCF) method is used. This method is based on a Cl expansion of the wave function in which both the coefficients of the Cl and those of the molecular orbitals are variationally determined. The most common approach is the Complete Active Space SCF (CASSCF) scheme, where the user selects the chemically important molecular orbitals (active space), within which a full Cl is done. 

With regard to the former, one would like to include as many important configurations as possible. Unfortunately, the definition of an important configuration is often debatable. One popular remedy is the full-valence complete active space SCF (CASSCF) approach in which configurations arising from all excitations from valence-occupied to valence-virtual orbitals are chosen. [29] Since this is equivalent to performing a full Cl within the valence space, the full-valence CASSCF method is limited to small systems. Nevertheless, the CASSCF approach using a well-chosen (often chemically motivated) subspace of the valence orbitals has been shown to yield a much improved depiction of the wave function at all points on a potential surface. Furthermore, the choice of an active space can be adjusted to describe excited state wave functions. 

For the sake of completeness, we recall that the idea of the single-root formalism exploiting the Hilbert space approach was also proposed by Banerjee and Simons [31] and Laidig and Bartlett [34,35]. In both approaches they start from the complete active space MC SCF wave function, however, in order to eliminate redundant cluster amplitudes they approximate the wave operator by... 

In ab initio methods the HER approximation is used for build-up of initial estimate for and which have to be further improved by methods of configurational interaction in the complete active space (CAS) [39], or by Mpller-Plesset perturbation theory (MPn) of order n, or by the coupled clusters [40,41] methods. In fact, any reasonable result within the ab initio QC requires at least minimal involvement of electron correlation. All the technical tricks invented to go beyond the HFR calculation scheme in terms of different forms of the trial wave function or various perturbative procedures represent in fact attempts to estimate somehow the second term of Eq. (5) - the cumulant % of the two-particle density matrix. 

In an MCSCF calculation, not only the coefficients of the multiple configurations in the MC wave function, but also the orbitals in them, are simultaneously optimized. An (n/m)MCSCF calculation, in which the n active electrons and m active orbitals are chosen in the manner described in the preceding paragraph, is called a complete active space (CAS)SCF calculation. ... 

Selection of orbitals to include in an MCSCF requires first and foremost a consideration of die chemistry being examined. For instance, in the TMM example above, a two-configuration wave function is probably not a very good choice in this system. When the orbitals being considered belong to a tt system, it is typically a good idea to include all of them, because as a rule they are all fairly close to one another in energy. Thus, a more complete active space for TMM would consider all four jt orbitals and the possible ways to distribute the four TT electrons within them. MCSCF active space choices are often abbreviated as (m,n) where m is the number of electrons and n is the number of orbitals, so this would be a (4,4) calculation. 

We shall not perform the somewhat elaborous calculation of the MC wave function in detail. A somewhat simpler example is the dissociation of a double bond and it is given as an exercise (exercise 2). Here we only note that the number of configuration state functions (CSF s) will increase very quickly with the number of active orbitals. In most cases we do not have to worry about the exact construction of the MC wave function that leads to correct dissociation. We simply use all CSFs that can be constructed by distributing the electrons among die active orbitals. This is the idea behind the Complete Active Space SCF (CASSCF) method. The total number of such CSFs is for N2 175 for a singlet wave function. A further reduction is obtained by imposing spatial symmetry. All these CSFs are not included in a wave... 

Hamiltonian proposed by Muller and Plesset gives rise to a very successful and efficient method to treat electron correlation effects in systems that can be described by a single reference wave function. However, for a multireference wave function the Moller-Plesset division can no longer be made and an alternative choice of B(0> is needed. One such scheme is the Complete Active Space See-ond-Order Perturbation Theory (CASPT2) developed by Anderson and Roos [3, 4], We will briefly resume the most important definitions of the theory one is referred to the original articles for a more extensive description of the method. The reference wave function is a CASSCF wave function that accounts for the largest part of the non-dynamical electron correlation. The zeroth-order Hamiltonian is defined as follows and reduces to the Moller-Plesset operator in the limit of zero active orbitals ... 

Amovilli et al. [20] presented a method to carry out VB analysis of complete active space-self consistent field wave functions in aqueous solution by using the DPCM approach [3], A Generalized Valence Bond perfect pairing (GVB-PP) level... 

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 idea of CASVB is based on the fact that the densities of variational wave functions are invariant under the transformations which hold the variational space unchanged. In the CASSCF case, a complete active space (CAS) is invariant under the linear transformation of active orbitals and also that of configuration state functions (CSFs). 

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