CAS-SCF function

The entity Hexp T ) denotes all terms obtained by joining the operators in H with those of 7 ". The notation, signifies that the operator inside the curly bracket has been written in normal order with respect to 4>v as the vacuum. The equality (8) follows from equation (7) because T" has no destruction operators in the hole-particle form and hence can be taken out of the normal order term of equation (7) from the left. Since T" always excites to the virtual manifold, it follows that, for a CAS-CI or a CAS-SCF function. 

Some algorithms exploit the relation between the modern valence bond wave-function and the complete active space self-consistent field (cas-scf) functions. This leads to the so-called CAS-vb approach [58-60]. 

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. ... 

SCF and CAS SCF calculations on mono and bimetallic transition metal hydride complexes are reported. The importance of including the non dynamical correlation elTects for the study of the cis-trans isomerism in dihydrido complexes and for the study of the CO insertion reaction into the metal hydride bond is stressed. The metal to metal hydrogen transfer in a class of bimetallic d — d hydride complexes is analyzed and the feasibility of the transfer discussed as a function of the coordination pattern around the two metal centers. 

CAS SCF calculations were therefore performed with the split valence basis set incremented by a p polarisation function on the hydrogen atoms. Two different sets of active orbitals were considered. The first one was designed to account for the d - n back donation and was therefore restricted to the n type valence orbitals. The three 3d orbitals, which are strongly occupied, were each correlated by two weakly occupied orbitals, owing to the mixed 4d and tt o character of these weakly occupied orbitals. This 3 + 6 set of active orbitals referred to as CAS SCF-6 is populated by 6 electrons. The second set, hereafter referred as CAS SCF-12, took into account both a and n correlation eficcts. Twelve electrons were correlated and... 

At the ab-initio level, the most obvious possibility is offered by CAS SCF or CAS FCI (i.e., Cl within the CAS or, equivalently, CAS SCF without the orbital reoptimization based on RHF orbitals, cf. [33, 34]) wave functions based on the smallest possible active-space that warrants the correct description of the dissociation channel at hand. This option was also suggested by Stolarczyk [29], although we are not aware of any concrete implementation. Our testing proved to be very encouraging [33, 34], particularly for open shell systems, in which case we employed the spin-adapted CCSD based on the unitary group approach (UGA) [16, 36]. Even in the case of triple bond breaking, the applicability of the CCSD approximation can be significantly extended, as will be shown in Sect. 4. Most recently, we have explored the MR CISD wave function as an external source, as described in the next section. 

The CAS FCI or SOCI corrected CCSD results represent a definite improvement over the standard SR CCSD results. Nonetheless, for a given choice of the active-space, the performance of the ASTQ CCSD approaches deteriorates with the increasing size of the basis set. This is easily understood, since the external source does not include excitations out of the active-space, which are more important for larger basis sets. We should also mention that computationally much more demanding CAS SCF wave functions are about as effective as CAS FCI ones (cf., e.g., Tables II and III of [33]). 

A CAS-CI or a CAS-SCF type of reference function, can be generically represented as a combination of reference determinants s... 

In a CAS-SCF or a CAS-CI function V o, all the n-body density matrices with hole labels factorize into antisymmetric products of one-body densities. As a result, the two and higher-body cumulants R Z are zero when all the labels a,b,c,d- ) in R Z. are holes. The only non-vanishing four- or higher-body cumulants are those with valence labels only. The mixed n-body densities with some holes and some valences are zero unless the number and indices of hole labels in the destruction operators match with those in the creation operators. In case they match, these density matrices factorize into antisymmetric products of one-body density matrices with hole labels and the various cumulants with valence labels. 

The results with cc-pVTZ and cc-pVQZ are shown in Table 20.2. First let us compare the results at the reference function (QCAS- and CAS-SCF) level. Although differences in the energy itself between QCAS-SCF and CAS-SCF are about 10 millihartree for both basis sets, the differences in the barrier height are 1.65 and 1.62 millihartree (1.0 and l.Okcal/mol) for cc-pVTZ and cc-pVQZ, respectively. The agreement of QCAS-SCF with CAS-SCF is very good. 

Recently, both valence bond (VB) and complete active space (CAS) SCF and CAS FCI wave functions were employed as a source of T3 and T< cluster amplitudes (11,13) (see also Ref. 12). The most satisfactory approach, however, that was developed very recently, relies on the MR CISD wave function, based on a relatively small model space. This approach is referred to as the reduced MR (RMR) CCSD method (15-17). 

Applications of the complete active space (CAS) SCF method and multiconfigurational second-order perturbation theory (CASPT2) in electronic spectroscopy are reviewed. The CASSCF/CASPT2 method was developed five to seven years ago and the first applications in spectroscopy were performed in 1991. Since then, about 100 molecular systems have been studied. Most of the applications have been to organic molecules and to transition metal compounds. The overall accuracy of the approach is better than 0.3 eV for excitation energies except in a few cases, where the CASSCF reference function does not characterize the electronic state with sufficient accuracy. 

A many-body perturbation theory (MBPT) approach has been combined with the polarizable continuum model (PCM) of the electrostatic solvation. The first approximation called by authors the perturbation theory at energy level (PTE) consists of the solution of the PCM problem at the Hartree-Fock level to find the solvent reaction potential and the wavefunction for the calculation of the MBPT correction to the energy. In the second approximation, called the perturbation theory at the density matrix level only (PTD), the calculation of the reaction potential and electrostatic free energy is based on the MBPT corrected wavefunction for the isolated molecule. At the next approximation (perturbation theory at the energy and density matrix level, PTED), both the energy and the wave function are solvent reaction field and MBPT corrected. The self-consistent reaction field model has been also applied within the complete active space self-consistent field (CAS SCF) theory and the eomplete aetive space second-order perturbation theory. ... 

CAS SCF Complete Active Space Self-Consistent Field An iterative and variational method of solving the Schrddinger equation with the variational wave function in the form of a linear combination of all the Slater determinants (coefficients and spinorbitals are determined variationally) that can be built from a limited set of the spinorbitals (forming the active space). 

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