Complete-active-space self-consistent-field CASSCF approach

It is evident that the approach described so far to derive the electronic structure of lanthanide ions, based on perturbation theory, requires a large number of parameters to be determined. While state-of-the-art ab initio calculation procedures, based on complete active space self consistent field (CASSCF) approach, are reaching an extremely high degree of accuracy [34-37], the CF approach remains widely used, especially in spectroscopic studies. However, for low point symmetry, such as those commonly observed in molecular complexes, the number of CF... 

In this chapter, the quantum chemical simulation of actinide and lanthanide complexes will be considered. The need for a multiconfigurational description of the wavefimction is discussed, and the complete-active-space self-consistent-field (CASSCF) approach, along with some related methods, is introduced and discussed. This approach, originally developed by Bj om Roos, allows for the strong static correlation present in these complexes due to a combination of electron-electron interactions and weak crystal field splittings to be taken into consideration in a systematic manner. Extensions to this approach, which also account for dynamical correlation will also be considered, fit the finally section, the application of the CASSCF approach will be illustrated with examples from the literature. 

The selection of configuration state functions to be included in MCSCF calculations is not a trivial task. Two approaches which can reduce the complexity of the problem are the complete active space self-consistent-field (CASSCF) [68] and the restricted active space self-consistent-field (RASSCF) [69] approach. Both are implemented in the Dalton program package [57] and are used in this study. Throughout the paper a CASSCF calculation is denoted by i active gactive RASSCF calculation by For the active spaces of HF, H2O, and CH4... 

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

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). 

A number of important trends can be drawn from Table 4.1, which are trends that have influenced how computational chemists approach related (and sometimes even largely unrelated) problems. Hartree-Fock (HF) self-consistent field (SCF) computations vastly overestimate the barrier, predicting a barrier twice as large as experiment. The omission of any electron correlation more seriously affects the transition state, where partial bonds require correlation for proper description, than the ground-state reactants. Inclusion of nondynamical correlation is also insufficient to describe this reaction complete active space self-consistent field (CASSCF) computations also overestimate the barrier by some 20 kcal... 

The major problem with MCSCF methods is selecting the necessary configurations to include for the property of interest. One of the most popular approaches is the Complete Active Space Self-consistent Field (CASSCF) method (also called Full Optimized Reaction Space, FORS). Here the selection of configurations is done by partitioning the MOs into active and inactive spaces. The active MOs will typically be some of the dii rest occupied and some of the lowest unoccupied MOs from a RHF calculationr Fhe-daactive MOs either have 2 or 0 electrons, i.e. always doubly occupied or empty. Within... 

However, a reasonable quantitative treatment for TM systems seems to require a fairly high level of theory. One particularly promising approach has been developed by Roos [28] based on the Complete Active Space Self Consistent Field (CASSCF) method with a second order perturbation treatment of the remaining (dynamical) electron correlation effects, CASPT2. 

A special case of full Cl is the complete active space self-consistent field (CASSCF) or fully optimized reaction space (FORS) approach in which one defines an active space of orbitals and corresponding electrons that are appropriate for a chemical process of interest [20]. The FORS wavefunction is then obtained as a linear combination of all possible electronic excitations (configurations) from the occupied to the unoccupied (virtual) orbitals in the active space, so a FORS wavefunction is a full Cl within the specified active space. Since a full Cl provides the exact wavefunction for a given atomic basis, there is no need to re-optimize the component molecular orbitals. On the other hand, a FORS wavefunction generally corresponds to an incomplete Cl, in the sense that only a subset of configuration (or determinant) space is included. Therefore, one also optimizes the molecular orbital coefficients to self-consistency. The calculation of a full Cl wavefunction is extremely computationally demanding, scaling exponentially with... 

Our group has already studied all the fluoro-, chloro-and bromo-carbenes [11-16]. We used the complete active space self-consistent field (CASSCF) method as well as complete active space second-order perturbation theory (CASPT2) and multi-reference configuration interaction (MRCI) approaches to compute the geometries, force constants, and vibrational frequencies of the (singlet) X and A states as well as the (triplet) a states. Our theoretical studies of most of these carbenes were carried out specifically to complement LIF studies that were pursued in our laboratories by Kable et al. [6]. In addition to the determination of spectroscopic constants, the spectroscopic and theoretical studies considered dynamics on the A surfaces, i.e. whether photodissociation or internal conversion to the ground state would occur. 

An important issue in MCSCF calculations is the selection of the configurations to be included in the wavefunction expansion. The most popular approach is the complete active space self-consistent field (CASSCF) method, also called full optimized reaction space (FORS). This approach starts Irom a zeroth order set of MOs, usually obtained via the Hartree-Fock method. The set of MOs is split into three subsets, as illustrated in Fig. 2.3. A first one containing occupied inactive orbitals, for which the occupation numbers are fixed to 2. A second one containing active orbitals, including both occupied and virtual orbitals of the reference Hartree-Fock configuration, in which all possible electron excitations are allowed. And a third one containing virtual inactive orbitals, for which the occupation numbers are fixed to 0. 

In this section we will introduce some wavefunction-based methods to calculate photoabsorption spectra. The Hartree-Fock method itself is a wavefunction-based approach to solve the static Schrodinger equation. For excited states one has to account for time-dependent phenomena as in the density-based approaches. Therefore, we will start with a short review of time-dependent Hartree-Fock. Several more advanced methods are available as well, e.g. configuration interaction (Cl), multireference configuration interaction (MRCI), multireference Moller-Plesset (MRMP), or complete active space self-consistent field (CASSCF), to name only a few. Also flavours of the coupled-cluster approach (equations-of-motion CC and linear-response CQ are used to calculate excited states. However, all these methods are applicable only to fairly small molecules due to their high computational costs. These approaches are therefore discussed only in a more phenomenological way here, and many post-Hartree-Fock methods are explicitly not included. 

In such an approach, the correlation energy is partitioned into a complete active space self-consistent field (CASSCF) part which describes the static correlations and a MR configuration interaction part for the dynamic correlations. In the case of metals we select an active space for the CASSCF calculation including the important bands around the Fermi level. The dynamic correlations are treated on top of the CASSCF wavefunction with an approximately size-extensive MR correlation method, i.e. an MR averaged coupled pair functional (MR-ACPF). 

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