Complete-active space self-consistent field wavefunction

II. Optimization of a Complete Active Space Self-consistent Field Wavefunction.405... 

II. OPTIMIZATION OF A COMPLETE ACTIVE SPACE SELF-CONSISTENT FIELD WAVEFUNCTION... 

The combination of modem valence bond theory, in its spin-coupled (SC) form, and intrinsic reaction coordinate calculations utilizing a complete-active-space self-consistent field (CASSCF) wavefunction, is demonstrated to provide quantitative and yet very easy-to-visualize models for the electronic mechanisms of three gas-phase six-electron pericyclic reactions, namely the Diels-Alder reaction between butadiene and ethene, the 1,3-dipolar cycloaddition of fulminic acid to ethyne, and the disrotatory electrocyclic ringopening of cyclohexadiene. 

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. 

To obtain geometries, 10-orbital 10-electron complete active space self-consistent field (CASSCF) [82-84] calculations were performed with the GAMESS-UK program [6], The occupied orbital order in an SCF for flat benzene is n,2c,2n. In the bent molecule, there is no clear distinction between a- and tt-orbitals and we want to include all the tt-orbitals in the CAS-space. Thus, 10 orbitals in the active space are required. Obviously, the 5 structure VB wavefunction would have been a preferable choice to use in the geometry optimisation. However, at that time, the VB gradients were not yet available. The energies of the VBSCF at the CASSCF geometries followed the CASSCF curve closely. 

For the construction of spin eigenfunctions see, for example, Ref. [22], There are obviously many parallels to the multiconfiguration self-consistent field (MCSCF) methods of MO theory, such as the restriction to a relatively small active space describing the chemically most interesting features of the electronic structure. The core wavefunction for the inactive electrons, 4>core, may be taken from prior SCF or complete active space self-consistent field (CASSCF) calculations, or may be optimised simultaneously with the and cat. 

Extension of reference wavefunctions—quasi-degenerate perturbation theory with quasi-complete active space self-consistent field reference functions (QCAS-QDPT) [35]... 

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... 

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. ... 

The electronic wavefunction was evaluated using the state averaged complete active space self-consistent field (SA-CASSCF) method [49, 50], as implemented in the Molpro [51] electronic structure package. We used the 6-31G basis set [52] and an active space of two electrons in two orbitals with equally weighted averaging over the lowest three singlet states, i.e., SA-3-CAS(2/2)/6-31G. ... 

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. 

It is important to emphasize from the outset that metal-metal bonds present a substantirJ challenge to electronic structure theory, particularly where diatomic overlap is weak and the electrons are highly correlated. The chromium dimer, Crj, for example, is a notoriously difficult case and has been the subject of debate for decades [13], Some progress toward a quantitative understanding of these correlation effects has been made through Complete Active Space Self Consistent Field (CASSCF) and related wavefunction-based techniques, but much of our qualitative understanding... 

Theoretical calculations were performed, initially with SCF-Xa-SW methods on a truncated model [16], and later with the complete active space self-consistent field (CASSCF) and mul-ticonfigurational complete active space second-order perturbation theory (CASPT2) methods on the full molecule [15]. The electronic structures from the two calculations were remarkably similar. The CASSCF/PT2 calculations predicted a single, dominant configuration (73%) with (a) (x) (x ) (a ) (8) (5 ). Although the formal bond order is 1.5, the effective bond order, which considers minor configurations that contribute to the ground-state wavefunction, is lower at 1.15. 

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). 

AMI Basis Sets Correlation Consistent Sets Complete Active Space Self-consistent Field (CASSCF) Second-order Perturbation Theory (CASPT2) Configuration Interaction Coupled-cluster Theory Density Functional Theory (DFT), Hartree-Fock (HF) and the Self-consistent Field Diradicals Electronic Wavefunctions Analysis G2 Theory M0ller-Plesset Perturbation Theory Natural Bond Orbital Methods Spin Contamination. 

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