Virtual configuration interaction

Tab. 3.1 Selected vibrational energies (cm ) of H5OJ calculated by a variety of methods and experiment for the OH-monomer stretches. 2d+2d are the adiabatic 4d calculations of Ref [56], 4d are the fully coupled 4d calculations of Ref [27], CC-VSCF are the correlation-consistent vibrational self-consistent field calculations of Ref [58] and VCI are the virtual configuration interaction calculations of Ref [27]. The potential used in these calculations is indicated after the back-slash.

Configuration Interaction (Cl) methods begin by noting that the exact wavefunction 4 cannot be expressed as a single determinant, as Hartree-Fock theory assumes. Cl proceeds by constructing other determinants by replacing one or more occupied orbitals within the Hartree-Fock determinant with a virtual orbital. 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

This reasoning was found to be paralleled in configuration interaction computations, and a similar triptych was obtained in which surfaces rather than MOs were drawn. Such a triptych has an avoided crossing and bifunnel only on branch A and the reasoning using this variation is parallel to that above. Such computations were used for virtually all of the bicycle-type reactions discussed above. 

The configuration interaction (Cl) treatment of electron correlation [83,95] is based on the simple idea that one can improve on the HF wavefunction, and hence energy, by adding on to the HF wavefunction terms that represent promotion of electrons from occupied to virtual MOs. The HF term and the additional terms each represent a particular electronic configuration, and the actual wavefunction and electronic structure of the system can be conceptualized as the result of the interaction of these configurations. This electron promotion, which makes it easier for electrons to avoid one another, is as we saw (Section 5.4.2) also the physical idea behind the Mpller-Plesset method the MP and Cl methods differ in their mathematical approaches. 

The HF (Hartree-Fock) Slater determinant is an inexact representation of the wavefunction because even with an infinitely big basis set it would not account fully for electron correlation (it does account exactly for Pauli repulsion since if two electrons had the same spatial and spin coordinates the determinant would vanish). This is shown by the fact that electron correlation can in principle be handled fully by expressing the wavefunction as a linear combination of the HF determinant plus determinants representing all possible promotions of electrons into virtual orbitals full configuration interaction. Physically, this mathematical construction permits the electrons maximum freedom in avoiding one another. 

A theoretical justification of the scaling procedure was given by Pupyshev et al [14]. They compared the force field Fhf obtained in the Hartree-Fock (HF) limit with the force-field Fa obtained in the configuration interaction (Cl) technique, where the electron correlation effects are taken into account by mixing the HF ground state function with electronic excitations from the occupied one-electron HF states to the virtual (unoccupied) HF states. It was assumed that the force constants F01 obtained in the Cl approximation simulate the exact harmonic force field while those, extracted from the HF approximation FHF cast the quantum-mechanical force field F1-"1. It was demonstrated that under conditions listed below ... 

VBCI Valence bond configuration interaction. A VB computational method that starts with a VBSCF wave function, which is further improved by CI. The Cl involves virtual orbitals that are localized on exactly the same regions as the respective active orbitals. There are a few VBCI levels that are denoted by the rank of excitation into the virtual orbitals, for example, VBCISD involves single and double excitations. 

In electron correlation treatments, it is a common procedure to divide the orbital space into various subspaces orbitals with large binding energy (core), occupied orbitals with low-binding energy (valence), and unoccupied orbitals (virtual). One of the reasons for this subdivision is the possibility to freeze the core (i.e., to restrict excitations to the valence and virtual spaces). Consequently, all determinants in a configuration interaction (Cl) expansion share a set of frozen-core orbitals. For this approximation to be valid, one has to assume that excitation energies are not affected by correlation contributions of the inner shells. It is then sufficient to describe the interaction between core and valence electrons by some kind of mean-field expression. 

The SCVB energy is, of course, just the result from this optimization. Should a more elaborate wave function be needed, the virtual orbitals are available for a more-or-less conventional, but non-orthogonal configuration interaction (Cl) that may be used to improve the SCVB result. Thus improving the basic SCVB result here may involve a wave function with many terms. 

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