Configuration interaction practical calculations

Full configuration interaction (Cl) calculations for the shieldings in H2 (50) and BH molecules have demonstrated that the CCSD(T) results are sufficiently close to the full Cl results, so that for practical purposes, the CCSD(T) calculations may be considered as the benchmark calculations for each molecule, against which other methods such as density functional methods may be tested. 

It is possible to use full or limited configuration interaction wavefunctions to construct poles and residues of the electron propagator. However, in practical propagator calculations, generation of this intermediate information is avoided in favor of direct evaluation of electron binding energies and DOs. 

The ideal calculation would use an infinite basis set and encompass complete incorporation of electron correlation (full configuration interaction). Since this is not feasible in practice, a number of compound methods have been introduced which attempt to approach this limit through additivity and/or extrapolation procedures. Such methods (e.g. G3 [14], CBS-Q [15] and Wl [16]) make it possible to approximate results with a more complete incorporation of electron correlation and a larger basis set than might be accessible from direct calculations. Table 6.1 presents the principal features of a selection of these methods. 

One of the drawbacks of DFT is its inability to treat multiconfigurational problems properly. In these cases, some type of configuration interaction (Cl) method is needed [60], Ideally, one would like to carry out full Cl calculations, but this is generally not possible for practical reasons. Thus, some kind of approximation is needed and, in this respect, the so-called complete active self-consistent field (CASSCF) procedure is often used. 

Basis sets for use in practical Hartree-Fock, density functional, Moller-Plesset and configuration interaction calculations make use of Gaussian-type functions. Gaussian functions are closely related to exponential functions, which are of the form of exact solutions to the one-electron hydrogen atom, and comprise a polynomial in the Cartesian coordinates (x, y, z) followed by an exponential in r. Several series of Gaussian basis sets now have received widespread use and are thoroughly documented. A summary of all electron basis sets available in Spartan is provided in Table 3-1. Except for STO-3G and 3 -21G, any of these basis sets can be supplemented with additional polarization functions and/or with diffuse functions. It should be noted that minimal (STO-3G) and split-valence (3-2IG) basis sets, which lack polarization functions, are unsuitable for use with correlated models, in particular density functional, configuration interaction and Moller-Plesset models. Discussion is provided in Section II. 

A number of methods have been proposed for calculations of the geometries of molecules in excited states. These include CIS (Configuration Interaction Singles) and variations on CIS to account for the effect of double substitutions, as well as so-called time dependent density functional models. Except for CIS (the simplest of the methods) there is very little practical experience. There is also very little solid experimental data on the geometries of excited-state molecules. 

Thus the one-particle basis determines the MOs, which in turn determine the JV-particle basis. If the one-paxticle basis were complete, it would at least in principle be possible to form a complete jV-particle basis, and hence to obtain an exact wave function variationally. This wave function is sometimes referred to as the complete Cl wave function. However, a complete one-paxticle basis would be of infinite dimension, so the one-paxticle basis must be truncated in practical applications. In that case, the iV-particle basis will necessarily be incomplete, but if all possible iV-paxticle basis functions axe included we have a full Cl wave function. Unfortunately, the factorial dependence of the iV-paxticle basis size on the one-particle basis size makes most full Cl calculations impracticably large. We must therefore commonly use truncated jV-paxticle spaces that axe constructed from truncated one-paxticle spaces. These two truncations, JV-particle and one-particle, are the most important sources of uncertainty in quantum chemical calculations, and it is with these approximations that we shall be mostly concerned in this course. We conclude this section by pointing out that while the analysis so fax has involved a configuration-interaction approach to solving Eq. 1.2, the same iV-particle and one-particle space truncation problems arise in non-vaxiational methods, as will be discussed in detail in subsequent chapters. 

Many-body calculations which go beyond the Hartree-Fock model can be performed in two ways, i.e. using either a variational or a perturbational procedure. There are a number of variational methods which account for correlation effects superposition-of-configurations (or configuration interaction (Cl)), random phase approximation with exchange, method of incomplete separation of variables, multi-configuration Hartree-Fock (MCHF) approach, etc. However, to date only Cl and MCHF methods and some simple versions of perturbation theory are practically exploited for theoretical studies of many-electron atoms and ions. 

Configuration-interaction calculations for most atoms are practicable only if they are treated in terms of a few active electrons and a singledeterminant closed-shell core, for which = 7 = 0. The core is inert in the sense that the configuration basis omits configurations with unoccupied core orbitals. 

As mentioned in section 1, the combination of the CI method and semiempirical Hamiltonians is an attractive method for calculations of excited states of large organic systems. However, some of the variants of the CI ansatz are not in practical use for large molecules even at the semiempirical level. In particular, this holds for full configuration interaction method (FCI). The truncated CI expansions suffer from several problems like the lack of size-consistency, and violation of Hellmann-Feynman theorem. Additionally, the calculations of NLO properties bring the problem of minimal level of excitation in CI expansion neccessary for the coirect description of electrical response calculated within the SOS formalism. 

An alternative method, named internally contracted Cl, was suggested by Meyer and was applied by Werner and Reinsch in the MCSCF self-consistent electron-pair (SCEP) approach. Here only one reference state is used, the entire MCSCF wavefunction. The Cl expansion is then in principle independent of the number of configurations used to build the MCSCF wavefunction. In practice, however, the complexity of the calculation also strongly depends on the size of the MCSCF expansion. A general configuration-interaction scheme which uses, for example, a CASSCF reference state, therefore still awaits development. Such a Cl wavefunction could preferably be used on the first-order interacting space, which for a CASSCF wavefunction can be obtained from single and double substitutions of the form ... 

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