Configuration interaction excitation level truncation

Practical configuration interaction methods augment the Hartree-Fock by adding only a limited set of substitutions, truncating the Cl expansion at some level of substitution. For example, the CIS method adds single excitations to the Hartree-Fock determinant, CID adds double excitations, CISD adds singles and doubles, CISDT adds singles, doubles, and triples, and so on. 

There is also a hierarchy of electron correlation procedures. The Hartree-Fock (HF) approximation neglects correlation of electrons with antiparallel spins. Increasing levels of accuracy of electron correlation treatment are achieved by Mpller-Plesset perturbation theory truncated at the second (MP2), third (MP3), or fourth (MP4) order. Further inclusion of electron correlation is achieved by methods such as quadratic configuration interaction with single, double, and (perturbatively calculated) triple excitations [QCISD(T)], and by the analogous coupled cluster theory [CCSD(T)] [8],... 

The exponential operator T creates excitations from 4>o according to T = l + 72 + 73 + , where the subscript indicates the excitation level (single, double, triple, etc.). This excitation level can be truncated. If excitations up to Tn (where N is the number of electrons) were included, vPcc would become equivalent to the full configuration interaction wave function. One does not normally approach this limit, but higher excitations are included at lower levels of coupled-cluster calculations, so that convergence towards the full Cl limit is faster than for MP calculations. 

As noted in equation (7), the Cl expansion is typically truncated according to excitation level in the vast majority of Cl studies, the expansion is truncated (for computational tractability) at doubly-substituted configurations. Since the Hamiltonian contains only two-body terms, only singles and doubles can interact directly with the reference this is a direct result of Slater s rules (cf. section 2.3.1). Furthermore, the matrix elements of singly substituted determinants (or CSFs) with the reference are zero when canonical SCF orbitals are used, according to Brillouin s theorem. Hence, one expects double excitations to make the largest contributions to the Cl wavefunction after the reference... 

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. 

Presently, the widely used post-Hartree-Fock approaches to the correlation problem in molecular electronic structure calculations are basically of two kinds, namely, those of variational and those of perturbative nature. The former are typified by various configuration interaction (Cl) or shell-model methods, and employ the linear Ansatz for the wave function in the spirit of Ritz variation principle (c/, e.g. Ref. [21]). However, since the dimension of the Cl problem rapidly increases with increasing size of the system and size of the atomic orbital (AO) basis set employed (see, e.g. the so-called Paldus-Weyl dimension formula [22,23]), one has to rely in actual applications on truncated Cl expansions (referred to as a limited Cl), despite the fact that these expansions are slowly convergent, even when based on the optimal natural orbitals (NOs). Unfortunately, such limited Cl expansions (usually truncated at the doubly excited level relative to the IPM reference, resulting in the CISD method) are unable to properly describe the so-called dynamic correlation, which requires that higher than doubly excited configurations be taken into account. Moreover, the energies obtained with the limited Cl method are not size-extensive. 

This requirement is satisfied by two of the methods most frequently used to calculate correlation effects Many body perturbation theory (MBPT) and coupled cluster theory (CC). In principle, configuration interaction (Cl) is also size consistent, but only if it is not truncated at a certain excitation level. Since untruncated Cl expansions become untractably large even for medium-size systems. Cl is not a method that can be used for the calculation of molecule/ surface interactions. 

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