Configuration interaction ansatz

Jeziorski B, Paldus J (1990) Valence universal exponential ansatz and the cluster structure of multireference configuration interaction wave function. J Chem Phys 90 2714-2731... 

In this section we examine some of the critical ideas that contribute to most wavefunction-based models of electron correlation, including coupled cluster, configuration interaction, and many-body perturbation theory. We begin with the concept of the cluster function which may be used to include the effects of electron correlation in the wavefunction. Using a formalism in which the cluster functions are constructed by cluster operators acting on a reference determinant, we justify the use of the exponential ansatz of coupled cluster theory. ... 

It is perhaps useful to compare the exponential ansatz of Eq. [31] with the analogous expansions of other wavefunctions. In the configuration interaction (Cl) approach, for example, a linear excitation operator is used instead of an exponential. 

In this section we present the extended geminal (EXGEM) ansatz, discuss the choice of root function, and comment on numerical models, i.e. different approximations to the full configuration interaction (FCI) equations defining the terms in the general EXGEM model. 

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. 

For a more accurate treatment, electron correlation has to be taken into account to this end methods akin to those used in a nonrelativistic description are employed. The total wave function may be considered as a superposition of (all possible) excitations. This can be expressed in terms of a configuration interaction (Cl) ansatz... 

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. 

Another important development was the reahzation of the importance of the size-consistency and size-extensivity in the studies of associative or dissociative chemical processes by Primas [67], as well as his clear delineation of the relationship between the configuration interaction and the exponential coupled-cluster Ansatze. 

In the following it will be outlined, how the parity violating potentials are computed within a sum-over-states approach, namely on the uncoupled Hartree-Fock (UCHF) level, and within the configuration interaction singles approach (CIS) which is equivalent to the Tamm-Dancoff approximation (TDA), that avoids, however, the sum over intermediate states. Then a further extension is discussed, namely the random phase approximation (RPA) and an implementation along similar lines within a density functional theory (DFT) ansatz, and finally a multi-configuration linear response approach is described, which represents a systematic procedure that... 

In the limit k oo, which implies that the number of one-particle functions also goes to 00, we obtain the exact solution of the electronic Schrodinger equation, which is the fuU configuration interaction (FCI) ansatz. A (stiU finite) expansion of the FCI wave function is only accomplishable for the smallest systems [21, 22] and further approximations to Eq. (8.6) are mandatory. 

The coupled-cluster electronic state is uniquely defined by the set of the cluster amplitudes and these amplitudes are used to obtain the coupled-cluster energy from Eq. (33). Due to the fact that the Ansatz of the coupled-cluster wave function has the exponential parametrization [Eq. (28)] the energy is size-extensive. This is an obvious advantage of the coupled-cluster formalism compared to some other techniques (e.g. configuration interaction). For a general discussion of coupled-cluster theory and the coupled-cluster equations see Refs. [5, 36]. 

There is clearly no extension of (1) aiming at the description of correlation effects among all possible pairs of electrons—or better (spin) orbitals—within a product ansatz for the total wavefunction. As a consequence, pair theories have developed in various directions and were not a really uniform undertaking. Their development was, of course, intimately tied to other techniques of electronic structure calculations, such as the configuration-interaction (Cl) or perturbation theory methods. 

In order to calculate nonadiabatic couplings in the framework of the TDDFT method a representation of the wavefunction based on Kohn-Sham (KS) orbitals is required. Since in the linear response TDDFT method the time-dependent electron density contains only contributions of single excitations from the manifold of occupied to virtual KS orbitals, a natural ansatz for the excited state electronic wavefunction is the configuration interaction singles (ClS)-Uke expansion ... 

The variational methods of the configuration interaction (Cl) type and the perturbative-type methods relying on the exponential coupled-cluster (CC) Ansatz are the most often used approaches in ab initio computations of highly accurate molecular properties, in particular of the potential energy surfaces (PESs) or curves (PECs) for the purposes of molecular dynamics [for recent reviews, see Refs. (1-4)]. In this latter case it is essential that the entire surface — or its various one- or multi-dimensional cuts — is available for a wide enough range of molecular geometries. 

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