Truncated configuration interaction methods

As explained above, the dynamic electronic correlation can be taken into account by the (truncated) configuration interaction method in the case of a single reference system. Similarly, the multi-reference configuration interaction (MRCI) method can be used for multi-reference systems. In this method, a CASSCF wavefunction is used as the zeroth order description of the system. A configuration interaction matrix... 

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. 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

The full configuration interaction method [34-36] is exact in the sense that after choosing appropriate atomic basis functions (defining the model in this way), the resulting many-electron wavefunction is an exact eigenfunction of the model Hamiltonian, the computational effort, nevertheless, increases in an exponential manner. Truncation of the full Cl expansion (especially after single and double excitations, CI-SD) considerably reduces the necessary computational resources, but leads unfortunately to the serious problem of nonsize-consistency [37, 38] which makes the results even for medium systems unrealistic. The coupled-cluster method [39, 40] theoretically properly describes extended systems as well, but numerous experiences show the enormous increase of computational work with the size of the system. 

Tgj is represented exactly and the exact electronic energy, which also includes dispersion effects correctly, is obtained. However, this comes with infinite computational costs. Hence, methods needed to be devised, which allow us to approximate the infinite expansion in Eq. (12.9) by a finite series to be as short as possible. A straightforward approach is the employment of truncated configuration interaction (CI) expansions. Note that (electronic) configuration refers to the set of molecular orbitals used to construct the corresponding Slater determinant. It is a helpful notation for the construction of the truncated series in a systematic manner and yields a classification scheme of Slater determinants with respect to their degree of excitation . Excitation does not mean physical excitation of the molecule but merely substitution of orbitals occupied in the Hartree-Eock determinant o by virtual, unoccupied orbitals. Within the LCAO representation of molecular orbitals the virtual orbitals are obtained automatically with the solution of the Roothaan equations for the occupied orbitals that enter the Hartree-Eock determinant. 

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 correlated methods such as truncated configuration interaction (CID or CISD), coupled cluster (CCD or CCSD), quadratic configuration interaction (QCISD) and Brueckner doubles (BD) (see Configuration Interaction and Coupled-cbister Theory), the energy and wavefunction can be written as... 

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 electron correlation problem remains a central research area for quantum chemists, as its solution would provide the exact energies for arbitrary systems. Today there exist many procedures for calculating the electron correlation energy (/), none of which, unfortunately, is both robust and computationally inexpensive. Configuration interaction (Cl) methods provide a conceptually simple route to correlation energies and a full Cl calculation will provide exact energies but only at prohibitive computational cost as it scales factorially with the number of basis functions, N. Truncated Cl methods such as CISD (A cost) are more computationally feasible but can still only be used for small systems and are neither size consistent nor size extensive. Coupled cluster... 

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. 

It is appropriate at this point to compare some formal properties of the three general approaches to dynamical correlation that we have introduced configuration interaction, perturbation theory, and the coupled-cluster approach. First, we note that taken fax enough (all degrees of excitation in Cl and CC, infinite order of perturbation theory) all three approaches will give the same answer. Indeed, in a complete one-paxticle basis all three will then give the exact answer. We axe concerned in this section with the properties of truncated Cl and CC methods and finite-order perturbation theory. 

The correspond to different electron configurations. In configuration interaction o is the Hartree-Fock function (or an approximation to it in a truncated basis set) and the other 4>t are constructed from virtual orbitals which are the by-product of the Hartree-Fock calculation. The coefficients Ci are found by the linear-variation method. Unfortunately, the so constructed are usually an inadequate basis for the part of the wavefunction not represented by [Pg.5]

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