CI expansion

The method has become known as the CASSCF State Interaction (CASSI) method and is also effective for long CAS-CI expansions and was recently extended to handle the integrals of the spin-orbit Flamiltonian.34... 

Since the triply and quadruply excited configurations are truncated separately, it is usually possible to shorten the Cl expansion somewhat further, if this should be desirable, by applying the procedure described in Section 2.3 to the total calculated truncated SDTQ-CI wavefunction obtained as described in this section. Thus, for the SDTQ[ 18/18] wavefunction of NCCN, an accuracy of 1 mh is achieved with 49,033 determinants by the estimation method, and with 43,038 determinants by the just mentioned additional improvement, reducing the truncation from 6.32% to 5.54% of the 776,316 determinants of the full SDTQ-CI expansion. 

Figure 2. Rate of convergence of truncated SDTQ-CI expansions based on a priori a posteriori ordering and error assessment. Filled circles Truncations based on anticipated a priori estimates. Open circles Truncations determined a posteriori from the full wavefunctions.

The final step is the orbital optimization for the truncated SDTQ-CI expansion. We used the Jacobi-rotation-based MCSCF method of Ivanic and Ruedenberg (55) for that purpose. Table 1 contains the results for the FORS 1 and FORS2 wavefiinctions of HNO and NCCN, obtained using cc-pVTZ basis sets (51). In all cases, the configurations were based on split-localized orbitals. For each case, four energies are listed corresponding to (i) whether the full or the truncated SDTQ-CI expansion was used and (ii) whether the split-localized orbitals were those deduced from the SD naturals orbitals or were eventually MCSCF optimized. It is seen that... 

A two-pronged approach has been discussed for dealing with electron correlation in large systems (i) An extension of zeroth-order full-valence type MCSCF calculations to larger systems by radical a priori truncations of SDTQ-CI expansions based on split-localized orbitals in the valence space and (ii) the recovery of the remaining dynamic correlation by means of a theoretically-based simple semi-empirical formula. 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

Even if the method sketched above works well for many cases, there are several difficulties connected with it. It cannot easily be extended to larger systems, since the necessary MR-CI expansion then becomes excessively large. 

The above sequence of steps is what enters a so called conventional MR-CI calculation. This is still a widely used method to solve the MR-CI equations and during the past decades many tricks have been developed to circumvent some of the inherent problems with this approach. These problems are mainly due to the storage of large data sets, in particular the storage of the Hamiltonian matrix elements or even worse the storage of the formula tape. Therefore, if this approach is used the MR-CI expansion has to be drastically truncated. The maximum number of configurations which can be handled is in practice 10 to 20 thousand terms. The Hamiltonian matrix will then contain on the order of a few million non-zero terms. Since an MR-CI expansion without truncation in normal applications is 10s to 107 configurations the adopted truncation scheme has to be extremely efficient if the final result should still be accurate. In the next section we will discuss an alternative approach by which it is possible to handle the non-truncated MR-CI expansion without approximations. 

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. 

A conceptually straightforward improvement on the CI approximation is to reoptimize the molecular orbitals for a truncated CI expansion. This approach is called multi-configuration self-consistent field method (MCSCF) and its most prominent variant is the complete active space SCF method (CASSCF) [64]. In the first generation of MCSCF methods [65, 66], the CI coefficients C/ in Eq. 

In order to improve on the result of a truncated CI expansion, one may set on top of a Cl-type calculation, a perturbation theory calculation to cover the missing small contribution to the correlation energy. Accordingly, CASSCF plus second-order perturbation theory is called CASPT2, CI plus second-order perturbation theory is called MR-MP2, and so on. 

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. 

If one uses only a single orthonormal set of orbitals say, then expansion (17) goes over to the MO-CI expansion for the total wavefunction. The first term of this will now contain doubly occupied orbitals ... 

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