TRANSFORMATION INDEX

The two-electron integrals involve the LCAO orbitals, and the time-consuming part of a traditional Cl calculation is the transformation of these to integrals involving the basis functions. This is often referred to as the four-index transformation. Not only that, it turns out that traditional Cl calculations are very slowly convergent we have to add a vast number of excited states in order to improve the energy significantly. 

The first-order energy involves only the perturbation operator and the unperturbed wavefunction. In an HF-LCAO treatment, the integrals would be over the LCAOs, and this implies a four-index transformation to integrals over the basis functions. 

The dependence is a consequence of performing the four index transformation with all four indices at once. As shown in Section 4.2.1, it is advantageous to perform the transformation one index at a time. 

Table 4. The isotropic indirect spin-spin coupling constant of calculated at various levels of theory. LL refers to the Levy-Leblond Hamiltonian, std refers to a full relativistic calculation using restricted (RKB) or unrestricted (UKB) kinetic balance, spf refers to calculations based on a spin-free relativistic Hamiltonian. Columns F, G and whether quaternion imaginary parts are deleted (0) or not (1) from the regular Fock matrix F prior to one-index transformation, from the two-electron Fock matrix G...

All other problems of index transformation can be dealt with in the same way, provided that the geometrical relations between the two alternative cells are clearly visualized. 

Direct evaluation of this four-fold sum would require 4didi dgdpjdgdftd c multiplications to form a symmetry block of R integrals. By constrast, sequential one-index transformations... 

Since spin-orbit coupling is very important in heavy element compounds and the structure of the full microscopic Hamiltonians is rather complicated, several attempts have been made to develop approximate one-electron spin-orbit Hamiltonians. The application of an (effective) one-electron spin-orbit Hamiltonian has several computational advantages in spin-orbit Cl or perturbation calculations (1) all integrals may be kept in central memory, (2) there is no need for a summation over common indices in singly excited Slater determinants, and (3) matrix elements coupling doubly excited configurations do not occur. In many approximate schemes, even the tedious four-index transformation of two-electron integrals ceases to apply. The central question that comes up in this context deals with the accuracy of such an approximation, of course. 

We are able to apply successively the relation in Eq. (13-66) and thereby we determine the higher-order index transformed integrals, e.g.,... 

We define a new set of operators using one-index transformed integrals [15]... 

All formulae until now are independent of the spin character of the perturbation. The spin complications enter at the level of one-index transformations. If k has been derived from an operator with spin, i.e. if it is given by Eq. (33) with the minus sign, the one-index transformation will change the spin properties of the Hamiltonian. In... 

A linear transformation of a configuration vector cb thus requires the construction of a configuration gradient with B> as the reference state [Eq. (94)], and the construction of an orbital gradient with a symmetric transition density matrix [Eq. (95)]. A linear transformation on an orbital vector °b requires the construction of a configuration gradient [Eq. (96)] and an orbital gradient [Eq. (97)] from the one-index transformed Hamiltonian K. 

After omitting all contributions to the MCSCF derivatives which arise from variations in the configuration space, the remaining terms either already appear in the AO basis (e.g., all terms some examples to make this point clear. 

Consider first the SCF response equations [Eq. (64)]. The construction of /differentiated integrals and from one-index transformed integrals. Both matrices may be calculated in the AO basis (see Appendix E). Furthermore, the solution of the response equations requires linear transformations of orbital trial vectors. Equation... 

The evaluation of F(n) requires the construction of Fock matrices from multiply one-index transformed integrals. For example, F(3) contains the term... 

The explicit calculation of C<0)/hf s avoided by modifying the densities Deff and deff by one-index transformed HF densities — DHF and... 

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