Four index transformation of two-electron integrals

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. 

All contributions in (49a) and (49c) are precomputed and stored, these being common to many matrix elements. Once this step is completed there is no longer any need to keep the cofactors D (i k/) and D (y /c/). Contribution (49c) is reduced by means of techniques similar to those employed in the four-index transformation of two-electron integrals. This can be accomplished in one, two, three or four passes, depending upon the amount of disk space and CPU time available. A two-pass program is probably a good compromise. In the one-pass case, contribution (49c) requires operations, correspond-... 

Foldy-Wouthuysen transformation, 215 Forbidden reaction, Woodward-Hoffmaim rules, 356 Force Field (FF), 6 Force field energy, 8 Force field parameters, 30 Four index transformation of two-electron integrals, 105... 

The four-index transformation of the two-electron integrals from the AO to the MO basis was carried out by a formalism given in equation (5.61), which divides the procedure, which is of 0(v ) when v is the number of basis functions, into four subsequent O(v ) summations. Similarly, the nonlinear part in equation (5.80), which is of OiN Ni) when No is the number of occupied and the number of virtual MOs, can be computed by two summations of 0(NlNy) and 0 NoN ), and two sums of 0 N N ) and 0 N Nl) (further details are given elsewhere ). In the computations one obtains from the first guess the MBPT(2) result, then one performs the iteration on the linear part of equation (5.80)... 

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. 

Indeed, it is this convenient fact which enables all the SCF methods outlined so far to be implemented compactly the underlying two-electron integral transformation (four-index multiplications) has been contained into the formation of J and K matrices and some one-electron (two-index matrix multiplications) transformations. However, if we use any method which demands the existence of the MO-based repulsion integrals with either i j or k or both)... 

The possibihty to ehminate the WFs tails may have a significant impact on the efficiency of LMP2 methods. For these methods the most expensive step is the evaluation and transformation of four-index two-electron integrals, where the computational cost is governed by the spatial extent of the AO support of the individual WFs. The details of the evaluation of these integrals can be found in [109]. 

The central problem in calculating E2 is to express the two-electron integrals, which were calculated when applying HF theory in an AO basis, in terms of the Bloch functions (5.20) (the so-called four-index transformation). Before attacking this problem one must employ the translational symmetry of the crystal (or chain) to simplify equation (5.36). For this purpose we shall consider the Coulomb integral... 

Such MO integrals are required for all electron correlation methods. The two-electron AO mtegrals are the most numerous and the above equation appears to involve a computational effect proportional to M AO integrals each multiplied by four sets of M MO coefficients). However, by performing the transformation one index at a time, the computational effort can be reduced to. ... 

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