Four index transformations

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. 

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. 

Procedures such as SCEP may be performed at 135 Mflops, whilst a four-index transformation of the 2-electron integrals will also proceed at 135 Mflops in large cases, indicating the CRAY to be between 5 and 25 CDC 7600 in power when given appropriate code. 

The four-index transformation is a good test case for parallel algorithm development of electronic structure calculations, because it has O(N ) operations, a low computation to data transfer ratio and is a compact piece of code. Distributed-memory algorithms were presented for a number of standard QC methods by Whiteside and co-workers Li52 special emphasis on the integral transformation. Details of their implementation on a 32-processor Intel hypercube were provided. 

Special emphasis has also been given by Bhusari, Bhate, and PaT jq the implementation of the four-index transformation on a Transputer network, as part of a general program of parallel ab initio algorithm development. Results were presented on a four-node machine. 

L. A. Covick and M. K. Sando,/ Comput. Chem., 11,1151 (1990). Four-Index Transformation on Distributed-Memory Parallel Computers. 

The dependence is a consequence of performing the four index transformation... 

S. Wilson, in Electron Correlation in Atoms and Molecules, Vol. 1 of Methods in Computational Chemistry, S. Wilson, Ed., Plenum Press, New York, 1987, pp. 251-309. Four-Index Transformations. 

Second-order Moller-Plesset perturbation theory (MP2) is the computationally least expensive and most popular ab initio electron correlation method [4,15]. Except for transition metal compounds, MP2 equilibrium geometries are of comparable accuracy to DFT. However, MP2 captures long-range correlation effects (like dispersion) which are lacking in present-day density functionals. The computational cost of MP2 calculations is dominated by the integral transformation from the atomic orbital (AO) to the molecular orbital (MO) basis which scales as 0(N5) with the system size. This four-index transformation can be avoided by introduction of the RI integral approximation which requires just the transformation of three-index quantities and reduces the prefactor without significant loss in accuracy [36,37]. This makes RI-MP2 the most efficient alternative for small- to medium-sized molecular systems for which DFT fails. 

Wilson170 provides a very clear and helpful survey of four-index transformation methods published before 1987. More recent work has focused attention on the sparsity of quantities in the AO basis. For example, Haser, Almlof, and Feyereisen have presented an integral-direct transformation algorithm which... 

Owing to the fact that the internal orbitals change in this optimization process, the operators /i , Jj and Kj also change. It is not necessary, however, to perform a second four-index transformation. Instead, the modified operators are obtained from the original ones by the much cheaper transformations... 

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-... 

It is also possible to separate the four-index transformation into two two-index transformations ... 

Two-electron (four-index) transformation routine. Does the full transformation and assumes that there is room in main memory for nbasis nbasis + l)/2 intermediate results in V. 

I. Evaluation of Integrals. M.-M. Rohmer, J. Demuynck, M. Benard, R. Wiest, C. Bachmann, C. Henriet, and R. Ernenwein, Comput. Phys. Commun., 60, 127 (1990). A Program System for Ab Initio MO Calculations on Vector and Parallel Processing Machines. II. SCF Closed-Shell and Open-Shell Iterations. R. Wiest, J. Demuynck, M. Benard, M.-M. Rohmer, and R. Ernenwein, Comput. Phys. Commun., 62,107 (1991). A Program System for Ab Initio MO Calculations on Vector and Parallel Processing Machines. III. Integral Reordering and Four-Index Transformation. 

Wong, A. T., R. J. Harrison, and A. P. Rendell. Parallel direct four-index transformations. Theor. Chim. Acta 93 317-331,1996. 

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