Speeding Up Hartree-Fock Calculations

For methods of searching structural databases for desired features, see Leach, Chapter 12 Y. C. Martin, M. G. Bures, and P. Willett, in K. Lipkowitz and D. B. Boyd (eds.). Reviews in Computational Chemistry, Vol. 1, VCH, Chapter 6. 

The calculation of the approximately b /S electron-repulsion integrals (ERIs) (rs tu) [Eq. (14.39)1 over the b basis functions consumes a major part of the time in an SCF MO calculation. Several methods are used to reduce the number of integrals evaluated. 

Molecular symmetry is used to identify integrals that are equal, so that only one of them need be evaluated. For example, in H2O, the integrals (HiD02j H2DH2D) and (H2D02i HiD HiD) are equal, provided the O—Hi and O—H2 bond distances are equal. Use of symmetry cuts the number of integrals to be evaluated in H2O approximately in half 

Many (rs tu) integrals involve basis functions representing inner-shell orbitals. These orbitals are little changed on molecule formation, and one can eliminate the need to explicitly represent them by using an effective core potential (ECP) or pseudopotential (Section 13.17). The ECP is a one-electron operator that replaces those two-electron Coulomb and exchange operators in the valence-electrons Hartree-Fock equation that arise 

The integrals (rs tu) not only must be calculated but also must be stored and then recalled from memory as their values are needed in each SCF iteration (recall the SCF example in Section 14.3). Typically, 5 to 50 iterations are needed to achieve SCF convergence. For the large basis sets used in modern ab initio calculations, the number of (rs tu) values to be stored for a large-molecule calculation may exceed the internal (core) memory capacity of the computer, and the (rs tu) values must be stored on external memory disk 

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In the previous section, the Roothaan method was introduced as a method for speeding up the Hartree-Fock calculation by expanding molecular orbitals with a basis set. The accuracy and computational time of the Roothaan calculations depend on the quality and number of basis functions, respectively. Therefore, it is necessary for reproducing accurate chemical reactions and properties to choose basis functions that give highly accurate molecular orbitals with a minimum number of functions. 

When applied to the exact exchange, the KLI scheme is as efficient as a Hartree-Fock calculation, and often only slightly less efficient than a GGA calculation. At this point one should nevertheless keep in mind that the KLI approximation only speeds up the calculation of Gk, but not that of the other ingredients of the OPM equation. The most time-consuming step in a KLI calculation is usually the evaluation of As soon as the exact... 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

Efficient use of symmetry can greatly speed up localized-orbital density-functional-exchange-and-correlation calculations. The local potential of density functional theory makes this process simpler than it is in Hartree-Fock-based methods. The greatest efficiency can be achieved by using non-Abelian point-group symmetry. Such groups have multidimensional irreducible representations. Only one member of each such representation need be used in the calculation. However efficient localized-orbital evaluation of the chosen matrix element requires the sum of the magnitude squared of the components of all the members on one of the symmetry inequivalent atoms, based on Eq. 13. 

A method to speed up MP2 calculations on large molecules is the local MP2 (LMP2) method of Saebp and Pulay [S. Saebd and P. Pulay, Annu. Rev. Phys. Chem., 44, 213 (1993)]. Here, instead of using canonical SCE MOs in the Hartree-Fock reference determinant d>o. one transforms to localized MOs (Section 15.8). Also, instead of using the virtual orbitals found in the SCF calculation as the orbitals a and b in (16.13) to which electrons are excited, one uses atomic orbitals that are orthogonal to the localized occupied MOs. Also, in (16.13), one includes only unoccupied orbitals a and b that are in the neighborhood of the localized MOs i and j. 

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