A new computational approach to Slater’s SCF–<i>X</i>α equation

Sambé, Hideo; Felton, R. H.

doi:10.1063/1.430555

Cited by 433 publications

(146 citation statements)

References 17 publications

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“…However, these algorithms currently only work efficiently for large systems in very small basis sets. When more accuracy is needed larger basis sets are essential and for these cases the resolution of identity (RI) method [1][2][3][4][5][6][7][8] has been used to reduce the computational effort. The RI method has worked very well for the calculation of the Coulomb contribution to the Fock matrix [9][10][11][12][13][14][15][16][17][18], but the exchange contribution is much more difficult to compute with the same benefits [15; 19-22].…”

Section: Introductionmentioning

confidence: 99%

Cholesky Decomposition Techniques in Electronic Structure Theory

Aquilante

Boman

Boström

et al. 2011

Challenges and Advances in Computational Chemistry and Physics

102

114

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We review recently developed methods to efficiently utilize the Cholesky decomposition technique in electronic structure calculations. The review starts with a brief introduction to the basics of the Cholesky decomposition technique. Subsequently, examples of applications of the technique to ab inito procedures are presented. The technique is demonstrated to be a special type of a resolution-of-identity or density-fitting scheme. This is followed by explicit examples of the Cholesky techniques used in orbital localization, computation of the exchange contribution to the Fock matrix, in MP2, gradient calculations, and so-called method specific

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Section: Introductionmentioning

confidence: 99%

Cholesky Decomposition Techniques in Electronic Structure Theory

Aquilante

Boman

Boström

et al. 2011

Challenges and Advances in Computational Chemistry and Physics

102

114

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“…If the dependence of the energy on the orbitals is not treated variationally, then the forces are not accurate [56]. This fundamental problem probably means that the proposal of Sambe and Felton [7] to fit the exchange-correlation (XC)…”

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confidence: 99%

The limitations of Slater’s element-dependent exchange functional from analytic density-functional theory

Zope

Dunlap

2006

The Journal of Chemical Physics

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Our recent formulation of the analytic and variational Slater-Roothaan (SR) method, which uses Gaussian basis sets to variationally express the molecular orbitals, electron density and the one body effective potential of density functional theory, is reviewed. Variational fitting can be extended to the resolution of identity method, where variationality then refers to the error in each two electron integral and not to the total energy. However, a Taylor series analysis shows that all analytic ab initio energies calculated with variational fits to two-electron integrals are stationary. It is proposed that the appropriate fitting functions be charge neutral and that all ab initio energies be evaluated using two-center fits of the two-electron integrals. The SR method has its root in the Slater's Xα method and permits an arbitrary scaling of the Slater-Gàspàr-Kohn-Sham exchange-correlation potential around each atom in the system. The scaling factors are the Slater's exchange parameters α. Of several ways of choosing these parameters, two most obvious are the Hartree-Fock (HF) α HF values and the exact atomic α EA values. The former are obtained by equating the self-consistent Xα energy and the HF energies, while the latter set reproduce exact atomic energies. In this work, we examine the performance of the SR method for predicting atomization energies, bond distances, and ionization potentials using the two sets of α parameters.The atomization energies are calculated for the extended G2 set of 148 molecules for different basis set combinations. The mean error (ME) and mean absolute error (MAE) in atomization energies are about 25 and 33 kcal/mol, respectively for the exact atomic, α EA , values. The HF values of exchange parameters, α HF , give somewhat better performance for the atomization energies with ME and MAE being about 15 and 26 kcal/mol, respectively. While both sets give performance better than the local density approximation or the HF theory, the errors in atomization energy are larger than the target chemical accuracy. To further improve the performance of the SR method for atomization energies, a new set of α values is determined by minimizing the MAE in atomization energies of 148 molecules. This new set gives atomization energies half as large (MAE ∼ 14.5 kcal/mol) and that are slightly better than those obtained by one of the most widely used generalized gradient approximations. Further improvements in atomization energies require going beyond Slater's element-dependent functional form for exchange employed in this work to allow exchange-correlation interactions between electrons of different spin. The MAE in ionization potentials of 49 atoms and molecules is about 0.5 eV and that in bond distances of 27 molecules is about 0.02 Å. The overall good performance of the computationally efficient SR method using any reasonable set of α values makes it a promising method for study of large systems.

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“…Density-fitting or resolution-of-the-identity (RI) techniques, the successors of early approximation schemes for evaluating two-electron integrals in atomic-orbital (AO) basis, [1][2][3] were pioneered by Whitten, [4] Baerends et al, [5] Sambe and Felton, [6] and Dunlap et al [7,8] in the 1970s. The work in the 1990s by Feyereisen et al [9,10] and by Ahlrichs and coworkers [11][12][13][14] was instrumental in turning the RI approach into the mainstream approximation it is today.…”

Section: Introductionmentioning

confidence: 99%

Attractive electron–electron interactions within robust local fitting approximations

et al. 2013

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An analysis of Dunlap's robust fitting approach reveals that the resulting two-electron integral matrix is not manifestly positive semidefinite when local fitting domains or non-Coulomb fitting metrics are used. We present a highly local approximate method for evaluating four-center two-electron integrals based on the resolution-of-the-identity (RI) approximation and apply it to the construction of the Coulomb and exchange contributions to the Fock matrix. In this pair-atomic resolutionof-the-identity (PARI) approach, atomic-orbital (AO) products are expanded in auxiliary functions centered on the two atoms associated with each product. Numerical tests indicate that in 1% or less of all Hartree-Fock and Kohn-Sham calculations, the indefinite integral matrix causes nonconvergence in the self-consistent-field iterations. In these cases, the two-electron contribution to the total energy becomes negative, meaning that the electronic interaction is effectively attractive, and the total energy is dramatically lower than that obtained with exact integrals. In the vast majority of our test cases, however, the indefiniteness does not interfere with convergence. The total energy accuracy is comparable to that of the standard Coulomb-metric RI method. The speed-up compared with conventional algorithms is similar to the RI method for Coulomb contributions; exchange contributions are accelerated by a factor of up to eight with a triple-zeta quality basis set. A positive semidefinite integral matrix is recovered within PARI by introducing local auxiliary basis functions spanning the full AO product space, as may be achieved by using Cholesky-decomposition techniques. Local completion, however, slows down the algorithm to a level comparable with or below conventional calculations.

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A new computational approach to Slater’s SCF–Xα equation

Cited by 433 publications

References 17 publications

Cholesky Decomposition Techniques in Electronic Structure Theory

Cholesky Decomposition Techniques in Electronic Structure Theory

The limitations of Slater’s element-dependent exchange functional from analytic density-functional theory

Attractive electron–electron interactions within robust local fitting approximations

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