Fourier representation method for accurate evaluation of the Coulomb and exchange lattice sums in extended chains

Abstract

By considering chains of hydrogen and helium atoms with one Gaussian function centered on each atomic sites, i.e., the simplest possible metallic and insulating model systems retaining all the characteristics of the restricted Hartree–Fock–Roothaan method for extended chains, a scheme is proposed to carry out accurately and efficiently both Coulomb and exchange lattice sums which occur in a Gaussian basis set environment.

Computational aspects of polymer band structure calculations by the Fourier space restricted Hartree-Fock method

“…In the Fourier space formulation of the Fock matrix elements F pq ( k ), the Coulomb and exchange parts are, respectively 9–12: and where \documentclass{article}\pagestyle{empty}\begin{document}$\mathbf{q}_{0}^{2}=q_{x}^{2}+q_{y}^{2}$\end{document} and q m + k − k ′ =( q x , q y , m + k − k ′).…”

“…The V pqrs ( k , k ′) terms present the following permutation symmetry with respect to the AO indices: Taking into account the permutation symmetry relation for the density matrix elements, the use of these symmetry relations reduces the number of terms to include in the calculation of V pq ( k ), i.e., In the buildup of the Fock matrix, each V pqrs ( k , k ′) term has to be added to the following Fock matrix element multiplied by the following density matrix elements: where the indices obey the following rules: Another choice is In the derivation of an explicit expression for the Coulomb term (4), the case where the summation index m is equal to 0 and the case where m ≠0 have to be distinguished 9–12. where Indeed, this expression includes a function that diverges at m =0 if the electron–nuclear attraction and the electron–electron repulsion terms are treated separately.…”

“…A Fourier space (FS), formulation of the RHF‐LCAO method 6 for systems with periodicity in one dimension 7–9 has been proposed to deal with the lattice summations of both the classical electrostatic and exchange terms. An approach based on a combination of the Poisson formula and the Ewald technique has been developed to compute these lattice summations consistently to within a specified numerical accuracy 10, 11. The method has been implemented in a prototype computer program using s ‐type Gaussian basis functions 12.…”

Fourier space for accurate ab initio RHF band structure calculations on chainlike systems

“…Very recently, it has been shown that combining the Poisson summation formula and the Ewald method in the context of gaussian basis sets [5] can be used to compute accurately these summations. Practical implementation, however, requires the development of efficient algorithms for the two-electron integrals.…”

Unrestricted Hartree–Fock band structure calculations for polymers: Application to a cross-talk system

“…The crudest approximation is the second one but this, however, allows the translational symmetry to be taken into account and enables the application of concepts encountered in condensed matter physics as well as specific techniques to account for the long-range Coulombic interactions. [6][7][8] Then an aperiodicity can in principle be introduced. A variety of ab initio polymer program packages based on Hartree-Fock, [9][10][11][12][13][14][15][16] density-functional methods, 17,18 and other correlated methods [19][20][21][22][23] are available.…”