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

Exchange contributions in the electronic structure of systems with 1D‐periodicity: Importance and computation

“…Applying the Fourier transform 11, 12, to the RHF equations yields different, but equivalent expressions of the RHF‐LCAO matrix elements 14–18. In particular, X pq ( k ), assumes the form 17, 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 ′).…”

“…With the RHF‐LCAO matrix elements expressed in direct space, an analysis of the convergence properties of the lattice sums is presented and the consequences thereof are stressed considering the simple case of the infinite chain of beryllium atoms (third section). In the fourth section, the same equations expressed in the Fourier space 11–18 are very briefly discussed to point out the virtue of the approach for computing efficiently the problematic exchange contributions. The paper ends with some considerations regarding the future of calculations on 1D periodic systems using the Fourier transform (FT) approach.…”

Virtues and potentialities of the Fourier transform method for electronic structure calculations of 1‐D periodic systems at the Hartree–Fock level and beyond

“…In that respect, the Fourier representation method appears to be capable of handling with the necessary accuracy all the terms, including exchange and correlation corrections found in MP2, needed for stable results. At the Hartree–Fock level, it has already been shown that this method can be turned into an effective scheme 16–18. However, it still awaits further development before it can be proposed for general use, including implementation for atomic functions of nonzero angular momentum and the MP2 option.…”

“…The calculations reported here have been carried out with the PLH program 19,20, a direct‐space implementation of the RHF LCAO formalism. Because the study case will be compared in Section 4 with results obtained with the Fourier approach using a prototype program (FTCHAIN) 16–18 at present limited to s ‐type Gaussian functions, 2 p functions have been simulated by a distributed basis set of s ‐type functions (DSGF). Accordingly, three s ‐type atomic functions are centered on the Be atom using the same contraction scheme (exponents and contraction coefficients) as in the standard 3‐21G basis.…”

“…These instabilities may appear at relatively large eigenvalues (> 10 −5 ) of the overlap matrices and are not related to a linear dependence in the basis sets. Considerable attention has been devoted to the calculation of the classic Coulomb contributions to the Fock matrix elements and efficient methods have been designed for their calculation in both the direct 9,10 and Fourier spaces 11–18. However, Hartree–Fock exchange contributions, improperly considered as rapidly decaying with respect to the number of interacting cells, can also create instability problems.…”