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Register a new userTight distance-dependent estimators for screening two-center and three-center short-range Coulomb integrals over Gaussian basis functions
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We derive distance-dependent estimators for two-center and three-center electron repulsion integrals over a short-range Coulomb potential, $textrm{erfc}(omega r_{12})/r_{12}$. These estimators are much tighter than one based on the Schwarz inequality and can be viewed as a complement to the distance-dependent estimators for four-center short-range Coulomb integrals and for two-center and three-center full Coulomb integrals previously reported. Because the short-range Coulomb potential is commonly used in solid-state calculations, including those with the HSE functional and with our recently introduced range-separated periodic Gaussian density fitting, we test our estimators on a diverse set of periodic systems using a wide range of the range-separation parameter $omega$. These tests demonstrate the robust tightness of our estimators, which are then used with integral screening to calculate periodic three-center short-range Coulomb integrals with linear scaling in system size.
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We extend the tight distance-dependent estimator proposed by Hollman et al. [J. Chem. Phys. 142, 154106 (2015)] for the three-center Coulomb integrals over Gaussian atomic orbitals to handle the two-center case. We also propose minor modifications of the original three-center estimator for the case of contracted ket Gaussians and concentric bra Gaussians.
The new combined formulas have been established for the complex and real rotation-angular functions arising in the evaluation of two-center overlap integrals over arbitrary atomic orbitals in molecular coordinate system. These formulas can be useful in the study of different quantum mechanical problems in both the theory and practice of calculations dealing with atoms, molecules, nuclei and solids when the integer and noninteger n complex and real atomic orbitals basis sets are emploed. This work presented the development of our previous paper (I.I. Guseinov, Phys. Rev. A, 32 (1985) 1864).
By using Poissons summation formula, we calculate periodic integrals over Gaussian basis functions by partitioning the lattice summations between the real and reciprocal space, where both sums converge exponentially fast with a large exponent. We demonstrate that the summation can be performed efficiently to calculate 2-center Gaussian integrals over various kernels including overlap, kinetic, and Coulomb. The summation in real space is performed using an efficient flavor of the McMurchie-Davidson Recurrence Relation (MDRR). The expressions for performing summation in the reciprocal space are also derived and implemented. The algorithm for reciprocal space summation allows us to reuse several terms and leads to significant improvement in efficiency when highly contracted basis functions with large exponents are used. We find that the resulting algorithm is only between a factor of 5 to 15 slower than that for molecular integrals, indicating the very small number of terms needed in both the real and reciprocal space summations. An outline of the algorithm for calculating 3-center Coulomb integrals is also provided.
There are many ways to numerically represent of chemical systems in order to compute their electronic structure. Basis functions may be localized in real-space (atomic orbitals), in momentum-space (plane waves), or in both components of phase-space. Such phase-space localized basis functions in the form of wavelets, have been used for many years in electronic structure. In this paper, we turn to a phase-space localized basis set first introduced by K. G. Wilson. We provide the first full study of this basis and its numerical implementation. To calculate electronic energies of a variety of small molecules and states, we utilize the sum-of-products form, Gaussian quadratures, and introduce methods for selecting sample points from a grid of phase-space localized Wilson basis. Both full configuration interaction and Hartree-Fock implementations are discussed and implemented numerically. As with many grid based methods, describing both tightly bound and diffuse orbitals is challenging so we have considered augmenting the Wilson basis set as projected Slater-type orbitals. We have also compared the Wilson basis set against the recently introduced wavelet transformed Gaussians (gausslets). Throughout, we give comments on the implementation and use small atoms and molecules to illustrate convergence properties of the Wilson basis.
We carried out numerical calculations by an extended-Hueckel program in order to check the analytical results reported in the preceding Part I and Part II. We typically consider alkali halide clusters composed of some tens of constituent atoms to calculate electronic energies under static conditions or versus the displacements of particular atoms. Among other things, the off-center displacement of substitutional Li+ impurity in most alkali halides is evidenced. The trigonometric profile of the rotational barriers is also confirmed for KCl.
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