Use of addition theorems in evaluation of multicenter nuclear-attraction and electron-repulsion integrals with integer and noninteger n Slater-type orbitals

Multicenter integrals appearing in the Hartree–Fock–Roothaan equations for molecules are calculated using different kinds of series expansion formulas obtained from the expansions of integer and noninteger n Slater-type orbitals, in terms of Ψ ^α -exponential-type orbitals (where α=1, 0, –1, –2,...) at a displaced center, that form complete orthonormal sets and are represented by linear combinations of integer n Slater-type orbitals. The convergence of these series is tested by calculating concrete cases. The accuracy of the results is quite high for quantum numbers, screening constants, and location of orbitals. Received: 13 February 2002 / Accepted: 11 March 2002 / Published online: 4 July 2002     

Efficient evaluation of Coulomb integrals in a mixed Gaussian and plane‐wave basis

An efficient way of calculation is presented for matrix elements between two plane waves interacting with a molecular Coulombic field. In concurrence with the absolute value of the momentum transfer vector, K = k₁ ? k ₂ , the most effective method of calculation is selected. The case of K = 0 requires special treatment. For 0 < |K| ≤ 0.3, it is profitable to evaluate the integrals by means of the multipole expansion, and for |K| > 0.3 the density fitting can be applied. For the large |K| the electronic part of the integral is much smaller than the nuclear part and the integral may be approximated by the nuclear contribution only. Some examples for testing the accuracy and time saved are presented. The primary purpose of this paper is to accelerate electron scattering calculations, but it also may be profitable for the electronic structure theory in attempts to use mixed Gaussian and plane‐wave basis sets. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Cuboidal basis functions

Summary An evaluation of the coulomb integral for a cuboid with uniform density is presented in analytic form, leading to the development of non-overlapping cube basis functions. The coulomb energy of the hydrogen molecule is determined with these functions fitted to the molecular orbital, and this result is compared with theab initio coulomb energy.     

Numerical Evaluation of Three-Center Two-Electron Coulomb and Hybrid Integrals Over B Functions Using the HD and H\overline D Methods and Convergence Properties

The difficulties of the numerical evaluation of three-center two-electron Coulomb and hybrid integrals over B functions, arise mainly from the presence of the hypergeometric series and semi-infinite very oscillatory integrals in their analytical expressions, which are obtained using the Fourier transform method.This work presents a general approach for accelerating the convergence of these integrals by first demonstrating that the hypergeometric function, involved in the analytical expressions of the integrals of interest, can be expressed as a finite sum and by applying nonlinear transformations for accelerating the convergence of the semi-infinite oscillatory integrals after reducing the order of the differential equation satisfied by the integrand.The convergence properties of the new approach are analysed to show that from the numerical point of view the method corresponds to the most ideal situation.The numerical results section illustrates the accuracy and unprecedented efficiency of evaluation of these integrals.     

On the evaluation of derivatives of Gaussian integrals

Summary We show that by a suitable change of variables, the derivatives of molecular integrals over Gaussian-type functions required for analytic energy derivatives can be evaluated with significantly less computational effort than current formulations. The reduction in effort increases with the order of differentiation.Dedicated to Prof. Klaus Ruedenberg     

A new one-electron model for extended Hückel type molecular orbital calculations

A new form of Coulomb and resonance integrals for extended Hückel type molecular orbital calculations was derived. In this model an electron moving in a molecule, when it stays in the neighborhood of the center of an atom in the molecule, sees the potential of the corresponding isolated atom. Compared with the well-known forms of Coulomb and resonance integrals, each integral derived from this model considers the influences from all atoms in the molecule, and includes the concept of atomic cell. The similarities of this model to other extended Hückel type formulations are also examined.     

Direct quantum chemical integral evaluation

Performing quantum chemical integral evaluation directly, without recursion and without direct coupling of angular momenta according to the rotation group is analyzed. The rotation group limits the structure of these closed‐form expressions. The result of all cross differentiation is a rotational invariant. Closed‐form expressions are obtained for the general three‐ and four‐center Gaussian integral. The solid harmonic addition formula can be used to express these integrals as sums of products of an exponent‐independent (angular) factor and a molecular‐orientation‐independent (exponential) factor in a variety of ways. The results are products of two such factors summed over the set of distinct, relevant polynomials of the exponents. The coefficients of these polynomials, angular factors, are complicated but common to all n‐center matrix elements and independent of any type of contraction. Derivatives must be obtained using the product rule. An implementation in the Solid Spherical Harmonic Gaussian (SSHG) computer code is outlined and preliminary comparison is made. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 373–383, 2001     

Efficient calculation of short-range Coulomb energies

An efficient algorithm for the calculation of short-range Coulomb energies is examined. The algorithm uses a boxing scheme and a prescreening for negligible integrals to evaluate the short-range Coulomb energy via computational work that scales only linearly with the size of the system. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 921–927, 1999     

Computation of two‐center Coulomb integrals over Slater‐type orbitals using elliptical coordinates

A general analytic formula is obtained for the two‐center Coulomb integrals over Slater‐type orbitals in elliptical coordinates. Finite series expansions are used in the evaluation of the radial part of the integrals. The analytic formula is expressed in terms of a product of the well‐known auxiliary functions A_k(p) and B_k(p) and incomplete gamma functions. Recursive relations for the computer evaluation of these functions are given as well. The recursive relations are stable and our computer results are in good agreement with the benchmark values given in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Analysis of environment response effects on excitation energies within subsystem-based time-dependent density-functional theory

We analyze the ability of subsystem time-dependent density-functional theory (sTDDFT) to describe environmental response effects. To this end, we utilize the recently proposed “exact” version of sTDDFT relying on projection-based embedding (PbE), which so far was applied only for the special case of two subsystems. We confirm that PbE-sTDDFT in combination with supersystem bases yields results equivalent to those of supermolecular TDDFT calculations for systems solvated by many solvent molecules, using the previously studied system of methylene-cyclopropene⋯(H₂O)₁₇ as an example. By means of this exact reference embedding framework, we are able to disentangle solvent effects introduced in terms of the embedding potential from those caused by solvent response couplings, both for the PbE variant and for sTDDFT with approximate non-additive kinetic energy functionals. Furthermore, we show that the use of a monomer basis introduces significant errors for the environmental response contribution. Employing a virtual-orbital localization strategy on top of PbE-sTDDFT, we can also directly assess the impact of inter-subsystem charge-transfer excitations on the entire solvent effect, which turn out to play a significant role for the environmental response. Finally, we analyze the response effects introduced by the individual solvent molecules and their interdependence, and show that a simple, pair-wise additive correction for solvent response yields excellent results in the present example.     

Evaluation of Multicenter Electronic Attraction, Electric Field and Electric Field Gradient Integrals with Screened and Nonscreened Coulomb Potentials over Integer and Noninteger n Slater Orbitals

By the use of complete orthonormal sets of -exponential-type orbitals, where ( = 1, 0, –1, –2,...) the multicenter electronic attraction (EA), electric field (EF) and electric field gradient (EFG) integrals of nonscreened and Yukawa-like screened Coulomb potentials are expressed through the two-center overlap integrals with the same screening constants and the auxiliary functions introduced in our previous paper (I.I. Guseinov, J. Phys. B, 3 (1970) 1399). The recurrence relations for auxiliary functions are useful for the calculation of multicenter EA, EF and EFG integrals for arbitrary integer and noninteger values of principal quantum numbers, screening constants, and location of slater-type orbitals. The convergence of the series is tested by calculating concrete cases.     

Density fitting for derivatives of Coulomb integrals in ab initio calculations using mixed Gaussian and plane‐wave basis

This article presents an efficient way for evaluation of Coulomb integrals of the type (k₁ (1)k₂ (1)|g₁ (2)g₂ (2)) and their derivatives with respect to nuclear coordinates by means of density fitting. Symbols k₁ andk₂ stand for plane‐wave functions and g₁ and g₂ for gaussians. The study was undertaken with the objective to accelerate electronic structure and electron scattering calculations in which a mixed Gaussian and plane‐wave basis set is used. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Strategies for an efficient implementation of the Gauss-Bessel quadrature for the evaluation of multicenter integral over STFs

In a previous work, a new Gauss quadrature was introduced with a view to evaluate multicenter integrals over Slater-type functions efficiently. The complexity analysis of the new approach, carried out using the three-center nuclear integral as a case study, has shown that for low-order polynomials its efficiency is comparable to the SD. The latter was developed in connection with multi-center integrals evaluated by means of the Fourier transform of B functions. In this work we investigate the numerical properties of the Gauss-Bessel quadrature and devise strategies for an efficient implementation of the numerical algorithms for the evaluation of multi-center integrals in the framework of the Gaussian transform/Gauss-Bessel approach. The success of these strategies are essential to elaborate a fast and reliable algorithm for the evaluation of multi-center integrals over STFs.     

On the Evaluation of Boys Functions Using Downward Recursion Relation

A new, simple, and efficient technique is presented for the accurate evaluation of the Boys functions F _m (x) (BFs) with integer and noninteger values of m appearing for the calculation of multicenter multielectron molecular integrals in a mixed Gaussian and plane-wave basis set. The extensive test calculations show that the proposed in this work algorithm is the most efficient in practical computations.     

New expansion of the Boys function

We propose a new expansion for the Boys function ∫₀¹t^2jexp(−r²t²) dt appearing in the calculation of molecular two-electron matrix elements if Gaussian basis sets are employed. This expansion involves a power series involving the terms C_{i, j}(τ) (r²−R²)ⁱ multiplied by exp(−τr²), where τ is an optimized parameter τ∈[0, 1]. The performances of the introduced expansion are discussed and illustrated by some numerical experiments. It appears that the proposed expansion is considerably shorter than the customary Taylor series, which in turn is the special case for τ=0. This is of some importance, particularly for higher j values. Further, the proposed expansion enables a single expression for calculating erf(x) for the whole range of variable x. The recursive relations for the expansion coefficients are derived and the truncation errors are estimated. A new method for calculating the Boys function by means of asymptotic series is represented too. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 305–315, 1998     

Lower Bounds to Ground-State Eigenvalues

A new lower bound method is presented for ground-state eigenvalues which relies on the Eckart inequality. The method bears similarity to the variational method and the Temple lower bound formula. Restrictions are that an exactly soluble base Hamiltonian must be available with a positive-semidefinite perturbation to the Hamiltonian of interest. A sample calculation shows that our method is able to sometimes best the Temple bound and also perform rigorously when the Temple bound does not.     

High level ab initio and density functional theory calculations of the dissociation energies,ionization energies,geometrical variations,and vibrational modes of ground and excited CO2, CO2+, and CO22+

The potential energy surface of CO₂²⁺ ( ³Σ_g^?) is investigated with HF, MP2, MP4, CBS‐Q, G1, G2MP2, G2, G3B3, and B3LYP/6‐311++G(3df,3pd) methods. Density functional theory shows the lowest dissociation channel of this compound to be the formation of CO⁺ ( ²Σ⁺)+O⁺ ( ⁴S_u) and to have a barrier of around 2 eV as well as a dissociation energy of around ?3.2 eV. Thus we propose that with enough correlation it is possible to accurately predict the energies of dissociation and barrier widths and heights to test for the stability of a particular molecular species. We also propose a refinement of current understanding by observing HOMO–LUMO gaps, Lowdin and Mulliken bond orders (to test for bond orbital overlap and hence qualitatively describe bonding and fragmentation in these complexes) and predicted spectrum for such studies as ZEKE spectroscopy (to study cationic states) and REMPI (to study the first excited states) of these class of molecules and, we hope, provide future insight into larger and more interesting systems. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Relativistic Gaussian basis sets for radon through plutonium

Summary Relativistic Gaussian basis sets of neutral atoms Rn-Pu and ions Th⁺⁴, U⁺³ and Pu⁺³ in the configurations of average energies are presented. The exponent parameters of the basis sets are determined by least-squares fitting to the numerical Dirac-Fock wave functions. The total energies obtained are within 0.155 a.u. of the Dirac-Fock limits and the qualities of the basis sets are between double-zeta and triple-zeta in the valence parts. Using the exponent parameters the Breit interaction energies have been calculated by perturbation theory and the self-consistent field treatment.