GEN1INT: A unified procedure for the evaluation of one‐electron integrals over Gaussian basis functions and their geometric derivatives

Abstract

ABSTRACT:We propose a unified procedure for evaluating a variety of one-electron integrals and their (arbitrary-order) geometric derivatives by using a generalized one-electron operator, which is formed as the product of four operators: (1) a scalar depending on the displacement of the two basis function centers A and B:The use of Hermite Gaussian functions enables us to evaluate both the integrals and their geometric derivatives on a common footing. This unified computational scheme has been implemented in an open-ended integral package GEN1INT, and interfaced to the DALTON program, using the Q5Cost library to ensure the portability of the code. Operators of the form f (|r − C|) = |r − C| −1 , |r − C| −2 , and Dirac delta function δ(r − C) have been implemented, and improvements in the evaluation of integrals involving the operator |r − C| −2 are proposed. The integral package GEN1INT can compute complicated one-electron property integrals and their arbitrary-order geometric derivatives, and is therefore expected to be a valuable tool when calculating higher order molecular properties, in particular, in combination with a recently proposed open-ended quasi-energy derivative approach (Thorvaldsen et al., J Chem Phys 2008, 129, 214108).

Molecular quantum mechanical gradients within the polarizable embedding approach—Application to the internal vibrational Stark shift of acetophenone

“…36 The required one-electron integrals are computed using the general one-electron integral framework Gen1Int by Gao, Thorvaldsen, and Ruud. 37,38 We note that molecular gradients in combination with polarizable and explicit environments have previously been reported in Refs. 39-41. In Subsection II A, we give a brief outline of the additional energy contributions introduced by the polarizable environment and proceed to describe our implementation of the firstorder geometric derivatives of the PE energy terms.…”

Analytic evaluation of the dipole Hessian matrix in coupled-cluster theory

“…Also, we note that our current implementation is not optimized with respect to memory requirements and it does not make use of point-group symmetry. Finally, we note that the computation of the dipole Hessian matrix requires no further integral derivatives than those needed for the calculation of the harmonic force constants apart from the second geometrical derivatives of the dipole integrals, which have been made available by interfacing the integral-derivative library GEN1INT 64 to CFOUR.…”

Analytic cubic and quartic force fields using density-functional theory

“…We use the DAL-TON program package 55 as a backend for the calculation of undifferentiated integrals and the unperturbed and perturbed density matrices, which are obtained with the linear response solver of Jørgensen et al 56 The calculation of properties associated with one-electron integrals was carried out using the GEN1INT library, 57 building on the flexible integral evaluation scheme of Gao and co-workers. 45 The differentiated twoelectron integrals were mainly calculated using Thorvaldsen's CGTO-DIFF-ERI code, 58 which uses the scheme of Reine et al for the evaluation of differentiated two-electron integrals using solid-harmonic Gaussians, 46 but some of the lowerorder contributions were calculated using DALTON. The differentiated exchange-correlation energy and potential contributions needed for the cubic and quartic force constants were calculated using the XCFUN library, 54 which uses automatic differentiation for evaluating the derivatives of the exchangecorrelation energy.…”

“…The geometrical derivatives of the one-electron integrals arising from the geometry dependence of the AOs are evaluated using the one-electron integral framework of Gao, Thorvaldsen, and Ruud. 45 The evaluation of geometrical derivatives of the two-electron repulsion integrals follows the approach of Reine, Tellgren, and Helgaker, 46 expanding solid-harmonic Gaussians directly in Hermite Gaussians. We furthermore extend automatic differentiation of exchangecorrelation kernels 47 to include corrections arising from the dependence of the AOs on the nuclear positions.…”