Calculation Of Molecular Integrals Research Articles

On the use of the Obara-Saika recurrence relations for the calculation of structure factors in quantum crystallography.

Modern methods of quantum crystallography are techniques firmly rooted in quantum chemistry and, as in many quantum chemical strategies, electron densities are expressed as two-centre expansions that involve basis functions centred on atomic nuclei. Therefore, the computation of the necessary structure factors requires the evaluation of Fourier transform integrals of basis function products. Since these functions are usually Cartesian Gaussians, in this communication it is shown that the Fourier integrals can be efficiently calculated by exploiting an extension of the Obara-Saika recurrence formulas, which are successfully used by quantum chemists in the computation of molecular integrals. Implementation and future perspectives of the technique are also discussed.

Exponentially more precise quantum simulation of fermions in the configuration interaction representation

We present a quantum algorithm for the simulation of molecular systems that is asymptotically more efficient than all previous algorithms in the literature in terms of the main problem parameters. As in Babbush et al (2016 New Journal of Physics 18, 033032), we employ a recently developed technique for simulating Hamiltonian evolution using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision. The algorithm of this paper involves simulation under an oracle for the sparse, first-quantized representation of the molecular Hamiltonian known as the configuration interaction (CI) matrix. We construct and query the CI matrix oracle to allow for on-the-fly computation of molecular integrals in a way that is exponentially more efficient than classical numerical methods. Whereas second-quantized representations of the wavefunction require qubits, where N is the number of single-particle spin-orbitals, the CI matrix representation requires qubits, where is the number of electrons in the molecule of interest. We show that the gate count of our algorithm scales at most as .

Reflections on the MOLECULE integral program (and its developer)

ABSTRACTIn this essay, I review some of the advances made by Jan Almlöf in the calculation of molecular integrals over Gaussian-type basis functions, and in the use of molecular symmetry to both produce integrals over symmetry-adapted combinations of basis functions and to ensure that these latter were constructed from the minimum possible number of distinct integrals over atom-centred functions. The survey is, as already noted, an essay, historical in nature, and I crave the reader's indulgence that it is structured as an essay and not as a conventional scientific manuscript.

Algorithm for the storage of expansion coefficients for the product of associated Legendre functions in elliptical coordinates useful for the calculations of molecular integrals

An algorithm is presented for the generation and storage of expansion coefficients for the product of two associated Legendre functions both with different centers introduced by the one of authors [J. Phys. B 3 (1970) 1399] for the calculation of multicenter integrals over STOs using auxiliary functions. The formulae for retrieving these coefficients in a non-sequential fashion are developed and presented. We believe that the use of formulae given in this work for the storage of expansion coefficients will have important contributions in reducing requirements for computer time in the calculation of molecular integrals with the aid of auxiliary functions.

Calculation of molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals and basic one-center Coulomb integrals.

The recurrence relations are established for the basic one-center Coulomb integrals over Slater-type orbitals (STOs). These formulae and the recurrence relations for basic overlap integrals are utilized for the calculation of multicenter electron-repulsion integrals. The calculations of multicenter electron-repulsion integrals are performed by the use of translation formulae for STOs obtained from the Lambda and Coulomb Sturmian exponential-type functions (ETFs). It is shown that these integrals show a faster convergence rate in the case of Coulomb Sturmian ETFs. The accuracy of the results is quite high for the quantum numbers of STOs and for the arbitrary values of internuclear distances and screening constants of atomic orbitals.

Computation of molecular integrals over Slater-type orbitals. X. Calculation of overlap integrals with integer and noninteger n Slater orbitals using complete orthonormal sets of exponential functions

Using two kinds of complete orthonormal sets of exponential-type orbitals, the series expansion formulas are derived for the overlap integrals with noninteger n ∗ Slater-type orbitals (NISTOs) in terms of overlap integrals over integer n Slater-type orbitals with the same and different screening constants. The convergence of the series is tested by calculating concrete cases for arbitrary values of noninteger principal quantum numbers and screening constants of NISTOs and internuclear distances.

Computation of molecular integrals over Slater-type orbitals. VIII. Calculation of overlap integrals with different screening parameters using series expansion formulas for Slater-type orbitals

Using expansion formulas for the Slater-type orbitals (STOs) obtained by the one of authors [Phys. Rev. A 31 (1985) 2851] the two-center overlap integrals with different screening parameters are expressed through the overlap integrals with the same screening parameter. The series representation of overlap integrals obtained in this paper is useful for small differences of the exponential parameters. For large differences of the exponential parameters the convergence of the infinite series becomes rather slow.

Computation of molecular integrals over Slater-type orbitals. Part VI. Calculation of overlap integrals with the same screening parameters using Gegenbauer coefficients

The overlap integrals with the same screening parameters of Slater-type orbitals were calculated using the Gegenbauer coefficients. The computer results are useful especially for large quantum numbers of overlap integrals, which occur in the series expansion formulae for multicenter molecular integrals.

Computation of molecular integrals over Slater-type orbitals. IV. Calculation of multicenter electron-repulsion integrals using recurrence relations for overlap integrals

Using formulas given by one of the authors [I.I. Guseinov, J. Mol. Struct. (Theochem) 417 (1997) 117], the multicenter electron-repulsion integrals with the arbitrary location and screening constants of Slater-type orbitals (STOs) are calculated for extremely large quantum numbers. Accuracy of the results is quite high for quantum numbers n, l and m of STOs.

Computation of molecular integrals over Slater-type orbitals. III. Calculation of multicenter nuclear-attraction integrals using recurrence relations for overlap integrals

Using recurrence relations for basic overlap integrals two- and three-center nuclear-attraction integrals are calculated for extremely large quantum numbers. The accuracy of the results is quite high for the principal quantum numbers of Slater functions and for the arbitrary values of internuclear distances and screening constants of atomic orbitals.

Computation of molecular integrals over Slater type orbitals I. Calculations of overlap integrals using recurrence relations

A new algorithm is presented for the calculation of two-center overlap integrals over Slater type orbitals, based on sets of recurrence relations and analytical formulas. The recurrence relations enable us to calculate some of overlap integrals by the use of which other overlap integrals are calculated analytically. The recurrence relations obtained in this work for these `basic overlap integrals' are especially useful for the calculation of any overlap integral for large quantum numbers. An accuracy of the computer results is satisfactory for the values of principal quantum numbers of Slater functions up to 50, and for the arbitrary values of screening constants of atomic orbitals and internuclear distances.

Calculation of Gaussian integrals using symbolic manipulation

The calculation of molecular integrals is extremely important for applications to such diverse areas as statistical mechanics and quantum chemistry. A careful derivation of a method for calculating primitive Gaussian integrals originally proposed by Obara and Saika is presented. The basic recursion relations for the two- and three-center overlap integrals is derived using a simple technique. Several new horizontal recursion relations are given. Finally, an innovative method for implementing these recursion relations is discussed. The recursion relations in this form are suited for programming using a symbolic manipulation language. There are several reasons why it is of interest to consider programming with symbolic manipulation. It has been found that it is possible to write algorithms that will generate values for Gaussian integrals for very large values of angular momentum automatically. Calculations can be done to arbitrary precision in Maple. Having these recursions programmed in Maple allows for the possibility of using the Maple programs to help in the writing of similar programs in other languages which are, numerically, much faster. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 557–570, 1997

Convergence accelerators in the computation of molecular integrals over Slater-type basis functions in the two-range one-center expansion method.

The so-called nonlinear transformations may be of substantial help in many practical problems including the so-called multicenter integrals. In this work Wynn's epsilon algorithm and the Levin's u transformation have been used to improve the convergence of the series representation of three-center two-electron integrals obtained via the so-called two-range one-center expansion method.

Calculation of molecular electronic structure with the help of the Σ-separation method: preliminary results and prospective development

Severe mathematical difficulties arise in the calculation of molecular integrals with exponentially decaying functions, particularly in the case of polyatomic molecules and heavy atom compounds. In an attempt to overcome this, the Σ-separation approach was chosen as the basis for a numerically advanced tabulation—restoration algorithm.

Computation of molecular integrals over partially generalized hermite-gaussian functions

Computation of molecular integrals over partially generalized hermite-gaussian functions

A study on universal Gaussian basis sets for first-row atoms

Analysis of various optimum and non-optimum Gaussian basis sets for firstrow elements have indicated that with a minimum increase of the basis set size and without loss of accuracy of the calculated total energy, a single “universal” Gaussian basis set may replace individually optimized Gaussian basis sets for a series of atoms. Such a universal Gaussian basis set may substantially reduce the computational work required for the calculation of molecular integrals in ab initio MO calculations.