On the evaluation of overlap integrals with exponential‐type basis functions

Homeier, Herbert H. H.; Steinborn, E. Otto

doi:10.1002/qua.560420416

Cited by 49 publications

(14 citation statements)

References 34 publications

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“…Then we obtain where n ³ l, n¢ ³ l¢+1, l¢ ³ 1, t = 0 and the coecients a l and b l are determined by following equations: The values of the Gaunt coecients and the Gegenbauer coecients should be known to calculate the radial part of overlap integrals with the same screening parameters, by using Eq. (9). Gegenbauer coecients were calculated from Eq.…”

Section: The Expression For a S Nlk;n H L H K In Terms Of Auxiliary Fmentioning

confidence: 99%

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Formulas and numerical table for the radial part of overlap integrals with the same screening parameters of Slater-type orbitals

Öztekin

Yavuz

Atalay

2001

Theoretical Chemistry Accounts: Theory, Computation, and Modeli

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The numerical properties of the radial part of overlap integrals with the same screening parameters in the form of polynomials in p = nR over Slater-type orbitals have been studied and obtained by using three dierent methods. For that purpose, the characteristics of auxiliary functions were used ®rst, then Fourier transform convolution theorem, and recurrence relations for the basic coecients of A s nlk;n H l H k were used. The calculations of the radial part of overlap integrals with the same screening parameters were made in the range 1 £ n £ 75, 1 £ n¢ £ 75, and 10 )6 £ p.

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Section: The Expression For a S Nlk;n H L H K In Terms Of Auxiliary Fmentioning

confidence: 99%

“…Some kind of ETOs, called B functions, in which radial parts are reduced to Bessel functions have been studied in Refs. [7,8,9,10,11,12,13] for the evaluation of overlap integrals.…”

Section: Introductionmentioning

confidence: 99%

Formulas and numerical table for the radial part of overlap integrals with the same screening parameters of Slater-type orbitals

Öztekin

Yavuz

Atalay

2001

Theoretical Chemistry Accounts: Theory, Computation, and Modeli

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“…An alternative approach which we propose here relies on the recursive relations (4) and (5). By inserting them into the definition (30) one arrives at…”

Section: A Spin-averaged Potentials Iaaa Typementioning

confidence: 99%

“…Nonetheless, a considerable interest has remained in the field of Slater-type orbitals (STOs) [3,4] or more general exponential-type orbitals (ETOs) [5,6]. This is motivated mainly by the superior analytical properties of STOs (i.e., fulfilment of the nuclear cusp condition [7] and correct long-range decay [8,9]) and their formal simplicity.…”

Section: Introductionmentioning

confidence: 99%

Combining Slater-type orbitals and effective core potentials

2017

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We present a general methodology to evaluate matrix elements of the effective core potentials (ECPs) within one-electron basis set of Slater-type orbitals (STOs). The scheme is based on translation of individual STO distributions in the framework of Barnett-Coulson method. We discuss different types of integrals which naturally appear and reduce them to few basic quantities which can be calculated recursively or purely numerically. Additionally, we consider evaluation of the STOs matrix elements involving the core polarisation potentials (CPP) and effective spin-orbit potentials. Construction of the STOs basis sets designed specifically for use with ECPs is discussed and differences in comparison with all-electron basis sets are briefly summarised. We verify the validity of the present approach by calculating excitation energies, static dipole polarisabilities and valence orbital energies for the alkaline earth metals (Ca, Sr, Ba). Finally, we evaluate interaction energies, permanent dipole moments and ionisation energies for barium and strontium hydrides, and compare them with the best available experimental and theoretical data.

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“…Since the enrichment functions are sharp and localized, we resorted to high-order tensor-product quadrature to obtain the results shown in Figure 6 [61]; however, for the method to be viable, efficiency in computing the stiffness matrix and force vector entries (Kd = f ) is paramount. For the integration of functions with peaks at or near a boundary over the interval, the Möbius transformation has been adopted in quantum chemistry [93][94][95]; for electronic-structure calculation with a 1/r term in the integrand of the Hamiltonian, Batcho [46] used the Duffy transformation [96,97]. In general, since the atom can be located anywhere inside the element, the above approaches do not meet our needs.…”

Section: Adaptive Quadraturementioning

confidence: 99%

The virtual element method for eigenvalue problems with potential terms on polytopic meshes

et al. 2018

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Over the course of the past two decades, quantum mechanical calculations have emerged as a key component of modern materials research. However, the solution of the required quantum mechanical equations is a formidable task and this has severely limited the range of materials systems which can be investigated by such accurate, quantum mechanical means. The current state of the art for large-scale quantum simulations is the planewave (PW) method, as implemented in now ubiquitous VASP, ABINIT, and QBox codes, among many others. However, since the PW method uses a global Fourier basis, with strictly uniform resolution at all points in space, and in which every basis function overlaps every other at every point, it suffers from substantial inefficiencies in calculations involving atoms with localized states, such as first-row and transition-metal atoms, and requires substantial nonlocal communications in parallel implementations, placing critical limits on scalability. In recent years, real-space methods such as finite-differences (FD) and finite-elements (FE) have been developed to address these deficiencies by reformulating the required quantum mechanical equations in a strictly local representation. However, while addressing both resolution and parallel-communications problems, such local real-space approaches have been plagued by one key disadvantage relative to planewaves: excessive degrees of freedom (grid points, basis functions) needed to achieve the required accuracies. And so, despite critical limitations, the PW method remains the standard today.In this work, we show for the first time that this key remaining disadvantage of real-space methods can in fact be overcome: by building known atomic physics into the solution process using modern partition-of-unity (PU) techniques in finite element analysis. Indeed, our results show order-of-magnitude reductions in basis size relative to state-of-the-art planewave based methods. The method developed here is completely general, applicable to any crystal symmetry and to both metals and insulators alike. We have developed and implemented a full self-consistent Kohn-Sham method, including both total energies and forces for molecular dynamics, and developed a full MPI parallel implementation for large-scale calculations. We have applied the method to the gamut of physical systems, from simple insulating systems with light atoms to complex d-and f-electron systems, requiring large numbers of atomic-orbital enrichments. In every case, the new PU FE method attained the required accuracies with substantially fewer degrees of freedom, typically by an order of magnitude or more, than the current state-of-the-art PW method. Finally, our initial MPI implementation has shown excellent parallel scaling of the most time-critical parts of the code up to 1728 processors, with clear indications of what will be required to achieve comparable scaling for the rest.Having shown that the key remaining disadvantage of real-space methods can in fact be overcome, the work has attrac...

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On the evaluation of overlap integrals with exponential‐type basis functions

Abstract: The numerical properties of a one-dimensional integral representation [H. P. Trivedi

Cited by 49 publications

References 34 publications

Formulas and numerical table for the radial part of overlap integrals with the same screening parameters of Slater-type orbitals

Formulas and numerical table for the radial part of overlap integrals with the same screening parameters of Slater-type orbitals

Combining Slater-type orbitals and effective core potentials

The virtual element method for eigenvalue problems with potential terms on polytopic meshes

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