Analytical evaluation of pseudopotential matrix elements with Gaussian-type solid harmonics of arbitrary angular momentum

Hu, Amanda; Staufer, Markus; Birkenheuer, U.; Igoshine, Valentin; Rösch, Notker

doi:10.1002/1097-461x(2000)79:4<209::aid-qua2>3.0.co;2-j

Cited by 12 publications

(6 citation statements)

References 63 publications

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“…Switching to spherical harmonic basis functions affects the summation over i , which is, however, not problematic: for a given λ, all (nonzero) a i ,λ have the same sign because they are nonzero only if i and λ have the same parity. The evaluation of ECP matrix elements with spherical harmonic Gaussians has been described in some detail,6 and we can provide further support for the above statement by analyzing the equations, and verified that also in that algorithm, the sum that is $\cal{O}(\alpha^{-(l+3/2)})$ contains terms of $\cal{O}(\alpha^{-3/2})$ [those with l 1 + l 2 + λ = 0 in eq. (34) of ref 6…”

Section: Are There Alternatives?supporting

confidence: 63%

Numerical instabilities in the computation of pseudopotential matrix elements

Wüllen

2005

J Comput Chem

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Steep high angular momentum Gaussian basis functions in the vicinity of a nucleus whose inner electrons are replaced by an effective core potential may lead to numerical instabilities when calculating matrix elements of the core potential. Numerical roundoff errors may be amplified to an extent that spoils any result obtained in such a calculation. Effective core potential matrix elements for a model problem are computed with high numerical accuracy using the standard algorithm used in quantum chemical codes and compared to results of the MOLPRO program. Thus, it is demonstrated how the relative and absolute errors depend an basis function angular momenta, basis function exponents and the distance between the off-center basis function and the center carrying the effective core potential. Then, the problem is analyzed and closed expressions are derived for the expected numerical error in the limit of large basis function exponents. It is briefly discussed how other algorithms would behave in the critical case, and they are found to have problems as well. The numerical stability could be increased a little bit if the type 1 matrix elements were computed without making use of a partial wave expansion.

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Section: Are There Alternatives?supporting

confidence: 63%

Numerical instabilities in the computation of pseudopotential matrix elements

Wüllen

2005

J Comput Chem

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“…53,54 Relativistic effects for the species containing a Pd atom were accounted for by either applying the Douglas-Kroll scalar relativistic approach to the Dirac-Kohn-Sham problem 55,56 or via relativistic pseudopotentials. 57,58 In some cases, ''Stuttgart'' pseudopotentials were used for the O 2Ϫ ions 59 and Mg 2ϩ centers. 60 The latter were also used for the (Mg pp *) 2ϩ centers, but without basis functions.…”

Section: B Computational Detailsmentioning

confidence: 99%

Cluster embedding in an elastic polarizable environment: Density functional study of Pd atoms adsorbed at oxygen vacancies of MgO(001)

Наслузов¹,

Rivanenkov²,

Gordienko³

et al. 2001

The Journal of Chemical Physics

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118

132

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Adsorption complexes of palladium atoms on F s , F s ϩ , F s 2ϩ , and O 2Ϫ centers of MgO͑001͒ surface have been investigated with a gradient-corrected ͑Becke-Perdew͒ density functional method applied to embedded cluster models. This study presents the first application of a self-consistent hybrid quantum mechanical/molecular mechanical embedding approach where the defect-induced distortions are treated variationally and the environment is allowed to react on perturbations of a reference configuration describing the regular surface. The cluster models are embedded in an elastic polarizable environment which is described at the atomistic level using a shell model treatment of ionic polarizabilities. The frontier region that separates the quantum mechanical cluster and the classical environment is represented by pseudopotential centers without basis functions. Accounting in this way for the relaxation of the electronic structure of the adsorption complex results in energy corrections of 1.9 and 5.3 eV for electron affinities of the charged defects F s ϩ and F s 2ϩ , respectively, as compared to models with a bulk-terminated geometry. The relaxation increases the stability of the adsorption complex Pd/F s by 0.4 eV and decreases the stability of the complex Pd/F s 2ϩ by 1.0 eV, but it only weakly affects the binding energy of Pd/F s ϩ . The calculations provide no indication that the metal species is oxidized, not even for the most electron deficient complex Pd/F s 2ϩ . The binding energy of the complex Pd/O 2Ϫ is calculated at Ϫ1.4 eV, that of the complex Pd/F s 2ϩ at Ϫ1.3 eV. The complexes Pd/F s and Pd/F s ϩ exhibit notably higher binding energies, Ϫ2.5 and Ϫ4.0 eV, respectively; in these complexes, a covalent polar adsorption bond is formed, accompanied by donation of electronic density to the Pd 5s orbital.

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“…Therefore, we will now discuss those properties of \documentclass{article}\pagestyle{empty}\begin{document}$\mathcal{Y}^{m}_{\ell}(\nabla)$\end{document} that are needed for the derivation of addition theorems for B functions. Other properties as well as numerous applications—mainly in the context of multicenter integrals—are described in the literature 6, 19, 29, 55, 57, 93, 105–135.…”

Section: The Translation Operator In Spherical Formmentioning

confidence: 99%

Addition theorems as three‐dimensional Taylor expansions. II.Bfunctions and other exponentially decaying functions

Weniger

2002

Int J of Quantum Chemistry

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Addition theorems can be constructed by doing three-dimensional Taylor expansions according to f (r + r ′ ) = exp(r ′ · ∇)f (r). Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form exp(x ′ ∂/∂x) exp(y ′ ∂/∂y) exp(z ′ ∂/∂z) would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator exp(r ′ · ∇) involving powers of the Laplacian ∇ 2 and spherical tensor gradient operators Y (2000)]. The application of the translation operator in its spherical form is particularly simple in the case of B functions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slatertype functions, bound-state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums of B functions, the addition theorems for these functions can be written down immediately.

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Analytical evaluation of pseudopotential matrix elements with Gaussian-type solid harmonics of arbitrary angular momentum

Cited by 12 publications

References 63 publications

Numerical instabilities in the computation of pseudopotential matrix elements

Numerical instabilities in the computation of pseudopotential matrix elements

Cluster embedding in an elastic polarizable environment: Density functional study of Pd atoms adsorbed at oxygen vacancies of MgO(001)

Addition theorems as three‐dimensional Taylor expansions. II.Bfunctions and other exponentially decaying functions

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