Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T15:00:26.477Z Has data issue: false hasContentIssue false
  • English
  • Français

Error Estimates of Some Numerical Atomic Orbitals in Molecular Simulations

Part of: Spectral theory and eigenvalue problems Partial differential equations, boundary value problems General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  03 July 2015

Huajie Chen  and
Reinhold Schneider
Huajie Chen*
Affiliation:
Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3,85747 Garching, Germany
Reinhold Schneider
Affiliation:
Institut fur Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany
*
*Corresponding author. Email addresses: chenh@ma.tum.de (H. Chen), schneidr@math.tu-berlin.de (R. Schneider)
Get access

Abstract

Numerical atomic orbitals have been successfully used in molecular simulations as a basis set, which provides a nature, physical description of the electronic states and is suitable for 𝒪(N) calculations based on the strictly localized property. This paper presents a numerical analysis for some simplified atomic orbitals, with polynomial-type and confined Hydrogen-like radial basis functions respectively. We give some a priori error estimates to understand why numerical atomic orbitals are computationally efficient in electronic structure calculations.

Keywords

Kohn-Sham density functional theorynumerical atomic orbitalsSlater-type orbitalsa priori error estimate

MSC classification

Secondary: 65N15: Error bounds 65N25: Eigenvalue problems 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 1 , July 2015 , pp. 125 - 146
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Almbladh, C.O. and von Barth, U., Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues, Phys. Rev. B, 31 (1985), pp. 3231–3244.Google Scholar
[2]Agmon, S., Lectures on the Exponential Decay of Solutions of Second-Order Elliptic Operators, Princeton University Press, Princeton, 1981.Google Scholar
[3]Anantharaman, A. and Cancès, E., Existence of minimizers for Kohn-Sham models in quantum chemistry, Ann. I. H. Poincaré-AN, 26 (2009), pp. 2425–2455.Google Scholar
[4]Averill, F.W. and Ellis, D.E., An efficient numerical multicenter basis set for molecular orbital calculations: application to FeCl4 J. Chem. Phys., 59 (1973), pp. 6412–6418.Google Scholar
[5]Bachmayr, M., Chen, H., and Schneider, R., Error estimates for Hermite and even-tempered Gaussian approximations in quantum chemistry, Numer. Math., 128 (2014), pp. 137–165.Google Scholar
[6]Babuška, I. and Rosenzweig, M., A finite element scheme for domains with corners, Numer. Math., 20 (1972), pp. 1–21.Google Scholar
[7]Braess, D., Asymptotics for the approximation of wave functions by sums of exponentials, J. Approximation Theory, 83 (1995), pp. 93–103.CrossRefGoogle Scholar
[8]Braess, D. and Hackbusch, W., Approximation of 1/x by exponential sums in [1,∞), IMA J. Numer. Anal., 25 (2005), pp. 685–697.Google Scholar
[9]Blum, V., Gehrke, R., Hanke, F., Havu, P., Havu, V., Ren, X., Reuter, K., Ab initio molecular simulations with numeric atom-centered orbitals, Comp. Phys. Commun., bf 180 (2009), pp. 2175–2196.Google Scholar
[10]Boys, S.F., Electron wave functions I. A general method for calculation for the stationary states of any molecular system, Proc. Roy. Soc. London, series A, 200 (1950), pp. 542–554.Google Scholar
[11]Grisvard, P., Singularities in boundary value problems, volume 22 of Research in Applied Mathematics, Masson, Paris, 1992.Google Scholar
[12]Cancès, E., Chakir, R., and Maday, Y., Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models, M2AN, 46 (2012), pp. 341–388.Google Scholar
[13]Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A., Spectral Methods for Fluid Dynamics, Springer-Verlag, 1988.Google Scholar
[14]Chen, H., Gong, X., He, L., Yang, Z., and Zhou, A., Numerical analysis of finite dimensional approximations of Kohn-Sham models, Adv. Comput. Math., 38 (2013), pp 225–256.Google Scholar
[15]Chen, H. and Schneider, R., Numerical analysis of augmented plane waves methods for full-potential electronic structure calculations, M2AN, DOI: http://dx.doi.org/10.1051/m2an/2014052.Google Scholar
[16]Ciarlet, P.G. ed., Handbook of Numerical Analysis, Vol. II. Finte Element Methods, North-Holland, Amsterdam, 1990.Google Scholar
[17]Delley, B., An all-electron numerical method for solving the local density functional for polyatomic molecules, J. Chem. Phys., 92 (1990), pp. 508–517.Google Scholar
[18]Egorov, Y.V. and Schulze, B.W., Pseudo-differential Operators, Singularities, Applications, Birkhäuse, Basel, 1997.Google Scholar
[19]Flad, H.J., Schneider, R., and Schulze, B.W., Asymptotic regularity of solutions to Hartree-Fock equations with Coulomb potential, Math. Meth. Appl. Sci., 31 (2008), pp. 2172–2201.Google Scholar
[20]Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., and Østergaard Sørensen, T., The electron density is smooth away from the nuclei, Communications in Mathematical Physics, 228 (2002), pp. 401–415.Google Scholar
[21]Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., and Østergaard Sørensen, T., Analyticity of the density of electronic wavefunctions, Arkiv för Matematik, 42 (2004), pp. 87–106.Google Scholar
[22]Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., and Østergaard Sørensen, T., Non-isotropic cusp conditions and regularity of the electron density of molecules at the nuclei, Annales Henri Poincaré, 8 (2007), pp. 731–748.Google Scholar
[23]Goedecker, S. and Colombo, L., Efficient linear scaling algorithm for tight binding molecular dynamics, Phys. Rev. Lett., 73 (1994), pp. 122–125.Google Scholar
[24]Gong, X., Shen, L., Zhang, D., and Zhou, A., Finite element approximations for Schrödinger equations with applications to electronic structure computations, J. Comput. Math., 23 (2008), pp. 310–327.Google Scholar
[25]Havu, V., Blum, V., Havu, P., Scheffler, M., Efficient O(N) integration for all-electron electronic structure calculation using numeric basis functions, J. Comput. Phys., 228 (2009), pp. 8367–8379.Google Scholar
[26]Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., and Østergaard Sørensen, T., Electron wavefunctions and densities for atoms, Annales Henri Poincaré, 2 (2001), pp. 77–100.Google Scholar
[27]Helgaker, T., Jorgensen, P., and Olsen, J., Molecular Electronic-Structure Theory, Wiley, 2000.Google Scholar
[28]Hohenberg, P. and Kohn, W., Inhomogeneous Electron Gas, Phys. Rev. B, 136 (1964), pp. 864–871.Google Scholar
[29]Junquera, J., Paz, O., Sanchez-Portal, D., and Artacho, E., Numerical atomic orbitals for linear-scaling calculations, Phys. Rev. B, 64 (2001), pp. 235111–1–23.Google Scholar
[30]Kohn, W. and Sham, L.J., Self-consistent equations including exchange and correlation effects, Phys. Rev. A, 140 (1965), pp. 1133–1138.Google Scholar
[31]Kutzelnigg, W., Theory of the expansion of wave functions in a Gaussian Basis, Int. J. Quantum Chem., 51 (1994), pp. 447–463.Google Scholar
[32]Kutzelnigg, W., Convergence of expansions in a Gaussian basis, in Strategies and Applications in Quantum Chemistry Topics in Molecular Organization and Engineering, Volume 14, 2 (2002), pp. 79–101.Google Scholar
[33]Bris, C. Le, ed., Handbook of Numerical Analysis, Vol. X. Special issue: Computational Chemistry, North-Holland, Amsterdam, 2003.Google Scholar
[34]Lieb, E.H. and Simon, B., The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), pp. 185–194.Google Scholar
[35]Lippert, G., Hutter, J., Ballone, P., and Parrinello, M., Response function basis sets: Application to density functional calculations, J. Phys. Chem., 100 (1996), pp. 6231–6235.Google Scholar
[36]Martin, R.M., Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, 2005.Google Scholar
[37]McWeeny, R.D. and Sutcliffe, B.T., Methods of Molecular Quantum Mechanics, second edition, Academic Press, New York, 1976.Google Scholar
[38]Osborn, J.E., Spectral approximation for compact operators, Mathematics of Computation, 29 (1975), pp. 712–725.Google Scholar
[39]Ozaki, T. and Kino, H., Numerical atomic basis orbitals from H to Kr, Phys. Rev. B 69 (2004), pp. 195113–1–19.Google Scholar
[40]Shen, J. and Wang, L., Some recent advances on spectral methods for unbounded domains, Commun. Comput. Phys., 5 (2009), pp. 195–241.Google Scholar
[41]Soler, J.M, Artacho, E., Gale, J.D., García, A., Junquera, J., Ordejón, P., and Sánchez-Portal, D., The SIESTA method for ab initio order-N materials simulation, J. Phys. Condens. Matter, 14 (2002), pp. 2745–2779.Google Scholar
[42]Szabo, A. and Ostlund, N.S., Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover, Mineola, New York, 1996.Google Scholar
[43]Yserentant, H., Regularity and Approximability of Electronic Wave Functions, Springer Heidelberg Dordrech London New York, 2000.Google Scholar
[44]Zunger, A. and Freeman, A.J., Self-consistent numerical-basis-set linear-combination-of-atomic-orbitals model for the study of solids in the local density formalism, Phys. Rev. B, 15 (1977), pp. 4716–4737.Google Scholar