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Error estimates for Hermite and even-tempered Gaussian approximations in quantum chemistry

Abstract

Atomic-like basis functions provide a natural, physically motivated description of electronic states, among which Gaussian-type orbitals are the most widely used basis functions in molecular simulations. This paper aims at developing a systematic analysis of numerical approximations based on linear combinations of Gaussian-type orbitals. We derive a priori error estimates for Hermite-type Gaussian bases as well as for even-tempered Gaussian bases. Numerical results are presented to support the theory.

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References

  1. 1.

    Almbladh, C.O., von Barth, U.: Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues. Phys. Rev. B 31, 3231–3244 (1985)

    Article  Google Scholar 

  2. 2.

    Agmon, S.: Lectures on the Exponential Decay of Solutions of Second-Order Elliptic Operators. Princeton University Press, Princeton (1981)

    Google Scholar 

  3. 3.

    Anantharaman, A., Cancès, E.: Existence of minimizers for Kohn–Sham models in quantum chemistry. Ann. I. H. Poincaré—AN 26, 2425–2455 (2009)

    Article  MATH  Google Scholar 

  4. 4.

    Babuška, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Finite Element Methods (Part 1), Handbook of Numerical Analysis, vol. 2, pp. 640–787. Elsevier Science Publishers, North-Holland (1991)

    Google Scholar 

  5. 5.

    Babuška, I., Rosenzweig, M.: A finite element scheme for domains with corners. Numer. Math. 20, 1–21 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Boys, S.F.: Electron wave functions I. A general method for calculation for the stationary states of any molecular system. Proc. R. Soc. Lond. Ser. A 200, 542–554 (1950)

    Article  MATH  Google Scholar 

  7. 7.

    Braess, D.: Asymptotics for the approximation of wave functions by sums of exponentials. J. Approx. Theory 83, 93–103 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Braess, D., Hackbusch, W.: Approximation of \(1/x\) by exponential sums in \([1,\infty )\). IMA J. Numer. Anal. 25, 685–697 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Cancès, E., Chakir, R., Maday, Y.: Numerical analysis of the planewave discretization of some orbital-free and Kohn–Sham models. M2AN 46, 341–388 (2012)

    Article  MATH  Google Scholar 

  10. 10.

    Chen, H., Gong, X., He, L., Yang, Z., Zhou, A.: Numerical analysis of finite dimensional approximations of Kohn–Sham models. Adv. Comput. Math. 38, 225–256 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Chen, H., Schneider, R.: Numerical analysis of augmented plane waves methods in full-potential electronic structure calculations. DFG SPP 1324 preprint, 116 (2012)

  12. 12.

    Egorov, Y.V., Schulze, B.W.: Pseudo-Differential Operators, Singularities, Applications. Birkhäuser, Basel (1997)

    Book  MATH  Google Scholar 

  13. 13.

    Flad, H.J., Schneider, R., Schulze, B.W.: Asymptotic regularity of solutions to Hartree–Fock equations with Coulomb potential. Math. Methods Appl. Sci. 31, 2172–2201 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergaard Sørensen, T.: The electron density is smooth away from the nuclei. Commun. Math. Phys. 228, 401–415 (2002)

    Article  MATH  Google Scholar 

  15. 15.

    Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergaard Sørensen, T.: Analyticity of the density of electronic wavefunctions. Arkiv för Matematik 42, 87–106 (2004)

    Article  MATH  Google Scholar 

  16. 16.

    Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergaard Sørensen, T.: Non-isotropic cusp conditions and regularity of the electron density of molecules at the nuclei. Ann. Henri Poincaré 8, 731–748 (2007)

    Article  MATH  Google Scholar 

  17. 17.

    Grisvard, P.: Singularities in boundary value problems. Research in Applied Mathematics, vol. 22. Masson, Paris (1992)

  18. 18.

    Hehre, W.J., Stewart, R.F., Pople, J.A.: Self-consistent molecular-orbital methods. I. Use of Gaussian expansions of Slater-type atomic orbitals. J. Chem. Phys. 51, 2657–2664 (1969)

    Google Scholar 

  19. 19.

    Helgaker, T., Jorgensen, P., Olsen, J.: Molecular Electronic-Structure Theory. Wiley, New York (2000)

    Book  Google Scholar 

  20. 20.

    Hill, R.N.: Rates of convergence and error estimation formulas for the Rayleigh–Ritz variational method. J. Chem. Phys. 83, 1173–1196 (1985)

    Google Scholar 

  21. 21.

    Hohenberg, P., Kohn, W.: Inhomogeneous electron gas. Phys. Rev. B 136, 864–871 (1964)

    Article  MathSciNet  Google Scholar 

  22. 22.

    Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergaard Sørensen, T.: Electron wavefunctions and densities for atoms. Ann. Henri Poincaré 2, 77–100 (2001)

    Article  Google Scholar 

  23. 23.

    Huzinaga, S.: Gaussian-type functions for polyatomic system. J. Chem. Phys. 42, 1293–1302 (1965)

    Google Scholar 

  24. 24.

    Klahn, B., Bingel, W.A.: The convergence of the Rayleigh–Ritz method in quantum chemistry II. Investigation of the convergence for special systems of Slater, Gauss and two-electron functions. Theor. Chim. Acta 44, 27–43 (1977)

    Article  MathSciNet  Google Scholar 

  25. 25.

    Klahn, B., Morgan, J.D.: Rates of convergence of variational calculations and of expectation values. J. Chem. Phys. 81, 410–433 (1984)

    Google Scholar 

  26. 26.

    Klopper, W., Kutzelnigg, W.: Gaussian basis sets and the nuclear cusp problem. J. Mol. Struct. Theochem 135, 339–356 (1986)

    Article  Google Scholar 

  27. 27.

    Kohn, W., Sham, L.J.: Self-consistent equations including exchange and correlation effects. Phys. Rev. A 140, 1133–1138 (1965)

    Article  MathSciNet  Google Scholar 

  28. 28.

    Kutzelnigg, W.: Theory of the expansion of wave functions in a Gaussian Basis. Int. J. Quantum Chem. 51, 447–463 (1994)

    Article  Google Scholar 

  29. 29.

    Kutzelnigg, W.: Convergence of expansions in a Gaussian basis. In: Ellinger, Y., Defranceschi, M. (eds.) Strategies and Applications in Quantum Chemistry. Topics in Molecular Organization and Engineering, XIV, vol. 14, pp. 463 (1996)

  30. 30.

    Le Bris, C. (ed.): Handbook of Numerical Analysis, vol. X. Special issue, Computational Chemistry, North-Holland, Amsterdam (2003)

  31. 31.

    Lieb, E.H., Simon, B.: The Hartree–Fock theory for Coulomb systems. Commun. Math. Phys. 53, 185–194 (1977)

    Article  MathSciNet  Google Scholar 

  32. 32.

    Martin, R.M.: Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  33. 33.

    McWeeny, R.D., Sutcliffe, B.T.: Methods of Molecular Quantum Mechanics, 2nd edn. Academic Press, New York (1976)

    Google Scholar 

  34. 34.

    Poularikas, A.D.(ed): Transforms and Applications Handbook, 3rd edn. The University of Alabama, Huntsville, CRC Press (2010)

  35. 35.

    Slater, J.C.: The Self-Consistent Field for Molecules and Solids. McGraw-Hill, New York (1974)

    Google Scholar 

  36. 36.

    Shen, J., Wang, L.: Some recent advances on spectral methods for unbounded domains. Commun. Comput. Phys. 5, 195–241 (2009)

    MathSciNet  Google Scholar 

  37. 37.

    Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York (1993)

    Book  MATH  Google Scholar 

  38. 38.

    Szabo, A., Ostlund, N.S.: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory. Dover, Mineola (1996)

    Google Scholar 

  39. 39.

    Werner, H.-J., Knowles, P.J.,Lindh, R., Manby, F.R., Schütz, M., et al.: MOLPRO, version 2010.1, a package of ab initio programs (2010). http://www.molpro.net

  40. 40.

    Yserentant, H.: Regularity and Approximability of Electronic Wave Functions. Springer, New York (2000)

    Google Scholar 

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Acknowledgments

The research for this paper has been enabled by the Alexander von Humboldt Foundation, whose support for the long term visit of H. Chen at Technische Universität Berlin is gratefully acknowledged.

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Correspondence to Reinhold Schneider.

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Bachmayr, M., Chen, H. & Schneider, R. Error estimates for Hermite and even-tempered Gaussian approximations in quantum chemistry. Numer. Math. 128, 137–165 (2014). https://doi.org/10.1007/s00211-014-0605-5

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Mathematics Subject Classification

  • 41A25
  • 65N35
  • 65Z05