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Numerical analysis of finite dimensional approximations of Kohn–Sham models

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Abstract

In this paper, we study finite dimensional approximations of Kohn–Sham models, which are widely used in electronic structure calculations. We prove the convergence of the finite dimensional approximations and derive the a priori error estimates for ground state energies and solutions. We also provide numerical simulations for several molecular systems that support our theory.

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References

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

    Google Scholar 

  2. 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 

  3. Arias, T.A.: Multiresolution analysis of electronic structure: semicardinal and wavelet bases. Rev. Mod. Phys. 71, 267–311 (1999)

    Article  Google Scholar 

  4. Bao, G., Zhou, A.: Analysis of finite dimensional approximations to a class of partial differential equations. Math. Methods Appl. Sci. 27, 2055–2066 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cancès, E., Chakir, R., Maday, Y.: Numerical analysis of nonlinear eigenvalue problems. J. Sci. Comput. 45, 90–117 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cancès, E., Chakir, R., Maday, Y.: Numerical analysis of the planewave discretization of some orbital-free and Kohn–Sham models. arXiv:1003.1612 (2010)

  7. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Springer, Berlin Heidelberg (2007)

    MATH  Google Scholar 

  8. Chen, H., Gong, X., Zhou, A.: Numerical approximations of a nonlinear eigenvalue problem and applications to a density functional model. Math. Methods Appl. Sci. 33, 1723–1742 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, H., He, L., Zhou, A.: Finite element approximations of nonlinear eigenvalue problems in quantum physics. Comput. Methods Appl. Mech. Eng. 200, 1846–1865 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, H., Zhou, A.: Orbital-free density functional theory for molecular structure calculations. Numer. Math. Theor. Meth. Appl. 1, 1–28 (2008)

    Google Scholar 

  11. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  12. Dai, X., Gong, X., Yang, Z., Zhang, D., Zhou, A.: Finite volume discretizations for eigenvalue problems with applications to electronic structure calculations. Multiscale Model. Simul. 9, 208–240 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dreizler, R.M., Gross, E.K.U.: Density Functional Theory. Springer, Heidelberg (1990)

    Book  MATH  Google Scholar 

  14. Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20, 303–353 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Genovese, L., Neelov, A., Goedecker, S., Deutsch, T., Ghasemi, S.A., Willand, A., Caliste, D., Zilberberg, O., Rayson, M., Bergman, A., Schneider, R.: Daubechies wavelets as a basis set for density functional pseudopotential calculations. J. Chem. Phys. 129, 014109–014112 (2008)

    Article  Google Scholar 

  16. 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 

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

    Article  MathSciNet  Google Scholar 

  18. Kato, T.: On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. 10, 151–177 (1957)

    Article  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  20. Langwallner, B., Ortner, C., Süli, E.: Existence and convergence results for the Galerkin approximation of an electronic density functional. M 3 AS, 12, 2237–2265 (2010)

    Google Scholar 

  21. Le Bris, C.: Quelques problèmes mathématiques en chimie quanntique moléculaire. Ph.D. thesis, Ècole Polytechnique (1993)

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

    Google Scholar 

  23. Maday, Y., Turinici, G.: Error bars and quadratically convergent methods for the numerical simulation of the Hartree–Fock equations. Numer. Math. 94, 739–770 (2000)

    MathSciNet  Google Scholar 

  24. Martin, R.M.: Electronic Structure: Basic Theory and Practical Method. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  25. Payne, M.C., Teter, M.P., Allan, D.C., Arias, T.A., Joannopoulos, J.D.: Iterative minimization techniques for ab-initio total-energy calculations: molecular dynamics and conjugategradients. Rev. Mod. Phys. 64, 1045–1097 (1992)

    Article  Google Scholar 

  26. Parr, R.G., Yang, W.T.: Density-Functional Theory of Atoms and Molecules. Clarendon Press, Oxford (1994)

    Google Scholar 

  27. Pulay, P.: Convergence acceleration of iterative sequences. The case of scf iteration. Chem. Phys. Lett. 73, 393–398 (1980)

    Article  Google Scholar 

  28. Saad, Y., Chelikowsky, J.R., Shontz, S.M.: Numerical methods for electronic structure calculations of materials. SIAM Rev. 52, 3–54 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schneider, R., Rohwedder, Neelov, T.A., Blauert, J.: Direct minimization for calculating invariant subspaces in density functional computations of the electronic structure. J. Comput. Math. 27, 360–387 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Simon, B.: Schrödinger operators in the twentieth century. J. Math. Phys. 41, 3523–3555 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Suryanarayana, P., Gavini, V., Blesgen, T., Bhattacharya, K., Ortiz, M.: Non-periodic finite-element formulation of Kohn–Sham density functional theory. J. Mech. Phys. Solids 58, 256–280 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Troullier, N., Martins, J.L.: A straightforward method for generating soft transferable pseudopotentials. Solid State Commun. 74, 613–616 (1990)

    Article  Google Scholar 

  33. Wang, Y.A., Carter, E.A.: Orbital-free kinetic-energy density functional theory. In: Schwartz, S.D. (ed.) Theoretical Methods in Condensed Phase Chemistry, pp. 117–184. Kluwer, Dordrecht (2000)

    Google Scholar 

  34. Zhang, D., Zhou, A., Gong, X.: Parallel mesh refinement of higher order finite elements for electronic structure calculations. Commun. Comput. Phys. 4, 1086–1105 (2008)

    Google Scholar 

  35. Zhou, A.: An analysis of finite-dimensional approximations for the ground state solution of Bose–Einstein condensates. Nonlinearity 17, 541–550 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhou, A.: Finite dimensional approximations for the electronic ground state solution of a molecular system. Math. Methods Appl. Sci. 30, 429–447 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhou, A.: Multi-level adaptive corrections in finite dimensional approximations. J. Comput. Math. 28, 45–54 (2010)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Aihui Zhou.

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Communicated by Zhongying Chen.

This work was partially supported by the National Science Foundation of China under grants 10871198 and 10971059, the Funds for Creative Research Groups of China under grant 11021101, and the National Basic Research Program of China under grant 2011CB309703.

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Chen, H., Gong, X., He, L. et al. Numerical analysis of finite dimensional approximations of Kohn–Sham models. Adv Comput Math 38, 225–256 (2013). https://doi.org/10.1007/s10444-011-9235-y

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