KSSOLV—a MATLAB toolbox for solving the Kohn-Sham equations

ABSTRACT

We describe the design and implementation of KSSOLV, a MATLAB toolbox for solving a class of nonlinear eigenvalue problems known as the Kohn-Sham equations. These types of problems arise in electronic structure calculations, which are nowadays essential for studying the microscopic quantum mechanical properties of molecules, solids, and other nanoscale materials. KSSOLV is well suited for developing new algorithms for solving the Kohn-Sham equations and is designed to enable researchers in computational and applied mathematics to investigate the convergence properties of the existing algorithms. The toolbox makes use of the object-oriented programming features available in MATLAB so that the process of setting up a physical system is straightforward and the amount of coding effort required to prototype, test, and compare new algorithms is significantly reduced. All of these features should also make this package attractive to other computational scientists and students who wish to study small- to medium-size systems.

REFERENCES

• 1: Wanda Andreoni , Alessandro Curioni, New advances in chemistry and materials science with CPMD and parallel computing, Parallel Computing, v.26 n.7-8, p.819-842, July 2000 [doi>10.1016/S0167-8191(00)00014-4]
• 2: Arias, T. A., Payne, M. C., and Joannopoulos, J. D. 1992. Ab initio molecular dynamics: Analytically continued energy functionals and insights into iterative solutions. Phys. Rev. Lett. 69, 1077--1080.
• 3: Ashcroft, N. W. and Mermin, N. D. 1976. Solid State Physics. Brooks Cole, Pacific Grove, CA.
• 4: Baroni, S., Corso, A. D., de Gironcoli, S., Giannozzi, P., Cavazzoni, C., Ballabio, G., Scandolo, S., Chiarotti, G., Focher, P., Pasquarello, A., Laasonen, K., Trave, A., Car, R., Marzari, N., and Kokalj, A. 2006. PWscf. http://www.pwscf.org/.
• 5: C. Bekas , E. Kokiopoulou , Yousef Saad, Computation of Large Invariant Subspaces Using Polynomial Filtered Lanczos Iterations with Applications in Density Functional Theory, SIAM Journal on Matrix Analysis and Applications, v.30 n.1, p.397-418, February 2008 [doi>10.1137/060675435]
• 6: Bloch, F. 1928. Über die Quantenmechanik der Elektronen in Kristallgittern. Z. Phys. 52, 555--600.
• 7: Cancès, E. 2001. Self-consistent field algorithms for Kohn-Sham models with fractional occupation numbers. J. Chem. Phys. 114, 10616--10622.
• 8: Cancès, E. and Le Bris, C. 2000a. Can we outperform the DIIS approach for electronic structure calculations? Int. J. Quant. Chem. 79, 82--90.
• 9: Cancès, E. and Le Bris, C. 2000b. On the convergence of SCF algorithm for the Hartree-Fock equations. Math. Models. Numer. Anal. 34, 749--774.
• 10: Davis, P. J. 1979. Circulant Matrices. Wiley, New York, NY.
• 11: James W. Demmel, Applied numerical linear algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997
• 12: Alan Edelman , Tomás A. Arias , Steven T. Smith, The Geometry of Algorithms with Orthogonality Constraints, SIAM Journal on Matrix Analysis and Applications, v.20 n.2, p.303-353, April 1999 [doi>10.1137/S0895479895290954]
• 13: Ewald, P. P. 1921. Die Berchnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 64, 253--287.
• 14: Gillan, M. J. 1989. Calculation of the vacancy formation in aluminum. J. Phys. Condens. Matter 1, 689--711.
• 15: Gene H. Golub , Charles F. Van Loan, Matrix computations (3rd ed.), Johns Hopkins University Press, Baltimore, MD, 1996
• 16: Gonze, X., Beuken, J.-M., Caracas, R., Detraux, F., Fuchs, M., Rignanese, G.-M., Sindic, L., Verstraete, M., Zerah, G., Jollet, F., Torrent, M., Roy, A., Mikami, M., Ghosez, P., Raty, J.-Y., and Allan, D. 2002. First-principles computation of material properties: The ABINIT software project. Computat. Mater. Sci. 25, 478--492.
• 17: Hohenberg, P. and Kohn, W. 1964. Inhomogeneous electron gas. Phys. Rev. B 136, 3B, B864--B871.
• 18: Ihm, J., Zunger, A., and Cohen, M. L. 1979. Momentum-space formalism for the total energy of solids. J. Phys. C: Sol. State Phys. 12, 4409--4422.
• 19: Kerker, G. P. 1981. Efficient iteration scheme for self-consistent pseudopotential calculations. Phys Rev. B 23, 3082--3084.
• 20: Kleinman, L. and Bylander, D. M. 1982. Efficacious form for model pseudopotentials. Phys. Rev. Lett. 48, 1425.
• 21: Andrew V. Knyazev, Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method, SIAM Journal on Scientific Computing, v.23 n.2, p.517-541, 2001 [doi>10.1137/S1064827500366124]
• 22: Kohn, W. and Sham, L. J. 1965. Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, 4A, A1133--A11388.
• 23: Kresse, G. and Furthmüller, J. 1996a. Efficiency of ab initio total energy calculations for metals and semiconductors using a plane-wave basis set. Computat. Mater. Sci. 6, 15--50.
• 24: Kresse, G. and Furthmüller, J. 1996b. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phy. Rev. B 54, 11169--11186.
• 25: Kronik, L., Makmal, A., Tiago, M. L., Alemany, M. M. G., Jain, M., Huang, X., Saad, Y., and Chelikowsky, J. R. 2006. PARSEC—the pseudopotential algorithm for real-space electronic structure calculations: Recent advances and novel applications to nano-structures. Phys. Stat. Sol. 5, 1063--1079.
• 26: Kudin, K. N., Scuseria, G. E., and Cances, E. 2006. A black-box self-consistent field convergence algorithm: One step closer. J. Chem. Phys. 116, 19, 8255--8261.
• 27: Le Bris, C. 2005. Computational chemistry from the perspective of numerical analysis. Acta Numerica 14, 363--444.
• 28: Mermin, N. D. 1965. Thermal properties of the inhomogeneous gas. Phys. Rev. A 137, 1441--1443.
• 29: Nogueira, F., Castro, A., and Marques, M. 2003. A Primer in Density Functional Theory. Springer, Berlin, Germany, Chapter A: Tutorial on Density Functional Theory, 218--256.
• 30: Nyquist, H. 1928. Certain topics in telegraph transmission theory. Trans. AIEE 47, 617--644.
• 31: Payne, M. C., Teter, M. P., Allen, D. C., Arias, T. A., and Joannopoulos, J. D. 1992. Iterative minimization techniques for ab initio total energy calculation: Molecular dynamics and conjugate gradients. Rev. Mod. Phys. 64, 4, 1045--1097.
• 32: Perdew, J. P. and Wang, Y. 1992. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 45, 13244--13249.
• 33: Perdew, J. P. and Zunger, A. 1981. Self-interaction correction to density-functional approximation for many-electron systems. Phys. Rev. B 23, 5048--5079.
• 34: Phillips, J. C. 1958. Energy-band interpolation scheme based on a pseudopotential. Phys. Rev. 112, 3, 685--695.
• 35: Phillips, J. C. and Kleinman, L. 1958. New method for calculating wave functions in crystals and molecules. Phys. Rev. 116, 2, 287--294.
• 36: Pickett, W. E. 1989. Pseudopotential methods in condensed matter applications. Comput. Phys. Rep. 9, 115--197.
• 37: Pulay, P. 1980. Convergence acceleration of iterative sequences: The case of SCF iteration. Chem. Phys. Lett. 73, 2, 393--398.
• 38: Pulay, P. 1982. Improved SCF convergence acceleration. J. Computat. Chem. 3, 4, 556--560.
• 39: Raczkowski, D., Canning, A., and Wang, L. W. 2001. Thomas-Fermi charge mixing for obtaining self-consistency in density functional calculations. Phys. Rev. B 64, 121101--1--4.
• 40: Ritz, W. 1908. Ueber eine neue methode zur lösung gewisser variationsproblem der mathematischen physik. J. Reine Angew. Math 135, 1--61.
• 41: Shao, Y., Molnar, L. F., Jung, Y., Kussmann, J., Ochsenfeld, C., Brown, S. T., Gilbert, A. T., Slipchenko, L. V., Levchenko, S. V., ONeill, D. P., Jr, R. A. D., Lochan, R. C., Wang, T., Beran, G. J., Besley, N. A., Herbert, J. M., Lin, C. Y., Voorhis, T. V., Chien, S. H., Sodt, A., Steele, R. P., Rassolov, V. A., Maslen, P. E., Korambath, P. P., Adamson, R. D., Austin, B., Baker, J., Byrd, E. F. C., Dachsel, H., Doerksen, R. J., Dreuw, A., Dunietz, B. D., Dutoi, A. D., Furlani, T. R., Gwaltney, S. R., Heyden, A., Hirata, S., Hsu, C.-P., Kedziora, G., Khalliulin, R. Z., Klunzinger, P., Lee, A. M., Lee, M. S., Liang, W. Z., Lotan, T., Nair, N., Peters, B., Proynov, E. I., Pieniazek, P. A., Rhee, Y. M., Ritchie, J., Rosta, E., Sherrill, C. D., Simmonett, A. C., Subotnik, J. E., Woodcock III, H. L., Zhang, W., Bell, A. T., Chakraborty, A. K., Chipman, D. M., Keil, F. J., Warshel, A., Hehre, W. J., Schaefer III, H. F., Kong, J., Krylov, A. I., Gill, P. M. W., and Head-Gordon, M. 2006. Advances in methods and algorithms in a modern quantum chemistry program package. Phys. Chem. Chem. Phys. 8, 3172--3191.
• 42: Teter, M. P., Payne, M. C., and Allan, D. C. 1989. Solution of Schrödinger's equation for large systems. Phys. Rev. B 40, 18, 12255--12263.
• 43: Trefethen, L. N. and Bau III, D. 1997. Numerical Linear Algebra. Siam, Philadelphia, PA.
• 44: Troullier, N. and Martins, J. L. 1991. Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B 43, 1993--2005.
• 45: Charles Van Loan, Computational frameworks for the fast Fourier transform, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992
• 46: VandeVondele, J. and Hutter, J. 2003. An efficient orbital transformation method for electronic structure calculations. J. Chem. Phys. 118, 4365--4369.
• 47: Voorhis, T. V. and Head-Gordon, M. 2002. A geometric approach to direct minimization. Molec. Phys. 100, 11, 1713--1721.
• 48: Wang, L. 2008. PETOT. http://hpcrd.lbl.gov/linwang/PEtot/PEtot.htm.
• 49: Weinert, M. and Davenport, J. W. 1992. Fractional occupations and density-functional energies and forces. Phys. Rev. B 45, 13709--13712.
• 50: Wentzcovitch, R. M., Martins, J. L., and Allen, P. B. 1992. Energy versus free-energy in first-principles molecular dynamics. Phys. Rev. B 45, 11372--11374.
• 51: Yang, C. 2007. KSSOLV User's Guide. Tech. rep. LBNL-63661. Lawrence Berkeley National Laboratory, Berkeley, CA.
• 52: Chao Yang , Juan C. Meza , Lin-Wang Wang, A constrained optimization algorithm for total energy minimization in electronic structure calculations, Journal of Computational Physics, v.217 n.2, p.709-721, 20 September 2006 [doi>10.1016/j.jcp.2006.01.030]
• 53: Chao Yang , Juan C. Meza , Lin-Wang Wang, A Trust Region Direct Constrained Minimization Algorithm for the Kohn-Sham Equation, SIAM Journal on Scientific Computing, v.29 n.5, p.1854-1875, September 2007 [doi>10.1137/060661442]
• 54: Yin, M. T. and Cohen, M. L. 1982. Theory of ab initio pseudopotential calculations. Phys. Rev. B 25, 12, 7403--7412.
• 55: Yunkai Zhou , Yousef Saad , Murilo L. Tiago , James R. Chelikowsky, Self-consistent-field calculations using Chebyshev-filtered subspace iteration, Journal of Computational Physics, v.219 n.1, p.172-184, 20 November 2006 [doi>10.1016/j.jcp.2006.03.017]

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
        G.1.10 Applications

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
      G.1.6 Optimization

General Terms:
Algorithms, Design

Keywords:
Kohn-Sham equations, Planewave discretization, density functional theory (DFT), direct constrained minimization (DCM), electronic structure calculation, nonlinear eigenvalue problem, pseudopotential, self-consistent field iteration (SCF)