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

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.

[1]  Arias,et al.  Ab initio molecular dynamics: Analytically continued energy functionals and insights into iterative solutions. , 1992, Physical review letters.

[2]  E. Cancès,et al.  Self-consistent field algorithms for Kohn–Sham models with fractional occupation numbers , 2001 .

[3]  F. Bloch.  Über die Quantenmechanik der Elektronen in Kristallgittern , 1929 .

[4]  M. L. Cohen,et al.  Ab initio pseudopotential theory , 1982 .

[5]  M. Gillan.  Calculation of the vacancy formation energy in aluminium , 1989 .

[6]  Harry Nyquist.  Certain Topics in Telegraph Transmission Theory , 1928 .

[7]  G. Kresse,et al.  Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set , 1996 .

[8]  A. Zunger,et al.  CORRIGENDUM: Momentum-space formalism for the total energy of solids , 1979 .

[9]  Y. Saad,et al.  PARSEC – the pseudopotential algorithm for real‐space electronic structure calculations: recent advances and novel applications to nano‐structures , 2006 .

[10]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[11]  G. V. Chester,et al.  Solid-State Physics , 1962, Nature.

[12]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[13]  Claude Le Bris,et al.  Computational chemistry from the perspective of numerical analysis , 2005, Acta Numerica.

[14]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[15]  Martins,et al.  Energy versus free-energy conservation in first-principles molecular dynamics. , 1992, Physical review. B, Condensed matter.

[16]  Fernando Nogueira,et al.  A Tutorial on Density Functional Theory , 2003 .

[17]  Andrew V. Knyazev,et al.  Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method , 2001, SIAM J. Sci. Comput..

[18]  J. VandeVondele,et al.  An efficient orbital transformation method for electronic structure calculations , 2003 .

[19]  Juan C. Meza,et al.  A Trust Region Direct Constrained Minimization Algorithm for the Kohn-Sham Equation , 2007, SIAM J. Sci. Comput..

[20]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[21]  G. Scuseria,et al.  A black-box self-consistent field convergence algorithm: One step closer , 2002 .

[22]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.

[23]  Martin Head-Gordon,et al.  Advances in Methods and Algorithms in a Modern Quantum Chemistry Program Package , 2006 .

[24]  Kresse,et al.  Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. , 1996, Physical review. B, Condensed matter.

[25]  Juan C. Meza,et al.  A constrained optimization algorithm for total energy minimization in electronic structure calculations , 2005, J. Comput. Phys..

[26]  Allan,et al.  Solution of Schrödinger's equation for large systems. , 1989, Physical review. B, Condensed matter.

[27]  M. Head‐Gordon,et al.  A geometric approach to direct minimization , 2002 .

[28]  P. Hohenberg,et al.  Inhomogeneous Electron Gas , 1964 .

[29]  C. Bris,et al.  Can we outperform the DIIS approach for electronic structure calculations , 2000 .

[30]  T. Arias,et al.  Iterative minimization techniques for ab initio total energy calculations: molecular dynamics and co , 1992 .

[31]  Leonard Kleinman,et al.  New Method for Calculating Wave Functions in Crystals and Molecules , 1959 .

[32]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[33]  Wang,et al.  Accurate and simple analytic representation of the electron-gas correlation energy. , 1992, Physical review. B, Condensed matter.

[34]  P. P. Ewald.  Die Berechnung optischer und elektrostatischer Gitterpotentiale , 1921 .

[35]  E. Cancès,et al.  On the convergence of SCF algorithms for the Hartree-Fock equations , 2000 .

[36]  P. Pulay.  Improved SCF convergence acceleration , 1982 .

[37]  J. C. Phillips,et al.  Energy-Band Interpolation Scheme Based on a Pseudopotential , 1958 .

[38]  Martins,et al.  Efficient pseudopotentials for plane-wave calculations. , 1991, Physical review. B, Condensed matter.

[39]  W. Ritz.  Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. , 1909 .

[40]  Matthieu Verstraete,et al.  First-principles computation of material properties: the ABINIT software project , 2002 .

[41]  H. Nyquist,et al.  Certain Topics in Telegraph Transmission Theory , 1928, Transactions of the American Institute of Electrical Engineers.

[42]  Y. Saad,et al.  Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Gene H. Golub,et al.  Matrix computations , 1983 .

[44]  Yousef Saad,et al.  Self-consistent-field calculations using Chebyshev-filtered subspace iteration , 2006, J. Comput. Phys..

[45]  Y. Saad,et al.  Polynomial filtered Lanczos iterations with applications in Density Functional Theory , 2005 .

[46]  Alessandro Curioni,et al.  New advances in chemistry and materials science with CPMD and parallel computing , 2000, Parallel Comput..

[47]  Davenport,et al.  Fractional occupations and density-functional energies and forces. , 1992, Physical review. B, Condensed matter.

[48]  Constantine Bekas,et al.  Computation of Large Invariant Subspaces Using Polynomial Filtered Lanczos Iterations with Applications in Density Functional Theory , 2008, SIAM J. Matrix Anal. Appl..

[49]  Marvin L. Cohen,et al.  Theory of ab initio pseudopotential calculations , 1982 .

[50]  A. Zunger,et al.  Self-interaction correction to density-functional approximations for many-electron systems , 1981 .

[51]  P. Pulay.  Convergence acceleration of iterative sequences. the case of scf iteration , 1980 .

[52]  Andrew Canning,et al.  Thomas-Fermi charge mixing for obtaining self-consistency in density functional calculations , 2001 .

[53]  C. Loan.  Computational Frameworks for the Fast Fourier Transform , 1992 .

[54]  Warren E. Pickett,et al.  Pseudopotential methods in condensed matter applications , 1989 .

[55]  Leonard Kleinman,et al.  Efficacious Form for Model Pseudopotentials , 1982 .

[56]  F. Nogueira,et al.  A primer in density functional theory , 2003 .

[57]  N. Mermin.  Thermal Properties of the Inhomogeneous Electron Gas , 1965 .

[58]  G. Kerker.  Efficient iteration scheme for self-consistent pseudopotential calculations , 1981 .