A spectral scheme for Kohn-Sham density functional theory of clusters. (English) Zbl 1351.81014

Summary: Starting from the observation that one of the most successful methods for solving the Kohn-Sham equations for periodic systems - the plane-wave method - is a spectral method based on eigenfunction expansion, we formulate a spectral method designed towards solving the Kohn-Sham equations for clusters. This allows for efficient calculation of the electronic structure of clusters (and molecules) with high accuracy and systematic convergence properties without the need for any artificial periodicity. The basis functions in this method form a complete orthonormal set and are expressible in terms of spherical harmonics and spherical Bessel functions. Computation of the occupied eigenstates of the discretized Kohn-Sham Hamiltonian is carried out using a combination of preconditioned block eigensolvers and Chebyshev polynomial filter accelerated subspace iterations. Several algorithmic and computational aspects of the method, including computation of the electrostatics terms and parallelization are discussed. We have implemented these methods and algorithms into an efficient and reliable package called ClusterES (Cluster Electronic Structure). A variety of benchmark calculations employing local and non-local pseudopotentials are carried out using our package and the results are compared to the literature. Convergence properties of the basis set are discussed through numerical examples. Computations involving large systems that contain thousands of electrons are demonstrated to highlight the efficacy of our methodology. The use of our method to study clusters with arbitrary point group symmetries is briefly discussed.


81-08 Computational methods for problems pertaining to quantum theory
82-08 Computational methods (statistical mechanics) (MSC2010)
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
81P68 Quantum computation
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
Full Text: DOI arXiv


[1] Martin, R. M., Electronic structure: basic theory and practical methods, (2004), Cambridge University Press · Zbl 1152.74303
[2] (Le Bris, C., Computational Chemistry, Handb. Numer. Anal., vol. X, (2003), North-Holland) · Zbl 1052.81001
[3] Saad, Y.; Chelikowsky, J. R.; Shontz, S. M., Numerical methods for electronic structure calculations of materials, SIAM Rev., 52, 1, 3-54, (2010) · Zbl 1185.82004
[4] Kresse, G.; Furthmuller, J., Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B, 54, 11169-11186, (1996)
[5] Kresse, G.; Furthmuller, J., Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci., 6, 1, 15-50, (1996)
[6] Marx, D.; Hutter, J., Ab initio molecular dynamics: basic theory and advanced methods, (2009), Cambridge University Press
[7] 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 conjugate gradients, Rev. Mod. Phys., 64, 1045-1097, (1992)
[8] Barnett, R. N.; Landman, U., Born-Oppenheimer molecular-dynamics simulations of finite systems: structure and dynamics of (H_{2}O)_{2}, Phys. Rev. B, 48, 4, 2081, (1993)
[9] 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.; Allan, D., First-principles computation of material properties: the ABINIT software project, Comput. Mater. Sci., 25, 3, 478-492, (2002)
[10] Segall, M. D.; Lindan, P. J.D.; Probert, M. J.; Pickard, C. J.; Hasnip, P. J.; Clark, S. J.; Payne, M. C., First-principles simulation: ideas, illustrations and the CASTEP code, J. Phys. Condens. Matter, 14, 11, 2717, (2002)
[11] Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I.; Corso, A. D.; de Gironcoli, S.; Fabris, S.; Fratesi, G.; Gebauer, R.; Gerstmann, U.; Gougoussis, C.; Kokalj, A.; Lazzeri, M.; Martin-Samos, L.; Marzari, N.; Mauri, F.; Mazzarello, R.; Paolini, S.; Pasquarello, A.; Paulatto, L.; Sbraccia, C.; Scandolo, S.; Sclauzero, G.; Seitsonen, A. P.; Smogunov, A.; Umari, P.; Wentzcovitch, R. M., QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials, J. Phys. Condens. Matter, 21, 39, (2009)
[12] Cancès, E.; Chakir, R.; Maday, Y., Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models, ESAIM: Math. Model. Numer. Anal., 46, 341-388, (2012) · Zbl 1278.82003
[13] Rappe, A. M.; Joannopoulos, J.; Bash, P., A test of the utility of plane-waves for the study of molecules from first principles, J. Am. Chem. Soc., 114, 16, 6466-6469, (1992)
[14] Hehre, W.; Stewart, R.; Pople, J., Self-consistent molecular-orbital methods. I. use of Gaussian expansions of Slater-type atomic orbitals, J. Chem. Phys., 51, 6, 2657-2664, (1969)
[15] Slater, J. C.; Koster, G. F., Simplified LCAO method for the periodic potential problem, Phys. Rev., 94, 1498-1524, (1954) · Zbl 0055.44404
[16] Soler, J.; Artacho, E.; Gale, J.; García, A.; Junquera, J.; Ordejón, P.; Sánchez-Portal, D., The SIESTA method for ab initio order-n materials simulation, J. Phys. Condens. Matter, 14, 11, 2745-2779, (2002)
[17] Chelikowsky, J. R.; Troullier, N.; Wu, K.; Saad, Y., Higher order finite difference pseudopotential method: an application to diatomic molecules, Phys. Rev. B, 50, 11355-11364, (1994)
[18] Castro, A.; Appel, H.; Oliveira, M.; Rozzi, C.; Andrade, X.; Lorenzen, F.; Marques, M.; Gross, E.; Rubio, A., Octopus: a tool for the application of time-dependent density functional theory, Phys. Status Solidi, B Basic Res., 243, 11, 2465-2488, (2006)
[19] Pask, J.; Sterne, P., Finite element methods in ab initio electronic structure calculations, Model. Simul. Mater. Sci. Eng., 13, 3, R71, (2005)
[20] 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, 2, 256-280, (2010) · Zbl 1193.81006
[21] Motamarri, P.; Nowak, M.; Leiter, K.; Knap, J.; Gavini, V., Higher-order adaptive finite-element methods for Kohn-Sham density functional theory, J. Comput. Phys., 253, 308-343, (2013) · Zbl 1349.74331
[22] Botti, S.; Castro, A.; Lathiotakis, N. N.; Andrade, X.; Marques, M. A.L., Optical and magnetic properties of boron fullerenes, Phys. Chem. Chem. Phys., 11, 4523-4527, (2009)
[23] Jing, X.; Troullier, N.; Chelikowsky, J. R.; Wu, K.; Saad, Y., Vibrational modes of silicon nanostructures, Solid State Commun., 96, 4, 231-235, (1995)
[24] Kronik, L.; Makmal, A.; Tiago, M. L.; Alemany, M. M.G.; Jain, M.; Huang, X.; Saad, Y.; Chelikowsky, J. R., Parsec the pseudopotential algorithm for real-space electronic structure calculations: recent advances and novel applications to nano-structures, Phys. Status Solidi B, 243, 5, 1063-1079, (2006)
[25] Zhou, Y.; Saad, Y.; Tiago, M. L.; Chelikowsky, J. R., Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration, Phys. Rev. E, 74, 066704, (2006)
[26] Scuseria, G. E., Ab initio calculations of fullerenes, Science, 271, 5251, 942-945, (1996)
[27] Gurin, V., Small metal clusters: ab initio calculated bare clusters and models within fullerene cages, (Buzaneva, E.; Scharff, P., Frontiers of Multifunctional Integrated Nanosystems, NATO Sci. Ser. II Math. Phys. Chem., vol. 152, (2005), Springer Netherlands), 31-38
[28] Gonzalez Szwacki, N.; Sadrzadeh, A.; Yakobson, B. I., \(\operatorname{B}_{80}\) fullerene: an ab initio prediction of geometry, stability, and electronic structure, Phys. Rev. Lett., 98, 166804, (2007)
[29] Hakkinen, H., Atomic and electronic structure of gold clusters: understanding flakes, cages and superatoms from simple concepts, Chem. Soc. Rev., 37, 1847-1859, (2008)
[30] Castleman, A. W.; Khanna, S. N., Clusters, superatoms, and building blocks of new materials, J. Phys. Chem. C, 113, 7, 2664-2675, (2009)
[31] Castleman, A. W., From elements to clusters: the periodic table revisited, J. Phys. Chem. Lett., 2, 9, 1062-1069, (2011)
[32] Iñiguez, M.; Lopez, M.; Alonso, J.; Soler, J., Electronic and atomic structure of na, mg, al and pb clusters, Z. Phys., D, At. Mol. Clust., 11, 2, 163-174, (1988)
[33] Mattei, G.; Toigo, F., Spherical averaged jellium model with norm-conserving pseudopotentials, Eur. Phys. J., D, At. Mol. Opt. Phys., 3, 3, 245-256, (1998)
[34] Yabana, K.; Bertsch, G. F., Electronic structure of C_{60} in a spherical basis, Phys. Scr., 48, 5, 633, (1993)
[35] Alasia, F.; Broglia, R. A.; Roman, H. E.; Serra, L.; Colo, G.; Pacheco, J. M., Single-particle and collective degrees of freedom in C_{60}, J. Phys. B, At. Mol. Opt. Phys., 27, 18, L643, (1994)
[36] Broglia, R.; Coló, G.; Onida, G.; Roman, H., Solid state physics of finite systems: metallic clusters, fullerenes, atomic wires, Adv. Texts Phys., (2004), Springer
[37] Brack, M., The physics of simple metal clusters: self-consistent jellium model and semiclassical approaches, Rev. Mod. Phys., 65, 677-732, (1993)
[38] Teodorescu, P. P.; Stanescu, N.-D.; Pandrea, N., Numerical analysis with applications in mechanics and engineering, (2013), John Wiley & Sons
[39] Schaeffer, N., Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations, Geochem. Geophys. Geosyst., 14, 3, 751-758, (2013)
[40] Hockney, R., Potential calculation and some applications, Methods Comput. Phys., 9, 135-211, (1970)
[41] Eastwood, J.; Brownrigg, D., Remarks on the solution of Poisson’s equation for isolated systems, J. Comput. Phys., 32, 1, 24-38, (1979) · Zbl 0407.65050
[42] Martyna, G. J.; Tuckerman, M. E., A reciprocal space based method for treating long range interactions in ab initio and force-field-based calculations in clusters, J. Chem. Phys., 110, 6, 2810-2821, (1999)
[43] Genovese, L.; Deutsch, T.; Neelov, A.; Goedecker, S.; Beylkin, G., Efficient solution of Poisson’s equation with free boundary conditions, J. Chem. Phys., 125, 7, 074105, (2006)
[44] Vömel, C.; Tomov, S. Z.; Marques, O. A.; Canning, A.; Wang, L.-W.; Dongarra, J. J., State-of-the-art eigensolvers for electronic structure calculations of large scale nano-systems, J. Comput. Phys., 227, 15, 7113-7124, (2008) · Zbl 1141.82346
[45] Knyazev, A., Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23, 2, 517-541, (2001) · Zbl 0992.65028
[46] Bottin, F.; Leroux, S.; Knyazev, A.; Zérah, G., Large-scale ab initio calculations based on three levels of parallelization, Comput. Mater. Sci., 42, 2, 329-336, (2008)
[47] F. Lorenzen, Massively-parallel eigensolver for the Octopus code, online article, 2006.
[48] Zhou, Y.; Saad, Y.; Tiago, M. L.; Chelikowsky, J. R., Self-consistent-field calculations using Chebyshev-filtered subspace iteration, J. Comput. Phys., 219, 172-184, (2006) · Zbl 1105.65111
[49] Gygi, F.; Draeger, E. W.; Schulz, M.; de Supinski, B. R.; Gunnels, J. A.; Austel, V.; Sexton, J. C.; Franchetti, F.; Kral, S.; Ueberhuber, C. W.; Lorenz, J., Large-scale electronic structure calculations of high-z metals on the bluegene/l platform, (Proceedings of the 2006 ACM/IEEE conference on Supercomputing, SC’06, (2006), ACM New York, NY, USA)
[50] Kohn, W.; Sham, L. J., Self-consistent equations including exchange and correlation effects, Phys. Rev. A, 140, 4, 1133-1138, (1965)
[51] Folland, G. B., Real analysis: modern techniques and their applications, (1999), Wiley · Zbl 0924.28001
[52] Troullier, N.; Martins, J. L., Efficient pseudopotentials for plane-wave calculations, Phys. Rev. B, 43, 3, 1993, (1991)
[53] Parr, R. G.; Yang, W., Density-functional theory of atoms and molecules, Int. Ser. Monogr. Chem., vol. 16, (1994), Oxford University Press USA
[54] Perdew, J. P.; Zunger, A., Self interaction correction to density functional approximations for many electron systems, Phys. Rev. B, 23, 10, 5048-5079, (1981)
[55] Ceperley, D. M.; Alder, B. J., Ground state of the electron gas by a stochastic method, Phys. Rev. Lett., 45, 7, 566-569, (1980)
[56] Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized gradient approximation made simple, Phys. Rev. Lett., 77, 3865-3868, (1996)
[57] Dederichs, P. H.; Zeller, R., Self-consistency iterations in electronic-structure calculations, Phys. Rev. B, 28, 5462-5472, (1983)
[58] Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T.; Ahlrichs, R.; Morgan, J., On the exponential fall off of wavefunctions and electron densities, (Osterwalder, K., Mathematical Problems in Theoretical Physics, Lect. Notes Phys., vol. 116, (1980), Springer Berlin/Heidelberg), 62-67 · Zbl 0446.35017
[59] Ahlrichs, R.; Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T.; Morgan, J. D., Bounds on the decay of electron densities with screening, Phys. Rev. A, 23, 5, 2106-2117, (1981)
[60] Jing, X.; Troullier, N.; Dean, D.; Binggeli, N.; Chelikowsky, J. R.; Wu, K.; Saad, Y., Ab initio molecular-dynamics simulations of si clusters using the higher-order finite-difference-pseudopotential method, Phys. Rev. B, 50, 16, 12234-12237, (1994)
[61] Banerjee, A. S., Harmonic analysis on isometry groups of objective structures and its applications to objective density functional theory, (2011), University of Minnesota Minneapolis, Master’s thesis
[62] Evans, L. C., Partial differential equations, Grad. Stud. Math., vol. 19, (1998), American Mathematical Society
[63] Kato, T., Perturbation theory for linear operators, Class. Math., (1995), Springer · Zbl 0836.47009
[64] Weisstein, E. W., CRC concise encyclopedia of mathematics, (2010), Chapman & Hall/CRC · Zbl 1006.00006
[65] Cooley, J. W.; Tukey, J. W., An algorithm for the machine calculation of complex Fourier series, Math. Comput., 19, 90, 297-301, (1965) · Zbl 0127.09002
[66] Mohlenkamp, M., A fast transform for spherical harmonics, J. Fourier Anal. Appl., 5, 2-3, 159-184, (1999) · Zbl 0935.65148
[67] Driscoll, J. R.; Healy, D., Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 2, 202-250, (1994) · Zbl 0801.65141
[68] Messiah, A., Clebsch-Gordan coefficients and 3j symbols, (Quantum Mechanics, vol. 2, (1962), North-Holland), 1054-1060, Appendix C
[69] Dhillon, I. S.; Parlett, B. N.; Vömel, C., The design and implementation of the MRRR algorithm, ACM Trans. Math. Softw., 32, 4, 533-560, (2006) · Zbl 1230.65046
[70] Saad, Y., Numerical methods for large eigenvalue problems, (2011), SIAM · Zbl 1242.65068
[71] Bylaska, E.; Tsemekhman, K.; Govind, N.; Valiev, M., Large-scale plane-wave-based density functional theory: formalism, parallelization and applications, (Reimers, J. R., Computational Methods for Large Systems: Electronic Structure Approaches for Biotechnology and Nanotechnology, (2011), John Wiley & Sons, Inc.), 77-116
[72] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods: fundamentals in single domains, Sci. Comput., (2006), Springer · Zbl 1093.76002
[73] Orszag, S. A., Transform method for the calculation of vector-coupled sums: application to the spectral form of the vorticity equation, J. Atmos. Sci., 27, 6, 890-895, (1970)
[74] Orszag, S. A., Numerical methods for the simulation of turbulence, Phys. Fluids, 12, 12, (1969), II-250 · Zbl 0217.25803
[75] Rasch, J.; Yu, A., Efficient storage scheme for precalculated Wigner 3 j, 6 j and gaunt coefficients, SIAM J. Sci. Comput., 25, 4, 1416-1428, (2004) · Zbl 1101.68542
[76] Genovese, L.; Neelov, A.; Goedecker, S.; Deutsch, T.; Ghasemi, S. A.; Willand, A.; Caliste, D.; Zilberberg, O.; Rayson, M.; Bergman, A., Daubechies wavelets as a basis set for density functional pseudopotential calculations, J. Chem. Phys., 129, 014109, (2008)
[77] Jackson, J. D., Classical electrodynamics, (1975), John Wiley & Sons New York · Zbl 0114.42903
[78] Löwdin, P. O., Quantum theory of cohesive properties of solids, Adv. Phys., 5, 17, 1-171, (1956)
[79] King-Smith, R.; Payne, M.; Lin, J., Real-space implementation of nonlocal pseudopotentials for first-principles total-energy calculations, Phys. Rev. B, 44, 23, 13063, (1991)
[80] Teter, M. P.; Payne, M. C.; Allan, D. C., Solution of Schrödinger’s equation for large systems, Phys. Rev. B, 40, 12255-12263, (1989)
[81] Bylander, D.; Kleinman, L.; Lee, S., Self-consistent calculations of the energy bands and bonding properties of \(\text{B}_{12} \text{C}_3\), Phys. Rev. B, 42, 2, 1394, (1990)
[82] Knyazev, A. V.; Neymeyr, K., A geometric theory for preconditioned inverse iteration, III: A short and sharp convergence estimate for generalized eigenvalue problems, Linear Algebra Appl., 358, 1, 95-114, (2003) · Zbl 1037.65039
[83] Yang, C.; Meza, J. C.; Wang, L.-W., A constrained optimization algorithm for total energy minimization in electronic structure calculations, J. Comput. Phys., 217, 2, 709-721, (2006) · Zbl 1102.81340
[84] Lin, L.; Shao, S.; E, W., Efficient iterative method for solving the Dirac-Kohn-Sham density functional theory, J. Comput. Phys., 245, 15, 205-217, (2013) · Zbl 1349.65168
[85] Knyazev, A. V.; Argentati, M. E.; Lashuk, I.; Ovtchinnikov, E., Block locally optimal preconditioned eigenvalue xolvers (BLOPEX) in HYPRE and petsc, SIAM J. Sci. Comput., 29, 5, 2224-2239, (2007) · Zbl 1149.65026
[86] A. Knyazev, private communication, 2013.
[87] Poulson, J.; Marker, B.; van de Geijn, R. A.; Hammond, J. R.; Romero, N. A., Elemental: a new framework for distributed memory dense matrix computations, ACM Trans. Math. Softw., 39, 2, 13, (2013), 24 pp · Zbl 1295.65137
[88] Blackford, L. S.; Choi, J.; Cleary, A.; D’Azevedo, E.; Demmel, J.; Dhillon, I.; Dongarra, J.; Hammarling, S.; Henry, G.; Petitet, A.; Stanley, K.; Walker, D.; Whaley, R. C., Scalapack users’ guide, (1997), Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 0886.65022
[89] Choi, J.; Dongarra, J. J.; Pozo, R.; Walker, D. W., Scalapack: a scalable linear algebra library for distributed memory concurrent computers, (Fourth Symposium on the Frontiers of Massively Parallel Computation, (1992), IEEE), 120-127
[90] van de Geijn, R. A., Using PLAPACK: parallel linear algebra package, (1997), The MIT Press
[91] Alpatov, P.; Baker, G.; Edwards, C.; Gunnels, J.; Morrow, G.; Overfelt, J.; van de Geijn, R.; Wu, Y.-J. J., PLAPACK: parallel linear algebra package design overview, (Proceedings of the 1997 ACM/IEEE Conference on Supercomputing, CDROM, (1997), ACM), 1-16
[92] Schatz, M. D.; Poulson, J.; van de Geijn, R. A., Scalable universal matrix multiplication algorithms: 2d and 3d variations on a theme, ACM Trans. Math. Softw., (2015), submitted for publication
[93] Banerjee, A. S., Density functional methods for objective structures: theory and simulation schemes, (2013), University of Minnesota Minneapolis, PhD thesis
[94] Gu, M., Single- and multiple-vector iterations, (Bai, Z.; Demmelc, J.; Dongarra, J.; Ruhe, A.; van der Vorst, H., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, (2000), SIAM Philadelphia)
[95] Stephan, U.; Drabold, D. A.; Martin, R. M., Improved accuracy and acceleration of variational order-n electronic-structure computations by projection techniques, Phys. Rev. B, 58, 20, 13472, (1998)
[96] Bekas, C.; Saad, Y.; Tiago, M. L.; Chelikowsky, J. R., Computing charge densities with partially reorthogonalized Lanczos, Comput. Phys. Commun., 171, 3, 175-186, (2005)
[97] Baroni, S.; Giannozzi, P., Towards very large-scale electronic-structure calculations, Europhys. Lett., 17, 6, 547, (1992)
[98] Zhou, Y.; Chelikowsky, J. R.; Saad, Y., Chebyshev-filtered subspace iteration method free of sparse diagonalization for solving the Kohn-Sham equation, J. Comput. Phys., 274, 770-782, (2014) · Zbl 1351.82098
[99] Daniel, J. W.; Gragg, W. B.; Kaufman, L.; Stewart, G., Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization, Math. Comput., 30, 136, 772-795, (1976) · Zbl 0345.65021
[100] Anderson, D. G., Iterative procedures for nonlinear integral equations, J. ACM, 12, 4, 547-560, (1965) · Zbl 0149.11503
[101] Broyden, C. G., A class of methods for solving nonlinear simultaneous equations, Math. Comput., 19, 92, 577-593, (1965) · Zbl 0131.13905
[102] Pulay, P., Convergence acceleration of iterative sequences: the case of scf iteration, Chem. Phys. Lett., 73, 2, 393-398, (1980)
[103] Johnson, D., Modified Broyden’s method for accelerating convergence in self-consistent calculations, Phys. Rev. B, 38, 18, 12807, (1988)
[104] Kudin, K. N.; Scuseria, G. E.; Cancès, E., A black box self consistent field convergence algorithm: one step closer, J. Chem. Phys., 116, 8255, (2002)
[105] Fang, H.; Saad, Y., Two classes of multisecant methods for nonlinear acceleration, Numer. Linear Algebra Appl., 16, 3, 197-221, (2009) · Zbl 1224.65134
[106] Kohanoff, J., Electronic structure calculations for solids and molecules: theory and computational methods, (2006), Cambridge University Press · Zbl 1113.81006
[107] Kresse, G.; Hafner, J., Ab initio molecular dynamics for liquid metals, Phys. Rev. B, 47, 558-561, (1993)
[108] Brent, R. P., Algorithms for minimization without derivatives, (1973), Courier Dover Publications · Zbl 0245.65032
[109] Galassi, M.; Davies, J.; Theiler, J.; Gough, B.; Jungman, G.; Alken, P.; Booth, M.; Rossi, F., GNU scientific library reference manual, (2009), Network Theory Ltd.
[110] Golub, G. H.; Welsch, J. H., Calculation of Gauss quadrature rules, Math. Comput., 23, 106, 221-230, (1969) · Zbl 0179.21901
[111] Goedecker, S.; Teter, M.; Hutter, J., Separable dual-space Gaussian pseudopotentials, Phys. Rev. B, 54, 1703-1710, (1996)
[112] Fiolhais, C.; Perdew, J. P.; Armster, S. Q.; MacLaren, J. M.; Brajczewska, M., Dominant density parameters and local pseudopotentials for simple metals, Phys. Rev. B, 51, 20, 14001, (1995)
[113] Chen, H.; Gong, X.; He, L.; Yang, Z.; Zhou, A., Numerical analysis of finite dimensional approximations of Kohn-Sham models, Adv. Comput. Math., 38, 2, 225-256, (2013) · Zbl 1278.35225
[114] Kotochigova, S.; Levine, Z. H.; Shirley, E. L.; Stiles, M.; Clark, C. W., Local-density-functional calculations of the energy of atoms, Phys. Rev. A, 55, 1, 191, (1997)
[115] Suryanarayana, P.; Bhattacharya, K.; Ortiz, M., A mesh-free convex approximation scheme for Kohn-Sham density functional theory, J. Comput. Phys., 230, 13, 5226-5238, (2011) · Zbl 1219.65058
[116] Nogueira, F.; Fiolhais, C.; He, J.; Perdew, J. P.; Rubio, A., Transferability of a local pseudopotential based on solid-state electron density, J. Phys. Condens. Matter, 8, 3, 287, (1996)
[117] Fiolhais, C.; Perdew, J. P.; Armster, S. Q.; MacLaren, J. M.; Brajczewska, M., Dominant density parameters and local pseudopotentials for simple metals, Phys. Rev. B, 53, 19, 13193, (1996), (Erratum)
[118] Hartwigsen, C.; Goedecker, S.; Hutter, J., Relativistic separable dual-space Gaussian pseudopotentials from H to rn, Phys. Rev. B, 58, 3641-3662, (1998)
[119] Fang, J.; Gao, X.; Zhou, A., A Kohn-Sham equation solver based on hexahedral finite elements, J. Comput. Phys., 231, 8, 3166-3180, (2012) · Zbl 1245.82007
[120] Yan, Q.-B.; Sheng, X.-L.; Zheng, Q.-R.; Zhang, L.-Z.; Su, G., Family of boron fullerenes: general constructing schemes, electron counting rule, and ab initio calculations, Phys. Rev. B, 78, 20, 201401, (2008)
[121] Green, W. H.; Gorun, S. M.; Fitzgerald, G.; Fowler, P. W.; Ceulemans, A.; Titeca, B. C., Electronic structures and geometries of \(\text{C}_{60}\) anions via density functional calculations, J. Phys. Chem., 100, 36, 14892-14898, (1996)
[122] Murray, J.; Sen, K., Molecular electrostatic potentials: concepts and applications, Theor. Comput. Chem., (1996), Elsevier
[123] Gunnarsson, O.; Harris, J.; Jones, R., Density functional theory and molecular bonding. I. first-row diatomic molecules, J. Chem. Phys., 67, 9, 3970-3979, (1977)
[124] Dickson, R. M.; Becke, A. D., Local density-functional polarizabilities and hyperpolarizabilities at the basis-set limit, J. Phys. Chem., 100, 40, 16105-16108, (1996)
[125] Guan, J.; Duffy, P.; Carter, J. T.; Chong, D. P.; Casida, K. C.; Casida, M. E.; Wrinn, M., Comparison of local-density and Hartree-Fock calculations of molecular polarizabilities and hyperpolarizabilities, J. Chem. Phys., 98, 6, 4753-4765, (1993)
[126] Chadi, D.; Cohen, M. L., Special points in the Brillouin zone, Phys. Rev. B, 8, 12, 5747, (1973)
[127] Evarestov, R. A.; Smirnov, V. P., Special points of the Brillouin zone and their use in the solid state theory, Phys. Status Solidi B, 119, 1, 9-40, (1983)
[128] Edmonds, A. R., Angular momentum in quantum mechanics, (1996), Princeton University Press · Zbl 0860.00016
[129] Folland, G. B., A course in abstract harmonic analysis, AMS/IP Stud. Adv. Math., (1994), Taylor & Francis
[130] Barut, A. O.; Raczka, R., Theory of group representations and applications, (1986), World Scientific Publishing Company · Zbl 0132.27901
[131] James, R. D., Objective structures, J. Mech. Phys. Solids, 54, 11, 2354-2390, (2006) · Zbl 1120.74312
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.