Mathematical and numerical analysis for molecular simulation: Accomplishments and challenges

International Congress of Mathematicians, ICM 2006

Volumn 3, Issue , 2006, Pages 1507-1522

  Le Bris, Claude ^a,b  

^a UNIVERSITÉ PARIS EST   (France)

^b INRIA ROCQUENCOURT   (France)

Author keywords

Computational chemistry; Density Functional Theory; Elliptic partial differential equations; Geometric integration; Hamiltonian dynamics; Hartree Fock theory; Markov chains; Molecular dynamics; Molecular simulation; Monte Carlo techniques; Nonlinear eigenvalue problems; Optimization; Reduced basis methods; Schr dinger equations; Sparse grids; Spectral methods; Spectral theory; Stochastic differential equations; Symplectic methods; Thomas Fermi theory

Indexed keywords

DINGER EQUATION; ELLIPTIC PARTIAL DIFFERENTIAL EQUATION; GEOMETRIC INTEGRATION; HAMILTONIAN DYNAMICS; HARTREE-FOCK THEORY; MOLECULAR SIMULATIONS; MONTE-CARLO TECHNIQUES; NONLINEAR EIGENVALUE PROBLEM; REDUCED BASIS METHODS; SPARSE GRID; SPECTRAL METHODS; SPECTRAL THEORY; STOCHASTIC DIFFERENTIAL EQUATIONS; SYMPLECTIC METHODS; THOMAS-FERMI;

COMPUTATIONAL CHEMISTRY; COMPUTER SIMULATION; DENSITY FUNCTIONAL THEORY; EIGENVALUES AND EIGENFUNCTIONS; HAMILTONIANS; MARKOV PROCESSES; MOLECULAR DYNAMICS; MOLECULAR STRUCTURE; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS;

MATHEMATICAL TECHNIQUES;

EID: 51849153912     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (79)

1. Allen, M. P., Tildesley, D. J., Computer simulation of liquids. Oxford Science Publications, Oxford 1988.
2. Ashcroft, N. W., Mermin, N. D., Solid-State Physics. Saunders College Publishing, New York 1976.
3. Auchmuty, G., Jia, W., Convergent iterative methods for the Hartree eigenproblem. RAIRO Modél. Math. Anal. Numér. 28 (1994), 575-610.
4. Bach, V., Error bound for the Hartree-Fock energy of atoms and molecules. Comm. Math. Phys. 147 (1992), 527-548.
5. Baffico, L., Benard, S., Maday, Y., Turinici, G., Zérah, G., Parallel in time molecular dynamics simulations. Phys. Rev. E. 66 (2002), 057701.
6. Bandrauk, A., Delfour, M., Le Bris, C. (eds.), Quantum control: mathematical and numerical challenges. CRM Proc. Lecture Notes 33, Amer. Math. Soc., Providence, RI, 2003.
7. Barrault, M., Cancès, E., Hager, W.W., Le Bris, C., Multilevel domain decomposition for electronic structure calculations. J. Comput. Phys., submitted
8. Benguria, R., Brezis, H., Lieb, E. H., The Thomas-Fermi- vonWeizsäcker theory of atoms and molecules. Comm. Math. Phys. 79 (1981), 167-180.
9. Blanc, X., Le Bris, C., Lions, P.-L., From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164 (2002), 341-381.
10. Blanc, X., Le Bris, C., Lions, P.-L., Atomistic to Continuum limits for computational materials science. Math. Model. Num. Anal., to appear.
11. Borneman, F. A. Homogenization in time of singularly perturbed mechanical systems. Lectures Notes in Math. 1697, Springer-Verlag, Berlin 1998.
12. Bowler, D. R., Miyazaki, T., Gillan, M. J., Recent progress in linear ab initio electronic structure techniques. J. Phys. Condens. Matter 14 (2002), 2781-2798.
13. Budd, Ch., Piggott, M. D., Geometric Integration and its application. In Handbook of Numerical Analysis XI, North-Holland, Amsterdam 2004, 35-139.
14. Cancès, E., Defranceschi, M., Kutzelnigg, W, Le Bris, C., Maday, Y., Computational Quantum Chemistry: a Primer. In Handbook of Numerical Analysis X, Special volume, Computational Chemistry, North-Holland, Amsterdam 2003, 3-270.
15. Cancès, E., Le Bris, C., Maday, Y., Turinici, G., Towards reduced basis approaches in ab initio electronic computations. J. Sci. Comput. 17 (1-4) (2002), 461-469.
16. Cancès, E., Le Bris, C., Maday, Y., Patera, T., Pau, G., Turinici, G., in preparation.
17. Cancès, E., Le Bris, C., On the time-dependent electronic Hartree-Fock equations coupled with a classical nuclear dynamics. Math. Models Methods Appl. Sci. 9 (1999), 963-990.
18. Cancès, E., Le Bris, C., On the convergence of SCF algorithms for the Hartree-Fock equations. M2AN Math. Model. Numer. Anal. 34 (2000), 749-774.
19. Cancès, E., Le Bris, C., Can we outperform the DIIS approach for electronic structure calculations. Int. J. Quantum Chem. 79 (2000), 82-90.
20. Cancès, E., Le Bris, C., Mennucci, B., Tomasi, J., Integral equation methods for molecular scale calculations in the liquid phase. Math. Models Methods Appl. Sci. 9 (1999) 35-44.
21. Cancès, E., Mennucci, B., Tomasi, J., A new integral formalism for the polarizable continuummodel: theoretical background and applications to isotropic and anisotropic dielectrics. J. Chem. Phys. 101 (1997), 10506-10517.
22. Cancès, E., Jourdain, B., Lelièvre, T., Quantum Monte-Carlo simulations of fermions: a mathematical analysis of the fixed-node approximation. Math. Models Methods Appl. Sci., to appear.
23. Cancès, E., Galicher, H., Lewin, M., Computing electronic structures: a new multiconfiguration approach for excited states. J. Comp. Phys 212 (2006), 73-98.
24. Catto, I., Le Bris, C., Lions, P.-L., Mathematical theory of thermodynamic limits: Thomas-Fermi type models. Oxford Math. Monogr., Oxford University Press, New York 1998.
25. Catto, I., Lions, P.-L., Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. I-IV. Comm. Partial Differential Equations 17 (1992), 1051-1110;
26. 18 (1993), 305-354, 381-429, 1149-1159.
27. Chadam, J. M., Glassey, R. T., Global existence of solutions to the Cauchy problem for time-dependent Hartree equations. J. Math. Phys. 16 (1975), 1122-1230.
28. Chandler, D., Barrier crossings: classical theory of rare but important events. In Computer Simulation of Rare Events and Dynamics of Classical and Quantum Condensed-Phase Systems - Classical and Quantum Dynamics in Condensed Phase Simulations (Berne et al., eds.), World Scientific, Singapore 1998, 3-23.
29. Coleman, A. J., Yukalov, V. I., Reduced density matrices. Lecture Notes in Chemistry 72, Springer-Verlag, Berlin 2000.
30. Darve, E., Wilson, M., Pohorille, A., Calculating Free Energies Using Scaled-force Molecular Dynamics Algorithm. Molecular Simulation 28 (2002), 113.
31. Defranceschi, M., Le Bris, C. (eds.), Mathematical models and methods for ab initio quantum chemistry. Lecture Notes in Chemistry 74, Springer-Verlag, Berlin 2001.
32. Deuflhard, P., et al. (eds.), Computational molecular dynamics: challenges, methods, ideas. Lecture Notes in Comput. Sci. Eng. 4, Springer-Verlag, Berlin 1999.
33. Deuflhard, P., Huisinga, W., Fischer, A., Schuette, Ch., Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Linear Algebra Appl. 315 (1-3) (2000), 39-59.
34. E, W., Ren, W., Vanden-Eijnden, E., String method for the study of rare events. Phys. Rev. B 66 (2002), 052301.
35. Dolbeault, J., Esteban, M. J., Séré, E., On the eigenvalues of operators with gaps. Application to Dirac operators. J. Funct. Anal. 174 (2000), 208-226.
36. Dolbeault, J., Esteban, M. J., Séré, E., A variational method for relativistic computations in atomic and molecular physics. Int. J. Quant. Chem. 93 (2003), 149-155.
37. Esteban, M. J., Séré, E., Solutions of the Dirac-Fock equations for atoms and molecules. Comm. Math. Phys. 203 (1999), 499-530.
38. Fefferman, Ch., The N-body problem in quantum mechanics Commun. Pure Appl. Math. 39 (1986), S67-S109.
39. Fefferman, Ch., Seco, L. A., On the energy of a large atom. Bull. Amer. Math. Soc. 23 (1990), 525-530.
40. Frenkel, D., Smit, B., Understanding molecular simulation. Second edition, Academic Press, San Diego, CA, 2002.
41. Friesecke, G., The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Ration. Mech. Anal. 169 (2003), 35-71.
42. Galli, G., Large scale electronic structure calculations using linear scaling methods. Phys. Stat. Sol. (b) 217 (2000), 231-249.
43. Goedecker, S., Linear scaling electronic structure methods. Rev. Mod. Phys. 71 (1999), 1085-1123.
44. Griebel, M., Hamaekers, J. Sparse grids for the Schrödinger equation, submitted.
45. Hackbusch, W., The efficient computation of certain determinants arising in the treatment of Schrödinger's equations. Computing 67 (2001), 35-56.
46. Hagedorn, G. A., Crossing the interface between Chemistry and Mathematics. Notices Amer. Math. Soc. 43 (3) (1996), 297-299.
47. Hagedorn, G. A., Joye, A., Molecular propagation through small avoided crossings of electron energy levels. Rev. Math. Phys. 11 (1999), 41-101.
48. Hairer, E., Lubich, Ch., Wanner, G., Geometric numerical integration. Springer Ser. Comput. Math. 31, Springer-Verlag, Berlin 2002.
49. Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Sørensen, T. Electronwavefunctions and densities for atoms. Ann. Henri Poincaré 2 (2001), 77-100.
50. Hunziker, W. and Sigal, I. M., The quantum N-body problem. J. Math. Phys. 41 (2000), 3348-3509.
51. Koch, O. and Lubich, Ch., Regularity of the multi-configuration time-dependent hartree approximation in quantum molecular dynamics, M2AN Math. Model. Numer. Anal., submitted.
52. Kudin, K., Scuseria, G. E., A fast multipole algorithm for the efficient treatment of the Coulomb problem in electronic structure calculations of periodic systems with Gaussian orbitals. Chem. Phys. Lett. 289 (1998), 611-616.
53. Kudin, K., Scuseria, G.E., Cancès, E., A black-box self-consistent field convergence algorithm: one step closer. J. Chem. Phys. 116 (2002), 8255-8261.
54. Le Bris, C. (ed.), Handbook of Numerical Analysis X. Special volume: Computational Chemistry, North-Holland, Amsterdam 2003.
55. Le Bris, C., Computational chemistry from the perspective of numerical analysis. Acta Numer. 14 (2005), 363-444.
56. Le Bris, C., Lions, P. L., From atoms to crystals: amathematical journey. Bull. Amer. Math. Soc. 42 (2005), 291-363.
57. Le Bris, C., Patera, A. T. (eds.), Computational chemistry.A special issue of M2AN Math. Model. Numer. Anal., to appear.
58. Le Bris, C., Lelièvre, T., Vanden-Eijnden, E., Work in progress.
59. Leimkuhler, B., Reich, S., Simulating Hamiltonian dynamics. Cambridge Monogr. Appl. Comput. Math. 14, Cambridge University Press, Cambridge 2004.
60. Lewin, M., Solutions of the Multiconfiguration Equations in Quantum Chemistry. Arch. Ration. Mech. Anal. 171 (1) (2004), 83-114.
61. Lieb, E. H., The stability of matter: from atoms to stars. Bull. Amer. Math. Soc. 22 (1990), 1-49.
62. Lieb, E. H., Simon, B., The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys. 53 (1977), 185-194.
63. Lieb, E. H., Simon, B., The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23 (1977), 22-116.
64. Lieb, E. H., Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53 (1981), 603-641.
65. Lions, P.-L., Hartree-Fock and related equations. In Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. IX (Paris, 1985-1986), Pitman Res. Notes Math. Ser. 181, Longman Sci. Tech., Harlow 1988, 304-333.
66. Lions, P.-L., Solutions of Hartree-Fock equations for Coulomb systems. Comm. Math. Phys. 109 (1987), 33-97.
67. Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. I, II. Ann. Inst. Henri PoincaréAnal. Non Linéaire 1 (1984), 109-145, 223-283.
68. Lions, P.-L., Remarks on mathematical modelling in quantum chemistry. Computational Methods in Appl. Sci., Wiley, 1996, 22-23.
69. McWeeny, R., Methods of molecular quantum mechanics. Second edition, Academic Press, 1992.
70. Maday, Y., Turinici, G., Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations. Numer. Math. 94 (4) (2003), 739-770.
71. Parr, R. G., Yang, W., Density functional theory of atoms and molecules. Oxford University Press, Oxford 1989.
72. Sanz-Serna, J. M., Calvo, M. P., Numerical Hamiltonian Problems. Appl. Math. Math. Comput. 7, Chapman and Hall, London 1994.
73. Schlick, T., et al. (eds.), Computational Molecular Biophysics. J. Comput. Phys. 151 (1999), 1-421.
74. Solovej, J. P., Universality in the Thomas-Fermi-von Weizsäcker model of atoms and molecules. Comm. Math. Phys. 129 (1990), 561-598.
75. Spruch, L., Pedagogic notes on Thomas-Fermi theory (and on some improvements): atoms, stars and the stability of bulk matter. Rev. Mod. Phys. 63 (1991), 151-209.
76. Teufel, S., Adiabatic perturbation theory in quantum dynamics. Lecture Notes in Math. 1821, Springer-Verlag, Berlin 2003.
77. Tuckerman, M. E., Ab initio molecular dynamics: Basic concepts, current trends and novel applications. J. Phys. Condens. Matter 14 (2002), R1297-R1355.
78. Voter, A. F., Sørensen, M. R., Accelerating atomistic simulations of defect dynamics: Hyperdynamics, parallel replica dynamics and temperature accelerated dynamics. Mat. Res. Soc. Symp. Proc. 538 (1999), 427-439.
79. Yserentant, H., On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math 98 (4) (2004), 731-759.