One particle equations for many particle quantum systems: The MCTHDF method

Bardos, Claude; Mauser, Norbert

doi:10.1090/S0033-569X-09-01181-7

Authors: Claude Bardos and Norbert J. Mauser
Journal: Quart. Appl. Math. 68 (2010), 43-59
MSC (2000): Primary 35Q40, 35Q55
DOI: https://doi.org/10.1090/S0033-569X-09-01181-7
Published electronically: October 19, 2009
MathSciNet review: 2598879
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Abstract: This contribution is devoted to the mathematical analysis of more or less sophisticated approximations of the time evolution of systems of $N$ quantum particles. New results for the Multiconfiguration Time Dependent Hartree-Fock (MCTDHF) method (which cover the material of the talk given by the first author at the “Nonlinear waves conference in honor of Walter Strauss”) are summarized and compared with the simpler Hartree and Hartree-Fock equations.

References

Riccardo Adami, François Golse, and Alessandro Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys. 127 (2007), no. 6, 1193–1220. MR 2331036, DOI https://doi.org/10.1007/s10955-006-9271-z

C. Bardos, I. Catto, N.J. Mauser and S. Trabelsi, Global-in-time existence of solutions to the multi-configuration time-dependent Hartree-Fock equations: A sufficient condition. Applied Math. Lett. 22, 147–152 (2009).

C. Bardos, I. Catto, N.J. Mauser and S. Trabelsi, Setting and analysis of the multi-configuration time-dependent Hartree-Fock equations. manuscript (2009).

Claude Bardos, Laszlo Erdős, François Golse, Norbert Mauser, and Horng-Tzer Yau, Derivation of the Schrödinger-Poisson equation from the quantum $N$-body problem, C. R. Math. Acad. Sci. Paris 334 (2002), no. 6, 515–520 (English, with English and French summaries). MR 1890644, DOI https://doi.org/10.1016/S1631-073X%2802%2902253-7

Claude Bardos, François Golse, Alex D. Gottlieb, and Norbert J. Mauser, Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, J. Math. Pures Appl. (9) 82 (2003), no. 6, 665–683 (English, with English and French summaries). MR 1996777, DOI https://doi.org/10.1016/S0021-7824%2803%2900023-0

Claude Bardos, François Golse, Alex D. Gottlieb, and Norbert J. Mauser, Accuracy of the time-dependent Hartree-Fock approximation for uncorrelated initial states, J. Statist. Phys. 115 (2004), no. 3-4, 1037–1055. MR 2054172, DOI https://doi.org/10.1023/B%3AJOSS.0000022381.86923.0a

Claude Bardos, François Golse, and Norbert J. Mauser, Weak coupling limit of the $N$-particle Schrödinger equation, Methods Appl. Anal. 7 (2000), no. 2, 275–293. Cathleen Morawetz: a great mathematician. MR 1869286, DOI https://doi.org/10.4310/MAA.2000.v7.n2.a2

M. Beck, A. H. Jäckle, G.A. Worth and H.-D. Meyer, The multi-configuration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagation wave-packets. Phys. Rep., 324, 1–105 (2000).

A. Bove, G. Da Prato, and G. Fano, On the Hartree-Fock time-dependent problem, Comm. Math. Phys. 49 (1976), no. 1, 25–33. MR 456066

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the $1/N$ limit of interacting classical particles, Comm. Math. Phys. 56 (1977), no. 2, 101–113. MR 475547

J. Caillat, J. Zanghellini, M. Kitzler, O. Koch, W. Kreuzer and A. Scrinzi, Correlated multi-electron systems in strong laser fields – An MCTDHF approach. Phys. Rev. A, 71, 012712 (2005).

F. Castella, $L^2$ solutions to the Schrödinger-Poisson system: existence, uniqueness, time behaviour, and smoothing effects, Math. Models Methods Appl. Sci. 7 (1997), no. 8, 1051–1083. MR 1487521, DOI https://doi.org/10.1142/S0218202597000530

J. M. Chadam and R. T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. Mathematical Phys. 16 (1975), 1122–1130. MR 413843, DOI https://doi.org/10.1063/1.522642

A. J. Coleman, Structure of fermion density matrices, Rev. Modern Phys. 35 (1963), 668–689. MR 0155637, DOI https://doi.org/10.1103/RevModPhys.35.668

Alexander Elgart, László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Nonlinear Hartree equation as the mean field limit of weakly coupled fermions, J. Math. Pures Appl. (9) 83 (2004), no. 10, 1241–1273 (English, with English and French summaries). MR 2092307, DOI https://doi.org/10.1016/j.matpur.2004.03.006

Alexander Elgart, László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Ration. Mech. Anal. 179 (2006), no. 2, 265–283. MR 2209131, DOI https://doi.org/10.1007/s00205-005-0388-z

László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 (2007), no. 3, 515–614. MR 2276262, DOI https://doi.org/10.1007/s00222-006-0022-1

L. Erdős and H.-T. Yau, Derivation of the nonlinear Schrödinger equation with Coulomb potential. Advances in theoretical math. physics. (2003)

J. Fröhlich, A. Knowles, and S. Schwarz, On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction. preprint arXiv: 0805.4299v1 [math-ph] 28 May 2008.

J. Fröhlich and A. Knowles, A Microscopic Derivation of the Time-Dependent Hartree-Fock Equation with Coulomb Two-Body Interaction. preprint arXiv:0810.4282v1

J. Frenkel, Wave Mechanics, Oxford University Press, Oxford (1934).

G. Friesecke, The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions, Arch. Ration. Mech. Anal. 169 (2003), no. 1, 35–71. MR 1996268, DOI https://doi.org/10.1007/s00205-003-0252-y

G. Friesecke, On the infinitude of non-zero eigenvalues of the single-electron density matrix for atoms and molecules, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2029, 47–52. MR 1993343, DOI https://doi.org/10.1098/rspa.2002.1027

Jean Ginibre and Giorgio Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z. 170 (1980), no. 2, 109–136. MR 562582, DOI https://doi.org/10.1007/BF01214768

F. Golse, On the mean field limit for large particle systems, Actes du colloque du GdR Equations aux derivées partielles, Forges-les-Eaux (2006).

A. Gottlieb and N.J. Mauser, A new measure of correlation for electron systems, Phys. Rev. Lett. 95 (12) 1230003 (2005).

A. Gottlieb and N.J. Mauser, Properties of non-freeness: An entropy measure of correlation of electrons, Int. J. of Quantum Information 5(6) (2007) 10 n:33.

Maxime Hauray and Pierre-Emmanuel Jabin, $N$-particles approximation of the Vlasov equations with singular potential, Arch. Ration. Mech. Anal. 183 (2007), no. 3, 489–524. MR 2278413, DOI https://doi.org/10.1007/s00205-006-0021-9

Claude Le Bris, A general approach for multiconfiguration methods in quantum molecular chemistry, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 4, 441–484 (English, with English and French summaries). MR 1287241, DOI https://doi.org/10.1016/S0294-1449%2816%2930183-4

Othmar Koch and Christian Lubich, Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics, M2AN Math. Model. Numer. Anal. 41 (2007), no. 2, 315–331. MR 2339631, DOI https://doi.org/10.1051/m2an%3A2007020

Mathieu Lewin, Solutions of the multiconfiguration equations in quantum chemistry, Arch. Ration. Mech. Anal. 171 (2004), no. 1, 83–114. MR 2029532, DOI https://doi.org/10.1007/s00205-003-0281-6

Elliott H. Lieb and Barry Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977), no. 3, 185–194. MR 452286

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), no. 1, 33–97. MR 879032

Per-Olov Löwdin, Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction, Phys. Rev. (2) 97 (1955), 1474–1489. MR 69061

Christian Lubich, On variational approximations in quantum molecular dynamics, Math. Comp. 74 (2005), no. 250, 765–779. MR 2114647, DOI https://doi.org/10.1090/S0025-5718-04-01685-0

N.J. Mauser and S. Trabelsi, $L^2$ analysis of the Multi-configuration Time-Dependent Equations. Preprint.

Heide Narnhofer and Geoffrey L. Sewell, Vlasov hydrodynamics of a quantum mechanical model, Comm. Math. Phys. 79 (1981), no. 1, 9–24. MR 609224

H. Neunzert, Neuere qualitative und numerische Methoden in der Plasmaphysik, Lecture Notes, Paderborn (1975).

Herbert Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys. 52 (1980), no. 3, 569–615. MR 578142, DOI https://doi.org/10.1103/RevModPhys.52.569

H. Spohn, On the Vlasov hierarchy, Math. Methods Appl. Sci. 3 (1981), no. 4, 445–455. MR 657065, DOI https://doi.org/10.1002/mma.1670030131

Saber Trabelsi, Solutions of the multiconfiguration time-dependent Hartree-Fock equations with Coulomb interactions, C. R. Math. Acad. Sci. Paris 345 (2007), no. 3, 145–150 (English, with English and French summaries). MR 2344813, DOI https://doi.org/10.1016/j.crma.2007.06.005

Sandro Zagatti, The Cauchy problem for Hartree-Fock time-dependent equations, Ann. Inst. H. Poincaré Phys. Théor. 56 (1992), no. 4, 357–374 (English, with English and French summaries). MR 1175475

J. Zanghellini, M. Kitzler, T. Brabec, and A. Scrinzi, Testing the multi-configuration time-dependent Hartree-Fock method. J. Phys. B: Atomic, Molecular, and Optical Phys., 37, 763–773 (2004).