Setting and analysis of the multi-configuration time-dependent Hartree-Fock equations. (English) Zbl 1229.35221
From the author’s abstract: We find an exposition of the mathematical analysis of the multi-configuration time-dependent Hartree-Fock approximation which is used in quantum physics for the dynamics of few electron problems. This model is a natural generalization of the time-dependent Hartree-Fock approximation, yielding a hierarchy of a model that should converge to the exact model. In fact, these Hartree-Fock equations for molecular systems with pairwise interaction are a set of coupled nonlinear partial differential equations and ordinary differential equations and are an approximation of the \(n\)-particle time-dependent Schrödinger equation based on time-dependent linear combinations of time-dependent Slater determinants. The authors establish existence and uniqueness of maximal solutions to the Cauchy problem in the energy space as the one-body density matrix is invertible.
Reviewer: M. Marin (Brasov)
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 18F15 | Abstract manifolds and fiber bundles (category-theoretic aspects) |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
Hartree-Fock equation; Schrödinger equation; Coulomb potential; standing wave; Strichartz estimateSoftware:
MCTDHReferences:
| [1] | Ando T.: Properties of fermions density matrices. Rev. Mod. Phys. 35(3), 690–702 (1963) · doi:10.1103/RevModPhys.35.690 |
| [2] | Baltuska A., Udem Th., Uiberacker M., Hentschel M., Gohle Ch., Holzwarth R., Yakovlev V., Scrinzi A., Hänsch T.W., Krausz F.: Attosecond control of electronic processes by intense light fields. Nature 421, 611 (2003) · doi:10.1038/nature01414 |
| [3] | Bardos C., Catto I., Mauser N.J., Trabelsi S.: Global-in-time existence of solutions to the multi-configuration time-dependent Hartree–Fock equations: a sufficient condition. Appl. Math. Lett. 22, 147–152 (2009) · Zbl 1163.35464 · doi:10.1016/j.aml.2007.12.033 |
| [4] | Bardos C., Golse F., Mauser N.J., Gottlieb A.: Mean-field dynamics of fermions and the time-dependent Hartree–Fock equation. J. Math. Pures Appl. 82, 665–683 (2003) · Zbl 1029.82022 |
| [5] | Beck M., Jäckle A.H., Worth G.A., Meyer H.-D.: The multi-configuration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagation wave-packets. Phys. Rep. 324, 1–105 (2000) · doi:10.1016/S0370-1573(99)00047-2 |
| [6] | Bove A., Da Prato G., Fano G.: On the Hartree–Fock time-dependent problem. Commun. Math. Phys. 49, 25–33 (1976) · Zbl 0303.34046 · doi:10.1007/BF01608633 |
| [7] | Caillat J., Zanghellini J., Kitzler M., Koch O., Kreuzer W., Scrinzi A.: Correlated multi-electron systems in strong laser fields–an MCTDHF approach. Phys. Rev. A 71, 012712 (2005) · doi:10.1103/PhysRevA.71.012712 |
| [8] | Cancès E., Le Bris C.: On the time-dependent Hartree–Fock equations coupled with a classical nuclear dynamics. Math. Models Methods Appl. Sci. 9, 963–990 (1999) · Zbl 1011.81087 · doi:10.1142/S0218202599000440 |
| [9] | Castella F.: L 2 solutions to the Schrödinger–Poisson system: existence, uniqueness, time behavior, and smoothing effects. Math. Models Methods Appl. Sci. 7(8), 1051–1083 (1997) · Zbl 0892.35141 · doi:10.1142/S0218202597000530 |
| [10] | Cazenave, T.: An introduction to nonlinear Schrödinger equations. Second edition, Textos de Métodos Mathemáticas 26, Universidade Federal do Rio de Janeiro,1993 |
| [11] | Cazenave, T., Haraux, A.: An introduction to semi-linear evolution equations. Oxford Lecture Series in Mathematics and Its Applications, Vol. 13. Oxford University Press, New York, 1998 · Zbl 0926.35049 |
| [12] | Chadam J.M., Glassey R.T.: Global existence of solutions to the Cauchy problem for the time-dependent Hartree equation. J. Math. Phys. 16, 1122–1230 (1975) · Zbl 0299.35084 · doi:10.1063/1.522642 |
| [13] | Coleman A.J.: Structure of Fermion density matrices. Rev. Mod. Phys. 35(3), 668–689 (1963) · doi:10.1103/RevModPhys.35.668 |
| [14] | Coleman, A.J., Yukalov, V.I.: Reduced Density Matrices: Coulson’s Challenge. Lectures Notes in Chemistry, Vol. 72, Springer, Berlin, 2000 · Zbl 0998.81506 |
| [15] | Dirac P.A.M.: Note on exchange phenomenon in the thomas atom. Proc. Cambridge Phil. Soc 26, 376 (1930) · JFM 56.0751.04 · doi:10.1017/S0305004100016108 |
| [16] | Frenkel J.: Wave Mechanics. Oxford University Press, Oxford (1934) · Zbl 0013.08702 |
| [17] | Friesecke G.: The multi-configuration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Rational Mech. Anal. 169, 35–71 (2003) · Zbl 1035.81069 · doi:10.1007/s00205-003-0252-y |
| [18] | Friesecke G.: 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(2029), 47–52 (2003) · Zbl 1046.81029 · doi:10.1098/rspa.2002.1027 |
| [19] | Gatti, F., Meyer, H.D., Worth, G.A. (eds): Multidimensional Quantum Dynamics: MCTDH Theory and Applications. Wiley-VCH, New York (2009) |
| [20] | Gottlieb A.D., Mauser N.J.: New measure of electron correlation. Phys. Rev. Lett. 95(12), 1230003 (2005) · doi:10.1103/PhysRevLett.95.123003 |
| [21] | Gottlieb, A.D., Mauser, N.J.: Properties of non-freeness: an entropy measure of electron correlation. Int. J. Quantum Inf. 5(6), 10–33 (2007). E-print arXiv:quant-ph/0608171v3 |
| [22] | Grobe R., Rzazewski K., Eberly J.H.: Measure of electron–electron correlation in atomic physics. J. Phys. B 27, L503–L508 (1994) · doi:10.1088/0953-4075/27/16/001 |
| [23] | Kato T., Kono H.: Time-dependent multi-configuration theory for electronic dynamics of molecules in an intense laser field. Chem. Phys. Lett. 392, 533–540 (2004) · doi:10.1016/j.cplett.2004.05.106 |
| [24] | Keel M., Tao T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998) · Zbl 0922.35028 · doi:10.1353/ajm.1998.0039 |
| [25] | Koch O., Kreuzer W., Scrinzi A.: Approximation of the time-dependent electronic Schrödinger equation by MCTDHF. Appl. Math. Comput. 173, 960–976 (2006) · Zbl 1088.65092 · doi:10.1016/j.amc.2005.04.027 |
| [26] | Koch O., Lubich C.: Regularity of the multi-configuration time-dependent hartree approximation in quantum molecular dynamics. M2AN Math. Model. Numer. Anal. 41, 315–331 (2007) · Zbl 1135.81380 · doi:10.1051/m2an:2007020 |
| [27] | Le Bris C.: A general approach for multi-configuration methods in quantum molecular chemistry. Ann. Inst. H. Poincaré Anal. Non Linéaire 11(4), 441–484 (1994) · Zbl 0837.35005 |
| [28] | Lewin M.: Solutions of the Multi-configuration Equations in Quantum Chemistry. Arch. Rational Mech. Anal. 171(1), 83–114 (2004) · Zbl 1063.81102 · doi:10.1007/s00205-003-0281-6 |
| [29] | Löwdin P.O.: Quantum theory of many-particles systems, I: physical interpretations by mean of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction. Phys. Rev. 97, 1474–1489 (1955) · Zbl 0065.44907 · doi:10.1103/PhysRev.97.1474 |
| [30] | Lubich C.: On variational approximations in quantum molecular dynamics. Math. Comp. 74, 765–779 (2005) · Zbl 1059.81188 · doi:10.1090/S0025-5718-04-01685-0 |
| [31] | Lubich C.: A variational splitting integrator for quantum molecular dynamics. Appl. Numer. Math. 48, 355–368 (2004) · Zbl 1037.81634 · doi:10.1016/j.apnum.2003.09.001 |
| [32] | Lubich, C.: From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. Edition EMS, 2008 · Zbl 1160.81001 |
| [33] | McWeeny R.: Methods of Molecular Quantum Mechanics, 2nd edn. Academic Press, London (1992) |
| [34] | Mauser, N.J., Trabelsi, S.: L 2 Analysis of the Multi-configuration Time-Dependent Equations. Math. Models Methods Appl. Sci. (2010, to appear) · Zbl 1206.35229 |
| [35] | Pazy A.: Semi-groups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983) · Zbl 0516.47023 |
| [36] | Segal I.: Non-linear semi-groups. Ann. Math. 78, 339–364 (1963) · Zbl 0204.16004 · doi:10.2307/1970347 |
| [37] | Trabelsi S.: Solutions of the multi-configuration time-dependent equations in quantum chemistry. C. R. Math. Acad. Sci. Paris 345(3), 145–150 (2007) · Zbl 1120.35077 · doi:10.1016/j.crma.2007.06.005 |
| [38] | Tsutsumi Y.: L 2olutions for nonlinear Schrödinger equation and nonlinear groups. Funk. Ekva. 30, 115–125 (1987) · Zbl 0638.35021 |
| [39] | Zagatti S.: The Cauchy problem for Hartree–Fock time dependent equations. Ann. Inst. H. Poincaré, Phys. Th. 56(4), 357–374 (1992) · Zbl 0763.35089 |
| [40] | Zanghellini J., Kitzler M., Fabian C., Brabec T., Scrinzi A.: An MCTDHF approach to multi-electron dynamics in laser fields. Laser Phys. 13(8), 1064–1068 (2003) |
| [41] | Zanghellini J., Kitzler M., Brabec T., Scrinzi A.: Testing the multi-configuration time-dependent Hartree–Fock method. J. Phys. B At. Mol. Phys. 37, 763–773 (2004) · doi:10.1088/0953-4075/37/4/004 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
