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2017 | Supplement | Buchkapitel

8. Introduction to Numerical Methods

verfasst von : Fabien Gatti, Benjamin Lasorne, Hans-Dieter Meyer, André Nauts

Erschienen in: Applications of Quantum Dynamics in Chemistry

Verlag: Springer International Publishing

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Abstract

The present chapter is dedicated to the numerical methods for solving the time-dependent Schrödinger equation for the nuclei.

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Vorheriges Kapitel Group Theory and Molecular Symmetry

Nächstes Kapitel Infrared Spectroscopy

A large part of this chapter is taken form the MCTDH lecture notes of H.-D. Meyer written by Daniel Pelaez-Ruiz. The authors sincerely thank Dr. Pelaez for letting them use his script.

The quadrature rule defined through a proper DVR, i.e. defined by Eqs. (8.30) and (8.31), is of Gaussian type, i.e. it yields an exact result for polynomials of degree 2\(n - 1\) or less.

For the primitive basis set of each coordinate, we use a DVR (see Sect. 8.1.3), more precisely HO DVR for \(R_1\) and a sine DVR for \(R_2\) with \(R_1\) \(\in \) [3.8, 5.6] a.u. and an unrestricted Legendre DVR for \(\theta \). Here, \(N_1\) = 36 for \(r_d\), \(N_2\) = 24 for \(r_v\), and \(N_3\) = 60 for \(\theta \).

For the primitive basis set of each coordinate, we use a DVR (see Sect. 8.1.3), more precisely a sine DVR for \(R_1\) and \(R_2\) with \(R_1\) \(\in \) [0.6, 6.24] and 48 functions and \(R_2\) \(\in \) [1.0, 9.04] a.u. and 68 functions, and a Legendre DVR for \(\theta \) with 31 functions.

A two-layers ML-MCTDH is identical to standard MCTDH

Actually, we are considering the wavefunction of a particular total angular momentum J (that makes the dissociate coordinate R one-dimensional). Hence \(\Psi \) should be replaced with \(\Psi ^J\), but for the sake of simplicity we suppress the total angular momentum label J.

The initial state is assumed to have no density beyond \(R_c\), i.e. \(\theta (R-R_c) |\Psi (R,{\varvec{q}},t=0)|^2 \equiv 0\).

Actually \(P_{\gamma \nu }\) commutes with \(F_\gamma \). Thus \(P_{\gamma \nu }\, F_\gamma = F_\gamma \, P_{\gamma \nu } = P_{\gamma \nu }\, F_\gamma \, P_{\gamma \nu }\). The extra projector is added for symmetry reasons only.

We have also suppressed some J-dependent phase factors in Eqs. (8.297, 8.299). They are irrelevant, because here we consider only the modulus of the S-matrix elements.

Here the strength parameter \(\eta \) is included in the definition of W, in contrast to Sect. 8.5.

Of course, mode combination can be used. Then \(h_r^{(\kappa )}\) operates on the \(\kappa \)-th particle in Eq. (8.333) and the integrals become low-dimensional rather than one-dimensional ones.

Note that the root-mean-square error is given by \({\textit{rmse}}=\sqrt{ \Delta ^2/N_{\text {tot}}}\), where \(N_{\text {tot}}\) is the total number of grid points.

Actually, we loop over the modes and update \(V^{\tiny {\text {ref}}}\) after each new SPP(k).

Cohen-Tannoudji C, Diu B, Laloe F (1992) Quantum mechanics. Wiley-VCH

Tannor DJ (2007) Introduction to quantum dynamics: a time-dependent perspective. University Science Books, Sausalito, CA

Dirac PAM (1930) Note on exchange phenomena in the Thomas atom. Proc Camb Philos Soc 26:376–385CrossRef

Frenkel J (1934) Wave mechanics. Clarendon Press, Oxford

Beck MH, Jäckle A, Worth GA, Meyer H-D (2000) The multi-configuration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wave packets. Phys Rep 324:1–105CrossRef

Kosloff D, Kosloff R (1983) A Fourier-method solution for the time-dependent Schrödinger equation as a tool in molecular dynamics. J Comput Phys 52:35CrossRef

Kosloff R, Tal-Ezer H (1986) A direct relaxation method for calculating eigenfunctions and eigenvalues of the Schrödinger equation on a grid. Chem Phys Lett 127:223CrossRef

Light JC, Hamilton IP, Lill JV (1985) Generalized discrete variable approximation in quantum mechanics. J Chem Phys 82:1400CrossRef

Lill JV, Parker GA, Light JC (1986) The discrete variable-finite basis approach to quantum scattering. J Chem Phys 85:900

10.

Corey GC, Lemoine D (1992) Pseudospectral method for solving the time-dependent Schrödinger equation in spherical coordinates. J Chem Phys 97:4115CrossRef

11.

Bramley MJ, Tromp JW, Carrington T Jr, Corey RC (1994) Efficient calculation of highly excited vibrational energy levels of floppy molecules: the band origins of H\(^+_3\) up to 35 000 cm\(^{-1}\). J Chem Phys 100:6175CrossRef

12.

Lemoine D (1994) The finite basis representation as the primary space in multidimensional pseudospectral schemes. J Chem Phys 101:10526CrossRef

13.

Kosloff R (1996) Quantum molecular dynamics on grids. In: Wyatt RE, Zhang JZH (eds) Dynamics of molecules and chemical reactions. Marcel Dekker, New York, pp 185–230

14.

Light JC, Carrington T Jr (2000) Discrete variable representations and their utilization. Adv Chem Phys 114:263

15.

Zare RN (1988) Angular momentum. Wiley, New York

16.

Worth GA, Beck MH, Jäckle A, Meyer HD (2007) The MCTDH Package, Version 8.2, (2000). Meyer HD (2002), Version 8.3, Version 8.4. Current version: 8.4.12 (2016). http://mctdh.uni-hd.de/

17.

Dawes R, Carrington T Jr (2004) A multidimensional discrete variable representation basis obtained by simultaneous diagonalization. J Chem Phys 121:726

18.

Li J, Carter S, Bowman JM, Dawes R, Xie D, Guo H (2014) High-level, first-principles, full-dimensional quantum calculation of the ro-vibrational spectrum of the simplest criegee intermediate (CH\(_2\)OO). J Phys Chem Lett 5:20364

19.

Harris DO, Engerholm GG, Gwinn GW (1965) Calculation of matrix elements for one-dimensional quantum-mechanical problems and the application to anharmonic oscillators. J Chem Phys 43:1515CrossRef

20.

Dickinson AS, Certain PR (1968) Calculation of matrix elements for one-dimensional quantum-mechanical problems. J Chem Phys 49:4209CrossRef

21.

Corey GC, Tromp JW, Lemoine D (1993) Fast pseudospectral algorithm in curvilinear coordinates. In: Cerjan C (ed) Numerical grid methods and their application to Schrödinger’s equation. Kluwer Academic Publishers, The Netherlands, pp 1

22.

Sukiasyan S, Meyer H-D (2001) On the effect of initial rotation on reactivity. A multi-configuration time-dependent Hartree (MCTDH) wave-packet propagation study on the H+D\(_2\) and D+H\(_2\) reactive scattering systems. J Phys Chem A 105:2604

23.

Echave J, Clary DC (1992) Potential optimized discrete variable representation Chem Phys Lett 190:225

24.

Bramley MJ, Handy NC (1993) Efficient calculation of rovibrational eigenstates of sequentially bonded four-atom molecules. J Chem Phys 98:1378

25.

Tremblay JC, Carrington T Jr (2006) Calculating vibrational energies and wave functions of vinylidene using a contracted basis with a locally reorthogonalized coupled two-term lanczos eigensolver. J Chem Phys 125:094311CrossRef

26.

Bowman JM, Carrington T Jr, Meyer H-D (2008) Variational quantum approaches for computing vibrational energies of polyatomic molecules. Mol Phys 106:2145

27.

Wang X, Carrington T Jr (2008) Using a nondirect product discrete variable representation for angular coordinates to compute vibrational levels of polyatomic molecules. J Chem Phys 128(19):194109CrossRef

28.

Wang X, Carrington T Jr (2008) Vibrational energy levels of CH\(_5^+\). J Chem Phys 129:234102CrossRef

29.

Wang X-G, Carrington T Jr, Dawes R, Jasper AW (2011) The vibration-rotation-tunneling spectrum of the polar and T-shaped-N-in isomers of (NNO)\(_2\). J Mol Spec 268:53

30.

Wang X-G, Carrington T Jr (2013) Computing rovibrational levels of methane with curvilinear internal vibrational coordinates and an Eckart frame. J Chem Phys 138:104106CrossRef

31.

Wang X-G, Carrington T Jr (2014) Rovibrational levels and wavefunctions of Cl\(^-\)H\(_2\)O. J Chem Phys 140:204306CrossRef

32.

Bowman J (1978) Self-consistent field energies and wavefunctions for coupled oscillators. J Chem Phys 68:608CrossRef

33.

Bowman J, Christoffel K, Tobin F (1979) Application of SCF-SI theory to vibrational motion in polyatomic molecules. J Phys Chem 83:905CrossRef

34.

Bowman JM, Carter S, Huang X (2003) MULTIMODE: a code to calculate rovibrational energies of polyatomic molecules. Int Rev Phys Chem 22:533

35.

Culot F, Laruelle F, Liévin J (1995) A vibrational CASSCF study of stretch-bend interactions and their influence on infrared intensities in the water molecule. Theory Chem Acc 92:211

36.

Bégué D, Gohaud N, Pouchan C, Cassam-Chenaï P, Liévin J (2007) A comparison of two methods for selecting vibrational configuration interaction spaces on a heptatomic system: ethylene oxide. J Chem Phys 127:164115CrossRef

37.

Heislbetz S, Rauhut G (2010) Vibrational multiconfiguration self-consistent field theory: implementation and test calculations. J Chem Phys 132:124102CrossRef

38.

Leforestier C, Bisseling RH, Cerjan C, Feit MD, Friesner R, Guldenberg A, Hammerich A, Jolicard G, Karrlein W, Meyer H-D, Lipkin N, Roncero O, Kosloff R (1991) A comparison of different propagation schemes for the time dependent Schrödinger equation. J Comput Phys 94:59CrossRef

39.

Feit MD, Fleck JA Jr, Steiger A (1982) Solution of the Schrödinger equation by a spectral method. J Comput Phys 47:412

40.

Feit MD, Fleck JA Jr (1983) Solution of the Schrödinger equation by a spectral method II: vibrational energy levels of triatomic molecules. J Chem Phys 78:301CrossRef

41.

Park TJ, Light JC (1986) Unitary quantum time evolution by iterative Lanczos reduction. J Chem Phys 85:5870CrossRef

42.

Arnoldi WE (1951) The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q Appl Math 9:17CrossRef

43.

Saad Y (1980) Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices. Linear Algebra Appl 34:269CrossRef

44.

Friesner RA, Tuckerman LS, Dornblaser BC, Russo TV (1989) J Sci Comput 4:327CrossRef

45.

Manthe U, Köppel H, Cederbaum LS (1991) Dissociation and predissociation on coupled electronic potential energy surfaces: a three-dimensional wave packet dynamical study. J Chem Phys 95:1708CrossRef

46.

Gear CW (1971) Numerical initial value problems in ordinary differential equations. Prentice-Hall, Englewood Cliffs

47.

Beck MH, Meyer H-D (1997) An efficient and robust integration scheme for the equations of motion of the multiconfiguration time-dependent Hartree (MCTDH) method. Z Phys D 42:113–129CrossRef

48.

Gerber RB, Ratner MA, Buch V (1982) Simplified time-dependent self-consistent field approximation for intramolecular dynamics. Chem Phys Lett 91:173CrossRef

49.

Bisseling RH, Kosloff R, Gerber RB, Ratner MA, Gibson L, Cerjan C (1987) Exact time-dependent quantum mechanical dissociation dynamics of I\(_2\)He: comparison of exact time-dependent quantum calculation with the quantum time-dependent self-consistent field (TDSCF) approximation. J Chem Phys 87:2760CrossRef

50.

Meyer H-D, Gatti F, Worth GA (eds) (2009) Multidimensional quantum dynamics: MCTDH theory and applications. Wiley-VCH, Weinheim

51.

Kotler Z, Nitzan A, Kosloff R (1988) Multiconfiguration time-dependent self-consistent field approximation for curve crossing in presence of a bath. A fast Fourier transform study. Chem Phys Lett 153:483CrossRef

52.

Makri N, Miller WH (1987) Time-dependent self-consistent (TDSCF) approximation for a reaction coordinate coupled to a harmonic bath: single and multiconfiguration treatments. J Chem Phys 87:5781CrossRef

53.

Meyer H-D, Manthe U, Cederbaum LS (1990) The multi-configurational time-dependent Hartree approach. Chem Phys Lett 165:73

54.

Manthe U, Meyer H-D, Cederbaum LS (1992) Wave-packet dynamics within the multiconfiguration Hartree framework: general aspects and application to NOCl. J Chem Phys 97:3199

55.

Szabo A, Ostlund NS (1996) Modern quantum chemistry. Dover, Mineola, NY

56.

Jensen F (1999) Introduction to computational chemistry. Wiley, Chichester

57.

Atkins PW (1983) Molecular quantum mechanics, 2nd edn. OUP, Oxford, UK

58.

Zanghellini J, Kitzler M, Fabian C, Brabec T, Scrinzi A (2003) An MCTDHF approach to multi-electron dynamics in laser fields. Laser Phys 13:1064

59.

Alon OE, Streltsov AI, Cederbaum LS (2008) Multiconfigurational time-dependent Hartree method for bosons: many-body dynamics of bosonic systems. Phys Rev A 77:033613CrossRef

60.

Hammerich AD, Kosloff R, Ratner MA (1990) Quantum mechanical reactive scattering by a multiconfigurational time-dependent self-consistent field (MCTDSCF) approach. Chem Phys Lett 171:97CrossRef

61.

Jäckle A, Meyer H-D (1998) Calculation of H+H\(_2\) and H+D\(_2\) reaction probabilities within the multiconfiguration time-dependent Hartree approach employing an adiabatic correction scheme. J Chem Phys 109:2614CrossRef

62.

Launay JM, Dourneuf ML (1989) Hyperspherical close-coupling calculation of integral cross sections for the reaction H+H\(_2\rightarrow \) H\(_2\)+H. Chem Phys Lett 163:178CrossRef

63.

Pack RT, Parker G (1987) Quantum reactive scattering in three dimensions using hyperspherical (aph) coordinates. theory. J Chem Phys 87:3888CrossRef

64.

Pack RT, Parker G (1989) Quantum reactive scattering in three dimensions using hyperspherical (aph) coordinates. III. Small \(\theta \) behavior and corrigenda. J Chem Phys 90:3511CrossRef

65.

Wang H, Thoss M (2003) Multilayer formulation of the multiconfiguration time-dependent Hartree theory. J Chem Phys 119:1289

66.

Wang H, Thoss M (2006) Quantum-mechanical evaluation of the Boltzmann operator in correlation functions for large molecular systems: a multilayer multiconfiguration time-dependent Hartree approach. J Chem Phys 124:034114CrossRef

67.

Wang H, Skinner DE, Thoss M (2006) Calculation of reactive flux correlation functions for systems in a condensed phase environment: a multilayer multi-configuration time-dependent hartree approach. J Chem Phys 125:174502CrossRef

68.

Wang H, Thoss M (2007) Quantum dynamical simulation of electron-transfer reactions in an anharmonic environment. J Phys Chem A 111:10369CrossRef

69.

Manthe U (2008) A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces. J Chem Phys 128:164116CrossRef

70.

Wang H, Thoss M (2009) Numerically exact quantum dynamics for indistinguishable particles: The multilayer multiconfiguration time-dependent Hartree theory in second quantization representation. J Chem Phys 131(2):024114CrossRef

71.

Vendrell O, Meyer H-D (2011) Multilayer multiconfiguration time-dependent Hartree method: Implementation and applications to a Henon-Heiles Hamiltonian and to pyrazine. J Chem Phys 134:044135CrossRef

72.

Wang H (2015) Multilayer multiconfiguration time-dependent Hartree theory. J Phys Chem A 119:7951CrossRef

73.

Westermann T, Brodbeck R, Rozhenko AB, Schoeller W, Manthe U (2011) Photodissociation of methyl iodide embedded in a host-guest complex: a full dimensional (189D) quantum dynamics study of CH\(_3\)I@resorc[4]arene. J Chem Phys 135:184102CrossRef

74.

Wang H, Shao J (2012) Dynamics of a two-level system coupled to a bath of spins. J Chem Phys 137:22A504CrossRef

75.

Manthe U (2006) On the integration of the multi-configurational time-dependent Hartree (MCTDH) equations of motion. Chem Phys 329:168

76.

Meyer H-D, Worth GA (2003) Quantum molecular dynamics: propagating wavepackets and density operators using the multiconfiguration time-dependent Hartree (MCTDH) method. Theory Chem Acc 109:251

77.

Meyer H-D, Le Quéré F, Léonard C, Gatti F (2006) Calculation and selective population of vibrational levels with the multiconfiguration time-dependent Hartree (MCTDH) algorithm. Chem Phys 329:179

78.

Joubert Doriol L, Gatti F, Iung C, Meyer HD (2008) Computation of vibrational energy levels and eigenstates of fluoroform using the multiconfiguration time-dependent Hartree method. J Chem Phys 129:224109CrossRef

79.

Vendrell O, Gatti F, Lauvergnat D, Meyer H-D (2007) Full dimensional (15D) quantum-dynamical simulation of the protonated water dimer I: Hamiltonian setup and analysis of the ground vibrational state. J Chem Phys 127:184302CrossRef

80.

Vendrell O, Brill M, Gatti F, Lauvergnat D, Meyer H-D (2009) Full dimensional (15D) quantum-dynamical simulation of the protonated water dimer III: mixed Jacobi-valence parametrization and benchmark results for the zero-point energy, vibrationally excited states and infrared spectrum. J Chem Phys 130:234305CrossRef

81.

Davidson E (1975) The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J Comput Phys 17:87CrossRef

82.

Olsen J, Jørgenson P, Simons J (1990) Passing the one-billion limit in full configuration-interaction (FCI) calculations. Chem Phys Lett 169:493CrossRef

83.

Riss UV, Meyer H-D (1993) Calculation of resonance energies and widths using the complex absorbing potential method. J Phys B 26:4503CrossRef

84.

Jolicard G, Austin E (1985) Optical potential stabilisation method for predicting resonance level. Chem Phys Lett 121:106CrossRef

85.

Jolicard G, Austin E (1986) Optical potential method of caculating resonance energies and widths. Chem Phys 103:295CrossRef

86.

Jolicard G, Leforestier C, Austin E (1988) Resonance states using the optical potential model. Study of Feshbach resonances and broad shape resonances. J Chem Phys 88:1026CrossRef

87.

Kosloff R, Kosloff D (1986) Absorbing boundaries for wave propagation problems. J Comput Phys 63:363CrossRef

88.

Neuhauser D, Baer M (1989) The time-dependent Schrödinger equation: application of absorbing boundary conditions. J Chem Phys 90:4351CrossRef

89.

Riss UV, Meyer H-D (1995) Reflection-free complex absorbing potentials. J Chem Phys 28:1475

90.

Riss UV, Meyer H-D (1996) Investigation on the reflection and transmission properties of complex absorbing potentials. J Chem Phys 105:1409CrossRef

91.

Jäckle A, Meyer H-D (1996) Time-dependent calculation of reactive flux employing complex absorbing potentials: general aspects and application within MCTDH. J Chem Phys 105:6778CrossRef

92.

Scheit S, Meyer H-D, Moiseyev N, Cederbaum LS (2006) On the unphysical impact of complex absorbing potentials on the Hamiltonian and its remedy. J Chem Phys 124:034102CrossRef

93.

Taylor JR (1972) Scattering theory: the quantum theory of nonrelativistic collisions. Wiley, New York

94.

Gatti F, Otto F, Sukiasyan S, Meyer H-D (2005) Rotational excitation cross sections of para-H\(_2\) + para-H\(_2\) collisions. A full-dimensional wave packet propagation study using an exact form of the kinetic energy. J Chem Phys 123:174311CrossRef

95.

Panda AN, Otto F, Gatti F, Meyer H-D (2007) Rovibrational energy transfer in ortho-H\(_2\) + para-H\(_2\) collisions. J Chem Phys 127:114310CrossRef

96.

Otto F, Gatti F, Meyer H-D (2008) Rotational excitations in para-H\(_2\) + para-H\(_2\) collisions: full- and reduced-dimensional quantum wave packet studies comparing different potential energy surfaces. J Chem Phys 128:064305CrossRef

97.

Otto F, Gatti F, Meyer H-D (2009) Erratum: "Rotational excitations in para-H\(_2\) + para-H\(_2\) collisions: full- and reduced-dimensional quantum wave packet studies comparing different potential energy surfaces". J Chem Phys 131:049901CrossRef

98.

Otto F, Gatti F, Meyer H-D (2012) Rovibrational energy transfer in collisions of H\(_2\) with D\(_2\). A full-dimensional wave packet propagation study. Mol Phys 110:619CrossRef

99.

Tannor DJ, Weeks DE (1993) Wave packet correlation function formulation of scattering theory: the quantum analog of classical \(S\)-matrix theory. J Chem Phys 98:3884CrossRef

100.

Weeks DE, Tannor DJ (1993) A time-dependent formulation of the scattering matrix using Møller operators. Chem Phys Lett 207:301CrossRef

101.

Weeks DE, Tannor DJ (1994) A time-dependent formulation of the scattering matrix for the collinear reaction H+H\(_2\,(\nu )\rightarrow \) H\(_2\,(\nu ^{\prime })\)+H. Chem Phys Lett 224:451CrossRef

102.

Schmidt E (1906) Zur Theorie der linearen und nichtlinearen Integralgleichungen. Math Ann 63:433CrossRef

103.

Peláez D, Meyer H-D (2013) The multigrid POTFIT (MGPF) method: grid representations of potentials for quantum dynamics of large systems. J Chem Phys 138:014108CrossRef

Titel: Introduction to Numerical Methods
verfasst von: Fabien Gatti
Benjamin Lasorne
Hans-Dieter Meyer
André Nauts
Verlag: Springer International Publishing
Buch: Applications of Quantum Dynamics in Chemistry
Print ISBN: 978-3-319-53921-8

Electronic ISBN: 978-3-319-53923-2

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-53923-2_8

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