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Applications of Quantum Dynamics in Chemistry pp 201–285Cite as

Introduction to Numerical Methods

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Part of the book series: Lecture Notes in Chemistry ((LNC,volume 98))

Abstract

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

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Notes

  1. 1.

    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.

  2. 2.

    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.

  3. 3.

    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 \(\theta \).

  4. 4.

    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.

  5. 5.

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

  6. 6.

    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.

  7. 7.

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

  8. 8.

    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.

  9. 9.

    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.

  10. 10.

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

  11. 11.

    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.

  12. 12.

    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.

  13. 13.

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

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Authors and Affiliations

  1. Institut des Sciences Moléculaires d’Orsay(ISMO), University of Paris-Sud, Orsay, Essonne, France

    Fabien Gatti

  2. Institute Charles Gerhardt, CNRS5253, Université de Montpellier, Montpellier, France

    Benjamin Lasorne

  3. Institute of Physical Chemistry, Theoretical Chemistry Group, University of Heidelberg, Heidelberg, Germany

    Hans-Dieter Meyer

  4. Faculté des Sciences, Ecole de Physique, and Institute of Condensed Matter and Nanosciences, Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    André Nauts

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  1. Fabien Gatti
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Gatti, F., Lasorne, B., Meyer, HD., Nauts, A. (2017). Introduction to Numerical Methods. In: Applications of Quantum Dynamics in Chemistry. Lecture Notes in Chemistry, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-53923-2_8

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