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Computing quantum dynamics in the semiclassical regime

Published online by Cambridge University Press:  30 November 2020

Caroline Lasser  and
Christian Lubich
Caroline Lasser
Affiliation:
Fakultät für Mathematik, Technische Universität München, 85748Garching bei München, Germany E-mail: [email protected]
Christian Lubich
Affiliation:
Mathematisches Institut, Universität Tübingen, 72076Tübingen, Germany E-mail: [email protected]
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Abstract

The semiclassically scaled time-dependent multi-particle Schrödinger equation describes, inter alia, quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimensions. This paper reviews and studies numerical approaches that are robust to the small semiclassical parameter. We present and analyse variationally evolving Gaussian wave packets, Hagedorn’s semiclassical wave packets, continuous superpositions of both thawed and frozen Gaussians, and Wigner function approaches to the direct computation of expectation values of observables. Making good use of classical mechanics is essential for all these approaches. The arising aspects of time integration and high-dimensional quadrature are also discussed.

Type
Research Article
Information
Acta Numerica , Volume 29 , May 2020 , pp. 229 - 401
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES 2

Abramowitz, M. and Stegun, I. A. (1964), Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables, Vol. 55 of National Bureau of Standards Applied Mathematics Series, US Government Printing Office.Google Scholar
Aistleitner, C. and Dick, J. (2015), ‘Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality’, Acta Arith. 167, 143–171.CrossRefGoogle Scholar
Arnal, A., Casas, F. and Chiralt, C. (2017), ‘On the structure and convergence of the symmetric Zassenhaus formula’, Comput. Phys. Commun. 217, 58–65.CrossRefGoogle Scholar
Auzinger, W., Kassebacher, T., Koch, O. and Thalhammer, M. (2016), ‘Adaptive splitting methods for nonlinear Schrödinger equations in the semiclassical regime’, Numer. Algorithms 72, 1–35.CrossRefGoogle Scholar
Bader, P., Iserles, A., Kropielnicka, K. and Singh, P. (2014), ‘Effective approximation for the semiclassical Schrödinger equation’, Found. Comput. Math. 14, 689–720.CrossRefGoogle Scholar
Bader, P., Iserles, A., Kropielnicka, K. and Singh, P. (2016), ‘Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential’, Proc. Roy. Soc. London A 472, 20150733.Google Scholar
Bambusi, D., Graffi, S. and Paul, T. (1999), ‘Long time semiclassical approximation of quantum flows: A proof of the Ehrenfest time’, Asymptot. Anal. 21, 149–160.Google Scholar
Bao, W., Jin, S. and Markowich, P. A. (2002), ‘On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime’, J. Comput. Phys. 175, 487–524.CrossRefGoogle Scholar
Bao, W., Jin, S. and Markowich, P. A. (2003), ‘Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes’, SIAM J. Sci. Comput. 25, 27–64.CrossRefGoogle Scholar
Begušić, T., Cordova, M. and Vaníček, J. (2019), ‘Single-Hessian thawed Gaussian approximation’, J. Chem. Phys. 150, 154117.CrossRefGoogle ScholarPubMed
Belyaev, A., Domcke, W., Lasser, C. and Trigila, G. (2015), ‘Nonadiabatic nuclear dynamics of the ammonia cation studied by surface hopping classical trajectory calculations’, J. Chem. Phys. 142, 104307.CrossRefGoogle ScholarPubMed
Berra, M., Bulai, I. M., Cordero, E. and Nicola, F. (2017), ‘Gabor frames of Gaussian beams for the Schrödinger equation’, Appl. Comput. Harmon. Anal. 43, 94–121.CrossRefGoogle Scholar
Berry, M. V. (1984), ‘Quantal phase factors accompanying adiabatic changes’, Proc. Roy. Soc. London A 392, 45–57.Google Scholar
Besse, C., Carles, R. and Méhats, F. (2013), ‘An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit’, Multiscale Model. Simul. 11, 1228–1260.CrossRefGoogle Scholar
Blanes, S. and Casas, F. (2016), A Concise Introduction to Geometric Numerical Integration, Monographs and Research Notes in Mathematics, CRC Press.CrossRefGoogle Scholar
Blanes, S. and Gradinaru, V. (2019), ‘High order efficient splittings for the semiclassical time-dependent Schrödinger equation’, J. Comput. Phys. 405, 109157.Google Scholar
Blanes, S., Casas, F. and Murua, A. (2008), ‘Splitting and composition methods in the numerical integration of differential equations’, Bol. Soc. Esp. Mat. Apl. SeMA 45, 89–145.Google Scholar
Born, M. and Fock, V. (1928), ‘Beweis des Adiabatensatzes’, Zeitschrift für Physik 51, 165–180.CrossRefGoogle Scholar
Bou-Rabee, N. and Sanz-Serna, J. M. (2018), Geometric integrators and the Hamiltonian Monte Carlo method. In Acta Numerica, Vol. 27, Cambridge University Press, pp. 113–206.Google Scholar
Bourquin, R., Gradinaru, V. and Hagedorn, G. A. (2012), ‘Non-adiabatic transitions near avoided crossings: Theory and numerics’, J. Math. Chem. 50, 602–619.CrossRefGoogle Scholar
Bouzouina, A. and Robert, D. (2002), ‘Uniform semiclassical estimates for the propagation of quantum observables’, Duke Math. J. 111, 223–252.Google Scholar
Brown, R. and Heller, E. (1981), ‘Classical trajectory approach to photodissociation: The Wigner method’, J. Chem. Phys. 75, 186–188.CrossRefGoogle Scholar
Brumm, B. (2015), ‘A fast matrix-free algorithm for spectral approximations to the Schrödinger equation’, SIAM J. Sci. Comput. 37, A2003–A2025.CrossRefGoogle Scholar
Bungartz, H.-J. and Griebel, M. (2004), Sparse grids. In Acta Numerica, Vol. 13, Cambridge University Press, pp. 147–269.Google Scholar
Caflisch, R. E. (1998), Monte Carlo and quasi-Monte Carlo methods. In Acta Numerica, Vol. 7, Cambridge University Press, pp. 1–49.Google Scholar
Cancès, E., Defranceschi, M., Kutzelnigg, W., Le Bris, C. and Maday, Y. (2003), Computational quantum chemistry: A primer. In Handbook of Numerical Analysis, Vol. X, North-Holland, pp. 3–270.Google Scholar
Cancès, E., Le Bris, C. and Maday, Y. (2006), Méthodes Mathématiques en Chimie Quantique: Une Introduction, Vol. 53 of Mathématiques & Applications (Berlin), Springer.CrossRefGoogle Scholar
Carles, R. (2008), Semi-Classical Analysis for Nonlinear Schrödinger Equations, World Scientific.CrossRefGoogle Scholar
Carles, R. (2013), ‘On Fourier time-splitting methods for nonlinear Schrödinger equations in the semiclassical limit’, SIAM J. Numer. Anal. 51, 3232–3258.CrossRefGoogle Scholar
Carles, R. and Fermanian-Kammerer, C. (2011), ‘Nonlinear coherent states and Ehrenfest time for Schrödinger equation’, Comm. Math. Phys. 301, 443–472.CrossRefGoogle Scholar
Carles, R. and Gallo, C. (2017), ‘On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit, II: Analytic regularity’, Numer. Math. 136, 315–342.CrossRefGoogle Scholar
Chartier, P., Le Treust, L. and Méhats, F. (2019), ‘Uniformly accurate time-splitting methods for the semiclassical linear Schrödinger equation’, ESAIM Math. Model. Numer. Anal. 53, 443–473.CrossRefGoogle Scholar
Coalson, R. D. and Karplus, M. (1990), ‘Multidimensional variational Gaussian wave packet dynamics with application to photodissociation spectroscopy’, J. Chem. Phys. 93, 3919–3930.CrossRefGoogle Scholar
Combescure, M. (1992), ‘The squeezed state approach of the semiclassical limit of the time-dependent Schrödinger equation’, J. Math. Phys. 33, 3870–3880.CrossRefGoogle Scholar
Combescure, M. and Robert, D. (1997), ‘Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow,’ Asymptot. Anal. 14, 377–404.CrossRefGoogle Scholar
Combescure, M. and Robert, D. (2012), Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics, Springer.CrossRefGoogle Scholar
Cordero, E., de Gosson, M. and Nicola, F. (2017), ‘Semi-classical time-frequency analysis and applications’, Math. Phys. Anal. Geom. 20, 26.CrossRefGoogle Scholar
de Gosson, M. A. (2011), Symplectic Methods in Harmonic Analysis and in Mathematical Physics, Vol. 7 of Pseudo-Differential Operators: Theory and Applications, Birkhäuser/Springer.CrossRefGoogle Scholar
Descombes, S. and Thalhammer, M. (2010), ‘An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime’, BIT 50, 729–749.CrossRefGoogle Scholar
Dick, J., Kuo, F. Y. and Sloan, I. H. (2013), High-dimensional integration: The quasi-Monte Carlo way. In Acta Numerica, Vol. 22, Cambridge University Press, pp. 133–288.Google Scholar
Dietert, H., Keller, J. and Troppmann, S. (2017), ‘An invariant class of wave packets for the Wigner transform’, J. Math. Anal. Appl. 450, 1317–1332.CrossRefGoogle Scholar
Dirac, P. A. M. (1930), ‘Note on exchange phenomena in the Thomas atom’, Proc. Cambridge Phil. Soc. 26, 376–385.CrossRefGoogle Scholar
Domcke, W., Yarkony, D. and Köppel, H., eds (2011), Conical Intersections: Theory, Computation and Experiment, Vol. 17 of Advanced Series in Physical Chemistry, World Scientific.CrossRefGoogle Scholar
Egorov, J. V. (1969), ‘The canonical transformations of pseudodifferential operators’, Uspehi Mat. Nauk 24, 235–236.Google Scholar
Fang, D., Jin, S. and Sparber, C. (2018), ‘An efficient time-splitting method for the Ehrenfest dynamics’, Multiscale Model. Simul. 16, 900–921.CrossRefGoogle Scholar
Faou, E. and Lubich, C. (2006), ‘A Poisson integrator for Gaussian wavepacket dynamics’, Comput. Vis. Sci. 9, 45–55.CrossRefGoogle Scholar
Faou, E., Gradinaru, V. and Lubich, C. (2009), ‘Computing semiclassical quantum dynamics with Hagedorn wavepackets’, SIAM J. Sci. Comput. 31, 3027–3041.CrossRefGoogle Scholar
Fermanian-Kammerer, C. (2014), Opérateurs pseudo-différentiels semi-classiques. In Chaos en Mécanique Quantique, Ed. Éc. Polytech., Palaiseau, pp. 53–100.Google Scholar
Fermanian-Kammerer, C. and Gérard, P. (2002), ‘Mesures semi-classiques et croisement de modes’, Bull. Soc. Math. France 130, 123–168.CrossRefGoogle Scholar
Fermanian Kammerer, C. and Lasser, C. (2008), ‘Propagation through generic level crossings: A surface hopping semigroup’, SIAM J. Math. Anal. 40, 103–133.CrossRefGoogle Scholar
Fermanian Kammerer, C. and Lasser, C. (2017), ‘An Egorov theorem for avoided crossings of eigenvalue surfaces’, Comm. Math. Phys. 353, 1011–1057.CrossRefGoogle Scholar
Fleck, J. A., Morris, J. R. and Feit, M. D. (1976), ‘Time-dependent propagation of high energy laser beams through the atmosphere’, Appl. Phys. 10, 129–160.CrossRefGoogle Scholar
Folland, G. B. (1989), Harmonic Analysis in Phase Space, Vol. 122 of Annals of Mathematics Studies, Princeton University Press.CrossRefGoogle Scholar
Frenkel, J. I. (1934), Wave Mechanics: Advanced General Theory, Clarendon Press.Google Scholar
Gabor, D. (1946), ‘Theory of communication’, J. Inst. Elect. Eng. 93, 429–457.Google Scholar
Gaim, W. and Lasser, C. (2014), ‘Corrections to Wigner type phase space methods’, Nonlinearity 27, 2951–2974.CrossRefGoogle Scholar
Gautschi, W. (1997), Numerical Analysis, Birkhäuser.Google Scholar
Gerstner, T. and Griebel, M. (1998), ‘Numerical integration using sparse grids’, Numer. Algorithms 18, 209–232.CrossRefGoogle Scholar
Giles, M. B. (2015), Multilevel Monte Carlo methods. In Acta Numerica, Vol. 24, Cambridge University Press, pp. 259–328.Google Scholar
Golse, F., Jin, S. and Paul, T. (2019), On the convergence of time splitting methods for quantum dynamics in the semiclassical regime. arXiv:1906.03546 Google Scholar
Gradinaru, V. (2007 a), ‘Fourier transform on sparse grids: Code design and the time dependent Schrödinger equation’, Computing 80, 1–22.CrossRefGoogle Scholar
Gradinaru, V. (2007 b), ‘Strang splitting for the time-dependent Schrödinger equation on sparse grids’, SIAM J. Numer. Anal. 46, 103–123.CrossRefGoogle Scholar
Gradinaru, V. and Hagedorn, G. A. (2014), ‘Convergence of a semiclassical wavepacket based time-splitting for the Schrödinger equation’, Numer. Math. 126, 53–73.CrossRefGoogle Scholar
Groenewold, H. J. (1946), ‘On the principles of elementary quantum mechanics’, Physica 12, 405–460.CrossRefGoogle Scholar
Grossmann, A. and Morlet, J. (1984), ‘Decomposition of Hardy functions into square integrable wavelets of constant shape’, SIAM J. Math. Anal. 15, 723–736.CrossRefGoogle Scholar
Hagedorn, G. A. (1980), ‘Semiclassical quantum mechanics, I: The limit for coherent states’, Comm. Math. Phys. 71, 77–93.CrossRefGoogle Scholar
Hagedorn, G. A. (1981), ‘Semiclassical quantum mechanics, III: The large order asymptotics and more general states’, Ann. Phys. 135, 58–70.CrossRefGoogle Scholar
Hagedorn, G. A. (1994), Molecular Propagation Through Electron Energy Level Crossings, Vol. 111, no. 536 of Memoirs of the American Mathematical Society.Google Scholar
Hagedorn, G. A. (1998 a), ‘Classification and normal forms for avoided crossings of quantum-mechanical energy levels’, J. Phys. A 31, 369–383.CrossRefGoogle Scholar
Hagedorn, G. A. (1998 b), ‘Raising and lowering operators for semiclassical wave packets’, Ann. Phys. 269, 77–104.CrossRefGoogle Scholar
Hagedorn, G. A. (2015), ‘Generating function and a Rodrigues formula for the polynomials in $d$ -dimensional semiclassical wave packets’, Ann. Phys. 362, 603–608.CrossRefGoogle Scholar
Hagedorn, G. A. and Joye, A. (1999 a), ‘Molecular propagation through small avoided crossings of electron energy levels’, Rev. Math. Phys. 11, 41–101.CrossRefGoogle Scholar
Hagedorn, G. A. and Joye, A. (1999 b), ‘Semiclassical dynamics with exponentially small error estimates’, Comm. Math. Phys. 207, 439–465.CrossRefGoogle Scholar
Hagedorn, G. A. and Joye, A. (2000), ‘Exponentially accurate semiclassical dynamics: Propagation, localization, Ehrenfest times, scattering, and more general states’, Ann. Henri Poincaré 1, 837–883.CrossRefGoogle Scholar
Hagedorn, G. A. and Lasser, C. (2017), ‘Symmetric Kronecker products and semiclassical wave packets’, SIAM J. Matrix Anal. Appl. 38, 1560–1579.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wanner, G. (2003), Geometric numerical integration illustrated by the Störmer–Verlet method. In Acta Numerica, Vol. 12, Cambridge University Press, pp. 399–450.Google Scholar
Hairer, E., Lubich, C. and Wanner, G. (2006), Geometric Numerical Integration, second edition, Vol. 31 of Springer Series in Computational Mathematics, Springer.Google Scholar
Hall, B. C. (2013), Quantum Theory for Mathematicians, Vol. 267 of Graduate Texts in Mathematics, Springer.CrossRefGoogle Scholar
Hardin, R. H. and Tappert, F. D. (1973), ‘Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations’, SIAM Rev. 15, 423.Google Scholar
Hastings, W. K. (1970), ‘Monte Carlo sampling methods using Markov chains and their applications’, Biometrika 57, 97–109.CrossRefGoogle Scholar
Heller, E. (1976 a), ‘Wigner phase space method: Analysis for semiclassical applications’, J. Chem. Phys. 65, 1289–1298.CrossRefGoogle Scholar
Heller, E. J. (1976 b), ‘Time dependent variational approach to semiclassical dynamics’, J. Chem. Phys. 64, 63–73.CrossRefGoogle Scholar
Heller, E. J. (1981), ‘Frozen Gaussians: A very simple semiclassical approximation’, J. Chem. Phys. 75, 2923–2931.CrossRefGoogle Scholar
Heller, E. J. (2018), The Semiclassical Way to Dynamics and Spectroscopy, Princeton University Press.Google Scholar
Herman, M. F. and Kluk, E. (1984), ‘A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations’, Chem. Phys. 91, 27–34.CrossRefGoogle Scholar
Hlawka, E. (1961), ‘Funktionen von beschränkter Variation in der Theorie der Gleichverteilung’, Ann. Mat. Pura Appl. (4) 54, 325–333.CrossRefGoogle Scholar
Hochman, G. and Kay, K. K. (2006), ‘Semiclassical corrections to the Herman–Kluk propagator’, Phys. Rev. A 73, 064102.CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R. (2013), Matrix Analysis, second edition, Cambridge University Press.Google Scholar
Hudson, R. L. (1974), ‘When is the Wigner quasi-probability density non-negative?’, Rep. Mathematical Phys. 6, 249–252.CrossRefGoogle Scholar
Hunziker, W. (1986), ‘Distortion analyticity and molecular resonance curves’, Ann. Inst. H. Poincaré Phys. Théor. 45, 339–358.Google Scholar
Hunziker, W. and Günther, C. (1980), ‘Bound states in dipole fields and continuity properties of electronic spectra’, Helv. Phys. Acta 53, 201–208.Google Scholar
Husimi, K. (1940), ‘Some formal properties of the density matrix’, Proc. Physico-Math. Soc. Japan (3) 22, 264–314.Google Scholar
Jahnke, T. and Lubich, C. (2000), ‘Error bounds for exponential operator splittings’, BIT 40, 735–744.CrossRefGoogle Scholar
Jensen, F. (2016), Introduction to Computational Chemistry, third edition, Wiley.Google Scholar
Jin, S., Markowich, P. and Sparber, C. (2011), Mathematical and computational methods for semiclassical Schrödinger equations. In Acta Numerica, Vol. 20, Cambridge University Press, pp. 121–209.Google Scholar
Jin, S., Sparber, C. and Zhou, Z. (2017), ‘On the classical limit of a time-dependent self-consistent field system: Analysis and computation’, Kinet. Relat. Models 10, 263–298.CrossRefGoogle Scholar
Kato, T. (1950), ‘On the adiabatic theorem of quantum mechanics’, J. Phys. Soc. Japan 5, 435–439.CrossRefGoogle Scholar
Kato, T. (1951), ‘Fundamental properties of Hamiltonian operators of Schrödinger type’, Trans. Amer. Math. Soc. 70, 195–211.Google Scholar
Kay, K. (2006), ‘The Herman–Kluk approximation: Derivation and semiclassical corrections’, Chem. Phys. 322, 3–12.CrossRefGoogle Scholar
Keller, J. (2019), ‘The spectrogram expansion of Wigner functions’, Appl. Comput. Harmon. Anal. 47, 172–189.CrossRefGoogle Scholar
Keller, J. and Lasser, C. (2013), ‘Propagation of quantum expectations with Husimi functions’, SIAM J. Appl. Math. 73, 1557–1581.CrossRefGoogle Scholar
Keller, J., Lasser, C. and Ohsawa, T. (2016), ‘A new phase space density for quantum expectations’, SIAM J. Math. Anal. 48, 513–537.CrossRefGoogle Scholar
Klaus, M. (1983), ‘On ${H}_2^{+}$ for small internuclear separation’, J. Phys. A 16, 2709–2720.CrossRefGoogle Scholar
Klein, C. (2008), ‘Fourth order time-stepping for low dispersion Korteweg–de Vries and nonlinear Schrödinger equations’, Electron. Trans. Numer. Anal. 29, 116–135.Google Scholar
Kluk, E., Herman, M. F. and Davis, H. L. (1986), ‘Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator’, J. Chem. Phys. 84, 326–334.CrossRefGoogle Scholar
Koksma, J. F. (1942), ‘A general theorem from the theory of uniform distribution modulo 1’, Math. Zutphen B 11, 7–11.Google Scholar
Kramer, P. and Saraceno, M. (1981), Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Vol. 140 of Lecture Notes in Physics, Springer.CrossRefGoogle Scholar
Lasser, C. and Röblitz, S. (2010), ‘Computing expectation values for molecular quantum dynamics’, SIAM J. Sci. Comput. 32, 1465–1483.CrossRefGoogle Scholar
Lasser, C. and Sattlegger, D. (2017), ‘Discretising the Herman–Kluk propagator’, Numer. Math. 137, 119–157.CrossRefGoogle Scholar
Lasser, C. and Swart, T. (2008), ‘Single switch surface hopping for a model of pyrazine’, J. Chem. Phys. 129, 034302.CrossRefGoogle Scholar
Lasser, C. and Teufel, S. (2005), ‘Propagation through conical crossings: An asymptotic semigroup’, Comm. Pure Appl. Math. 58, 1188–1230.CrossRefGoogle Scholar
Lasser, C. and Troppmann, S. (2014), ‘Hagedorn wavepackets in time-frequency and phase space’, J. Fourier Anal. Appl. 20, 679–714.CrossRefGoogle Scholar
Lasser, C., Schubert, R. and Troppmann, S. (2018), ‘Non-Hermitian propagation of Hagedorn wavepackets’, J. Math. Phys. 59, 082102, 35.CrossRefGoogle Scholar
Le Bris, C. (2005), Computational chemistry from the perspective of numerical analysis. In Acta Numerica, Vol. 14, Cambridge University Press, pp. 363–444.Google Scholar
Lee, H. and Scully, M. (1980), ‘A new approach to molecular collisions: Statistical quasiclassical method’, J. Chem. Phys. 73, 2238–2242.CrossRefGoogle Scholar
Leimkuhler, B. and Reich, S. (2004), Simulating Hamiltonian Dynamics, Vol. 14 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.Google Scholar
Lin, L., Lu, J. and Ying, L. (2019), Numerical methods for Kohn–Sham density functional theory. In Acta Numerica, Vol. 28, Cambridge University Press, pp. 405–539.Google Scholar
Liu, H., Runborg, O. and Tanushev, N. M. (2013), ‘Error estimates for Gaussian beam superpositions’, Math. Comp. 82, 919–952.CrossRefGoogle Scholar
Liu, H., Runborg, O. and Tanushev, N. M. (2016), ‘Sobolev and max norm error estimates for Gaussian beam superpositions’, Commun. Math. Sci. 14, 2037–2072.CrossRefGoogle Scholar
Lu, J. and Zhou, Z. (2018), ‘Frozen Gaussian approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms’, Math. Comp. 87, 2189–2232.CrossRefGoogle Scholar
Lubich, C. (2008), From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS).CrossRefGoogle Scholar
Martinez, A. (2002), An Introduction to Semiclassical and Microlocal Analysis, Universitext, Springer.CrossRefGoogle Scholar
Martinez, A. and Sordoni, V. (2002), ‘A general reduction scheme for the time-dependent Born–Oppenheimer approximation’, C. R. Math. Acad. Sci. Paris 334, 185–188.CrossRefGoogle Scholar
Martinez, A. and Sordoni, V. (2009), Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules, Vol. 200, no. 936 of Memoirs of the American Mathematical Society.Google Scholar
Marčuk, G. I. (1968), ‘Some application of splitting-up methods to the solution of mathematical physics problems’, Apl. Mat. 13, 103–132.CrossRefGoogle Scholar
Mastroianni, G. and Monegato, G. (1994), Error estimates for Gauss–Laguerre and Gauss–Hermite quadrature formulas. In Approximation and Computation (West Lafayette, IN, 1993), Vol. 119 of International Series of Numerical Mathematics, Birkhäuser, pp. 421–434.Google Scholar
McLachlan, R. I. and Quispel, G. R. W. (2002), Splitting methods. In Acta Numerica, Vol. 11, Cambridge University Press, pp. 341–434.Google Scholar
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953), ‘Equation of state calculations by fast computing machines’, J. Chem. Phys. 21, 1087–1092.CrossRefGoogle Scholar
Miller, W. H. (1974 a), ‘Classical-limit quantum mechanics and the theory of molecular collisions’, Adv. Chem. Phys. 25, 69–177.Google Scholar
Miller, W. H. (1974 b), ‘Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants’, J. Chem. Phys. 61, 1823–1834.CrossRefGoogle Scholar
Ohsawa, T. (2015 a), ‘The Siegel upper half space is a Marsden–Weinstein quotient: Symplectic reduction and Gaussian wave packets’, Lett. Math. Phys. 105, 1301–1320.CrossRefGoogle Scholar
Ohsawa, T. (2015 b), ‘Symmetry and conservation laws in semiclassical wave packet dynamics’, J. Math. Phys. 56, 032103.CrossRefGoogle Scholar
Ohsawa, T. (2018), ‘Symplectic semiclassical wave packet dynamics, II: Non-Gaussian states’, Nonlinearity 31, 1807.CrossRefGoogle Scholar
Ohsawa, T. (2019), ‘The Hagedorn–Hermite correspondence’, J. Fourier Anal. Appl. 25, 1513–1552.CrossRefGoogle Scholar
Ohsawa, T. and Leok, M. (2013), ‘Symplectic semiclassical wave packet dynamics’, J. Phys. A 46, 405201.CrossRefGoogle Scholar
Park, T. J. and Light, J. C. (1986), ‘Unitary quantum time evolution by iterative Lanczos reduction’, J. Chem. Phys. 85, 5870–5876.CrossRefGoogle Scholar
Reed, M. and Simon, B. (1975), Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness, Academic Press (Harcourt Brace Jovanovich).Google Scholar
Reed, M. and Simon, B. (1980), Methods of Modern Mathematical Physics, I: Functional Analysis, second edition, Academic Press (Harcourt Brace Jovanovich).Google Scholar
Richings, G. W., Polyak, I., Spinlove, K. E., Worth, G. A., Burghardt, I. and Lasorne, B. (2015), ‘Quantum dynamics simulations using Gaussian wavepackets: The vMCG method’, Int. Rev. Phys. Chem. 34, 269–308.CrossRefGoogle Scholar
Riesz, F. (1930), ‘Über die linearen Transformationen des komplexen Hilbertschen Raumes’, Acta Litt. Sci. Math. Szeged 5, 23–54.Google Scholar
Robert, D. (2007), Propagation of coherent states in quantum mechanics and applications. In Partial Differential Equations and Applications, Vol. 15 of Sémin. Congr., Soc. Math. France, Paris, pp. 181–252.Google Scholar
Robert, D. (2010), ‘On the Herman–Kluk semiclassical approximation’, Rev. Math. Phys. 22, 1123–1145.CrossRefGoogle Scholar
Siegel, C. L. (1943), ‘Symplectic geometry’, Amer. J. Math. 65, 1–86.CrossRefGoogle Scholar
Simon, B. (1983), ‘Holonomy, the quantum adiabatic theorem, and Berry’s phase’, Phys. Rev. Lett. 51, 2167.CrossRefGoogle Scholar
Smolyak, S. A. (1963), ‘Quadrature and interpolation formulas for tensor products of certain classes of functions’, Doklady Akademii Nauk 148, 1042–1045 (Soviet Math. Dokl. 4 (1963), 240–243).Google Scholar
Sogge, C. D. (2017), Fourier Integrals in Classical Analysis , second edition, Vol. 210 of Cambridge Tracts in Mathematics, Cambridge University Press.CrossRefGoogle Scholar
Soto, F. and Claverie, P. (1983), ‘When is the Wigner function of multidimensional systems nonnegative?’, J. Math. Phys. 24, 97–100.CrossRefGoogle Scholar
Spohn, H. and Teufel, S. (2001), ‘Adiabatic decoupling and time-dependent Born–Oppenheimer theory’, Comm. Math. Phys. 224, 113–132.CrossRefGoogle Scholar
Stone, M. H. (1929), ‘Linear transformations in Hilbert space’, Proc. Nat. Acad. Sci. USA 15, 198–200 and 423–425.CrossRefGoogle ScholarPubMed
Strang, G. (1968), ‘On the construction and comparison of difference schemes’, SIAM J. Numer. Anal. 5, 506–517.CrossRefGoogle Scholar
Suzuki, Y., Suryanarayana, G. and Nuyens, D. (2019), ‘Strang splitting in combination with rank-1 and rank- $r$ lattices for the time-dependent Schrödinger equation’, SIAM J. Sci. Comput. 41, B1254–B1283.CrossRefGoogle Scholar
Swart, T. and Rousse, V. (2009), ‘A mathematical justification for the Herman–Kluk propagator’, Comm. Math. Phys. 286, 725–750.CrossRefGoogle Scholar
Szabo, A. and Ostlund, N. S. (1996), Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, revised edition, Dover.Google Scholar
Teufel, S. (2003), Adiabatic Perturbation Theory in Quantum Dynamics, Vol. 1821 of Lecture Notes in Mathematics, Springer.CrossRefGoogle Scholar
Thaller, B. (2000), Visual Quantum Mechanics, Springer–TELOS.CrossRefGoogle Scholar
Tully, J. (1990), ‘Molecular dynamics with electronic transitions’, J. Chem. Phys. 93, 1061.CrossRefGoogle Scholar
Uspensky, J. V. (1928), ‘On the convergence of quadrature formulas related to an infinite interval’, Trans. Amer. Math. Soc. 30, 542–559.CrossRefGoogle Scholar
Vaníček, J. and Begušić, T. (2020), Ab initio semiclassical evaluation of vibrationally resolved electronic spectra with thawed Gaussians. In Molecular Spectroscopy and Quantum Dynamics (Quack, M. and Marquardt, R., eds), Elsevier.Google Scholar
von Neumann, J. (1930), ‘Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren’, Math. Ann. 102, 49–131.CrossRefGoogle Scholar
Wang, H., Sun, X. and Miller, W. (1998), ‘Semiclassical approximations for the calculation of thermal rate constants for chemical reactions in complex molecular systems’, J. Chem. Phys. 108, 9726–9736.CrossRefGoogle Scholar
Weyl, H. (1927), ‘Quantenmechanik und Gruppentheorie’, Z. Physik 46, 1–46.CrossRefGoogle Scholar
Wigner, E. (1932), ‘On the quantum correction for thermodynamic equilibrium’, Phys. Rev. 40, 749–759.CrossRefGoogle Scholar
Xie, W. and Domcke, W. (2017), ‘Accuracy of trajectory surface-hopping methods: Test for a two-dimensional model of the photodissociation of phenol’, J. Chem. Phys. 147, 184114.CrossRefGoogle ScholarPubMed
Xie, W., Sapunar, M., Došclić, N., Sala, M. and Domcke, W. (2019), ‘Assessing the performance of trajectory surface hopping methods: Ultrafast internal conversion in pyrazine’, J. Chem. Phys. 150, 154119.CrossRefGoogle ScholarPubMed
Zenger, C. (1991), Sparse grids. In Parallel Algorithms for Partial Differential Equations (Kiel, 1990), Vol. 31 of Notes on Numerical Fluid Mechanics, Vieweg, pp. 241–251.Google Scholar
Zheng, C. (2014), ‘Optimal error estimates for first-order Gaussian beam approximations to the Schrödinger equation’, SIAM J. Numer. Anal. 52, 2905–2930.CrossRefGoogle Scholar
Zworski, M. (2012), Semiclassical Analysis, Vol. 138 of Graduate Studies in Mathematics, American Mathematical Society.CrossRefGoogle Scholar