Frozen Gaussian approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms

Lu, Jianfeng; Zhou, Zhennan

doi:10.1090/mcom/3310

Frozen Gaussian approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms
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by Jianfeng Lu and Zhennan Zhou PDF

Math. Comp. 87 (2018), 2189-2232 Request permission

Abstract:

We develop a surface hopping algorithm based on frozen Gaussian approximation for semiclassical matrix Schrödinger equations, in the spirit of Tully’s fewest switches surface hopping method. The algorithm is asymptotically derived from the Schrödinger equation with rigorous approximation error analysis. The resulting algorithm can be viewed as a path integral stochastic representation of the semiclassical matrix Schrödinger equations. Our results provide mathematical understanding to and shed new light on the important class of surface hopping methods in theoretical and computational chemistry.

References

Weizhu Bao, Shi Jin, and Peter A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys. 175 (2002), no. 2, 487–524. MR 1880116, DOI 10.1006/jcph.2001.6956
M.J. Bedard-Hearn, R.E. Larsen, and B.J. Schwartz, Mean-field dynamics with stochastic decoherence (mf-sd): A new algorithm for nonadiabatic mixed quantum/classical molecular-dynamics simulations with nuclear-induced decoherence, J. Chem. Phys. 123 (2005), no. 23, 234106.
Lihui Chai, Shi Jin, Qin Li, and Omar Morandi, A multiband semiclassical model for surface hopping quantum dynamics, Multiscale Model. Simul. 13 (2015), no. 1, 205–230. MR 3301305, DOI 10.1137/140967842
Clotilde Fermanian Kammerer and Caroline Lasser, Propagation through generic level crossings: a surface hopping semigroup, SIAM J. Math. Anal. 40 (2008), no. 1, 103–133. MR 2403314, DOI 10.1137/070686810
George A. Hagedorn, Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps, Comm. Math. Phys. 136 (1991), no. 3, 433–449. MR 1099690
George A. Hagedorn and Alain Joye, Landau-Zener transitions through small electronic eigenvalue gaps in the Born-Oppenheimer approximation, Ann. Inst. H. Poincaré Phys. Théor. 68 (1998), no. 1, 85–134 (English, with English and French summaries). MR 1618922
S. Hammes-Schiffer and J.C. Tully, Proton transfer in solution: Molecular dynamics with quantum transitions, J. Chem. Phys. 101 (1994), no. 6, 4657–4667.
G. Hanna and R. Kapral, Quantum-classical Liouville dynamics of nonadiabatic proton transfer, J. Chem. Phys. 122 (2005), no. 24, 244505.
Eric J. Heller, Frozen Gaussians: a very simple semiclassical approximation, J. Chem. Phys. 75 (1981), no. 6, 2923–2931. MR 627226, DOI 10.1063/1.442382
M.F. Herman and E. Kluk, A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations, Chem. Phys. 91 (1984), no. 1, 27–34.
I. Horenko, C. Salzmann, B. Schmidt, and Ch. Schütte, Quantum-classical Liouville approach to molecular dynamics Surface hopping Gaussian phase-space packets, J. Chem. Phys. 117 (2002), no. 24, 11075–11088.
Shi Jin, Peter Markowich, and Christof Sparber, Mathematical and computational methods for semiclassical Schrödinger equations, Acta Numer. 20 (2011), 121–209. MR 2805153, DOI 10.1017/S0962492911000031
Shi Jin, Peng Qi, and Zhiwen Zhang, An Eulerian surface hopping method for the Schrödinger equation with conical crossings, Multiscale Model. Simul. 9 (2011), no. 1, 258–281. MR 2801205, DOI 10.1137/090774185
S. Jin and Z. Zhou, A semi-Lagrangian time splitting method for the Schrödinger equation with vector potentials, Commun. Inform. Syst. 13 (2013), 247–289.
R. Kapral and G. Ciccotti, Mixed quantum-classical dynamics, J. Chem. Phys. 110 (1999), no. 18, 8919–8929.
K. Kay, Integral expressions for the semi-classical time-dependent propagator, J. Chem. Phys. 100 (1994), no. 6, 4377–4392.
K. Kay, The Herman-Kluk approximation: derivation and semiclassical corrections, Chem. Phys. 322 (2006), no. 1, 3–12.
L. Landau, Zur theorie der energieubertragung. II., Physics of the Soviet Union 2 (1932), no. 2, 46–51.
B.R. Landry and J.E. Subotnik, How to recover Marcus theory with fewest switches surface hopping: Add just a touch of decoherence, J. Chem. Phys. 137 (2011), no. 22, 22A513.
Caroline Lasser, Torben Swart, and Stefan Teufel, Construction and validation of a rigorous surface hopping algorithm for conical crossings, Commun. Math. Sci. 5 (2007), no. 4, 789–814. MR 2375047
Jianfeng Lu and Xu Yang, Frozen Gaussian approximation for high frequency wave propagation, Commun. Math. Sci. 9 (2011), no. 3, 663–683. MR 2865800, DOI 10.4310/CMS.2011.v9.n3.a2
Jianfeng Lu and Xu Yang, Convergence of frozen Gaussian approximation for high-frequency wave propagation, Comm. Pure Appl. Math. 65 (2012), no. 6, 759–789. MR 2903799, DOI 10.1002/cpa.21384
Gianluca Panati, Herbert Spohn, and Stefan Teufel, The time-dependent Born-Oppenheimer approximation, M2AN Math. Model. Numer. Anal. 41 (2007), no. 2, 297–314. MR 2339630, DOI 10.1051/m2an:2007023
O.V. Prezhdo, Mean field approximation for the stochastic Schrödinger equation, J. Chem. Phys. 111 (1999), no. 18, 8366–8377.
Herbert Spohn and Stefan Teufel, Adiabatic decoupling and time-dependent Born-Oppenheimer theory, Comm. Math. Phys. 224 (2001), no. 1, 113–132. Dedicated to Joel L. Lebowitz. MR 1868994, DOI 10.1007/s002200100535
J.E. Subotnik, W. Ouyang, and B.R. Landry, Can we derive Tully’s surface-hopping algorithm from the semiclassical quantum Liouville equation? Almost, but only with decoherence, J. Chem. Phys. 139 (2011), no. 21, 214107.
J.E. Subotnik and N. Shenvi, A new approach to decoherence and momentum rescaling in the surface hopping algorithm, J. Chem. Phys. 134 (2011), no. 2, 024105.
Torben Swart and Vidian Rousse, A mathematical justification for the Herman-Kluk propagator, Comm. Math. Phys. 286 (2009), no. 2, 725–750. MR 2472042, DOI 10.1007/s00220-008-0681-4
J.C. Tully, Molecular dynamics with electronic transitions, J. Chem. Phys. 93 (1990), no. 2, 1061–1071.
J.C. Tully and R.K. Preston, Trajectory surface hopping approach to nonadiabatic molecular collisions: the reaction of $h^+$ with $d_2$, J. Chem. Phys. 55 (1971), no. 2, 562–572.
Y. Wu and M.F. Herman, Nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation method revisited: Applications to Tully’s three model systems, J. Chem. Phys. 123 (2005), no. 14, 144106.
Y. Wu and M.F. Herman, A justification for a nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation of the time evolution operator, J. Chem. Phys. 125 (2006), no. 15, 154116.
Y. Wu and M.F. Herman, On the properties of a primitive semiclassical surface hopping propagator for nonadiabatic quantum dynamics, J. Chem. Phys. 127 (2007), no. 4, 044109.
C. Zener, Non–adiabatic crossing of energy levels, Proc. R. Soc. London A 137 (1932), no. 883, 696–702.