求解带有锥形交叉区域薛定谔方程的改进界面跃迁格式

doi:10.19596/j.cnki.1001-246x.7773

计算物理 ›› 2019, Vol. 36 ›› Issue (1): 113-126.DOI: 10.19596/j.cnki.1001-246x.7773

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求解带有锥形交叉区域薛定谔方程的改进界面跃迁格式

李新春

上海交通大学数学科学学院, 上海 200240

收稿日期:2017-09-26 修回日期:2017-11-05 出版日期:2019-01-25 发布日期:2019-01-25
作者简介:李新春(1988-),男,汉族,山东,博士生,研究方向为计算数学,E-mail:neo.lee@sjtu.edu.cn
基金资助:
国家自然科学基金（91330203）资助项目

A Modified Surface Hopping Method for Schrödinger Equation with Conical Crossings

LI Xinchun

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

Received:2017-09-26 Revised:2017-11-05 Online:2019-01-25 Published:2019-01-25

摘要/Abstract

摘要： 改进一种欧拉框架下的界面跃迁格式.原格式是为求解带有多个能级，不同能级之间存在锥形交叉区域的薛定谔方程提出的.石墨烯中的电子输运与上述过程相似.Kammerer等人提出带有跳跃算子的数值格式能够更好地保持石墨烯中电子输运的能量守恒.引入这种跳跃算子可以改进原有的界面跃迁格式.改进格式的数值结果取得了预期的效果.

关键词: 薛定谔方程, 刘维尔方程, 界面跃迁格式

Abstract: We present a modified surface hopping method for Schrödinger equation with conical crossings. At the crossing manifold, Landau-Zener formula gives probability that electrons hop to another energy level. Kammerer and Mehats raised a scheme with jump operators ensuring good energy conservation for transport of electrons in a graphene layer which is similar to the physics above. We employ these jump operators to improve a surface hopping method and obtain accurate numerical results.

Key words: Schrödinger equation, Liouville equation, surface hopping method

中图分类号:

O413.1

李新春. 求解带有锥形交叉区域薛定谔方程的改进界面跃迁格式[J]. 计算物理, 2019, 36(1): 113-126.

LI Xinchun. A Modified Surface Hopping Method for Schrödinger Equation with Conical Crossings[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2019, 36(1): 113-126.

参考文献

[1] LANDAU L D, LIFSCHITZ E M. Quantum mechanics:Non-relativistic theory[M]. 3rd ed. Oxford:Pergamon Press, 1977.
[2] BORN M, OPPEMJEO,ER R. Zur quantentheorie der molekeln[J]. Annalen der Physik, 1927, 389(20):457-484.
[3] COMBES J M, DUCLOS P, SEILER R. The Born-Oppenheimer approximation[M]//Rigorous Atomic and Molecular Physics. New York:Plenum Press, 1981:185-213.
[4] HAGEDORN G A. High order corrections to the time-independent Born-Oppenheimer approximation I:Smooth potentials[J]. Annales de l'IHP Physique théorique, 1987, 47:1-16.
[5] HAGEDORN G A, JOYE A. A time-dependent Born-Oppenheimer approximation with exponentially small error estimates[J]. Communications in Mathematical Physics, 2001, 223(3):583-626.
[6] KLEIN M, MARTINEZ A, SEILER R, WANG X P. On the Born-Oppenheimer expansion for polyatomic molecules[J]. Communications in Mathematical Physics, 1992, 143(3):607-639.
[7] SPOHN H, Teufel S. Adiabatic decoupling and time-dependent Born-Oppenheimer theory[J]. Communications in Mathematical Physics, 2001, 224(1):113-132.
[8] BITLER L J. Chemical reaction dynamics beyond the Born-Oppenheimer approximation[J]. Annual Review of Physical Chemistry, 1998, 49(1):125-171.
[9] BAER M, ENGLMAN R. A study of the diabatic electronic representation within the Born-Oppenheimer approximation[J]. Molecular Physics, 1992, 75(2):293-303.
[10] WORTH G A, CEDERBAUM L S. Beyond Born-Oppenheimer:Molecular dynamics through a conical intersection[J]. Annu Rev Phys Chem, 2004, 55:127-158.
[11] TULLY J C, PRESTON R K. Trajectory surface hopping approach to nonadiabatic molecular collisions:The reaction of h+ with d2[J]. The Journal of Chemical Physics, 1971, 55(2):562-572.
[12] TULLY J C. Molecular dynamics with electronic transitions[J]. The Journal of Chemical Physics, 1990, 93(2):1061-1071.
[13] VORONIN A I, MARQUES J M C, VARANDAS A J C. Trajectory surface hopping study of the li+ li2(x1σg+) dissociation reaction[J]. The Journal of Physical Chemistry A, 1998, 102(30):6057-6062.
[14] SHOLL D S, TULLY J C. A generalized surface hopping method[J]. The Journal of Chemical Physics, 1998, 109(18):7702-7710.
[15] DRUKKER K. Basics of surface hopping in mixed quantum/classical simulations[J]. Journal of Computational Physics, 1999, 153(2):225-272.
[16] HORENKO I, SALZMANN C, SCHMIDT B,SCHVTTE C. Quantum-classical Liouville approach to molecular dynamics:Surface hopping Gaussian phase-space packets[J]. The Journal of Chemical Physics, 2002, 117(24):11075-11088.
[17] LASSER C, TEUFEL S. Propagation through conical crossings:An asymptotic semigroup[J]. Communications on Pure and Applied Mathematics, 2005, 58(9):1188-1230.
[18] KAMMERER C F, LASSER C. Wigner measures and codimension two crossings[J]. Math Phys, 2002.
[19] LASSER C, SWART T, TEUFUL S, et al. Construction and validation of a rigorous surface hopping algorithm for conical crossings[J]. Communications in Mathematical Sciences, 2007, 5(4):789-814.
[20] KUBE S, LASSER C, WEBER M. Monte Carlo sampling of Wigner functions and surface hopping quantum dynamics[J]. Journal of Computational Physics, 2009, 228(6):1947-1962.
[21] JIN S, QI P, ZHANG Z W. An Eulerian surface hopping method for the Schrödinger equation with conical crossings[J]. Multiscale Modeling & Simulation, 2011, 9(1):258-281.
[22] JIN S, WEN X. Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials[J]. Communications in Mathematical Sciences, 2005, 3(3):285-315.
[23] JIN S, WEN X. Computation of transmissions and reflections in geometrical optics via the reduced Liouville equation[J]. Wave Motion, 2006, 43(8):667-688.
[24] JIN S, NOVAK K A. A semiclassical transport model for two-dimensional thin quantum barriers[J]. Journal of Computational Physics, 2007, 226(2):1623-1644.
[25] JIN S, LIAO X M. A Hamiltonian-preserving scheme for high frequency elastic waves in heterogeneous media[J]. Journal of Hyperbolic Differential Equations, 2006, 3(04):741-777.
[26] NETO A C, GUINEA F, PERES N M, NOVOSELOV K S, GEIM A K. The electronic properties of graphene[J]. Reviews of Modern Physics, 2009, 81(1):109.
[27] SARMA S D, ADAM S, HWANG E H, ROSSI E. Electronic transport in two-dimensional graphene[J]. Reviews of Modern Physics, 2011, 83(2):407.
[28] KAMMERER C F, MÉHATS F. A kinetic model for the transport of electrons in a graphene layer[J]. Journal of Computational Physics, 2016, 327:450-483.
[29] HAGEDORN G A. Molecular propagation through electron energy level crossings[J]. American Mathematical Soc, 1994, 536.
[30] WIGNER E. On the quantum correction for thermodynamic equilibrium[J]. Physical Review, 1932, 40(5):749.
[31] GÉRARD P, MARKOWICH P A, MAUSER N J, POUPAUD F. Homogenization limits and Wigner transforms[J]. Communications on Pure and Applied Mathematics, 1997, 50(4):323-379.
[32] PAUL T, LIONS P L. Sur les mesures de Wigner[J]. Revista Matemática Iberoamericana, 1993, 9(3):553-618.
[33] LANDAU L D. Zur theorie der energieubertragung, Ⅱ[J]. Physics of the Soviet Union, 1932, 2(46):46-51.
[34] ZENER C. Non-adiabatic crossing of energy levels[C]//Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences, London:The Royal Society, 1932, 137:696-702.
[35] LEVEQUE R J. Finite volume methods for hyperbolic problems[M]. Cambridge:Cambridge University Press, 2002.
[36] JIN S, WEN X. A Hamiltonian-preserving scheme for the Liouville equation of geometrical optics with partial transmissions and reflections[J]. SIAM Journal on Numerical Analysis, 2006, 44(5):1801-1828.
[37] JIN S, YIN D S. Computational high frequency waves through curved interfaces via the Loiuville equation and geometric theory of diffraction[J]. Journal of Computational Physics, 2008, 227:6106-6139.
[38] KAMMERER C F, LASSER C. Single switch surface hopping for molecular dynamics with transitions[J]. The Journal of Chemical Physics, 2008, 128(14):144102.
[39] THIEL S, KLÜNER T, FREUND H J. Interference-effects in the laser-induced desorption of small molecules from surfaces:A model study[J]. Chemical Physics, 1998, 236(1):263-276.
[40] BAO W, JIN S, MARKOWICH P A. On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime[J]. Journal of Computational Physics, 2002, 175(2):487-524.

求解带有锥形交叉区域薛定谔方程的改进界面跃迁格式

A Modified Surface Hopping Method for Schrödinger Equation with Conical Crossings

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