December  2011, 4(4): 1097-1120. doi: 10.3934/krm.2011.4.1097

A hybrid Schrödinger/Gaussian beam solver for quantum barriers and surface hopping

1. 

Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, United States

2. 

Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, United States

Received  May 2011 Revised  October 2011 Published  November 2011

In this paper, we propose a hybrid method coupling a Schrödinger solver and the Gaussian beam method for the numerical simulation of quantum tunneling through potential barriers or surface hopping across electronic potential energy surfaces. The idea is to use a Schrödinger solver near potential barriers or zones where potential energy surfaces cross, and the Gaussian beam method--which is much more efficient than a direct Schrödinger solver--elsewhere. Buffer zones are used to convert data between the Schrödinger solver and the Gaussian beam solver. Numerical examples show that this method indeed captures quantum tunneling and surface hopping accurately, with a computational cost much lower than a direct quantum solver in the entire domain.
Citation: Shi Jin, Peng Qi. A hybrid Schrödinger/Gaussian beam solver for quantum barriers and surface hopping. Kinetic and Related Models, 2011, 4 (4) : 1097-1120. doi: 10.3934/krm.2011.4.1097
References:
[1]

M. Baer, "Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections,'' Wiley, Hoboken, New Jersey, 2006. doi: 10.1002/0471780081.

[2]

W. Bao, S. Jin and P. Markowich, On the time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524. doi: 10.1006/jcph.2001.6956.

[3]

M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Phys., 84 (1927), 457-484. doi: 10.1002/andp.19273892002.

[4]

N. Ben Abdallah, P. Degond and I. M. Gamba, Coupling one-dimensional time-dependent classical and quantum transport models, J. Math. Phys., 43 (2002), 1-24. doi: 10.1063/1.1421635.

[5]

M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond Ser. A, 392 (1984), 45-57. doi: 10.1098/rspa.1984.0023.

[6]

V. Cerveny, M. M. Popov and I. Psencik, Computation of wave fields in inhomogeneous media-Gaussian beam approach, Geophys. J. R. Astr. Soc., 70 (1982), 109-128.

[7]

K. Drukker, Basics of surface hopping in mixed quantum/classical simulations, J. Comp. Phys., 153 (1999), 225-272. doi: 10.1006/jcph.1999.6287.

[8]

P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Communications on Pure and Applied Mathematics, 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[9]

E. J. Heller, Cellular dynamics: A new semiclassical approach to time-dependent quantum mechanics, J. Chem. Phys., 94 (1991), 2723-2729. doi: 10.1063/1.459848.

[10]

N. R. Hill, Gaussian beam migration, Geophys., 55 (1990), 1416-1428. doi: 10.1190/1.1442788.

[11]

I. Horenko, C. Salzmann, B. Schmidt and C. Schutte, Quantum-classical Liouville approach to molecular dynamics: Surface hopping Gaussian phase-space packets, J. Chem. Phys., 117 (2002), 11075-11088. doi: 10.1063/1.1522712.

[12]

S. Jin, P. Markowich and C. Sparber, Mathematical and computational methods for semiclassical Schrödinger equations, Acta Numerica, 20 (2011), 211-289.

[13]

S. Jin and K. Novak, A semiclassical transport model for thin quantum barriers, Multiscale Modeling and Simulation, 5 (2006), 1063-1086. doi: 10.1137/060653214.

[14]

S. Jin and K. Novak, A semiclassical transport model for two-dimensional thin quantum barriers, J. Comp. Phys., 226 (2007), 1623-1644. doi: 10.1016/j.jcp.2007.06.006.

[15]

S. Jin and K. Novak, A coherent semiclassical transport model for pure-state quantum scattering, Comm. Math. Sci., 8 (2010), 253-275.

[16]

S. Jin, P. Qi and Z. Zhang, An Eulerian surface hopping method for the Schrödinger equation with conical crossings, SIAM Multiscale Modeling & Simulation, 9 (2011), 258-281.

[17]

S. Jin, H. Wu and X. Yang, Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Comm. Math. Sci., 6 (2008), 995-1020.

[18]

S. Jin and D. Yin, Computation of high frequency wave diffraction by a half plane via the Liouville equation and geometric theory of diffraction, Communications in Computational Physics, 4 (2008), 1106-1128.

[19]

L. Landau, Zur Theorie der Energiebertragung II, Physics of the Soviet Union, 2 (1932), 46-51.

[20]

C. F. Kammerer and C. Lasser, Wigner measures and codimension two crossings, Jour. Math. Phys., 44 (2003), 507-527. doi: 10.1063/1.1527221.

[21]

C. F. Kammerer and C. Lasser, Single switch surface hopping for molecular dynamics with transitions, J. Chem. Phys., 128 (2008), 5-9.

[22]

C. Lasser, T. Swart and S. Teufel, Construction and validation of a rigorous surface hopping algorithm for conical crossings, Comm. Math. Sci., 5 (2007), 789-814.

[23]

S. Leung, J. Qian and R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics, 72 (2007), 61-76. doi: 10.1190/1.2752136.

[24]

J. Lu and X. Yang, Frozen Gaussian approximation for high frequency wave propagation, Commun. Math. Sci., 9 (2011), 663-683.

[25]

M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition, Wave Motion, 47 (2010), 421-439. doi: 10.1016/j.wavemoti.2010.02.001.

[26]

W. H. Miller and Thomas F. George, Semiclassical theory of electronic transitions in low energy atomic and molecular collisions involving several nuclear degrees of freedom, Journal of Chemical Physics, 56 (1972), 5637-5652. doi: 10.1063/1.1677083.

[27]

B. N. Parlett, "The Symmetric Eigenvalue Problem,'' Classics ed., SIAM, Philadelphia, PA, USA, 1998.

[28]

M. M. Popov, A new method of computation of wave fields using Gaussian beams, Wave Motion, 4 (1982), 85-97. doi: 10.1016/0165-2125(82)90016-6.

[29]

J. Qian and L. Ying, Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation, J. Comput. Phys., 229 (2010), 7848-7873. doi: 10.1016/j.jcp.2010.06.043.

[30]

J. Ralston, Gaussian beams and the propagation of singularities, in "Studies in PDEs," MAA. Stud. Math., 23, Math. Assoc. America, Washington, DC, (1982), 206-248.

[31]

D. Sholla and J. Tully, A generalized surface hopping method, J. Chem. Phys., 109 (1998), 7702. doi: 10.1063/1.477416.

[32]

J. Tully and R. Preston, Trajectory surface hopping approach to nonadiabatic molecular collisions: The reaction of $H^+$ with $D_2$, J. Chem. Phys., 55 (1971), 562-572. doi: 10.1063/1.1675788.

[33]

J. Tully, Molecular dynamics with electronic transitions, J. Chem. Phys., 93 (1990), 1061. doi: 10.1063/1.459170.

[34]

N. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6 (2008), 449-475.

[35]

N. Tanushev, J. Qian and J. Ralston, Mountain waves and Gaussian beams, SIAM J. Multiscale Model. Simul., 6 (2007), 688-709. doi: 10.1137/060673667.

[36]

N. M. Tanushev, R. Tsai and B. Engquist, Coupled finite difference-Gaussian beam method for high frequency wave propagation, UCLA CAM Reports, (2010), 10-40.

[37]

N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields, Journal of Computational Physics, 228 (2009), 8856-8871. doi: 10.1016/j.jcp.2009.08.028.

[38]

A. Voronin, J. Marques and A. Varandas, Trajectory surface hopping study of the $Li$ + $Li_2$ ($X^1\Sigma_g^+$) Dissociation reaction, J. Phys. Chem. A, 102 (1998), 6057-6062. doi: 10.1021/jp9805860.

[39]

D. Wei and X. Yang, Gaussian beam method for high frequency wave propagation in heterogeneous media, preprint 2011.

[40]

W. R. E. Weiss and G. A. Hagedorn, Reflection and transmission of high freuency pulses at an interface, Transport Theory and Statistical Physics, 14 (1985), 539-565. doi: 10.1080/00411458508211692.

[41]

D. Yin and C. Zheng, Gaussian beam formulations and interface conditions for the one-dimensional linear Schrödinger equation, Wave Motion, 48 (2011), 310-324. doi: 10.1016/j.wavemoti.2010.11.006.

[42]

D. Yin and C. Zheng, Composite Gaussian beam approximation method for multi-phased wave functions, preprint, 2011.

[43]

C. Zener, Non-adiabatic crossing of energy levels, Proceedings of the Royal Society of London, Series A, 137 1932, 692-702.

show all references

References:
[1]

M. Baer, "Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections,'' Wiley, Hoboken, New Jersey, 2006. doi: 10.1002/0471780081.

[2]

W. Bao, S. Jin and P. Markowich, On the time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524. doi: 10.1006/jcph.2001.6956.

[3]

M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Phys., 84 (1927), 457-484. doi: 10.1002/andp.19273892002.

[4]

N. Ben Abdallah, P. Degond and I. M. Gamba, Coupling one-dimensional time-dependent classical and quantum transport models, J. Math. Phys., 43 (2002), 1-24. doi: 10.1063/1.1421635.

[5]

M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond Ser. A, 392 (1984), 45-57. doi: 10.1098/rspa.1984.0023.

[6]

V. Cerveny, M. M. Popov and I. Psencik, Computation of wave fields in inhomogeneous media-Gaussian beam approach, Geophys. J. R. Astr. Soc., 70 (1982), 109-128.

[7]

K. Drukker, Basics of surface hopping in mixed quantum/classical simulations, J. Comp. Phys., 153 (1999), 225-272. doi: 10.1006/jcph.1999.6287.

[8]

P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Communications on Pure and Applied Mathematics, 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[9]

E. J. Heller, Cellular dynamics: A new semiclassical approach to time-dependent quantum mechanics, J. Chem. Phys., 94 (1991), 2723-2729. doi: 10.1063/1.459848.

[10]

N. R. Hill, Gaussian beam migration, Geophys., 55 (1990), 1416-1428. doi: 10.1190/1.1442788.

[11]

I. Horenko, C. Salzmann, B. Schmidt and C. Schutte, Quantum-classical Liouville approach to molecular dynamics: Surface hopping Gaussian phase-space packets, J. Chem. Phys., 117 (2002), 11075-11088. doi: 10.1063/1.1522712.

[12]

S. Jin, P. Markowich and C. Sparber, Mathematical and computational methods for semiclassical Schrödinger equations, Acta Numerica, 20 (2011), 211-289.

[13]

S. Jin and K. Novak, A semiclassical transport model for thin quantum barriers, Multiscale Modeling and Simulation, 5 (2006), 1063-1086. doi: 10.1137/060653214.

[14]

S. Jin and K. Novak, A semiclassical transport model for two-dimensional thin quantum barriers, J. Comp. Phys., 226 (2007), 1623-1644. doi: 10.1016/j.jcp.2007.06.006.

[15]

S. Jin and K. Novak, A coherent semiclassical transport model for pure-state quantum scattering, Comm. Math. Sci., 8 (2010), 253-275.

[16]

S. Jin, P. Qi and Z. Zhang, An Eulerian surface hopping method for the Schrödinger equation with conical crossings, SIAM Multiscale Modeling & Simulation, 9 (2011), 258-281.

[17]

S. Jin, H. Wu and X. Yang, Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Comm. Math. Sci., 6 (2008), 995-1020.

[18]

S. Jin and D. Yin, Computation of high frequency wave diffraction by a half plane via the Liouville equation and geometric theory of diffraction, Communications in Computational Physics, 4 (2008), 1106-1128.

[19]

L. Landau, Zur Theorie der Energiebertragung II, Physics of the Soviet Union, 2 (1932), 46-51.

[20]

C. F. Kammerer and C. Lasser, Wigner measures and codimension two crossings, Jour. Math. Phys., 44 (2003), 507-527. doi: 10.1063/1.1527221.

[21]

C. F. Kammerer and C. Lasser, Single switch surface hopping for molecular dynamics with transitions, J. Chem. Phys., 128 (2008), 5-9.

[22]

C. Lasser, T. Swart and S. Teufel, Construction and validation of a rigorous surface hopping algorithm for conical crossings, Comm. Math. Sci., 5 (2007), 789-814.

[23]

S. Leung, J. Qian and R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics, 72 (2007), 61-76. doi: 10.1190/1.2752136.

[24]

J. Lu and X. Yang, Frozen Gaussian approximation for high frequency wave propagation, Commun. Math. Sci., 9 (2011), 663-683.

[25]

M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition, Wave Motion, 47 (2010), 421-439. doi: 10.1016/j.wavemoti.2010.02.001.

[26]

W. H. Miller and Thomas F. George, Semiclassical theory of electronic transitions in low energy atomic and molecular collisions involving several nuclear degrees of freedom, Journal of Chemical Physics, 56 (1972), 5637-5652. doi: 10.1063/1.1677083.

[27]

B. N. Parlett, "The Symmetric Eigenvalue Problem,'' Classics ed., SIAM, Philadelphia, PA, USA, 1998.

[28]

M. M. Popov, A new method of computation of wave fields using Gaussian beams, Wave Motion, 4 (1982), 85-97. doi: 10.1016/0165-2125(82)90016-6.

[29]

J. Qian and L. Ying, Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation, J. Comput. Phys., 229 (2010), 7848-7873. doi: 10.1016/j.jcp.2010.06.043.

[30]

J. Ralston, Gaussian beams and the propagation of singularities, in "Studies in PDEs," MAA. Stud. Math., 23, Math. Assoc. America, Washington, DC, (1982), 206-248.

[31]

D. Sholla and J. Tully, A generalized surface hopping method, J. Chem. Phys., 109 (1998), 7702. doi: 10.1063/1.477416.

[32]

J. Tully and R. Preston, Trajectory surface hopping approach to nonadiabatic molecular collisions: The reaction of $H^+$ with $D_2$, J. Chem. Phys., 55 (1971), 562-572. doi: 10.1063/1.1675788.

[33]

J. Tully, Molecular dynamics with electronic transitions, J. Chem. Phys., 93 (1990), 1061. doi: 10.1063/1.459170.

[34]

N. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6 (2008), 449-475.

[35]

N. Tanushev, J. Qian and J. Ralston, Mountain waves and Gaussian beams, SIAM J. Multiscale Model. Simul., 6 (2007), 688-709. doi: 10.1137/060673667.

[36]

N. M. Tanushev, R. Tsai and B. Engquist, Coupled finite difference-Gaussian beam method for high frequency wave propagation, UCLA CAM Reports, (2010), 10-40.

[37]

N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields, Journal of Computational Physics, 228 (2009), 8856-8871. doi: 10.1016/j.jcp.2009.08.028.

[38]

A. Voronin, J. Marques and A. Varandas, Trajectory surface hopping study of the $Li$ + $Li_2$ ($X^1\Sigma_g^+$) Dissociation reaction, J. Phys. Chem. A, 102 (1998), 6057-6062. doi: 10.1021/jp9805860.

[39]

D. Wei and X. Yang, Gaussian beam method for high frequency wave propagation in heterogeneous media, preprint 2011.

[40]

W. R. E. Weiss and G. A. Hagedorn, Reflection and transmission of high freuency pulses at an interface, Transport Theory and Statistical Physics, 14 (1985), 539-565. doi: 10.1080/00411458508211692.

[41]

D. Yin and C. Zheng, Gaussian beam formulations and interface conditions for the one-dimensional linear Schrödinger equation, Wave Motion, 48 (2011), 310-324. doi: 10.1016/j.wavemoti.2010.11.006.

[42]

D. Yin and C. Zheng, Composite Gaussian beam approximation method for multi-phased wave functions, preprint, 2011.

[43]

C. Zener, Non-adiabatic crossing of energy levels, Proceedings of the Royal Society of London, Series A, 137 1932, 692-702.

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