American Institute of Mathematical Sciences

Advanced
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

Shi Jin 1, and  Peng Qi 2,

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.
Keywords: Gaussian beam, Surface hopping, Time-splitting spectral method, quantum tunneling..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Shi Jin, Peng Qi. A hybrid Schrödinger/Gaussian beam solver for quantum barriers and surface hopping. Kinetic & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[42]

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

[43]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[42]

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

[43]

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

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