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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 |
References:
[1] |
M. Baer, "Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections,'', Wiley, (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.
doi: 10.1006/jcph.2001.6956. |
[3] |
M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln,, Ann. Phys., 84 (1927), 457.
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.
doi: 10.1063/1.1421635. |
[5] |
M. V. Berry, Quantal phase factors accompanying adiabatic changes,, Proc. R. Soc. Lond Ser. A, 392 (1984), 45.
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. Google Scholar |
[7] |
K. Drukker, Basics of surface hopping in mixed quantum/classical simulations,, J. Comp. Phys., 153 (1999), 225.
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.
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.
doi: 10.1063/1.459848. |
[10] |
N. R. Hill, Gaussian beam migration,, Geophys., 55 (1990), 1416.
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.
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. Google Scholar |
[13] |
S. Jin and K. Novak, A semiclassical transport model for thin quantum barriers,, Multiscale Modeling and Simulation, 5 (2006), 1063.
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.
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.
|
[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.
|
[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.
|
[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. Google Scholar |
[19] |
L. Landau, Zur Theorie der Energiebertragung II,, Physics of the Soviet Union, 2 (1932), 46. Google Scholar |
[20] |
C. F. Kammerer and C. Lasser, Wigner measures and codimension two crossings,, Jour. Math. Phys., 44 (2003), 507.
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. 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.
|
[23] |
S. Leung, J. Qian and R. Burridge, Eulerian Gaussian beams for high frequency wave propagation,, Geophysics, 72 (2007), 61.
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. Google Scholar |
[25] |
M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition,, Wave Motion, 47 (2010), 421.
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.
doi: 10.1063/1.1677083. |
[27] |
B. N. Parlett, "The Symmetric Eigenvalue Problem,'', Classics ed., (1998). Google Scholar |
[28] |
M. M. Popov, A new method of computation of wave fields using Gaussian beams,, Wave Motion, 4 (1982), 85.
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.
doi: 10.1016/j.jcp.2010.06.043. |
[30] |
J. Ralston, Gaussian beams and the propagation of singularities,, in, 23 (1982), 206.
|
[31] |
D. Sholla and J. Tully, A generalized surface hopping method,, J. Chem. Phys., 109 (1998).
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.
doi: 10.1063/1.1675788. |
[33] |
J. Tully, Molecular dynamics with electronic transitions,, J. Chem. Phys., 93 (1990).
doi: 10.1063/1.459170. |
[34] |
N. Tanushev, Superpositions and higher order Gaussian beams,, Commun. Math. Sci., 6 (2008), 449.
|
[35] |
N. Tanushev, J. Qian and J. Ralston, Mountain waves and Gaussian beams,, SIAM J. Multiscale Model. Simul., 6 (2007), 688.
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. 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.
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.
doi: 10.1021/jp9805860. |
[39] |
D. Wei and X. Yang, Gaussian beam method for high frequency wave propagation in heterogeneous media,, preprint 2011., (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.
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.
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). Google Scholar |
[43] |
C. Zener, Non-adiabatic crossing of energy levels,, Proceedings of the Royal Society of London, 137 (1932), 692. Google Scholar |
show all references
References:
[1] |
M. Baer, "Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections,'', Wiley, (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.
doi: 10.1006/jcph.2001.6956. |
[3] |
M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln,, Ann. Phys., 84 (1927), 457.
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.
doi: 10.1063/1.1421635. |
[5] |
M. V. Berry, Quantal phase factors accompanying adiabatic changes,, Proc. R. Soc. Lond Ser. A, 392 (1984), 45.
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. Google Scholar |
[7] |
K. Drukker, Basics of surface hopping in mixed quantum/classical simulations,, J. Comp. Phys., 153 (1999), 225.
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.
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.
doi: 10.1063/1.459848. |
[10] |
N. R. Hill, Gaussian beam migration,, Geophys., 55 (1990), 1416.
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.
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. Google Scholar |
[13] |
S. Jin and K. Novak, A semiclassical transport model for thin quantum barriers,, Multiscale Modeling and Simulation, 5 (2006), 1063.
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.
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.
|
[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.
|
[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.
|
[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. Google Scholar |
[19] |
L. Landau, Zur Theorie der Energiebertragung II,, Physics of the Soviet Union, 2 (1932), 46. Google Scholar |
[20] |
C. F. Kammerer and C. Lasser, Wigner measures and codimension two crossings,, Jour. Math. Phys., 44 (2003), 507.
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. 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.
|
[23] |
S. Leung, J. Qian and R. Burridge, Eulerian Gaussian beams for high frequency wave propagation,, Geophysics, 72 (2007), 61.
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. Google Scholar |
[25] |
M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition,, Wave Motion, 47 (2010), 421.
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.
doi: 10.1063/1.1677083. |
[27] |
B. N. Parlett, "The Symmetric Eigenvalue Problem,'', Classics ed., (1998). Google Scholar |
[28] |
M. M. Popov, A new method of computation of wave fields using Gaussian beams,, Wave Motion, 4 (1982), 85.
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.
doi: 10.1016/j.jcp.2010.06.043. |
[30] |
J. Ralston, Gaussian beams and the propagation of singularities,, in, 23 (1982), 206.
|
[31] |
D. Sholla and J. Tully, A generalized surface hopping method,, J. Chem. Phys., 109 (1998).
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.
doi: 10.1063/1.1675788. |
[33] |
J. Tully, Molecular dynamics with electronic transitions,, J. Chem. Phys., 93 (1990).
doi: 10.1063/1.459170. |
[34] |
N. Tanushev, Superpositions and higher order Gaussian beams,, Commun. Math. Sci., 6 (2008), 449.
|
[35] |
N. Tanushev, J. Qian and J. Ralston, Mountain waves and Gaussian beams,, SIAM J. Multiscale Model. Simul., 6 (2007), 688.
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. 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.
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.
doi: 10.1021/jp9805860. |
[39] |
D. Wei and X. Yang, Gaussian beam method for high frequency wave propagation in heterogeneous media,, preprint 2011., (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.
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.
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). Google Scholar |
[43] |
C. Zener, Non-adiabatic crossing of energy levels,, Proceedings of the Royal Society of London, 137 (1932), 692. Google Scholar |
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