- -

Exponential propagators for the Schrödinger equation with a time-dependent potential

RiuNet: Institutional repository of the Polithecnic University of Valencia

Share/Send to

Cited by

Statistics

Exponential propagators for the Schrödinger equation with a time-dependent potential

Show full item record

Bader, P.; Kopylov, N.; Blanes Zamora, S. (2018). Exponential propagators for the Schrödinger equation with a time-dependent potential. The Journal of Chemical Physics. 149(24):1-7. https://doi.org/10.1063/1.5036838

Por favor, use este identificador para citar o enlazar este ítem: http://hdl.handle.net/10251/120355

Files in this item

Item Metadata

Title: Exponential propagators for the Schrödinger equation with a time-dependent potential
Author: Bader, Philipp Kopylov, Nikita Blanes Zamora, Sergio
UPV Unit: Universitat Politècnica de València. Departamento de Matemática Aplicada - Departament de Matemàtica Aplicada
Issued date:
Abstract:
[EN] We consider the numerical integration of the Schrodinger equation with a time-dependent Hamiltonian given as the sum of the kinetic energy and a time-dependent potential. Commutator-free (CF) propagators are exponential ...[+]
Copyrigths: Reserva de todos los derechos
Source:
The Journal of Chemical Physics. (issn: 0021-9606 )
DOI: 10.1063/1.5036838
Publisher:
American Institute of Physics
Publisher version: http://doi.org/10.1063/1.5036838
Project ID:
info:eu-repo/grantAgreement/MINECO//MTM2016-77660-P/ES/NUEVOS RETOS EN INTEGRACION NUMERICA: FUNDAMENTOS ALGEBRAICOS, METODOS DE ESCISION, METODOS DE MONTECARLO Y OTRAS APLICACIONES/
info:eu-repo/grantAgreement/GVA//GRISOLIA%2F2015%2FA%2F137/
Thanks:
We wish to acknowledge Fernando Casas for his help in the construction of the methods Upsilon<INF>3</INF><SUP>[6]</SUP>. The authors acknowledge Ministerio de Economia y Competitividad (Spain) for financial support through ...[+]
Type: Artículo

References

Bader, P., Iserles, A., Kropielnicka, K., & Singh, P. (2014). Effective Approximation for the Semiclassical Schrödinger Equation. Foundations of Computational Mathematics, 14(4), 689-720. doi:10.1007/s10208-013-9182-8

Bader, P., Iserles, A., Kropielnicka, K., & Singh, P. (2016). Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2193), 20150733. doi:10.1098/rspa.2015.0733

Lubich, C. (2008). From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. doi:10.4171/067 [+]
Bader, P., Iserles, A., Kropielnicka, K., & Singh, P. (2014). Effective Approximation for the Semiclassical Schrödinger Equation. Foundations of Computational Mathematics, 14(4), 689-720. doi:10.1007/s10208-013-9182-8

Bader, P., Iserles, A., Kropielnicka, K., & Singh, P. (2016). Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2193), 20150733. doi:10.1098/rspa.2015.0733

Lubich, C. (2008). From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. doi:10.4171/067

Feit, M. ., Fleck, J. ., & Steiger, A. (1982). Solution of the Schrödinger equation by a spectral method. Journal of Computational Physics, 47(3), 412-433. doi:10.1016/0021-9991(82)90091-2

Tremblay, J. C., & Carrington, T. (2004). Using preconditioned adaptive step size Runge-Kutta methods for solving the time-dependent Schrödinger equation. The Journal of Chemical Physics, 121(23), 11535-11541. doi:10.1063/1.1814103

Sanz‐Serna, J. M., & Portillo, A. (1996). Classical numerical integrators for wave‐packet dynamics. The Journal of Chemical Physics, 104(6), 2349-2355. doi:10.1063/1.470930

Kormann, K., Holmgren, S., & Karlsson, H. O. (2008). Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian. The Journal of Chemical Physics, 128(18), 184101. doi:10.1063/1.2916581

Peskin, U., Kosloff, R., & Moiseyev, N. (1994). The solution of the time dependent Schrödinger equation by the (t,t’) method: The use of global polynomial propagators for time dependent Hamiltonians. The Journal of Chemical Physics, 100(12), 8849-8855. doi:10.1063/1.466739

Castro, A., Marques, M. A. L., & Rubio, A. (2004). Propagators for the time-dependent Kohn–Sham equations. The Journal of Chemical Physics, 121(8), 3425-3433. doi:10.1063/1.1774980

Blanes, S., Casas, F., & Murua, A. (2017). Symplectic time-average propagators for the Schrödinger equation with a time-dependent Hamiltonian. The Journal of Chemical Physics, 146(11), 114109. doi:10.1063/1.4978410

Blanes, S., Casas, F., & Murua, A. (2015). An efficient algorithm based on splitting for the time integration of the Schrödinger equation. Journal of Computational Physics, 303, 396-412. doi:10.1016/j.jcp.2015.09.047

Gray, S. K., & Verosky, J. M. (1994). Classical Hamiltonian structures in wave packet dynamics. The Journal of Chemical Physics, 100(7), 5011-5022. doi:10.1063/1.467219

McLachlan, R. I., & Quispel, G. R. W. (2002). Splitting methods. Acta Numerica, 11, 341-434. doi:10.1017/s0962492902000053

Neuhauser, C., & Thalhammer, M. (2009). On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential. BIT Numerical Mathematics, 49(1), 199-215. doi:10.1007/s10543-009-0215-2

Thalhammer, M. (2008). High-Order Exponential Operator Splitting Methods for Time-Dependent Schrödinger Equations. SIAM Journal on Numerical Analysis, 46(4), 2022-2038. doi:10.1137/060674636

Thalhammer, M. (2012). Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations. SIAM Journal on Numerical Analysis, 50(6), 3231-3258. doi:10.1137/120866373

Gray, S. K., & Manolopoulos, D. E. (1996). Symplectic integrators tailored to the time‐dependent Schrödinger equation. The Journal of Chemical Physics, 104(18), 7099-7112. doi:10.1063/1.471428

Sanz-Serna, J. M., & Calvo, M. P. (1994). Numerical Hamiltonian Problems. doi:10.1007/978-1-4899-3093-4

Saad, Y. (1992). Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator. SIAM Journal on Numerical Analysis, 29(1), 209-228. doi:10.1137/0729014

Park, T. J., & Light, J. C. (1986). Unitary quantum time evolution by iterative Lanczos reduction. The Journal of Chemical Physics, 85(10), 5870-5876. doi:10.1063/1.451548

Blanes, S., & Moan, P. C. (2000). Splitting methods for the time-dependent Schrödinger equation. Physics Letters A, 265(1-2), 35-42. doi:10.1016/s0375-9601(99)00866-x

Magnus, W. (1954). On the exponential solution of differential equations for a linear operator. Communications on Pure and Applied Mathematics, 7(4), 649-673. doi:10.1002/cpa.3160070404

Blanes, S., Casas, F., Oteo, J. A., & Ros, J. (2009). The Magnus expansion and some of its applications. Physics Reports, 470(5-6), 151-238. doi:10.1016/j.physrep.2008.11.001

Blanes, S., & Moan, P. C. (2006). Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems. Applied Numerical Mathematics, 56(12), 1519-1537. doi:10.1016/j.apnum.2005.11.004

Thalhammer, M. (2006). A fourth-order commutator-free exponential integrator for nonautonomous differential equations. SIAM Journal on Numerical Analysis, 44(2), 851-864. doi:10.1137/05063042

Alvermann, A., & Fehske, H. (2011). High-order commutator-free exponential time-propagation of driven quantum systems. Journal of Computational Physics, 230(15), 5930-5956. doi:10.1016/j.jcp.2011.04.006

Auer, N., Einkemmer, L., Kandolf, P., & Ostermann, A. (2018). Magnus integrators on multicore CPUs and GPUs. Computer Physics Communications, 228, 115-122. doi:10.1016/j.cpc.2018.02.019

Munthe–Kaas, H., & Owren, B. (1999). Computations in a free Lie algebra. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 357(1754), 957-981. doi:10.1098/rsta.1999.0361

Bader, P., Blanes, S., Ponsoda, E., & Seydaoğlu, M. (2017). Symplectic integrators for the matrix Hill equation. Journal of Computational and Applied Mathematics, 316, 47-59. doi:10.1016/j.cam.2016.09.041

P.V. Koseleff, “Formal calculus for Lie methods in Hamiltonian mechanics,” Ph.D. thesis, Lawrence Berkeley Laboratory, 1994.

Chin, S. A. (1997). Symplectic integrators from composite operator factorizations. Physics Letters A, 226(6), 344-348. doi:10.1016/s0375-9601(97)00003-0

Omelyan, I. P., Mryglod, I. M., & Folk, R. (2002). Construction of high-order force-gradient algorithms for integration of motion in classical and quantum systems. Physical Review E, 66(2). doi:10.1103/physreve.66.026701

Walker, R. B., & Preston, R. K. (1977). Quantum versus classical dynamics in the treatment of multiple photon excitation of the anharmonic oscillator. The Journal of Chemical Physics, 67(5), 2017. doi:10.1063/1.435085

Iserles, A., Kropielnicka, K., & Singh, P. (2018). Magnus--Lanczos Methods with Simplified Commutators for the Schrödinger Equation with a Time-Dependent Potential. SIAM Journal on Numerical Analysis, 56(3), 1547-1569. doi:10.1137/17m1149833

[-]

recommendations

 

This item appears in the following Collection(s)

Show full item record