zbMATH — the first resource for mathematics

Quantum dynamics with the parallel transport gauge. (English) Zbl 1440.65113
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q55 NLS equations (nonlinear Schrödinger equations)
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
81V70 Many-body theory; quantum Hall effect
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35B25 Singular perturbations in context of PDEs
35Q49 Transport equations
82M36 Computational density functional analysis in statistical mechanics
78A60 Lasers, masers, optical bistability, nonlinear optics
81V55 Molecular physics
Full Text: DOI
[1] D. G. Anderson, Iterative procedures for nonlinear integral equations, J. Assoc. Comput. Mach., 12 (1965), pp. 547-560. · Zbl 0149.11503
[2] J. E. Avron and A. Elgart, Adiabatic theorem without a gap condition, Comm. Math. Phys., 203 (1999), pp. 445-463, https://doi.org/10.1007/s002200050620. · Zbl 0936.47047
[3] W. Bao, S. Jin, and P. A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), pp. 487-524. · Zbl 1006.65112
[4] F. A. Bornemann and C. Schütte, On the singular limit of the quantum-classical molecular dynamics model, J. Appl. Math., 59 (1999), pp. 1208-1224. · Zbl 0926.34073
[5] H. Candy and W. Rozmus, A symplectic integration algorithm for separate Hamiltonian functions, J. Comput. Phys., 92 (1991), pp. 230-256. · Zbl 0709.70012
[6] A. Castro, M. Marques, and A. Rubio, Propagators for the time-dependent Kohn-Sham equations, J. Chem. Phys., 121 (2004), pp. 3425-3433.
[7] Z. Chen and E. Polizzi, Spectral-based propagation schemes for time-dependent quantum systems with application to carbon nanotubes, Phys. Rev. B, 82 (2010), 205410.
[8] D. Cohen, T. Jahnke, K. Lorenz, and C. Lubich, Numerical integrators for highly oscillatory Hamiltonian systems: A review, in Analysis, Modeling and Simulation of Multiscale Problems, Springer, New York, 2006, pp. 553-576. · Zbl 1367.65191
[9] H. D. Cornean, D. Monaco, and S. Teufel, Wannier functions and \({\mathbb{Z}}_2\) invariants in time-reversal symmetric topological insulators, Rev. Math. Phys., 29 (2017), 1730001. · Zbl 1370.81081
[10] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Courier Corp., North Chelmsford, MA, 2003. · Zbl 1191.70001
[11] E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer-Verlag, Berlin, Heidelberg, 2006. · Zbl 1094.65125
[12] E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equation I: Nonstiff Problems, Springer Ser. Comput. Math. 8, Springer, Berlin, Heidelberg, 1987. · Zbl 0638.65058
[13] E. Hairer and G. Wanner, Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems, Springer Ser. Comput. Math. 14, Springer, Berlin, Heidelberg, 1991. · Zbl 0729.65051
[14] D. R. Hamann, Optimized norm-conserving Vanderbilt pseudopotentials, Phys. Rev. B, 88 (2013), 085117.
[15] W. Hu, L. Lin, and C. Yang, DGDFT: A massively parallel method for large scale density functional theory calculations, J. Chem. Phys., 143 (2015), 124110.
[16] W. Humphrey, A. Dalke, and K. Schulten, VMD – Visual Molecular Dynamics, J. Molec. Graphics, 14 (1996), pp. 33-38.
[17] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge Texts Appl. Math. 44, Cambridge University Press, Cambridge, UK, 2009. · Zbl 1171.65060
[18] T. Jahnke and C. Lubich, Numerical integrators for quantum dynamics close to the adiabatic limit, Numer. Math., 94 (2003), pp. 289-314. · Zbl 1029.65069
[19] J. Jia and J. Huang, Krylov deferred correction accelerated method of lines transpose for parabolic problems, J. Comput. Phys., 227 (2008), pp. 1739-1753. · Zbl 1134.65064
[20] C. F. Kammerer and A. Joye, Nonlinear Quantum Adiabatic Approximation, preprint, https://arxiv.org/abs/1906.11069, 2019.
[21] A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), pp. 1214-1233, https://doi.org/10.1137/S1064827502410633. · Zbl 1077.65105
[22] T. Kato, On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Japan, 5 (1950), pp. 435-439.
[23] C. T. Kelley, Iterative Methods for Optimization, Front. Appl. Math. 18, SIAM, Philadelphia, 1999, https://doi.org/10.1137/1.9781611970920.
[24] O. Koch and C. Lubich, Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434-454, https://doi.org/10.1137/050639703. · Zbl 1145.65031
[25] L. Lin, J. Lu, L. Ying, and W. E, Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation, J. Comput. Phys., 231 (2012), pp. 2140-2154. · Zbl 1251.82008
[26] L. Lin and C. Yang, Elliptic preconditioner for accelerating self-consistent field iteration in Kohn-Sham density functional theory, SIAM J. Sci. Comp., 35 (2013), pp. S277-S298, https://doi.org/10.1137/120880604. · Zbl 1284.82009
[27] C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), pp. 2141-2153. · Zbl 1198.65186
[28] M. Nakahara, Geometry, Topology and Physics, CRC Press, Boca Raton, FL, 2003. · Zbl 1090.53001
[29] G. Nenciu, Linear adiabatic theory. Exponential estimates, Comm. Math. Phys., 152 (1993), pp. 479-496, https://doi.org/10.1007/BF02096616. · Zbl 0768.34038
[30] P. Nettesheim, F. A.Bornemann, B. Schmidt, and C. Schütte, An explicit and symplectic integrator for quantum-classical molecular dynamics, Chem. Phys. Lett., 256 (1996), pp. 581-588.
[31] P. Nettesheim and C. Schütte, Numerical integrators for quantum-classical molecular dynamics, in Computational Molecular Dynamics: Challenges, Methods, Ideas (Berlin, 1997), Lect. Notes Comput. Sci. Eng. 4, Springer, Berlin, 1999, pp. 396-411. · Zbl 0966.81064
[32] G. Onida, L. Reining, and A. Rubio, Electronic excitations: Density-functional versus many-body Green’s-function approaches, Rev. Mod. Phys., 74 (2002), 601.
[33] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett., 77 (1996), pp. 3865-3868.
[34] E. Runge and E. K. U. Gross, Density-functional theory for time-dependent systems, Phys. Rev. Lett., 52 (1984), 997.
[35] A. Russakoff, Y. Li, S. He, and K. Varga, Accuracy and computational efficiency of real-time subspace propagation schemes for the time-dependent density functional theory, J. Chem. Phys., 144 (2016), 204125.
[36] A. Schleife, E. W. Draeger, Y. Kanai, and A. A. Correa, Plane-wave pseudopotential implementation of explicit integrators for time-dependent Kohn-Sham equations in large-scale simulations, J. Chem. Phys., 137 (2012), 22A546.
[37] C. Sparber, Weakly nonlinear time-adiabatic theory, Ann. Henri Poincaré, 17 (2016), pp. 913-936. · Zbl 1337.81056
[38] S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, Springer-Verlag, Berlin, Heidelberg, 2003. · Zbl 1053.81003
[39] Z. Wang, S.-S. Li, and L.-W. Wang, Efficient real-time time-dependent density functional theory method and its application to a collision of an ion with a 2D material, Phys. Rev. Lett., 114 (2015), pp. 1-5.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.