Unitary quantum time evolution by iterative Lanczos reduction

Park, Tae Jun; Light, John C.

doi:10.1063/1.451548

Cited by 877 publications

(563 citation statements)

References 29 publications

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“…Letting m/M → 0 in the numerical method, it can be shown that the solution given by (21) tends to a small perturbation of the Verlet method formally applied to that equation:…”

Section: Quantum-classical Molecular Dynamicsmentioning

confidence: 99%

“…Integration schemes employing matrix functions apparently have not hitherto been used in practice, except in a few special cases where direct diagonalization is possible. However, since the mid-eighties, starting with a paper by Park and Light [21] on quantum propagators, Lanczos' method has been put to good use in approximating matrix-function vector products. More recently, the excellent convergence properties of this approach have been clarified in [7,12].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Bunch of Time Integrators for Quantum/Classical Molecular Dynamics

Hochbruck¹,

Lubich²

1999

Computational Molecular Dynamics: Challenges, Methods, Ideas

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Abstract. We present novel time integration schemes for Newtonian dynamics whose fastest oscillations are nearly harmonic, for constrained Newtonian dynamics including the Car-Parrinello equations of ab initio molecular dynamics, and for mixed quantum-classical molecular dynamics. The methods attain favorable properties by using matrix-function vector products which are computed via Lanczos' method. This permits to take longer time steps than in standard integrators.

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“…Letting m/M → 0 in the numerical method, it can be shown that the solution given by (21) tends to a small perturbation of the Verlet method formally applied to that equation:…”

Section: Quantum-classical Molecular Dynamicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Bunch of Time Integrators for Quantum/Classical Molecular Dynamics

Hochbruck¹,

Lubich²

1999

Computational Molecular Dynamics: Challenges, Methods, Ideas

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“…38,113,[115][116][117][118] The numerical solution of the time-dependent Schödinger equation has been carried out with an orthogonalised Krylov subspace method. 29,119 …”

Section: Coherent Hole Transfer In Dnamentioning

confidence: 99%

Vibronic couplings and coherent electron transfer in bridged systems

Borrelli

Capobianco

Landi

et al. 2015

Phys. Chem. Chem. Phys.

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A discrete state approach to the dynamics of coherent electron transfer processes in bridged systems, involving three or more electronic states, is presented. The approach is based on a partition of the Hilbert space of the time independent basis functions in subspaces of increasing dimensionality, which allows for checking the convergence of the time dependent wave function.Vibronic coupling are determined by Duschinsky analysis carried out over the normal modes of the redox partners obtained at high DFT computational level.

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“…1. This approach applies even better to evolving heavy-particle wave packets because the Chebyshev [16,23] or Lanczos [16,24] schemes can be used. These schemes are, first, more efficient than the split method and, second, allow one to take full advantage of the sparse structure of the Hamiltonian in the wavelet basis (see below).…”

Section: Introductionmentioning

confidence: 99%

“…For instance, in our example φ n,j |H|φ n,i = 0, if |i − j| ≥ 20. Finally, the time evolution can be computed by standard techniques such as split, Lanczos, or Chebychev methods [16,23,24]. If after the projection on wavelet towers, the Hamiltonian matrix is small enough for a direct diagonalization, Eq.…”

Section: Introductionmentioning

confidence: 99%

The wave packet propagation using wavelets

Borisov

Shabanov

2002

Chemical Physics Letters

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It is demonstrated that the wavelets can be used to considerably speed up simulations of the wave packet propagation in multiscale systems. Extremely high efficiency is obtained in the representation of both bound and continuum states. The new method is compared with the fast Fourier algorithm. Depending on ratios of typical scales of a quantum system in question, the wavelet method appears to be faster by a few orders of magnitude.1. Owing to the fast development of the computational tools the direct solution of the time-dependent Schroedinger equation has become one of the basic approaches to study the evolution of quantum systems. Thus, the wave packet propagation (WPP) method is successfully applied to time dependent and time-independent problems in gas-surface interactions, molecular and atomic physics, and quantum chemistry [1,2,3,4]. One of the main issues of the numerical approaches to the time-dependent Schroedinger equation is the representation of the wave function |Ψ(t) of the system. Development of the pseudospectral global grid representation approaches yielded a very efficient way to tackle this problem. The discrete variable representation (DVR) [5] and the Fourier grid Hamiltonian method (FGH) [6,7] have been widely used in time-dependent molecular dynamics [1,2,8], S-matrix [9,10] or eigenvalue [11] calculations. The FGH method based on the Fast Fourier Transform (FFT) algorithm is especially popular. This is because for the mesh of N points the evaluation of the kinetic energy operator requires only NlogN operations and can be easily implemented numerically. In the standard FGH method the wave function is represented on a grid of equidistant point in coordinate and momentum space. It is well suited when the the local de Broglie wavelength does not change much over the physical region spanned by the wave function of the system [11]. There are, however, lot of examples where the system under consideration spans the physical regions with very different properties. Consider, for example, the scattering of a slow particle on a narrow and deep potential well. Despite the short wave lengths occur only in the well, in the FGH method they would determine the lattice mesh over entire physical space leading to high computational cost. In fact, pseudospectral global grid representation approaches are difficult to use in multiscale problems. This is why much of work has been devoted recently to the development of the mapping procedures in order to enhance sampling efficiency in the regions of the rapid variation of the wave function [8,11,12,13,14,15]. Though mapping procedure, based on the variable change x = f (ξ) is very efficient in 1D, it is far from being universal. One obvious drawback 1 is that it is difficult to implement in higher dimensions. In the case of several variables, there is no simple procedure to define the mapping functions f so that the lattice would be fine only in some designated regions of space. Next, the topology of the new coordinate surfaces can be different from that ...

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Unitary quantum time evolution by iterative Lanczos reduction

Cited by 877 publications

References 29 publications

A Bunch of Time Integrators for Quantum/Classical Molecular Dynamics

A Bunch of Time Integrators for Quantum/Classical Molecular Dynamics

Vibronic couplings and coherent electron transfer in bridged systems

The wave packet propagation using wavelets

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