18,078 research outputs found
Global integration of the Schr\"odinger equation: a short iterative scheme within the wave operator formalism using discrete Fourier transforms
A global solution of the Schr\"odinger equation for explicitly time-dependent
Hamiltonians is derived by integrating the non-linear differential equation
associated with the time-dependent wave operator. A fast iterative solution
method is proposed in which, however, numerous integrals over time have to be
evaluated. This internal work is done using a numerical integrator based on
Fast Fourier Transforms (FFT). The case of a transition between two potential
wells of a model molecule driven by intense laser pulses is used as an
illustrative example. This application reveals some interesting features of the
integration technique. Each iteration provides a global approximate solution on
grid points regularly distributed over the full time propagation interval.
Inside the convergence radius, the complete integration is competitive with
standard algorithms, especially when high accuracy is required.Comment: 25 pages, 14 figure
Calculation of Densities of States and Spectral Functions by Chebyshev Recursion and Maximum Entropy
We present an efficient algorithm for calculating spectral properties of
large sparse Hamiltonian matrices such as densities of states and spectral
functions. The combination of Chebyshev recursion and maximum entropy achieves
high energy resolution without significant roundoff error, machine precision or
numerical instability limitations. If controlled statistical or systematic
errors are acceptable, cpu and memory requirements scale linearly in the number
of states. The inference of spectral properties from moments is much better
conditioned for Chebyshev moments than for power moments. We adapt concepts
from the kernel polynomial approximation, a linear Chebyshev approximation with
optimized Gibbs damping, to control the accuracy of Fourier integrals of
positive non-analytic functions. We compare the performance of kernel
polynomial and maximum entropy algorithms for an electronic structure example.Comment: 8 pages RevTex, 3 postscript figure
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