kazim Opening Report on boundary coinditions.pptx

IMPLEMENTATION OF TRANSPARENT(EXACT) BOUNDARY
CONDITIONS IN REACTIVE SCATTERING PROBLEMS
(Opening Report)
State Key Laboratory of Reaction Molecular Dynamics &
Centre for Theoretical and Computational Chemistry
Name: Syed Kazim Usman
Supervisor: Prof. Dr. Zhigang Sun
Group: 1102

Contents
 A high-accurate transparent boundary conditions for the time-dependent
quantum wave packet method by Yin Huang, Hailin Zhao, Syed Kazim
Usman, Zhigang Sun (2019)
Literature Review
 Discrete Transparent Boundary Conditoions for Schrodinger Equation by
Mathias Ehrhardt, Anton Arnold
 Absorbing Boundary Conditions for the Schrodinger Equation on Finite
Intervals by J. P. Kuska (1992)
 Almost Exact Boundary Condition for 1-D Schrodinger Equation
by Gang Pang, Lei Bian, and Shaqiang Tang (2012)
 Numerov Extension of Transparent Boundary Conditons for 1-D
Schrodinger Equation by Curt. A Moyer (2003)

Literature Review
Simulation of reactive scattering processes through time-
dependent wave packet method involves the following
steps:
 Define an initial wave packet (function)
 Choose a suitable grid representation in a certain
coordinate system
 Choose an efficient time propagator to evolve the wave
packet. From the evolving wave packet, we can extract
all the scattering information.

Cont.
There are three approaches to the discretization of the spatial coordinates
of the Hamiltonian:
 Local methods, such as finite difference methods
 Global methods, includes all of the spectral methods and the
corresponding grids methods
 The Spectral element methods
 Besides these three spatial discretization methods, we also have higher
order finite difference method, spectral difference method, and the
distributed approximation functional (DAF) method
 The Time-dependent methods can calculate the total reaction
probability by evaluating the reactive flux at the dividing and simply use
an absorbing potential to absorb the wavefunction beyond the
transition state region.
Following are the methods for time Propagation.
 Crank Nicolson Method
 Split operator method
 Chebyshev polynomial Method
 Short iterative Lanczos method
 Gaussian Wavepackets

Schrodinger Equation
The motion of a particle with the mass in the interval may be
described by the one-dimensional Schrodinger equation as

Boundary Conditions
In a solution to time dependent Schrodinger Equation using spatial
discretization method, the spatial grid range has to be limited.
However, during the propagation time, the wave function usually can
extend out of the grid. We have to artificially impose the boundary
condition, which is capable of damping the wave function before reaching
the end of the grid, but without affecting the interior wave function. A
usual choice is by including the absorbing potential in the potential
operator or by imposing damping function to the wave function.
Smooth exterior complex scaling method was reported to be a more
accurate way to impose the boundary conditions in a numerical solution
to TD Schrodinger equation, which has been applied in a calculation of
electronic dynamics included by ultra-short laser pulses.

Methods For Boundary Treatment
In order to avoid affection of the undesirable spurious reflections at the
boundaries in the long time period, in a modern time-dependent wave
packet calculations, usually a complex absorbing potential (CAP) of
the for or and equivalent damping function is introduced near the
edge of the grid to attenuate the wave function gradually.
Perfectly Matched layers (PML)
exterior complex Scaling (ECS)
smooth exterior scaling (SES)
Drawback
When the distribution of the translational energy in the wave packet is of
broad range, especially involving ultra-slow translational energy, the
aforementioned techniques become very inefficient.

High-Accurate Transparent Boundary Conditions For
Time-Dependent Quantum Wave Packet Method

Cont.
The wave function (Blue lines) at
a series of time (A) propagated
under the potential energy (B),
travelling from right to left. The
red lines in the boundary regions
are calculated by the CG-TBC,
which are used to update the
wave function around
boundaries. The grey area in
panel (A) indicates the CAP

Cont.
A) and C): Total transmission probabilities as a function of collision energy,
calculated using the flux method (M1), amplitude method (M2) and
correlation function method (M3) with the CG-TBC and transmission
probabilities energy 0.36912 eV calculated using the CG-TBC and CAP.
B) and D): The relative error in log scale as a function of propagation time

Cont.
Comparison of the total transmission
probabilities: (A), the real (B) and
imaginary part (C) of the S-matrix, as
a function of collision energy
calculated using the CG-TBC and the
CAP with boundary regions of length
of 50 a.u. The enlarged plots in the
subsets of panel A) clearly demonstrate
the rationality of the CAP strength

Discrete Transparent Boundary Conditoions For
Schrodinger Equation By Mathias Ehrhardt & Anton Arnold

Cont.
The discretized Schrodinger equation reads:
With the uniform grid points , and the approximations .
The crank-Nicolson scheme (3.1) can be written in the form:
With

Cont.…
Theorem
The left (at ) and right (at ) discrete transparent boundary conditions for the Crank-Nicolson discretization
(3.1) if the 1D Schrodinger equation are respectively
With
(A)
, ,
, ,
denotes the Legendre polynomials ans the Kronecker symbol.
are called convolution coefficients.
The recurrence formula for the summed coefficients is as follows for :
These can be used after calculating the first values for by the formula (A).

Kuska Transparent Boundary Conditions

Almost Exact Transparent Boundary Conditions

Plan-I
 Spectral element (FE) DVR method for angular degree of
freedom in triatomic scattering.
 The method usually results a well banded Hamiltonian matrix
for angular kinetic operator.
 Another feature of the method is to have high accuracy.
 It will be assured that it more effective than using the DVR
method with Legendre polynomials
 Relative performance of the method will be checked and
compared with other existing methods

kazim Opening Report on boundary coinditions.pptx

References
[1]. R. T Pack, G. A. Parker, JCP (87) 3888 (1987)
[2].J. Crank and P. Nicolson, Proc. Cambridge Philos. Soc 43, 50 (1947). [3].R. Kosloff, J.
Phys. Chem. 92, 2087 (1988)
[4]. J. P. Boyd, Chebyshev and Fourier Spectral Methods, Lecture Notes in Engineering, vol. 49
(Springer, New York, 1989)
[5]. D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and
Applications (SIAM, Philadelphia, PA, 1997)
[6]. J. A. F. Jr., J. R. Morris, and M. D. Feit, Ann. Phys. 10, 129 (1976).
[7] Z. G. Sun, D. H. Zhang, and W. T. Yang, Phys. Chem. Chem. Phys. 14, 1817 (2012).
[8]. S. K. Gray and G. G. Balint-Kurti, J. Chem. Phys. 108, 950 (1998).
[9]. J. Park and J. C. Light, J. Chem. Phys. 85, 5870 (1986)
[10] J. Crank and P. Nicolson, Proc. Cambridge Philos. Soc 43, 50 (1947)
[11] B. Podolsky, Phys. Rev. 32, 812 (1928)
[12] J. Makarewicz, J. Phys. B 21, 1803 (1988)

kazim Opening Report on boundary coinditions.pptx

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