FBTS_MPI¶
Purpose of Module¶
The FBTSMPI module implements the ForwardBackward Trajectory Solution (FBTS) to the quantumclassical Liouville equation [KapralCiccotti1999] developed by Hsieh and Kapral [HsiehKapral2012], [HsiehKapral2013].
In the case of a manybody system that can be partitioned into a quantum subsystem and classical environment, this module can be used in the calculation of timedependent observables. The purpose of this module is to provide an efficient and approximate method to study the nonadiabatic dynamics of these systems.
Background Information¶
In this approximate quantum dynamics method both the quantum subsystem and classicallike environment are transformed into a continuous phase space representation. This is achieved through a partial Wigner transform over the environmental degrees of freedom and a mapping representation for the quantum subsystem, wherein the subsystem degrees of freedom are represented by coherent state variables: .
Classicallike equations of motion are then used to evolve an ensemble of Monte Carlo sampled trajectories through time and the matrix elements of the average value of a timedependent operator, (having undergone the Wigner transform):
is calculated by the FBTS method using,
where and .
Applications¶
The particular system that this FBTSMPI module has been built for is in the study of excitation energy transfer in biological light harvesting systems, socalled proteinpigment complexes, through the use of the Frenkel exciton model. [IshizakiFleming2009PNAS] The total Hamiltonian of this system is: .
In this model the quantum subsystem of interest, , is the electronic excited states of the pigment molecules, the surrounding vibrational environment, , is represented as a collection of harmonic oscillatorsand the interaction between the two, , is characterized by the spectral density.
Specifically, the subsystem Hamiltonian is built such that the diagonal elements is the site energy, of a particular pigment, j, with the coupling between the pigments on the diagonals, :
The Hamiltonian of the bath is written as, where N is the total number of bath oscillators:
Lastly, the coupling Hamiltonian:
In this module an approximate form of the spectral density is used, known as the Debye spectral density given below:
The initial application for this module is in examining the mechanisms of exciton transport, which can be studied through the timedependent exciton site populations for a given lightharvesting complex. The approximate nature of this dynamics method combined with the parallelization of the trajectory ensemble allows one to model exciton transport in large systems with many pigments that would otherwise be prohibitively expensive to simulate.
Building and Testing¶
In order to compile this module, two files are required, FBTS_MPI.f90
and luxury.f90
, one contains
the FBTS method and the other returns a random number. Both of these files are located
in the ./source
subdirectory and can be compiled using:
mpifort FBTS_MPI.f90 luxury.f90 o FBTS_MPI.x
Upon successful compilation of the code execution of the code requires two input files, one containing relevant information concerning the simulation and the subsystem Hamiltonian matrix in units of wavenumbers.
The file Input_Data.dat
contains the simulation parameters and can be easily modified.
The number of states of the system, the state in which the initial excitation will occur
and the number of trajectories this module will complete can be changed.
The influence of the bath can also be adjusted through the parameters that will define the
Debye spectral density, the characteristic frequency of the bath, :math:omega_c,
the reorganization energy and the number of bath oscillators.
There are three parameters that concern the time length of the simulation, num_timestep
,
timestep
and timestep_block
. The total time length of the simulation is determined
by: num_timestep * timestep
. The parameter timestep_block
determines at what interval
the timedependent observables will be calculated and collected.
An example of this Input_Data.dat
file and subsystem Hamiltonian matrix can be found in
the ./tests/Dimer_Model
subdirectory. In order to test the code move the executable
to the this subdirectory and compare the output site populations against the exact results
from [IshizakiFleming2009] Figure 4(b).
Remember that the output provided by the module is given in atomic units of time and must be converted
to femtoseconds to compare.
Another model is provided for testing, the light harvesting complex known as the FennaMatthewsOlson (FMO) complex that contains 7 states, the exact results are from [WilkinsDattani2015].
The output from the FBTS_MPI module should be in good agreement to the exact results.
Source Code¶
The FBTS_MPI module source code is located at: FBTS_MPI.
References¶
[KapralCiccotti1999] 

[HsiehKapral2012] 

[HsiehKapral2013] 

[IshizakiFleming2009] 

[IshizakiFleming2009PNAS] 

[WilkinsDattani2015] 
