Partial Linearised Density Matrix (PLDM)

A collection of serial, vectorized and parallel version of PLDM dynamics. The reduced density matrix is propagated by propagating the MMST variables. The individual MMST variables form a forward ($|\psi_F\rangle$) and a backward wavefunction ($\langle\psi_B|$), which can be used to decompose the $ρ_{sys}$ dynamics ($\mathcal{O}$($N^2$)) into propagation of two wavefucntions ($\mathcal{O}$($N$)) which exponentially reduces the computational cost of large systems. We apply this on some model polaritonic HTC hamiltonians for large number of molecules and show the speedup of the current implementation compared to previous PLDM implementations. The code is structured as

Bath

• initBathParams.py generate the bath parameters $c_j$ and $\omega_j$ from a spectral density
• initBathMPI.py generates the the initial nuclei position and momenta for a trajectory

System

• model.py contains the system functions
  • system Hamiltonian (Hψ)
  • Dipole (μψ)
  • system-bath interation hamiltonian (H_SB)
• parameters.py contains the parameters of the system and also simulation parameters

Dynamics

• pldm.py contains the dynamics functions
  • Force → calculates the force on the bath DOF's from the system dynamics according to $$F(R_i)=c_j^i\sum_i\rho_{ii}$$
  • Umap → propagates the mapping variables according to chebyshev expansion $$e^{-i\frac{\hat{H}\Delta}{\hbar}}\ket{\Psi}=b_0(z)|\Psi^{(0)}\rangle+\sum_{n=1}^{\infty} \phi_n(z) b_n(z) |\Psi^{(n)}\rangle$$
  • velver → propagates the nuclei
• trajClass.py defines a trajectory class

Run

• operatorDynamics.py lets you define your arbitrary operator for dynamics calculation.
• initBathMPI.py generates the initial nuclei position and momentum coordiantes for each trajectory in a seperate trajectory folder within the "Data/" folder.
• runDynamics.sh and runInitBath.sh are the SLURM script files to submit the jobs

Data

• The folder where the trajectory simulation data will be generated. After the dynamics simulation is done, there will be one folder for each trajectory containing the nuclei R, P and mapping variables stored with time.

PostProcess

• postProcess.py combines the individual trajectory data and generates the ensemble averages initial density matrix.
• plotDynamics.py plots the population and coherence elements of the reduced density matrix -runPlot.sh is the SLRUM script to run plotDynamics.py.

Tests

• testHel.py tests the validity of $Hψ$ operation
• test_SB.py tests the validity of system-bath interaction part of dynamics
• umapTest.py tests the validity of chebyshev propagation

Dependencies

The code requires the mandatory intallation of following python packages:

• numpy
• scipy
• numba

Additional (optional) dependencies which can be used to efficiently vectorize and parallelize the dynamics:

• mpi4py
• cupy
• pyTorch
• cuda