fiend 1.0.2

Solving the Schrödinger equation

The time dependent Schrödinger equation (TDSE) (in Hartree atomic units) is

where is the time-evolving state, the initial state, the time-dependent Hamiltonian operator whose time-independent part is given by with the stationary potential .

The stationary states of the time-independent Hamiltonian can be solved from the time-independent Schrödinger equation (TISE)

where and are the ith eigenenergy and eigenstate.

In a typical setup, the initial state for the time propagation, , is one of the stationary states.

Theoretical background

For description on how TISE and TDSE have been discretized, please see MyArticle.

In short, we describe our system in cylindrical coordinates assuming zero magnetic quantum number . This allows us to describe the system in a 2D slice of the cylindrical coordiante system. We select simulation domains and for TISE and TDSE and mesh them with unstructured triangular meshes. Note that the simulation domains can be different so you can, e.g., increase the simulation domain for TDSE calculations if need be.

After meshing, one selects the basis functions. They are Lagrange polynomials up to some degree n with compact support on the mesh cells. Meshing and choosing the basis allows us to discretize the Schrödinger equations to

and

where are vectors of the real-valued expansion coefficients of the stationary states, is a vector of the complex-valued expansion coefficients of the time-dependent state, and the system matrices in the Lagrange polynomial basis are given by

and

Here the natural inner product is

Note that we have emposed continuity boundary condition at and either zero Dirichlet or zero Neumann boundary conditions elsewhere on the boundary.

Interactions

By default we implement three types of laser-matter interactions :

the dipole approximation in the length gauge for linearly polarized vector potentials

the dipole approximation in the length gauge for linearly polarized vector potentials ,

and nonhomogeneous vector potentials

Step 1. Solving TISE

Time-independent system should always be solved first. Make sure that you have loaded environment with a FEniCS installation supporting HDF5, PETSc, and SLEPc. In the pre-installed Docker image, you can use the command ml petsc/real.

TISE can be solved with the script fiend_linpol_tise. It solves an atomic system within the single active electron approximation, and the atomic potential can be selected with the command line option --atom_type.

For a full list of options run fiend_linpol_tise --help. Examples can be found in demos/hhg/step1.sh and demos/hhg/step2.sh.

New static potentials can be implemented in fiend.utils.predefined_potentials, please see the module source for further details.

solve_tise saves the following files:

• data/tise_mesh.h5 which includes the mesh used for solving TISE.

• data/tise_eigenvalues which is a list of eigenenergies

• data/tise_states.h5 which contains the expansion coefficients for all the stationary states

• data/config which includes the simulation parameters used for solving the TISE

Step 2. Preparing system matrices for TDSE

After solving TISE, you should set up matrices for time dependent simulations. For atomic systems in linearly polarized fields this can be accomplished with the script fiend_linpol_prepare_tdse, but for more complex setups see demos/nanotip/. Make sure that you have loaded a FEniCS installation with hdf5, petsc, and slepc enabled. In the Docker image this can be achieved with ml petsc/real.

The main feature of this step is that it can change the mesh from the TISE simulation. This allows you to solve TISE in a small simulation domain and use a larger simulation domain for time-dependent simulation. A full list of options can be obtained with fiend_linpol_prepare_tdse --help. If some of the options are not given, the values used in solving the TISE are used. So, e.g., if you only want to increase the radius of the meshed domain, supply only --radius X where X is the new radius.

Complex absorbing potentials (CAP) can be included in the time propagation simulations with options

• --cap_width which sets the width of the absorber from the domain boundary

• --cap_height which sets the strength of the absorber

Also other absorbers such as the smooth exterior complex scaling could easily be implemented.

prepare_tdse saves the following files using PETSc binary format

• data/tdse_CAP* containing the imaginary part of the CAP matrix

• data/tdse_H0* containing the time-independent part of the Hamiltonian matrix

• data/tdse_S* containing the overlap matrix

• data/tdse_rho* containing the -component of the dipole matrix

• data/tdse_Z* containing the z-component of the dipole matrix

• data/tdse_partialRho* containing matrix elements of

• data/tdse_partialZ* containing matrix elements of

• data/tdse_state_N* containing the vector representation of the stationary states in the TDSE mesh

In addition, fiend_linpol_prepare_tdse saves the new mesh in data/tdse_mesh.h5 and appends the new configuration parameters in data/config.

NOTE: There are three versions of the overlap matrix, S, S0, and S1. S is the pure overlap matrix, S0 has the rows corresponding to the Dirichlet boundaries zeroed, and S1 is like S0 but it has 1s on the diagonals of the zeroed rows. For all other matrices, the rows corresponding to the Dirichlet boundaries are always zeroed.

Step 3. Propagation

Now you must load Python packages petsc4py, slepc4py, and mpi4py with support for complex numbers. In the docker image this can be done with ml petsc/complex. Unfortunately, currently (as of version 2018.1.0) FEniCS doesn’t support PETSc with complex numbers so we must resort to two different versions of the python packages, but hopefully in the near future we can remedy this.

Time propagation can be achieved with the command fiend_linpol_propagate. It reads the matrices prepared in step 2 and uses those for time propagation. A complete list of options can be printed out with fiend_linpol_propagate --help.

Note that you can request saving frequency with --save_interval. E.g., --save_interval 10 saves every 10th time-step. Time-propagation saves data to

• data/tdse_observables.npz which includes an array for each observable (except the density)

• data/tdse_wavefunction/realpart_iteration_N_real which is a PETSc vector of the expansion coefficients for the real part of the wavefunction at saveslot N

• data/tdse_wavefunction/imagpart_iteration_N_real which is a PETSc vector of the expansion coefficients for the imaginary part of the wavefunction at saveslot N

• data/config where it appends new configuration options

Step 4. Post-processing

--save option for the post-processing scripts saves the figures to data/figures and data to data/postprocessing.

PETSc and SLEPc options

PETSc and SLEPc have numerous options to fine-tune their operation. All arguments passed to scripts after PETSC_ARGS are used to initialize the PETSc options database and the arguments before that remain in sys.argv and are used by the argumentparser of FIEND.

Using Docker

We recommend using Docker images for running simulations on your personal computer. For supercomputers and clusters, we recommend either installing the entire package from source or with pip.

To use the prepared Docker image, first install Docker <https://www.docker.com>_ to your PC. The docker image for fiend <https://hub.docker.com/r/solanpaa/fiend>_ can fetched with the command docker pull solanpaa/fiend. Please note that the image is a few GB in size.

By default, running the docker image for interactive process,

docker run -it fiend

drops you in a Unix shell. All the scripts are pre-installed, and you can switch between the complex and real PETSc installations with commands ml petsc/real and ml petsc/complex.

To enable GUIs for the scripts, you should launch the docker container with the commands executed on the HOST machine (works on Linux)

$ xhost +local:docker

$ docker run -e DISPLAY=$DISPLAY -v /tmp/.X11-unix:/tmp/.X11-unix:ro -it fiend

As you probably wish to save the datafiles computed within the container, you should create a directory data on your HOST and mount it to the container with

$ docker run -e DISPLAY=$DISPLAY -v /tmp/.X11-unix:/tmp/.X11-unix:ro -v `pwd`/data:/home/fiend/data:Z -it fiend

This also allows you to provide the laser parameters by saving it to a file within data on the HOST and passing the argument --vecpot data/filename to docker run fiend fiend_linpol_propagate.

For non-interactive use, you can pass the above command arguments corresponding to the script you would like to execute together with its arguments, e.g.,

$ docker run -e DISPLAY=$DISPLAY -v /tmp/.X11-unix:/tmp/.X11-unix:ro -v `pwd`/data:/home/fiend/data:Z fiend fiend_linpol_tise --radius 10 --how_many 3

Parallelization of the linear algebra backend can be achieved with the flag -e OPENBLAS_NUM_THREADS N of docker run. MPI-parallelized simulations can be achieved with the environment variable NMPIPROC, e.g.,

$ docker run -e NMPIPROC=3 -e DISPLAY=$DISPLAY -v /tmp/.X11-unix:/tmp/.X11-unix:ro -v `pwd`/data:/home/fiend/data:Z fiend fiend_linpol_tise --radius 10 --how_many 3

Installation with pip

pip3 install fiend should do the trick. Note that you have to manually install real and complex PETSc, SLEPc, petsc4py, slepc4py, and FEniCS suite.

Installation from sources

The fiend package is hosted at GitLab <https://gitlab.com/qcad.fi/fiend>_.

python3 setup.py install should install the package. Note that you have to manually install real and complex PETSc, SLEPc, petsc4py, slepc4py, and FEniCS suite.