Examples

There are four example scripts that come with quantumGrid: ecs_femdvr_time_indep_h2 for a time independent calculation and ecs_femdvr_time_dep_h2 for a time dependent calculation and two examples that calculate vibrational states of \(H_2\) and CO called femdvr_vib_states_h2 and femdvr_vib_states_co, respectively. Once quantumGrid is installed in your local environment, these scripts can be called without typing “python”. To see command line options for both scripts, just use the help command:

$ ecs_femdvr_time_dep_h2 --help

For example, to turn on plotting for the time independent example run the script with the plotting option:

$ ecs_femdvr_time_indep_h2 --want_to_plot=True

ecs_femdvr_time_indep_h2

For time-independent potential, this example implements Exterior Complex Scaling on the FEM-DVR contour. The value of R0 and the complex scale factor \(e^{I*theta}\) are specified. The representation of the potential must be able to be evaluated on the complex part of the contour.

Example:

Finds all eigenvalues of complex scaled Hamiltonian and plots any one of them, specified by n_Plot

Args:

  1. want_to_plot (bool): Optional command that turns on plotting; default is false.

Potentials defined here:

  1. Morse potential for \(H_2\)

  2. Bernstein fit of Kolos and Wolneiwicz potential with \(\frac{1}{R^6}\), \(\frac{1}{R^8}\), \(\frac{1}{R^{10}}\) asymptotic behavior – Gives near spectroscopic accuracy used in [1], results there are reproduced by this code.

ecs_femdvr_time_dep_h2

Finds all eigenvalues of complex scaled \(H_2\) Hamiltonian for nuclear motion plots any one of them, specified by n_Plot Then starts a Gaussian packet in the well (e.g. with \(j=17\)) and follows it as it separates into a part that is bound in the well and a part that dissociates and vanishes on the ECS contour.

Args:

  1. number_of_time_intervals (int): First command line argument. Number of time intervals to perform the Crank-Nicolson propagation; defaults to 300.

  2. time_step (int): Second command line argument. Time step in the propagator to relax or restrict the calculation as needed; defaults to 0.1 atu.

  3. want_to_plot_animate (bool): Optional command that turns on plotting; default is false.

Potentials defined here:

  1. Morse potential for \(H_2\)

  2. Bernstein fit of Kolos and Wolneiwicz potential with \(\frac{1}{R^6}\), \(\frac{1}{R^8}\), \(\frac{1}{R^{10}}\) asymptotic behavior – Gives near spectroscopic accuracy used in [1], results there are reproduced by this code.

femdvr_vib_states_h2

\(H_2\) vibrational states using CI singles and doubles potential curve from Psi4. This potential yields a \(n = 0 \rightarrow 1\) excitation energy within a few wavenumbers of the value using the NIST values for constants of diatomic molecules for \(H_2\) in the formula \(E_n = (n+\frac{1}{2})w_e - (n+\frac{1}{2})^2 w_ex_e\), which is \(4158 cm^{-1}\).

Shows how to

  1. Read in and interpolate a potential function known at discrete points

  2. Use FEMDVR class to build FEM-DVR grid

  3. Use FEMDVR class to build Hamiltonian in DVR basis

  4. Find eigenvalues and eigenvectors of Hamiltonian

  5. Plot eigenfunctions of the Hamiltonian

Args:

  1. want_to_plot (bool): Optional command that turns on plotting; default is false.

File read for this example:

  1. potcurve_CISD_H2_ccpvTZ.dat

femdvr_vib_states_co

CO vibrational states using CI singles, doubles and triples potential curve from Psi4. This potential gives a dissociation energy of \(~12.2 eV\), not very good by comparison to the \(~11.1 eV\) experimental value. It yields a \(n = 0 \rightarrow 1\) excitation energy of \(2207 cm^{-1}\) compared with the value using the NIST values for constants of diatomic molecules for \(H_2\) in the formula \(E_n = (n+\frac{1}{2})w_e - (n+\frac{1}{2})^2 w_ex_e\), which is \(2143 cm^{-1}\) So not quite spectroscopic accuracy.

Shows how to

  1. Read in and interpolate a potential function known at discrete points

  2. Use FEMDVR class to build FEM-DVR grid

  3. Use FEMDVR class to build Hamiltonian in DVR basis

  4. Find eigenvalues and eigenvectors of Hamiltonian

  5. Plot eigenfunctions of the Hamiltonian

Args:

  1. want_to_plot (bool): Optional command that turns on plotting; default is false.

File read for this example:

  1. potcurve_CISDT_CO_ccpvDZ.dat

time_dep_two_potential_excitation

In this example we demonstrate excitation from one potential curve (electronic state) to another in a diatomic molecule by a finite pulse using the FEM-DVR grid. The dipole matrix element between the two states is assumed to be constant as a function of internuclear distance. Potential curves are Morse oscillator functions with different well depths and shapes shifted by 0.15 hartrees. Reduced mass is reduced mass of H2. Note that a denser DVR grid may be necessary for heavier masses. Pulse has Sin^2 envelope with 3 femtosecond duration and is centered at 0.2 hartrees

Output includes an animation of the wave packet, plots of the potentials and initial wave packet, and text output of grid parameters, Hamiltonian eigenvalues, and properties of wave packets during the propagation at the plotting intervals in time.

Plot output, including an mp4 file of the animation is placed in the directory Plot_Output/

Uses the basic FEM-DVR functions in the FEM_DVR() class and also the functions specific to the case of nuclear motion on two potential surfaces:

  1. Hamiltonian_Two_States: constructs the two Hamiltonian matrices and coupling

  2. Crank_Nicolson_Two_States: propagates the two component wave function with coupling between them

  3. Plot_Psi_Two_States: plots both components of the wave function (on the two potentials)

Args:

  1. number_of_time_intervals (int): First command line argument. Number of time intervals to perform the Crank-Nicolson propagation; defaults to 300.

  2. time_step (int): Second command line argument. Time step in the propagator to relax or restrict the calculation as needed; defaults to 0.1 atu.

  3. want_to_plot_animate (bool): Optional command that turns on plotting; default is false.

two_electron

Temkin-Poet (s-wave limit) or colinear model of a two-electron atom: H- anion or He bound and autoionizing states

Finds all eigenvalues of complex scaled 2D Hamiltonian and plots any one of them, specified by n_state_plot. NB Diagonalization does not take advantage of sparse banded nature of the 2D Hamiltonian matrix in the FEM-DVR. Larger scale practical calculations must do so.

As an example the \(2s^2\) autoionizing state of He is chosen and the plot of the wave function shows the localized resonant state in the middle and autoionization decay down the sides parallel to the \(r_1\) and \(r_2\) axes. Numerical check: \(E_{gnd} = -2.8790288\) for He in s-wave limit singlet S autoionizing state \(E_{res} = -0.7228 - 0.001199 i\)

Note that the basis in this example is a simple product basis of DVR functions in \(r_1\) and \(r_2\), so both singlet (symmetric) and triplet (antisymmetric) spatial wave functions appear as eigenfunctions of the 2D Hamiltonian.

Modifying Scripts

The actual names of these four example scripts are ECS_FEMDVR_diatomic_time_indep_vibration_H2.py, ECS_FEMDVR_diatomic_time_dep_vibration_H2.py, H2_vib_states_FEM_DVR.py, CO_vib_states_FEM_DVR.py, Time-dep_excitation_2_potential_curves.py, and Two-electron_ECS.py. If you downloaded the source package from github, then these examples are in the examples directory. If quantumgrid was installed using the conda instruction then the scripts should be in /Path/to/Anaconda/envs/YOUR_ENVIRONMENT_NAME/lib/python3.7/site-packages/quantumgrid_examples. If you are in a Unix environment then you can simply find them with the following command:

$ locate ECS_FEMDVR_diatomic_time_dep_vibration_H2.py

At any rate, once found you can modify your script however you like!

References

1(1,2)

Julia Turner and C.William McCurdy. The application of exterior complex scaling in calculations on resonances in nuclear motion in molecular systems. Chemical Physics, 71(1):127 – 133, 1982. URL: http://www.sciencedirect.com/science/article/pii/0301010482870122, doi:https://doi.org/10.1016/0301-0104(82)87012-2.