QMol-grid: A MATLAB package for quantum-mechanical simulations in atomic and molecular systems

The QMol-grid package provides a suite of routines for performing quantum-mechanical simulations in atomic and molecular systems in one spatial dimension. All simulations use an underlying Cartesian-grid discretization scheme. The QMol-grid package is written using MATLAB's object-oriented features and handle classes.

The QMol-grid package provides support for the following types of computations:

• DFT Ground- and excited-state density-functional theory, both using a Cartesian grid or basis-set discretization
• HF Ground- and excited-state Hartree Fock, both using a Cartesian grid or basis-set discretization
• SE Ground- and excited-state Schrödinger equation, both using a Cartesian grid or basis-set discretization
• TDDFT Real-time time-dependent density-functional theory, using a Cartesian grid
• TDSE Time-dependent Schrödinger equation, using a Cartesian grid

The time-propagators are computed using symplectic-split operators (2nd, 4th, and 6th order in time, and spectral in space). They support field-free and laser-driven simulations (in the dipole approximation) with the following on-the-fly features, each specifying their own time sampling

• Checkpointing, with the creation of a restart MATLAB file that can be used to resume a calculation that was stopped before it was finished
• Calculation and storage of the dipole, dipole velocity, and dipole acceleration signals
• Calculation and storage of the wave function(s)/Kohn-Sham orbitals and Hamiltonian-component energies
• Storage of the wave function(s) (Schrödinger), and Kohn-Sham orbitals and one-body density (DFT)
• Calculation and storage of the ionization signal, keeping track of how much electronic density is absorbed at the domain boundaries
• Calculation and storage of the results of installable output functions of the wave function(s) (Schrödinger), and Kohn-Sham orbitals or one-body density (DFT)
• Saving the intermediate Schrödinger- or DFT-model objects in separate MATLAB files (.mat)

---

Table of Contents

• Installation
• Documentation
• Example 1: Schrödinger-equation ground state
• Example 2: Time-dependent density-functional theory
• Reference

---

Installation

The QMol-grid package can be installed directly from MATLAB or this GitHub repository.

From MALAB:

• In the Add-Ons explorer, search for "QMol-grid" and the add the package to MATLAB. This will install the latest release version of QMol-grid. The package documentation will then be accessible in MATLAB's, in the "Supplemental Software" section

From this GitHub repository:

• include the entire content of the folder /src/, including all its subfolders, in a folder of the type <user>/Documents/MATLAB/QMol-grid
  • from a release: download the zipped file in the release, and unzip the content to get the /src/ folder
  • from the github repository (development version): download the entire content of the folder /src/, including all its subfolders
• add permanently the folder <user>/Documents/MATLAB/QMol-grid (without its subfolders) to MATLAB path
• after successful installation, the package documentation will be accessible in MATLAB's, in the "Supplemental Software" section

Tests

Display the list of components in the QMol-grid package

QMol_doc.showComponents;

The QMol-grid package provides a suite of tests that performs low-level checks on the various properties that are defined throughout its components. The test suite can be used to check basic functions after installing the package or modifying its components.

Run all available unit tests using

QMol_test.test;

Limitations

• The current release only supports one-dimensional computations
• Time-dependent Hartree Fock is currently not available
• Time propagation on basis sets is currently not available

↑

---

Documentation

A full description of the QMol-grid package, including all possible input parameters and calculation features is included in the MATLAB documentation provided with the package. After installation, the package documentation is accessible in MATLAB, in the "Supplemental Software" section. The documentation includes a series of tutorials, starting with SE ground-state calculations, and going through TDSE, DFT, and TDDFT calculations to help new users getting familiarized with setting up calculations, input parameters, and output variables. Throughout, the documentation also includes many script samples illustrating how one can use the various features. Finally, the documentation discusses the required class structure for advanced users who wish to add their own functionalities to the package and inherit common interface methods to the QMol-grid package.

The wiki provides a copy of the QMol-grid package documentation.

↑

---

Example 1: Schrödinger-equation ground state

Here we illustrate how to use the QMol-grid package to calculate the ground-state wave function of a one-dimensional hydrogen-like atom. The Schrödinger-equation ground-state corresponds to the lowest-energy solution to the eigenvalue problem $\hat{\mathcal{H}}\psi(x)=E\psi(x)$, where $\hat{\mathcal{H}}$ is the Schrödinger-equation Hamiltonian operator, $\psi$ is the wave function, and $E$ its associated energy. In atomic units, the Hamiltonian operator is $\hat{\mathcal{H}} = -\frac{\Delta}{2} + \hat{\mathcal{V}}$.

Specifically, it walks through defining (i) the domain and grid discretization over which the Schrödinger-equation and wave function are calculated, the (ii) atomic potential and (iii) Schrödinger-equation model, and calculating (iv) the ground state associated with these properties.

We model the one-dimensional hydrogen model atom using a soft-Coulomb potential $V(x)=-1/\sqrt{x^2+a^2}$ with

H = QMol_Va_softCoulomb('softeningParameter',sqrt(2));

where 'softeningParameter' specifies the value for the parameter $a$. Here we choose the softening parameter $a=\sqrt{2}$ to match H's ground state energy. By default, the atom is located at the origin $x=0$. Note that H only corresponds to the atomic model, which is shared with molecular systems and various quantum frameworks. Thus, it must be turned into a valid Schrödinger-equation potential, using

V = QMol_SE_V('atom',H);

Here 'atom' indicates to the QMol_SE_V object that the list of atomic centers is provided next -- here a single H effective potential.

The simulation domain must be a Cartesian grid -- with all increasing, equally spaced discretization points -- and should be wide enough and with small enough of a discretization step to properly capture the wave function. In our case we select a domain ranging -15 to 15 a.u., with a discretization steps of 0.1 a.u.

x = -15:.1:15;

We now have all the elements to define a Schrödinger-equation model object with the potential and domain defined above

SE = QMol_SE(                        ...
         'xspan',                x,  ...
         'potential',            V);

Like above, when creating the SE object, we recognize the definition of the discretization domain and effective potential with the keywords 'xspan' and 'potential', respectively.

We next move to calculating its associated ground-state wave function and energy using the two commands

GSS = QMol_SE_eigs;
GSS.computeGroundState(SE);

The first line creates the eigen-state solver while the second performs the actual ground-state calculation on the Schrödinger-equation object SE.

At the end of the calculation, the ground-state wave function is stored in the input object SE, together with relevant information such as the domain discretization. For instance, solely relying on SE, one can plot the ground-state wave function with

figure
    plot(SE.xspan,SE.waveFunction.waveFunction,'-','LineWidth',2)
    set(gca,'box','on','FontSize',12,'LineWidth',2)
    xlabel('x (a.u.)')
    ylabel('wave function (a.u.)')
    xlim(SE.xspan([1 end]))

producing

From the plot command line, we see that the domain-discretization grid may be recovered using the xspan property in the object SE (using the standard object-oriented dot notation SE.xspan). On the other hand, the wave function is nested inside another object, which explains the consecutive dots SE.waveFunction.waveFunction. Other properties in the object SE.waveFunction are used by ground/excited-state and TDSE calculations; we refer to the QMol_SE_wfcn documentation page for further details.

↑

---

Example 2: Time-dependent density-functional theory

For a given set of initial Kohn-Sham orbitals $\phi_{k}$, the TDDFT dynamics is described by the nonlinear system of partial differential equations, in atomic units (a.u.) $$i\partial_t \phi_k({\bf x};t) =\hat{\mathcal{H}}_{\rm DFT}[{\phi_k}_k;t]({\bf x};t)\ \phi_k({\bf x};t) .\quad\quad\quad (2.1)$$ The QMol-grid package relies on the canonical Hamiltonian structure of TDDFT Mauger 2024 to integrate the dynamics of equation (2.1). In this example, we illustrate how to use the QMol-grid package to integrate the TDDFT dynamics of an open-shell one-dimentional molecular ion model with 3 atomic centers and 5 active electrons.

Initial condition

In the QMol-grid package, TDDFT simulations are decoupled from setting up the initial condition, which must be done independently. Similar to example 1, we build the molecular model out of 3 one-dimensional atomic models, each contributing 2 electrons to the molecule, using soft-Coulomb potentials. For our example, we start by calculating the neutral-molecule ground state:

% Molecular model
V_1     =   QMol_Va_softCoulomb('atom','X_1','charge',2,'position',-3);
V_2     =   QMol_Va_softCoulomb('atom','X_2','charge',2,'position', 0);
V_3     =   QMol_Va_softCoulomb('atom','X_3','charge',2,'position', 3);
 
% DFT model
Vext    =   QMol_DFT_Vext('atom',{V_1,V_2,V_3});
Vh      =   QMol_DFT_Vh_conv;
Vxc     =  {QMol_DFT_Vx_LDA_soft,QMol_DFT_Vc_LDA_soft};
 
DFT     =   QMol_DFT_spinPol(                                       ...
                'xspan',                       -50:.1:50,           ...
                'occupation',                  {[1 1 1],[1 1 1]},   ...
                'externalPotential',            Vext,               ...
                'HartreePotential',             Vh,                 ...
                'exchangeCorrelationPotential', Vxc,                ...
                'selfInteractionCorrection',    'ADSIC'             );
 
% DFT ground state
SCF     =   QMol_DFT_SCF_Anderson;
SCF.solveSCF(DFT);

The "% Molecular model" block defines the atomic effective potential, specifying the name, bare charge, and location of each atomic center, respectively. The "% DFT model" block first defines the molecular potential Vext, followed by the DFT functionals Vh and Vxc to be used in the (TD)DFT calculations -- see the documentation's ground-state DFT tutorial for further details regarding the model parameters. The final block "% DFT ground state" first creates the eigen-state DFT solver, here an Anderson mixing scheme, and performs the ground-state self-consistent field (SCF) calculation.

Next, we manually induce an excitation in the molecular cation by successively (i) replacing one of the Kohn-sham orbitals by a superposition of molecular-orbital states (excitation part) and (ii) removing an electron, going from 3 to 2, from the down-spin Kohn-Sham orbitals (ionization part).

% Induce excitation
DFT.orbital.set('orbitalDown',[DFT.KSO.KSOdw(:,1) (DFT.KSO.KSOdw(:,2)+DFT.KSO.KSOdw(:,3))/sqrt(2)]);

% Induce ionization
DFT.set('occupation',{[1 1 1],[1 1]});

We now have a non-stationary set of Kohn-Sham orbitals, leading to field-free dynamics under the flow of equation (2.1).

TDDFT simulation

With the DFT molecular model, including the initial condition, in hand, we now move to integrating the subsequent field-free TDDFT dynamics. For this, we select a fourth-order Forest Ruth symplectic split-operator scheme (see Mauger 2024 for more details). Note that, here the field-free TDDFT dynamics does not lead to any ionization and therefore no boundary conditions need be specified at the edges of the domain. For field-driven simulations, absorbing boundary conditions can be specified to avoid spurious boundary effects.

TDDFT   =   QMol_TDDFT_SSO_4FR(                     ...
                'time',                 0:10:100,   ...
                'timeStep',             2e-2,       ...
                'saveDensity',          true,       ...
                'saveDensityTime',      1);

In our example, the TDDFT object is created with:

• The first pair of arguments specifies that the integration should start at time t=0 and end at t=100 a.u. The step of 10 a.u., is unrelated to the propagation time step and instead specifies the time intervals to use in progress display.
• The second pair of arguments specifies the (fixed) time step for the propagation.
• The third pair of arguments indicates that the one-body density should be saved periodically, with the period specified by the fourth pair of arguments, i.e., every 1 a.u. in our case.

Then, we launch the TDDFT integration with

TDDFT.propagate(DFT);

At the end of the simulation, the DFT object has been updated to contain the Kohn-Sham orbitals at t=100 a.u. The time-dependent one-body density is stored in the TDDFT object itself.

Plotting the result

Next we recover calculated observables out of the TDDFT object. Each set of observable is stored in a separate structure property in the TDDFT object, which containts (i) the exact time vector at which the quantity has been saved and (ii) the observable itself. In our case, the structure of interest is TDDFT.outDensity with the up- and down-spin densities respectively stored in the fields totalUp and totalDown. The densities are matrices with columns corresponding to the successive saved times. To plot the spin density, defined as the difference between the up- and down-spin one-body densities, we use

figure
    imagesc(TDDFT.outDensity.time,DFT.xspan,TDDFT.outDensity.totalUp-TDDFT.outDensity.totalDown)
    set(gca,'box','on','FontSize',12,'LineWidth',2,'YDir','normal')
    xlim(TDDFT.outDensity.time([1 end]))
    ylim([-10 10])
    xlabel('time (a.u.)')
    ylabel('position (a.u.)')
    title('spin density')
    colorbar vert

producing

Profiling (estimating the memory footprint)

Before running the TDDFT calculation, the user has the possibility to check how much memory the simulation requires to run and store the requested one-body densities. Using the same calculation workflow as above, right after creating the TDDFT object, the memory footprint is obtained with

TDDFT.initialize(DFT);
QMol_DFT_profiler(TDDFT,'memory');

In our case, the estimated total TDDFT-object size is 1.8MB with 1.5MB for the saved electron density. Saving the TDDFT and DFT object in a MATLAB file at the end of the propagation produces a 1.6~MB .mat file. We mostly attribute the slight difference with the profiler estimate to run-time memory overhead associated with internal variables that are not stored in the saved objects.

↑

---

Reference

• F. Mauger and C. Chandre, QMol-grid: A MATLAB package for quantum-mechanical simulations in atomic and molecular systems, arXiv:2406.17938

@unpublished{mauger2024,
  title = {QMol-grid: A MATLAB package for quantum-mechanical simulations in atomic and molecular systems},
  author = {Mauger, Fran\c{c}ois and Chandre, Cristel},
  year = {2024},
  URL = {https://arxiv.org/abs/2406.17938}
}

For more information: fmauger@lsu.edu

Acknowledgments

The original development of the QMol-grid package, and its (TD)DFT features, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0012462. Addition of the (TD)SE features was supported by the National Science Foundation under Grant No. PHY-2207656.

↑