Ground-state Schrödinger-equation (SE)

Ground-state Schrödinger equation corresponds to the lowest-energy solution of the eigen-value problem

$$ \hat{\mathcal{H}} \psi (x)=E\psi (x),~~~~~~(1) $$

where $\hat{\mathcal{H}}$ is the Hamiltonian operator, $\psi$ the (ground state) wave function, and $E$ its associate energy. The Hamiltonian is defined as the sum of the kinetic and potential operators

$$ \hat{\mathcal{H}} =-\frac{\Delta }{2}+\hat{\mathcal{V}} .~~~~~~(2) $$

The QMol-grid package provides support both for ground-state Schrödinger equation with the QMol_SE class.

Schrödinger-equation components

For clarity and streamlined future development of Schrödinger-equation capabilities in the QMol-grid package, we sort SE component (classes) into different topical groups.

Model discretization

The following classes are used to discretize Schrödinger-equation systems

• QMol_SE_wfcn is the Cartesian-grid discretization of the wave function.
• QMol_SE_V is the Cartesian-grid discretization of the potential operator $\hat{\mathcal{V}}$ .

For basis-set discretization models, with QMol_disc_basis, the description of the wave function is overloaded by the QMol_SE_wfcn_basis class.

Memory and execution-time profiler

The choice of the domain discretization can greatly affect the resources, both in terms of memory and execution time, required to carry out simulations.

• QMol_SE_profiler provides estimates of the memory and run-time requirements for a given Schrödinger-equation model or components.

Keep in mind that differentiation are performed via fast-Fourier transforms. Thus, as a rule of thumb, domain discretization with small prime factor number of grid points tend to produce faster results -- for best performance, prime factors not greater than 7.

Ground-state computation

The Schrödinger-equation ground state, or more generally some SE eigen state, is computed by numerically solving Eq. (1). In the QMol-grid package, this can be achieved with

• QMol_SE_eigs is the eigen solver to be for grid-based discretizations.
• QMol_SE_eig_basis is the eigen solver to be used for basis-set discretizations.

Notes for developers

SE model implementation

The QMol_SEq abstract class defines the common interface and run-time documentation for Schrödinger-equation models.

Notes

• Schrödinger-equation features were introduced in version 01.20