ofdft

Subpackage containing OFDFT implementation.

The electronic energy is given by the following functional:

\[E[\rho] = E_{kin}[\rho] + E_{nuc}[\rho] + E_{H}[\rho] + E_{xc}[\rho]\]

where \(E_{kin}[\rho]\) is the kinetic energy functional, \(E_{nuc}[\rho]\) the nuclear electron attraction energy, \(E_{H}[\rho]\) the Hartree energy (electron-electron coulom repulsion) and \(E_{xc}[\rho]\) the exchange correlation energy.

The nuclear electron attraction energy is given by:

\[E_{nuc}[\rho] = \int v_{nuc}(r) \rho(r) \mathrm{d}r\]

where \(v_{nuc}(r)\) is the nuclear electron attraction potential.

The Hartree energy is given by:

\[E_{H}[\rho] = \frac{1}{2} \int \int \frac{\rho(r) \rho(r')}{|r-r'|} \mathrm{d}r \mathrm{d}r'\]

To represent the electron density [M-OFDFT] we use the Linear Combination of Atomic Basis Functions (LCAB) approach. The density is then given by:

\[\rho(r) = \sum_\mu p_\mu \omega_\mu(r)\]

where \(p_\mu\) are the coefficients of the atomic basis and \(\omega_\mu(r)\) are the basis functions, for which we use the usual gaussian type ‘atomic orbitals’.

To obtain the groundstate density and energy we need to minimize the energy functional w.r.t. the electron density. This is done by minimizing the energy functional w.r.t. the coefficients of the density expansion.

\[\min_{\mathbf{p}} E[\rho_p]\]

which we solve by gradient descent.

\[\nabla E[\rho_p] = \nabla E_{kin}[\rho_p] + \nabla E_{xc}[\rho_p] + \nabla E_{H}[\rho_p] + \nabla E_{nuc}[\rho_p]\]

Additionally, the particle number needs to be preserved. To achieve this we project the gradient onto the subspace of constant particle number. Which results in the following update rule:

\[p^{(n+1)} = p^{(n)} - \epsilon \left( \mathbf{1} - \frac{ww^T}{w^Tw} \right) \nabla E[\rho_p]\]

with \(w\) being the normalization vector of the basis functions \(w_\mu = \int \omega_\mu (r) \mathrm{d}r\) and \(\epsilon\) being the learning rate/step length of the gradient descent.

Modules

basis_integrals	Compute analytic integrals over contracted Gaussian-type orbitals (GTO).
callbacks	Callback classes for the minimization.
density_optimization
energies	Energies class for storing calculated energies.
functional_factory	Provides energy functionals.
initial_guess	Module for generating initial guesses for the electronic structure.
libxc_functionals	This module deals with libxc kinetic, exchange and correlation functionals.
ofstate	Contains the class for storing the state of the OFDFT calculation.
optimizer
plot_ofdft
run_density_optimization
torch_functionals	Implementation of functionals in PyTorch.