Glossary

Variable Names and Meanings

This section provides a summary of the variables and arguments used in the package, along with their meanings.

Mesh Grid

Math Symbol	Variable Name	Meaning
\(\boldsymbol{r}\)	r_vector_grid	Grid in real space of the unit cell. Shape: [x, y, z, 3]
\(\boldsymbol{G}\)	g_vector_grid (g_vecs)	Grid in reciprocal space of the unit cell. Shape: [x, y, z, 3]
\(\boldsymbol{k}\)	kpts	k-point path sampling in reciprocal space. Shape: [num_kpts, 3]
\(\boldsymbol{w}_k\)	kpts_weight	Weights for k-point integration. Shape: [num_kpts, ]

Physical Quantities

The physical quantities in the table below are evaluated on the mesh grid in real or reciprocal space.

Math Symbol	Variable Name	Meaning
\(\bf{A}\)	cell_vectors	Real-space unit cell translation vectors
\(\bf{B}\)	cell_vectors_reciprocal	Reciprocal lattice vectors
\(\rho(\bf{r})\)	density_grid	Electronic charge density in real space
\(\tilde\rho(\bf{G})\)	density_grid_reciprocal	Electronic charge density in reciprocal space
\(\psi({\bf{r}})\)	wave_grid	Electronic wavefunction in real space
\(\tilde\psi(\bf{G})\)	wave_grid_reciprocal	Electronic wavefunction in reciprocal space
\(\boldsymbol{\varepsilon}_{i}\)	eigenvalues	Single-particle eigenvalues of the Hamiltonian matrix
\(h_{ij}\)	hamiltonian	Single-particle Hamiltonian matrix
\(c\)	coefficient, (coeff)	Plane wave expansion coefficients. Shape: [spin, kpts, band, x, y, z]
\(c\)	coeff_dense	Dense format of plane wave coefficients. Shape: [spin, kpts, g_vec, band]
\(w\)	weight_real, & weight_imaginary	Trainable neural network parameters
-	mask	Binary mask for g_vector_grid that enforces the plane wave cutoff energy
\(o\)	occupation	Electronic occupation numbers for each state

Notes:

• Any variable ending with _vector_grid has the shape (x, y, z, 3), representing 3D vectors on a grid.

• Variables ending with _grid have the shape (x, y, z), representing scalar quantities on a grid.

• The postfix _real is always omitted. If a variable does not have the postfix _reciprocal, it is in real space.

Arguments and Intermediate Variables

Variable Name	Meaning
cutoff_energy	Energy cutoff for plane wave basis expansion
grid_sizes	Number of grid points along each direction for both \(k\) and \(G\) points
k_grid_sizes	Number of k-points along each direction for Brillouin zone sampling
position	Atomic positions in real-space coordinates
charge	Nuclear charge
ewald_eta	Ewald summation convergence parameter \(\eta\)
ewald_cut	Cutoff energy for Ewald summation. The higher the cutoff energy, the more accurate the Ewald summation, but the slower the calculation.
num_band	Number of electronic bands (usually equals the number of electrons)
num_k	Total number of k-points in the calculation
num_g	Total number of G-vectors in the plane wave basis
vol	Volume of the crystal unit cell
spin	Number of spin channels (1 for non-spin-polarized, 2 for spin-polarized calculations)