grid.coulomb module

Coulomb potential module for Gaussian charge densities.

Provides exact analytical formulas for evaluating the electrostatic potential of s-type and p-type Gaussian functions.

grid.coulomb.coulomb_gaussian_p(r: ndarray, alpha: float, normalized: bool = True) → ndarray

Compute the exact Coulomb potential of a p-type radial Gaussian charge density.

If normalized is True, the charge density is: .. math:

\rho(r) = \frac{2}{3} \frac{\alpha^{5/2}}{\pi^{3/2}} r^2 e^{-\alpha r^2}

and the potential is: .. math:

V(r) = \frac{\text{erf}(\sqrt{\alpha} r)}{r}
+ \frac{4}{3} \sqrt{\frac{\alpha}{\pi}} e^{-\alpha r^2}

If normalized is False, the charge density is: .. math:

\rho(r) = r^2 e^{-\alpha r^2}

and the potential is: .. math:

V(r) = \frac{3 \pi^{3/2}}{2 \alpha^{5/2}} \frac{\text{erf}(\sqrt{\alpha} r)}{r}
+ \frac{2\pi}{\alpha^2} e^{-\alpha r^2}

Parameters

rnp.ndarray

Radial distances from the center of the Gaussian.

alphafloat

Gaussian exponent.

normalizedbool, default=True

Whether to compute the potential of a normalized p-type Gaussian.

Returns

np.ndarray

Coulomb potential evaluated at the radial distances as a 1D array (scalar input returns shape (1,)).

grid.coulomb.coulomb_gaussian_s(r: ndarray, alpha: float, normalized: bool = True) → ndarray

Compute the exact Coulomb potential of an s-type Gaussian charge density.

If normalized is True, the charge density is: .. math:

\rho(r) = \left(\frac{\alpha}{\pi}\right)^{3/2} e^{-\alpha r^2}

and the potential is: .. math:

V(r) = \frac{\text{erf}(\sqrt{\alpha} r)}{r}

If normalized is False, the charge density is: .. math:

\rho(r) = e^{-\alpha r^2}

and the potential is: .. math:

V(r) = \left(\frac{\pi}{\alpha}\right)^{3/2} \frac{\text{erf}(\sqrt{\alpha} r)}{r}

Parameters

rnp.ndarray

Radial distances from the center of the Gaussian.

alphafloat

Gaussian exponent.

normalizedbool, default=True

Whether to compute the potential of a normalized s-type Gaussian.

Returns

np.ndarray

Coulomb potential evaluated at the radial distances.

grid.coulomb.coulomb_potential(points: ndarray, centers_s: ndarray, coeffs_s: ndarray, alphas_s: ndarray, centers_p: ndarray | None = None, coeffs_p: ndarray | None = None, alphas_p: ndarray | None = None, normalized: bool = True) → ndarray

Compute the total Coulomb potential at evaluation points from a set of Gaussians.

Parameters

pointsnp.ndarray

Evaluation points, shape (N, 3).

centers_snp.ndarray

Centers of the s-type Gaussians, shape (Ks, 3).

coeffs_snp.ndarray

Coefficients of the s-type Gaussians, shape (Ks,).

alphas_snp.ndarray

Exponents of the s-type Gaussians, shape (Ks,).

centers_pnp.ndarray, optional

Centers of the p-type Gaussians, shape (Kp, 3).

coeffs_pnp.ndarray, optional

Coefficients of the p-type Gaussians, shape (Kp,).

alphas_pnp.ndarray, optional

Exponents of the p-type Gaussians, shape (Kp,).

normalizedbool, default=True

Whether the coefficients correspond to normalized Gaussians.

Returns

np.ndarray

The computed electrostatic potential, shape (N,).