grid.poisson module

Poisson solver module.

This module solves the following Poisson equation:

\[\nabla^2 V(r) = -4\pi \rho(r),\]

for some Coulomb potential \(V(r)\) and charge density \(\rho(r)\) over a centered atomic grid. It is recommended to use the boundary value problem for handing singularities near the origin of the atomic grid.

grid.poisson.solve_poisson_bvp(molgrid: MolGrid | AtomGrid, func_vals: ndarray, transform: BaseTransform, boundary: float | None = None, include_origin: bool = True, remove_large_pts: float = 1000000.0, ode_params: dict | None = None)

Return interpolation of the solution to the Poisson equation solved as a boundary value problem.

The Poisson equation solves for function \(g\) of the following:

\[\nabla^2 g = (-4\pi) f,\]

for a fixed function \(f\), where \(\nabla^2\) is the Laplacian. This is transformed to an set of ODE problems as a boundary value problem. [1]

If boundary is not provided, then the boundary of \(g\) for large r is set to \(\int \int \int f(r, \theta, \phi) / r\). The solution \(g\) is assumed to be zero at the origin \(g(0, \theta, \phi) = 0\). Use solve_poisson_ivp if this assumption isn’t needed.

Parameters

molgridMolGrid or AtomGrid

Molecular or atomic grid that is used for integration and expanding func into real spherical harmonic basis.

func_valsndarray(N,)

The function values evaluated on all \(N\) points on the molecular grid.

transformBaseTransform, optional

Transformation from infinite domain \(r \in [0, \infty)\) to another domain that is a finite.

boundaryfloat, optional

The boundary value of \(g\) in the limit of r to infinity.

include_originbool, optional

If true, will add r=0 point when solving for the ode only. If false, it is recommended to have many radial points near the origin.

remove_large_ptsfloat, optional

If true, will remove any points larger than remove_large_pts when solving for the ode only.

ode_paramsdict, optional

The parameters for the ode solver. See grid.ode.solve_ode_bvp for all options.

Returns

callable(ndarray(N, 3) -> float) :

The solution to Poisson equaiton/potential \(g : \mathbb{R}^3 \rightarrow \mathbb{R}\).

References

[1]

Becke, A. D., & Dickson, R. M. (1988). Numerical solution of Poisson`s equation in polyatomic molecules. The Journal of chemical physics, 89(5), 2993-2997.

grid.poisson.solve_poisson_ivp(molgrid: MolGrid | AtomGrid, func_vals: ndarray, transform: BaseTransform, r_interval: tuple = (1000, 1e-05), ode_params: dict | None = None)

Return interpolation of the solution to the Poisson equation solved as an initial value problem.

The Poisson equation solves for function \(g\) of the following:

\[\nabla^2 g = (-4\pi) f,\]

for a fixed function \(f\), where \(\nabla^2\) is the Laplacian. This is transformed to a set of ODE problems as an initial value problem.

Ihe initial value problem is chosen so that the boundary of \(g\) for large r is set to \(\int \int \int f(r, \theta, \phi) / r\). Depending on \(f\), this function has difficulty in capturing the origin \(r=0\) region, and is recommended to keep the final interval \(a\) close to zero.

Parameters

molgridMolGrid or AtomGrid

Molecular or atomic grid that is used for integration and expanding func into real spherical harmonic basis.

func_valsndarray(N,)

The function values evaluated on all \(N\) points on the molecular grid.

transformBaseTransform, optional

Transformation from infinite domain \(r \in [0, \infty)\) to another domain that is a finite.

r_intervaltuple, optional

The interval \((b, a)\) of \(r\) for which the ODE solver will start from and end, where \(b>a\). The value \(b\) should be large as it determines the asymptotic region of \(g\) and value \(a\) is recommended to be small but not zero depending on \(f\).

ode_paramsdict, optional

The parameters for the ode solver. See grid.ode.solve_ode_ivp for all options.

Returns

callable(ndarray(N, 3) -> float) :

The solution to Poisson equaiton/potential \(g : \mathbb{R}^3 \rightarrow \mathbb{R}\).