grid.poisson module
Poisson solver module.
This module solves the following Poisson equation:
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.
- interpolate_laplacian(molgrid, func_vals)[source]
Return a function that interpolates the Laplacian of a function.
\[\nabla^2 f = \frac{1}{r}\frac{\partial^2 rf}{\partial r^2} - \frac{\hat{L}}{r^2},\]such that the angular momentum operator satisfies \(\hat{L}(Y_l^m) = l (l + 1) Y_l^m\). Expanding f in terms of spherical harmonic expansion, we get that
\[\nabla^2 f = \sum_l \sum_m \bigg[ \frac{\partial^2 \rho_{lm}(r)}{\partial r^2} + \frac{2}{r} \frac{\partial \rho_{lm}(r)}{\partial r} - \frac{l(l+1)}{r^2}\rho_{lm}(r) \bigg] Y_l^m,\]where \(\rho_{lm}^f\) is the lth, mth radial component of function f.
- Parameters
- Returns
Function that interpolates the Laplacian of a function whose input is Cartesian points. The float value is the cutoff where radial points smaller than the cutoff are replaced with the cutoff. Computing the Laplacian at r=0 can cause problems depending on the function provided.
- Return type
callable[ndarray(M,3), float -> ndarray(M,)]
Warning
Since \(\rho_{lm}\) and its derivatives are being interpolated and due to division by powers of \(r\), it is recommended to be very careful of having values near zero.
- solve_poisson_bvp(molgrid, func_vals, transform, boundary=None, include_origin=True, remove_large_pts=1000000.0, ode_params=None)[source]
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
molgrid (MolGrid or AtomGrid) – Molecular or atomic grid that is used for integration and expanding func into real spherical harmonic basis.
func_vals (ndarray(N,)) – The function values evaluated on all \(N\) points on the molecular grid.
transform (BaseTransform, optional) – Transformation from infinite domain \(r \in [0, \infty)\) to another domain that is a finite.
boundary (float, optional) – The boundary value of \(g\) in the limit of r to infinity.
include_origin (bool, 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_pts (float, optional) – If true, will remove any points larger than remove_large_pts when solving for the ode only.
ode_params (dict, optional) – The parameters for the ode solver. See grid.ode.solve_ode_bvp for all options.
- Returns
The solution to Poisson equaiton/potential \(g : \mathbb{R}^3 \rightarrow \mathbb{R}\).
- Return type
callable(ndarray(N, 3) -> float)
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.
- solve_poisson_ivp(molgrid, func_vals, transform, r_interval=(1000, 1e-05), ode_params=None)[source]
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
molgrid (MolGrid or AtomGrid) – Molecular or atomic grid that is used for integration and expanding func into real spherical harmonic basis.
func_vals (ndarray(N,)) – The function values evaluated on all \(N\) points on the molecular grid.
transform (BaseTransform, optional) – Transformation from infinite domain \(r \in [0, \infty)\) to another domain that is a finite.
r_interval (tuple, 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_params (dict, optional) – The parameters for the ode solver. See grid.ode.solve_ode_ivp for all options.
- Returns
The solution to Poisson equaiton/potential \(g : \mathbb{R}^3 \rightarrow \mathbb{R}\).
- Return type
callable(ndarray(N, 3) -> float)