AlphaFMM
AlphaFMM
Section Hamiltonian::Poisson
Type float
Default 0.291262136
Dimensionless parameter for the correction of the self-interaction of the
electrostatic Hartree potential, when using PoissonSolver = FMM.
Octopus represents charge density on a real-space grid, each point containing a value $\rho$ corresponding to the charge density in the cell centered in such point. Therefore, the integral for the Hartree potential at point $i$, $V_H(i)$, can be reduced to a summation:
$V_H(i) = \frac{\Omega}{4\pi\varepsilon_0} \sum_{i \neq j} \frac{\rho(\vec{r}(j))}{|\vec{r}(j) - \vec{r}(i)|} + V_{self.int.}(i)$ where $\Omega$ is the volume element of the mesh, and $\vec{r}(j)$ is the position of the point $j$. The $V_{self.int.}(i)$ corresponds to the integral over the cell centered on the point $i$ that is necessary to calculate the Hartree potential at point $i$:
$V_{self.int.}(i)=\frac{1}{4\pi\varepsilon_0} \int_{\Omega(i)}d\vec{r} \frac{\rho(\vec{r}(i))}{|\vec{r}-\vec{r}(i)|}$
In the FMM version implemented into Octopus, a correction method
for $V_H(i)$ is used
(see García-Risueño et al., J. Comp. Chem. 35, 427 (2014)).
This method defines cells neighbouring cell $i$, which
have volume $\Omega(i)/8$ (in 3D) and charge density obtained by
interpolation. In the calculation of $V_H(i)$, in order to avoid
double counting of charge, and to cancel part of the errors arising
from considering the distances constant in the summation above, a
term $-\alpha_{FMM}V_{self.int.}(i)$ is added to the summation (see
the paper for the explicit formulae).