# Advanced Poisson solvers¶

The `PoissonSolver`

with default parameters uses zero boundary conditions on
the cell boundaries. This becomes a problem in systems involving large dipole
moment, for example (due to, e.g., plasmonic charge oscillation on a
nanoparticle). The potential due to the dipole is long-ranged and, thus, the
converged potential requires large vacuum sizes.

However, in LCAO approach large vacuum size is often unnecessary. Thus, to
avoid using large vacuum sizes but get converged potential, one can use two
approaches or their combination: 1) use multipole moment corrections or 2)
solve Poisson equation on a extended grid. These two approaches are
implemented in `ExtendedPoissonSolver`

and `ExtraVacuumPoissonSolver`

.
Also regular `PoissonSolver`

in GPAW has the option `remove_moment`

.

In any nano-particle plasmonics calculation, it is necessary to use multipole correction. Without corrections more than 10Å of vacuum is required for converged results.

## Multipole moment corrections¶

The boundary conditions can be improved by adding multipole moment corrections to the density so that the corresponding multipoles of the density vanish. The potential of these corrections is added to the obtained potential. For a description of the method, see [1].

This can be accomplished by following solver:

```
from gpaw.poisson_extended import ExtendedPoissonSolver
poissonsolver = ExtendedPoissonSolver(eps=eps,
moment_corrections=4)
```

This corrects the 4 first multipole moments, i.e., \(s\), \(p_x\), \(p_y\), and
\(p_z\) type multipoles. The range of multipoles can be changed by changing
`moment_corrections`

parameter. For example, `moment_correction=9`

includes in addition to the previous multipoles, also \(d_{xx}\), \(d_{xy}\),
\(d_{yy}\), \(d_{yz}\), and \(d_{zz}\) type multipoles.

This setting suffices usually for spherical-like metallic nanoparticles, but more complex geometries require inclusion of very high multipoles or, alternatively, a multicenter multipole approach. For this, consider the advanced syntax of the moment_corrections. The previous code snippet is equivalent to:

```
from gpaw.poisson_extended import ExtendedPoissonSolver
poissonsolver = ExtendedPoissonSolver(eps=eps,
moment_corrections=[{'moms': range(4), 'center': None}])
```

Here `moment_corrections`

is a list of dictionaries with following
keywords: `moms`

specifies the considered multipole moments, e.g.,
`range(4)`

equals to \(s\), \(p_x\), \(p_y\), and \(p_z\) multipoles, and
`center`

specifies the center of the added corrections in atomic units
(`None`

corresponds to the center of the cell).

As an example, consider metallic nanoparticle dimer where the nanoparticle
centers are at `(x1, y1, z1)`

Å and `(x2, y2, z2)`

Å. In this case, the
following settings for the `ExtendedPoissonSolver`

may be tried out:

```
import numpy as np
from ase.units import Bohr
from gpaw.poisson_extended import ExtendedPoissonSolver
moms = range(4)
center1 = np.array([x1, y1, z1]) / Bohr
center2 = np.array([x2, y2, z2]) / Bohr
poissonsolver = ExtendedPoissonSolver(eps=eps,
moment_corrections=[{'moms': moms, 'center': center1},
{'moms': moms, 'center': center2}])
```

When multiple centers are used, the multipole moments are calculated on non-overlapping regions of the calculation cell. Each point in space is associated to its closest center. See Voronoi diagrams for analogous illustration of the partitioning of a plane.

[1] | A. Castro, A. Rubio, and M. J. Stott,
Solution of Poisson’s equation for finite systems using plane-wave methods,
Can. J. Phys. 81, 1151 (2003).
doi:10.1139/p03-078 |

## Adding extra vacuum to the Poisson grid¶

The multipole correction scheme is not always successful for complex system
geometries. For these cases, one can use a separate large grid just for
solving the Hartree potential. Such a large grid can be set up by using
`ExtraVacuumPoissonSolver`

wrapper:

```
from gpaw.poisson import PoissonSolver
from gpaw.poisson_extravacuum import ExtraVacuumPoissonSolver
poissonsolver = ExtraVacuumPoissonSolver(gpts=(256, 256, 256),
poissonsolver_large=PoissonSolver(eps=eps))
```

This uses the given \(poissonsolver_large\) to solve the Poisson equation on
a large grid defined by the number of grid points \(gpts\).
The size of the grid is given **in the units of the Poisson grid**
(this is usually the same as the fine grid).
It is important to **use grid sizes that are divisible by high powers of 2 to
accelerate the multigrid scheme** used in `PoissonSolver`

.

To speed up the calculation of the Hartree porential on the large grid, one can apply additional coarsening:

```
from gpaw.poisson import PoissonSolver
from gpaw.poisson_extravacuum import ExtraVacuumPoissonSolver
poissonsolver = ExtraVacuumPoissonSolver(gpts=(256, 256, 256),
poissonsolver_large=PoissonSolver(eps=eps),
coarses=1,
poissonsolver_small=PoissonSolver(eps=eps))
```

The `coarses`

parameter describes how many times the given large grid
is coarsed before the \(poissonsolver_large\) is used solve the Poisson equation
there. With the given value `coarses=1`

, the grid is coarsed once and
the actual calculation grid is of size `(128, 128, 128)`

with the grid
spacing twice as large compared to the original one.
The obtained coarse potential is used to correct the boundary conditions
of the potential calculated on the original small and fine
grid by \(poissonsolver_small\).

As `ExtraVacuumPoissonSolver`

is wrapper, it can be combined with any
`PoissonSolver`

instance. For example, one can define multiple subsequently
larger grids via:

```
from gpaw.poisson import PoissonSolver
from gpaw.poisson_extravacuum import ExtraVacuumPoissonSolver
poissonsolver0 = ExtraVacuumPoissonSolver(gpts=(256, 256, 256),
poissonsolver_large=PoissonSolver(eps=eps),
coarses=1,
poissonsolver_small=PoissonSolver(eps=eps))
poissonsolver = ExtraVacuumPoissonSolver(gpts=(256, 256, 256),
poissonsolver_large=poissonsolver0,
coarses=1,
poissonsolver_small=PoissonSolver(eps=eps))
```

See `poissonsolver.get_description()`

or the `txt`

output for
the corresponding grids.