# 8. Electrostatics¶

The Coulomb (or electrostatic) interaction is defined as follows. For a pair of particles at distance $$r$$ with charges $$q_1$$ and $$q_2$$, the interaction is given by

$U_C(r)=C \cdot \frac{q_1 q_2}{r}.$

where

(1)$C=\frac{1}{4\pi \epsilon_0 \epsilon_r}$

is a prefactor which can be set by the user. The commonly used Bdrm length $$l_B = e_o^2 / (4 \pi \epsilon_0 \epsilon_r k_B T)$$ is the length at which the Coulomb energy between two unit charges is equal to the thermal energy $$k_B T$$. Based on the this length, the prefactor is given by $$C=l_B k_B T$$.

Computing electrostatic interactions is computationally very expensive. ESPResSo features some state-of-the-art algorithms to deal with these interactions as efficiently as possible, but almost all of them require some knowledge to use them properly. Uneducated use can result in completely unphysical simulations.

Coulomb interactions have to be added to the list of active actors of the system object to become active. This is done by calling the add-method of espressomd.system.System.actors. Only one electrostatics method can be active at any time.

Note that using the electrostatic interaction also requires assigning charges to the particles via the particle property espressomd.particle_data.ParticleHandle.q.

This example shows the general usage of an electrostatic method <SOLVER>. All of them need the Bjerrum length and a set of other required parameters. First, an instance of the solver is created and only after adding it to the actors list, it is activated. Internally the method calls a tuning routine on activation to achieve the given accuracy:

import espressomd
from espressomd import electrostatics

system = espressomd.System()
solver = electrostatics.<SOLVER>(prefactor=C, <ADDITIONAL REQUIRED PARAMETERS>)


where the prefactor $$C$$ is defined as in Eqn. (1)

## 8.1. Coulomb P3M¶

espressomd.electrostatics.P3M

Required parameters:
• prefactor
• accuracy

For this feature to work, you need to have the fftw3 library installed on your system. In ESPResSo, you can check if it is compiled in by checking for the feature FFTW with espressomd.features. P3M requires full periodicity (1 1 1). Make sure that you know the relevance of the P3M parameters before using P3M! If you are not sure, read the following references [Ewa21][HE88][KP92][DH98a][DH98b][DHL00][Des00][CBLCHolm08].

### 8.1.1. Tuning Coulomb P3M¶

The tuning method is called when the handle of the Coulomb P3M is added to the actor list. At this point, the system should already contain the charged particles. Set parameters are fixed and not changed by the tuning algorithm. This can be useful to speed up the tuning during testing or if the parameters are already known.

To prevent the automatic tuning, set the tune parameter to False. To manually tune or retune P3M, call espresso.electrostatics.P3M.Tune(). Note, however, that this is a method the P3M object inherited from espressomd.electrostatics.ElectrostaticInteraction. All parameters passed to the method are fixed in the tuning routine. If not specified in the Tune() method, the parameters prefactor and accuracy are reused.

It is not easy to calculate the various parameters of the P3M method such that the method provides the desired accuracy at maximum speed. To simplify this, it provides a function to automatically tune the algorithm. Note that for this function to work properly, your system should already contain an initial configuration of charges and the correct initial box size. Also note that the provided tuning algorithms works very well on homogeneous charge distributions, but might not achieve the requested precision for highly inhomogeneous or symmetric systems. For example, because of the nature of the P3M algorithm, systems are problematic where most charges are placed in one plane, one small region, or on a regular grid.

The function employs the analytical expression of the error estimate for the P3M method [HE88] and its real space error [KP92] to obtain sets of parameters that yield the desired accuracy, then it measures how long it takes to compute the Coulomb interaction using these parameter sets and chooses the set with the shortest run time.

After execution the tuning routines report the tested parameter sets, the corresponding k-space and real-space errors and the timings needed for force calculations. In the output, the timings are given in units of milliseconds, length scales are in units of inverse box lengths.

### 8.1.2. Coulomb P3M on GPU¶

espressomd.electrostatics.P3MGPU

Required parameters:
• prefactor
• accuracy

The GPU implementation of P3M calculates the far field portion on the GPU. It uses the same parameters and interface functionality as the CPU version of the solver. It should be noted that this does not always provide significant increase in performance. Furthermore it computes the far field interactions with only single precision which limits the maximum precision. The algorithm does not work in combination with the electrostatic extensions Dielectric interfaces with the ICC\star algorithm and Electrostatic Layer Correction (ELC).

## 8.2. Debye-Hückel potential¶

For a list of all parameters see espressomd.electrostatics.DH Uses the Debye-Hückel electrostatic potential defined by

$U^{C-DH} = C \cdot \frac{q_1 q_2 \exp(-\kappa r)}{r}\quad \mathrm{for}\quad r<r_{\mathrm{cut}}$

where $$C$$ is defined as in Eqn. (1). The Debye-Hückel potential is an approximate method for calculating electrostatic interactions, but technically it is treated as other short-ranged non-bonding potentials. For $$r>r_{\mathrm cut}$$ it is set to zero which introduces a step in energy. Therefore, it introduces fluctuations in energy.

For $$\kappa = 0$$, this corresponds to the plain Coulomb potential.

## 8.3. Dielectric interfaces with the ICC$$\star$$ algorithm¶

The ICC$$\star$$ algorithm allows to take into account arbitrarily shaped dielectric interfaces and dynamic charge induction. For instance, it can be used to simulate a curved metallic boundary. This is done by iterating the charge on a set of spatially fixed ICC particles until they correctly represent the influence of the dielectric discontinuity. All ICC particles need a certain area, normal vector and dielectric constant to specify the surface. ICC relies on a Coulomb solver that is already initialized. So far, it is implemented and well tested with the Coulomb solver P3M. ICC is an ESPResSo actor and can be activated via:

icc = ICC(<See the following list of ICC parameters>)


Parameters are:

• first_id:
ID of the first ICC Particle.
• n_icc:
Total number of ICC Particles.
• convergence:
Abort criteria of the iteration. It corresponds to the maximum relative change of any of the interface particle’s charge.
• relaxation:
SOR relaxation parameter.
• ext_field:
Homogeneous electric field added to the calculation of dielectric boundary forces.
• max_iterations:
Maximal number of iterations.
• eps_out:
Relative permittivity of the outer region (where the particles are).
• normals:
List of size n_icc with normal vectors pointing into the outer region.
• areas
List of size n_icc with areas of the discretized surface.
• sigmas
List of size n_icc with an additional surface charge density in absence of any charge induction
• epsilons
List of size n_icc with the dielectric constant associated to the area.

The ICC particles are setup as normal ESPResSo particles. Note that they should be fixed in space and need an initial nonzero charge. The following usage example sets up parallel metallic plates and activates ICC:

# Set the ICC line density and calculate the number of
# ICC particles according to the box size
l = 3.2
nicc = int(box_l / l)
nicc_per_electrode = nicc * nicc
nicc_tot = 2 * nicc_per_electrode
iccArea = box_l * box_l / nicc_per_electrode
l = box_l / nicc

# Lists to collect required parameters
iccNormals = []
iccAreas = []
iccSigmas = []
iccEpsilons = []

# Add the fixed ICC particles:

# Left electrode (normal [0,0,1])
for xi in xrange(nicc):
for yi in xrange(nicc):
system.part.add(pos=[l * xi, l * yi, 0], q=-0.0001, fix=[1, 1, 1], type=icc_type)
iccNormals.extend([0, 0, 1] * nicc_per_electrode)

# Right electrode (normal [0,0,-1])
for xi in xrange(nicc):
for yi in xrange(nicc):
system.part.add(pos=[l * xi, l * yi, box_l], q=0.0001, fix=[1, 1, 1], type=icc_type)
iccNormals.extend([0, 0, -1] * nicc_per_electrode)

# Common area, sigma and metallic epsilon
iccAreas.extend([iccArea] * nicc_tot)
iccSigmas.extend([0] * nicc_tot)
iccEpsilons.extend([100000] * nicc_tot)

icc = ICC(first_id=0,
n_icc=nicc_tot,
convergence=1e-4,
relaxation=0.75,
ext_field=[0, 0, 0],
max_iterations=100,
eps_out=1.0,
normals=iccNormals,
areas=iccAreas,
sigmas=iccSigmas,
epsilons=iccEpsilons)



With each iteration, ICC has to solve electrostatics which can severely slow down the integration. The performance can be improved by using multiple cores, a minimal set of ICC particles and convergence and relaxation parameters that result in a minimal number of iterations. Also please make sure to read the corresponding articles, mainly [ALK+ed][TSuzenS+10][KSH11] before using it.

## 8.4. MMM2D¶

Note

Required features: ELECTROSTATICS, PARTIAL_PERIODIC.

MMM2D is an electrostatics solver for explicit 2D periodic systems. It can account for different dielectric jumps on both sides of the non-periodic direction. MMM2D Coulomb method needs periodicity 1 1 0 and the layered cell system. The performance of the method depends on the number of slices of the cell system, which has to be tuned manually. It is automatically ensured that the maximal pairwise error is smaller than the given bound. Note that the user has to take care that the particles don’t leave the box in the non-periodic z-direction e.g. with constraints. By default, no dielectric contrast is set and it is used as:

mmm2d = electrostatics.MMM2D(prefactor=C, maxPWerror=1e-3)


where the prefactor $$C$$ is defined in Eqn. (1). For a detailed list of parameters see espressomd.electrostatics.MMM2D. The last two, mutually exclusive parameters dielectric and dielectric_constants_on allow to specify dielectric contrasts at the upper and lower boundaries of the simulation box. The first form specifies the respective dielectric constants in the media, which however is only used to calculate the contrasts. That is, specifying $$\epsilon_t=\epsilon_m=\epsilon_b=\text{const}$$ is always identical to $$\epsilon_t=\epsilon_m=\epsilon_b=1$$:

mmm2d = electrostatics.MMM2D(prefactor=C, maxPWerror=1e-3, dielectric=1,
top=1, mid=1, bot=1)


The second form specifies only the dielectric contrasts at the boundaries, that is $$\Delta_t=\frac{\epsilon_m-\epsilon_t}{\epsilon_m+\epsilon_t}$$ and $$\Delta_b=\frac{\epsilon_m-\epsilon_b}{\epsilon_m+\epsilon_b}$$. Using this form allows to choose $$\Delta_{t/b}=-1$$, corresponding to metallic boundary conditions:

mmm2d = electrostatics.MMM2D(prefactor=C, maxPWerror=1e-3, dielectric_contrast_on=1,
delta_mid_top=-1, delta_mid_bot=-1)


Using const_pot allows to maintain a constant electric potential difference pot_diff between the xy-planes at $$z=0$$ and $$z=L$$, where $$L$$ denotes the box length in $$z$$-direction:

mmm2d = electrostatics.MMM2D(prefactor=100.0, maxPWerror=1e-3, const_pot=1, pot_diff=100.0)


This is done by countering the total dipole moment of the system with the electric field $$E_{induced}$$ and superposing a homogeneous electric field $$E_{applied} = \frac{U}{L}$$ to retain $$U$$. This mimics the induction of surface charges $$\pm\sigma = E_{induced} \cdot \epsilon_0$$ for planar electrodes at $$z=0$$ and $$z=L$$ in a capacitor connected to a battery with voltage pot_diff. Using 0 is equivalent to $$\Delta_{t/b}=-1$$.

Finally, the far cutoff setting should only be used for testing reasons, otherwise you are more safe with the automatic tuning. If you even don’t know what it is, do not even think of touching the far cutoff. For details on the MMM family of algorithms, refer to appendix The MMM family of algorithms. Please cite [AH02b] when using MMM2D.

A complete (but unphysical) sample script for a plate capacitor simulated with MMM2D can be found in /samples/visualization_mmm2d.py.

## 8.5. Electrostatic Layer Correction (ELC)¶

ELC can be used to simulate charged system with 2D periodicity. In more detail, is a special procedure that converts a 3D electrostatic method to a 2D method in computational order N. Currently, it only supports P3M. This means, that you will first have to set up the P3M algorithm before using ELC. The algorithm is definitely faster than MMM2D for larger numbers of particles ($$>400$$ at reasonable accuracy requirements). The periodicity has to be set to 1 1 1 still, ELC cancels the electrostatic contribution of the periodic replica in z-direction. Make sure that you read the papers on ELC ([AdeJoannisH02][TAH08]) before using it. ELC is an ESPResSo actor and is used with:

elc = electrostatic_extensions.ELC(gap_size=box_l * 0.2, maxPWerror=1e-3)

Parameters are:
• gap_size:
The gap size gives the height of the empty region between the system box and the neighboring artificial images. ESPResSo does not make sure that the gap is actually empty, this is the users responsibility. The method will compute fine if the condition is not fulfilled, however, the error bound will not be reached. Therefore you should really make sure that the gap region is empty (e.g. with wall constraints).
• maxPWerror:
The maximal pairwise error sets the least upper bound (LUB) error of the force between any two charges without prefactors (see the papers). The algorithm tries to find parameters to meet this LUB requirements or will throw an error if there are none.
• delta_mid_top/delta_mid_bot:
ELC can also be used to simulate 2D periodic systems with image charges, specified by dielectric contrasts on the non-periodic boundaries ([TAH08]). Similar to MMM2D, these can be set with the keywords delta_mid_bot and delta_mid_top, setting the dielectric jump from the simulation region (middle) to bottom (at z<0) and from middle to top (z > box_l[2] - gap_size). The fully metallic case delta_mid_top=delta_mid_bot=-1 would lead to divergence of the forces/energies in ELC and is therefore only possible with the const_pot option.
• const_pot:
As described, setting this to 1 leads to fully metallic boundaries and behaves just like the mmm2d parameter of the same name: It maintains a constant potential pot_diff by countering the total dipole moment of the system and adding a homogeneous electric field according to pot_diff.
• pot_diff:
Used in conjunction with const_pot set to 1, this sets the potential difference between the boundaries in the z-direction between z=0 and z = box_l[2] - gap_size.
• far_cut:
The setting of the far cutoff is only intended for testing and allows to directly set the cutoff. In this case, the maximal pairwise error is ignored.
• neutralize:
By default, ELC just as P3M adds a homogeneous neutralizing background to the system in case of a net charge. However, unlike in three dimensions, this background adds a parabolic potential across the slab [BAC09]. Therefore, under normal circumstance, you will probably want to disable the neutralization for non-neutral systems. This corresponds then to a formal regularization of the forces and energies [BAC09]. Also, if you add neutralizing walls explicitly as constraints, you have to disable the neutralization. When using a dielectric contrast or full metallic walls (delta_mid_top != 0 or delta_mid_bot != 0 or const_pot=1), neutralize is overwritten and switched off internally. Note that the special case of non-neutral systems with a non-metallic dielectric jump (eg. delta_mid_top or delta_mid_bot in ]-1,1[) is not covered by the algorithm and will throw an error.

## 8.6. MMM1D¶

Note

Required features: ELECTROSTATICS, PARTIAL_PERIODIC for MMM1D, the GPU version additionally needs the features CUDA and MMM1D_GPU.

from espressomd.electrostatics import MMM1D
from espressomd.electrostatics import MMM1DGPU


Please cite [AH05] when using MMM1D.

See espressomd.electrostatics.MMM1D or espressomd.electrostatics.MMM1DGPU for the list of available parameters.

mmm1d = MMM1D(prefactor=C, far_switch_radius=fr, maxPWerror=err, tune=False,
bessel_cutoff=bc)
mmm1d = MMM1D(prefactor=C, maxPWerror=err)


where the prefactor $$C$$ is defined in Eqn. (1). MMM1D Coulomb method for systems with periodicity 0 0 1. Needs the nsquared cell system (see section Cellsystems). The first form sets parameters manually. The switch radius determines at which xy-distance the force calculation switches from the near to the far formula. The Bessel cutoff does not need to be specified as it is automatically determined from the particle distances and maximal pairwise error. The second tuning form just takes the maximal pairwise error and tries out a lot of switching radii to find out the fastest one. If this takes too long, you can change the value of the setmd variable timings, which controls the number of test force calculations.

mmm1d_gpu = MMM1DGPU(prefactor=C, far_switch_radius=fr, maxPWerror=err,
tune=False, bessel_cutoff=bc)
mmm1d_gpu = MMM1DGPU(prefactor=C, maxPWerror=err)


MMM1D is also available in a GPU implementation. Unlike its CPU counterpart, it does not need the nsquared cell system. The first form sets parameters manually. The switch radius determines at which xy-distance the force calculation switches from the near to the far formula. If the Bessel cutoff is not explicitly given, it is determined from the maximal pairwise error, otherwise this error only counts for the near formula. The second tuning form just takes the maximal pairwise error and tries out a lot of switching radii to find out the fastest one.

For details on the MMM family of algorithms, refer to appendix The MMM family of algorithms.

## 8.7. Scafacos Electrostatics¶

Espresso can use the electrostatics methods from the SCAFACOS Scalable fast Coulomb solvers library. The specific methods available depend on the compile-time options of the library, and can be queried using espressomd.scafacos.available_methods

To use SCAFACOS, create an instance of espressomd.electrostatics.Scafacos and add it to the list of active actors. Three parameters have to be specified:

• method_name: name of the SCAFACOS method being used.
• method_params: dictionary containing the method-specific parameters
• prefactor: Coulomb prefactor as defined in (1).

The method-specific parameters are described in the SCAFACOS manual. Additionally, methods supporting tuning have the parameter tolerance_field which sets the desired root mean square accuracy for the electric field

To use the, e.g., ewald solver from SCAFACOS as electrostatics solver for your system, set its cutoff to $$1.5$$ and tune the other parameters for an accuracy of $$10^{-3}$$, use:

from espressomd.electrostatics import Scafacos
scafacos = Scafacos(prefactor=1, method_name="ewald",
method_params={"ewald_r_cut": 1.5, "tolerance_field": 1e-3})