Electrostatics and Poisson Solvers

The electrostatic boundary conditions are part of the physical model, not just a numerical setting. They determine which periodic images interact, whether a charged calculation needs a compensating charge, and which total-energy differences are meaningful. Set them explicitly before converging the cell size.

Periodicity settings

Two settings have to be distinguished:

• CELL/PERIODIC controls the periodicity used for the atomic system, including geometry and pair lists.

• POISSON/PERIODIC selects the periodic directions of the electrostatic Green’s function.

They should normally describe the same physical system. CP2K permits different values because some specialized workflows need them, but a mismatch can make different parts of a calculation use different images. Do not use a mismatch merely to suppress interactions in a vacuum direction.

The generalized IMPLICIT solver has an additional, independent BOUNDARY_CONDITIONS keyword. Its default is PERIODIC, so POISSON_SOLVER IMPLICIT does not become a 1D or 2D solver solely because POISSON/PERIODIC was set to X, XY, or another reduced periodicity. Configure the IMPLICIT subsection explicitly when nonperiodic, mixed, or dielectric boundary conditions are required.

For example, a slab that is periodic in the \(x\) and \(y\) directions can use the analytic 2D Green’s function as follows:

&SUBSYS
  &CELL
    ABC 10.0 10.0 20.0
    PERIODIC XY
  &END CELL
  # ...
&END SUBSYS

&DFT
  &POISSON
    PERIODIC XY
    POISSON_SOLVER ANALYTIC
  &END POISSON
  # ...
&END DFT

Solver overview

Choose POISSON_SOLVER from the intended periodicity and boundary conditions. The input reference is the authoritative list of currently supported combinations.

Solver	Supported periodicity	Main considerations
PERIODIC	3D	Standard fully periodic reciprocal-space solution.
ANALYTIC	0D, 1D, 2D	Analytic reciprocal-space Green’s functions; convergence can be slow.
MT	0D, 2D	Martyna–Tuckerman image decoupling. The cell must be at least twice the extent of the complete electronic density. Martyna1999
MULTIPOLE	0D	Fits the total charge with atom-centred Gaussians to decouple periodic images. Blöchl1995
WAVELET	0D, 2D, 3D	Requires FFTSG-compatible grid dimensions. The default EXTENDED_FFT_LENGTHS setting, .FALSE., guarantees this independently of the selected FFT library. The density must decay to zero at nonperiodic cell faces. Genovese2006 Genovese2007
IMPLICIT	Generalized 0D–3D conditions	Supports periodic, Neumann, and mixed Dirichlet/periodic conditions and spatially varying dielectrics. Configure its boundary conditions explicitly. BaniHashemian2016

Warning

MT can return completely wrong results when the electronic density reaches beyond half the cell. The relevant extent includes diffuse density tails, not only the nuclear coordinates.

For an isolated, localized molecule or cluster, MT, WAVELET, or MULTIPOLE are common 0D choices; ANALYTIC and IMPLICIT provide further options. A slab normally needs a genuine 2D solver or explicitly constructed mixed boundary conditions. For a wire, use a solver that supports 1D periodicity. Fully periodic bulk calculations normally use PERIODIC.

Systems with a net charge

The electrostatic energy of a non-neutral periodic cell is not defined without an additional convention. With the 3D PERIODIC solver, CP2K uses a homogeneous compensating background and omits the singular zero reciprocal-space component. The electronic density, ionic charge, and background therefore form one electrostatic problem; separate electron–background or ion–background energies are not independently meaningful.

Increasing the cell volume reduces the background density but also changes the interaction with periodic images. Consequently, a charged 3D calculation has a cell-size- and cell-shape-dependent finite-size error that usually decays only algebraically. CP2K does not infer a universal physical correction merely from CHARGE. Compare energies only between calculations that use the same boundary-condition and potential-zero convention, or apply a correction appropriate to the physical model.

An isolated 0D solver can give a finite energy for a localized charged cluster without a periodic background, provided that the density is contained in the numerical cell. The situation differs for partially periodic systems: a net charge per unit length or area produces a long-ranged field. Without an explicit countercharge, electrode, electrolyte, or other compensating model, the energy of a charged 1D wire or 2D slab generally has no vacuum-independent isolated limit. Increasing the vacuum can then change the reported energy rather than converge it.

Important

Total energies from different Poisson solvers are not automatically on the same reference for a charged system. A constant shift of the electrostatic potential is irrelevant for a neutral cell but changes the energy assigned to a cell with nonzero net charge. Solver-to-solver agreement is therefore not, by itself, a valid correctness test.

Implicit solvent, planar countercharges, and electrode models can make a charged interface a well-defined physical problem, but they describe different systems. Select such a model from the experimental or thermodynamic boundary conditions rather than as a numerical correction applied afterwards.

Converging vacuum and cell size

Use a convergence series that preserves the physical model:

1. Choose the periodic directions, Poisson solver, charge-compensation model, and potential reference first.

2. Converge CUTOFF and REL_CUTOFF separately.

3. For a 0D system, enlarge every nonperiodic direction. For a slab or wire, enlarge only the nonperiodic direction or directions. Keep the structure centred and keep the basis, functional, SCF thresholds, and periodic cell vectors fixed.

4. Record energy differences relative to the largest cell together with forces and, where relevant, the planar-averaged potential and charge density. Check that the density is negligible at every nonperiodic boundary required by the solver.

5. Repeat at a higher grid cutoff if changes in the FFT grid could be confused with a vacuum-size trend.

For a neutral localized system, the total energy should reach a plateau once image interactions, density truncation, and grid changes are below the target accuracy. For a charged 3D supercell, analyse the finite-size trend under the homogeneous-background convention. For a charged 1D or 2D system, do not interpret the absence of a plateau as a numerical failure until the electrostatic model contains the physically required compensation.

Diffuse basis functions can create states concentrated in the vacuum, especially for charged surfaces. Inspect orbitals and density when adding vacuum or diffuse functions: changing the cell may otherwise change the electronic state as well as its electrostatic finite-size error.