Theoretical Background

Generally, the PEEC method divides the entire, periodic and infinite system into two parts, the inner (I) part, or so called cluster, and the outer (O) part which describes its environment. Thus, unlike "true" periodic quantum mechanical methods, PEECM primarily aims at calculations of structure and properties of localized defects in dominantly ionic crystals. The innermost part of the cluster is treated quantum mechanically (QM), whereas in the remaining cluster part cations are replaced by effective core potentials (ECPs) and anions by ECPs or by simply point charges. Such an "isolating" outer ECP shell surrounding the actual QM part is necessary in order to prevent artificial polarization of the electron density by cations which would otherwise be in a direct contact with the QM boundary. The outer part or the environment of the cluster is described by a periodic array of point charges, representing cationic and anionic sites of a perfect ionic crystal.

The electronic Coulomb energy term arising from the periodic field of point charges surrounding the cluster has the following form

$$\sum_{{\mu \nu}}^{} \sum_{k}^{{N \in \text{UC}}} \sum_{{\vec L \in \text{O}}}^{{\infty}} \int {\frac{{\mu(\vec r)\nu(\vec r)}}{{\vert\vec r - \vec R_k - \vec L\vert}}} \vec{r}\,$$

where UC denotes the unit cell of point charges, D_μν are elements of the density matrix, μ, ν are basis functions, q_k, $\vec{R}_{k}^{}$ denote charges and positions of point charges, and $\vec{L}\,$ denote direct lattice vectors of the outer part O. It is evaluated using the periodic fast multipole method (PFMM) [68] which, unlike the Ewald method [69], defines the lattice sums entirely in the direct space. In general, PFMM yields a different electrostatic potential then the Ewald method, but the difference is merely a constant shift which depends on the shape of external infinite surface of the solid (i.e. on the way in which the lattice sum converges toward the infinite limit). However, this constant does not influence relative energies which are the same as obtained using the Ewald method, provided that the total charge of the cluster remains constant. Additionally, since the electrostatic potential within a solid is not a well defined quantity, both the absolute total energies and orbital energies have no meaning (i.e. you cannot compare energies of neutral and charged clusters!).