The Periodic Electrostatic Embedded Cluster Method (PEECM) functionality [205] provides electronic embedding of a finite, quantum mechanical cluster in a periodic, infinite array of point charges. It is implemented within HF and DFT energy and gradient TURBOMOLE modules: dscf, grad, ridft, rdgrad, and escf. Unlike embedding within a finite set of point charges the PEEC method always yields the correct electrostatic (Madelung) potential independent of the electrostatic moments of the point charges field. It is also significantly faster than the traditional finite point charges embedding.
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
|
where UC denotes the unit cell of point charges, Dμν are elements of the density matrix, μ, ν are basis functions, qk, k denote charges and positions of point charges, and denote direct lattice vectors of the outer part O. It is evaluated using the periodic fast multipole method (PFMM) [206] which, unlike the Ewald method [207], defines the lattice sums entirely in the direct space. In general, PFMM yields a different electrostatic potential than 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!).
There are three key steps in setting up a PEECM calculation. In the first step the periodic field of point charges has to be defined by specifying the point charges unit cell. Next step is the definition of the part infinite of point charges field that will be replaced by the explicit quantum mechanical cluster. Finally, the quantum mechanical cluster together with surrounding ECPs representing cationic sites as well as point charges representing anions is defined and put in place of the point charges. The input preparation steps can be summarized as follows
The following two examples show the definition of the point charges unit cells.
Example 1. Ca4F19 cluster embedded in bulk CaF2
In this example a QM cluster with the composition Ca4F19, surrounded by 212 ECPs and 370 explicit point charges, representing Ca2+ cations and F- anions is embedded in a periodic field of point charges (+2 for Ca and -1 for F) corresponding to the CaF2 fluorite lattice.
First, the program has to know that this is a three-dimensional periodic system. This is specified by the keyword periodic 3, meaning periodicity in three dimensions. The dimensions of the unit cell for bulk CaF2 are given in the subsection cell of the $embed keyword. By default, the unit cell dimensions are specified in atomic units and can be changed to Å using cell ang. The positions of the point charges in the unit cell are specified in the subsection content. In this example positions are given in fractional crystal coordinates (content frac). You can change this by specifying content for Cartesian coordinates in atomic units or content ang for Cartesian coordinates in Å. The values of point charges for Ca and F are given in the subsection charges.
The above input defines a periodic, perfect, and infinite three-dimensional lattice of point charges corresponding to the bulk CaF2 structure. In order to use this lattice for PEECM calculation we have to make “space” for our QM cluster and the isolating shell. This is done by specifying the part of the lattice that is virtually removed from the perfect periodic array of point charges to make space for the cluster. The positions of the removed point charges are specified in the subsection cluster of the $embed keyword. Note, that the position of the QM cluster and the isolating shell must exactly correspond to the removed part of the crystal, otherwise positions of the cluster atoms would overlap with positions of point charges in the periodic lattice, resulting in a “nuclear fusion”.
repeated for Ca216F389
end
By default, the positions of point charges are specified in atomic units as Cartesian coordinates. You can change this by specifying cluster frac for fractional crystal coordinates or cluster ang for Cartesian coordinates in Å.
Finally, you have to specify the coordinates of the QM cluster along with the surrounding ECPs representing cationic sites and explicit point charges representing anions. This is done in the usual way using the $coord keyword.
repeated for Ca216F389
$end
This is the standard TURBOMOLE syntax for atomic coordinates. The actual distinction between QM cluster, ECP shell, and explicit point charges is made in the $atoms section.
In the example above the F atoms 1 and 6-23 as well Ca atoms 2-5 are defined as QM atoms with def-TZVP basis sets. The Ca atoms 24-235 are pure ECPs and have no basis functions (basis =none) and F atoms 236-605 are explicit point charges with charge -1, with no basis functions and no ECP.
This step ends the input definition for the PEECM calculation.
Example 2. Al8O12 cluster embedded in α-Al2O3 (0001) surface
In this example a QM cluster with the composition Al8O12, surrounded by 9 ECPs representing Al3+ cations is embedded in a two-dimensional periodic field of point charges (+3 for Al and -2 for O) corresponding to the (0001) surface of α-Al2O3.
As in the first example, the program has to know that this is a two-dimensional periodic system and this is specified by the keyword periodic 2. The dimensions of the unit cell for the (0001) α-Al2O3 surface are given in the subsection cell of the $embed keyword. The aperiodic direction is always the z direction, but you have to specify the unit cell as if it was a 3D periodic system. This means that the third dimension of the unit cell must be large enough to enclose the entire surface in this direction. The unit cell dimensions are specified in Å using cell ang. The positions of the point charges in the unit cell are specified as Cartesian coordinates in Å (content ang). The values of point charges for Al and O are given in the subsection charges.
The above input defines a periodic, perfect, and infinite two-dimensional lattice of point charges corresponding to the (0001) α-Al2O3 surface. In order to use the lattice for PEECM calculation we have to make “space” for our QM cluster and the surrounding ECP shell. This is done by specifying the part of the lattice that is virtually removed from the perfect periodic array of point charges to make space for the cluster. The positions of the removed point charges are specified in the subsection cluster of the $embed keyword. Note, that the position of the QM cluster must exactly correspond to the removed part of the crystal, otherwise positions of the cluster atoms would overlap with positions of point charges in the periodic lattice, resulting in a “nuclear fusion”.
The positions of point charges are specified in Å as Cartesian coordinates.
Finally, you have to specify the coordinates of the QM cluster along with the surrounding ECPs. This is done in the usual way using the $coord keyword.
This is the standard TURBOMOLE syntax for atomic coordinates. The actual distinction between QM cluster and ECP shell is made in the $atoms section.
In the example above the Al atoms 1-8 and O atoms 9-20 are defined as QM atoms with def-SV(P) basis sets. The Al atoms 21-29 are pure ECPs and have no basis functions (basis =none).
This step ends the input definition for the PEECM calculation.