In the subsystem formulation of the density-functional theory a large system is decomposed into several constituting fragments that are treated individually. This approach offers the advantage of focusing the attention and computational cost on a limited portion of the whole system while including all the remaining enviromental effects through an effective embedding potential. Here we refer in particular to the (fully-variational) Frozen Density Embedding (FDE)[139] with the Kohn-Sham Constrained Electron Density (KSCED) equations[140,141].
In the FDE/KSCED method the embedding potential required by an embedded subsystem with density ρ_{A} to account for the presence of another (frozen) subsystem with density ρ_{B} is:
Using freeze-and-thaw [142] cycles, the role of the frozen and the embedded subsystem is iteratively exchanged, till convergence. If expressions (15.2) and (15.3) are computed exactly, then the density ρ_{A} + ρ_{B} will coincide with the exact density of the total system.
Because the FDE/KSCED was originally developed in the Kohn-Sham framework, using standard GGA approximations for E_{xc}[ρ], the non-additive exchange-correlation potential ( δE_{xc}^{nadd}/δρ_{A}(r)) can be computed exactly as a functional of the density, leaving the expression of the non-additive kinetic energy term as the only approximation (with respect the corresponding GGA calculation of the total system), because the exact explicit density dependence of T_{s} from the density is not known. Using GGA approximations for the kinetic energy functional ( T_{s} ) we have:
The FDE total energy of total system is:
E^{FDE}[,] | = | T_{s}[] + T_{s}[] + T_{s}^{nadd}[;] | |
+ | V_{ext}[A + B] + J[ + ] + E_{xc}[ + ] . | (15.6) |
Using the Generalized Kohn-Sham (GKS) theory, also hybrid exchange-correlation functionals can be used in embedding calculations. To obtain a practical computational method, the obtained embedding potential must be approximated by a local expression as shown in Ref. [143]. This corresponds to performing for each subsystem hybrid calculations including the interaction with other subsystems through an embedding potential derived at a semilocal level of theory. When orbital dependent exchange-correlation functionals (e.g. hybrid functional and LHF) are considered within the FDE method, the embedding potential includes a non-additive exchange-correlation term of the form
E^{nadd}_{xc}[ρ_{A};ρ_{B}] = E_{xc}[Φ^{A+B}[ρ_{A} + ρ_{B}]] - E_{xc}[Φ^{A}[ρ_{A}]] - E_{xc}[Φ^{B}[ρ_{B}]] | (15.7) |
E^{nadd}_{xc}[ρ_{A};ρ_{B}] E^{GGA}_{xc}[ρ_{A} + ρ_{B}] - E^{GGA}_{xc}[ρ_{A}] - E^{GGA}_{xc}[ρ_{B}] . | (15.8) |