Embedding potential, approximants

In principle, testing the accuracy of a given approximation to T ad[pA, Pb would require exact reference data for T ad[pA, Pb - In general, such reference data for a given pair of electron densities pA and ps can be obtained by means of the Levy constrained-search procedure and the definition of T ad[pA,Pb]- In the embedding potential of Eq. 53, however, not T ad[pA, Pb] but its functional derivative is used. It is, therefore, useful to start with the analysis of the accuracy of approximations to STrd[pA,ps]... 

The LSGF method on the other hand is an order-IV method for calculation of the electronic system. It is based on a supercell (which may just be one unit cell) with periodic boundary conditions, see Fig.(4.5), and the concept of a Local Interaction Zone (LIZ), which is embedded in an effective medium, usually chosen to be the Coherent Potential Approximation medium (see next chapter). For each atom in the supercell, one uses the Dyson equation to solve the electronic structure problem as an impurity problem in the effective medium. The ASA is employed as well as the ASA+M correction described above. The total energy is defined to... 

The latter is equivalent to an electron in the conduction band and a hole in the valence band, i.e., with a radius large compared to the lattice parameters. Therefore, bonding of the electron is weak similar to a positronium atom embedded in a dielectric with dielectric constant e and an interaction potential approximated by V(r)=-e /er. Such a system has energy states similar to the hydrogen atom following the formula ... 

Density-based embeddings This approach starts with a calculation on the whole system to construct an approximate yet accurate representation of the total density Plot by performing a periodic DFT calculation. Then, a guess density of the cluster is constructed from a calculation on the isolated unit or using some simple embedding scheme as described above. The total density is now divided in two parts ptot = Pi+ P2 and the one-electron embedding potential is constructed from the functional derivative of interaction energy with respect to the cluster density pi. 

The difference between the Kohn-Sham results for the isolated radical and for the complex arises om the intermolecular interactions. The closer are the KSCED and the supermolecule KS results the better is the applied KSCED embedding potential. The KSCED calculations are performed using super-molecular expansion of the subsystem s electron densities (i.e. the electron densities of the subsystems are allowed to spread over the whole complex). In such KSCED calculations in which the electron density of each subsystem is expanded using all basis functions of the complex and which apply the freeze-and-thaw cycle, any differences between the supermolecule Kohn-Sham results and the KSCED ones can be attributed to the accuracy of the applied approximation for the non-additive kinetic energy. 

Approximations to the exact (in the Hartre Fock approximation) separable embedding potential were introduced in [490] that enable one to incorporate this potential into existing molecular calculation packages. The test calculations on the (CH3)20 molecule were performed that showed good accuracy of the potential. 

In both the FDET and supermolecular TDDFT calculations, we used the SAOP scheme [129] to approximate the relevant exchange-correlation potential contributions. To examine the possible dependence of the complexation-induced shifts in the excitation energy of the cis-7HQ chromophore on the approximations exploited for one of the nonelectrostatic components of the embedding potential... 

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