The frozen density energy

The first intermediate state is called the frozen density state, where the two fragments, A and B, are allowed to approach each other without distorting the densities around them. As a result the density of the frozen state is simply a superimposition of the original densities around A and B p[p + /Og] = P + Pb)- The energy difference between the initial state and this frozen state is called the frozen density energy (AEfrz), which consists of both electrostatic and van der Waals interactions  

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The implementation of DEDA represents a critical difference from other EDA methods in that energies of all intermediate states are variationally determined, which to the best of our knowledge has not been achieved before. For the frozen density energy, the optimization is done through a constrained search formulation in DFT, i.e., [/o + = min E [p], and implemented with the Wu-... 

The most important distinction between DEDA and other wave function-based EDA approaches [4-15] lies in the calculation of the frozen density energy. We have explained above how DEDA uses constrained search to variationally calculate the energy of the frozen density state where fragments densities are superimposed without distortions. This approach not only yields an optimal... 

Wu, Q. Variational nature of the frozen density energy in density-based energy decomposition analysis and its application to torsional potential./ Chem. Phys., 140, 244109 (20143. 

Wesolowski T A and Warshel A 1994 Ab initio free energy perturbation calculations of solvation free energy using the frozen density functional approach J. Phys. Chem. 98 5183... 

With the frozen density ansatz all terms in parenthesis in the last equation will be zero. The only contribution from F to the adsorption energy difference is therefore the non-local electrostatic energy,... 

In Equation (1.161) (or equivalently in Equation (1.165) for the nonequilibrium case) we have shown that excited state free energies can be obtained by calculating the frozen-PCM energy E s and the relaxation term of the density matrix, PA (or P 1) where the calculation of the relaxed density matrices requires the solution of a nonlinear problem in which the solvent reaction field is dependent on such densities. 

Wesolowski, T. and Warshell A., Ab Initio Free Energy Perturbation Calculations of Solvation Free Energy Using the Frozen Density Functional Approach. J.Phys.Chem (1994) 98 5183-5187. 

Alternatively, there have been other approaches earlier that, currently, seem to be less frequently used than those discussed above. Among those is the frozen-density-functional approach of Wesolowski and Warshel. This is an explicit approach where solvent molecules are directly included in the calculations. It is based on density-functional theory and a separation of the total electron density into that of the solute, pi, and that of the solvent, p2- According to the density-functional theory, we may write the total electronic energy as a functional of... 

In Table 1 we summarize their main findings. For the frozen-density calculations they considered two different approaches, one where the solvent-molecule density was kept fixed and one where it was allowed to relax. In the table we have only shown the results for the latter, which according to the authors led to an improved accuracy. The table shows that the dipole and the quadrupole moments are very similar for both approaches, which is to a lesser extent the case for the excitation energies and the static (hyper)polarizabilities. The latter were calculated using time-dependent density-functional theory. In order to understand this discrepancy the authors used also a supermolecule approach with just two solvent molecules. By comparing with results from calculations with the frozen-density and the polarizable-molecule approaches on the same system they concluded that the frozen-density approach was the more accurate one in calculating the responses to electromagnetic fields. 

In the second intermediate state, the density on each fragment is allowed to relax to the extent that the number of elertrons on each fragment is constrained to be the same as in the frozen density (IV and Alg, respectively) that is, no charge transfer. The energy difference between the two intermediate states I and II leads to the polarization component (A poi). 

AFfrz separated from the density relaxation terms (AEpoi and AEct) but also allows a clean separation of electrostatic and Pauli repulsion terms. Similar intermediate states in wave function-based EDA approaches are represented by the HL antisymmetrization of two fragments wave functions, [which is] necessary because molecular orbitals from different fragments are not orthogonal. This antisymmetrized wave function, however, deforms the frozen density [12] that is to say, its density does not correspond to the sum of fragments densities. Such ambiguity makes it difficult to separate electrostatic and Pauli repulsion terms in other EDA approaches. In addition, a one-step antisymmetrization of the wave functions means its energy is not variational. 

Figure 4.3 Comparision of EDA frozen energies in kcal/mol along an angle for water dimer [upper] and formamide dimer [lower] with DEDA [blue curves] and MO-EDA [red curves] The MO-EDA employs the Heitler-London [HL] antisymmetrization of two fragments wave functions to represent the frozen density state. Reprinted with permission from Lu, Z., Zhou, N., Wu, Q. and Zhang, Y. Directional dependence of hydrogen bonds A density-based energy decomposition analysis and its implications on force field development./Chem Theory Comput 7, 4038-4049 [2011]. Copyright [2011] American Chemical Society.

The accuracy of the frozen energy term is also largely influenced by the van der Waals (vdW) term, which is "the other" part of the frozen density interaction. One of the main challenges with the force field development is to model this vdW interaction. Currently the Lennard-Jones 12-6 term [78], - Bij/Rfj),... 

Cortona embedded a DFT calculation in an orbital-free DFT background for ionic crystals [183], which necessitates evaluation of kinetic energy density fiinctionals (KEDFs). Wesolowski and Warshel [184] had similar ideas to Cortona, except they used a frozen density background to examine a solute in solution and examined the effect of varying the KEDF. Stefanovich and Truong also implemented Cortona s method with a frozen density background and applied it to, for example, water adsorption on NaCl(OOl) [185]. 

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