11 Molecules in Complex Environments: Solvent Models, QM/MM and QM/EFP Features, Density Embedding

11.7 Frozen-Density Embedding Theory

(June 30, 2021)

Frozen-Density Embedding Theory 1166 Wesolowski T. A., Warshel A.
J. Phys. Chem.
(1993), 97, pp. 8050.
Link
, 1167 Wesolowski T. A.
Phys. Rev. A
(2008), 77, pp. 012504.
Link
(FDET) provides a formal framework in which the whole system is described by means of two independent quantities: the embedded wave function (interacting or not) and the density associated with the environment.

The total energy equation in frozen density embedding theory for a wave function in state I embedded in a environment density ρB(𝐫) reads (for definitions see Table 11.9):

EABtot[ΨAI,ρB] =ΨAI|H^A|ΨAI+VBnuc[ρAI]+Jint[ρAI,ρB]+Excnad[ρAI,ρB]
+Tsnad[ρAI,ρB]+EvBHK[ρB]+VAnuc[ρB] (11.107)

The embedding operator v^emb, which is added to the Hamiltonian of subsystem A (H^A), is given in the form of a potential:

vemb[ρAI,ρB,vB](𝐫)=vB(𝐫)+ρB(𝐫)|𝐫-𝐫|d𝐫+δExc,Tnad[ρAI,ρB]δρAI(𝐫) (11.108)

The last term (non-electrostatic component) in equation 11.108 causes the embedding potential to be ρA-dependent, which in return induces an inconsistency between the potential and the energy. In the canonical form of FDET (conventional FDET) this inconsistency is addressed by performing macrocycles in which the embedding potential is repeatedly constructed using the current (embedded) density ρAcurr(𝐫) after each cycle until self-consistency is reached. Each calculation performed with FDE-Man accounts for just one cycle, so self-consistency can only be reached by running multiple calculations, where the densities are updated using the importing options for the density matrices described in Section 11.7.2.2. Self-consistent macrocycles include the mutual polarization procedure: Freeze-and-Thaw.

However, in linearized FDET the non-additive energy functionals (for abbreviation denoted as Exc,Tnad[ρAI,ρB]) are each approximated by a functional which is linear in ρA(𝐫). The approximation is constructed as a Taylor expansion of the non-additive energy functional at a reference density ρAref(𝐫) with the series being truncated after the linear term.

Exc,Tnad[ρAI,ρB]Exc,Tnad[ρAref,ρB]+(ρAI(𝐫)-ρAref(𝐫))δExc,Tnad[ρAref,ρB]δρAref(𝐫)d𝐫 (11.109)

In contrast to conventional FDET, the embedding potential then becomes ρA-independent and macrocycles are no longer necessary. Another consequence of the linearization is that orthogonality between states is maintained since the same potential is used for all states.