# 11.7 Frozen-Density Embedding Theory

(June 30, 2021)

Frozen-Density Embedding Theory , (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 $\rho_{B}(\mathbf{r})$ reads (for definitions see Table 11.9):

 $\displaystyle E_{AB}^{\mathrm{tot}}[\Psi_{A}^{I},\rho_{B}]$ $\displaystyle=\langle\Psi_{A}^{I}|\hat{H}_{A}|\Psi_{A}^{I}\rangle+V_{B}^{% \mathrm{nuc}}[\rho_{A}^{I}]+J_{\mathrm{int}}[\rho_{A}^{I},\rho_{B}]+E_{xc}^{% \mathrm{nad}}[\rho_{A}^{I},\rho_{B}]$ $\displaystyle\quad+T_{s}^{\mathrm{nad}}[\rho_{A}^{I},\rho_{B}]+E_{v_{B}}^{% \mathrm{HK}}[\rho_{B}]+V_{A}^{\mathrm{nuc}}[\rho_{B}]$ (11.107)

The embedding operator $\hat{v}_{\mathrm{emb}}$, which is added to the Hamiltonian of subsystem A $\left(\hat{H}_{A}\right)$, is given in the form of a potential:

 $v_{\mathrm{emb}}[\rho_{A}^{I},\rho_{B},v_{B}](\mathbf{r})=v_{B}(\mathbf{r})+% \int\frac{\rho_{B}(\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}% \mathrm{d}\mathbf{r}+\frac{\delta E_{xc,T}^{\mathrm{nad}}[\rho_{A}^{I},\rho_{B% }]}{\delta\rho_{A}^{I}(\mathbf{r})}$ (11.108)

The last term (non-electrostatic component) in equation 11.108 causes the embedding potential to be $\rho_{A}$-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 $\rho_{A}^{\mathrm{curr}}(\mathbf{r})$ 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 $E_{xc,T}^{\mathrm{nad}}[\rho_{A}^{I},\rho_{B}]$) are each approximated by a functional which is linear in $\rho_{A}(\mathbf{r})$. The approximation is constructed as a Taylor expansion of the non-additive energy functional at a reference density $\rho_{A}^{\mathrm{ref}}(\mathbf{r})$ with the series being truncated after the linear term.

 $E_{xc,T}^{\mathrm{nad}}[\rho_{A}^{I},\rho_{B}]\approx E_{xc,T}^{\mathrm{nad}}[% \rho_{A}^{\mathrm{ref}},\rho_{B}]+\int\left(\rho_{A}^{I}(\mathbf{r})-\rho_{A}^% {\mathrm{ref}}(\mathbf{r})\right)\frac{\delta E_{xc,T}^{nad}[\rho_{A}^{\mathrm% {ref}},\rho_{B}]}{\delta\rho_{A}^{\mathrm{ref}}(\mathbf{r})}\mathrm{d}\mathbf{r}$ (11.109)

In contrast to conventional FDET, the embedding potential then becomes $\rho_{A}$-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.