11.7 Frozen-Density Embedding Theory 11.7.5 Examples 11.8 Polarizable Embedding Model

11.7.6 FDE-Man output

(April 13, 2024)

In general the FDE-Man output indicates all important stages of the embedding calculation, which are:

1.

Generation of $\rho_{A}^{\mathrm{ref}}(\mathbf{r})$ ,
2.

Generation of $\rho_{B}(\mathbf{r})$ ,
3.

Construction of the embedding potential,
4.

Start of FDE-Man embedding calculation and
5.

Final FDE-Man summary.

In the following table definitions of the terms printed to the output are collected. These quantities are printed for every state, i.e. for every $\rho_{A}^{I}(\mathbf{r})$ . In addition, the non-electrostatic interactions with respect to the reference density $\rho_{A}^{\mathrm{ref}}(\mathbf{r})$ are printed at the top of the FDE-Man summary.

Subsystem Energies
Embedded system (A)	$\langle\Psi_{A}^{I}\|\hat{H}_{A}+v_{\mathrm{emb}}^{\mathrm{lin}}\|\Psi_{A}^{I}\rangle$
Environment (B)	$E_{B}=E_{v_{B}}^{\mathrm{HK}}[\rho_{B}]\;\text{or}\;E_{\mathrm{HF}}$
Electrostatic Interactions
rho_A <-> rho_B	$J_{\mathrm{int}}[\rho_{A},\rho_{B}]=\int\int\frac{\rho_{A}(\mathbf{r})\rho_{B}% (\mathbf{r}^{\prime})}{\|\mathbf{r}-\mathbf{r}^{\prime}\|}\mathrm{d}\mathbf{r}% \mathrm{d}\mathbf{r}^{\prime}$
rho_A <-> Nuc_B	$V_{B}^{\mathrm{nuc}}[\rho_{A}]=\int\rho_{A}(\mathbf{r})v_{B}(\mathbf{r})% \mathrm{d}\mathbf{r}$
rho_B <-> Nuc_A	$V_{A}^{\mathrm{nuc}}[\rho_{B}]=\int\rho_{B}(\mathbf{r})v_{A}(\mathbf{r})% \mathrm{d}\mathbf{r}$
Nuc_A <-> Nuc_B	$V_{N_{A}N_{B}}=\sum_{i}\sum_{j}\frac{Z_{i}Z_{j}}{\|R_{i}-R_{j}\|}$
Non-Electrostatic Interactions
non-additive E_xc	$E_{xc}^{\mathrm{nad}}[\rho_{A}^{I},\rho_{B}]$
non-additive T_s	$T_{s}^{\mathrm{nad}}[\rho_{A}^{I},\rho_{B}]$
integrated v_xc nad	$\int\rho_{A}^{I}(\mathbf{r})v_{xc}^{\mathrm{nad}}(\mathbf{r})\mathrm{d}\mathbf% {r}$
integrated v_T nad	$\int\rho_{A}^{I}(\mathbf{r})v_{T}^{\mathrm{nad}}(\mathbf{r})\mathrm{d}\mathbf{r}$
Final FDET energies
Delta_Lin	$\int(\rho_{A}^{I}(\mathbf{r})-\rho_{A}^{\mathrm{ref}}(\mathbf{r}))v_{xc,T}^{% \mathrm{nad}}(\mathbf{r})\mathrm{d}\mathbf{r}$
Final Energy (A)	$E_{A}^{\mathrm{emb}}[\Psi_{A}^{I},\rho_{B}]=\langle\Psi_{A}^{I}\|\hat{H}_{A}\|% \Psi_{A}^{I}\rangle+J_{\mathrm{int}}[\rho_{A}^{I},\rho_{B}]+V_{B}^{\mathrm{nuc% }}[\rho_{A}^{I}]$
	$+E_{xc,T}^{\mathrm{nad}}[\rho_{A}^{\mathrm{ref}},\rho_{B}]+\Delta^{\mathrm{lin% }}[\rho_{A}^{I},\rho_{A}^{\mathrm{ref}},\rho_{B}]+V_{A}^{\mathrm{nuc}}[\rho_{B% }]+V_{N_{A}N_{B}}$
Final Energy (A+B)	$E_{A}^{\mathrm{emb}}[\Psi_{A}^{I},\rho_{B}]+E_{B}$

Table 11.11: Definition of output terms.

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11.7.6 FDE-Man output