Frozen-Density Embedding Theory^{973, 974} (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 ${\rho}_{B}$ reads
(for definitions see Table 12.9):

${E}_{AB}^{\mathrm{tot}}[{\mathrm{\Psi}}_{A}^{I},{\rho}_{B}]$ | $=\u27e8{\mathrm{\Psi}}_{A}^{I}|{\widehat{H}}_{A}|{\mathrm{\Psi}}_{A}^{I}\u27e9+{V}_{B}^{\mathrm{nuc}}[{\rho}_{A}^{I}]+{J}_{\mathrm{int}}[{\rho}_{A}^{I},{\rho}_{B}]+{E}_{xc}^{\mathrm{nad}}[{\rho}_{A}^{I},{\rho}_{B}]$ | |||

$\mathrm{\hspace{1em}}+{T}_{s}^{\mathrm{nad}}[{\rho}_{A}^{I},{\rho}_{B}]+{E}_{{v}_{B}}^{\mathrm{HK}}[{\rho}_{B}]+{V}_{A}^{\mathrm{nuc}}[{\rho}_{B}]$ | (12.82) |

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

$${v}_{\mathrm{emb}}[{\rho}_{A}^{I},{\rho}_{B},{v}_{B}](\mathbf{r})={v}_{B}(\mathbf{r})+\int \frac{{\rho}_{B}({\mathbf{r}}^{\prime})}{|\mathbf{r}-{\mathbf{r}}^{\prime}|}d\mathbf{r}+\frac{\delta {E}_{xc,T}^{\mathrm{nad}}[{\rho}_{A}^{I},{\rho}_{B}]}{\delta {\rho}_{A}^{I}(\mathbf{r})}$$ | (12.83) |

The last term (non-electrostatic component) in equation 12.83 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.

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.

$${E}_{xc,T}^{\mathrm{nad}}[{\rho}_{A}^{I},{\rho}_{B}]\approx {E}_{xc,T}^{\mathrm{nad}}[{\rho}_{A}^{\mathrm{ref}},{\rho}_{B}]+\int \left({\rho}_{A}^{I}(\mathbf{r})-{\rho}_{A}^{\mathrm{ref}}(\mathbf{r})\right)\frac{\delta {E}_{xc,T}^{nad}[{\rho}_{A}^{\mathrm{ref}},{\rho}_{B}]}{\delta {\rho}_{A}^{\mathrm{ref}}(\mathbf{r})}d\mathbf{r}$$ | (12.84) |

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.