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FDET energy functional for the ground state: ${E}_{{v}_{\mathit{AB}}}^{\mathit{FDET}}[{\mathrm{\Psi}}_{A},{\rho}_{B}]$
In FrozenDensity Embedding Theory (FDET), an ${N}_{\mathit{AB}}$ electron density is represented by means of two independent variables: i) an ${N}_{A}$electron wavefunction (${\mathrm{\Psi}}_{A}$) and ii) an nonnegative real function (${\rho}_{B}(r)$) such that $\int {\rho}_{B}(r)dr={N}_{B}$, where ${N}_{B}$ is an integer such that ${N}_{\mathit{AB}}={N}_{A}+{N}_{B}$.
The FDET total energy functional (${E}_{{v}_{\mathit{AB}}}^{\mathit{FDET}}[{\mathrm{\Psi}}_{A},{\rho}_{B}]$) satisfies the following relation with the HohenbergKohn energy functional (${E}_{{v}_{\mathit{AB}}}^{\mathit{HK}}[\rho ]$):
$$\begin{array}{rcll}\underset{{\mathrm{\Psi}}_{A}\to {N}_{A}}{\mathrm{min}}{E}_{{v}_{\mathit{AB}}}^{\mathit{FDET}}[{\mathrm{\Psi}}_{A},{\rho}_{B}]={E}_{{v}_{\mathit{AB}}}^{\mathit{FDET}}[{\mathrm{\Psi}}_{A}^{o},{\rho}_{B}]={E}_{{v}_{\mathit{AB}}}^{\mathit{HK}}[{\rho}_{A}^{o}+{\rho}_{B}],& & & \text{}\end{array}$$
where ${v}_{\mathit{AB}}(r)$ is a given external potential and ${\rho}_{A}^{o}(r)=<{\mathrm{\Psi}}_{A}^{o}\left{\sum}_{i=1}^{{N}_{A}}\delta \left({r}_{i}r\right)\right{\mathrm{\Psi}}_{A}^{o}>$. The optimal embedded wavefunction is obtained from the FDET eigenvalue equation:
$$\begin{array}{rcll}\left({\u0124}_{{v}_{\mathit{AB}}}+{\widehat{v}}_{\mathit{emb}}^{\mathit{FDET}}[{\rho}_{A}^{o},{\rho}_{B}]\right){\mathrm{\Psi}}_{A}^{o}={\lambda}^{o}{\mathrm{\Psi}}_{A}^{o}& & & \text{}\end{array}$$
The expressions for ${E}_{{v}_{\mathit{AB}}}^{\mathit{FDET}}[{\mathrm{\Psi}}_{A},{\rho}_{B}]$ and the corresponding embedding potential are given in [Wesolowski, Phys. Rev. A 77, 012504 (2008)] for various choices for the form of the embedded wavefunction.
For the embedded noninteracting reference system of electrons, see
[Wesolowski & Warshel, J. Phys. Chem. 97, 8050 (1993)].
For the embedded oneparticle density matrix, see
[Pernal & Wesolowski, Int. J. Quant. Chem. 109, 2520 (2009)]
FDET beyond groundstate:
1) For noninteracting reference embedded wavefunctions (LRTDDFT way):
[Wesolowski, J. Am. Acad. Sci. 126, 11444 (2004)]
2) For interaction embedded wavefunctions (orthogonal stationarystate way):
[Wesolowski, J. Chem. Phys. 140, 18A530 (2014)]
FDET equalities for nonvariational embedded wavefunctions:
[ Wesolowski, J. Chem. Theor. & Comput. 16, 68806885 (2020) ]
In FDET, the constrained optimisation of the HohenbergKohn energy functional E_{vAB}^{HK}[ ρ ] is performed using Ψ_{A} and ρ_{B} as independent variables. FDET provides, therefore, the formal basis of various possible multiscale/multilevel simulation methods combining the quantum mechanical level of description for the embedded species (Ψ_{A}) with methods applying various physical laws to generate ρ_{B}.

FDET with ρ_{B} and Ψ_{A} from different quantum mechanical methods

– ρ_{B} as the superposition of electron densities of isolated molecules in the environment
[Wesolowski & Warshel, J. Phys. Chem., 98, 5183 (1994)]; [Zhou et al, J. Am. Chem. Soc., 136, 2723 (2014)] for instance

– ρ_{B} as the superposition of electron densities of isolated atoms in the environment
[Zbiri et al., Chem. Phys. Lett. , 397, 441 (2004)] or [HumbertDroz et al., Theor. Chem. Acc., 132, 1405 (2013)] for instance 
– ρ_{B} as the density of the whole environment
[Zech et al., J. Chem. Theor. & Comput., 14 4028 (2018)] for instance 
– ρ_{B} as the density of the whole environment perturbed by the interaction with the embedded species
[Zbiri et al., Chem. Phys. Lett. , 397, 441 (2004)] or [HumbertDroz et al., Theor. Chem. Acc., 132, 1405 (2013)] for instance


FDET with ρ_{B} from methods beyond quantum mechanics
The FDET relations hold for any admissible ρ_{B} obtained from physical laws appropriate for the characteristic length and timescale for the environment.

A continuum function (< ρ_{B} > (r) corresponding to the electron density averaged over configurations of nuclei in a statistical ensemble can be also used as ρ_{B} in all FDET equations. We have used various techniques (3DRISM, Molecular DFT, and explicit molecular dynamics simulations) to generate < ρ_{B} > and use it to estimate the solvent effect on absorption (see [Kaminski et al., J. Phys. Chem., 114, 6082 (2010); Zhou et al., Phys. Chem. Chem. Phys., 13 10565 (2011) ; Laktionov et al, Phys. Chem. Chem. Phys., 18 21069 (2016) ; GonzalezEspinoza et al., J. Chem. Theor. & Comput., 18 1072 (2022)] ) or emission [Shedge & Wesolowski, ChemPhysChem, 15, 3291 (2014)] spectra.

Recently, we have shown that the experimental electron density obtained from Xray diffraction data on molecular crystals used as ρ_{B} in FDET equations yields excitation energies of embedded chromophores comparable to the ones obtained using ρ_{B} from ab initio calculations [Ricardi et al. Acta Crystallographica A  Foundations and Advances, 76, 571 (2020)].

The density functional for the kinetic energy (T_{s}[ρ])
is a key quantity in both, the embedding methods based on FDET and in methods based on the orbitalfree formulation of DFT (OFDFT).
(For an overview of OFDFT methods, see Recent Progress in OrbitalFree DensityFunctional Theory, T.A. Wesolowski & Y.A. Wang, Eds., World Scientific, 2013). Any approximation used for T_{s}[ρ] and its functional derivative applicable in OFDFT methods can be also used to approximate the nonadditive kinetic energy:
T_{s}^{nad}[ρ_{A},ρ_{B}]=
T_{s}[ρ_{A}+ρ_{B}]T_{s}[ρ_{A}]T_{s}[ρ_{B}]
and its functional derivative with respect to ρ_{A} (nonadditive kinetic otential), which is one of the components of the FDET embedding potential.
Such a "topdown" strategy to approximate this derivative proved to be prone to surprising failures in FDET.
For example, although addition of the secondorder expansion correction to ThomasFermi functional is known to improve the kinetic energy,
the addition of the corresponsing correction to the nonadditive kinetic energy and potential worsens the FDET results as shown already in the our original work reporting the first FDET based embedding method
[Wesolowski & Warshel, J. Phys. Chem. 97, 8050 (1993)].
Subsequent dedicated studies, have shown that this failure is due to the incorrect behaviour of the derivative of T_{s}^{nad}[ρ_{A},ρ_{B}] rather than the T_{s}^{nad}[ρ_{A},ρ_{B}]
contribution to the energy [Wesolowski & Weber, Intl. J. Quant. Chem. 61, 303 (1997)].
We pursued two strategies in developeding approximations for the nonadditive kinetic potential and energy. The decomposable one, in which the second order contributions are smoothly dupmed (GGA97 approximation  see below) or the nondecomposable strategy ("bottomup") in which the exact mathematical properties are imposed on the nonadditive kinetic potential without constructing the "parent" approximation for T_{s}[ρ] (NDCS abd NDSD functionals  see below).
Approximations to the bifunctional of the nonadditive kinetic energy and/or potential:
 Generalized Gradient Approximation for the nonadditive kinetic potential (GGA97): [Wesolowski et al., J. Chem. Phys. 105, 9182 (1996)] and [Wesolowski, J. Chem. Phys. 106, 8516 (1997) ],
 nondecomposable approximation for the nonadditive kinetic potential (NDSD): [Garcia Lastra et al., J. Chem. Phys. 129, 074107 (2008)],
 improved nondecomposable approximation for the nonadditive kinetic potential (NDCS): [Polak et al., J. Chem. Phys. 156, 044103 (2022)]
Subsystem formulation of DFT traces its origin to the methods used in the solidstate physics community to model ionic and rare gas crystals [Senatore & Subbaswamy, Phys. Rev. B 34, 5754 (1986)]. In 1991, Cortona gave such methods a formal foundations by relating the underlying relations for energy and density to the HohenbergKohn theorems [Cortona, Phys. Rev. B 44, 8454 (1991)] and applied a method based on subsystem DFT to model structure and elastic properties of ionic solids.
II. Our contributions:In 1996, we started exploring
the applicability of methods based on subsystem DFT in the field of intermolecular interactions [Wesolowski & Weber, Chem. Phys. Lett. 248, 71 (1996)].
This lead us to the development of:
Efficient algorithms:
 "freezeandthaw" optimisation of subsystem densities: [Wesolowski & Weber, Chem. Phys. Lett. 248, 71 (1996)]
 simultaneous optimisation of density and nuclear coordinates:[Dulak et al., J. Chem. Theor. & Comput. 3, 735 (2007)];
 linearisation of the nonelectrostatic components of the FDET embedding potential [Dulak & Wesolowski, J. Chem. Theory. & Comput. 2 1538 (2006)],
and
Approximations to the bifunctional of the nonadditive kinetic energy and/or potential:
 Generalized Gradient Approximation for the nonadditive kinetic potential (GGA97): [Wesolowski et al., J. Chem. Phys. 105, 9182 (1996)] and
[Wesolowski, J. Chem. Phys. 106, 8516 (1997) ],
 nondecomposable approximation for the nonadditive kinetic potential (NDSD):
[Garcia Lastra et al., J. Chem. Phys. 129, 074107 (2008)],
 improved nondecomposable approximation for the nonadditive kinetic potential (NDCS): [Polak et al., J. Chem. Phys. 156, 044103 (2022)]
In 19962007 we applied subsytem DFT based mathod to various types of intermolecular complexes which allowed us to determine the domain of applicability of the semilocal approximations for the density functionals approximated in subsystem DFT . For a benchmarking studies, see [Dulak & Wesolowski, J. Molecular Modeling 13, 631 (2007)] for interaction energies, and [Dulak et al., J. Chem. Theor. & Comput. 3, 735 (2007)] for equilibrium complexes.
III. Extension of subsystem DFT to excited states:In 2004, in collaboration with Mark E. Casida, we generalized subsystem formulation of DFT as a ground state formalism to excited states using the general framework of linearresponse timedependent DFT. [M.E. Casida & T.A. Wesolowski, Intl. J. Quant. Chem. 96 577 (2004)]
IV. Subsystem DFT: current interestsThe overview of such developments till 2006 is given in the review [Tomasz A. Wesolowski, Oneelectron Equations for Embedded Electron Density: Challenge for Theory and Practical Payoffs in MultiLevel Modeling of Complex Polyatomic Systems, in: Computational Chemistry: Reviews of Current Trends  Vol. 10, World Scientific, 182 (2006)]. Since the publication of the benchmarking results in 2006 2007, we are using subsystem DFT based methods only occasionally: either as a tool to determine the accuracy of the developed approximations for the nonadditive kinetic energy bifunctional (and its derivative) [Polak et al., J. Chem. Phys. 156, 044103 (2022)] or as one of many possible techniques to generate the frozen density in FrozenDensity Embedding Theory based multilevel/multiscale simulations.