# Research

**FDET energy functional for the ground state:**
${E}_{{v}_{\mathit{AB}}}^{\mathit{FDET}}[{\mathrm{\Psi}}_{A},{\rho}_{B}]$

In Frozen-Density 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 non-negative 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 Hohenberg-Kohn 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 non-interacting reference system of electrons, see

[Wesolowski & Warshel, *J. Phys. Chem. * ** 97**, 8050 (1993)].

For the embedded one-particle density matrix, see

[Pernal & Wesolowski, *Int. J. Quant. Chem. * **109**, 2520 (2009)]

**Extensions of FDET for excited states: **

1) For non-interacting reference embedded wavefunctions (LR-TDDFT way):

[Wesolowski, *J. Am. Acad. Sci. * ** 126**, 11444 (2004)]

2) For interaction embedded wavefunctions (orthogonal stationary-state way):

[Wesolowski, *J. Chem. Phys. * **140**, 18A530 (2014)]

**Extensions of FDET for non-variational embedded wavefunctions:**

[ Wesolowski, *J. Chem. Theor. & Comput. ***16**, 6880-6885 (2020) ]

Subsystem formulation of DFT traces its origin to the methods used in the solid-state 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 Hohenberg-Kohn 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.

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 and approximations to the bi-functional of the non-additive kinetic energy:
[Wesolowski et al., *J. Chem. Phys.* **105**, 9182 (1996)],

[Wesolowski, *J. Chem. Phys.* **106**, 8516 (1997) ],

[Garcia Lastra et al., *J. Chem. Phys.* **129**, 074107 (2008)].

The domain of applicability of the semi-local approximations for the density functionals approximated in subsystem DFT based methods was established in benchmarking
studies on representative intermolecular complexes, 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.

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 linear-response time-dependent DFT.
The overview of such developments till 2006 is given in the review [Tomasz A. Wesolowski, *One-electron Equations for Embedded Electron Density:
Challenge for Theory and Practical Payoffs in Multi-Level Modeling of Complex Polyatomic Systems*,
in: *Computational Chemistry: Reviews of Current Trends - Vol.* **10**, World Scientific, 1-82 (2006)].

Since the publication of the benchmarking results in 2006 -2007, we are using subsystem DFT only occasionally:
Either as a tool to determine the accuracy of the developed approximations for the non-additive kinetic energy bi-functional (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 Frozen-Density Embedding Theory based simulations.
We are, however, still interested in some formal aspects of subsystem DFT such as: the lack of unfitness of the solution of subsystem DFT equations,
relation of the effective potential in subsystem DFT to the Phillips-Kleinman pseudopotential, etc.