11.8.1 Introduction

The polarizable embedding (PE) model is a fragment-based quantum-classical explicit embedding scheme to model molecular properties in complex heterogeneous environments ⁹⁴⁹ Olsen J. M. H., Aidas K., Kongsted J.
J. Chem. Theory Comput.
(2010), 6, pp. 3721. Link ^{, 950} Olsen J. M. H., Kongsted J.
Adv. Quantum Chem.
(2011), 61, pp. 107. Link . The theory is explained thorougly in literature ⁹⁴⁹ Olsen J. M. H., Aidas K., Kongsted J.
J. Chem. Theory Comput.
(2010), 6, pp. 3721. Link ^{, 950} Olsen J. M. H., Kongsted J.
Adv. Quantum Chem.
(2011), 61, pp. 107. Link ^{, 790} List N. H., Olsen J. M. H., Kongsted J.
Phys. Chem. Chem. Phys.
(2016), 18, pp. 20234. Link . In essence, the environment is represented by a multi-center multipole expansion to model electrostatic interactions, whereas polarization is taken into account by dipole-dipole polarizabilities placed at the expansion points. Polarization effects can thus be treated fully self-consistently by mutual polarization of the environment and the quantum region.

A recent tutorial review on how to prepare PE calculations in general (creating embedding potentials) is also available ¹²¹⁹ Steinmann C. et al.
Int. J. Quantum Chem.
(2019), 119, pp. 1. Link . For automated generation of embedding potentials, please refer to the PyFraME tool ¹¹ 1 https://gitlab.com/FraME-projects/PyFraME which is also explained in the aforementioned review.

PE can be used for Hartree-Fock and density-functional theory ground-state SCF methods. In addition, PE has been combined with the algebraic-diagrammatic construction for the polarization propagator (ADC) ¹¹²⁷ Scheurer M. et al.
J. Chem. Theory Comput.
(2018), 14, pp. 4870. Link , explained in the subsequent section.

The combined scheme of the PE model and ADC (PE-ADC) ¹¹²⁷ Scheurer M. et al.
J. Chem. Theory Comput.
(2018), 14, pp. 4870. Link is built on top of a PE-HF ground-state calculation and takes into account perturbative corrections of the excitation energies in a density-driven manner. That is, after the Hartree-Fock ground-state calculations, the induced dipole moments in the environment are kept frozen and an ADC calculation is performed as usual. Thereafter, perturbative corrections of the electronic excitation energies are calculated based on i) the transition density (perturbative linear-response-type correction, ptLR), and ii) the difference density (perturbative state-specific correction, ptSS) for each excited state.