Chapter 8: Hohenberg–Kohn Theorems as a basis for Multi-scale Simulations: Frozen-density Embedding Theory

In frozen-density embedding theory (FDET) overviewed in this chapter, the Hohenberg–Kohn theorems are used to i) interpret the eigenvalue problem solved in various multi-level simulation methods as the constrained optimisation of the total electron density, and ii) derive the exact expressions for the total energy and the effective potential. The key issues for practical applications of FDET in multi-scale simulations such as the state of development of the approximation used for the bi-functional E$xctnad$[ρ_A,ρ_B] and the localisation of the embedded wavefunction are discussed in dedicated sections. A short overview of the current directions in the applications of various FDET-based simulations is given in the last section.