Foundations

In Hartree--Fock and Kohn--Sham density functional theory, the reference state is described by a single Slater determinant, also referred to as the self-consistent field (SCF) approximation. The detailed derivation of response functions in this approximation can be found in Norman et al. (2018), but the key steps will be indicated below.

Runge-Gross theorem¶

The basics of TDDFT comes from the work of Work of Runge and Gross, who presented a theory which underlies TDDFT:

For every single-particle potential $V(r,t)$, which can be expanded in a Taylor series around $t_0$, there exists a one-to-one mapping $G:V(r,t) \rightarrow \rho (r,t)$. This mapping is defined by solving the time-dependent Schrödinger equation with a fixed initial state $\phi$ and calculating the corresponding densities $\rho (r,t)$.

They also presented schemes for practical schemes, the most important being:

The exact time-dependent density can be expressed as a sum over states: $\rho (r,t) = \sum | \phi (r,t)|^2$

We calculate time-dependent orbitals from:

This requires knowledge of exact time-dependent xc-functional, which is even less known than the ground-state correspondance.

Adiabatic approximation¶

In the adiabatic approximation one siplifies the xc-functional as one which do not change over time, such that standard functionals can be used. This is a reasonable approximations for systems changing slowly in time, for which the time-dependence of the energy then changes through the time-dependence of the density.

It can be note that the search for non-adiabatic functional may be somewhat futile, as it would technically require the solution of the full time-dependent Schrödinger equation, as has been discussed.