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 [NRS18], 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:

\[ i \frac{\partial}{\partial t} \phi_i (r,t) = \left( -\frac{1}{2} \nabla^2 + V_{\textrm{eff}} (r,t) \right) \phi_i (r,t) \]

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.