# 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 [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.