# 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:

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.

## Parameterization#

SCF states are parameterized as:

where the creation, \(\hat{a}^\dagger\), and annihilation, \(\hat{a}\), operators act on unoccupied, secondary, \(a\) and occupied, inactive, \(i\) molecular orbitals.

## Equation of motion#

The time evolution of the SCF state is determined from the Ehrenfest theorem:

We require that this equation is fulfilled for the set of time-transformed electron-transfer operators:

as well as the corresponding Hermitian conjugate. There will be two equations associated with each single-electron excited determinant, so there are twice as many equations as there are unknown *complex* parameters, \(\kappa_{ai}(t)\), with independent real and imaginary parts in the time-dependent phase-isolated wave function, \(| \bar{\psi}(t) \rangle\).

At this point, we adopt perturbation theory:

which, after finding a first-order solution in the frequency domain, will eventually results in

The vector \(\boldsymbol{\kappa}^{(1)}\) collects the set of parameters and their complex conjugate and matrices \(\mathbf{E}^{[2]}\) and \(\mathbf{S}^{[2]}\) are known as the electronic Hessian and overlap matrices, respectively, and \(\mathbf{V}^{\omega, [1]}\) is known as the property gradient of the perturbation operator.

## Response functions#

The response functions of an observable \(\hat{\Omega}\) are defined by:

and we are able to identify the following expression for the linear response function

where \(\boldsymbol{\Omega}^{[1]}\) (analogously to \(\mathbf{V}^{\omega, [1]}\)) is known as the property gradient associated with the observable. This expression for the linear response functions is also known as the random phase approximation (RPA) equation in the case a Hartree–Fock reference state. It is the principal equation to be solved in time-dependent Hartree–Fock (TDHF) and time-dependent density functional theory (TDDFT) theories.