# 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.

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:

$\begin{equation*} \label{psi-kappa-param} | \bar{\psi}(t) \rangle = e^{-i\hat{\kappa}(t)} |0\rangle ; \quad \hat{\kappa}(t) = \sum_a^\mathrm{unocc} \sum_i^\mathrm{occ} \left[ \kappa_{ai}(t) \hat{a}^\dagger_a \hat{a}_i + \kappa_{ai}^*(t) \hat{a}^\dagger_i \hat{a}_a \right] , \end{equation*}$

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:

$\begin{equation*} \label{ehren-equiv} \frac{\partial}{\partial t} \langle \bar{\psi}(t) | \hat{\Omega}_{pq}^\dagger | \bar{\psi}(t) \rangle - \langle \bar{\psi}(t) | \frac{\partial \hat{\Omega}_{pq}^\dagger}{\partial t} | \bar{\psi}(t) \rangle = \frac{1}{i\hbar} \langle \bar{\psi}(t) | [\hat{\Omega}_{pq}^\dagger, \hat{H}] | \bar{\psi}(t) \rangle \end{equation*}$

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

$\begin{equation*} \hat{\Omega}_{ai}^\dagger = e^{-i\hat{\kappa}} \, \hat{a}^\dagger_a \hat{a}_i \, e^{i\hat{\kappa}} \end{equation*}$

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:

$\begin{equation*} \kappa_{ai}(t) = \kappa_{ai}^{(1)} + \kappa_{ai}^{(2)} + \kappa_{ai}^{(3)} + \cdots , \end{equation*}$

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

$\begin{equation*} \boldsymbol{\kappa}^{(1)}(\omega) = - i \left(\mathbf{E}^{[2]} - \hbar \omega \mathbf{S}^{[2]}\right)^{-1} \mathbf{V}^{\omega, [1]} . \end{equation*}$

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:

$\begin{equation*} \langle \bar{\psi}(t) | \hat{\Omega} | \bar{\psi}(t) \rangle = \langle 0 | \hat{\Omega} | 0 \rangle + \sum_{\omega} \langle \! \langle \hat{\Omega}; \hat{V}^{\omega} \rangle \! \rangle F^{\omega} e^{-i\omega t} + \cdots \end{equation*}$

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

$\begin{equation*} \langle \! \langle \hat{\Omega}; \hat{V}^{\omega} \rangle \! \rangle = - \left[\boldsymbol{\Omega}^{[1]}\right]^\dagger \left(\mathbf{E}^{[2]} - \hbar \omega \mathbf{S}^{[2]}\right)^{-1} \mathbf{V}^{\omega, [1]} , \end{equation*}$

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.