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.

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:

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

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.