# Casida equation, excited states and response theory

I have several doubts about the characterization of excited states within the framework of TD-DFT (Time dependent Density Functional Theory) / Response Theory.

Just to give some context and fix notation, I will explain very quickly the general picture.

Context:

Basic Linear Response Theory (that is, 1st Order Response Theory) applied to the Kohn-Sham hamiltonian gives: $$n_1(r,t)=\int \int \chi(r,r^\prime,t-t^\prime)v_1(r^\prime,t^\prime) dt^\prime dr^\prime,$$ where $\chi$ is the Kohn-Sham density-density response function, computable from a Ground State DFT calculation and $v_1$ is the effective Kohn-Sham one-electron potential approximated to first order. It is useful to define the so-called XC-Kernel $f_{xc}(r,r^\prime,t,t^\prime)= \dfrac{\delta v_{xc}[n](r,t)}{\delta n(r^\prime,t^\prime)}\vert_{n_0(r)},$ for which there are a lot of possible approximations. After going over to frequency space and after tons of algebra (details can be found for example in "Density Functional Theory: An Advanced Course" by Engel & Dreizler or in "Time-Dependent Density-functional theory" by Ullrich; both are excelent and quite pedagogical references for this material, in a beginner's opinion!) one reaches the so-called Casida equation which is an eigenvalue problem: $$C\vec{Z}=\Omega^2 \vec{Z},$$ where $\Omega$ are the frequencies corresponding to the excitation energies and $C$ is a matrix which depends on the Ground-state KS states and on the XC Kernel. This Kernel, in principle, depends on the frequency $\Omega$ (at least if one whishes to go beyond the adiabatic approximation), and so this equation is not solvable by a single diagonalization, but rather a self-consistent approach is needed! The eigenvectors $\vec{Z}$ contain all the information needed to construct de excited-states densities to first order, $n_1(r,\Omega).$

Question 1: This is not a TD-DFT approach, right? I don't see any time progagation going on here, so I'm a bit confused because I thougth this was all part of TD-DFT, but it seems to me that one "just" has to perform a ground state DFT and afterwards solve Casida equation, so one does not need with the Time Dependent Schrödinger equation, right? Or am I wrong?. Okey, strictly speaking, one needs Runge-Gross and Van Leeuwen theorems to give a rigorous justification to all of this, but the time-dependent Schrödinger equation, time propagation and numerical integration schemes (such as Crank-Nicolson integrator) don't come into play in all of this at any point,.... right? Or am I misunderstanding something fundamental about the method?

Question 2: Say we want to study the dynamics of nuclei in a excited state. So, what do we do? Imagine $\Omega_0$ is the lowest eigenvalue in the Casida problem above. From that we can get the excited state density corresponding to $\Omega_0,$ $n_1(r,\Omega_0).$ But how do we compute the forces? We cannot just plug this density in the Energy functional, that wouldn't make sense, since that functional is defined for ground state densities, right? We could of course calculate the first-order correction to the effective Kohn-Sham potential to first order, but nothing else. How to compute the XC energy in the excited state? Or what about the kinetic energy; how to compute it from the density in a DFT-style if one doesn't have a representation of the excited state as a Slater determinant?

Question 3: Imagine on whished to go beyond linear responde theory, to N-th order response theory. The equations can be formally written, but then I have no idea as to what would be the strategy to compute the excitation energies or the excited-state densities. You wouldn't have that sort of eigenvalue problem. So, it is not only that the problem would become much harder. The thing is that even at the most basic conceptual level I wouldn't know how one would define excitation energies and related properties within a Higher-order Response Theory.

If all of this is of any interest, I will post later more questions and doubts related to these problems.

1. You are right. There is no time anymore in Casida's equation, though one uses the time-dependent Schroedinger equation and the whole machinery of TD-DFT in its derivation. In practice you just do a ground state calculation with DFT and then use the orbitals and Kohn-Sham energies to set up Casida's equation.

The term TD-DFT is a bit confusing as people use it both for the linear response methods like Casida's equation and the explicit solution of the time-dependent Schroedinger equation through the time-dependent Kohn-Sham scheme. It would be nice to distinguish these rather different things by name, and some people refer to the former as LR-TD-DFT where LR stands for linear response. But that is unfortunately not standard and I'm afraid you'll have to live with the confusion.

1. Let $$E_0 = E[\rho_0]$$ be the total energy you got from your ground state DFT calculation. If you call your excitation energy $$\Omega_n$$, then the energy of the excited state $$E_1$$ is just given by $$E_n = E_0 + \Omega_n \; .$$ The gradients in the excited state are then just $$\vec \nabla E_1 = \vec \nabla E_0 + \vec \nabla \Omega_n \; .$$ Here $$\vec \nabla E_0$$ is just the normal gradient from DFT. There are analytical expressions for the gradient of the excitation energy $$\vec \nabla \Omega_n$$, but they are rather nasty. Check this paper.

I don't know the answer to your 3rd question. Maybe someone else can jump in.

• Thank you for the answer! Thanks for the reference; it is not very detailed, but at least is something to star with. I am guessing that is type of thing people do (extracting analytical expresions for the gradients of the $\Omega_n$) when they need excited state forces to perform a surface hopping simulation, for example. Commented Jun 26, 2017 at 12:42