It seems that TD-DFT is a common choice for a computationally efficient method to describe excited states. But if DFT is a single-configurational method, and hence a ground state method, how does the TD-DFT method treat the excited states?
(...) the Runge-Gross theorem shows that for a many-body system evolving from a given initial wavefunction, there exists a one-to-one mapping between the potential (or potentials) in which the system evolves and the density (or densities) of the system.
The Runge–Gross theorem provides the formal foundation of time-dependent density functional theory. It shows that the density can be used as the fundamental variable in describing quantum many-body systems in place of the wavefunction, and that all properties of the system are functionals of the density.
This means that any state of the system can be described by an electron density, not just the ground state. If I understand this correctly, the R-G theorem is more general than the H-K theorem, which can be derived from the R-G theorem by removing the explicit time dependence.
Here is a paywalled link to their original paper.
An interesting comment from the Wikipedia proof is
The proof relies heavily on the assumption that the external potential can be expanded in a Taylor series about the initial time. The proof also assumes that the density vanishes at infinity, making it valid only for finite systems.
In the case of applied fields, such as a slowly varying or oscillating electric field, these may be treated with perturbation theory rather than self-consistently; this is the same thing as the Taylor expansion. A consequence is that TD-DFT may not be valid in the presence of strong applied fields or a strong external potential.
This is a complicated question you ask. You will most likely get an answer like the one provided by azag0. I would be very surprised if this helped you in any way.
I want to rephrase your question: "How does one get excitation energies from TD-DFT?"
I'm gonna assume that this is completely equivalent with the question you've asked, maybe with less of an philosophical touch.
If you ask me, the best path to understanding (in principle) what TD-DFT does is to understand CIS (TDA). You can see TD-DFT as an extension to CIS. This is not how you find it in text books, but the reason why you ask this question is somewhat related to the problem that you don't find it like this in text books. The way you asked the question tells me you have access to publications, maybe you are studying at an university, so I highly recommend a publication by Dreuw and Head-Gordon called "Single-Reference ab Initio Methods for the Calculation of Excited States of Large Molecules", Chem. Rev. 2005, 105 (11), 4009–4037. DOI: 10.1021/cr0505627.
If you do not understand this publication, I would come back and ask simpler questions first, questions one might be able to answer in a simple post on an online board.
TD-DFT is a complete time-dependent theory of the motion of many electrons. It is to the time-dependent Schrödinger equation what ordinary DFT is to the time-independent Schrödinger equation.
What is most commonly referred to as TD-DFT is in fact a particular application of TD-DFT. Irrespective of a method, the excitation energies of a molecule or a solid can be in general calculated as frequencies at which the linear density response function (or, really, any linear response function) diverges. (Density response function says how a density of a system at some point changes because of a potential change at a different point.) This can be understood as frequencies at which the electronic motion resonates with the oscillating electromagnetic field; that is, the field that excites the molecule. (This also partially explains the correspondence between energy and frequency.) Because TD-DFT is a full theory, it can be also used to calculate the linear density response, and hence the excitation energies. Formally, this translates into finding the poles of the density response function. This calculation is what is most often referred to as TD-DFT. (Poles are particular types of singularities in complex functions.)