# Closed-form expression for excitation energies, given the exact XC functional

In an answer to my question regarding the theoretical rigour in computing excitation energies using only the Kohn-Sham orbital energies, the rigour turned out to be nonexistent.

After looking this matter up, I began to wonder if the exact excitation energies (whose definition is explicitly written in my question, whose answer is linked above, and shall not be re-stated here) could be written in closed form using the relevent K-S orbital energies AND the exact XC functional, for example as the sum of a constant times the K-S "band gap" plus another constant times the derivative of the exact XC applied to the exact density(rough guess and potentially true, potentially false example to express the spirit I'm having here)

Does such a closed-form expression, ignoring facts like how "the exact XC functional is currently unknown", exist at least in theory?

## 2 Answers

Do you consider the exact frequency-dependent XC kernel as part of "the exact XC functional"? And do you allow the use of additional ingredients like two-electron integrals?

If both of your answers are "yes", then you can construct the Casida equation with these ingredients, and you can solve it to get the excitation energies. However, in this case the equation to be solved is a non-linear eigenvalue problem, which in general must be solved iteratively. So in general there is no closed-form expression of excitation energies (indeed, even if the eigenvalue problem is linear, the eigenvalues are still in general not expressible in closed form as a function of the matrix elements). Of course, there is the theoretical possibility that the exact frequency-dependent XC kernel has some peculiar property that allows the eigenvalue problem to be solved in closed form, but if such property exists, it would likely have such a very deep physical reason, that someone has already discovered it. In practice, however, the reverse seems to be true.

So, while mathematically we probably cannot rule out the possibility of a (possibly extremely convoluted) closed-form expression of excitation energies, like we probably can never rule out that $$\pi+e$$ is a rational number, I hope that the above analysis will convince you that such a closed-form expression is extremely unlikely to exist.

No, they cannot be obtained from a DFT calculation. As the first paper I linked in the previous response says, DFT is a ground state theory, and it should not be expected to get excited state properties correct. For excited state properties you can use TDDFT, GW/BSE, and also wave function based methods that are common in quantum chemistry. You can even use these methods to rigorously get core level excitations (which is something that you alluded to in your previous question).

These methods all can yield values that are meant to be compared with excited state properties, and TDDFT is formally exact in a similar sense that DFT is for ground state properties.

You mentioned not having access to the journals. Here’s a pdf from one of the co-author’s website http://vintage.fh.huji.ac.il/~roib/Postscripts/JCTC_Perspectives%20-%20final.pdf

• This is true, but some excitation energies may be expressed simply as the ground state of a different system, or a function of several ground states, e.g. the ground state energy of the system with an electron added or removed. This is mentioned in the linked answer here mattermodeling.stackexchange.com/a/9686/403 Commented Sep 23, 2022 at 0:19
• Yes, I wrote that answer. From this question it seemed to me like they were referring to methods besides adding or removing electrons and computing differences in total energies
– AGS
Commented Sep 23, 2022 at 1:55