Excited-states properties with traditional density functional theory

Recently I was shown the book "Density Functional Theory II, Relativistic and Time Dependent Extensions" by Nalewajski in which a chapter from E. K. U. Gross et al. is printed. [1] This chapter contains an interesting statement in it's introduction:



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> Excited-state properties, however, are notoriously difficult to calculate within the traditional density functional framework and time-dependent phenomena are not accessible at all.
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What are those excited-state properties that are "notoriously difficult to calculate" and what makes it hard to calculate them without the TDDFT framework?




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Literature:  

[1] [Gross, E. K. U.; Dobson, J. F.; Petersilka, M. Density Functional Theory of Time-Dependent Phenomena. In *Density Functional Theory II*; Nalewajski, R. F., Ed.; Topics in Current Chemistry 181; Springer Berlin Heidelberg, 1996; pp 81–172.](http://dx.doi.org/10.1007/BFb0016643)




Conventional density functional theory (DFT) is strictly limited to describing the electronic density of ground electronic states. This is because the [Hohenberg–Kohn theorems](https://en.wikipedia.org/wiki/Density_functional_theory#Hohenberg.E2.80.93Kohn_theorems), on which it is based, are restricted to non-degenerate ground states (with no magnetic field). Also, because DFT aims at solving the time-independent Schrödinger equation (i.e. it explicitly finds stationary solutions), time-dependent phenomena are not accessible.


From the ground state density, any property of the system *including excited state properties* could in theory be derived given an appropriate functional[1] $X[\rho]$ that maps the ground state density $\rho$ onto the property $X$. However, **in practice, no known functional exists that maps the ground state density to excitation energies or other excited-state properties**. Thus, excited states and their properties are *in practice* unreachable by DFT: I suppose this is what is meant by Gross when stating them as *“notoriously difficult to calculate”*. I myself would have said *impossible*.


Finally, to give a short list of properties that can be described with TDDFT but not with DFT — are thus all excited state properties. In particular: bandplots/bandgaps, excitation energies, transition moments, photo absorption spectra, frequency-dependent response properties (optical & dielectric properties), etc.


Specifically, one has to be careful about band plots and band gaps, which are routinely reported from time-independent DFT calculations: they are a representation of the Kohn–Sham energies, i.e., the energies of the fictive non-interacting Kohn–Sham system, which has no physical interpretation at all. 




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[1] Note here that we are talking about a functional in the generic, mathematical meaning of the word. It is not the exchange–correlation functional.


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There are quite a few papers out there highlighting some examples. I particularly like the following paper titled "[Failures of TDDFT in describing the lowest intramolecular charge-transfer excitation in *para*-nitroaniline](http://www.tandfonline.com/doi/abs/10.1080/00268976.2013.793841)" because this is TDDFT getting it absolutely wrong for small molecule like *para*-nitroaniline.


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