Interpretation of TD-DFT results

Question

For the first time I'm doing TD-DFT calculations (wB97XD functional) in Gaussian 09 for an open-shell system and the results look like the hell of a mess for me. The molecule is rather big, so I started with just 10 excited states to see how it would work.

First off, the ground electronic state is $^2A_2$, orbital 453 is HOMO and 454 is SOMO. Now to the excited states. In the output I found, for example, the following piece that looks strange to my eyes

Excited State   3:  2.139-B2     0.9431 eV 1314.65 nm  f=0.0816 <S**2>=0.894    
453B -> 454B       0.98895  
453B <- 454B       0.17190

1. The first line is fine: it says that the third excited state is $^2B_2$ (taking spin contamination into account).
2. The second line is also fine: it tells that the determinant corresponding to excitation from HOMO to SOMO has the largest coefficients in the CI expansion.
3. But what is the meaning of the third line?

Answer 1

Time-dependent DFT can be used to predict excitation energies through a linear-response formulation.

In this Gaussian result, beyond the first line, you are looking at the largest coefficients in the configuration-interaction (CI) style expansion.

(It's not strictly CI, but the implementation of time-dependent HF or RPA is essentially the same for TDDFT or Tamm-Dancoff Approximation TDDFT.)

Anyway, since you're performing a calculation on an open-shell system, the formulation for the state will include all one-electron excitations AND all one-electron de-excitations. That state has a de-excitation from 454B into 453B (i.e., the SOMO will fill the hole left by the excitation). Now you may think "but that doesn't accomplish anything." Remember that Gaussian only prints the dominant coefficients.

For this reason, Rich Martin created the Natural Transition Orbitals method as a nice way to express the excitation, rather than a set of coefficients and the ground-state orbitals.

Oftentimes, there is no dominant configuration in the list of excitation amplitudes, thereby making a straightforward interpretation of the excited state difficult. This is particularly unsatisfactory when attempting to determine the qualitative nature of an excited state. An additional complication in the DFT case is that in principle all orbitals in DFT but the HOMO are devoid of physical significance. However, chemical intuition is built on the orbital construct, and a simple orbital interpretation of "what got excited to where" is important.

This is available in Gaussian from the Population keyword and in other codes.

Incidentally, there's a nice review of TDDFT in: "Progress in Time-Dependent Density-Functional Theory" Annual Review of Physical Chemistry Vol. 63: 287-323.