# Geometry Optimizations for Excited States

Electronic excitations are calculated vertically according to the Frank—Condon principle, this means that the geometry does not change upon the excitation and we merely calculate the energy required to reach the next electronic state. But for some instances, say calculating not only the absorption spectra but also the emission, it is important to know what the geometry minimum of this final state looks like, or if it even exists at all (Figure 1). Optimizing the geometry of a given excited state requires the prior calculation of the vertical excitations whether via a multireference method, quantum Monte Carlo, or the Time Dependent Density Functional Theory, TD-DFT, which due to its lower computational cost is the most widespread method.

Most single-reference treatments, ab initio or density based, yield good agreement with experiments for lower states, but not so much for the higher excitations or process that involve the excitation of two electrons. Of course, an appropriate selection of the method ensures the accuracy of the obtained results, and the more states are considered, the better their description although it becomes more computationally demanding in turn.

In Gaussian 09 and 16, the argument to the ROOT keyword selects a given excited state to be optimized. In the following example, five excited states are calculated and the optimization is requested upon the second excited state. If no ROOT is specified, then the optimization would be carried out by default on the first excited state (Where L.O.T. stands for Level of Theory).

`#p opt TD=(nstates=5,root=2) L.O.T.`

Gaussian16 includes now the calculation of analytic second derivatives which allows for the calculation of vibrational frequencies for IR and Raman spectra, as well as transition state optimization and IRC calculations in excited states opening thus an entire avenue for the computation of photochemistry.

If you already computed the excited states and just want to optimize one of them from a previous calculation, you can read the previous results with the following input :

`#p opt TD=(Read,Root=N) L.O.T. Density=Current Guess=Read Geom=AllCheck`

Common problems. The following error message is commonly observed in excited state calculations whether in TD-DFT, CIS or other methods:

`No map to state XX, you need to solve for more vectors in order to follow this state.`

This message usually means you need to increase the number of excited states to be calculated for a proper description of the one you’re interested in. Increase the number N for nstates=N in the route section at higher computational cost. A rule of thumb is to request at least 2 more states than the state of interest. This message can also reflect the fact that during the optimization the energy ordering changes between states, and can also mean that the ground state wave function is unstable, i.e., the energy of the excited state becomes lower than that of the ground state, in this case a single determinant approach is unviable and CAS should be used if the size of the molecule allows it. Excited state optimizations are tricky this way, in some cases the optimization may cross from one PES to another making it hard to know if the resulting geometry corresponds to the state of interest or another. Gaussian recommends changing the step size of the optimization from the default 0.3 Bohr radius to 0.1, but obviously this will make the calculation take longer.

`Opt=(MaxStep=10)`

If the minimum on the excited state potential energy surface (PES) doesn’t exist, then the excited state is not bound; take for example the first excited state of the H2 molecule which doesn’t show a minimum, and therefore the optimized geometry would correspond to both H atoms moving away from each other indefinitely (Figure 2). Nevertheless, a failed optimization doesn’t necessarily means the minimum does not exist and further analysis is required, for instance, checking the gradient is converging to zero while the forces do not.

Computational and theoretical chemist in his early forties, in love with life, science, baseball, and literature. Science literacy makes us responsible citizens, it is therefore a scientific duty to talk, write, and engage with the general public; as Feynman said, if you find science boring, your learning from the wrong teacher. "Make like a molecule and react!"

Posted on January 26, 2021, in Computational Chemistry, DFT, Fluorescence, Gaussian, Photochemistry, Spectroscopy, TD-DFT, White papers and tagged , , , , , , , , , , , , , . Bookmark the permalink. 9 Comments.

1. Kjell Jorner

Thanks for the excellent post! A technical detail that I found important is that Gaussian defaults to non-equilibrium solvation for single points and equilibrium solvation for geometry optimisations and frequencies. This makes sense most of the time, but if you want equilibrium solvation for a single point, you can add the keyword “iop(9/73=2)”. I used this to get SPs at higher level along excited state reaction paths for example.

2. MANAJIT DAS

Thank you sir. It is helpful to me.

On Tue, Jan 26, 2021 at 10:48 PM Dr. Joaquin Barroso’s Blog wrote:

> joaquinbarroso posted: ” Electronic excitations are calculated vertically > according to the Frank—Condon principle, this means that the geometry > does not change upon the excitation and we merely calculate the energy > required to reach the next electronic state. But for some inst” >

3. Srinath Kompella

Thank you, Dr. Joaquin Barroso. I find your blogs very helpful. I have a problem relevant to this blog, and I will be grateful if you could provide suggestions for solving it.

I have a specific conformation of the dye molecule (taken from MD simulations) for which I am trying to obtain the vibronically resolved UV-Vis spectrum.

During the ground state geometry optimization of that conformation, I freeze certain torsions so that the key features of the conformation are retained. Fortunately, the subsequent frequency calculation does not show any imaginary frequencies.

If I freeze the same torsions during the excited state geometry optimization, the resultant geometry has one imaginary frequency, because of which the vibronically resolved spectrum does not make sense. If the torsions are not frozen during this step, the dye molecule takes a completely different conformation that is of no interest to me.

I understand that the torsional preferences of the excited state geometry could be slightly different from those of the ground state geometry, which is why freezing leads to an imaginary frequency in the former. But how do I find the nearest local minimum of the excited state? Kindly provide your feedback/suggestions.

• Hello Srinath,

Very interesting question. Could it be that this imaginary frequency is indicative of your molecule getting broken upon excitation? That is, maybe your molecule is not stable in the first excited state at the vertical optimization. Having said that, maybe unfreezing those torsion angles one by one could lead you to a sensible result.

I hope this helps

4. Srinath Kompella

Thank you for the response, Dr. Joaquin Barroso.

I thought the same and therefore performed a scan of the first excited state potential energy surface (ES1-PES) with respect to two torsions. Those two torsions essentially determine the dye conformation, and I was freezing them during the optimizations of the ground state and the first excited state. The ES1-PES does not show any minimum around the torsional values corresponding to the conformation of my interest, which tells me that it is not stable in the first excited state.

I did try unfreezing one of the two torsional angles, but that led to a (completely different) conformation that is not relevant to me.

Could you please suggest any other way of calculating vibrationally resolved UV-Vis and ECD spectrum in this scenario? Thank you.

• In this case if the ES1 PES doesn’t have a minimum you can’t assume the molecule isn’t stable because the PES is restricted by the freezing of the molecule.
Have you tried exploring a different excited state? That might help

5. Srinath Kompella

Thank you, Dr. Joaquin Barroso.

I did try exploring one more excited state, and the scenario is similar there. The vertical excitation calculation shows that the transitions to other excited states either have very low oscillator strengths or occur at much shorter wavelengths than the range of our interest. Therefore, I have not probed them.

Currently, I am trying the “Vertical Gradient” approach wherein the excited state equilibrium geometry and its Hessian are not needed.

I will be happy to try any other suggestions from your end. Thanks a lot.

6. Felix Leon Garcia

Thank you Dr. Barroso,

I wonder if you could help me with a little problem I am facing. I have calculated the first 10 excited states (triplets and singlets) of a copper complex that I am working with in the lab TDA=(Nstates=10,50-50). Then, I have optimised the first singlet excited state (opt TD=(Nstates=4,50-50,root=2)). It took quite a few attempts and a lot of computational time, but eventually I got the optimised structure.

My problem comes when I go to chemcraft and check the MO diagram, I see a totally normal HOMO and LUMO, having the HOMO to pair electrons as per normal, while I was expecting to see one electron in a higher energy orbital (and opposite spin). DId I do something wrong?.

Felix

7. Roumany Israil

Hi,

I am having problems optimizing already computed excited states.
This is my input: