# Computing accurate vibrational and rotational contributions to the free energy of transition states and loosely bound complexes

Recently I have been dealing with a lot of transition states and relatively loosely bound ion-dipole complexes and I have some trouble figuring out how to make sure that the rotational and vibrational contributions to the Gibbs free energy are as accurate as they reasonably can. (I am using Gaussian 09 rev C)

I am using MP2/aug-cc-pVTZ, based on literature results it should be in close agreement with even better methods for the type of molecules I am working with. First I have optimized the geometry to verytight convergence, then reoptimized it with verytight, CalcAll to make sure the geometry is as converged as possible, and get the thermochemistry output at the same time.
I have once read in a CCL post, that tightening SCF convergence can increase the accuracy of analytic derivatives, so I am performing all calculations with SCF=Conver=12.

My concerns are mostly related to having a number of low frequency modes, and the possibility of coupling between vibrations and rotations.
For example, in the case of the transition state between $\ce{MeCl + F-}$ and $\ce{MeF + Cl-}$, I have three low frequency modes: 258.4867 $cm^{-1}$, 258.4867 $cm^{-1}$ 303.2302 $cm^{-1}$ (and of course an imaginary frequency). Sure enough, Gaussian warns that:

Warning -- explicit consideration of 3 degrees of freedom as vibrations may cause significant error

Based from what I gathered from various sources, this means that these low frequencies have significant uncertainties in them, and they may or may not be internal/hindered rotations. I have ran a calculation with Freq=hindrot but it did not identify any of the modes as rotations. However, the Gaussian manual says that the hindered rotor analysis is not always successful at finding rotations when transition states are involved.

My other concern is the validity of the rigid rotor approximation for molecules having low frequency modes (ie. low force constants). My intuitions tell me, that the lower the force constants in a molecule, the worse are the errors caused by the centrifugal distortions of the molecule.

So my actual questions are:

1.What is the proper way to accurately handle low frequency modes? How does one make sure that they are not some sort of rotation, when hindered rotor analysis does nothing? If they legit low frequency vibrations, are they reliable? If they are not reliable how can they be corrected?

2.Do I have to be concerned about low frequencies also implying that the rotational contributions are also affected due to centrifugal distortion?

• I forget.. does Gaussian "automatically" remove the 6 rotational and translational modes for you? (Some programs do.) The mode energies you gave don't seem especially low. – Geoff Hutchison Dec 1 '15 at 16:06
• You can also convert between the rotational constants in the Gaussian output (usually in GHz) and wavenumbers to compare with the frequencies you mention. – Geoff Hutchison Dec 1 '15 at 16:07
• Yes, the translational and rottational modes are handled automatically. – uLoop Dec 1 '15 at 20:12
• @GeoffHutchison I think I saw somewhere that both the T+R unprojected and projected frequencies are output. Dunno if there's a flag you have to set for that, though. – hBy2Py Dec 1 '15 at 20:42
• If you are trying to be that accurate, you may have significant error from sources you have not considered: e.g. anharmonicity of the potential energy surface, esp for the soft nodes. – Greg Dec 2 '15 at 0:51

Given the accuracy it sounds like you're looking for, and the systems you're studying, this indeed appears to be a Very Hard Problem. I think your intuitions are well founded.

1) How to handle the low-frequency modes?

A lot of manual exploration of the PES, doing things like running normal mode scans and optimizing intermediate geometries along suspected internal rotations. This gives you an idea how the various energetic barriers to non-harmonic motions compare to a Boltzmann distribution of states.

• If the barriers are high, the harmonic approximation is probably decent and you can just use Gaussian's results
• If barriers are low, then you have ~free internal rotation.
• If intermediate, then you're facing hindered internal rotation.

There's literature on evaluating enthalpy/entropy contributions of free and hindered internal rotations. Some can be found in my answer to a different question, here. Other potentially relevant resources include:

• Czaszar (WIREs Comput Molec Sci 2 273, 2012)
Overview of anharmonic force fields and their computation.

• Bloino (J Chem Theor Comput 8(3) 1015, 2012)
Overview of the VPT2 anharmonic computational system implemented in recent versions of Gaussian.

• Carbonniere (Chem Phys Lett 392 365, 2004)
Description of the vibrational-rotational (Coriolis) coupling anharmonic correction and its potential significance for some systems. More recent references are likely available.

• Meal (J Chem Phys 24(6) 1119, 1956)
Meal (J Chem Phys 24(6) 1126, 1956)
Classic discussions of Coriolis coupling.

You should be able to turn up quite a bit more information with some quick further searches, if need be.

2) Do I need to be worried about contributions from centrifugal distortion?

Very likely. Unfortunately, I don't know of very many examples where people have attempted corrections such as this. Certainly you're looking at manually developing a suitable analysis; I can't imagine any software packages out there at present are capable of automatically chugging through this for you.