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Let's assume I have already relaxed a molecular structure for a non-trivial molecule and obtained a geometry that is close to the one resulting from X-ray crystallography. This can be achieved in a number of ways, e.g. trivially (relaxation in vacuum) if the molecule is not feeling a significant distortion from its environment, or with ad-hoc tricks such as surrounding it with a frozen local environment. In any case, I have access to a set of vibrational normal modes, that, in a first approximation and for my purposes, may resemble the actual vibrations of the molecules in the crystal. This can be done with a variety of packages, let's say we are using Gaussian.

Independently, we have information on the phonon spectrum. Again, this can be done with a variety of packages, let's say we are using Quantum Espresso.

The question is: is there a good procedure to map (or, at least, to relate) the molecular normal modes with the lattice phonons? In particular I'm interested in complicated (C1 symmetry) molecules, so any way that does not require relying on symmetry arguments would be preferred.

To put the question into perspective: the goal would be to facilitate the molecular design of systems with improved properties. Designing and obtaining molecules is hard enough as it is, but designing specific crystal packings is adding a layer of complexity, hence the utility of projecting (most of) the key behavior into the single molecule level, whenever it is possible.

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    $\begingroup$ This is a very good question! $\endgroup$
    – Nike Dattani
    Apr 30, 2020 at 14:07
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    $\begingroup$ In the first step when you relaxed your molecular structure and extracted its normal modes, did you used periodic boundary conditions for whatever DFT model that you used or you treated it as a single molecule without periodicity? If no, I think you can't use those molecular normal modes for studying lattice vibrations. Why you don't extract phononic band structure for your molecule placed in a lattice based on your specific symmetry? Is there any problem with that? $\endgroup$
    – Mithridates the Great
    Apr 30, 2020 at 16:47
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    $\begingroup$ How do you get the X-ray structure for a molecule? The X-ray crystallographic will work for single crystal or powder (in any case, crystal ==> periodic structure). When you "remove" the molecule from the crystal, you broke the symmetry of your system and you eliminate the interactions between the molecule and it neighbors. As for the same molecule you can have multiples (infinite?) crystal polymorphs, I think you can not make a map in a suitable form. $\endgroup$
    – Camps ♦
    May 1, 2020 at 1:26
  • $\begingroup$ I don't know how you could "relate/map" lattice modes to molecular modes, since vibrational frequency between them can be different. Nonetheless, I know a commercial software of Materials Design, which has this functionality - for solids and molecular species. It involves combining "somehow" phonon dispersion curves at gamma point and combine it with dielectric tensor calculations (in the case of periodic solids say in VASP using LEPSILON tag). The animation feature is packed in this software where you can pick which normal mode branch you want to see the vibrational animation for. $\endgroup$
    – gogo
    May 22, 2020 at 3:41

1 Answer 1

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If you index the molecule and the atoms in the crystallographic unit cell the same, you could extract the displacement from the phonon eigenvectors and the displacement from the normal modes. By projecting each normal mode displacement onto each phonon displacement, you can get a qualitative relationship between a molecular deformation and a lattice vibration. This could be readily visualized in a heat map. I can't think of any quantitative relationship; however, this offers a quantitative view on relating favorable deformations between a given crystal polymorph and molecule.

Details of the displacement and eigenvectors are here: Dove, M. T. Structure and Dynamics: An Atomic View of Materials (2002)

with a more thorough work on phonons here: Dove, M. T. Introduction to Lattice Dynamics (1993).

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  • $\begingroup$ Are there open source programs implemented that do this? $\endgroup$
    – gogo
    May 22, 2020 at 12:41
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    $\begingroup$ What about the molecular environment? The isolated molecule has more degree of freedom than the same molecule in the crystal. Also, as there are polymorphs, how to deal with them? $\endgroup$
    – Camps ♦
    May 22, 2020 at 12:46
  • $\begingroup$ @gogo I'm unaware of anything that does this automatically, but if the indices are the same it is just a permutation of dot products. $\endgroup$
    – Phil Maffettone
    May 22, 2020 at 13:42
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    $\begingroup$ @I.Camps, absolutely. The isolated molecule with have 3N DOF, and assuming there is only one molecule per unit cell, the entire crystal will have 3N phonon eigenvectors. I would however anticipate that some of the molecular distortions in the optical modes will contain a subset of molecular normal modes, while other lattice distortions keep the molecules relatively rigid. The second point hit's the nail on the head. I'm not sure what amount of predictability this approach can have, and it would have to be run for each combination molecule and polymorph. But it's an interesting post analysis. $\endgroup$
    – Phil Maffettone
    May 22, 2020 at 13:48

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