Drawing the vibrational modes depends on its shape and symmetry . Knowing the wave nomber ,you should check with Atlas of Spectroscopy

Thank you for your quick reply!!! Assuming that the symmetry of the molecule is known is there some kind of algorithm to follow? (I don't know if the question makes any sense but I am trying to understand...)

I agree with the very useful answers of Hatem and Gert.

However, when you post a question, please give more infos about " the molecule" e.g. (Inorganic, organic, metallcomplexes....a.s.o....). On other part, a comparison with experiment needs a solid background in group theory, quantum chemistry and molecular spectroscopy .

Please note that the molecular symmetry group for a given molecule is composed of selected elements. The fundamental reason why a molecule has symmetry is that it contains two or more identical atoms.

Good luck when you ´ ll deal and read with mathematics, group theory and molecular spectroscopy books.

JRG

Thank you for your helpful suggestions!!! As it seems I will need a lot of luck... The reason why I haven't referred to a specific molecule is that I was wondering if there is a general way to draw the normal modes of a molecule by knowing its symmetry...

The question is quite general and thus accepts different answer. I think the previous posts are very good approaches to the issue.

As fundamental aspects seem to be required, before talking about normal mode, the (vibrational) degrees of freedom are needed to be introduced. Physicochemical textbooks can be consulted, the following webpage is very clear also:

http://chemwiki.ucdavis.edu/Physical_Chemistry/Spectroscopy/Vibrational_Spectroscopy/Vibrational_Modes

The symmetry of the molecule has to be considered and the irreducible representation obtained, as already suggested. How to connect the symmetry properties with the vibrational one is not so obvious, but the usefulness of the "normal mode" concept appears right here, connected with the notion of “internal coordinates” (For instance: bond length as stretching, bond angles as deformations, dihedral angles as torsions, etc). For this topic I would like to recommend the following book:

Cotton, F. A. „Chemical Applications of Group Theory"

See also the following site:

http://www4.ncsu.edu/~franzen/public_html/CH795N/lecture/XIV/XIV.html

The “prediction” of the frequency wavenumber for each normal mode is a separate issue. In my opinion, the deepness of the analysis should depend on the system under study (gas, liquid or solid) and on the available experimental data (infrared, Raman, etc.).

Maybe it is best to refer this topic as band “assignment”. The most basic analysis is the well-known "group frequency", describing the typical values for the most important group in the molecule. The comparison with similar species is also desirable. As already suggested, quantum chemical calculations are nowadays feasible, especially for isolated molecules.

Finally, a Potential Energy Distribution (PED) can be also performed, i.e. the potential energy contribution of the each of the internal coordinates to the normal modes. For doing that, several programs have been developed (for the “Asym40” code see Hedberg, L.; Mills, I. M. J. Mol. Spectrosc. 2000, 203, 82)

You can use Jmol (an open source and free software) for the representation of vibration modes, you can see a demo here: http://jmol.sourceforge.net/demo/vibration/

To predict the vibration modes you can minimize the structure with a sofware like Gaussian or Spartan and then open the output file with Jmol.

Well, you have two options:

1. as already suggested, use a QC software to calculate and visualize the normal modes of a molecule

2. use the molecular symmetry information, construct the projection operators and apply them on internal coordinates. This way you will obtain the symmetry coordinates (linear combinations of internal coordinates) as basis for the irreducible representations of the point group to whom your molecule belongs. The combinations of internal coordinates will tell you the way in which the molecule is vibrating.

This is not straightforward but for very simple molecules.

As an example, for the water molecule one can define three internal coordinates (dr1, dr2 and dtheta, describing the stretches of the two OH bonds and the change of the HOH angle). Applying the P^A1 projection operator on the two stretching coordinates one obtains:

P^A1 dr1 = dr1+dr2

P^A1 dr2 = dr1+dr2

that is, the symmetry coordinate S1=dr1+dr2 (or, normalized to 1/sqrt(2) * (dr1+dr2)) is a basis for the irreducible representation A1. In other words, the vibration defined by dr1+dr2, which is a symmetric stretch of the two OH bonds of water, has A1 symmetry.

Applying the P^B2 operator on either dr1 or dr2 gives the combination (dr1-dr2, normalized by 1/sqrt(2)) which points to an asymmetric stretching vibration.

Finally, the dtheta internal coordinate (which describes the change of the HOH angle in water) is a basis for A1 by itself.

For the most general answer, you would want to use an electronic structure calculation (molecular orbital calculation) with software like Gaussian, Spartan, or GAMESS to name a few. If you are looking for approximate but still normal modes of vibration, you can also use molecular mechanics approach using force constants for bond stretching, bending, torsion, etc. (In fact, the Cartesian coordinate output of the vibrational normal modes from software like Gaussian can be mapped onto coordinates in terms of stretches, bending, etc, that Wilson, Decius and Cross explains in detail in their classic book).

Polymers require a bit different treatment, but are essentially modeled as infinitely long helical chains (in the case of modeling frequencies for the crystal form of polymers). The most accurate treatment is found in Turrell's "Infrared and Raman Spectra of Crystals."

If you are just looking for a sketch of normal modes of known symmetric molecules (say, substituted aromatics) along with the actual vibrational frequencies, then I have found a book like Structure Determination of Organic Compounds by Pretsch / Buhlmann / Affolter to be very helpful.

This field of study hasn't been altogether completely automated (although it's coming close for small molecules), so a more specific question would probably get an easier-to-implement answer. If you have an example, you will probably get a good answer from this forum of experts.

Dear Dimitris, From a practical point of view, I would use Gaussian or some open source code to calculate the frequencies and forces for the molecule you wish to obtain a drawing of normal modes for, and then use my vibrational analysis package to produce a figure for the displacement vectors of each, or of a selected few. This software is available here:

https://www.researchgate.net/publication/260145630_rwilliams-vib-software-12feb2014?ev=prf_pub

and an example, in Ortep divergent stereo, of the normal mode drawings that it can provide can be seen on page 605 here:

https://www.researchgate.net/publication/20891303_Relation_between_calculated_amide_frequencies_and_solution_structure_in_Ala-X_peptides?ev=prf_pub

where they lead to some insight about coupling. These can be useful where molecules are small, but they become hopelessly complicated in molecules much larger than L-Ala-L-Ala here.

This is not commercial software; it was developed for research, but it builds out of the box on an Ubuntu Linux system. No warranty is attached, but if you, or anyone else uses it, please enjoy. Best wishes, Bob

Data rwilliams-vib-software-12feb2014

Article Relation between calculated amide frequencies and solution s...