Introduction to Vibrational Frequencies

What are Vibrational Frequencies?

Molecules are not static entities; they are in constant motion. One of the fundamental types of molecular motion is vibration, where atoms within a molecule oscillate about their equilibrium positions. The rate at which these atoms vibrate is described by vibrational frequencies. These frequencies, typically expressed in wavenumbers \(   \mathrm{cm^{-1}}  \), provide insight into the nature and strength of bonds within the molecule.

Why is it Important?

Characterizing Molecules: Vibrational frequencies can be directly compared to experimental infrared (IR) and Raman spectra, allowing for the characterization and identification of molecules.

Understanding Bond Strengths: The frequency of a vibration is related to the strength of the bond involved; stronger bonds tend to vibrate at higher frequencies.

Identifying Reaction Pathways: Vibrational frequencies can help identify transition states in reaction mechanisms. A transition state will typically have one imaginary frequency corresponding to the reaction coordinate.

Evaluating Molecular Stability: The presence of imaginary frequencies (negative eigenvalues) in a vibrational frequency calculation indicates that the molecule is not at a minimum on the potential energy surface.

How Does it Work?

Optimized Geometry: Before calculating vibrational frequencies, the geometry of the molecule is typically optimized to ensure it is at an energy minimum or transition state.

Hessian Matrix Calculation: The electronic structure program calculates the Hessian matrix, which is a matrix of second derivatives of the energy with respect to atomic displacements. This matrix provides information about the curvature of the potential energy surface.

Diagonalization: By diagonalizing the Hessian matrix, the vibrational frequencies of the molecule are obtained.

Assignments: Each vibrational frequency can be associated with a specific movement or distortion of the molecule. Visualization tools can help in understanding these vibrations.

Points to Remember:

A vibrational frequency calculation provides both the frequencies and the intensities of the vibrations, the latter of which can be directly compared to experimental IR or Raman spectra.

Imaginary frequencies indicate points on the potential energy surface that are not true minima or maxima. In the context of a transition state search, one (and only one) imaginary frequency is expected.

The accuracy of vibrational frequencies depends on the chosen quantum mechanical method and basis set, with some methods providing frequencies closer to experimental values than others.

Practical part 1 - Calculation of freqencies with XTB


In this practical exercise, we will use ultra fast GFN2-xTB method implemented in the xTB program to calculate frequencies. The purpose of this exercise is to learn how to check whether you are dealing with the ground state geometry of a molecule or not. Remember, the true ground state geometry of a molecule is characterized by the absence of imaginary (negative) frequencies, while transition states are characterized by a single imaginary frequency. We will again use the structures of non optimized and optimized phenol, and compare the obtained results.

3D structure of non optimized phenol:

3D structure of optimized phenol:

Procedure for calculating frequencies with the xTB program

Step 1 – Download molecular structures of non optimized and optimized phenol given above
Step 2 – Open “Online xtb calculator” to perform calculations of vibrational frequencies on both structures
Step 3 – Load the structure of non optimized phenol
Step 4 – Select task: Frequencies
Step 5 – Select method: GFN2-xTB
Step 6 – Press “RUN XTB” button 
Step 7 – Inspect the output file. Because we run frequency calculations on non optimized structure, a negative frequencies will be produced. To check the values of frequencies, place your cursor in the “Output file” text area press “Ctrl+F”, type “Frequency Printout” and press “Enter”. This will take you to the part of the output file containing frequencies. You will see many negative frequencies.
Step 8 – When you get negative frequencies, xTB produces a slightly distorted geometry to be used for further geometrical optimizations. In this case, within, you will see that an additional window opened, containing this structure in .xyz format. You can download it and use for further optimizations.
Step 9 – Now repeat steps from 3 to 7, but this time using the optimized structure of phenol. This time you will get only positive frequencies.
Step 10 – Download the structure of non optimized ibuprofen molecule.
Step 11 – In online xTB program, load the structure of non optimized ibuprofen molecule
Step 12 – Select task: Optimization + Frequencies, this will perform optimization and then automatically calculation of vibrational frequencies on optimized structure, which saves a lot of time 🙂
Step 13 – Select method: GFN2-xTB
Step 14 – Press “RUN XTB” button
Step 15 – By inspecting the output file, answer the following: a) was optimization succesfull and b) which frequency has the highest intensity

Practical part 2 - Visualization of frequencies


Visualization of frequencies can be tricky to be performed online, so in this part we will use Avogadro program for molecular visualization. Instead of using results produced the xTB program, we will use output file produced with the ORCA modeling package using the DFT method with B3LYP density functional and with SVP basis set. Since we are limited with time, will not cover now how to produce input file for ORCA and how to run a calculation with this package. Instead, you will use the already produced output file that can be downloaded by using the button bellow.

To download, right-click and click “Save Link As”. When downloading .out files, WordPress changes the extension from .out to .txt. Revert it manually to .out.

Now please follow the procedure to get information about frequencies:

  1. Download the .out file that contains output from the DFT calculation by the ORCA program on the structure of ibuprofen. In this calculation, it was instructed to ORCA to perform optimization and then a subsequent frequency calculations on ibuprofen molecule.
  2. Open the Avogadro program on your computer
  3. In Avogadro, load the .out file you just downloaded by going to File/Open or by simply draging .out file to Avogadro.
  4. On the right side of Avogadro, two panels should automatically load, one for the frequencies and one for the molecular orbitals. If they are not opened, go to Settings/Toolbars/Vibrations. This should open the Vibrations window. Click on any vibration that has nonzero intensity (I > 0). 
  5. Click Animate to animate vibrations. Animate the most intensive vibration (you should get the result as in the following video)