# Modes of vibration of the benzene molecule¶

Author

Javier Junquera and Andrei Postnikov

The goal of this exercise is to compute the vibrational frequency of a molecule (benzene).

Before running the calculation to compute the vibrational frequencies, the first step is to relax the geometrical structure of the system under study.

So, to start with, we run a conjugate gradient minimization to relax the atomic positions. The input file has been prepared for you in the file benzene.relax.fdf. See how the structure of the benzene molecule has been introduced in the Z-matrix format (especification of internal variables, such as distances, angles, and torsional angles). This allows the minimization including some constraints in the symmetry in a trivial way. We just allow the C-C and C-H distances to relax:

%block Zmatrix
molecule
2 0 0 0  xm1 ym1  zm1   0 0 0
2 1 0 0  CC  90.0 60.0  0 0 0
2 2 1 0  CC  CCC  90.0  0 0 0
2 3 2 1  CC  CCC  0.0   0 0 0
2 4 3 2  CC  CCC  0.0   0 0 0
2 5 4 3  CC  CCC  0.0   0 0 0
1 1 2 3  CH  CCH  180.0 0 0 0
1 2 1 7  CH  CCH  0.0   0 0 0
1 3 2 8  CH  CCH  0.0   0 0 0
1 4 3 9  CH  CCH  0.0   0 0 0
1 5 4 10 CH  CCH  0.0   0 0 0
1 6 5 11 CH  CCH  0.0   0 0 0
constants
ym1 5.00
zm1 0.00
CCC 120.0
CCH 120.0
variables
CC 1.390
CH 1.090
constraints
xm1 CC -1.0 3.903229
%endblock Zmatrix


We run Siesta to carry out the relaxation:

siesta  benzene.relax.fdf > benzene.relax.out


## Computing the force constants in real space¶

• We have prepared an input file, called benzene.ifc.fdf to run Siesta and compute the interatomic force constants in real space:

• We have copied the relaxed coordinates and unit cell from the benzene.XV generated after the relaxation to the AtomicCoordinatesAndAtomicSpecies block.

• We have included the atomic masses after the coordinates of each atom. This will be useful for the next step with the vibra code

Here are the relevant sections:

LatticeConstant     1.0 Bohr
%block LatticeVectors
20.932528150       0.000000000       0.000000000
0.000000000      19.551203193       0.000000000
0.000000000       0.000000000      10.714661844
%endblock LatticeVectors

AtomicCoordinatesFormat NotScaledCartesianBohr
%block AtomicCoordinatesAndAtomicSpecies
4.738724869       9.448634389       0.000000000     2    12.0107
6.057380810      11.732613477      -0.000000000     2    12.0107
8.694692693      11.732613477      -0.000000000     2    12.0107
10.013348634       9.448634389      -0.000000000     2    12.0107
8.694692693       7.164655301      -0.000000000     2    12.0107
6.057380810       7.164655301      -0.000000000     2    12.0107
2.647979028       9.448634389      -0.000000000     1     1.00794
5.012007889      13.543252488      -0.000000000     1     1.00794
9.740065613      13.543252488      -0.000000000     1     1.00794
12.104094475       9.448634389      -0.000000000     1     1.00794
9.740065613       5.354016289      -0.000000000     1     1.00794
5.012007889       5.354016289      -0.000000000     1     1.00794
%endblock AtomicCoordinatesAndAtomicSpecies


To compute the interatomic force constant in real space, we have to run Siesta:

siesta < benzene.ifc.fdf > benzene.ifc.out


The interatomic force constant matrix in real space is stored in a file with name of the form SystemLabel.FC.

## Computing the dynamical matrix at the Gamma point, and (phonon) modes¶

Once the interatomic force constants in real space have been computed, a discrete Fourier transform is performed to compute the dynamical matrix in reciprocal space. Then, the dynamical matrix is diagonalized and its eigenfrequencies and eigenvectors are computed. This is done using the vibra code.

In the case of a molecule, only the Gamma point is relevant. It is specified in the same way as to compute the electronic band structure, in the same file benzene.ifc.fdf:

Eigenvectors    .true.        # Compute both phonon eigenvalues and eigenvectors
BandLinesScale  pi/a
%block BandLines
1   0.0   0.0   0.0   \Gamma  # Only the Gamma point (enough for a molecule)
%endblock BandLines


To compute the vibrational frequencies:

Your_siesta_directory/Util/Vibra/Src/vibra < benzene.ifc.fdf > vibra.out


The output of this code is:

• SystemLabel.bands: with the different mode frequencies (in cm^-1). They are stored in the same way as the electronic band structure.

• SystemLabel.vectors: with the eigenmodes at Gamma (the format is self-explained).

Note

Some of the modes might have negative frequencies. How could that be?

## How to visualize the normal modes¶

After getting the .vectors file (calculated by vibra) and the .XV file (computed in Siesta), run the vib2xsf program.

You have to answer a few question on the fly, regarding the name of the files where the .vectors are stored, the units to be used to introduce the lattice vectors (Bohrs or Angstroms), the zero of coordinates, the unit cell lattice vectors, the first mode to visualize, the last mode to visualize, the amplitude of the modes to be visualized, and the number of steps in the movie.

You can play a little bit, but to save time we have prepared all the answers in the file vib2xsf.dat for you. Just run:

vib2xsf < vib2xsf.dat


This will produce two files per mode:

• .XSF file: contains a static structures (as in .XV), with arrors added to each atom to indicate displacement pattern.

• .AXSF file: contains the animation of a phonon, for a (user-chosen) amplitude and number of steps.

They can be visualized using XCRYSDEN:

xcrysden

Select "File"
Open Structure
Open AXSF (Animation XCrySDen Structure File)


The same can be done to visualize the XSF file, but just choosing:

Select "File"
Open Structure
Open XSF file (XCrySDen Structure File)


Note

It might be interesting to analyze those modes with negative or small frequencies.