Raman Spectrum of carbon chains: How to identify the vibrational modes

Question

So I am using NWChem to calculate the raman spectra of C2H2 and C4H4. For example for the first I obtained the following frequencies:

#  freq [ 1/cm]  S [Ang**4/amu]
------------------------------------------------------------
  1   644.63        0.0087
  2   679.18        0.0076
  3   792.83        0.0001
  4  2146.53        8.0917
  5  3506.05        0.0007
  6  3645.57       37.8189

and for example, when comparing with the expected results from http://www.chem.purdue.edu/gchelp/vibs/c2h2.html , I could say that my mode number four corresponds to C-C stretching. So now, when I perform the same calculation for C4H4, I get the following:

#  freq [ 1/cm]  S [Ang**4/amu]
------------------------------------------------------------
  1   216.77        2.3873
  2   305.29        0.7775
  3   548.05        0.8095
  4   643.94        1.7042
  5   677.47        0.5218
  6   700.29        6.8906
  7   893.88        0.1890
  8   928.98        0.5379
  9  1033.68        0.1554
  10  1134.09        1.5802
 11  1357.84       16.9774
 12  1504.93       31.2415
 13  1715.33       17.1094
 14  2269.08       38.4659
 15  3293.55       19.5975
 16  3313.05       78.1132
 17  3449.94       44.9757
 18  3578.50       24.6781

My question is this, how could I identify the vibrational mode corresponding to C-C stretching now? And if I performed the same calculation for C6H6, how could I identify the C-C vibrational mode? Is there a way of identifying what each mode corresponds to? not only in calculation but also in experiments?

thank you

Answer 1

When you perform a IR or Raman calculation, the primary step is the diagonalization of the mass-weighted Hessian:

$$ \begin{align} H_{ij,ab}^{\mathrm{mw}} &= \frac{1}{\sqrt{m_{i}m_{j}}} \frac{\partial^{2} E}{\partial a_{i} \partial b_{j}} \\ \mathbf{Hq} &= \lambda \mathbf{q} \end{align} $$

where $H_{ij,ab}^{\mathrm{mw}}$ is a $3N\times3N$ Hermitian matrix, $i,j$ are indices running over the atoms, $m_{i}$ is the mass of atom $i$, and $a,b \in x,y,z$ (the $3N$ Cartesian molecular coordinates). The eigenvalue $\lambda_{i}$, multiplied by some conversion factors, is a frequency in the harmonic oscillator approximation, and the corresponding eigenvector $\vec{q}_{i}$ is the normal mode, the collective atomic displacements, for that frequency.

(A calculation of Raman intensities involves extra work, namely the calculation of $\frac{\partial^{3} E}{\partial \varepsilon_{\alpha} \partial \varepsilon_{\beta} \partial a_{i}}$, but the normal modes are identical.)

Aside from performing literature searches for normal mode frequency assignments on related systems, a good alternative is the visualization of these normal modes. Many outputs from frequency calculations can be opened in Avogadro:

Clicking on a frequency in the right-hand side pane (only IR intensities are displayed) can display both the displacement vectors and an animation of the displacement (see the lower right corner box). Avogadro is able to read this portion of the NWChem output file:

          -------------------------------------------------
          NORMAL MODE EIGENVECTORS IN CARTESIAN COORDINATES
          -------------------------------------------------
             (Projected Frequencies expressed in cm-1)
...
                    7           8           9          10          11          12

 P.Frequency      252.66      366.45      606.76      773.89      790.45      808.06

           1     0.06344     0.00000    -0.02127     0.00000     0.00000     0.00096
           2    -0.10753     0.00000    -0.09540     0.00000     0.00000    -0.00247
           3     0.00000    -0.03639     0.00000    -0.01614     0.01014     0.00000
           4    -0.08820     0.00000    -0.09326     0.00000     0.00000     0.00127
           5     0.02291     0.00000    -0.01064     0.00000     0.00000    -0.00238
           6     0.00000     0.00615     0.00000     0.14073    -0.07465     0.00000
           7    -0.13091     0.00000     0.17533     0.00000     0.00000     0.05512
           8     0.01443     0.00000     0.09744     0.00000     0.00000     0.01777
           9     0.00000     0.19419     0.00000    -0.08894     0.11256     0.00000
          10     0.11294     0.00000    -0.04324     0.00000     0.00000    -0.12834
          11     0.07728     0.00000     0.04956     0.00000     0.00000    -0.02839
          12     0.00000    -0.10132     0.00000    -0.04711    -0.12489     0.00000
          13     0.29675     0.00000    -0.23722     0.00000     0.00000     0.83429
          14     0.12532     0.00000     0.00136     0.00000     0.00000     0.22209
          15     0.00000    -0.33078     0.00000     0.53191     0.69392     0.00000
          16     0.24449     0.00000     0.20104     0.00000     0.00000     0.01418
          17    -0.04794     0.00000    -0.02443     0.00000     0.00000     0.00175
          18     0.00000    -0.37449     0.00000    -0.49495     0.26769     0.00000
          19     0.03319     0.00000    -0.05962     0.00000     0.00000    -0.00066
          20    -0.29762     0.00000    -0.33571     0.00000     0.00000    -0.01410
          21     0.00000     0.25757     0.00000     0.27323    -0.15332     0.00000
          22    -0.06568     0.00000    -0.11340     0.00000     0.00000    -0.00250
          23     0.13575     0.00000    -0.12891     0.00000     0.00000    -0.02550
          24     0.00000    -0.29795     0.00000    -0.17367     0.10673     0.00000
...

So, to find the C-C stretching mode (or any vibration of interest), clicking through the list in Avogadro is more convenient than eyeballing the displacements in the output file. With the advent of accurate computational methods, the assignment of peaks in experimental spectra is usually aided by calculations identical to these. It's even possible to plot the computed spectra with some artificial broadening: