Symmetric Tops

So far, we have only looked at linear molecules as they have a very simple description. In this chapter, we will look into the next class of molecules, the symmetrip top molecules.

Theory

Each molecule has three principal axes, meaning they are oriented in such a manner that the moment of inertia tensor is fully diagonal with the three diagonal components being $I_a, I_b$ and $I_c$. By convention, the moments of inertia are ordered to fulfill $$ I_a \leq I_b \leq I_c $$ In the literature, often the moments of inertia are specified indirectly via the inversely proportional rotational constants $$ A/B/C = \frac{h}{8 \pi^2 I_{a/b/c}} $$ For a symmetric top, two of the three moments of inertia are equal. This means there can either be so-called prolate tops with $$ I_a \lt I_b = I_c $$ or so-called oblate tops with $$ I_a = I_b \lt I_c $$ Prolate tops have a more elongated shape, like a cigar, and oblate tops have a more flat shape, like a plate.

In the following, the theory will be shown at the example of a prolate symmetric top but the formulas for an oblate top can be found analogously by exchanging $A$ with $C$ (and $a$ with $c$).

The rotational energy of a symmetric top is obtained by using $I_b = I_c$ and $$ E_\text{rot} = \frac{J_a^2}{2I_a} + \frac{J_b^2}{2I_b} + \frac{J_c^2}{2I_c} $$ This yields $$ \begin{align} E_{p}&=\frac{J_{a}^{2}}{2I_{a}}+\frac{J_{b}^{2}+J_{c}^{2}}{2I_{b}}\\ &=\frac{J_{a}^{2}}{2I_{a}}+\frac{J^{2}-J_{a}^{2}}{2I_{b}}\\ &=J^{2}\frac{1}{2I_{b}}+J_{a}^{2}\left(\frac{1}{2I_{a}}-\frac{1}{2I_{b}}\right) \end{align} $$ The corresponding rotational Hamiltonian is $$ \mathbf{H}=\mathbf{J}^{2}\frac{1}{2I_{b}}+\mathbf{J}_{a}^{2}\left(\frac{1}{2I_{a}}-\frac{1}{2I_{b}}\right) $$ When using this equation and the rotational constants $A$, $B$ and $C$ instead of the moments of inertia, the energies are found to be $$ E_{p}/h =BJ(J+1) + (A-B)K_{a}^{2} $$ Analogously, the energies of an oblate symmetric top are found to be $$ E_{o}/h =BJ(J+1) + (C-B)K_{c}^{2} $$

Here, the total angular momentum quantum number $J$ and its projections on the respective molecule axes $K_a$ and $K_c$ are used. The projection quantum numbers have to fulfill $J \geq K_a$ and $J \geq K_c$ as a projection cannot be larger than the quantity itself. The equations for the energy levels are similar to the linear molecule case but have a $K_a$ or $K_c$ dependent offset. For a prolate symmetric top, the energy offset increases with $K_a$ whereas it decreases for the oblate symmetric top with $K_c$ (from $A \geq B \geq C$ it follows that $A-B \geq 0$ and $C-B \leq 0$). This is shown in the following figure where the energy term diagrams are compared to a linear molecule.

Simulated energy term diagrams for a prolate symmetric top (red), a linear molecule (black), and an oblate symmetric top (blue). For all molecules, $B$ was chosen as 8 GHz and for the prolate symmetric top $A=12\text{ GHz}$ while for the oblate symmetric top $C=4\text{ GHz}$. Only levels up to $J_\text{max} = 8$ and $K_\text{max} = 5$ are shown.

The selection rules for a symmetric top molecule are $$ \begin{align} \Delta J &= \pm 1 \\ \Delta K &= 0 \end{align} $$ resulting in the same transition frequencies as for a rigid linear molecule $$ \nu_{J+1,K\leftarrow J,K}=2B(J+1) $$ Only when adding centrifugal distortion effects, the degeneracy for transitions with different $K_a$ values is lifted $$ \begin{align} \begin{split} E_{p}/h = & BJ(J+1) - D_{J}(J(J+1))^2 \\ + & (A-B)K_{a}^2-D_{K}K_{a}^4-D_{JK}J(J+1)K_{a}^2 + \text{...} \end{split} \end{align} $$ and accordingly the transition frequencies become $$ \begin{align} \nu_{J+1,K\leftarrow J,K}=2B(J+1)-4D_{J}(J+1)^{3}-2D_{JK}(J+1) K_a^{2} + \text{...} \end{align} $$ Two things should be noted here. First, each $K_a$ series can be seen as a linear rotor spectrum with $B_\text{eff} = (B - D_{JK} K_a^2)$. Secondly, the parameter $D_K$ is not determinable from the spectrum as the transition frequencies do not depend on it. This may not be a problem if only the frequency positions of the spectrum are of interest, but if the complete energy term diagram is of interest (e.g., for a partition function) this parameter has to be estimated or calculated.

Implementation in SPFIT/SPCAT

Setting up a symmetric top in SPFIT/SPCAT requires a little care. We will look exemparily at methyl cyanide (CH$_3$CN), the favorite molecule of a dear colleague of mine. The *.par file is a slight modification from the file available in the CDMS:

	
CH3CN
  15  110    5    0    0.0000E+000    1.0000E+003    1.0000E+000 1.0000000000
s  -1    1   0   30   0   6   2   2     0    1   0
         1000  1.489000744965263E+005 1.00000000E+035 /A-B
          100  9.198899102551335E+003 1.00000000E+035 /B
         2000 -2.825727761558185E+000 1.00000000E+035 /-DK
         1100 -1.774063654292496E-001 1.00000000E+035 /-DJK
          200 -3.807510349803836E-003 1.00000000E+035 /-DJ
         3000  5.100000000000003E-005 1.00000000E-035 /HK
         2100  6.065576462396010E-006 1.00000000E+035 /HKJ
         1200  1.024780274614346E-006 1.00000000E+035 /HJK
          300 -2.595095183127352E-010 1.00000000E+035 /HJ
         3100 -4.538665856120687E-010 1.00000000E+035 /LKKJ
         2200 -5.285920186154858E-011 1.00000000E+035 /LJK
         1300 -7.746795214251004E-012 1.00000000E+035 /LJJK
          400 -1.586445135171180E-015 1.00000000E+035 /LJJK
         2300  5.696570048228503E-016 1.00000000E+035 /PJK
         1400  4.815222180909658E-017 1.00000000E+035 /PJJK

The first line gives the title of our project (CH3CN). Then, in the second row, the number of parameters is specified (15), the number of assigned lines (110), the number of iterations of the fitting algorithm (5), and so on. In the third line, the first parameter is a comment (for the choice of parameter names), then the SPIND parameter (-1) specifies via its magnitude that the degeneracy of spins in our molecule is one (no hyperfine-structure, ...) and via its sign (-) that we use the symmetric rotor quanta. Additionally, the IAX parameter is set to 6 to indicate that the axis for statistical weight is a 3-fold top axis. From the fourth row on, the rotational constants and parameters are specified. They are equivalent to the linear molecule case, but now we also use the fourth last digit $k$ to specify the power of $N_z^2$ (third last digit specifies the power of $N(N+1)$, and the last two digits specify the vibrational state).

From this, we can reconstruct the equation for all the rotational energy levels: $$ E_\text{Rot} = (A-B) N_z^2 + B N(N+1) + ... + P_{JJK} N_z^2 \ (N(N+1))^4 $$

The *.int file is adapted to methyl cyanide as well but follows the same principle for the linear molecule case:

	
CH3CN
1  41505    14683.6324       0    135   -7.0    -8.5    1807
  001  3.92197

If you create a *.var file from the *.par file, save the *.int file, and run SPCAT, you will see another interesting detail. Transitions with $K_a \neq 0$ are doubly degenerate:

	
   73577.4511  0.0003 -3.9822 3   48.3748 18  41505 202 4-3         3 3         
   73577.4511  0.0003 -3.9822 3   48.3748 18  41505 202 4 3         3-3         
   73584.5429  0.0002 -3.9973 3   23.5474  9  41505 202 4 2         3 2         
   73584.5429  0.0002 -3.9973 3   23.5474  9  41505 202 4 2         3 2

Here, the last four numbers are the quantum numbers of the transitions. The first two rows are the transitions from $J_{K_a} = 3_3$ to $J_{K_a} = 4_3$ and similarly the last two lines are the transitions from $J_{K_a} = 3_2$ to $J_{K_a} = 4_2$. If $K_a$ is a multiple of three, the upper and lower values of $K_a$ have opposite signs. Otherwise, the signs are positive. If you want to merge these entries, either run CALMRG from the CALPGM suite on the output, or write a simple script to merge these entries. For the *.lin file, the sign of $K_a$ values which are multiples of 3 can be chosen freely. This means, there is no difference between the following lines

	
 18  3 17  3                             331014.2959     .0010  /GC & CP 2006
 18 -3 17  3                             331014.2959     .0010  /GC & CP 2006
 18  3 17 -3                             331014.2959     .0010  /GC & CP 2006
 18 -3 17 -3                             331014.2959     .0010  /GC & CP 2006

Try Yourself

With the information you have been given, download the experimental spectrum (530-590 GHz) of methyl cyanide and try to assign the ground vibrational state and see if you can find any isotopologues or vibrationally excited states. The experimental spectrum was recorded in Cologne with a spectrometer using frequency modulation and a 2f-demodulation of the detector signal, resulting in lineshapes that look similar to the second derivative of a Voigt-profile. Also, you will see that the spectrum has quite some standing waves (periodic almost sinus-like features on the baseline) which result from reflections in the experimental setup. You can remove most of the standing waves by FFT-filtering the spectrum, e.g. with the FFTFilter package in Python. Lastly, some of the strongest lines are saturated to facilitate the detection of weaker features, resulting in weird lineshapes for these strong lines.

In LLWP, the number of quantum numbers should be set automatically to two after loading the *.cat file. Choose the series in the Reference Series window to comply with the selection rules ($\Delta K_a = 0$), e.g.:

Example for the choice of series in LLWP.

After you have assigned all the lines in the spectrum and modeled them nicely, you can double check your results in the CDMS by searching for the most abundant isotopologues and vibrationally excited states of methyl cyanide. Compare the parameters from the CDMS with the parametes obtained from your own analysis.

Vibrational States

Asymmetric Tops