Symmetric Tops

So far, we have focused on linear molecules because they have a very simple spectroscopic pattern. In this chapter, we will examine the next class of molecules: symmetrip top molecules.

Theory

Each molecule has three principal axes. These axes are oriented such that the three orthogonal axes that result in the moment of inertia tensor being fully diagonal with the three diagonal components $I_a, I_b$ and $I_c$. By convention, the moments of inertia are ordered to fulfill $$ I_a \leq I_b \leq I_c $$ 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. Prolate symmetric tops have equal moments of inertia along the $b$- and $c$-axes $$ I_a \lt I_b = I_c $$ making them elongated, like a cigar. Meanwhile, oblate symmetric tops have equal moments of inertia along the $a$- and $b$-axes $$ I_a = I_b \lt I_c $$ giving them a rather flat shape, like a plate.

In the following, the formulas for the prolate symmetric top are derived 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 from the general expression $$ E_\text{rot} = \frac{J_a^2}{2I_a} + \frac{J_b^2}{2I_b} + \frac{J_c^2}{2I_c} $$ and exploiting $I_b = I_c$. 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 $$ \mathbf{H}=\mathbf{J}^{2}\frac{1}{2I_{b}}+\mathbf{J}_{a}^{2}\left(\frac{1}{2I_{a}}-\frac{1}{2I_{b}}\right) $$ The energies are the eigenvalues and are expressed in the following formula with the rotational constants $A$, $B$ and $C$ instead of the moments of inertia $$ E_{p}/h =BJ(J+1) + (A-B)K_{a}^{2} $$ Analogously, the energies of an oblate symmetric top are found to be $$ E_{o}/h =BJ(J+1) + (C-B)K_{c}^{2} $$

In these formulae, $J$ is the total angular momentum quantum number and $K_a$ and $K_c$ are its projections on the respective molecular axes. The projection quantum numbers fulfill $J \geq K_a$ and $J \geq K_c$ as a projection is never 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 the case of a linear molecule.

Simulated energy term diagrams for a prolate symmetric top (red), a linear molecule (black), and an oblate symmetric top (blue). $B$ is 8 GHz for all molecules 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 $$ \begin{align} \Delta J &= \pm 1 \\ \Delta K &= 0 \end{align} $$ Due to the symmetry of the symmetric tops, their dipole moment is fully aligned with their symmetry axis and thus $K$ cannot change in electric dipole interactions. When neglecting centrifugal distortion, this results in the same transition frequencies as for a linear molecule $$ \nu_{J+1,K\leftarrow J,K}=2B(J+1) $$ Only when centrifugal distortion effects are considered, 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 $B_\text{eff} = (B - D_{JK} K_a^2)$. Second, the parameters $A$ and $D_K$ cannot be determined from the pure-rotational spectrum since the transition frequencies do not depend on them. 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 of for reliable relative intensities) these parameters have to be determined from other measurements or calculated.

Implementation in SPFIT/SPCAT

Setting up a symmetric top in SPFIT/SPCAT is only slightly different from setting up a linear molecule. Here, we will use the methyl cyanide (CH$_3$CN) as an example since it is the favorite molecule of a dear colleague of mine. The following *.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 states the title of our project. In the second line, 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 line on, the rotational constants and parameters are specified. Their coding is equivalent to the linear molecule case. The final two digits specify the state (here always 00 for the ground vibrational state), the third last digit specifies the power of $N(N+1)$, and the fourth last digit specifies the power of $N_z^2$.

We can use the *.par file to reconstruct the used expression for the 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 also similar to 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 an 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

In the above output, the last four numbers are the quantum numbers of the transitions. The first two lines 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, you can either run CALMRG from the CALPGM suite on the output, or write a simple script to merge these entries. For the *.lin file, the signs for the $K_a$ values that 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 it Yourself

Download the experimental spectrum (340-500 GHz and 530-590 GHz) of methyl cyanide and assign the ground vibrational state. See then if you can find any isotopologues or vibrationally excited states. The experimental spectrum was recorded in Cologne with a spectrometer employgin frequency modulation and a 2f-demodulation of the detector signal, resulting in lineshapes that look similar to the second derivative of a Voigt-profile. 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. Most of the standing waves can be removed by FFT-filtering the spectrum, e.g. with the FFTFilter package in Python. Also, the sensitivity of the spectrometer is very frequency dependent, making it difficult to compare absolute intensities (especially if the transitions are far apart in frequency). Last, some of the strongest lines in the 530-590 GHz spectrum 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 2 after loading the *.cat file. Otherwise you can go to View > Config and change the value of series_qns to 2. Make sure to choose a series in the Reference Series window that complies 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 to experimental uncertainty, 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 CDMS parameters with the parametes obtained from your own analysis.

Vibrational States

Asymmetric Tops