Asymmetric Tops
Asymmetric top molecules have three distinct moment of inertia $$ I_a < I_b < Ic $$ and are the class that most molecules belong to. As you will see in the following, there is no general analytical solution. Therefore, we will try to build up some intution with the help of the limiting cases, the prolate and oblate symmetric tops (see previous chapter).
Theory
As mentioned above, the energy expression for asymmetric tops has no general analytical solution as all three moment of inertia components are different $I_{a} < I_{b} < I_{c}$. Thus, the problem is solved numerically which is typically done in matrix formalism. For this, the Hamiltonian is reformulated as $$ \begin{align} \frac{h}{4 \pi^2} \mathbf{H}_\text{rot} &=A \mathbf{J}_{a}^{2}+B \mathbf{J}_{b}^{2}+C \mathbf{J}_{c}^{2} \\ &=\left(\frac{A+B}{2}\right)\left(\mathbf{J}_{a}^{2}+\mathbf{J}_{b}^{2}\right)+C \mathbf{J}_{c}^{2}+\frac{(A-B)}{2}\left(\mathbf{J}_{a}^{2}-\mathbf{J}_{b}^{2}\right) \\ &=\left(\frac{A+B}{2}\right) \mathbf{J}^{2}+\left(C-\frac{A+B}{2}\right) \mathbf{J}_{c}^{2}+\left(\frac{A-B}{4}\right)\left(\mathbf{J}_{+}^{2}+\mathbf{J}_{-}^{2}\right) \end{align} $$ and the following identities are used $$ \begin{align} \left\langle J, K\left|\mathbf{J}^{2}\right| J, K\right\rangle &=\hbar^{2} J(J+1) \\ \left\langle J, K\left|\mathbf{J}_{c}^{2}\right| J, K\right\rangle &=\hbar^{2} K^{2} \\ \left\langle J, K+2\left|\mathbf{J}_{-}^{2}\right| J, K\right\rangle &=\hbar^{2}((J-K)(J+K+1)(J-K-1)(J+K+2))^{1/2} \\ \left\langle J, K-2\left|\mathbf{J}_{+}^{2}\right| J, K\right\rangle &=\hbar^{2}((J+K)(J-K+1)(J+K-1)(J-K+2))^{1/2} \end{align} $$ As all matrix elements are diagonal in $J$, the energy levels for each $J$ value can be calculated independently. This greatly reduces the size of the matrices and improves the efficiency of the calculations.
Toward even smaller matrices
The process is explained here for $J=1$, as this case can be solved analytically. A total of three states has to be considered due to the $2J + 1$ different values of $K$, which is the projection quantum number of $J$ on the molecular $z$-axis. The states are written, as in the identities above, in the format $|J,K\rangle$. The system is then described with the following matrix $$ \begin{equation} \mathbf{H}/h = \begin{pmatrix} C+\frac{A+B}{2} & 0 & \frac{A-B}{2} \\ 0 & A+B & 0 \\ \frac{A-B}{2} & 0 & C+\frac{A+B}{2} \\ \end{pmatrix} \end{equation} $$ The energy values are found as the eigenvalues of the matrix $$ \begin{align} \lambda_{1} &= A+B \\ \lambda_{2} &= A+C \\ \lambda_{3} &= B+C \end{align} $$ To label the corresponding states the correlation diagram of prolate and oblate symmetric tops is introduced and Ray's asymmetry parameter $$ \kappa = \frac{2B-A-C}{A-C} $$ is used.
The limiting cases are the prolate symmetric top on the left-hand side and the oblate symmetric top on the right-hand side. For each limiting case, the corresponding energy levels are shown. The curves connecting prolate and oblate states display the correlation. The levels for the asymmetric top are labeled by $J_{K_{a}, K_{c}}$, with $K_{a}$ stemming from the prolate symmetric top and $K_{c}$ from the oblate symmetric top. It is important, that $K_{a}$ and $K_{c}$ are only good quantum numbers in their respective limiting cases, but not in between. Important restrictions on $K_{a}$ and $K_{c}$ are, that their sum is equal to $J$ or $J+1$ and equivalently to the symmetric top case $ J \geq K_{a}$ and $J \geq K_{c}$.
States with same $J$ value are ordered in energy by $\tau=K_{a}-K_{c}$. As a result the eigenvalues are connected to the three possible states in the following manner: $E(1_{1,0})/h=A+B$ as this is the highest energy, $ E(1_{1,1})/h=A+C$, and $E(1_{0,1})/h=B+C$, as this is the lowest energy.
The figure shows the energy term diagrams for three molecules with different values of $\kappa$ (ordered from left to right by increasing values of $\kappa$). On the left hand side, glycidaldehyde with $\kappa = -0.98$ is close to the prolate limit for which reason $K_a$ is given on the $x$-axis. Accordingly, its energy term diagram matches qualitatively the prolate symmetric top energy term diagram (see previous chapter). Differences are, that for each pair of $J$ and $K_a$ values, two levels exist for the asymmetric top as $K_a$ is no longer a good quantum number and levels with both $J+1=K_a + K_c$ and $J=K_a + K_c$ are possible, and that the energy offset with $K_a$ is much more pronounced than for the simulated symmetric top in the previous chapter. The splitting between the asymmetry components (levels with same $J$ and $K_{a}$ value but different values of $J - K_a - K_c$) increases with $J$ but decreases with $K_a$. For low $J$ values, levels with the same $J$ and $K_a$ values are paired in energy (so-called prolate pairing) whereas at high $J$ values energy levels with same $J$ and $K_c$ values are paired in energy (so-called oblate pairing). On the right hand side, the energy term diagram of cyclopentadiene is shown, which is quite close to the oblate limit ($\kappa = 0.90$) for which reason $K_c$ is given on the $x$-axis. The qualitative agreement with the oblate symmetric top energy term diagram (see previous chapter) is very high. The asymmetry splitting increases with $J$ and decreases with $K_c$, similar to glycidaldehyde just that $K_a$ and $K_c$ are exchanged. Thus, also the pairing is inverted, going from oblate pairing for low $J$ values to prolate pairing for high $J$ values. Acetone-$^{13}$C$_1$ is quite asymmetric ($\kappa = 0.32$) which is also resembled by the strong asymmetry splitting in its energy term diagram.
These kinds of patterns are essential for the analyses of asymmetric molecules. Which patterns are present for each molecule depends heavily on the dipole moments and the selection rules tied to them.
Selection Rules
The selection rules for $J$ and $M$, the projection of $J$ on the fixed space axis, are $\Delta J=0,\pm 1$ and $\Delta M =0,\pm1$ due to the maximum angular momentum $\hbar$ that a photon may carry. Additionally, selection rules exist for $K_{a}$ and $K_{c}$ that are coupled to the dipole moment components $\mu_{a}$, $\mu_{b}$ and $\mu_{c}$.
Origin of selection rules
Every non-zero dipole moment component is responsible for a set of transitions with a set of selection rules. Accordingly, the designation is $a$-, $b$- and $c$-type transitions, with the corresponding component of the dipole moment being non-zero. \newline $a$-type transitions follow the selection rules $$ \begin{align} \Delta K_{a}&= 0 \ (\pm2, \pm4,...) \\ \Delta K_{c}&=\pm1 \ (\pm3, \pm 5,...) \end{align} $$ $b$-type transitions follow the selection rules $$ \begin{align} \Delta K_{a}&=\pm1 \ (\pm3, \pm 5,...) \\ \Delta K_{c}&=\pm1 \ (\pm3, \pm 5,...) \end{align} $$ $c$-type transitions follow the selection rules $$ \begin{align} \Delta K_{a}&=\pm1 \ (\pm3, \pm5,...) \\ \Delta K_{c}&=0 \ (\pm2, \pm 4,...) \end{align} $$ Values in brackets indicate, that these values are possible but less likely than the first value. The different transition types are shown in the following figure.
Consequences
In contrast to linear or symmetric top molecules, there are a lot more potential transitions in asymmetric tops. This typically results in a much more dense but weaker spectrum (given similar dipole moments).
Where to start?
Typically low $J$ and $K_a$ values offer a good starting point to begin the analysis. By gradually increasing in $J$ and $K_a$, the deviations from one series to another can be easily followed in the Loomis-Wood plots.
Whereas it is straight forward to assign all transitions of linear molecules and symmetric tops in Loomis-Wood plots, asymmetric tops require more care due to the more extensive selection rules. There are several approaches to tackle this problem. LLWP offers the Series Finder window which shows all predicted transitions and has the option to exclude already assigned transitions. By filtering the predictions for the coverage of the experimental spectrum, this leaves only unassigned transitions which can be used to check that all predicted transitions have been assigned (or that any remaining transitions are too weak in intensity to show up in the experimental spectrum).
On a similar note, LLWP provides a Peakfinder window to identify all peaks in the experimental spectrum. Similar to the Series Finder window, already assigned lines can be excluded which leaves only unassigned lines. If even the final model does not predict these lines, this can be a hint to isotopologues, vibrational states, or even impurifications.
Blended Lines
Another problem arises from the dense spectrum. Often multiple transitions are blended in the spectrum, meaning they are overlapping each other. If the two lines are still separable, the Blended Lines module can fit multiple lineprofiles to the experimental spectrum to find separate center frequencies. If the two lines are not separable, they can be assigned together as a blend. This means, they are assigned to the same center frequency and SPFIT will understand that these lines are blended. To do so, set the Blend Width in the Reference Series window to an appropriate value and fit the center position of the blended peak. A dialog will open and prompt you to select which lines should be assigned together as a blend. Only predictions within the Blend Width value from the predicted position of the reference transition will show up in the dialog.
Reduction and Represenation
The effective asymmetric top Hamiltonian requires the choice of a reduction (basically a set of parameters to use in your effective Hamiltonian). In the $S$-reduction, the centrifugal distortion Hamiltonian takes the form $$ \mathbf{H}_\text{cd} = -D_J \mathbf{J}^4 - D_{JK} \mathbf{J}^2 \mathbf{J}^2_z - D_K \mathbf{J}_z^4 + d_1 \mathbf{J}^2 \left( \mathbf{J}_+^2 + \mathbf{J}_-^2 \right) + d_2 \left( \mathbf{J}_+^4 + \mathbf{J}_-^4 \right) + ... $$ whereas in the $A$-reduction the centrifugal distortion Hamiltonian takes the form $$ \mathbf{H}_\text{cd}^{(4)} = -\Delta_J \mathbf{J}^4 - \Delta_{JK} \mathbf{J}^2 \mathbf{J}^2_z - \Delta_K \mathbf{J}_z^4 - \frac{1}{2} \left\{ \delta_{J} \mathbf{J}^2 + \delta_{K} \mathbf{J}_z^2, \mathbf{J}_{\pm}^2 \right\} \\ $$ where $\mathbf{J}_\pm = \mathbf{J}_x \pm i \mathbf{J}_y $ are the raising and lowering operators. The connection between the $a$, $b$, $c$ and the $x$, $y$, $z$ labels for the axes is given by the chosen representation. The $S$-reduction is typically advantageous for very symmetric molecules ($\kappa$ close to +1 or -1).
Representations
Adding Parameters
The large number of possible parameters makes it also complicated to find the right parameters to add to your model. Some important rules for choosing the right parameter sets are
- Make sure to only add parameters that are included in your reduction (not every parameter that you can code into the *.par/*.var file is physically meaningful)
- Try to keep the parameter set as small as possible to not overfit your model
- Add lower order parameters before adding higher-order parameters
- Check if parameters have physically meaningful values (this can be hard, especially as a beginner; compare with quantum chemical calculations, other isotopologues or vibrational states, or similar molecules)
- Check if all parameters have values that are significantly larger than their uncertainties (that all parameters are actually well determined)
- Adding a parameter to the model should considerably improve the quality of the model
pyckett --sreduction yourfilename
The output will look similar to
>>> pyckett_add yourfilename --sreduction
ID | RMS [kHz] | RejLines | Diverging | Init Value | Final Value
-----------------------------------------------------------------------
40100 | 190.94 | 0 | NEVER | 1.00e-37 | 6.28e-06
200 | 255.92 | 0 | NEVER | 1.00e-37 | -1.35e-06
1100 | 265.94 | 0 | NEVER | 1.00e-37 | -1.06e-06
50000 | 234.18 | 0 | NEVER | 1.00e-37 | 2.42e-06
2000 | 271.79 | 0 | NEVER | 1.00e-37 | -7.75e-07
Initial values were an RMS of 277.10 kHz, 0 rejected lines, and
diverging NEVER.
Best run is parameter 40100 with a final parameter value
of 6.28e-06, RMS of 190.94 kHz, 0 rejected lines, and diverging NEVER.
The first line is how to invoke the command. pyckett_add is the command name, yourfilename the path (without the file extension) of the *.par and *.lin files of the corresponding analysis, and --sreduction specifies that the $S$-reduction should be used. The following lines are the output generated by the program, representing a table. It is evident, that the parameter $d_1$ with the parameter ID 40100 would improve the model the most (as indicated by its root-mean-square (RMS) value in the first row). The parameter is a good choice to add to the model because its final parameter value is physically reasonable and its addition does not lead to the rejection of any lines nor to the fit diverging. Furthermore, its addition would improve the RMS value much more significantly than the addition of any other parameter. It is important to note that although Pyckett greatly improves the workflow by eliminating tedious manual and error-prone work, it does not eliminate the necessity of checking the results for spectroscopic meaningfulness. Similarly, the influence of omitting already added parameters can be tested with pyckett_omit.
A simple Example
In the following, the *.par, *.lin, and *.int files for the ground vibrational state of cyclopentadiene are shown. They should give you a sound starting point for similar molecules.
The *.par file is given in the following where sextic and higher parameters are excluded to keep the file concise.
c-C5H6
8 3510 10 0 0.0000E+00 1.0000E+01 -1.0000E+00 1.0000000000
s 1 -1 0 80 0 2 7 9 0 -1 0
10000 8.426108830069525E+03 1.00000000E+37 /A
20000 8.225640353653858E+03 1.00000000E+37 /B
30000 4.271437289845152E+03 1.00000000E+37 /C
200 -2.692727318837336E-03 1.00000000E+37 /-DJ
1100 4.059604568020397E-03 1.00000000E+37 /-DJK
2000 -1.682728070664547E-03 1.00000000E+37 /-DK
40100 -4.221893683527950E-05 1.00000000E+37 /d1
50000 -6.011886308978298E-07 1.00000000E+37 /d2
To understand the following explanations, it is useful to also open the CRIB sheet and go to the *.par file section. The first line specifies the name of the project, and the second line specifies the number of parameters (8), the number of lines in the *.lin file (3510), the number of iterations for the fit (10), the number of excluded parameters (0; counting from the end), the initial Marquardt-Levenburg parameter (0), the maximum (obs-calc)/error value before excluding lines from the fit (10; set this parameter pretty high in the beginning to not exclude lines when your model is not good yet), the FRAC parameter (-1; basically gives you standard errors), and the scaling for infrared line frequencies (1).
The third row indicates that our reduction (s), the degenracy of spins (1; no hyper-fine structure, ...), that we want to use the III$^l$ representation (negative sign of third parameter), and only a single vibrational state (magnitude of third parameter). The next values specify the minimum (0) and maxmimum values (80) for $K$. The next zero indicates that there are no $\Delta N \neq 0$ interactions. Afterwards, the weights are specified. The 2 means that the $b$-axis is used for the statitical weights and it is specified that the even states have a weight of 7 and the odd states of 9. This results from the symmetry of cyclopentadiene which leads to ortho and para states with the respective weights of 7:9.
The following lines define the parameters. The first number specifies the operators to multiply the parameter with, the second column are the current values, the third column specifies how much they should change (here you can float (1e+37) or fix parameters (1e-37)), and the fourth column is a comment. You can see, that $-D_J$, $-D_{JK}$, and $-D_K$ are given with a negative sign as their respective terms have a minus sign in the rotational Hamiltonian.
Parameter Coding
| EX | FF | I2 | I1 | NS | TYP | KSQ | NSQ | V2 | V1 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Digits | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1-3 | 1-3 |
V2 and V1 specify the (vibrational) state the parameter belongs to. Depending on the number of defined states (defined by the NVIB parameter, the 3rd parameter in the third row) they take 1-3 digits. For less than 10 states, one digit is used, for less than 100 states, two digits are used. It is important to note here, that the highest possible value (9 for one digit, 99 for two digits, 999 for three digits) has a special meaning as these parameter values will be applied to all states. This can be handy when defining e.g., vibrational states via rotation-vibration interaction parameters. NSQ (short for N squared) is the power of $N(N+1)$ which the parameter value is multiplied with and similarly KSQ specifies the power of $N_z^2$. Here, $N$ denotes the total rotational angular momentum quantum numbers (excluding any spins! This means $N + S = J$). TYP specifies the projection type where 1-3 are reserved for the projections onto the $a$, $b$, and $c$ axes, respectively. Subsequently, powers of $N_+^{2n} + N_-^{2n}$ are specified, with TYP = 3 corresponding to $N_+^{2} + N_-^{2}$, TYP = 4 to $N_+^{4} + N_-^{4}$, and so on till TYP = 10.
It is important that not all parameters that can be coded also should be used. Depending on the Hamiltonian you are using, only a specific subset of parameters should be used. See the information on A- and S-reduction e.g., on the PROSPE page.
When new parameters have to be added to the model, you typically want to test the parameters that increase either NSQ, KSQ, or TYP by 1 from already included parameters. After you have added a parameter to your model, always check its value and its uncertainty. If the value is unphysical (e.g., $D$ is larger than $B$) or is not properly determined, this can hint toward an effective of overfitting model.
An excerpt of the *.lin file looks like the following
2 1 1 2 0 2 11869.9900 0.0500 1.0000 /Scharpen and Laurie, 1965
1 1 1 0 0 0 12697.5615 0.0200 1.0000 /Benson and Flygare, 1970
4 2 2 4 1 3 20186.9900 0.0500 1.0000 /Scharpen and Laurie, 1965
3 1 2 3 0 3 20239.6800 0.0500 1.0000 /Scharpen and Laurie, 1965
3 2 2 3 1 3 20276.5900 0.0500 1.0000 /Scharpen and Laurie, 1965
4 3 2 4 2 3 20296.5300 0.0500 1.0000 /Scharpen and Laurie, 1965
5 4 2 5 3 3 20336.6800 0.0500 1.0000 /Scharpen and Laurie, 1965
2 0 2 1 1 1 21032.5000 0.0500 1.0000 /Scharpen and Laurie, 1965
2 1 2 1 0 1 21240.4100 0.0500 1.0000 /Scharpen and Laurie, 1965
2 2 1 1 1 0 29549.7000 0.0500 1.0000 /Scharpen and Laurie, 1965
3 0 3 2 1 2 29678.2700 0.0500 1.0000 /Scharpen and Laurie, 1965
3 1 3 2 0 2 29685.8800 0.0500 1.0000 /Scharpen and Laurie, 1965
The first six numbers are the $J$, $K_a$, $K_c$ values of the upper and lower levels, then the frequency in MHz, the uncertainty in MHz, their weight in a blend, and a comment.
The *.int file has the following content
c-C5H6
1 66520 50808.4397 0 80 -8 -8.5 510
002 0.419
The first line is the name of the project. In the second line, the first value is a flag to configure the output, then an ID for the *.cat file is specified (this ID is important when incorporating your final predictions into a larger database), the partition function of the molecule, the min (0) and max (80) values for $F = $, the logarithmic cutoff strengths for the *.cat file (STR0 = -8 and STR1 = -8.5). The first value is a static cutoff, meaning lines with LOGINT $\lt$ STR0 are not included in the *.cat file. The second value depends on the frequency of the transition as well, transitions with LOGINT $\lt$ STR1 * (FRQ/300GHZ)**2 will not be added to the *.cat file. The last value is the frequency limit in GHz (510 GHz). If you want to predict the spectrum at a temperature different to 300 K, you can add the temperature as the ninth parameter.
The last line specifies the dipole moment of the molecule. Cyclopentadiene has a single non-zero dipole moment component, $\mu_b = 0.419 \text{ D}$. The code 002 specifies the initial and final vibrational states (both 0 for the ground vibrational state) and the axis (a=1, b=2, c=3). The second value specifies the dipole moment in Debye.
It is highly recommended to begin by adapting the files of a similar molecule (e.g., from the Numerous Examples from the CDMS) and to read the CRIB sheet carefully. The CDMS also provides examples on many more complicated cases, including radicals, tunneling, vibration-rotation interactions, and many more.
Publicly available high-resolution spectra of asymmetric tops can be found for example on the Patterson Group page. They also provided rotational constants and centrifugal distortion parameter values for these species. You can choose some of this data to test your understanding of the concepts presented in this chapter.