Interpreting the Microwave Spectra of Diatomic Molecules

: A brief review of the theory of the rigid rotor and its application to microwave spectroscopy is given. By careful selection of examples, procedures are given for the analysis of successively more complicated spectra, and the theory is extended to the harmonic nonrigid rotor and anharmonic nonrigid rotor when needed. The microwave spectra of carbon monoxide, and of some alkali halides, provide excellent examples for analysis and for student exercises.


Introduction
Microwave radiation corresponds to the frequency range from approximately 0.3 to 300 GHz and is important in modern satellite and cell phone communication. Microwaves lie between radio waves of lower frequency and infrared radiation of higher frequency and have a wavelength from 0.001 to 1 m [1]. Microwave spectroscopy gives information about the dipole moment and moments of inertia of a gas-phase molecule because the photon energies correspond to transitions between different rotational levels [2]. Radio telescopes can measure this radiation in space, supporting accurate estimates of the age of the universe through the cosmic microwave background, detecting the presence of small molecules in cosmic dust clouds, and inferring the presence of dark matter by measuring the rotational speed of galaxies via the hydrogen line at 21 cm. On Earth, microwave spectrometers have made their way into a few undergraduate laboratories [3,4], as have specialized computer programs for their prediction and analysis [5][6][7][8]. In some cases, Fourier-transform infrared spectrometers can be used to observe rotational bands [9]. The simplest spectra to analyze are molecules containing only two atoms. If performed correctly, very accurate bond distances, devoid of solvent or crystal packing effects, can be calculated.

Basic Theory
A diatomic molecule can be regarded as two masses, m 1 and m 2 , separated by a distance r. If we assume that the distance r is fixed, then we refer to this as the rigid rotor approximation. This problem can be shown to be equivalent to a single mass µ rotating at the same distance r about an origin located at the center of mass of the molecule. This reduced mass µ is given as follows: The moment of inertia of this system, I, is given by constant used therein. Carbon monoxide has a longer bond length in the gas phase than in the solid. The two pedagogical goals achieved here are (1) demonstrating the assignment of rotational quantum number J using extraspectroscopic information (XRD), and (2) confirming isotopologue assignment by comparing B ratios to µ ratios.

Alkali Halides
The alkali metal to halogen distances in the alkali halides in both the crystal and gas-phase, as measured by X-ray [29] and/or electron diffraction [30], respectively, are given in Table 1. Even though the crystal measurements are at a lower temperature, the distances are longer than the gas phase by an average of 12% for the NaCl-like structures and 17% for the CsCl-like structures. The major reason is that in the gas phase, the atoms only have one nearest neighbor, whereas in the solid, the atoms have six (NaCl-like) or eight (CsCl-like) equidistant nearest neighbors. The missing gas-phase results can be estimated for the purposes of predicting the rotational constants if needed.  3 X-ray diffraction (CsCl-like). 4 Electron diffraction (gas), 1000-1300 • C.

Cesium Iodide (CsI)
Both cesium and iodine are monoisotopic, so complications from multiple isotopologues are avoided. The predicted value of 2B from the electron diffraction value is 1340 MHz (see Supplementary Material, Excel file CsI.xlsx). The spectrum of cesium iodide in the region 22-26 GHz was measured [31] at 640 • C, and the frequencies observed (error 0.1 MHz) are given in Figure 1. Inspection of the nine frequencies (n = 9) demonstrates that there appear to be three major groupings (clusters) of frequencies. One-dimensional cluster analysis has recently been applied by the author to confirm this [32]. This tells us that the primary variation in the spectra is due to three distinct values of an independent variable, which we identify as J.
R PEER REVIEW 4 cluster analysis has recently been applied by the author to confirm this [32]. This tells us that the primary variation in the spectra is due to three distinct values of an independent variable, which we identify as J. We now turn to determining J for these clusters. The cluster averages (MHz) for N = 3 are given as 22,600, 23,942, and 25,387, with differences of 1343 and 1445 (average 1394 ± 51). The predicted value of 2B is 1340 MHz, and it is therefore reasonable to assume that these clusters are due to successive values of J. The differences between the highest frequency datum (head) of each cluster are 1414.14 and 1414.13 MHz. Dividing the k = 3 cluster averages by 1340 MHz gives 16.87, 17.87, and 18.94, which suggests that these bands should be assigned as J = 16, 17, and 18, respectively. However, dividing the cluster averages by the average separation between clusters gives 16.21, 17.18, and 18.22, suggesting J = 15, 16, and 17, respectively. Dividing the heads by the head difference gives 16.00431, 17.00432, and 18.00431, which would suggest that this is the most accurate way to estimate the correct values of J + 1 (J = 15, 16, and 17).
The two pedagogical goals achieved here are (1) introducing the utility of cluster analysis, and (2) if more than one value if J is believed to be present, comparing the assignment of J using cluster averages/predicted 2B (extraspectroscopic), cluster averages/cluster differences, and head/head difference (preferred).
With the rotational quantum numbers J assigned, we may now calculate the effective rotational constants for each point of the spectrum. These are shown in Figure 2. The variation within each frequency cluster (or effective rotational constants) is not yet explained; however, it appears quite regular. Cluster analysis may also be carried on the effective rotational constants to show that there are four clusters [32]. There is also a very slight decrease in effective rotational constant with increasing J. We must return to the theory to explain this. We now turn to determining J for these clusters. The cluster averages (MHz) for N = 3 are given as 22,600, 23,942, and 25,387, with differences of 1343 and 1445 (average 1394 ± 51). The predicted value of 2B is 1340 MHz, and it is therefore reasonable to assume that these clusters are due to successive values of J. The differences between the highest frequency datum (head) of each cluster are 1414.14 and 1414.13 MHz. Dividing the k = 3 cluster averages by 1340 MHz gives 16.87, 17.87, and 18.94, which suggests that these bands should be assigned as J = 16, 17, and 18, respectively. However, dividing the cluster averages by the average separation between clusters gives 16.21, 17.18, and 18.22, suggesting J = 15, 16, and 17, respectively. Dividing the heads by the head difference gives 16.00431, 17.00432, and 18.00431, which would suggest that this is the most accurate way to estimate the correct values of J + 1 (J = 15, 16, and 17).
The two pedagogical goals achieved here are (1) introducing the utility of cluster analysis, and (2) if more than one value if J is believed to be present, comparing the assignment of J using cluster averages/predicted 2B (extraspectroscopic), cluster averages/cluster differences, and head/head difference (preferred).
With the rotational quantum numbers J assigned, we may now calculate the effective rotational constants for each point of the spectrum. These are shown in Figure 2. The variation within each frequency cluster (or effective rotational constants) is not yet explained; however, it appears quite regular. Cluster analysis may also be carried on the effective rotational constants to show that there are four clusters [32]. There is also a very slight decrease in effective rotational constant with increasing J. We must return to the theory to explain this.

Advanced Theory
A diatomic molecule can vibrate as well as rotate. If the oscillation is harmonic, then for small oscillations x about the internuclear distance re, the Schrödinger equation is where k is the spring constant. If we scale x by the constant α 1/2 to give the dimensionless ξ, where then the solutions to Equation (6) are a product of a Gaussian and a Hermite polynomial Hv: where v is the vibrational quantum number [33]. The energy levels are given by where = ( / ) 1/2 . The energy levels of the molecule as a whole (ignoring electronic states) are the sum of the vibrational and rotational terms: If the molecule is not a harmonic oscillator, but still obeys the Born-Oppenheimer approximation, then the potential can be written as a general function ( ). The Schrödinger equation can still be separated into rotational and vibrational components, and the solutions to the rotational component are still the familiar spherical harmonics. The radial equation becomes

Advanced Theory
A diatomic molecule can vibrate as well as rotate. If the oscillation is harmonic, then for small oscillations x about the internuclear distance r e , the Schrödinger equation is where k is the spring constant. If we scale x by the constant α 1/2 to give the dimensionless ξ, where then the solutions to Equation (6) are a product of a Gaussian and a Hermite polynomial H v : where v is the vibrational quantum number [33]. The energy levels are given by where ω = (k/µ) 1/2 . The energy levels of the molecule as a whole (ignoring electronic states) are the sum of the vibrational and rotational terms: If the molecule is not a harmonic oscillator, but still obeys the Born-Oppenheimer approximation, then the potential can be written as a general function U(r). The Schrödinger equation can still be separated into rotational and vibrational components, and the solu-tions to the rotational component are still the familiar spherical harmonics. The radial equation becomes The last term is related to the potential energy associated with a "centrifugal" force [34] due to angular momentum J. With the substitution Equation (11) may be simplified to If we consider a harmonic oscillator then we may transform the independent variable and Equation (13) becomes If x is small and can be ignored in the denominator of the last term, then Equation (16) becomes This is simply the harmonic oscillator equation with the rotational energy separated out, and the energy will simply be the sum of the rotational and vibrational terms: We note that the vibrational frequency ν e and rotational constant B e correspond to the minimum in the potential energy (the subscript e refers to equilibrium). What happens if x becomes too large to ignore? In this case, Equation (16) is equivalent to which can be approximated, as long as |x| < r e , as One way to solve this problem is by using perturbation theory on the zeroth-order harmonic oscillator equation, which we will discuss later. We examine an alternate approach. If the infinite series is truncated after the linear or quadratic terms, we obtain and Both of these equations are exactly soluble, because the effective potential is still quadratic. Equation (21) becomes (after rearranging and completing the square) If we substitute then we obtain This is just the normal harmonic oscillator equation with usual solution, except an extra term is taken out, which is formally identified as the centrifugal distortion term [35]: The harmonic oscillator solutions are no longer centered about x = 0, but increase with rotational constant by δ 1 .
If we now include the quadratic term as well, Equation (22) becomes Expanding and collecting terms, we obtain If we let If we substitute then we obtain Some third-and fourth-order (in J(J + 1)) pure rotational terms have been subtracted out, and we are left with the usual harmonic oscillator problem, but with a different force constant k . The main effect is on the vibrational energy levels: and, therefore, this effect is called the vibration-rotation interaction. It can be approximated as We may let When combined with the significant rotational terms, we obtain If we let the effective rotational constant for vibrational level v be then we can see that each vibrational level will have a different effective rotational constant, and that the effective rotational constants are approximately evenly spaced. [36] Although Equation (35) predicts that α e is positive, in practice it is negative because of anharmonicity (below). The variation in the effective rotational constant, as seen in Figure 2, is thus explained as being due to the vibrational quantum number v. Alternatively, we may apply perturbation theory to the harmonic oscillator, with the following perturbation: First-order perturbation theory will have no contribution from terms of odd order in x. The first-order correction to the energy (from terms of even order in x) is where f is a quadratic function in v + 1 2 . The second-order correction to the energy contains the centrifugal distortion correction (from the linear term in x), a vibrational anharmonicity term involving (U (3) ) 2 (v + 1 2 ) 2 , and additional terms involving [34,37]. The j = 1 term here is usually larger than the corresponding first-order term and is opposite in sign, explaining why α e is usually negative. The first-order correction involving the quadratic function f and the j = 2 term of the second-order correction can be related to γ e below.
The pedagogical goal achieved here is to show what happens to the energy levels when the molecule cannot be treated as a harmonic oscillator and rigid rotor, in order to explain some of the finer structure in the spectra of alkali halides.

Cesium Iodide (CsI, Reprise)
Returning to Figure 2, it is now clear that the four clusters represent different vibrational states, and we assign them to v = 0 − 3, with the highest frequency corresponding to v = 0. We may either model the effective rotational constants via Equation (37), or model the rotation spectra itself using Model 1: These will give slightly different results under equal weighting least squares. We obtain B e = 708.266 and α e = 2.040 MHz, sum of squares error (SSE) = 0.7213 (Model 1, Table 2). In Figure 3, the error in this model is plotted as a function of the quantum numbers v and J. The error is up to five times the estimated error (0.1 MHz) in each individual measurement, which suggests that improvements can be made to the model. In addition, the error appears to have a quadratic dependence on v and a linear dependence on J. Model 1 may be improved by adding a quadratic term in v + 1 2 to give Model 2: or by adding a centrifugal distortion term to give Model 3: or by adding a centrifugal distortion term to give Model 3:   (3) the incorporation of either the D e and γ e terms lead to an approximately equal (but small) reduction in error, whereas incorporation of both leads to a much better fit.

Cesium Bromide (CsBr)
Bromine has two isotopes in approximately a 1:1 ratio, so two isotopologues should be observable in natural samples. The predicted value of 2B from the electron diffraction value is 2100 MHz (see Supplementary Material, Excel file CsBr.xlsx). The spectrum of cesium bromide in the region 21-26 GHz was measured [31] at 690 • C, and the frequencies observed (error 0.1-0.2 MHz) are given in Figure 4. Inspection of the 24 frequencies demonstrates that there are three clusters of frequencies centered at 21,382, 23,334, and 25,630 MHz [32]. The average difference of these is 2124 ± 150 MHz. The highest frequency per cluster differences is 2113 ± 45 MHz. These all suggest transitions corresponding to J = 9, 10, and 11.
OR PEER REVIEW 11 When the effective rotational constants (×2) are plotted ( Figure 5), it can be seen that they naturally divide themselves into two series of equally spaced progressions, with the exception of a point at 2137.4 MHz corresponding to the observed frequency at 25,648.95 MHz. Fitting of these to Equation (37) gives Be values of 1081.27 and 1064.51 MHz. This was performed by fitting the data to both isotopologues and choosing the smaller of the two sum-of-squared errors. The deviation in 2B for the "wrong" isotopologue is an approximately constant 33 MHz. This procedure also enables separation by isotopologue. The isotopologue mass ratio is 1.015736, and the Be ratio is 1.015744, an agreement to five significant figures. This is strong evidence that the higher frequency progression (Isotopologue 1) is 133 Cs 79 Br, and the lower frequency progression (Isotopologue 2) is 133 Cs 81 Br. We can analyze these progressions separately. Using Model 1, the residual error for 133 Cs 79 Br is plotted in Figure 6. The "bad" point corresponds to v = 3, J = 11. While a Q-test  When the effective rotational constants (×2) are plotted ( Figure 5), it can be seen that they naturally divide themselves into two series of equally spaced progressions, with the exception of a point at 2137.4 MHz corresponding to the observed frequency at 25,648.95 MHz. Fitting of these to Equation (37) gives B e values of 1081.27 and 1064.51 MHz. This was performed by fitting the data to both isotopologues and choosing the smaller of the two sum-of-squared errors. The deviation in 2B for the "wrong" isotopologue is an approximately constant 33 MHz. This procedure also enables separation by isotopologue. The isotopologue mass ratio is 1.015736, and the B e ratio is 1.015744, an agreement to five significant figures. This is strong evidence that the higher frequency progression (Isotopologue 1) is 133 Cs 79 Br, and the lower frequency progression (Isotopologue 2) is 133 Cs 81 Br. We can analyze these progressions separately. Using Model 1, the residual error for 133 Cs 79 Br is plotted in Figure 6. The "bad" point corresponds to v = 3, J = 11. While a Q-test on the residuals can be used to exclude this point, it is important to note that this point appears to be 10 MHz too high. We suspect that this is a printing error, and that this value should actually be 25,638.95 MHz instead of 25,648.95 MHz. We will use this value henceforth as it reduces the SSE 20-fold. The results of fitting to all four models are given in Table 3. In all cases, the B e ratio matches the expected isotopic mass ratio by five significant figures. For Model 4, the r e values for the two isotopologues, 3.072274 and 3.072287 Å, are in excellent agreement.     The pedagogical goals achieved here show that for CsBr, calculating effective rotational constants using Model 1 can (1) help assign points to different isotopologues, (2) identify a "bad" point, and (3) confirm isotopologue assignments by a calculation of Be ratios.

Cesium Chloride (CsCl)
Chlorine has two isotopes in approximately a 3:1 ratio, so two isotopologues should be observable in natural samples. The predicted value of 2B from the electron diffraction value is 4000 MHz (See Supplementary Materials, Excel file CsCl.xlsx). The spectrum of The pedagogical goals achieved here show that for CsBr, calculating effective rotational constants using Model 1 can (1) help assign points to different isotopologues, (2) identify a "bad" point, and (3) confirm isotopologue assignments by a calculation of B e ratios.

Cesium Chloride (CsCl)
Chlorine has two isotopes in approximately a 3:1 ratio, so two isotopologues should be observable in natural samples. The predicted value of 2B from the electron diffraction value is 4000 MHz (See Supplementary Materials, Excel file CsCl.xlsx). The spectrum of cesium chloride in the region 24.3-25.4 GHz was measured [38] at 715 • C, and the frequencies observed (error 1.5 MHz) are given in Figure 7. Inspection of the eight frequencies demonstrates that there appears to be just one major grouping of frequencies. The estimated value of 2B suggests transitions corresponding to J = 5. There are two apparent gaps at about 24,450 and 24,880 MHz, which appear to be due to a missing transition and a break between two vibrational progressions, respectively. We note that because we do not have multiple values of J, there is no way to estimate the centrifugal distortion constant. We also note that the error in each measurement is fairly large compared to the previous two cesium salts analyzed.
R PEER REVIEW 14 not have multiple values of J, there is no way to estimate the centrifugal distortion constant. We also note that the error in each measurement is fairly large compared to the previous two cesium salts analyzed. Under the assumption that the head of each progression corresponds to v = 0, we find that the higher frequency progression is well described by Model 1, with Be = 2113.31 and αe = 9.96, whereas the lower frequency progression gives Be = 2071.45 and αe = 9.60 (all MHz). The expected mass ratio is 1.04468, but the Be ratio is 1.02020. Therefore, the vibrational assignments are incorrect, and we must revise our assumption. It is unlikely that we would miss the head of the progression of the heavier isotopologue, as it falls within the range of other observed bands, so we assume that we have assigned this correctly and that we have misassigned the lighter isotopologue because it falls out of the observed range. With this model, any misassignment of v results in the same αe, but different Be. The following Be ratios are calculated for head assignments v = 1-5: 1.025, 1.029, 1.034, 1.039, and 1.044. This suggests that the assignments of the lighter isotopologue correspond to v = 5-8 instead of v = 0-3, and that the isotopologues are 133 Cs 35 Cl and 133 Cs 37 Cl. This was confirmed by a later measurement at 720 °C [31], where the missing transitions (v = 0-4) were seen. The newer results were all systematically lower in frequency by 30 MHz, but the precision was improved to 0.1-0.5 MHz (Figure 8). The model parameters are given in Table 4. For Model 2, the re values for the two isotopologues, 2.906337 and 2.906362 Å , Under the assumption that the head of each progression corresponds to v = 0, we find that the higher frequency progression is well described by Model 1, with B e = 2113.31 and α e = 9.96, whereas the lower frequency progression gives B e = 2071.45 and α e = 9.60 (all MHz). The expected mass ratio is 1.04468, but the B e ratio is 1.02020. Therefore, the vibrational assignments are incorrect, and we must revise our assumption. It is unlikely that we would miss the head of the progression of the heavier isotopologue, as it falls within the range of other observed bands, so we assume that we have assigned this correctly and that we have misassigned the lighter isotopologue because it falls out of the observed range. With this model, any misassignment of v results in the same α e , but different B e . The following B e ratios are calculated for head assignments v = 1-5: 1.025, 1.029, 1.034, 1.039, and 1.044. This suggests that the assignments of the lighter isotopologue correspond to v = 5-8 instead of v = 0-3, and that the isotopologues are 133 Cs 35 Cl and 133 Cs 37 Cl. This was confirmed by a later measurement at 720 • C [31], where the missing transitions (v = 0-4) were seen. The newer results were all systematically lower in frequency by 30 MHz, but the precision was improved to 0.1-0.5 MHz (Figure 8). The model parameters are given in The pedagogical goal achieved here is showing that the vibrational assignments should always be checked by comparison of the predicted Be ratios. Figure 8. Comparison of microwave spectra of cesium chloride from two sources (Grouping 0 (orange) [38], Grouping 1 (blue) [31]).

Cesium Fluoride (CsF)
Fluorine is monoisotopic. The predicted value of 2B from the electron diffraction value is 10,000 MHz (See Supplementary Materials, Excel file CsF.xlsx). The spectra of cesium fluoride in the region 21.4-22.1 GHz was measured [31] at 700 °C, and the frequencies observed (error 0.2-1.0 MHz) are given in Figure 9. Inspection of the five frequencies demonstrates that there appears to be just one major grouping of frequencies. The observed spectrum corresponds to J = 1, v = 0-4. Fitting gives Be = 5527.123, αe = 35.072 (Model 1), and Be = 5527.259, αe = 35.218, γe = 0.0279 (Model 2), all MHz. The bond distance re is 2.3454 Å . This problem makes an excellent student exercise (Appendix A) without the complicating effects of multiple J and/or isotopes.  The pedagogical goal achieved here is showing that the vibrational assignments should always be checked by comparison of the predicted B e ratios.

Cesium Fluoride (CsF)
Fluorine is monoisotopic. The predicted value of 2B from the electron diffraction value is 10,000 MHz (See Supplementary Materials, Excel file CsF.xlsx). The spectra of cesium fluoride in the region 21.4-22.1 GHz was measured [31] at 700 • C, and the frequencies observed (error 0.2-1.0 MHz) are given in Figure 9. Inspection of the five frequencies demonstrates that there appears to be just one major grouping of frequencies. cies observed (error 0.2-1.0 MHz) are given in Figure 9. Inspection of the five frequencies demonstrates that there appears to be just one major grouping of frequencies. The observed spectrum corresponds to J = 1, v = 0-4. Fitting gives Be = 5527.123, αe = 35.072 (Model 1), and Be = 5527.259, αe = 35.218, γe = 0.0279 (Model 2), all MHz. The bond distance re is 2.3454 Å . This problem makes an excellent student exercise (Appendix A) without the complicating effects of multiple J and/or isotopes. Microwave Spectra of CsF Figure 9. The microwave spectrum of cesium fluoride.

Rubidium Iodide (RbI)
Rubidium has two isotopes, 85 Rb and 87 Rb, present in approximately a 3:1 ratio. The predicted value of 2B from electron diffraction is 1880 MHz (see Supplementary Materials, Excel file RbI.xlsx). The spectra of rubidium iodide in the region 21.5-25.6 GHz was measured [31] at 660 • C, and the frequencies observed (error 0.1-0.2 MHz) are given in Figure 10. Inspection of the 13 frequencies demonstrates that there appears to be three major groupings (clusters) of frequencies. The centers of the clusters are at 21,617.58, 23,464.75, and 25,238.83 MHz, with an average difference of 1810(37) MHz. The average difference between the highest frequency components is 1965(80) MHz. Using the estimated 2B or cluster averages would suggest either J = 10, 11, 12 (Assignment # 2) or J = 11, 12, 13 (Assignment #1), whereas using the highest frequency differences (assuming v = 0) would suggest J = 10, 11, 12, with an anomaly for J = 11. While the preference is for J = 10, 11, 12, the assignment is somewhat less definitive, so we have plotted the effective rotational constants for both scenarios (Figure 11). For assignment #1, all 13 points are still visible in nine subclusters, apparently in two vibrational progressions. This would imply that the centrifugal distortion constants are quite large, and that there are at least four points with the same v, but with multiple values of J. In addition, the B e ratio is 1.009306. For assignment #2, only 10 clusters are visible, and these can (barely) be separated into two vibrational progressions with a B e ratio of 1.013951. The expected mass ratio is 1.013961, which agrees with the second assignment to five significant figures. We therefore assume that the transitions are as assignment #2. The effective rotational constants for J = 11 correspond to v = 1-2, not to v = 0-1, which explains why the highest frequency method for J determination did not work as well.
What would happen if we assumed that the spectrum was due to a single isotopologue, but that some lines were doubled due to some unknown effect? In that case, we may analyze the effective rotational constants as usual using Model 1. The residual error of this model is shown in Figure 12. There is a clear separation of the residual into two groups with this double assignment, and the least-squares process tries to fit both, resulting in a fairly large error. If we eliminate all points currently assigned to v = 4-6 (Model 1 Elim 1), the error drops significantly, and one set of the excluded points fits the expected residual error pattern. We can then re-include those points in the fit (Model 1 Elim 2). The points excluded belong to a second species. We then re-assign the vibrational quantum numbers of the second species to obtain a reasonable B e ratio, from v = 4-6 (1.000736) to v = 0-2 (1.013950). We then proceed as usual to obtain r e of 3.176993 and 3.176997 Å for the two isotopologues ( Table 5).
The pedagogical goals achieved here show that that, for RbI, (1) plotting effective rotational constants for different J assignments can assist in determining J, and (2) plotting residuals can help identify incorrect assumptions. Rubidium has two isotopes, 85 Rb and 87 Rb, present in approximately a 3:1 ratio. The predicted value of 2B from electron diffraction is 1880 MHz (see Supplementary Materials, Excel file RbI.xlsx). The spectra of rubidium iodide in the region 21.5-25.6 GHz was measured [31] at 660 °C, and the frequencies observed (error 0.1-0.2 MHz) are given in Figure  10. Inspection of the 13 frequencies demonstrates that there appears to be three major groupings (clusters) of frequencies. The centers of the clusters are at 21,617.58, 23,464.75, and 25,238.83 MHz, with an average difference of 1810(37) MHz. The average difference between the highest frequency components is 1965(80) MHz. Using the estimated 2B or cluster averages would suggest either J = 10, 11, 12 (Assignment # 2) or J = 11, 12, 13 (Assignment #1), whereas using the highest frequency differences (assuming v = 0) would suggest J = 10, 11, 12, with an anomaly for J = 11. While the preference is for J = 10, 11, 12, the assignment is somewhat less definitive, so we have plotted the effective rotational constants for both scenarios (Figure 11). For assignment #1, all 13 points are still visible in nine subclusters, apparently in two vibrational progressions. This would imply that the centrifugal distortion constants are quite large, and that there are at least four points with the same v, but with multiple values of J. In addition, the Be ratio is 1.009306. For assignment #2, only 10 clusters are visible, and these can (barely) be separated into two vibrational progressions with a Be ratio of 1.013951. The expected mass ratio is 1.013961, which agrees with the second assignment to five significant figures. We therefore assume that the transitions are as assignment #2. The effective rotational constants for J = 11 correspond to v = 1-2, not to v = 0-1, which explains why the highest frequency method for J determination did not work as well.   What would happen if we assumed that the spectrum was due to a single isotopologue, but that some lines were doubled due to some unknown effect? In that case, we may analyze the effective rotational constants as usual using Model 1. The residual error of this model is shown in Figure 12. There is a clear separation of the residual into two Figure 11. The rotational constants of rubidium iodide for two different J assignments. The orange points are the original spectrum, the blue points are separation into isotopologues. = 0-2 (1.013950). We then proceed as usual to obtain re of 3.176993 and 3.176997 Å for the two isotopologues ( Table 5).
The pedagogical goals achieved here show that that, for RbI, (1) plotting effective rotational constants for different J assignments can assist in determining J, and (2) plotting residuals can help identify incorrect assumptions.

Rubidium Bromide (RbBr)
Because both rubidium and bromine are diisotopic, there may be a total of four isotopologues that could be observed. The predicted value of 2B from electron diffraction is 2650 MHz (see Supplementary Materials, Excel file RbBr.xlsx). The spectrum of rubidium bromide in the region 22.5-26.0 GHz was measured [31] at 730 • C, and the frequencies observed (error 0.1 MHz) are given in Figure 13. Inspection of the 12 frequencies demonstrates that there appear to be two major groupings (clusters) of frequencies, a single frequency at 22,752.29 MHz and a cluster centered at 25,285.29 MHz. The cluster difference is 2533 MHz, but the highest frequency components differ by 2843.70 MHz. Using the estimated 2B suggests either J = 7,8 or J = 8,9. Using the cluster average would suggest J = 8,9, whereas using the highest frequency differences (assuming v = 0) would suggest J = 7,8. With the highest frequency differences usually being more accurate, and along with the 2B clustering favoring this as well, we assign these to J = 7,8, with three different vibrational progressions ( Figure 14).
The The pedagogical goal achieved here shows that, for RbBr, more than two isotopologues can be successfully analyzed.

Model
Be αe γe

Rubidium Bromide (RbBr)
Because both rubidium and bromine are diisotopic, there may be a total of four isotopologues that could be observed. The predicted value of 2B from electron diffraction is 2650 MHz (see Supplementary Materials, Excel file RbBr.xlsx). The spectrum of rubidium bromide in the region 22.5-26.0 GHz was measured [31] at 730 °C, and the frequencies observed (error 0.1 MHz) are given in Figure 13. Inspection of the 12 frequencies demonstrates that there appear to be two major groupings (clusters) of frequencies, a single frequency at 22,752.29 MHz and a cluster centered at 25,285.29 MHz. The cluster difference is 2533 MHz, but the highest frequency components differ by 2843.70 MHz. Using the estimated 2B suggests either J = 7,8 or J = 8,9. Using the cluster average would suggest J = 8,9, whereas using the highest frequency differences (assuming v = 0) would suggest J = 7,8. With the highest frequency differences usually being more accurate, and along with the 2B clustering favoring this as well, we assign these to J = 7,8, with three different vibrational progressions ( Figure 14).  The lightest isotopologue ( Rb Br) is also the most abundant. The mass ratios for the heavier isotopologues are 1.011195 ( 87 Rb 79 Br), 1.012963 ( 85 Rb 81 Br), and 1.024452 ( 87 Rb 81 Br). The Be ratios for Model 1 are calculated as 1.011188 and 1.012955 and for Model 2 as 1.011202 and 1.012965, which demonstrate that the other two isotopologues observed are 87 Rb 79 Br and 85 Rb 81 Br ( Table 6). The values of re are calculated as 2.944776, 2.944820, and 2.944823 Å . Figure 14. The rotational constants of rubidium bromide. The orange points are the original spectrum, the blue points are separation into isotopologues. The pedagogical goal achieved here shows that, for RbBr, more than two isotopologues can be successfully analyzed.

Potassium Iodide (KI)
Potassium has two isotopes, 39 K and 41 K, present in approximately a 14:1 ratio. The predicted value of 2B from electron diffraction is 3300 MHz (see Supplementary Materials, Excel file KI.xlsx). The spectrum of potassium iodide in the region 18-26 GHz was measured [31] at 690 °C, and the frequencies observed (error 0.1-0.3 MHz) are given in Figure  15. Inspection of the 14 frequencies demonstrates that there appears to be three major groupings (clusters) of frequencies. Analysis of this spectrum is left for the reader as a more challenging exercise (Appendix B). The results for 39

Potassium Iodide (KI)
Potassium has two isotopes, 39 K and 41 K, present in approximately a 14:1 ratio. The predicted value of 2B from electron diffraction is 3300 MHz (see Supplementary Materials, Excel file KI.xlsx). The spectrum of potassium iodide in the region 18-26 GHz was measured [31] at 690 • C, and the frequencies observed (error 0.1-0.3 MHz) are given in Figure 15. Inspection of the 14 frequencies demonstrates that there appears to be three major groupings (clusters) of frequencies. Analysis of this spectrum is left for the reader as a more challenging exercise (Appendix B). The results for 39  The value of re is calculated as 3.047776 Å . The value of r0 for the heavier isotopologue 41 K 127 I can be calculated as 3.051148 Å . Figure 15. The microwave spectrum of potassium iodide, and the 3-cluster averages.

Potassium Chloride (KCl)
Potassium chloride has four naturally occurring isotopologues: 39 [39] at 715 °C, and the four frequencies observed (error 10 MHz) could either correspond to a vibrational progression, with a gap, or possibly to a mixture of up to four isotopologues. This determination is left to the reader (see Appendix C). A more precise spectrum was measured [40] at 700 °C and is also left as an exercise (see Appendix D).

Sodium Chloride (NaCl)
Sodium is monoisotopic. The predicted value of 2B from electron diffraction is 11,600 MHz (see Supplementary Materials, Excel file NaCl.xlsx). The spectrum of sodium chloride in the region 25.0-26.1 GHz was measured [38] at 775 °C, and the seven frequencies observed (error 0.75 MHz) correspond to two vibrational progressions. The assignment is left to the reader (Appendix E).

Conclusions
The microwave spectra of CO, CsI, CsBr, CsCl, CsF, RbI, RbBr, KI, KCl, and NaCl are presented and procedures for their analysis are discussed. By the use of either additional information (electron or X-ray diffraction) or by cluster analysis of the spectra, the approximate value of 2B can be determined, followed by the assignment of the frequencies to rotational quantum number J (Model 0). Once assigned, effective rotational constants for each transition can be assigned to specific vibrational quantum numbers v and/or isotopologues, to obtain the rotational constant Be, bond length re, and vibration-rotation inter- Microwave Spectra of KI Figure 15. The microwave spectrum of potassium iodide, and the 3-cluster averages.

Potassium Chloride (KCl)
Potassium chloride has four naturally occurring isotopologues: 39 [39] at 715 • C, and the four frequencies observed (error 10 MHz) could either correspond to a vibrational progression, with a gap, or possibly to a mixture of up to four isotopologues. This determination is left to the reader (see Appendix C). A more precise spectrum was measured [40] at 700 • C and is also left as an exercise (see Appendix D).

Sodium Chloride (NaCl)
Sodium is monoisotopic. The predicted value of 2B from electron diffraction is 11,600 MHz (see Supplementary Materials, Excel file NaCl.xlsx). The spectrum of sodium chloride in the region 25.0-26.1 GHz was measured [38] at 775 • C, and the seven frequencies observed (error 0.75 MHz) correspond to two vibrational progressions. The assignment is left to the reader (Appendix E).

Conclusions
The microwave spectra of CO, CsI, CsBr, CsCl, CsF, RbI, RbBr, KI, KCl, and NaCl are presented and procedures for their analysis are discussed. By the use of either additional information (electron or X-ray diffraction) or by cluster analysis of the spectra, the approximate value of 2B can be determined, followed by the assignment of the frequencies to rotational quantum number J (Model 0). Once assigned, effective rotational constants for each transition can be assigned to specific vibrational quantum numbers v and/or isotopologues, to obtain the rotational constant B e , bond length r e , and vibration-rotation interaction constant α e (Model 1). In some cases, other spectroscopic constants (D e , γ e ) can be determined. Data Availability Statement: All data used in this article can be found in either the references cited or in the Supplementary Material.