# Interpreting the Microwave Spectra of Diatomic Molecules

*Spectroscopy Journal*)

## Abstract

**:**

## 1. Introduction

## 2. Basic Theory

_{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:

_{l}discussed in the solution to the hydrogen atom, since the solution is the same. Because the selection rule for rotational transitions [12,13,14] is $\Delta J=\pm 1$, and because we normally deal with absorption (J corresponds to the lower state), the transitions are observed at

^{−1}). We shall refer to this simplest possible model (Equation (3)) as Model 0. Because B depends on the moment of inertia, and thus the mass of the atoms, different isotopologues will have different values of the rotational constant.

## 3. Materials and Methods

## 4. Results

#### 4.1. Carbon Monoxide (CO)

^{12}C

^{16}O and at 110,201.1 ± 0.4 MHz for

^{13}C

^{16}O (isotopically enriched to 14%

^{13}C). The X-ray crystal structure of carbon monoxide gives a distance of 1.0629 Å [27,28]. Using this distance, the predicted value of 2B of the lighter isotopologue is 130,500 MHz (see Supplementary Material, Excel file CO.xlsx). The agreement between the predicted value of 2B and the observed value (to within 15%) suggests that the observed transition corresponds to J = 0→J = 1. This illustrates one method for assigning J, which is normally the first step. If we have two or more isotopologues, then we can check the isotopic assignments by comparing the ratio of the rotational constants to the inverse of the ratio of the reduced masses, assuming the bond length of the isotopologues is the same. In this case, B

_{12}/B

_{13}= 1.04600, whereas μ

_{13}/μ

_{12}= 1.04612. The agreement is reasonable. The bond distances are calculated to be 1.130895 Å (

^{12}C

^{16}O) and 1.130832 Å (

^{13}C

^{16}O). These differ from each other slightly because they correspond to the first vibrational state r

_{0}, and they differ from those reported in ref. [26] because of the slightly smaller value of Planck’s constant used therein. Carbon monoxide has a longer bond length in the gas phase than in the solid.

#### 4.2. Alkali Halides

#### 4.3. Cesium Iodide (CsI)

## 5. Advanced Theory

_{e}, the Schrödinger equation is

^{1/2}to give the dimensionless ξ, where

_{v}:

_{e}and rotational constant B

_{e}correspond to the minimum in the potential energy (the subscript e refers to equilibrium).

_{1}.

_{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.

_{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.

## 6. More Results

#### 6.1. Cesium Iodide (CsI, Reprise)

_{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 + ½ to give Model 2:

_{e}= 708.269, α

_{e}= 2.045, γ

_{e}= 0.0015 MHz, SSE = 0.6840), whereas Model 3 keeps it (B

_{e}= 708.280, α

_{e}= 2.040, D

_{e}= 2.57 × 10

^{−5}MHz, SSE = 0.5194). The error is reduced to an acceptable level only by including both terms (Model 4):

_{e}= 708.362, α

_{e}= 2.043, γ

_{e}= 0.0011, D

_{e}= 1.62 × 10

^{−4}MHz, SSE = 0.0102). The estimated Cs-I bond distance is 3.3151 Å, which is about 0.1 Å shorter than the electron diffraction result. This microwave result corresponds to r

_{e}, whereas the electron diffraction result would be a Boltzmann average over many vibrational and rotational states. Even so, each unit increase in vibrational quantum number adds only about 0.005 Å to r

_{v}for CsI.

_{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.

#### 6.2. Cesium Bromide (CsBr)

_{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.

_{e}ratios.

#### 6.3. Cesium Chloride (CsCl)

_{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 Table 4. For Model 2, the r

_{e}values for the two isotopologues, 2.906337 and 2.906362 Å, are in excellent agreement.

_{e}ratios.

#### 6.4. Cesium Fluoride (CsF)

_{e}= 5527.123, α

_{e}= 35.072 (Model 1), and B

_{e}= 5527.259, α

_{e}= 35.218, γ

_{e}= 0.0279 (Model 2), all MHz. The bond distance r

_{e}is 2.3454 Å. This problem makes an excellent student exercise (Appendix A) without the complicating effects of multiple J and/or isotopes.

#### 6.5. Rubidium Iodide (RbI)

^{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.

_{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).

#### 6.6. Rubidium Bromide (RbBr)

^{85}Rb

^{79}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 B

_{e}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 r

_{e}are calculated as 2.944776, 2.944820, and 2.944823 Å.

#### 6.7. Potassium Iodide (KI)

^{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}K

^{127}I are Model 1, B

_{e}= 1824.840, α

_{e}= 7.941 MHz; Model 2, B

_{e}= 1824.962, α

_{e}= 8.047, γ

_{e}= −0.0141 MHz; Model 3, B

_{e}= 1825.023, α

_{e}= 7.944, D

_{e}= 0.0022 MHz; Model 4, B

_{e}= 1825.059, α

_{e}= 8.040, γ

_{e}= −0.0130, D

_{e}= 0.0013 MHz. The value of r

_{e}is calculated as 3.047776 Å. The value of r

_{0}for the heavier isotopologue

^{41}K

^{127}I can be calculated as 3.051148 Å.

#### 6.8. Potassium Chloride (KCl)

^{39}K

^{35}Cl (70.65%),

^{39}K

^{37}Cl (22.61%),

^{41}K

^{35}Cl (5.10%),

^{41}K

^{37}Cl (1.63%). The predicted value of 2B from electron diffraction is 7100 MHz (see Supplementary Materials, Excel file KCl.xlsx). The spectra of potassium chloride in the region 22.4–23.2 GHz was measured [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).

#### 6.9. Sodium Chloride (NaCl)

## 7. Conclusions

_{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.

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Analysis of the Microwave Spectrum of Cesium Fluoride (CsF)

_{e}and α

_{e}. (d) Give a precise value of r

_{e}.

_{e}= 5527.259, α

_{e}= 35.218 (d) r

_{e}= 2.3454 Å]

## Appendix B. Analysis of the Microwave Spectrum of Potassium Iodide (KI)

_{e}, α

_{e}, γ

_{e}, D

_{e}where possible. (d) Give a precise value of r

_{e}.

^{39}K

^{127}I (J = 4, v = 0,1; J = 5, v = 0–3, 5–7; J = 6, v = 0–3) except 21,036.78 MHz (

^{41}K

^{127}I, J = 5, v = 0) (c)

^{39}K

^{127}I Model 4 (MHz): B

_{e}= 1825.059, α

_{e}= 8.040, γ

_{e}= −0.013, D

_{e}= 0.00129 (d) r

_{e}= 3.047776 Å]

## Appendix C. Analysis of the Microwave Spectrum of Potassium Chloride (KCl)

_{e}= 3858.1, α

_{e}= 27.2, γ

_{e}= 0.7]

## Appendix D. Analysis of the Microwave Spectrum of Potassium Chloride (KCl)

^{39}K

^{35}Cl (head 23,067.5 MHz, v = 0–3) and

^{39}K

^{37}Cl (head 22,410.3, v = 0,1), Model 1 (MHz):

^{39}K

^{35}Cl, B

_{e}= 3856.278, α

_{e}= 23.512;

^{39}K

^{37}Cl, B

_{e}= 3746.075, α

_{e}= 22.050 (d) r

_{e}= 2.6667Å]

## Appendix E. Analysis of the Microwave Spectrum of Sodium Chloride (NaCl)

^{23}Na

^{35}Cl, head 26,051.1 MHz, v = 0–4 and

^{23}Na

^{37}Cl, head 25,493.9 MHz, v = 0–2. Model 1 (MHz):

^{23}Na

^{35}Cl, B

_{e}= 6536.70, α

_{e}= 48.07;

^{23}Na

^{37}Cl, B

_{e}= 6396.86, α

_{e}= 46.7 (d) r

_{e}= 2.3609 Å]

## References

- Ewing, G.W. Microwave Absorption Spectroscopy. J. Chem. Educ.
**1966**, 43, A683–A722. [Google Scholar] - Townes, C.H.; Schawlow, A.L. Microwave Spectroscopy; Dover: New York, NY, USA, 1975. [Google Scholar]
- Schwendeman, R.H.; Volltrauer, H.N.; Laurie, V.W.; Thomas, E.C. Microwave Spectroscopy in the Undergraduate Laboratory. J. Chem. Educ.
**1970**, 47, 526–532. [Google Scholar] - Dyke, T.R.; Muenter, J.S. A Stark Modulation Absorption Cell for Student Microwave Spectrometers. J. Chem. Educ.
**1974**, 51, 33. [Google Scholar] [CrossRef] - Pollnow, G.F.; Hopfinger, A.J. A Computer Experiment in Microwave Spectroscopy. J. Chem. Educ.
**1968**, 45, 528–531. [Google Scholar] [CrossRef] - Pollnow, G.F.; Chung, C.S.C. An Interactive Computer Program for Microwave Spectroscopy. J. Chem. Educ.
**1973**, 50, 794. [Google Scholar] - McNaught, I.J.; Moore, R. Microwave Spectroscopy Tutor. J. Chem. Educ.
**1995**, 72, 993–994. [Google Scholar] [CrossRef] - McNaught, I.J.; Moore, R. Winspec: A Microwave Spectroscopy Tutor. J. Chem. Educ.
**1996**, 73, 523.az. [Google Scholar] - Woods, R.; Henderson, G. FTIR Rotational Spectroscopy. J. Chem. Educ.
**1987**, 64, 921–924. [Google Scholar] [CrossRef] - McQuarrie, D.A. Quantum Chemistry, 2nd ed.; University Science Books: Herndon, VA, USA, 2008; Chapter 6. [Google Scholar]
- Pye, C.C. On the Solution of the Quantum Rigid Rotor. J. Chem. Educ.
**2006**, 83, 460–463. [Google Scholar] [CrossRef] - Moynihan, C.T. Rationalization of the ΔJ=±1 Selection Rule for Rotational Transitions. J. Chem. Educ.
**1969**, 46, 431. [Google Scholar] - Foss, J.G. Photonic Angular Momentum and Selection Rules for Rotational Transitions. J. Chem. Educ.
**1970**, 47, 778–779. [Google Scholar] [CrossRef] - Chattaraj, P.K.; Sannigrahi, A.B. A Simple Group-Theoretical Derivation of the Selection Rules for Rotational Transitions. J. Chem. Educ.
**1990**, 67, 653–655. [Google Scholar] - Laidler, K.J.; Meiser, J.H. Physical Chemistry, 3rd ed.; Houghton Mifflin: Boston, MA, USA, 1999; Chapter 13. [Google Scholar]
- Engel, T.; Reid, P. Physical Chemistry; Pearson: San Francisco, CA, USA, 2006; Chapter 19. [Google Scholar]
- House, J.E. Fundamentals of Quantum Mechanics; Academic Press: San Diego, CA, USA, 1998; Chapter 7. [Google Scholar]
- Ratner, M.A.; Schatz, G.C. Introduction to Quantum Mechanics in Chemistry; Prentice-Hall: Upper Saddle River, NJ, USA, 2001; Chapter 4. [Google Scholar]
- Atkins, P.; de Paula, J. Physical Chemistry, 10th ed.; Freeman: New York, NY, USA, 2014; Chapter 12. [Google Scholar]
- Berry, R.S.; Rice, S.A.; Ross, J. Physical Chemistry, 2nd ed.; Oxford University Press: New York, NY, USA, 2000; Chapter 7. [Google Scholar]
- Alberty, R.A.; Silbey, R.J. Physical Chemistry, 2nd ed.; Wiley: New York, NY, USA, 1997; Chapter 13. [Google Scholar]
- Mortimer, R.G. Physical Chemistry, 2nd ed.; Harcourt: San Diego, CA, USA, 2000; Chapter 19. [Google Scholar]
- Zielinski, T.J. Fostering Creativity and Learning Using Instructional Symbolic Mathematics Documents. J. Chem. Educ.
**2009**, 86, 1466–1467. [Google Scholar] - Microsoft Office Professional Plus (Excel) 2016, Microsoft Corporation: Redmond, WA, USA, 2016.
- Solver, Microsoft Office Add-In, Frontline Systems: Incline Village, NV, USA, 2016.
- Gilliam, O.R.; Johnson, C.M.; Gordy, W. Microwave Spectroscopy in the Region from Two to Three Millimeters. Phys. Rev.
**1950**, 78, 140–144. [Google Scholar] - Vegard, L. Struktur und Leuchtfahigkeit von festem Kohlenoxyd. Z. Phys.
**1930**, 61, 185–190. [Google Scholar] [CrossRef] - Crystallography Open Database. Available online: http://www.crystallography.net (accessed on 31 March 2018).
- Wyckoff, R.W.G. Crystal Structures; Wiley: New York, NY, USA, 1963; Chapter 3; pp. 85–237. [Google Scholar]
- Maxwell, L.R.; Hendricks, S.B.; Mosley, V.M. Interatomic Distances of the Alkali Halide Molecules by Electron Diffraction. Phys. Rev.
**1937**, 52, 968–972. [Google Scholar] - Honig, A.; Mandel, M.; Stitch, M.L.; Townes, C.H. Microwave Spectra of the Alkali Halides. Phys. Rev.
**1954**, 96, 629–642. [Google Scholar] [CrossRef] - Pye, C.C. One-Dimensional Cluster Analysis and its Application to Chemistry. Chem. Educ.
**2020**, 25, 50–57. [Google Scholar] - Pye, C.C. Intuitive Solution to Quantum Harmonic Oscillator at Infinity. J. Chem. Educ.
**2004**, 81, 830–831. [Google Scholar] - McHale, J.L. Molecular Spectroscopy; Prentice-Hall: Upper Saddle River, NJ, USA, 1999; Chapters 8–9. [Google Scholar]
- Dykstra, C.E. Introduction to Quantum Chemistry; Prentice-Hall: Englewood Cliffs, NJ, USA, 1994; Chapter 5. [Google Scholar]
- Hollenberg, J.L. Energy States of Molecules. J. Chem. Educ.
**1970**, 47, 2–14. [Google Scholar] [CrossRef] - Brown, J.M. Molecular Spectroscopy; Oxford Science Publications: Oxford, UK, 1999; Chapter 5. [Google Scholar]
- Stitch, M.L.; Honig, A.; Townes, C.H. Microwave Spectra at High Temperature—Spectra of CsCl and NaCl. Phys. Rev.
**1952**, 86, 813–814. [Google Scholar] [CrossRef] - Stitch, M.L.; Honig, A.; Townes, C.H. High Temperature Microwave Spectroscopy—Spectrum of KCl, TlCl. Phys. Rev.
**1952**, 86, 607. [Google Scholar] - Tate, P.A.; Strandberg, M.W.P. Stark Effect in the Microwave Spectra of KCl and NaCl. J. Chem. Phys.
**1954**, 22, 1380–1383. [Google Scholar] [CrossRef]

**Figure 5.**The effective rotational constants (×2) of cesium bromide. Different J values are offset by 0.2 units. The orange points are the original spectrum, the blue points are separation into isotopologues.

**Figure 7.**The microwave spectrum of cesium chloride, and separation into isotopologues. The orange points are the original spectrum, the blue points are separation into isotopologues.

**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.

**Figure 12.**The Model 1 residuals of the effective rotational constants of rubidium iodide, assuming a single species. The top graph includes all points, the middle graph eliminates points assigned as v = 4–6 from the fit, and the bottom graph only eliminates the leftmost v = 4–6 points.

**Figure 14.**The rotational constants of rubidium bromide. The orange points are the original spectrum, the blue points are separation into isotopologues.

MX | R (Å) ^{1} | R (Å) ^{4} | Ratio |
---|---|---|---|

LiF | 2.0086 | ||

LiCl | 2.56477 | ||

LiBr | 2.7506 | ||

LiI | 3.000 | ||

NaF | 2.310 | ||

NaCl | 2.8203 | 2.51 | 1.124 |

NaBr | 2.98662 | 2.64 | 1.131 |

NaI | 3.2364 | 2.90 | 1.116 |

KF | 2.6735 | ||

KCl | 3.14647 | 2.79 | 1.128 |

KBr | 3.3000 | 2.94 | 1.122 |

KI | 3.53278 | 3.23 | 1.093 |

RbF | 2.82 | ||

RbCl | 3.2905 | 2.89 | 1.139 |

RbBr | 3.427 | 3.06 | 1.120 |

RbI | 3.671 | 3.26 | 1.126 |

CsF | 3.004 | ||

CsCl | 3.51 ^{2} | 3.06 | 1.147 |

3.5706 ^{3} | 1.167 | ||

CsBr | 3.7118 ^{3} | 3.14 | 1.182 |

CsI | 3.9549 ^{3} | 3.41 | 1.159 |

^{1}X-ray diffraction (NaCl-like).

^{2}450 °C.

^{3}X-ray diffraction (CsCl-like).

^{4}Electron diffraction (gas), 1000–1300 °C.

Model | B_{e} | α_{e} | γ_{e} | D_{e}^{1} | SSE |
---|---|---|---|---|---|

1 | 708.266 | 2.040 | n/a | n/a | 0.7213 |

2 | 708.269 | 2.045 | 0.0015 | n/a | 0.6840 |

3 | 708.280 | 2.040 | n/a | 25.7 | 0.5194 |

4 | 708.362 | 2.043 | 0.0011 | 162 | 0.0102 |

^{1}Centrifugal distortion constants have been multiplied by 10

^{6}.

**Table 3.**Model parameters for cesium bromide (MHz) (

^{133}Cs

^{79}Br, n = 17;

^{133}Cs

^{81}Br, n = 7).

Model | B_{e} | α_{e} | γ_{e} | D_{e}^{1} | SSE | |
---|---|---|---|---|---|---|

^{133}Cs^{79}Br | 1 | 1081.242 | 3.692 | n/a | n/a | 3.998 |

2 | 1081.283 | 3.720 | 0.0033 | n/a | 1.426 | |

3 | 1081.281 | 3.692 | n/a | −0.162 | 2.520 | |

4 | 1081.313 | 3.718 | 0.0030 | −0.138 | 0.361 | |

^{133}Cs^{81}Br | 1 | 1064.512 | 3.617 | n/a | n/a | 0.263 |

2 | 1064.521 | 3.632 | 0.0032 | n/a | 0.177 | |

3 | 1064.538 | 3.618 | n/a | −0.097 | 0.130 | |

4 | 1064.552 | 3.635 | 0.0037 | −0.107 | 0.018 |

^{1}Centrifugal distortion constants have been multiplied by 10

^{6}.

Model | B_{e} | α_{e} | γ_{e} | SSE | |
---|---|---|---|---|---|

^{133}Cs^{35}Cl [38], n = 4 | 1 | 2163.119 | 9.963 | n/a | 0.115 |

[31], n = 9 | 1 | 2161.029 | 10.021 | n/a | 4.195 |

2 | 2161.152 | 10.10 | 0.0091 | 0.522 | |

^{133}Cs^{37}Cl [38], n = 4 | 1 | 2071.458 | 9.600 | n/a | 8.940 |

2 | 2071.155 | 9.238 | −0.0703 | 0.002 | |

[31], n = 3 | 1 | 2068.713 | 9.436 | n/a | 0.035 |

2 | 2068.682 | 9.378 | −0.0192 | exact |

**Table 5.**Model parameters for rubidium iodide (MHz) (

^{85}Rb

^{127}I, n = 10;

^{87}Rb

^{127}I, n = 3).

Model | B_{e} | α_{e} | γ_{e} | D_{e}^{1} | SSE | |
---|---|---|---|---|---|---|

^{85}Rb^{127}I | 1 | 984.221 | 3.261 | n/a | n/a | 1.013 |

2 | 984.245 | 3.284 | 0.0033 | n/a | 0.237 | |

3 | 984.307 | 3.259 | n/a | 289 | 0.644 | |

4 | 984.236 | 3.279 | 0.0027 | −81 | 0.113 | |

^{87}Rb^{127}I | 1 | 970.681 | 3.204 | n/a | n/a | 0.014 |

2 | 970.672 | 3.188 | −0.0056 | n/a | exact |

^{1}Centrifugal distortion constants have been multiplied by 10

^{6}.

**Table 6.**Model parameters for rubidium bromide (MHz) (

^{85}Rb

^{79}Br, n = 6;

^{87}Rb

^{79}Br, n = 3;

^{85}Rb

^{81}Br, n = 3).

Model | B_{e} | α_{e} | γ_{e} | D_{e}^{1} | SSE | |
---|---|---|---|---|---|---|

^{85}Rb^{79}Br | 1 | 1424.768 | 5.542 | n/a | n/a | 0.432 |

2 | 1424.798 | 5.583 | 0.0086 | n/a | 0.041 | |

3 | 1424.822 | 5.538 | n/a | −0.79 | 0.268 | |

4 | 1424.822 | 5.576 | 0.0076 | −0.40 | 0.005 | |

^{87}Rb^{79}Br | 1 | 1409.003 | 5.456 | n/a | n/a | 0.011 |

2 | 1409.014 | 5.478 | 0.0072 | n/a | exact | |

^{85}Rb^{81}Br | 1 | 1406.546 | 5.450 | n/a | n/a | 0.020 |

2 | 1406.562 | 5.479 | 0.0097 | n/a | exact |

^{1}Centrifugal distortion constants have been multiplied by 10

^{6}.

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**MDPI and ACS Style**

Pye, C.C.
Interpreting the Microwave Spectra of Diatomic Molecules. *Spectrosc. J.* **2023**, *1*, 3-27.
https://doi.org/10.3390/spectroscj1010002

**AMA Style**

Pye CC.
Interpreting the Microwave Spectra of Diatomic Molecules. *Spectroscopy Journal*. 2023; 1(1):3-27.
https://doi.org/10.3390/spectroscj1010002

**Chicago/Turabian Style**

Pye, Cory C.
2023. "Interpreting the Microwave Spectra of Diatomic Molecules" *Spectroscopy Journal* 1, no. 1: 3-27.
https://doi.org/10.3390/spectroscj1010002