2. Basic Theory
A diatomic molecule can be regarded as two masses,
m1 and
m2, 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
The Schrödinger equation, applied to this system, gives the well-known energy levels [
10]:
where
J is the rotational quantum number, and
B is the rotational constant, given by
where
h is Planck’s constant. The solution of the Schrödinger equation is rather tedious and is skipped in many texts [
11]. The energy levels are 2
J + 1-fold degenerate, corresponding to different values of a second quantum number
m. The quantum numbers
J and
m are analogous to
l and
ml 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
, and because we normally deal with absorption (
J corresponds to the lower state), the transitions are observed at
A series of equally spaced lines, separated by 2B, should be observed. The rotational constant is often reported as a frequency (B/h, Hz) or as a wavenumber (B/hc, cm−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.
The above treatment is the deepest extent to which many texts treat the rotational spectroscopy of a diatomic molecule [
15,
16,
17,
18]. This simple approach does work for some diatomic molecules in which only the lowest vibrational level is significantly populated. We shall see below that the centrifugal distortion term and the vibration–rotation interaction are also needed to understand other spectra. Other texts acknowledge the existence of one [
19,
20] or both [
10,
21,
22] of these terms, but without derivation. We provide these derivations below.
4. Results
4.1. Carbon Monoxide (CO)
The microwave spectrum of carbon monoxide was measured by Gilliam, Johnson, and Gordy [
26]. Transitions were observed at 115,270.56 ± 0.25 MHz for
12C
16O and at 110,201.1 ± 0.4 MHz for
13C
16O (isotopically enriched to 14%
13C). The X-ray crystal structure of carbon monoxide gives a distance of 1.0629 Å [
27,
28]. Using this distance, the predicted value of 2
B of the lighter isotopologue is 130,500 MHz (see
Supplementary Material, Excel file CO.xlsx). The agreement between the predicted value of 2
B 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,
B12/
B13 = 1.04600, whereas
μ13/
μ12 = 1.04612. The agreement is reasonable. The bond distances are calculated to be 1.130895 Å (
12C
16O) and 1.130832 Å (
13C
16O). These differ from each other slightly because they correspond to the first vibrational state
r0, 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.
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.
4.2. 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.
4.3. Cesium Iodide (CsI)
Both cesium and iodine are monoisotopic, so complications from multiple isotopologues are avoided. The predicted value of 2
B 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.
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.
5. 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
α1/2 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
. 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
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
νe and rotational constant
Be 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
, 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
δ1.
If we now include the quadratic term as well, Equation (22) becomes
Expanding and collecting terms, we obtain
If we let
then
If we substitute
then we obtain
Some third- and fourth-order (in
) 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
αe 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
. The second-order correction to the energy contains the centrifugal distortion correction (from the linear term in
x), a vibrational anharmonicity term involving
, 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
αe 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
γe 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.
7. 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 interaction constant αe (Model 1). In some cases, other spectroscopic constants (De, γe) can be determined.