Infrared spectroscopy is a standard technique, which depends upon the interaction between infrared radiation and the vibrating dipole moments of molecules. Except for homonuclear diatomics and noble gases, a characteristic spectrum for each substance is generated in the vapor phase. This individual spectrum is derived from the exclusive physical properties of each molecule that is studied. Infrared spectroscopy serves as a remarkable tool for obtaining the values of these physical properties, especially the molecular structure and force constants of the chemical bonds. An ideal example to demonstrate the principles is the phase spectrum of the heteronuclear diatomic molecule hydrogen chloride (HCl).
When measuring the infrared radiation absorbed by a liquid or solid, welldefined regions of absorption can be attributed to the vibrational transitions. Assuming that the resolution of the instrument is sufficient, the IR spectra of the gas phase molecules do not exhibit such smooth and structureless absorption bands but rather absorption regions of significant complexity. The fine structure of these absorption bands can be correlated to changes in the molecule’s rotational energy accompanied by a vibrational transition.
FTIR spectrum of hydrogen chloride in the gas phase. Insert: The rovibrational lines are split due to the chlorine 35 and chlorine 37 natural isotopic ratio.
Simple Theory
In order to describe molecular motions, one begins with simple physical models in which the atoms are considered to be points of mass held together by weightless elastic springs. The diatomic one, which has two mass points connected by a spring, is the simplest molecule to be built this way. Classical mechanics gives the corresponding vibrational equation of this molecule; the solution is represented by a sinusoidal motion of two masses at a frequency determined by their masses and the restoring force of the spring. Although the potential and the kinetic energy of the system continually interconvert during the vibration, the sum of both energies keeps constant as in a pendulum.
Figure 1. Mechanical model of the vibrating diatomic molecule. k the spring constant corresponds to the molecular binding force, m_{1} and m_{2} are the atomic mass and r_{0} the equilibrium distance.
Figure 2. The potential energy of the twomass model can be assumed to be a parabola, that is harmonic. The further the masses are displaced from their equilibrium position, the higher the potential energy of the system.
The resulting classical vibrational frequency of such an oscillator is given by:
The quantum mechanical description of the harmonic oscillator can be used to describe the vibrational motion of a heteronuclear diatomic molecule. The quantized energy levels are given by:
E_{v} = hν (v+1/2)
Vibrational and rotational energy levels for a diatomic molecule such as HCl. The spacing between V=0 and V=1 is much more than depicted. The R branch rovibrational lines are shown on the left and the P branch rovibrational lines are shown on the right. Only the first few J levels are shown.
Where h is Planck’s constant, ν is the classical vibrational frequency, and E_{v} is the energy of the vibrational state with quantum number v. v can take any integer value starting from 0.
Under regular conditions, the vibrational transitions occur from the ground state (v=0) to the first excited state (v=1). This simple vibrational model predicts just one single vibrational band for a diatomic molecule. However, a series of closely spaced bands as for the spectrum of HCl can be observed in the figure. These lines are because of the rotational substructure, and a further quantum number J for the rotational energy is introduced. J also takes integer values starting from 0. Therefore, every vibrational level has a stack of rotational levels and transitions take place from v=0 to v=1 with J changing by –1 (so called P branch lines) and J changing by +1 (so called R branch lines).
A few examples where lines are assigned and the number in brackets refers to the J number of the v=0 level are shown in the figure for the HCl spectrum. It turns out that the Q branch line with no change in J and expected at the band center is missing because of the operation of selection rules. This leads to a gap between R(0) and P(1), see the figure.
The rotational energies are given by BJ(J+1), where B is called the rotational constant. Knowledge of B leads to the molecular structure determination or, in this simple case, the bond length. To a first approximation, the rotational constants B of the v=1 (upper) and v=0 (lower) vibrational levels are determined as follows. For every J, the differences in cm^{1} positions of the pairs of lines shown on the left side of the two equations are taken and divided by (2J+1).
R(J1) – P(J+1) = 2B_{lower} (2J+1)
R(J) – P(J) = 2B_{upper} (2J+1)
J 
R(J) 
P(J) 
R(J1)P(J+1) 
R(J)P(J) 
2J+1 
2B_{lower} 
2B_{upper} 
(2J+1)**2 
0 
2906.25 







1 
2925.9 
2865.09 
62.63 
60.81 
3 
20.877 
20.27 
9 
2 
2944.92 
2843.62 
104.35 
101.3 
5 
20.87 
20.26 
25 
3 
2963.29 
2821.55 
146 
141.74 
7 
20.857 
20.249 
49 
4 
2981.01 
2798.92 
187.55 
182.09 
9 
20.839 
20.232 
81 
5 
2998.05 
2775.74 
229 
222.31 
11 
20.818 
20.21 
121 
6 
3014.42 
2752.01 
270.29 
262.41 
13 
20.792 
20.185 
169 
7 
3030.1 
2727.76 
311.44 
302.34 
15 
20.763 
20.156 
225 
8 
3045.07 
2702.98 
352.39 
342.09 
17 
20.729 
20.123 
289 
9 
3059.32 
2677.71 
393.13 
381.61 
19 
20.691 
20.085 
361 
10 
3072.85 
2651.94 
433.6 
420.91 
21 
20.648 
20.043 
441 
11 
3085.66 
2625.72 
473.86 
459.94 
23 
20.603 
19.997 
529 
12 

2598.99 






J 
R(J) 
P(J) 
R(J1)P(J+1) 
R(J)P(J) 
2J+1 
2B_{lower} 
2B_{upper} 
(2J+1)**2 
0 
2904.08 
1 
2923.71 
2862.99 
62.52 
60.72 
3 
20.84 
20.24 
9 
2 
2942.7 
2841.56 
104.17 
101.14 
5 
20.834 
20.228 
25 
3 
2961.05 
2819.54 
145.75 
141.51 
7 
20.821 
20.216 
49 
4 
2978.74 
2796.95 
187.24 
181.79 
9 
20.804 
20.199 
81 
5 
2995.76 
2773.81 
228.62 
221.95 
11 
20.784 
20.177 
121 
6 
3012.11 
2750.12 
269.84 
261.99 
13 
20.757 
20.153 
169 
7 
3027.76 
2725.92 
310.9 
301.84 
15 
20.727 
20.123 
225 
8 
3042.72 
2701.21 
351.76 
341.51 
17 
20.692 
20.089 
289 
9 
3056.92 
2676 
392.44 
380.92 
19 
20.655 
20.048 
361 
10 
3070.51 
2650.28 
432.79 
420.23 
21 
20.609 
20.011 
441 
11 

2624.13 


23 


529 
B = Rotational constant; Delta J = +1 is the R branch, Delta J = 1 is the P branch
This can be carried out for both chlorine isotopes (Cl35 and Cl 37), see next two tables.
Extended Theory
Examination of the B constants determined in this simple way displays a distinct curvature with J number. This emerges from the neglect of the centrifugal distortion of the HCl molecule, which increases with J rotational quantum number. As the molecule stretches at higher rotational energy, its moment of inertia I increases. B and the moment of inertia are inversely related to, and therefore B decreases with J. Moreover, B has a slightly lower value in the upper (v=1) vibrational state.
Note: I = μr^{2} where μ = reduced mass and r = internuclear distance
μ = M_{H}M_{Cl}/(M_{H} + M_{CL}) where M = atomic masses
More detailed theory leads to the revised equations:
R(J1) – P(J+1) = (2B_{lower} – 3D_{lower})(2J+1) – D_{lower} (2J+1)^{3}
R(J) – P(J) = (2B_{upper} – 3D_{upper})(2J+1) – D_{upper}(2J+1)^{3}
Where D = centrifugal distortion constant, which has a small value
A plot of the previously determined 2B values versus (2J+1)^{2} will result in a straight line. The intercept will be (2B3D) and the slope – D.
Calculation of the Internuclear Distance
The rotational constants should be extrapolated to the equilibrium state when the molecule would reside hypothetically at the bottom of the potential well.

HCL35 
Literature 
HCL37 
B lower 
10.442 
10.440254 
10.424 
B upper 
10.138 
10.136228 
10.122 
D lower 
0.00053 
0.00052828 
0.000536 
D upper 
0.000524 
0.00052157 
0.00053 
Values in cm^{1 }
Literature values for HCl35 taken from very highresolution spectroscopy – Journal of Molecular Spectroscopy, vol. 17, 1965, p. 122.
B_{lower} = B_{equilibrium} – 0.5 α
B_{upper} = B_{equilibrium} – 1.5 α

HCL35 
HCL37 
B equilibrium 
10.594 
10.576 
alpha 
0.304 
0.302 
Values in cm^{1}
Where α = vibration – rotation interaction constant (α = alpha in the table)
I = 16.8575/B gives the moment of inertia I, where B is in units of cm^{1} and I has units of amu Angstroms^{2} (1 Angstrom = 10^{8} cm). The number in this equation derives from the quantity h/8π^{2}c where c = velocity of light.
The literature value for the HCl equilibrium bond length determined by microwave spectroscopy is 1.27455 Angstroms (Phys. Rev., vol 136, A1229, (1964)). From a theoretical standpoint, these bond length values for HCl35 and HCl37 should be the same.
This experiment with the ALPHA FTIR spectrometer shows excellent agreement with these facts.

HCL35 
HCL37 
B equilibrium 
10.594 
10.576 
I (amu Angstroms**2) 
1.5912 
1.594 
Reduced mass (amu) 
0.979569 
0.981054 
Internuclear distance (Angstroms) 
1.2745 
1.2747 
Derivation of the Vibrational Band Origin V_{01}
V_{01} = R(0) – (2B_{upper} – 4D_{upper}) 
Equation 1 
V_{01} = P(1) + (2B_{lower} – 4D_{lower}) 
Equation 2 
The effect of D can be neglected.
Assuming a harmonic oscillator, the band origins for HCl37 and HCl35 are related by:
V_{01} (HCl37) = V_{01} (HCl35) * √ (reduced mass HCl35/reduced mass HCl37)
Using values of the reduced mass for HCl35 and HCl37 from the table, the prediction for HCl37 is 2883.78 cm^{1} when compared to the experimental value of 2883.84 cm^{1}.

HCL35 
HCL37 
eqn 1 
2885.97 
2883.84 
eqn 2 
2885.97 
2883.84 
Values in cm^{1}, literature value for HCl35 is 2885.9775 cm^{1}, J. Mol. Spectroscopy vol. 17, 1965, p.122
Conclusions
The ALPHA II is an economical and compact FTIR spectrometer that is ideally suited for both teaching applications and QA/QC. With the highresolution option for the ALPHA II FTIR spectrometer, one can obtain excellent HCl vapor spectra measured in a gas cell. Analysis of the HCl spectrum, making full use of the OPUS peak pick software installed on the ALPHA II TouchPC or Windows PC, gives molecular constants that are fully consistent with the values published in the spectroscopic research literature.
These outstanding experimental results are possible, thanks to the excellent signal to noise ratio, good line shape, and precise frequency registration of the ALPHA II FTIR spectrometer.
This information has been sourced, reviewed and adapted from materials provided by Bruker Optics.
For more information on this source, please visit Bruker Optics.