Analysis of Two Infrared Bands of CH₂D₂

Abstract

Two infrared absorption bands of CH₂D₂ have been analyzed in the semirigid rotor approximation. These are the A-type band at 2671.67 cm⁻¹ and the C-type band at 4425.61 cm⁻¹. The A-type band has previously been assigned as v₃+v₉, and the C-type band is tentatively assigned as v₃+v₆ The upper state of the A-type band is perturbed presumably by the close lying level 2v₅. This interaction has not been investigated. The following values were found for the rotational constants of the ground vibrational state: A₀=4.303 cm⁻¹, B₀= 3.504 cm⁻¹, C₀= 3.049 cm⁻¹.

1. Introduction

The rotational-vibrational spectra of all the deuterated species of methane except CH₂D₂ have been well investigated [1].1

Methane and CD₄ are spherical tops while CH₃D and CD₃H are symmetric rotors; on the other hand CH₂D₂ is an asymmetric rotor. It seemed of interest to determine the rotational constants of CH₂D₂ in order to have a set of constants for each of the species. Fortunately, high resolution spectra could be obtained for both an A-type and a C-type band for this molecule. As recently discussed [2], this is sufficient data to enable a good determination of the ground state constants using the complementary $\mathrm{\Delta }{F}_2^{″}$ values obtained from the two bands. These two bands have been analyzed in the semirigid approximation to yield good values of the ground state constants. Unfortunately a perturbation in the excited state of the A-type band introduces an uncertainty in the effective constants for this band that is larger than can be justified for the precision of the data. Only transitions involving levels with low values of the rotational quantum numbers have been used in the analysis in order to minimize the effect of centrifugal distortion.

2. Experimental Procedure

The spectra were recorded with the grating instrument of the Infrared Spectroscopy Section [3] using a 10,000 lines/in. grating with a ruled area of about 6×8 in. A cooled PbS photoconductive cell was used as the detector.

Both the A- and C-type bands were recorded using a multiple reflection cell with a total path length of 6 m, filled with CH₂D₂ to a pressure of 2 cm of Hg. The C-type band was observed in the second order of the grating and the A-type band in the first. The CH₂D₂ obtained from Merck & Co., Ltd., had a stated minimum purity of 98%.

The wavelengths of the lines were measured using higher order infrared emission lines of the rare gases as standards, and interpolating between them through the use of the fringes of a Fabry-Perot interferometer as previously described [4].

3. Theory

Preliminary to the actual analysis of the spectra the mean square values of the angular momenta about the three principal axes of inertia and the intensities were calculated.

Assuming tetrahedral geometry and identical bond lengths of 1.094 A for CH₂D₂ the moments of inertia, reciprocal moments in cm⁻¹ units, and the asymmetry parameter κ were obtained.

For the calculated value of κ=−0.27, α, β, and γ, as defined by Allen [2], were obtained by linear interpolation in published tables [5] of E(κ) for each energy level.

α, β, and γ may be shown to be identical with $\langle P_z^2\rangle$, $\langle P_x^2\rangle$, and $\langle P_y^2\rangle$, respectively, in an I^r representation [6]. The energy of a rotational level in a given vibrational state, neglecting centrifugal distortion, may be written as [2]:

$$E(J_{K_{-1,}{K}_1})=\alpha A_v+\beta B_v+\gamma C_v,$$

and the difference between two rotational levels in the same vibrational state as:

$$\mathrm{\Delta }F=\mathrm{\Delta }\alpha A_v+\mathrm{\Delta }\beta B_v+\mathrm{\Delta }\gamma C_v.$$

The Δα’s, Δβ’s and Δγ’s for each of the ΔF₂’s observable from strong transitions in the A- and C-type band were calculated.

The strong transitions for the A-type band are those which satisfy the selection rules:

$$\mathrm{\Delta }J=0,\pm 1\mathrm{\Delta }{K}_{-1}=0\mathrm{\Delta }{K}_1=\pm 1,$$

and for the C-type band the strong transitions are those for which

$$\mathrm{\Delta }J=0,\pm 1\mathrm{\Delta }{K}_{-1}=\pm 1\mathrm{\Delta }{K}_1=0.$$

The relative intensities of the transitions were calculated by combining the Boltzmann factor, calculated using the estimated moments of inertia, and nuclear spin statistics with the appropriate line strengths from published tables [7].

The nuclear spins of the equivalent pairs of hydrogen and deuterium atoms in CH₂D₂ give rise to degeneracies of the rotational levels. For the ground vibrational state the statistical weight factors are 15 for the symmetric (A) rotational levels and 21 for the antisymmetric (B) levels.

The A-type band at 2672 cm⁻¹ has been previously observed by Wilmshurst and Bernstein [8] who assigned it to the combination v₃+v₉. As the observed band type is consistent with this assignment and there should be no other bands of this type near 2670 cm⁻¹, there appears to be no reason to doubt the assignment.

The C-type band at 4425 cm⁻¹ can best be assigned at v₃+v₆ as this seems to be the only combination which would give a C-type band in the region 4400–4500 cm⁻¹.

4. Analysis

As each of the bands was isolated from others of comparable intensity, initial assignments, with the aid of the calculated intensities, could be made by inspection. From the initial assignments ΔF₂ values were obtained, and with the calculated Δα’s, Δβ’s and Δγ’s were used to refine the values of A, B, and C in the ground and upper vibrational states. From these values of A, B, and C, α’s, β’s, and γ’s pertinent to the value of κ in each vibrational state were calculated as before.

A trial spectrum was then calculated from the expression:

$$v=v_0+{\alpha }^{\prime }{A}_v+{\beta }^{\prime }{B}_v+{\gamma }^{\prime }{C}_v-\alpha A_0-\beta B_0-\gamma C_0.$$

From the trial spectrum and intensities more transitions could be assigned, enabling further refinement of the reciprocal moments of inertia.

While this iterative procedure worked well with the C-type band, Coriolis perturbations in the A-type band caused some difficulty in definitely locating some of the transitions.

The lowest observably perturbed level in the v₃+v₉ vibrational state is the 4₁₃ level which is pushed down by 0.24 cm⁻¹. For J=5, the levels 5₀₅, 5₁₄, and 5₂₃ are all perturbed, and for J= 6 over half of the levels are perturbed.

This perturbation has not been investigated in detail. It probably arises through interaction with the vibrational state 2v₅, the fundamental of which is theoretically inactive in the infrared, but appears to have been observed at 1329 cm⁻¹ [8], the transitions becoming allowed through Coriolis perturbation. v₅ has apparently been observed in the Raman also at 1333 cm⁻¹ [9].

No account was taken of the effect of centrifugal distortion in this analysis. Since only levels with low J values were used in the analysis, the effect of this correction on the rotational constants was minimized. No systematic differences between the observed and calculated spectra were noticed until rather high J values were reached. In these regions of the absorption serious overlapping of transitions make the unique assignment of transitions to observed absorption peaks doubtful.

Although no statistical analysis of the data was made, the excellent agreement between the observed and calculated ΔF₂ values for the ground state, and the sensitivity of the calculated ΔF₂ values to values of the rotational constants seem to indicate a probable error of the order of ±0.002 cm⁻¹ for each of the ground state constants. The agreement between the calculated and observed values of ΔF₂ for the ground state may be seen in table 1.

Table 1.

Ground state $\mathrm{\Delta }{F}_2^{″}$

A-type band			C-type band
$\mathrm{\Delta }{F}_2^{″}$	Calculated	Observed	$\mathrm{\Delta }{F}_2^{″}$	Calculated	Observed
	cm−1	cm−1		cm−1	cm−1
000–202	19.513	19.55	000–220	23.911	23.92
101–303	32.103	32.10	101–321	37.524	37.53
111–313	31.547	31.55	111–331	41.285	41.29
110–312	33.791	33.79	110–330	40.862	40.88
202–404	44.373	44.38	202–422	51.644	51.64
212–414	43.976	43.98	212–432	55.068	55.06
211–413	46.958	46.96	211–431	53.908	53.91
221–423	45.754	45.76	221–441	58.305	58.27
220–422	47.277	47.25	220–440	58.165	58.16
303–505	56.501	56.50	303–523	66.378	66.37
313–515	56.300	56.28	313–533	69.201	69.20
312–514	59.715	59.69	312–532	67.212	69.23
322–524	58.539	58.54	322–542	71.733	71.73
321–523	60.967	60.96	321–541	71.127	71.12
331–533	59.463	59.47	331–551	75.463	75.43
330–532	60.141	60.13	330–550	75.431	75.43
404–606	68.637	68.61	404–624	81.651	81.64
414–616	68.559	68.53	414–634	83.676	83.63
413–615	72.042	71.98	413–633	81.042	81.03
423–625	71.147	71.11	423–643	85.399	85.38
422–624	74.350	74.33	422–642	83.950	83.93
432–634	72.585	72.55	432–652	88.728	88.65
431–633	74.092	74.07	431–651	88.532	88.52
441–643	72.848	72.81	441–661	92.665	92.56
440–642	73.062	73.02	440–660	92.660	92.56

The constants for the excited states of these bands cannot be determined as precisely as those for the ground state with the available ΔF₂ values, but ±0.005 cm⁻¹ would seem to be a generous estimate of the probable error in the constants for these states. The calculated and observed ΔF₂ values for the excited states are compared in tables 2 and 3.

Table 2. C-type band $\mathrm{\Delta }{F}_2^{\prime }$.

A=4.255 cm⁻¹ B=3.590 cm⁻¹ C=3.151 cm⁻¹

ΔF′2	Calc	Obs
	cm−1	cm−1
000–220	23.918	23.96
101–321	37.927	37.94
111–331	41.086	41.12
110–330	40.685	40.70
202–422	52.436	52.44
212–432	55.238	55.25
211–431	54.163	54.18
221–441	57.925	57.91
220–440	57.778	57.80
303–523	67.521	67.51
313–533	69.726	69.74
312–532	67.944	67.93
322–542	71.753	71.74
321–541	71.131	71.15
331–551	74.891	74.82
330–550	74.854	74.82
404–624	83.064	82.95
414–634	84.534	84.48
413–633	82.268	82.15
423–643	85.784	85.82
422–642	84.431	84.42
432–652	88.552	88.51
431–651	88.324	88.32
441–061	91.899	91.83
440–660	91.890	91.83

Table 3. A-type band $\mathrm{\Delta }{F}_2^{\prime }$.

A=4.254 cm⁻¹ B=3.654 cm⁻¹ C=3.019 cm⁻¹

$\mathrm{\Delta }{F}_2^{\prime }$	Calc	Obs
	cm−1	cm−1
000–202	19.715	19.70
101–303	33.132	32.11
111–313	31.617	31.62
110–312	34.688	34.68
202–404	44.163	44.17
212–414	43.916	43.91
211–413a	47.748	47.51
221–423	46.516	46.54
220–422	48.979	48.96
303–505a	56.159	56.51
313–515	56.080	56.07
312–514a	60.078	59.35
322–524	59.144	59.11
321–523a	62.809	62.70
331–533	60.862	60.86
330–532	62.271	62.59

^aPerturbed levels.

The constants determined for the three vibrational levels are given in table 4. The band origins were determined from the best fit between the observed and calculated spectra for low J values.

Table 4.

Rotational and vibrational constants

	Ground state	v3+v9	v3+v6
	cm−1	cm−1	cm−1
A	4.303	4.254	4.255
B	3.504	3.654	3.590
C	3.049	3.019	3.151
v0		2671.67	4425.61

Figure 1. The A-type hand of CH₂D₂ at 2672 cm⁻¹.

The identification given is the ground state designation of $J_{K_{-1},K_1}$ for R_0,1 transitions to the high wave number side of the band origin, and for P₀, $\overline{1}$ transitions to the low wave number side of the band origin.

Figure 2. The C-type band of CH₂D₂ at 4426 cm⁻¹.

The identification given is the ground state designation of $J_{K_{-1},K_1}$ for R_1,0 transitions to the high wave number side of the band origin, and for $P_{\overline{1}}$, 0 transitions to the low wave number side of the band origin.