LABORATORY OPTICAL SPECTROSCOPY OF THE THIOPHENOXY RADICAL AND ITS PROFILE SIMULATION AS A DIFFUSE INTERSTELLAR BAND BASED ON ROTATIONAL DISTRIBUTION BY RADIATION AND COLLISIONS

Shibuya et al. (1988) studied the thiophenoxy radical in the gas phase by laser-induced fluorescence and reported bands between 5200 and 4932 Å. The observed bands were presumed to be the ²A₂ ← ²B₂ electronic transition based on the theoretical calculation of the phenoxy radical (C₆H₅O) reported by Chang et al. (1975). However, Lim et al. (2006) reported the ground state of the thiophenoxy radical as ²B₁. We calculated the energy levels of the first (²B₂), second (²A₂), third (²B₁), fourth (²A₁), and fifth (²B₁) excited states of the thiophenoxy radical to be 0.43, 2.46, 3.01, 3.95, and 4.22 eV, respectively, using the theoretical calculation of TD-B3LYP/cc-pVTZ by Gaussian 03 (Frisch et al. 2003). The first excited state is optically forbidden from the ground state, and the transitions to the second and third excited states are b-type and a-type, respectively. Shibuya et al. (1988) reported origin and vibronic bands at 5172 and 5046 Å in air, respectively. Although a stellar line at the wavelength of the vibronic band is an interference factor, the origin band is suitable for comparison with a DIB. We observed the absorption spectrum of the origin band using the CRD spectrometer, as shown in Figure 2. The rotational profile of the band was of b-type; thus, the present band can be assigned to the ²A₂ – X²B₁ electronic transition.

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To analyze the rotational profile, we employed the ground-state rotational constants obtained by B3LYP/cc-pVTZ and assumed a rotational temperature of 300 K in the discharge because the rotational temperature of C₂ was measured to be 300 K in the same hollow cathode glow discharge cell. When these constants and temperature are fixed, the observed rotational profile gives the constants of ΔA and Δ$\overline B$ ($\overline B$ = (B + C)/2), which correspond to the excited-state rotational constants as differences from the ground-state constants. However, simultaneous determination of ΔA and Δ$\overline B$ could not be converged in contour fitting with PGOPHER (version 8.0, Western). To solve this difficulty in fitting we calculated the excited-state rotational constants by B3LYP/cc-pVTZ and obtained the calculated values of ΔA and Δ$\overline B$. Using the fixed ratio of ΔA to Δ$\overline B$, i.e., ${{\Delta A} /{\Delta \overline B }}$, we determined ΔA, Δ$\overline B$, and the transition frequency T₀₀ by manual manipulation of the constants by comparing the simulated and observed spectra. The obtained constants are listed in Table 1, and the profile comparison is shown in Figure 2.

Table 1. Obtained Molecular Constants of C₆H₅S in cm⁻¹

State	Ground a	Excited b
	X2B1	2A2
T00	0	19327.7(3)c
A	0.1893
ΔAd		0.0073(5)c
$\bar{B}$ e	0.04848
Δ$\bar{B}$ e,d		−0.0017(1)c
B – C	0.01222

Notes. ^aThe ground-state rotational constants were obtained by theoretical calculation of B3LYP/cc-pVTZ. ^bThe excited-state constants were determined by rotational profile simulation as shown in Figure 2. ^cValues in parentheses denote the uncertainties and apply to the last digits of the constants. ^dΔA = A' − A, and $\Delta\bar{B}$ is also similar. ^e$\bar{B}$ = (B + C)/2.

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A simulation of a rotational profile, which depends on the rotational distribution of molecules, is essential to compare a laboratory band with DIBs. In a diffuse cloud, where the number density is low, spontaneous emission between J and J − 1 is not negligible. The rotational distribution is determined based on their interaction with the environment through radiation and collisions. They are influenced by the radiative temperature T_r and the kinetic temperature T_k. When radiation is included along with collisions, the detailed balancing between J and J − 1 levels takes the form

$$\begin{equation} n(J)\left({A^J + B_{J - 1}^ \downarrow \rho + C_{J - 1}^ \downarrow } \right) = n({J - 1})\left({B_{J - 1}^ \uparrow \rho + C_{J - 1}^ \uparrow } \right), \end{equation} \tag{ 1 }$$

where n(J) is the number density of molecules with the rotational quantum number J, and Einstein A and B coefficients express spontaneous emission and induced radiative effect, respectively (Oka et al. 2013). The coefficients are expressed by

$$\begin{equation} A^J = \frac{{8\pi h\nu ^3 }}{{c^3 }}B_{J - 1}^ \downarrow \end{equation} \tag{ 2 }$$

and

$$\begin{equation} B_{J - 1}^ \downarrow = \frac{{2\pi ^2 }}{{3\varepsilon _0 h^2 }}\mu _{J \to J - 1} ^2 \end{equation} \tag{ 3 }$$

in the MKSA system, where ν is the transition frequency. The $C_{J - 1}^ \downarrow$ and $C_{J - 1}^ \uparrow$ constants in Equation (1) are the rates for collision-induced rotational transitions (Oka 1974) and can be approximated using Equation (7) in the report by Oka et al. (2013). Using Planck's formula $\rho = ({{{8\pi h\nu ^3 } /{c^3 }}})({e^{{{h\nu } /{kT_r }}} - 1})^{ - 1}$ and ${{B_{J - 1}^ \uparrow } / {B_{J - 1}^ \downarrow }} = {{({2J + 1})} /{({2J - 1})}}$, the relation between n(J) and n(J − 1) is derived as follows:

$$\begin{eqnarray} &&n(J)\left[ {A^J \left({1 + \frac{1}{{\exp ({{{h\nu } /{kT_r }}}) - 1}}} \right) + C_{J - 1}^ \downarrow } \right]\nonumber\\ &&\quad= n({J - 1})\left[ {A^J \frac{{2J + 1}}{{2J - 1}}\frac{1}{{\exp ({{{h\nu } /{kT_r }}}) - 1}} + C_{J - 1}^ \uparrow } \right] \end{eqnarray} \tag{ 4 }$$

(Oka et al. 2013).

Consider the case of a near prolate C_2v molecule. As the permanent dipole moment is along the a-axis, emission and absorption for rotational transitions are limited to the a-type transition, i.e., ΔK_a is even and ΔK_c is odd. As an assumption, we limited the transition to ΔK_a = 0 and ΔK_c = ±1 in the present analysis because the transitions of |ΔK_a| ⩾ 2 are weak. Thus, transitions in each K_a level are similar to the case of a linear molecule, although J < K_a levels are missing, as shown in Figure 3. In this assumption, rotation along the a-axis cannot be cooled by radiation, and the a-axis can be regarded as a "hot axis." Rotation along the hot axis can result in a wide electronic transition profile due to the population of high K_a levels.

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In the case of the J → J − 1 transition in each K_a series, the transition dipole is given as

$$\begin{equation} \mu _{J \to J - 1} ^2 = \mu ^2 \frac{S}{{2J + 1}}, \end{equation} \tag{ 5 }$$

where μ is the permanent dipole moment and S is the transition strength. Using Equation (2) together with Equations (3), (5), and $\nu = 2\overline B J$ in the ΔK_a = 0 transition except for the Q-branch, we have

$$\begin{eqnarray} A^J &=& \frac{{8\pi h\nu ^3 }}{{c^3 }}\frac{{2\pi ^2 }}{{3\varepsilon _0 h^2 }}\mu ^2 \frac{S}{{2J + 1}} = \frac{{2^7 \pi ^3 \overline B ^3 S\mu ^2 }}{{3\varepsilon _0 hc^3 }}\frac{{J^3 }}{{2J + 1}}\nonumber\\ &=& \alpha \overline B ^3 \mu ^2 \frac{{J^3 S}}{{2J + 1}}, \end{eqnarray} \tag{ 6 }$$

where α = 2⁷π³/3ε₀hc³. Thus, from Equations (4) and (6) we obtain

$$\begin{eqnarray} &&n(J) = n(0)\prod\limits_{m = K_a + 1}^J\nonumber\\ &&\times\left[ \frac{{\alpha \overline B ^3 \mu ^2 \frac{{m^3 S}}{{2m - 1}}\frac{1}{{\exp ({{{2h\overline B m} /{kT_r }}}) - 1}} + C\sqrt {\frac{{2m + 1}}{{2m - 1}}} \exp ({{{ - h\overline B m} /{kT_k }}})}}{{\alpha \overline B ^3 \mu ^2 \frac{{m^3 S}}{{2m + 1}}\left({1 + \frac{1}{{\exp ({{{2h\overline B m} /{kT_r }}}) - 1}}} \right) + C\sqrt {\frac{{2m - 1}}{{2m + 1}}} \exp ({{{h\overline B m} /{kT_k }}})}} \right].\nonumber\\ && \end{eqnarray} \tag{ 7 }$$

Since the number densities n(J) in our Equation (7) cannot be expressed analytically, we calculate them numerically. They are calculated as a function of the radiative temperature T_r (= 2.73 K) in a diffuse cloud due to the cosmic blackbody radiation, the kinetic temperature T_k, the rotational constant $\overline B ({ {=} {{({B + C})} /2}})$, the permanent dipole moment μ, and the collision rate C.

At first, the population is distributed to each K_a series based on a simple Boltzmann population at the kinetic temperature T_k. Then, in each K_a series, relative populations for each J level are redistributed using our Equation (7). The calculated new distribution is input into PGOPHER, which can individually import the numerical population for each rotational level. The rotational profile is simulated using PGOPHER.

For simplicity, we employed a singlet approximation for the rotational structure, i.e., no spin splitting, to calculate the number densities n(J) for the thiophenoxy radical. The rotational constants and the permanent dipole moment obtained were $\overline B$ = 1453.5 MHz and μ = 3.1 D by the theoretical calculation of B3LYP/cc-pVTZ. The S values were derived from the rotational constant A, B, and C obtained from the calculation. In the diffuse clouds toward HD 204827, the excitation temperature of ∼40 K is presumed from the rotational structure of the C₂ spectrum (Oka et al. 2003). As the excitation temperature of a molecule with no permanent dipole moment can correspond to the kinetic temperature, we used T_k = 40 K toward HD 204827. The collision rate was assumed to be C = 10⁻⁷ s⁻¹ as a typical value of diffuse clouds (Oka & Epp 2004; Oka et al. 2013). Toward HD 204827, the K I line is resolved into at least four components (Pan et al. 2004), and the two strong components are separated by approximately 5 km s⁻¹, i.e., 0.09 Å at the 7172 Å region. The reported DIB spectrum (Hobbs et al. 2008) was obtained by the resolving power R = 38,000, which gave a resolution of 0.14 Å at 7172 Å. To estimate the rotational profile as a DIB toward HD 204827, we used a resolution of 0.22 Å as a summation of the two components' separation and the resolution. The thiophenoxy radical was simulated for T_k = 40 K and T_r = 2.73 K, hereafter case (b), as shown in Figure 4(b). For comparison, the profiles with the extreme cases, T_k = T_r = 2.73 K (case (a)) and the simple Boltzmann population at 40 K (case (c)), were also simulated, as shown in Figures 4(a) and (c), respectively. Not only case (c) but also case (b) shows a wide width for the profile, although the structures are not similar.

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Next, we considered the contribution of the ΔK_a = ±2 (ΔK_c = ±1) transitions, which can transfer population from high K_a to low K_a by radiative cooling in T_r < T_k. The ν³S values of the ΔK_a = ±2 transitions in this molecule are smaller by one order than those of the ΔK_a = 0 transitions; thus, the values of A^J(ΔK_a = ±2) are not so far from that of the collision rate C. This radiative cooling by the ΔK_a = ±2 transitions may compete with collisional heating. However, a simulation including the cooling and the heating for ΔK_a = ±2 is difficult because a proper value of C is difficult to estimate. The case (b) simulation can be suitable if the relation C ≫ A^J(ΔK_a = ±2) holds. The actual rotational profile of the DIB may be between case (a) and case (b) in Figure 4.

The observed electronic transition of the thiophenoxy radical (5173 Å in air) lies near two reported DIBs at 5170 and 5176 Å (Hobbs et al. 2008). To check whether the small wavelength differences are an effect of rotational structure, a comparison was made between a DIB spectrum in the 5150–5192 Å range and the simulated spectrum of the thiophenoxy radical in case (b). The results are shown in Figure 5, and it is clear that the simulated spectrum does not fit the DIB absorptions for both width and wavelength.

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An upper limit of the column density for the thiophenoxy radical toward HD 204827 can be estimated using the procedure used by Motylewski et al. (2000). It was assumed that a signal-to-noise ratio of 5 is required to detect DIB absorption. The FWHM of the band is approximately 2 Å, as shown in Figure 4(b), and the detection limit of the equivalent width in the data presented by Hobbs et al. (2008) is 0.015 Å. The oscillator strength of the ²A₂ – X²B₁ transition for the thiophenoxy radical that is theoretically obtained by TD-B3LYP/cc-pVTZ is 0.003. These values then lead to 2 × 10¹³ cm⁻² as the upper limit of the column density for the thiophenoxy radical toward HD 204827. This upper limit is relatively larger than that of C₆H ((1.09–1.43) × 10¹² cm⁻²) reported by Motylewski et al. (2000) because of the small oscillator strength of the thiophenoxy radical. The two transitions from the ground state to the third (²B₁) and fifth (²B₁) excited states have strong oscillator strengths of 0.057 and 0.054, respectively, obtained theoretically. The two transitions may correspond to the broad emission band at the 4000–5000 Å region in the solution reported by Russell (1975) and the absorption band at ∼3100 Å in the vapor phase reported by Porter & Wright (1955). However, these reported bands are not high resolution. Since the line width of a typical DIB is ∼1 Å, the transition to the long-lifetime excited state can be a candidate for an origin of the typical DIB. To have a lifetime broadening of <1 Å at 4000 Å, the excited-state lifetime of >1 ps is necessary. The lifetime broadening of the observed band in this work as shown in Figure 2 is less than ∼0.1 cm⁻¹, which corresponds to the lifetime of >50 ps. To compare the two transitions with DIBs and to determine the upper limits of the column densities of the two transitions, high-resolution observations of laboratory spectra that can show magnitudes of lifetime broadening are necessary.

The rotational profile of a linear molecule is correlated with the radiative temperature T_r in a diffuse cloud as reported by Oka et al. (2013). In this study, it is suggested that the rotational profile of the near prolate C_2v molecule can depend not only on the radiative temperature but also on the collision rate due to the non-negligible ΔK_a = ±2 transitions. Band widths of some DIBs depend on the line of sight (e.g., Oka et al. 2013). It is possible that a near prolate C_2v molecule with a hot axis is a potential molecular origin of a DIB with such line-of-sight dependence, because the collision rate varies depending on a cloud.

The thiophenoxy radical was produced in the hollow cathode glow discharge of thiophenol. The ²A₂ – X²B₁ electronic transition of the thiophenoxy radical was observed using a newly constructed cavity ringdown spectrometer. To compare the observed laboratory data with the reported DIB toward HD 204827 (Hobbs et al. 2008), we derived the model of rotational distribution by radiation and collisions for the near prolate C_2v molecule in a diffuse cloud. The rotational profile of the thiophenoxy radical as a DIB was simulated by the model. Although the comparison with the reported DIB suggests no agreement, the upper limit of the column density was estimated to be 2 × 10¹³ cm⁻². The reported profile simulation raises the possibility that DIB with a variable line width dependent on the line of sight originates from a near prolate C_2v molecule.