Photodissociation of CS from Excited Rovibrational Levels

In a similar manner to our recent molecular structure work on the SiO molecule (Forrey et al. 2016; Cairnie et al. 2017), which is iso-electronic to CS, the potential energy curves and transition dipole matrix (TDM) elements for several of the low-lying electronic states are calculated. We use a state-averaged-multi-configuration-self-consistent-field (SA-MCSCF) approach, followed by multi-reference configuration interaction (MRCI) calculations together with the Davidson correction (MRCI+Q; Helgaker et al. 2000). The SA-MCSCF method is used as the reference wave function for the MRCI calculations.

Potential energy curves (PECs) and TDMs as a function of internuclear distance R are calculated starting from a bond separation of R = 1.8 Bohr extending out to R = 12 Bohr. At bond distances beyond this value we use a multipole expansion, detailed below, to represent the long-range part of the potentials. The basis sets used in our work are the augmented correlation consistent polarized sextuplet (aug-cc-pV6Z (AV6Z)) Gaussian basis sets. The use of such large basis sets is well known to recover 98% of the electron correlation effects in molecular structure calculations (Helgaker et al. 2000). All the PEC and TDM calculations for the CS molecule were performed with the quantum chemistry program package MOLPRO 2015.1 (Werner et al. 2015), running on parallel architectures.

For molecules with degenerate symmetry, an Abelian subgroup is required to be used in MOLPRO. For a diatomic molecule like CS with ${{\rm{C}}}_{\infty v}$ symmetry, it will be substituted by C_2v symmetry with the order of irreducible representations being (A₁, B₁, B₂, A₂). When symmetry is reduced from ${{\rm{C}}}_{\infty v}$ to C_2v, the correlating relationships are $\sigma \to {a}_{1}$, $\pi \to$ (b₁, b₂), and $\delta \to$ (a₁, a₂). In order to take account of short-range interactions, we employed the non-relativistic state-averaged complete active-space-self-consistent-field (SA-CASSCF)/MRCI method available within the MOLPRO (Werner et al. 2012, 2015) quantum chemistry suite of codes.

For the CS molecule, eight molecular orbitals (MOs) are put into the active space, including four a₁, two b₁, and two b₂ symmetry MOs which correspond to the 3s3p shell of sulfur and 2s2p shell of carbon. The rest of the electrons in the CS molecule are put into closed-shell orbitals, including four a₁, one b₁ and one b₂ symmetry MOs. The molecular orbitals for the MRCI procedure were obtained using the SA-MCSF method, for which we carried out the averaging processes on the lowest three ¹Σ⁺ (¹A₁), three ¹Π (¹B₁), three ³Σ⁺ (³A₁), three ³Π (³B₁), two ¹Δ (¹A₂) and two ³Δ (³A₂) molecular states. The fourteen MOs (8a₁, 3b₁, 3b₂, 0a₂), i.e., (8, 3, 3, 0), were then used to perform all the PEC and TDM calculations for the electronic states of interest in the MRCI+Q approximation. Table 1 compares theoretical results for the permanent dipole moment μ_X of the X ¹Σ⁺ ground state at various levels of approximation with experiment to demonstrate the accuracy of the MRCI+Q approximation applied here. As can be seen from Table 1 our MRCI+Q results for the permanent dipole moment of CS, for the ground state, at the equilibrium geometry, are within 4% of the experimental value (Winnewisser & Cook 1968). Table 2 compares the equilibrium distance R_e (Å) and the dissociation energy D_e (eV) at various level of approximation for the X ¹Σ⁺, A^{' 1}Σ⁺, and A ¹Π states of CS. We note that for the X ¹Σ⁺ ground state, the early experimental work of Crawford & Shurcliff (1934) determined values R_e = 1.2851 Å and D_e (eV) = 7.752 eV, in less favorable agreement with our present ab initio work or that of other high level molecular structure calculations. As shown in Table 2 the use of polarized-core-valence basis sets by Li et al. (2013) provides spectroscopically accurate results for R_e (Å) and D_e (eV), respectively, being within 0.03 pm and 0.032 eV compared with the available experiment. We find that our TDMs differ slightly in magnitude but agree in trend with those presented by Li et al. (2013) on the range they are computed.

Beyond a bond separation of R = 12 Bohr, a multipole expansion is smoothly fitted to the PECs and TDMs up to R = 100 Bohr. For the PECs this has the form

$$\begin{eqnarray}&&V(R)=-\displaystyle \frac{{C}_{5}}{{R}^{5}}-\displaystyle \frac{{C}_{6}}{{R}^{6}},\end{eqnarray} \tag{ 1 }$$

where C₅ and C₆ are coefficients for each electronic state shown in Table 3. For R < R_min, down to a bond length of 1.5 a₀, a short-range interaction potential of the form $V(R)=A\exp (-{BR})+C$ was fitted to the ab initio potential curves.

Table 3.  CS Electronic States

Molecular	Separated Atom					United Atom
State	Atomic State	Energy (eV)a	Energy (eV)b	C5c	C6d	State
X 1Σ+	C(2s22p2 3P) + S(3s23p4 3P)	0.0	0.0	27.34	58.02	3d24s2a1D
A 1Π	${\rm{C}}(2{s}^{2}2{p}^{2}{}^{3}P)+{\rm{S}}(3{s}^{2}3{p}^{4}{}^{3}P)$	0.0	7.95(−4)	0.0	58.02	3d24s2a1D
A' 1Σ+	C(2s22p2 3P) + S(3s23p4 3P)	0.0	1.14(−3)	0.0	58.02	3d24s2a1G
2 1Π	C(2s22p2 3P) + S(3s23p4 3P)	0.0	6.52(−4)	−18.23	58.02	3d24s2a1G
B 1Σ+	C(2s22p2 1D) + S(3s23p4 1D)	2.3812287	2.38172	27.26	55.56	3d34s b1G
3 1Π	C(2s22p2 1D) + S(3s23p4 1D)	2.3812287	2.38652	10.06	55.56	3d34s b1G
4 1Π	C(2s22p2 1D) + S(3s23p4 1D)	2.3812287	2.39969	−11.81	55.56	3d34s a1H

Notes.

^aExperimental data from NIST Atomic Spectra Database (Kramida et al. 2016). ^bCurrent theory extrapolated to the asymptotic limit with Equation (1). ^cEstimated following Chang (1967). See the text for details. ^dEstimated from the London formula. See the text for details.

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A method to estimate the value of the quadrupole–quadrupole coefficient C₅ for an electronic state of a diatomic molecule like CS is given by Chang (1967). In order to compute the long-range dispersion coefficient C₆, the London formula

$$\begin{eqnarray}&&{C}_{6}=\displaystyle \frac{3}{2}\displaystyle \frac{{{ \mathcal I }}_{{\rm{C}}}{{ \mathcal I }}_{{\rm{S}}}}{[{{ \mathcal I }}_{{\rm{C}}}+{{ \mathcal I }}_{{\rm{S}}}]}{\alpha }_{{\rm{C}}}{\alpha }_{{\rm{S}}}\end{eqnarray} \tag{ 2 }$$

is applied, where α is the dipole polarizability and ${ \mathcal I }$ is the ionization energy of each of the atoms in a given atomic state. The ionization energies are taken from the NIST Atomic Spectra Database (Kramida et al. 2016). For the sulfur atom, the dipole polarizabilities of α_S = 18.8 and α_S = 19.5, respectively, are used for the ground state 3s²3p^{4 3}P and the excited state 3s²3p^{4 1}D (Mukherjee & Ohno 1989). For the ground state 2s²2p^{2 3}P of atomic carbon, a dipole polarizability of α_C = 10.39 is used (Miller & Kelly 1972). An estimated value of α_C = 10.78 is used for the excited state 2s²2p^{2 1}D of carbon: this value was obtained by scaling the ground state polarizability to match the ratio of the ³P and ¹D polarizabilities of sulfur.

The potentials for the excited states of the CS molecule were shifted so that the asymptotic energies as $R\to \infty$ agree with the separated atom energy differences found in the NIST Atomic Spectra Database (Kramida et al. 2016) shown in Table 3. Except for the 4 ¹Π state, shifts are less than ∼5 meV indicating the reliability of the MRCI+Q calculations within the uncertainty of the estimated dispersion coefficients. The potential curves for CS are shown in Figure 1.

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The TDMs for the CS molecule are similarly extended to long and short-range internuclear bond distances. For R > R_max a functional fit of the form $D(R)=a\exp (-{bR})+c$ is applied, while in the short-range R < R_min a quadratic fit of the form D(R) = a' R² + b' R + c' is adopted. We deduce from the atomic states of C and S that the long-range $R\to \infty$ limit of each TDM is zero. Similarly, the united-atom limit (which is the Ti atom) as R → 0 of each TDM is zero as well (see Table 3). The TDMs are shown in Figure 2.

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The wave functions of the bound rovibrational levels are computed by solving the radial Schrödinger equation for nuclear motion on the X ¹Σ⁺ potential curve. The wave functions are obtained numerically using the standard Numerov method (Cooley 1961; Johnson 1977) with a step size of 0.001 Bohr. We find 85 vibrational levels with a total of 14,908 rovibrational levels. This covers nearly the full range of rovibrational levels in the X ¹Σ⁺ state.

Here, we present a brief overview of the state-resolved photodissociation cross section calculation; further details are given in previous work (Miyake et al. 2011). In units of cm², the state-resolved cross section for a bound–free transition from initial rovibrational level v'' J'' is

$$\begin{eqnarray}{\sigma }_{v^{\prime\prime} J^{\prime\prime} }({E}_{\mathrm{ph}}) & = & 2.689\times {10}^{-18}\\ & & \times {E}_{\mathrm{ph}}g\displaystyle \sum _{J^{\prime} }\left(\displaystyle \frac{1}{2J^{\prime\prime} +1}{S}_{J^{\prime} }| {D}_{k^{\prime} J^{\prime} ,v^{\prime\prime} J^{\prime\prime} }{| }^{2}\right)\end{eqnarray} \tag{ 3 }$$

(Kirby & van Dishoeck 1988), where k'J' are the continuum states of the final electronic state. The Hönl–London factors, S_J'(J'') (Watson 2008), are expressed for a ${\rm{\Sigma }}\leftarrow {\rm{\Sigma }}$ electronic transition as

$$\begin{eqnarray}{S}_{J^{\prime} }(J^{\prime\prime} )=\left\{\begin{array}{ll}J^{\prime\prime} , & J^{\prime} =J^{\prime\prime} -1\,({\rm{P \mbox{-} branch}})\\ J^{\prime\prime} +1, & J^{\prime} =J^{\prime\prime} +1\,({\rm{R \mbox{-} branch}}),\end{array}\right.\end{eqnarray} \tag{ 4 }$$

and for a ${\rm{\Pi }}\leftarrow {\rm{\Sigma }}$ transition as

$$\begin{eqnarray}{S}_{J^{\prime} }(J^{\prime\prime} )=\left\{\begin{array}{ll}(J^{\prime\prime} -1)/2, & J^{\prime} =J^{\prime\prime} -1\,({\rm{P \mbox{-} branch}})\\ (2J^{\prime\prime} +1)/2, & J^{\prime} =J^{\prime\prime} \,({\rm{Q \mbox{-} branch}})\\ (J^{\prime\prime} +2)/2, & J^{\prime} =J^{\prime\prime} +1\,({\rm{R \mbox{-} branch}}).\end{array}\right.\end{eqnarray} \tag{ 5 }$$

The matrix element of the electric TDM for absorption from v'' J'' to the continuum k'J' is

$$\begin{eqnarray}&&{D}_{k^{\prime} J^{\prime} ,v^{\prime\prime} J^{\prime\prime} }=\langle {\chi }_{k^{\prime} J^{\prime} }(R)| D(R)| {\chi }_{v^{\prime\prime} J^{\prime\prime} }(R)\rangle ,\end{eqnarray} \tag{ 6 }$$

with the integration taken over R where D(R) is the appropriate TDM function. The bound rovibrational wave functions χ_v''J'' and continuum wave functions χ_k'J'(R) are computed using the standard Numerov method with a step size of 0.001 Bohr. They are normalized such that they behave asymptotically as

$$\begin{eqnarray}&&{\chi }_{k^{\prime} J^{\prime} }(R)\sim \sin \left(k^{\prime} R-\displaystyle \frac{\pi }{2}J^{\prime} +{\eta }_{J^{\prime} }\right),\end{eqnarray} \tag{ 7 }$$

where η_J' is the single-channel phase shift of the upper electronic state. Finally, the degeneracy factor g is given by

$$\begin{eqnarray}&&g=\displaystyle \frac{2-{\delta }_{0,{\rm{\Lambda }}^{\prime} +{\rm{\Lambda }}^{\prime\prime} }}{2-{\delta }_{0,{\rm{\Lambda }}^{\prime\prime} }},\end{eqnarray} \tag{ 8 }$$

where Λ' and Λ'' are the angular momenta projected along the nuclear axis for the final and initial electronic states, respectively.

Predissociation is also possible through an intermediate transition to a bound level of an excited state. In units of cm², the predissociation cross section is

$$\begin{eqnarray}&&\sigma =8.85\times {10}^{-22}{\lambda }^{2}{x}_{{\ell }}{f}_{u{\ell }}{\eta }^{d}\end{eqnarray} \tag{ 9 }$$

(Heays et al. 2017), where λ is the photon wavelength in Å and f_uℓ is the oscillator strength of the transition from lower state ℓ to upper state u. We approximate the ground-state fractional population x_ℓ and the upper level tunneling probability η^d to both be 1 to give an upper limit to the predissociation cross section.

In LTE, a Boltzmann population distribution is assumed for the rovibrational levels in the electronic ground state. The total quantum-mechanical photodissociation cross section as a function of both temperature T and wavelength λ is

$$\begin{eqnarray}&&\sigma (\lambda ,T)=\displaystyle \frac{{\sum }_{v^{\prime\prime} }\,{\sum }_{J^{\prime\prime} }\,{g}_{{iv}^{\prime\prime} J^{\prime\prime} }\exp [-({E}_{00}-{E}_{v^{\prime\prime} J^{\prime\prime} })/{k}_{b}T]{\sigma }_{v^{\prime\prime} J^{\prime\prime} }}{{\sum }_{v^{\prime\prime} }\,{\sum }_{J^{\prime\prime} }\,{g}_{{iv}^{\prime\prime} J^{\prime\prime} }\exp [-({E}_{00}-{E}_{v^{\prime\prime} J^{\prime\prime} })/{k}_{b}T]},\end{eqnarray} \tag{ 10 }$$

where g_iv''J'' = 2J'' + 1 is the total statistical weight, E_v''J'' is the magnitude of the binding energy of the rovibrational level v'' J'', and k_b is the Boltzmann constant. The denominator is the rovibrational partition function.

The photodissociation rate for a molecule in an ultraviolet radiation field is given by

$$\begin{eqnarray}&&k=\int \sigma (\lambda )I(\lambda )d\lambda ,\end{eqnarray} \tag{ 11 }$$

where σ(λ) is the photodissociation cross section and I(λ) is the photon radiation intensity summed over all incident angles. The photon radiation intensity emitted by a blackbody with temperature T is

$$\begin{eqnarray}&&I(\lambda ,T)=\displaystyle \frac{8\pi c/{\lambda }^{4}}{\exp ({hc}/{k}_{{b}}{T}\lambda )-1},\end{eqnarray} \tag{ 12 }$$

where h is the Planck constant and c is the speed of light.

We also compute the photodissociation rate in the unattenuated ISRF, as given by Draine (1978), but modified for λ > 2000 Å by Heays et al. (2017), using Equation (11). In an interstellar cloud the radiation field is attenuated by dust reducing the photodissociation rate as a function of depth into the cloud, or parameterized as the visual extinction A_V. Assuming a plane-parallel, semi-infinite slab, with both sides of the cloud exposed isotropically to the ISRF, we applied the radiative transfer code of Roberge et al. (1991) to compute the photodissociation rate as a function of A_V and fit the rate to the forms

$$\begin{eqnarray}&&k({A}_{{\rm{V}}})={a}_{1}\exp (-{a}_{2}{A}_{{\rm{V}}}+{a}_{3}{A}_{{\rm{V}}}^{2}),\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&k({A}_{{\rm{V}}})={a}_{4}{E}_{2}({a}_{5}{A}_{{\rm{V}}}),\end{eqnarray} \tag{ 14 }$$

where E₂ is the second-order exponential integral. The grain model of Draine & Lee (1984) that was adopted corresponds to the galactic average of the total-to-selective extinction R_V = 3.1.

State-resolved photodissociation cross sections have been computed for transitions from 14,908 initial rovibrational levels in the X ¹Σ⁺ ground electronic state to the six considered excited electronic states. Cross sections are computed for photons with wavelengths starting at 500 Å up to at most 50,000 Å in 1 Å increments, typically stopping at the relevant threshold. A smaller wavelength step size is used near thresholds to resolve appropriate resonances. In Figure 3, a comparison of the state-resolved cross sections from the ground rovibrational level v'', J'' = 0, 0 for each transition is shown. The 2 ¹Π and A' ¹Σ⁺ (2 ¹Σ⁺) transitions have the dominant cross sections from the ground rovibrational level, while the transition to the A ¹Π state makes very little contribution. The behavior of the current cross sections are significantly different from those adopted in Heays et al. (2017).

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Predissociation is possible following bound–bound transitions to the B ¹Σ⁺, 3 ¹Π, and 4 ¹Π states. Estimates of predissociation cross sections are computed for transitions to a wide range of bound rovibrational levels. Cross sections for transitions from v'', J'' = 0, 0 are shown in Figure 3 computed using Equation (9). We find that the line cross sections due to predissociation are much smaller than the direct cross sections for the 4 ¹Π state. However, while predissociation through the B ¹Σ⁺ and 3 ¹Π states give cross sections comparable to those of their direct continuum cross sections, the continuum cross section for the 2 ¹Π dominates the predissociation lines by more than an order of magnitude over the relevant wavelength range. Predissociation does not appear to be important for the photodestruction of CS and is therefore not considered further.

The ${{\rm{A}}}^{{\prime} }{}^{1}{{\rm{\Sigma }}}^{+}\,\leftarrow$ X ¹Σ⁺ transition generally has large state-resolved cross sections; so a sampling of cross sections are displayed in Figure 4. Cross sections are plotted for several rotational levels of the ground vibrational level v'' = 0, and for several vibrational levels at their respective lowest rotational level, J'' = 0. State-resolved cross sections for the other five electronic transitions have also been computed (not shown).

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LTE cross sections have been computed for each transition using the state-resolved cross sections from 1000 K to 10,000 K in 1000 K intervals. A comparison of LTE cross sections for each transition as a function of photon wavelength at 3000 K is displayed in Figure 5. The ${{\rm{A}}}^{{\prime} }{}^{1}{{\rm{\Sigma }}}^{+}\,\leftarrow$ X ¹Σ⁺ transition is the dominant transition at longer wavelengths, while the $4{}^{1}{\rm{\Pi }}\,\leftarrow$ X ¹Σ⁺ transition dominates for short wavelengths. Since the ${{\rm{A}}}^{{\prime} }{}^{1}{{\rm{\Sigma }}}^{+}\,\leftarrow$ X ¹Σ⁺ transition is dominant for the majority of wavelengths, LTE cross sections for this transition at several temperatures are shown in Figure 6.

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Photodissociation rates for transitions from v'', J'' = 0, 0 for all six electronic transitions have been computed for the unattenuated ISRF and for the attenuated ISRF into interstellar clouds with total visual extinction. These values are listed and compared with those of Heays et al. (2017) and the UMIST compilation (McElroy et al. 2013) in Table 4. We consider fiducial diffuse and dense clouds with total visual extinctions of A_V = 1 and 20, respectively. Consistent with the cross section magnitudes, the ISRF photodissociation rates are dominated by the A${}^{{\prime} }{}^{1}{{\rm{\Sigma }}}^{+}\,\leftarrow$ X ¹Σ⁺ and 2 ¹Π $\leftarrow$ X ¹Σ⁺ transitions, which leave the two atoms in their ground states. However, about 10% of the photodissociation yield results in both C and S in their ¹D metastable states through the 4 ¹Π $\leftarrow$ X ¹Σ⁺ transition. Using reliable CS photodissociation cross sections, the current unattenuated ISRF rates are about a factor of ∼2.5–3 smaller than the estimates adopted by Heays et al. (2017) and McElroy et al. (2013).

Table 4.  Interstellar CS Photodissociation Rate Fits^a

Source	ISRF	Dense Cloud			Diffuse Cloud		Products
	k(s−1)	a1(s−1)	a2	a1(s−1)	a2	a3	C + S
		[a4(s−1)]	[a5]
A 1Π $\leftarrow$ X	1.50(−21)	5.43(−22)	2.085	8.29(−22)	3.73	4.00	3P + 3P
2 1Π $\leftarrow$ X	1.35(−10)	5.11(−11)	2.50	7.29(−11)	4.16	4.37
A' 1Σ+ $\leftarrow$ X	1.94(−10)	7.59(−11)	2.19	1.08(−10)	3.12	3.86
3 1Π $\leftarrow$ X	5.28(−15)	1.86(−15)	3.16	2.75(−15)	5.55	5.55	1D + 1D
B 1Σ+ $\leftarrow$ X	9.56(−13)	3.53(−13)	2.84	5.05(−13)	4.89	5.00
4 1Π $\leftarrow$ X	4.05(−11)	1.47(−11)	3.02	2.12(−11)	5.25	5.30
Total	3.70(−10)	1.48(−10)	2.32	2.163(−10)	3.98	4.21	⋯
	⋯	[2.13(−10)]	[1.69]	⋯	⋯	⋯	⋯
Heaysb	9.49(−10)	5.41(−10)	2.49	⋯	⋯	⋯	⋯
	⋯	[9.49(−10)]	[1.95]	⋯	⋯	⋯	⋯
UMISTc	9.70(−10)	9.70(−10)	2.00	⋯	⋯	⋯	⋯

Notes.

^aFits to Equations (13) and (14). ^bHeays et al. (2017). ^cMcElroy et al. (2013).

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We computed photodissociation rates for a blackbody radiation field; Figure 7 shows a plot of the photodissociation rates when the molecule is initially in the v'', J'' = 0, 0 rovibrational level for each final electronic state versus the blackbody temperature for a wide range of temperatures. Blackbody photodissociation rates were also obtained by Heays et al. (2017), but they were normalized to reproduce the ISRF energy density from 912 to 2000 Å, as opposed to the normalization inherent in Equation (12) adopted here. Appropriate scale factors, e.g., geometric dilution, should be applied for the relevant astrophysical environment. At the highest temperatures, the current photorates should be taken as a lower limit as photoionization and photodissociation through high-lying Rydberg states will begin to become important.

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Finally, we consider a situation where a gas containing CS is in LTE at a certain temperature and is immersed in a radiation field generated by a blackbody at the same temperature (i.e., equal gas kinetic and radiation temperatures). The photodissociation rates of CS in such a situation are computed using the LTE cross sections; a plot of these rates against the blackbody/gas temperature is shown in Figure 8.

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