HIGH-ACCURACY QUARTIC FORCE FIELD CALCULATIONS FOR THE SPECTROSCOPIC CONSTANTS AND VIBRATIONAL FREQUENCIES OF 1¹A' l-C₃H⁻: A POSSIBLE LINK TO LINES OBSERVED IN THE HORSEHEAD NEBULA PHOTODISSOCIATION REGION

Recent work by Huang et al. (2013b) has questioned the attribution of lines observed in the Horsehead nebula photodissociation region (PDR) to l-C₃H⁺. Quartic force fields (QFFs) computed from high-level ab initio quantum mechanical energies analyzed using perturbation theory at second order (Papousek & Aliev 1982) are known to produce highly accurate spectroscopic constants. Even though the B₀ computed by Huang et al. (2013b; 11 262.68 MHz) is within 0.16% of the B-type rotational constant derived from the observations by Pety et al. (2012; 11 244.9474 MHz), the computed D_e of 4.248 kHz differs by 44.5% from the observed D value of 7.652 kHz. This "error" is more than an order of magnitude larger than any other error for a computed D_e of a cation (using similar levels of theory) as compared to known high-resolution experimental data. Furthermore, the sextic distortion constant, H, differs by three orders of magnitude. As a result, it is unlikely that l-C₃H⁺ corresponds to the lines observed by Pety et al. (2012).

This result motivates the question, "What is the carrier of these lines?" If these observed lines are, in fact, related to one another, certain inferences can be made about the molecular carrier. To match the rotational constants derived from the transition energies corresponding to the observed lines, the carrier is either linear or quasi-linear, almost certainly composed of three carbon atoms as well as a single hydrogen atom, and closed-shell since there are no splittings in the lines as required for the rotational spectra of open-shell molecules (M. C. McCarthy 2013, private communication). All of these criteria are, in fact, met by l-C₃H⁺, but this cation's difference between observational and high-accuracy theoretical rotational constants, especially the D constant, discussed above and by Huang et al. (2013b), probably rules it out. As a result, the quasi-linear anion, 1 ¹A' l-C₃H⁻, remains the most likely candidate carrier of the Horsehead nebula PDR rotational lines of interest, especially since anions have been shown to be more abundant in the interstellar medium (ISM) than originally thought (Cordiner et al. 2013), and there has been reason to suspect the presence of C₆H⁻ in the Horsehead nebula PDR (Agúndez et al. 2008).

Even though the most stable singlet isomer of C₃H⁻ is the cyclic form, c-C₃H⁻, the barrier to isomerization is high enough ( 45 kcal mol⁻¹) for the quasi-linear C_s isomer to be kinetically stable (Lakin et al. 2001). Various mechanisms for interstellar synthesis of this anion are possible (Millar et al. 2007; Larsson et al. 2012; Senent & Hochlaf 2013) and are probably related to those responsible for the creation of the related C_2nH⁻ for n = 2–4 anions previously detected in the ISM (McCarthy et al. 2006; Cernicharo et al. 2007; Brünken et al. 2007b). Furthermore, radical C₃H in both the linear and cyclic forms has also been detected in the ISM (Thaddeus et al. 1985; Yamamoto et al. 1987), suggesting the possible interstellar existence of the anion.

Additionally, C₃H⁻ is of astronomical interest since it has been computationally shown by Fortenberry (2013) to possess not only a rare dipole-bound singlet excited electronic state (the 2 ¹A' state), but also an even more rare valence excited state (1 ¹A'') below the electron binding or electron detachment energy. In fact, the valence electronically excited state is the only such state thus far proposed to exist for an anion of this size which also contains only first-row atoms (Fortenberry & Crawford 2011b, 2011a; Fortenberry 2013). The valence excited state and the bent structure of C₃H⁻ are both the result of an unfilled π orbital. The two components of the HOMO π-type orbital split when the additional electron in the anion spin-pairs with the lone electron in the radical's π-type HOMO. A carbene and bent structure are thus created. The valence (1 ¹A'') state of C₃H⁻ is then the product of an excitation from the occupied portion of the split π orbital into the unoccupied portion, an uncommon process not present in the C_2nH⁻ anions. Furthermore, anions have been proposed as carriers of some diffuse interstellar bands (DIBs; Sarre 2000; Cordiner & Sarre 2007; Fortenberry et al. 2013a), and the two electronically excited states of C₃H⁻ may be of importance to the DIBs and to the chemistry of PDRs, as well.

The spectroscopic constants and fundamental vibrational frequencies of 1 ¹A' l-C₃H⁻ are computed through the established means of QFFs (Huang & Lee 2008). Starting from a restricted Hartree–Fock (RHF; Scheiner et al. 1987) coupled cluster (Lee & Scuseria 1995; Shavitt & Bartlett 2009; Crawford & Schaefer 2000) singles, doubles, and perturbative triples (CCSD(T); Raghavachari et al. 1989) aug-cc-pV5Z (Dunning 1989; Kendall et al. 1992; Dunning et al. 2001) geometry further corrected for core correlation effects from the Martin–Taylor basis set (Martin & Taylor 1994), a grid of 743 symmetry-unique points is generated. Simple-internal coordinates for the bond lengths and ∠H−C−C are coupled to linear LINX and LINY (Allen et al. 2005) coordinates exactly as those defined in Fortenberry et al. (2012b) for HOCO⁺. Displacements of 0.005 Å for the bond lengths, 0.005 rad for the bond angle, and 0.005 for the LINX and LINY coordinates and the associated energies computed at each point define the QFF, which is of the form:

$$\begin{equation} V=\frac{1}{2}\sum _{ij}F_{ij}\Delta _i\Delta _j + \frac{1}{6}\sum _{ijk}F_{ikj}\Delta _i\Delta _j\Delta _k + \frac{1}{24}\sum _{ijkl}F_{ikjl}\Delta _i\Delta _j\Delta _k\Delta _l, \end{equation} \tag{ 1 }$$

where Δ_i are the displacements and F_ij... are force constants (Huang & Lee 2008).

At each point, CCSD(T)/aug-cc-pVXZ (where X = T, Q, 5) energies are computed and extrapolated to the complete basis set (CBS) limit via a three-point formula (Martin & Lee 1996). Additionally, energy corrections are made to the CBS energy for core correlation and for scalar relativistic effects (Douglas & Kroll 1974). The resulting QFF is denoted as the CcCR QFF for the CBS energy, core correlation correction, and scalar relativistic correction, respectively, (Fortenberry et al. 2011). The augmented Dunning basis sets have been shown by Skurski et al. (2000) to be reliable for computations of anionic properties. An initial least-squares-fit of the CcCR energy points leads to a minor transformation of the reference geometry such that the gradients are identically zero. This geometry and the resulting force constants are then employed in the rovibrational computations. All electronic structure computations make use of the MOLPRO 2010.1 quantum chemical package (Werner et al. 2010), and all employ the Born–Oppenheimer approximation making the QFFs identical for the isotopologues.

The QFF is fit from the 805 redundant total energy points with a sum of squared residuals on the order of 3 × 10⁻¹⁷ a.u.² Cartesian derivatives are then computed from the QFF with the INTDER program (Allen et al. 2005). From these, the SPECTRO program (Gaw et al. 1991) employs second-order vibrational perturbation theory (VPT2) to generate the spectroscopic constants (Papousek & Aliev 1982) and vibrational frequencies (Mills 1972; Watson 1977). After transforming the force constants into the Morse-cosine coordinate system so that the potential possesses proper limiting behavior (Dateo et al. 1994; Fortenberry et al. 2013b), vibrational configuration interaction (VCI) computations with the MULTIMODE program (Carter et al. 1998; Bowman et al. 2003) also produce vibrational frequencies. The VCI computations make use of similar basis set configurations as those utilized by Fortenberry et al. (2012a, 2012b) in similar quasi-linear tetra-atomic systems.

The force constants computed in this study are listed in Table 1. The CcCR geometrical parameters and spectroscopic constants are given in Table 2 for both 1 ¹A' l-C₃H⁻ and the deuterated isotopologue. The equilibrium dipole moment is computed with respect to the center-of-mass with CCSD(T)/aug-cc-pV5Z to be 2.16 D. The C−C−C R_α vibrationally-averaged bond angle is nearly collinear at 174540 while the vibrationally-averaged ∠H−C−C is 109491. These values are in line with those computed by Lakin et al. (2001). As has been discussed by Fortenberry (2013) for C₃H⁻, the C₁ carbon atom adjacent to the hydrogen atom shown in Figure 1 is a carbene-type carbon containing a lone pair which leads to a longer C₁ −C₂ bond length compared to the shorter C₂ −C₃ bond length. Even though this result differs from the CCSD(T) results from Lakin et al. (2001), their reported CASSCF and HF results give bond lengths similar to ours, leading us to conclude that the CCSD(T) C−C bond lengths are mislabeled in the paper by Lakin et al. (2001). The vibrationally averaged geometrical parameters change slightly upon deuteration. Similar bond angles of the heavy atoms have been computed for the trans-HOCO⁺, HOCS⁺, and HSCO⁺ systems (Fortenberry et al. 2012a, 2012b) with very good agreement present for known experimental data.

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Table 2. The Zero-Point (R_α Vibrationally-averaged) and Equilibrium Structures, Rotational Constants, CCSD(T)/aug-cc-pV5Z Dipole Moment, Vibration-Rotation Interaction Constants, and Quartic and Sextic Distortion Constants of 1 ¹A' l-C₃H⁻ and the Deuterated form with the CcCR QFF

	C3H−	Previousa	C3D−
r0(C1 −H)	1.119 438 Å		1.116 446 Å
r0(C1 −C2)	1.351 595 Å		1.351 753 Å
r0(C2 −C3)	1.282 845 Å		1.282 620 Å
∠0(H−C1 −C2)	109491		109530
∠0(C1 −C2 −C3)	174540		174643
A0	529 134.2 MHz		295 539.6 MHz
B0	11 339.66 MHz		10 626.03 MHz
C0	11 087.35 MHz		10 238.74 MHz
DJ	4.954 kHz		4.544 kHz
DJK	0.702 MHz		0.316 MHz
DK	217.543 MHz		94.897 MHz
d1	−0.112 kHz		−0.253 kHz
d2	−0.023 kHz		−0.052 kHz
HJ	3.344 mHz		16.516 mHz
HJK	3.221 Hz		2.151 Hz
HKJ	−3.229 kHz		−0.745 kHz
HK	358.867 kHz		90.731 kHz
H1	0.132 mHz		0.634 mHz
H2	0.203 mHz		0.612 mHz
H3	0.037 mHz		0.133 mHz
τaaaa	−873.001 MHz		−380.872 MHz
τbbbb	−0.021 MHz		−0.021 MHz
τcccc	−0.019 MHz		−0.017 MHz
τaabb	−2.766 MHz		−1.619 MHz
τaacc	−0.081 MHz		0.319 MHz
τbbcc	−0.020 MHz		−0.018 MHz
Φaaa	355 640.661 Hz		89 988.504 Hz
Φbbb	0.001 Hz		0.004 Hz
Φccc	0.000 Hz		0.001 Hz
Φaab	390.158 Hz		703.204 Hz
Φabb	4.265 Hz		3.112 Hz
Φaac	−3 614.354 Hz		−1 445.590 Hz
Φbbc	0.000 Hz		0.001 Hz
Φacc	−0.271 Hz		0.151 Hz
Φbcc	0.001 Hz		0.002 Hz
Φabc	4.570 Hz		3.618 Hz
αA 1	27 922.5 MHz		11 662.9 MHz
αA 2	−725.5 MHz		−917.5 MHz
αA 3	484.8 MHz		170.2 MHz
αA 4	−35 092.1 MHz		−16 226.3 MHz
αA 5	−3 103.1 MHz		−4 597.5 MHz
αA 6	12 333.4 MHz		9 042.1 MHz
αB 1	4.2 MHz		6.9 MHz
αB 2	83.5 MHz		77.2 MHz
αB 3	45.1 MHz		40.3 MHz
αB 4	−12.0 MHz		−8.4 MHz
αB 5	−47.1 MHz		−48.4 MHz
αB 6	−48.6 MHz		−45.9 MHz
αC 1	14.8 MHz		18.3 MHz
αC 2	78.6 MHz		70.1 MHz
αC 3	38.4 MHz		39.3 MHz
αC 4	16.0 MHz		12.9 MHz
αC 5	−16.1 MHz		−15.2 MHz
αC 6	−78.5 MHz		−69.8 MHz
re(C1 −H)b	1.106 939 Å	1.110 Å	⋅⋅⋅
re(C1 −C2)	1.349 832 Å	1.289 Å	⋅⋅⋅
re(C2 −C3)	1.281 900 Å	1.363 Å	⋅⋅⋅
∠e(H−C1 −C2)	109529	1092	⋅⋅⋅
∠e(C2 −C3 −C4)	174571	1712	⋅⋅⋅
Ae	530 044.3 MHz	524.5 GHz	295 106.5 MHz
Be	11 352.05 MHz	11.2 GHz	10 636.73 MHz
Ce	11 114.02 GHz	10.9 MHz	10 266.68 MHz
μc	2.16 D	⋅⋅⋅	⋅⋅⋅
μx	1.63 D	⋅⋅⋅	⋅⋅⋅
μy	1.41 D	⋅⋅⋅	⋅⋅⋅

Notes. ^aCCSD(T)/aug-cc-pVQZ QFF results from Lakin et al. (2001). ^bThe equilibrium geometries are identical among isotopologues from the use of the Born–Oppenheimer approximation. ^cThe C₃H⁻ coordinates (in Å with the center-of-mass at the origin) used to generate Born–Oppenheimer dipole moment components are: H, 1.733414, −0.910473, 0.000000; C₁, 1.276456, 0.098036, 0.000000; C₂, −0.069613, −0.016965, 0.000000; C₃, −1.352424, −0.004605, 0.000000.

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The most notable values in Table 2 are the rotational constants and the quartic centrifugal distortion (D-type) constants. For 1 ¹A' l-C₃H⁻, the B₀ rotational constant is 11 339.66 MHz while C₀ is 11 087.35 MHz. The equilibrium constants are slightly larger, but both sets are in reasonable agreement with those computed by Lakin et al. (2001). The D-type constants have not been vibrationally-averaged, and D_J, most prominently, is 4.954 kHz.

Direct comparison between these explicitly computed values and those deduced from the Horsehead nebula PDR spectrum observed by Pety et al. (2012) is not possible since the isomer of C₃H⁻ of interest here is not perfectly linear. Pety et al. (2012) assume a linear structure in order to fit the effective rotational constant, B_eff, and the effective centrifugal distortion constant, D_eff and use the second-order fitting equation,

$$\begin{equation} \nu _{J+1\rightarrow J}=2B(J+1)-4D(J+1)^3, \end{equation} \tag{ 2 }$$

to compute the affiliated rotational constants. C₃H⁻ is non-linear and requires the following related equation from McCarthy et al. (1997):

$$\begin{equation} \nu _{J+1\rightarrow J}=(B+C)(J+1)-\left\lbrace 4D_J+\frac{(B-C)^2}{c \left[ A-\frac{(B+C)}{2} \right] } \right\rbrace (J+1)^3, \end{equation} \tag{ 3 }$$

with the assumption that K = 0 forcing c = 8. As such, we can set Equation (2) equal to Equation (3). The (J + 1) term in Equation (3) is equal to 2 B_eff, and the (J + 1)³ term in Equation (3) is equal to 4 D_eff. Using the CcCR computed A₀, B₀, C₀, and D_J values, where D_J is the only equilibrium constant, B_eff is computed to be 11 213.51 MHz, and D_eff is 8.795 kHz. Hence, direct comparison between the CcCR C₃H⁻ derived effective rotational constants and those obtained from the lines observed by Pety et al. (2012) is possible.

The second-order fit of the lines observed by Pety et al. (2012) indicates that the carrier must have a B-type constant that is very close to 11 244.9474 MHz and a D-type quartic distortion constant that is around 7.652 kHz. The B_eff computed with the A₀, B₀, and C₀ rotational constants by the above approach is very close, off by 31.44 MHz or 0.28%. This is roughly the same difference between the observed B and that of l-C₃H⁺ (Huang et al. 2013b). However, the 8.795 kHz D_eff for 1 ¹A' l-C₃H⁻ is much closer to the 7.652 kHz D derived from the lines observed by Pety et al. (2012) in the Horsehead nebula than the linear cation (Huang et al. 2013b). Even so, this D_eff of 8.795 kHz differs from the observation by 1.14 kHz or 14.93%.

Table 3 provides some insight into the accuracies that can be expected for calculated rotational constants of similar molecules. Related quasilinear molecules studied previously have all been cations. Hence, within Table 3, the cation B and D-type constants listed are more correctly understood to be B_eff and D_eff as is the case for C₃H⁻ (i.e., Equation (3) is used). Calculation of the vibrationally-averaged B_eff values incorporate B₀ and C₀ while the equilibrium B_eff values incorporate B_e and C_e. Calculation of D_eff for each of the bent, quasilinear systems utilizes A₀, B₀, and C₀ and the equilibrium D_J value since vibrational averaging is not available for the D-type constants. The lone exception to this definition of D_eff is the C₃H⁻ D_eff computed with A_e, B_e, and C_e given in the second line of Table 3, which actually lowers the C₃H⁻ D_eff value to 8.366 kHz, a difference of 0.714 kHz or 9.34% from that determined by Pety et al. (2012). Finally, since all of the anions observed in the ISM have been linear, directly comparable B₀, B_e, and D_e constants have been computed explicitly and are listed in Table 3. Note that all of the experimental B and D values contained in Table 3 correspond to vibrationally averaged constants.

From Table 3, the quasi-linear cations listed below C₃H⁻ show strong correlation between the computed B_eff from the use of B₀ and C₀ and the B_eff derived from the various experiments. Additionally, the D_eff values computed the same way with the equilibrium D_J also show good, albeit not as strong, correlation between theory and experiment. Unfortunately, C₃H⁻ has errors that are larger than this. However, this probably results from a combination of basis set incompleteness and higher-order correlation effects. Even though aug-cc-pVXZ basis sets used at the CCSD(T) level of theory have been shown to be effective in the computation of anionic properties (Skurski et al. 2000; Fortenberry & Crawford 2011b), higher-order properties such as the D-type constants are more susceptible to even the smallest errors. This is clear for the cations as well, where the D_eff values are not as accurate as the B_eff values.

The known interstellar anions and the related C₂H⁻ system, which has not yet been detected in the ISM, are linear and have B and D computed directly, either as B₀ or B_e and D_e. Note that the theoretical rotational constants are not as accurate for the anions as they are for the cations. Most notably, the B_e/B₀ and D_e values computed with a CCSD(T)/aug-cc-pCV5Z cubic force field for C₅N⁻ by Botschwina & Oswald (2008) are directly used in the identification of this anion in the ISM (Cernicharo et al. 2008). As listed in Table 3, agreement between computed B values and that necessary to match the observed rotational lines actually worsens when B₀ is used instead of B_e, more than doubling the percent error. This is the same behavior as what is currently found for C₃H⁻. Additionally, the D_e percent error for C₅N⁻, as compared to observation, is 9.1%, almost exactly what it is for C₃H⁻ when using the equilibrium rotational constants. The force field employed by Botschwina & Oswald (2008) also includes core correlation like the CcCR QFF. Hence, the present rotational constants are in the same accuracy range for C₃H⁻ as those used to detect C₅N⁻ in the ISM. Furthermore, the calculated D_e values compared to experiment for C₆H⁻ and C₈H⁻ actually have a larger percent error than D_eff for C₃N⁻, C₅N⁻, or even C₃H⁻.

Comparison of the sextic distortion constant, H_eff, is not as straightforward. There is a dearth of data on how the computation of this value for anions compares to experiment. H_J, which is an equilibrium value, is not exactly H_eff, but they are probably related. Even though H obtained by Pety et al. (2012) is 560 mHz and H_J for C₃H⁻ is 3.344 mHz, this is an order of magnitude closer agreement than this same H compared to the H_e for l-C₃H⁺, 0.375 mHz (Huang et al. 2013b). Additionally, the same basis set and correlation errors for anions that affect the calculation of D will be present for H. As a result, we can only say here that as far as H is concerned for comparison to the lines observed in the Horsehead nebula by Pety et al. (2012), 1 ¹A' C₃H⁻ is a better candidate than l-C₃H⁺.

Even though lower levels of theory have been used to reproduce rotational constants of the detected, linear interstellar anions, C₃H⁻ is the only anion examined here that is not linear. It is known that basis set effects can be pronounced in the computation of bond angles in anions (Lee & Schaefer 1985; Huang & Lee 2009) where the average change in a bond angle computed with a standard basis set and one augmented to include diffuse functions is around 10. For example, the equilibrium ∠C−C−C in C₃H⁻ from a simple CCSD(T)/cc-pVTZ QFF is 17332, while this same angle is 17420 with a CCSD(T)/aug-cc-pVTZ QFF. In fact, the resulting 088 difference by simply adding diffuse functions to the standard basis set is actually larger than the equilibrium ∠C−C−C difference between the CCSD(T)/aug-cc-pVTZ QFF and that from the CcCR QFF, 038. As a result, B_eff for C₃H⁻ is slower to converge with respect to the basis set chosen relative to the linear anions. This is made clear in that the CCSD(T)/cc-pVTZ QFF vibrationally averaged B_eff is 11 056.74 MHz whereas the corresponding CcCR B_eff is 11 213.51 MHz, an increase of 156.77 MHz. The linear anions are able to use lower level levels of theory in order to approach the experimental rotational constants, but higher levels of theory are required for the non-linear anion. The fact that B_eff computed with the equilibrium rotational constants is closer to the B derived from the observations by Pety et al. (2012) than B_eff computed with the vibrationally averaged rotational constants is coincidental. However, the important point is that the C₃H⁻ vibrationally averaged B_eff approaches the corresponding observed value as more accurate QFFs are employed, and the remaining error is typical.

The harmonic and anharmonic vibrational frequencies for both 1 ¹A' l-C₃H⁻ and l-C₃D⁻ are given in Table 4. Positive anharmonicities are present in both isotopologues for the ν₅ C₁ −C₂ −C₃ bending and the ν₆ torsional modes. VPT2 and VCI produce fundamental vibrational frequencies from the CcCR QFF that are quite consistent. The largest deviation between the methods, 1.0 cm⁻¹, is found for the ν₄ H−C₁ −C₂ bending mode. Comparison of the C₃H⁻ CcCR QFF vibrational frequencies, whether computed using VPT2 or VCI, to those computed by Lakin et al. (2001) is roughly consistent for ν₁-ν₄. The ν₅ anharmonic frequencies differ by more than 50 cm⁻¹, though the ω₅ harmonic frequencies are very similar (i.e., the difference in the ν₅ fundamental frequency is mostly due to differences in the anharmonic correction). The torsional mode is nearly identical between the two studies, though in this case the harmonic frequencies differ by more than 50 cm⁻¹. It is hoped that the present QFF computations of the fundamental vibrational frequencies provided here will assist in the characterization of this anion in current and future studies of the ISM or simulated laboratory experiments at infrared wavelengths in addition to studies in the sub-millimeter spectral region.

The lines observed by Pety et al. (2012) are present in the Horsehead nebula PDR but not in the dense core. Typically, a PDR is defined in terms of shells starting from the exterior shell dominated by an influx of far-ultraviolet (FUV) photons. In this region, the photons are most often absorbed by polycyclic aromatic hydrocarbons (PAHs) and dust particles. However, electrons are also produced in these regions from various mechanisms involving the aforementioned larger molecular particles as well as from interactions with atoms or small molecules. As the FUV flux is reduced from shielding resulting from the PAHs and dust, the H₂ shell is formed. Moving further in to the region, CO begins to form, and, finally, O₂ formation is present in the dense core when the photon shielding is high enough (Tielens 2005; Wolfire 2011). In fact, PDRs are believed to be a major cache of the interstellar molecular abundance due to the stability of the dense cores.

It could be assumed that such a large flux of high-energy photons in the outer shells would remove any excess electron from an anion or even from many neutral radicals. However, this same process results in a veritable sea of elections that could attach to neutrals and actually lead to the creation of anions even in the Horsehead nebula PDR (Millar et al. 2007). Additionally, many anions are known to be surprisingly stable (Hammer et al. 2003; Simons 2008, 2011; Fortenberry & Crawford 2011b, 2011a; Fortenberry 2013), and electron attachment rates are also believed to be quite high in these regions (Millar et al. 2007). Several anions have also been shown to possess dipole-bound excited states, or threshold resonances, which may play a significant role in the creation and recreation of interstellar anions (Güthe et al. 2001; Carelli et al. 2013). The mechanism of radiative attachment (RA), outlined by Carelli et al. (2013) as radiative stabilization, describes attachment of an electron to a neutral species, A, through creation of the excited electronic state of the resultant anion:

$$\begin{equation} \mathrm{A}+e^- \rightarrow \mathrm{[A^-]^*} \rightarrow \mathrm{A^-} + h\nu. \end{equation} \tag{ 4 }$$

Relaxation can take place such that the electronic ground state of the anion would be present (Carelli et al. 2013; Millar et al. 2007; Herbst & Osamura 2008).

The dipole-bound (and only) excited state of a small anion may function as the necessary excited state for RA. Dipole-bound states are known to exist for each of C₄H⁻, C₆H⁻, and C₈H⁻ (Pino et al. 2002). In order for such a state to be present, the dipole moment of the corresponding neutral, a radical for these systems, must be on the order of 2 D or larger (Simons 2008, 2011). For the ²Π ground states of C₆H and C₈H, the dipole moments are large enough to support a singlet dipole-bound excited state. C₄H, ²Σ⁺ in its ground state (Fortenberry et al. 2010), has a relatively small dipole moment at 0.8 D (Graf et al. 2001). Hence, in order for C₄H⁻ to form, the radical must either excite out of the weakly dipolar $\tilde{X}\ ^2\Sigma ^+$ state into the large-dipole A ²Π state before undergoing RA, or it must form through another manner besides RA. As discussed by Gupta et al. (2007) and McCarthy & Thaddeus (2008), the need for radical excitation followed by RA could explain the very low [C₄H⁻/C₄H] ratio observed toward various interstellar objects (Agúndez et al. 2008; Cordiner et al. 2013). Even though these two states of C₄H are "nearly degenerate" (Taylor et al. 1998), some additional energy is required to populate the A ²Π state, which, in turn, lowers the probability of electron attachment. Furthermore, C₂H is also ²Σ⁺ in its ground state, but the excitation energy into the large-dipole A ²Π state is more than double its counterpart in C₄H (Fortenberry et al. 2010), which may shed light on the even lower [C₂H⁻/C₂H] ratio proposed by Agúndez et al. (2008). C₆H⁻ and C₈H⁻ could be present in the Horsehead nebula PDR, as has been suggested from observations and modeling by Agúndez et al. (2008), but these longer anions may only be accessible from their dipole-bound excited states. However, C₃H⁻ has more than just a dipole-bound excited state.

A few rare anions possess valence excited electronic states between the dipole-bound state and the ground electronic state (Fortenberry & Crawford 2011a, 2011b). As mentioned in the  Introduction, 1 ¹A' C₃H⁻ is, thus far, the only anion composed solely of first-row atoms (and hydrogen) to possess a valence singlet excited state (Fortenberry 2013). The presence of two excited electronic states with the same spin multiplicity should increase the production of C₃H⁻ since multiple relaxation pathways exist. Beginning from the dipole-bound state, the excited anion can relax within the RA mechanism to the ground electronic state either directly or via the valence excited state first. Enough C₃H⁻ may then exist in a steady state to counterbalance the destructive photons present in this region.

If the Horsehead nebula PDR abundances of l-C₃H⁺ from Pety et al. (2012) can be inferred to actually be 1 ¹A' C₃H⁻, the [C₃H⁻/C₃H] ratio could be as high as 0.30 in the Horsehead nebula PDR. This is not as high as the upper limit proposed for [C₆H⁻/C₆H] at 8.9, but it is an order of magnitude larger than [C₄H⁻/C₄H] (Agúndez et al. 2008) as can be expected since the ground electronic state of C₃H is strongly dipolar and that of C₄H is not. The amount of C₃H⁻ should decrease as the observations move toward the dense molecular core due to the higher reactivity of this anion. The reaction cross-section of anions is much larger than in neutrals (Eichelberger et al. 2007), and C₃H⁻ could go through various destructive processes (Millar et al. 2007; Larsson et al. 2012) as the molecular density increases. Alternatively, this anion could exist within the observed sightline but on the outer edge of the PDR where the photon flux is small enough for a measurable population to be stable. In this region a longer path length of such material is also present away from the high A_V dense core. Either way, the existence of 1 ¹A' C₃H⁻ in the Horsehead nebula PDR is feasible.

Since the link between l-C₃H⁺ and the lines observed in the Horsehead nebula PDR by Pety et al. (2012) has recently been strongly questioned by Huang et al. (2013b), another viable candidate is necessary. The rotational lines seem to require a closed-shell quasi-linear structure composed of three carbon atoms along with a hydrogen atom. 1 ¹A' C₃H⁻ appears to be the most likely candidate. Here, the CcCR QFF has determined a B_eff for this anion to be in error by 0.28% from that required to fit the observed lines. The use of the equilibrium rotational constants fortuitously lowers the error to 0.11%. However, the error reduction and error magnitudes themselves are in line with the computed C₅N⁻ rotational constants used in its interstellar detection. Additionally, the discrepancy between the A_e, B_e, and C_e computed C₃H⁻ D_eff and the D_eff deduced from the observed interstellar rotational lines is similar to the D_e errors for C₄H⁻, C₃N⁻, and C₅N⁻ and less than that of C₆H⁻, which are all reported for CCSD(T) computations, i.e., similar levels of theory. Hence, the consistency of the errors for C₃H⁻ with other anions previously detected in the ISM coupled with its matching the required spectral criteria and the rationale for its existence involving its valence and dipole-bound excited states, make this anion the strongest candidate carrier for the Horsehead nebula PDR lines and, potentially, the seventh and most recent anion detected in the ISM. It would also be the first detected interstellar odd-numbered carbon monohydrogen chain anion.

R.C.F. is currently supported on a NASA Postdoctoral Program Fellowship administered by Oak Ridge Associated Universities. NASA/SETI Institute Cooperative Agreement NNX12AG96A has funded the work undertaken by X.H. Support from NASA's Laboratory Astrophysics "Carbon in the Galaxy" Consortium Grant (NNH10ZDA001N) is gratefully acknowledged. The U.S. National Science Foundation (NSF) Multi-User Chemistry Research Instrumentation and Facility (CRIF:MU) award CHE-0741927 provided the computational hardware, and award NSF-1058420 has supported T.D.C. The CheMVP program was used to create Figure 1. The authors also acknowledge many others for their contributions to our astronomical understanding of this subject. These include, most notably: Dr. Michael C. McCarthy of the Harvard-Smithsonian Center for Astrophysics, Dr. Naseem Rangwala of the University of Colorado, Dr. Lou Allamandola of the NASA Ames Research Center, and Dr. Christiaan Boersma of the NASA Ames Research Center and San Jose State University.