We have studied photodetachment of the amidogen anion $NH2\u2212$ as a function of photon energy near the detachment threshold. The detachment spectrum is obtained over the energy range of 6190–6355 cm^{−1} from the loss rate of the anions from a cryogenic radiofrequency multipole ion trap. By modeling all accessible rotational state-to-state photodetachment transitions, we can assign rotational state-specific thresholds to the measured spectrum. In this way, we have determined the electron affinity of NH_{2} to be 6224 ± 1 cm^{−1}.

## I. INTRODUCTION

Electron-atom and electron-molecule collision dynamics are important for many fundamental physical processes. They also play a role in technical environments, such as in plasma etching processes, in gas lasers, in combustion and fusion chemistry, and in astrophysics and atmospheric and biological processes. In interstellar models, simple nitrogen hydrides, such as NH_{2}, also known as amidogen, often appear in the first steps of the nitrogen chemical network^{1,2} and the existence of $NH2\u2212$ in interstellar space is a subject of discussion.^{3,4} Collisions with electrons play an important role in the formation (radiative electron attachment)^{5} and destruction (photodetachment) of interstellar anions.^{6,7}

Amidogen has been extensively investigated in the past. Nevertheless there is still a lack of high resolution studies on the low energy collision dynamics for NH_{2}. Investigating the photodetachment process in $NH2\u2212$ is an interesting approach to investigate low energy electron collision dynamics. Photodetachment excitations can also serve as a probe of the rotational state population of molecular ions.^{8} This can be used for rotational spectroscopy^{9} as well as for inelastic collision rate measurements.^{10} Such experiments impose strict requirements on the uncertainty of the electron affinity (EA) of the neutral. Namely, the photon energy has to be precisely set in order to satisfy certain criteria: only molecular anions on or above a specific excited rotational state may be neutralized; the other molecules remain unaffected. In this way, photodetachment excitations provide rotational resolution. The best currently available value for the electron affinity of NH_{2} (0.771 ± 0.005 eV^{11} or 6220 ± 40 cm^{−1}) contains an uncertainty extending over transitions between different rotational levels and cannot be used in precision studies.

In this article, we present the partially rotationally resolved threshold photodetachment spectrum of $NH2\u2212$ confined in a 22-pole ion trap. To determine the value of the electron affinity of amidogen, we fit the measured relative photodetachment cross section to a model in order to assign the observed spectra to electronic transitions between the different rotational states of the ion and the neutral. We also analyze the effect of the rotational temperature and the ortho-to-para ratio of the anions on the photodetachment rates. In Sec. II, we present the experimental procedure. Then we provide a detailed description of the relevant photodetachment transitions. In Sec. IV, we show and discuss the experimental results, the modeled photodetachment cross section, and the fit of the model to the data. Our summary is presented in Sec. V.

## II. EXPERIMENTAL PROCEDURE

A detailed description of the experimental setup and procedures can be found in previous publications^{9,12,13} and only the main features are outlined here. $NH2\u2212$ ions, along with other species, are produced in a pulsed jet of the gas mixture He/NH_{3}(90%, 10%) in a plasma discharge. Assuming that the anions are predominantly formed in the reaction NH_{3} + $e\u2212\u2009\u2192\u2009NH2\u2212$ + H, we expect an ortho to para ratio of 3.^{14} The 22-pole ion trap^{15,16} is loaded with the selected ion using a Wiley-McLaren time of flight mass spectrometer and an appropriate timing of the trap loading. For the measurements, the trap, together with its thermal shield, is cooled to 9(1) K using a closed-cycle refrigerator. For measurements at a slightly higher temperature of 28(2) K, heating elements are used to lift the trap temperature. He buffer gas thermalizes with the trap shield and cools ions internally and translationally by collisions. Typical buffer gas densities of about 1 to 4 × 10^{13} cm^{−3} guarantee complete thermalisation on the millisecond time scale.

The photodetachment process is investigated by measurements of the loss rate *k*_{PD} of the ions in the trap, induced by the absorption of an infrared photon from the detachment laser. Due to photodetachment, the ion intensity in the trap drops exponentially with exposure time $[NH2\u2212]\u221dexp(\u2212kPDt)$. To obtain a photodetachment spectrum, we perform loss rate measurements at different photon energies. For each measurement, ions are stored for a set of storage times between 0.1 and 12 s, with the maximum storage time chosen according to the observed decay rate at the specific wavelength. The ions are then unloaded from the trap, and their integrated intensity is measured on a microchannel plate detector. The fitted loss rates are normalized to the power of the detachment laser. The result is directly proportional to the photodetachment cross section.^{8,13}

The photodetachment laser is a home built extended cavity laser in Littrow configuration. It is based on the Single Angled Facet (SAF) gain chip SAF1118H (Covega) and Spectrogon grating with a gold coating (900 grooves/mm, optimized for 1550 nm). The scanning range of the system is about 6000–6750 cm^{−1}. Its wavelength is measured with a near-infrared sensitive wavemeter (Burleigh WA-1000) with an accuracy that is limited due to the wavementer’s multi-mode HeNe reference laser to an accuracy of ±500 MHz or about ±1 ppm. The detachment laser has two output beams. The one from the angled facet (reflectivity of about 0.005%) is coupled to the wavemeter through a free optical path window. The second, exiting from the normal facet (reflectivity 10%), is guided to the trap setup by a multimode fiber and aligned with the trap axis. The power of this laser beam entering the setup was between 4 and 8 mW. During the measurements, we continuously monitored the relative incident photon flux entering the trap by means of a photo diode.

Strictly speaking, the ion loss in the trap has to be described by a double exponential decay

where the square brackets denote the number of ions in the trap and $kpdo$ and $kpdp$ are the photodetachment rates for ortho and para species, respectively. At short exposure times ($t<1/kpdo$ and $t<kpdp$), an approximate expression can be used to evaluate the photodetachment rate of all $NH2\u2212$ ions,

where $Go/p=[NH2\u2212]0o/p/[NH2\u2212]0$ is the initial fraction of ortho/para $NH2\u2212$, respectively (*G*_{o} + *G*_{p} = 1). A single exponential fit to the experimental data was indeed found to be a good approximation for these measurements.

## III. PHOTODETACHMENT TRANSITIONS

In the following, we describe the rotational energy levels of $NH2\u2212$ and NH_{2} in their electronic and vibrational ground states and the single-photon allowed detachment transitions (see Fig. 1). Both the anion and the neutral belong to the C_{2ν} symmetry group and their ground electronic states are X ^{1}A_{1} for $NH2\u2212$ and X^{2}B_{1} for NH_{2}.

The rotational spectrum of $NH2\u2212$ has been barely studied in the past. Rotational constants with accuracies of about 200 MHz have been obtained from vibrational spectroscopy.^{17} Recently we have been able to perform direct rotational spectroscopy on the two lowest rotational transitions of $NH2\u2212$ with an accuracy of less than 3 MHz.^{4} The spectroscopic constants and the energy level structure of NH_{2} including hyperfine effects are outlined in many publications.^{18–25}

### A. NH_{2}

The energy structure of NH_{2} is complicated by the presence of the unpaired electron (electron spin *S* = 1/2). Each rotational level is identified with the quantum numbers *N*_{τ} (*τ* = *K*_{a} − *K*_{c}), where the rotational angular momentum *N* is split by the spin-rotation interaction into two sub-levels with quantum numbers *J* = *N* ± 1/2. The further hyperfine splitting of fine structure levels appears due to the ^{14}N nuclear spin (*I*_{N} = 1). These levels are labeled with the *F*_{1} quantum number (**F**_{1} **= J + I**_{N}). Due to the presence of equivalent hydrogen nuclei, we distinguish ortho rotational levels, characterized by even *τ* values, and para rotational levels, characterized by odd *τ*. The hyperfine sub levels belonging to the ortho states additionally split into the 3 sub levels due to the hydrogen nuclear spin *I*_{H} = 1 and are characterized by a quantum number *F* (**F = F**_{1} **+ I**_{H}). The fine structure splitting of the neutral is about 4-16 GHz and its hyperfine levels are separated by few tens of MHz.^{19,26}

### B. $NH2\u2212$

$NH2\u2212$ is a closed shell molecule; therefore, only hyperfine splitting is present. The rotational levels of $NH2\u2212$ were calculated using the molecular constants from vibrational spectroscopy^{17} and Watson’s Hamiltonian.^{27} Compared to the neutral case, the ortho and para states are interchanged: Rotational levels characterized by even *τ* are para states and rotational levels with odd *τ* are ortho states.

The hyperfine splitting of the ion is described by quadrupole constants, which are known to be about 0.1-2 MHz for the NH_{2} molecule. Due to the limited resolution of the present photodetachment spectroscopy measurements, hyperfine splittings are unresolved in the data, but they have to be accounted for in the calculation of the transition intensities.

### C. Selection rules for photodetachment transitions

A photodetachment transition is allowed if the product

of irreducible representations of the $C2v$ symmetry group contains the totally symmetric representation *A*_{1}. In the above expression, Γ^{de} is the irreducible representation of the detached electron, $\Gamma NH2$ is the irreducible representation of the NH_{2} state (given by the direct product of rotational, vibrational, and electronic representations), $\Gamma NH2\u2212$ is the irreducible representation of the $NH2\u2212$ state (again given by the direct product of rotational, vibrational, and electronic representations), and Γ^{dip} is the irreducible representation of the dipole moment operator. Γ^{de} and Γ^{dip} are the representations in the laboratory frame of reference. The molecules are described in I^{r} notation. The non-zero component of the dipole (along the symmetry axis of the molecule) then transforms as the *A*_{2} irreducible representation in the lab reference frame.

The irreducible representation of the rotational states with *N* = 0, 1, 2 are summarized in Table I. It is easy to see that the departing electron has A_{1} symmetry (*s*-type photodetachment) for the transitions between the following rotational states: A_{1} → B_{2}, A_{2} → B_{1}, B_{1} → A_{2}, B_{2} → A_{1}. The departing electron has A_{2} symmetry (*p*-type photodetachment) for the remaining transitions: A_{1} → B_{1}, A_{2} → B_{2}, B_{1} → A_{1}, B_{2} → A_{2}. Although we specify here only rotational state representations, the selection rules were derived by taking into account the total states of the molecules, $\Gamma NH2$ and $\Gamma NH2\u2212$.

N . | K_{a} − K_{c}
. | Γ_{rot}
. | N . | K_{a} − K_{c}
. | $\Gamma rot\u2032$ . | hν − EA (cm^{−1})
. | H . |
---|---|---|---|---|---|---|---|

$NH2\u2212$ . | NH_{2}
. | . | |||||

0 | 0 | A1 | 1 | 1 | B2 | 36.65 | 2 |

1 | −1 | B1 | 1 | 0 | A2 | 10.68 | 3 |

1 | −1 | B1 | 2 | 0 | A2 | 62.49 | 3 |

1 | 0 | A2 | 1 | −1 | B1 | −10.05 | 3 |

1 | 0 | A2 | 2 | 1 | B1 | 84.73 | 3 |

1 | 1 | B2 | 0 | 0 | A1 | −36.12 | 2 |

1 | 1 | B2 | 2 | −2 | A1 | 25.98 | 0.491 |

1 | 1 | B2 | 2 | 2 | A1 | 81.05 | 3.509 |

2 | −2 | A1 | 1 | 1 | B2 | −25.49 | 0.491 |

2 | −2 | A1 | 2 | −1 | B2 | 7.2 | 5.848 |

2 | −2 | A1 | 3 | −1 | B2 | 137.44 | 3.636 |

2 | −2 | A1 | 3 | 3 | B2 | 137.44 | 0.025 |

2 | −1 | B2 | 2 | −2 | A1 | −6.5 | 5.848 |

2 | −1 | B2 | 2 | 2 | A1 | 48.57 | 0.819 |

2 | −1 | B2 | 3 | 0 | A1 | 110.67 | 3.333 |

2 | 0 | A2 | 1 | −1 | B1 | −62.31 | 3 |

2 | 0 | A2 | 2 | 1 | B1 | 32.46 | 1.667 |

2 | 0 | A2 | 3 | −3 | B1 | 69.57 | 0.564 |

2 | 0 | A2 | 3 | 1 | B1 | 69.57 | 4.769 |

2 | 1 | B1 | 1 | 0 | A2 | −81.52 | 3 |

2 | 1 | B1 | 2 | 0 | A2 | −29.72 | 1.667 |

2 | 1 | B1 | 3 | −2 | A2 | 71.84 | 0.152 |

2 | 1 | B1 | 3 | 2 | A2 | 71.84 | 5.181 |

2 | 2 | A1 | 1 | 1 | B2 | −78.18 | 3.509 |

2 | 2 | A1 | 2 | −1 | B2 | −45.49 | 0.819 |

2 | 2 | A1 | 3 | −1 | B2 | 84.75 | 0.421 |

2 | 2 | A1 | 3 | 3 | B2 | 84.75 | 5.251 |

3 | −3 | B1 | 2 | 0 | A2 | −68.37 | 0.564 |

3 | −3 | B1 | 3 | −2 | A2 | 33.19 | 9.483 |

3 | −3 | B1 | 3 | 2 | A2 | 33.19 | 0.029 |

3 | −2 | A2 | 2 | 1 | B1 | −66.1 | 0.152 |

3 | −2 | A2 | 3 | −3 | B1 | −28.98 | 9.483 |

3 | −2 | A2 | 3 | 1 | B1 | −28.98 | 0.751 |

3 | −1 | B2 | 2 | −2 | A1 | −134.73 | 3.636 |

3 | −1 | B2 | 2 | 2 | A1 | −79.66 | 0.421 |

3 | −1 | B2 | 3 | 0 | A1 | −17.56 | 3.400 |

3 | 0 | A1 | 2 | −1 | B2 | −107.62 | 3.333 |

3 | 0 | A1 | 3 | −1 | B2 | 22.62 | 3.400 |

3 | 0 | A1 | 3 | 3 | B2 | 22.62 | 1.267 |

3 | 1 | B1 | 2 | 0 | A2 | −68.37 | 4.769 |

3 | 1 | B1 | 3 | −2 | A2 | 33.19 | 0.751 |

3 | 1 | B1 | 3 | 2 | A2 | 33.19 | 1.404 |

3 | 2 | A2 | 2 | 1 | B1 | −66.1 | 5.181 |

3 | 2 | A2 | 3 | −3 | B1 | −28.98 | 0.029 |

3 | 2 | A2 | 3 | 1 | B1 | −28.98 | 1.404 |

3 | 3 | B2 | 2 | −2 | A1 | −134.73 | 0.025 |

3 | 3 | B2 | 2 | 2 | A1 | −79.66 | 5.251 |

3 | 3 | B2 | 3 | 0 | A1 | −17.5592 | 1.267 |

N . | K_{a} − K_{c}
. | Γ_{rot}
. | N . | K_{a} − K_{c}
. | $\Gamma rot\u2032$ . | hν − EA (cm^{−1})
. | H . |
---|---|---|---|---|---|---|---|

$NH2\u2212$ . | NH_{2}
. | . | |||||

0 | 0 | A1 | 1 | 1 | B2 | 36.65 | 2 |

1 | −1 | B1 | 1 | 0 | A2 | 10.68 | 3 |

1 | −1 | B1 | 2 | 0 | A2 | 62.49 | 3 |

1 | 0 | A2 | 1 | −1 | B1 | −10.05 | 3 |

1 | 0 | A2 | 2 | 1 | B1 | 84.73 | 3 |

1 | 1 | B2 | 0 | 0 | A1 | −36.12 | 2 |

1 | 1 | B2 | 2 | −2 | A1 | 25.98 | 0.491 |

1 | 1 | B2 | 2 | 2 | A1 | 81.05 | 3.509 |

2 | −2 | A1 | 1 | 1 | B2 | −25.49 | 0.491 |

2 | −2 | A1 | 2 | −1 | B2 | 7.2 | 5.848 |

2 | −2 | A1 | 3 | −1 | B2 | 137.44 | 3.636 |

2 | −2 | A1 | 3 | 3 | B2 | 137.44 | 0.025 |

2 | −1 | B2 | 2 | −2 | A1 | −6.5 | 5.848 |

2 | −1 | B2 | 2 | 2 | A1 | 48.57 | 0.819 |

2 | −1 | B2 | 3 | 0 | A1 | 110.67 | 3.333 |

2 | 0 | A2 | 1 | −1 | B1 | −62.31 | 3 |

2 | 0 | A2 | 2 | 1 | B1 | 32.46 | 1.667 |

2 | 0 | A2 | 3 | −3 | B1 | 69.57 | 0.564 |

2 | 0 | A2 | 3 | 1 | B1 | 69.57 | 4.769 |

2 | 1 | B1 | 1 | 0 | A2 | −81.52 | 3 |

2 | 1 | B1 | 2 | 0 | A2 | −29.72 | 1.667 |

2 | 1 | B1 | 3 | −2 | A2 | 71.84 | 0.152 |

2 | 1 | B1 | 3 | 2 | A2 | 71.84 | 5.181 |

2 | 2 | A1 | 1 | 1 | B2 | −78.18 | 3.509 |

2 | 2 | A1 | 2 | −1 | B2 | −45.49 | 0.819 |

2 | 2 | A1 | 3 | −1 | B2 | 84.75 | 0.421 |

2 | 2 | A1 | 3 | 3 | B2 | 84.75 | 5.251 |

3 | −3 | B1 | 2 | 0 | A2 | −68.37 | 0.564 |

3 | −3 | B1 | 3 | −2 | A2 | 33.19 | 9.483 |

3 | −3 | B1 | 3 | 2 | A2 | 33.19 | 0.029 |

3 | −2 | A2 | 2 | 1 | B1 | −66.1 | 0.152 |

3 | −2 | A2 | 3 | −3 | B1 | −28.98 | 9.483 |

3 | −2 | A2 | 3 | 1 | B1 | −28.98 | 0.751 |

3 | −1 | B2 | 2 | −2 | A1 | −134.73 | 3.636 |

3 | −1 | B2 | 2 | 2 | A1 | −79.66 | 0.421 |

3 | −1 | B2 | 3 | 0 | A1 | −17.56 | 3.400 |

3 | 0 | A1 | 2 | −1 | B2 | −107.62 | 3.333 |

3 | 0 | A1 | 3 | −1 | B2 | 22.62 | 3.400 |

3 | 0 | A1 | 3 | 3 | B2 | 22.62 | 1.267 |

3 | 1 | B1 | 2 | 0 | A2 | −68.37 | 4.769 |

3 | 1 | B1 | 3 | −2 | A2 | 33.19 | 0.751 |

3 | 1 | B1 | 3 | 2 | A2 | 33.19 | 1.404 |

3 | 2 | A2 | 2 | 1 | B1 | −66.1 | 5.181 |

3 | 2 | A2 | 3 | −3 | B1 | −28.98 | 0.029 |

3 | 2 | A2 | 3 | 1 | B1 | −28.98 | 1.404 |

3 | 3 | B2 | 2 | −2 | A1 | −134.73 | 0.025 |

3 | 3 | B2 | 2 | 2 | A1 | −79.66 | 5.251 |

3 | 3 | B2 | 3 | 0 | A1 | −17.5592 | 1.267 |

The angular momentum has to be conserved, which implies restrictions on allowed Δ*J* values. The photon has unit angular momentum and the detached electron carries at least the spin angular moment of 1/2 (“*s*-wave” photodetachment). This requires Δ*J* = ±1/2, ±3/2 (Δ*N* = 0, ±1) for an *s*-wave electron (A_{1}), and Δ*J* = ±1/2, ±3/2, ±5/2 (Δ*N* = 0, ±1, ±2) for a *p*-wave electron (A_{2}).

The presence of two nuclear spin states only allows for transitions between ortho or between para states (ortho to para transitions are forbidden). This automatically satisfies selection rules between unresolved fine and hyperfine sub-levels.

## IV. RESULTS AND DISCUSSION

### A. Photodetachment spectra

Two measured photodetachment spectra are plotted in Fig. 2 as a function of the photon energy of the detachment laser. The data were obtained at trap temperatures of 9(1) K and 28(2) K. Qualitatively, differences in the two spectra directly reflect different rotational temperatures of the ion ensembles, as we found earlier for near-threshold detachment of OH^{−}.^{8} Such differences are seen clearly near 6230 cm^{−1} and with less statistical significance near 6250 cm^{−1}. This enhancement of the photodetachment cross section for higher rotational excitation was recently used to study the fundamental rotational transitions of $NH2\u2212$.^{4}

Besides the overall increase with photon energy and despite the finite signal-to-noise ratio, a few steps may be observed, where the cross section grows more rapidly. Such steps indicate the opening of *s*-wave photodetachment channels because only for these transitions, the cross section grows with excess energy to the power of 1/2 or less. We can clearly assign one *s*-wave threshold, which appears at 6235(1) cm^{−1}. This is marked by the arrow and shown in the inset in Fig. 2. Based on the signal-to-noise ratio, we estimate the accuracy of the threshold assignment to be about 1 cm^{−1}. Further steps may be visible at 20 cm^{−1}, 30 cm^{−1}, and 60 cm^{−1} relative to the observed step. Note that a direct assignment of these steps to rotational transitions is not possible due to the large uncertainty of the current electron affinity of NH_{2} of 40 cm^{−1}. This is further analyzed below.

### B. Photodetachment model

Here we model the dependence of the photodetachment cross section on photon energy at different rotational temperatures in order to extract the electron affinity of NH_{2}. The general treatment of the photodetachment process as a reaction between two particles was given by Wigner.^{28} In the case of the photodetachment of an electron from a negative atomic ion, the threshold behavior is expressed as $\sigma \u221d(h\nu \u2212Eth)l+1/2$, where *l* is the angular momentum of the detached electron, *hν* is the photon energy, and *E*_{th} is the threshold energy. This behavior is known to be different for a molecule possessing a permanent dipole moment.^{29} The interaction between the departing electron and the molecular dipole leads to the appearance of a long range dipole potential, which depends on the orientation of the dipole. As a result, the exponent of the threshold law *p*_{N} deviates from *l* + 1/2 and can be different for different rotational states $\sigma \u221d(h\nu \u2212Eth)pN$.

The general expression for the photodetachment cross section is given by the sum over all allowed rotational state-to-state transitions,

where *P*_{ψ} is the Boltzmann factor that gives the population of $NH2\u2212$ rotational state, *I*_{ψ→ψ′} is the transition strength for the channel *ψ* → *ψ*′, *hν* refers to the photon energy, *E*_{ψ→ψ′}, the threshold energy for this transition, and Θ(*hν* − *E*_{ψ→ψ′}) is the Heaviside step function. *p*_{ψ→ψ′} is the exponent of the Wigner threshold law that depends on the symmetry of the detached electron. The rotational wave functions *ψ* and *ψ*′ of $NH2\u2212$ and NH_{2} can be expressed as

The transition strengths *I*_{ψ→ψ′} are proportional to the corresponding Hönl-London factors *H*_{Nτ→N′τ′}. These were calculated using Eqs. (6.117) and (6.123) in Ref. 30. Since Eq. (4) distinguishes states of a different quantum number *M*, but the Hönl London factors are obtained by summing over all possible *M* and *M*′ states, we have to use *I* ∝ *H*_{Nτ→N′τ′}/(2*N* + 1)/(2*N*′ + 1).

Near threshold transitions leading to electrons with A_{1} character (*s*-type transitions) are much stronger than those for A_{2}-type electrons (*p*-type transitions) because of the different exponents in the threshold law. Only far from threshold, the stronger increase of the cross section with excess energy for *p*-type transitions can make it dominate over *s*-type photodetachment. There is no hint of such a behavior in the experimental data (see Fig. 2). In fact, the photodetachment spectrum flattens at higher photon energy. Based on this, we neglect *p*-type transitions in our model for near-threshold detachment.

The exact value of the *p*_{ψ→ψ′} exponent for the *s*-wave detachment (to be referred as *p*_{s}) depends on the permanent dipole moment of the NH_{2} molecule, and on the initial and final rotational states of $NH2\u2212$ and NH_{2}, respectively. The *p*_{s}-factor was calculated for a linear molecule with a dipole moment equivalent to the one of bent NH_{2} at the equilibrium geometry. It was found to be either 0.12 or 0.5 depending on the rotational state. The correct *p*_{s} for NH_{2} has to fall between those values because its rotational state can be described by a superposition of the rotational states of a linear molecule.

It is relatively easy to evaluate the exponent for linear molecules using the known expression for the electron-dipole interaction in the molecular reference frame and transforming the interaction potential into the lab frame, where the polarization direction of the photodetachment laser is defined. The result of this calculation for an interval of different possible dipole moments is shown in Fig. 3. In the calculation, it was assumed that the anion is a closed-shell molecule, such as OH^{−} or C_{n}N^{−} (*n* = 1, 3, 5).

The calculation for the $C2v$ molecules is more involved and was not performed for this study. In fact, the rotational states of the $C2v$ molecules (such as $NH2\u2212$ and NH_{2}) are superpositions of the ones of linear molecules. It means that for photodetachment transitions to the lowest rotational states of the NH_{2} molecule, the dominant term in the photodetachment cross section is determined by the lowest value *p*_{s} relevant to a particular superposition of linear-molecule states. The dipole moment of NH_{2} is about 0.8 *ea*_{0}.^{32} From Fig. 3, the *p*_{s} exponent can range from 0.12 for transitions to the *J* = 3/2, Ω = 3/2 rotational states to almost 0.5 for the *J* = 3/2, Ω = 1/2 states.

In our model, we neglect the rotational state-dependence of the exponent, as we cannot resolve the fine structure of the *J* = 1/2 and *J* = 3/2 states, and use an average value of *p*_{s} = 0.2. This yields the relative detachment cross section

Here we have summed over the transitions that are not resolved in the experiment, i. e., over the states with different *J*, *F*_{1}, *F*, *M*, which yields the degeneracies *g*, *g*′,

The partition function is given by

The result of the model is shown in Fig. 4, separately for para and ortho $NH2\u2212$ and for different rotational temperatures of the anions. The threshold energies were calculated based on spectroscopic constants in Refs. 17 and 19. Our model includes all the transitions listed in Table I. At 40 K, the total population in the rotational states with *N* ≥ 3 does not exceed 4% for para and 6% for ortho. Therefore, to adequately describe our experimental data up to 40 K, it is safe to consider only rotational levels up to *N*′ = 2 of the ion and up to *N* = 3 of the neutral. Transitions of the type *N*′ → *N* = 2 → 3 already lead to excess energies of *hν* − *EA* ∼ 100 cm^{−1}, which are quite far above threshold.

Depending on the ortho-to-para ratio of $NH2\u2212$ in the experiment, the steps for ortho-transitions may appear more or less prominent with respect to the para-steps (see Fig. 4). The most prominent steps appear at about −36 cm^{−1} (1_{1} → 0_{0}), −10 cm^{−1} (1_{0} → 1_{−1}), 10 cm^{−1} (1_{−1} → 1_{0}), 36 cm^{−1} (0_{0} → 1_{1}). The transition 0_{0} → 0_{0}, corresponding to the electron affinity, is forbidden by the selection rules. Instead, we can consider transition 0_{0} → 1_{1} with *hν* − *EA* ≈ 37 cm^{−1}. Above this energy, all anionic rotational states are involved in photodetachment and a smoother increase of the cross section is observed.

### C. Assignment of photodetachment transitions

In order to assign specific detachment transitions, listed in Table I, to the measured spectra, we compare the measured spectrum (Fig. 2) with the modeled cross section (Fig. 4). The measured photodetachment rate shows an overall increase by a factor of about 15-30 over the energy range of 150 cm^{−1}. This allows us to exclude the energy range with *hν* − *EA* < −30 cm^{−1} from the analysis (note the log scale in Fig. 4). We therefore restrict the assignment of the most prominent step appearing in the experimental spectrum to one of three pronounced transitions, 1_{−1} → 1_{0} (*hν* − *EA* ≈ 11 cm^{−1}), 1_{0} → 1_{−1} (*hν* − *EA* ≈ −10 cm^{−1}), 0_{0} → 1_{1} (*hν* − *EA* ≈ 37 cm^{−1}).

Further evidence for the assignment is taken from the effect of trap temperature on the photodetachment rate. At higher trap temperatures, increased photodetachment rates below and above the step are observed. Qualitatively the experimental curve is quite adequately reproduced by the para cross section. There is an increase in the photodetachment rates just below and above the step 1_{0} → 1_{−1} for higher trap temperatures, as was observed in the experiment. This would lead to an assigned electron affinity of about 6245 cm^{−1}. However, the ortho step 1_{−1} → 1_{0} also needs to be considered, as for a mixture of ortho and para states in the experiment, the measured rate would still show an increase on both sides of this step. Thus, alternatively an electron affinity of about 6224 cm^{−1} is to be considered. In the simulation, the third transition (0_{0} → 1_{1}) does not show a cross section change with temperature similar to the experiment and is excluded.

To resolve which of the two transitions is responsible for the measured step near 6236 cm^{−1}, we compare the measured spectra in Fig. 2 with the result of the relative cross sections presented in Eq. (6). We have carried out the simulations for the fixed ortho-to-para ratios, a value of 3, which is expected from the ion source. The rotational temperature and an overall scaling factor are treated as fit parameters with boundaries of ±10 K and ±5 around their respective initial set value. The electron affinity of NH_{2} was either chosen to be 6224 cm^{−1} or 6245 cm^{−1} with an allowed variation of ±1 cm^{−1}. The parameters were then optimized according to the *χ*^{2}-method.

With an electron affinity of 6224 cm^{−1}, a fit in very good agreement is achieved for both trap temperatures, which is shown in Fig. 5 as the solid line. We estimate the accuracy of this electron affinity to be better than 1 cm^{−1}. The spectrum measured for 9(1) K trap temperature is reproduced best with a simulation for a rotational temperature of 20(1) K. This indicates that rotational thermalization is not complete, an effect that was seen in previous measurements for OH^{−}.^{33} The spectrum for 28(2) K is found to agree well with a simulation with rotational temperature of 27(1) K. The alternative value of 6245 cm^{−1} can only be made to fit the measured spectrum to some degree for an unreasonably high rotational temperature of 40 K and an about sixfold higher *χ*^{2} value. Furthermore, this fit (plotted as the dashed line in the upper panel of Fig. 5) does not reproduce the shape of the observed step at 9 K. This value for the electron affinity is therefore excluded.

## V. SUMMARY

In this work, we have presented the photodetachment spectrum of $NH2\u2212$ and a model for all allowed detachment transitions covering the wavelength range from 6190 cm^{−1} to 6355 cm^{−1}. A clear threshold has been observed in the spectrum that is assigned to a rotational state-to-state transition using a refined model of the relative photodetachment cross section. The rotational temperature of the ions was found to be well described by the trap temperature at 28 K. At 9 K trap temperature, a rotational temperature of 20 K was needed to model the spectrum. This shows that at the lowest trap temperatures, the rotational temperature is elevated, an effect that was found previously for OH^{−} and has not been explained, yet.^{33} Based on the cross section model, we can assign the most visible threshold to the ortho transition 1_{−1} → 1_{0} and find the electron affinity of NH_{2} to be 6224 ± 1 cm^{−1} (771.7 ± 0.1 meV), with a forty-fold improvement in accuracy based on the previous measurement.^{11}

The near threshold detachment spectroscopy has already allowed us to perform high resolution terahertz spectroscopy of the lowest two rotational transitions in $NH2\u2212$.^{4} First evidence has been obtained on the influence of the photodetachment process on the ortho-to-para ratio of the trapped ions.^{14} In the future, the near-threshold detachment spectroscopy can be used to investigate rotational state-dependent inelastic and reactive collisions of para and ortho $NH2\u2212$ with hydrogen and helium in the cryogenic environment of the ion trap. Furthermore, this approach may be extended to other polyatomic molecular anions, such as H_{3}O^{−} or $H3O2\u2212$.

## ACKNOWLEDGMENTS

This work has been supported by the European Research Council under ERC Grant Agreement No. 279898; by the Austrian Science Fund (FWF) through Project No. P29558-N36; and through the Doctoral Programme Atoms, Light, and Molecules, Project No. W1259-N27. V.K. acknowledges support from the National Science Foundation Grant No. PHY-15-06391 and the Austrian-American Educational Commission.

## REFERENCES

_{2}- under photodetachment in cold ion traps