Electron scattering cross sections from NH3: a comprehensive study based on R-matrix method

In this study, a comprehensive cross-sectional computation on the electron scattering from NH3 was performed using the frame of the R-matrix method. Cross section sets for total elastic, differential, momentum transfer, total ionization, and electronic excitation are presented. The electron-impact dissociation of NH3 to NH 2+ H and NH+H2 was considered by summing up the cross sections for the molecular dissociative excitations to the correlated dissociation channels. The reported cross sections for vibrational excitation of NH3 remain controversial and are unable to distinguish the N–H symmetric and asymmetric stretching modes of close frequencies. Here, the excitation cross section for each vibrational mode of NH3 was computed by applying the theoretical approach combining the fixed-nuclei R-matrix method, normal mode approximation and the vibrational frame transformation. The uncertainties of the cross section sets computed in this study were estimated for convenience of use in the plasma modeling.


Introduction
Ammonia (NH 3 ) could be used as an energy carrier owing to its well-developed synthesis technology, safe storage, and * Authors to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. transport characteristics [1][2][3]. Therefore, burning NH 3 as a carbon-free fuel for future power generation to reduce greenhouse-gas emissions has gained special attention in recent years [4,5]. However, the 'drop-in' substitution of traditional fossil fuels by NH 3 presents a significant challenge due to its poor combustion qualities [6]. Recently, impressive progress was achieved experimentally to enhance the combustion performance of NH 3 by utilizing cold plasma. The overall efficiency of the combustion was improved and the ignition of the flame speeded up [7,8]. Nevertheless, the mechanism underlying plasma-assisted combustion is still unknown. Cold plasma is known to generate a large number of energetic electrons driving the plasma-based technologies. One of the fundamental processes in this context is the collision between these electrons and NH 3 , which plays a crucial role in understanding the dynamics behind the plasma-assisted combustion. Accordingly, it is necessary to obtain the electron-impact cross section sets for NH 3 to construct the plasma model of the combustion.
Cross sections for different electron-NH 3 scattering processes have been explored in experiment and theory during the past decades. The total cross section (TCS) for NH 3 was first measured by Brüche [9] in the energy range of 1-20 eV. Later, several TCSs up to medium or high electron energies were measured using different equipment [10][11][12][13][14][15]. In 1992, Alle et al [16] conducted crossed electron-molecular beam experiments on total elastic, differential (DCS), and momentum transfer cross section (MTCS) of electron-NH 3 collisions in the energy range of 2-30 eV. Homem et al [17] recently reported the measurements extending these cross sections to medium and high scattering energies in conjunction with a corresponding theoretical validation. From a theoretical perspective, calculations have been consistently performed to produce the cross sections for elastic scatterings until very recently [18][19][20][21][22][23][24][25][26][27]. For instance, in 1991, Gianturco [19] computed the electron-impact total elastic and DCS of NH 3 using a parameter-free model. Recently, Akter et al [27] employed the independent atom model (IAM) with screening correction to calculate those cross sections as well as MTCS and total ionization cross sections within a broad energy range of 1 to 1 M eV.
Cross sections for inelastic scatterings including excitation, ionization and dissociation are also considered, but are more challenging to obtain. Regarding the vibrational excitation of NH 3 , there is only one set of cross sections evaluated by the Born method reported by Itikawa [28] for the infrared (IR) active mode of vibration. It should be noted that all of the vibrational modes of NH 3 are IR active. Another unpublished data source given by Hayashi [29] could be found in the LXCat database [30]. However, the two data sets are of great difference with more than one order of magnitude. In addition, they are both unable to distinguish the N-H symmetric and asymmetric stretching modes of close frequencies. To the best of our knowledge, no advanced theoretical study on the vibrational excitation of NH 3 was documented due to the multidimensional nature of the problem. The electronic excitation cross sections of NH 3 were only calculated by Munjal and Baluja [24] using the R-matrix method but no experimental measurement is available. The ionization cross section is the most important parameter for modeling the discharge of plasma. It was measured and calculated in many experimental and theoretical studies for NH 3 [31][32][33][34]. Electrons could induce dissociation through dissociative excitation, dissociative ionization and dissociative electron attachment (DEA). Hamilton et al [35] estimated the cross sections for dissociative ionization of NH 3 (see figure 1 of [35]) to test the proposed semi-empirical method based on the binary encounter Bethe (BEB) theory [36] in 2017. Excellent agreement with the experimental measurements [32] were found for the dominant fragments NH + 3 and NH + 2 . Cross sections for neutral dissociation of NF x (x = 1,2,3) by electron impact were also estimated [35], which provided an idea to cope with the unreported electron-impact dissociation of NH 3 . The DEA of NH 3 had been experimentally investigated [37,38] but the reported absolute cross sections exhibit considerable differences. Theoretical and further experimental studies are required to verify the obtained DEA cross sections. A review of all the recent progress for the cross sections of electron collisions with NH 3 can be found in the compilations of Karwasz et al [39] in 2001 and Itikawa [40] in 2017. For the convenience of reference, a summary of the investigations reporting different types of cross sections for electron-NH 3 collisional processes is given chronologically in table A1.
Despite the extensive studies on electron-impact cross sections of NH 3 which have been undertaken, the plasma modeling still suffered from the following limitations (1) the available cross sections of electron-NH 3 scattering are fragmented and inconsistent due to the different setups in experiments and varying degrees of approximations, (2) cross sections of some electron scattering processes, such as the neutral dissociation of NH 3 , are absent, and (3) electron-impact cross sections for the fragments produced in the NH 3 plasma are rarely documented. In order to model the NH 3 -containing plasma kinetics, a comprehensive theoretical study for electron-NH 3 scattering through an unified framework is imperative [41]. Therefore, we used the frame of the R-matrix method [42] to produce directly the total elastic, DCSs, MTCSs, and electronic excitation of NH 3 by electron impact in the present study. The total ionization cross sections are evaluated by the BEB method embedded in the Quantemol electron collisions (Quantemol-EC) interface [43]. With the assumption that electronic excitations to dissociative states or those having curvecrossing with the dissociative states lead to dissociation [35], we estimated the cross sections for electron-impact dissociation of NH 3 for the first time. In addition, for the inconclusive cross sections for vibrational excitation of NH 3 , we described the excitation from the ground vibrational state to its first vibrational excited state using a theoretical approach based on the fixed-nuclei R-matrix method, normal mode approximation and the vibrational frame transformation [44,45]. The calculated cross section data will provide a new theoretical reference.
The following section 2 of this article demonstrates the theoretical approach used for the scattering cross section calculation. The calculation details are described in section 3. In section 4, the calculated electron-impact cross section sets of NH 3 are presented and discussed. The uncertainty estimations of the present computed cross section sets are given in section 5. Finally, we conclude the study in section 6.

Theoretical approach
The R-matrix scattering calculation was performed by running UKRmol+ [46] in Quantemol-EC suite [43,47]. Here we overview briefly the theoretical approach. A detailed description of the method and its technical implementation can be found in a review article [42] and the references therein.
In the R-matrix approach, the configuration space is separated into the inner region by a sphere with radius 'a' around the molecular center-of-mass and the outer region of the rest part. In the inner region, the incident electron is indistinguishable from the N electrons of the target molecule, so that all the short-range effects among the (N + 1)-electron system such as exchange and correlations must be considered. The internal wavefunction is typically constructed through the close-coupling (CC) approximation and specified as: where Φ N i is the wavefunction of the ith electronic state of the N-electron target. u ij denotes the continuum orbitals introduced to describe the scattering electron that associates the ith target state of a particular symmetry. x i specifies the space-spin coordinates of the electrons. The antisymmetrization operator A is used for the (N+1)-electron system to obey the Pauli principle. Therefore, the first summation in the above expression generates the configurations of the target plus continuum through running over all the retained target's electronic states. The short-range effects of polarization arising from the occupation of the penetrated electron in the virtual orbitals of the target are represented by the second summation through running over the χ N+1 i configurations. a ijk and b ik are the variational coefficients and could be obtained by diagonalizing the N + 1 electronic Hamiltonian.
The problem in the outer region is greatly simplified as the exchange interactions with any of the target electrons are neglected. Only the long-range multipolar potential remains and imposes on the incident electron moving around the target. The solutions of the radial wavefunction of the scattering electron to those asymptotic channels are given by Gailitis expansion techniques [48]. Treatment of the inner region builds the Rmatrix as a function of the scattering energy at the boundary of the two regions. It is then propagated outwards to the outer region until matching with the asymptotic channels to finally extract the K-matrix. All the scattering observables of interest can be derived from the K-matrix, such as the eigenphase sum and the scattering matrix (S-matrix). The position and width of the resonances can be determined by fitting the eigenphase sums with the Breit-Wigner formula. The S-matrix is defined in terms of the K-matrix asŜ =1 +iK 1−iK based on which cross sections are calculated. It should be noted that the S-matrix discussed above is calculated in the fixed-nuclei approximation, namely the vibrational degrees of freedom are neglected in the R-matrix model.
In order to compute the cross sections for vibrational excitation, we performed the vibrational frame transformation integrating over a set of nuclear geometries to obtain the S-matrix describing the picture of molecular vibrations by: where the coordinates of normal vibrational mode i of the target are collectively represented by q i . η νi and η ν ′ i correspond to the wavefunctions of the initial and final vibrational state ν i and ν ′ i of mode i. S l ′ λ ′ ,lλ is the fixed-nuclei Smatrix element with the electron angular momentum and its projection on the molecular axis lλ and l ′ λ ′ as the initial and exit channels, respectively. The physical matrix constructed by equation (2) describes the scattering amplitude from the vibrational state ν i to ν ′ i . As previously discussed [45,49], this formalism is in principle suitable only if the fixed-nuclei S l ′ λ ′ ,lλ is smooth regarding to the scattering energy. If there is no resonance in the energy range we are mostly interested in, the vibrational excitation cross section of mode i can be written as: where m is the reduced mass of the electron-molecule scattering system and E el is the scattering electron energy.

Target representation
At the equilibrium position, the polar covalent NH 3 molecule belongs to the C 3v point group with a permanent dipole moment of 1.47 D [50]. The configuration of its ground electronic state is: For the purpose of a sufficient description of the target in the scattering calculations, we conducted several tests using different models to calculate the geometric and electric properties of the target using the MOLPRO package [51]. We started by optimizing the equilibrium position and calculating the vibrational frequencies. The dipole moment of the molecule is also essential as it represents the major contribution to the interaction between the projectile electron and the target. All the target properties calculated with different basis sets and complete active space (CAS) are tabulated in table 1 and compared with the experimental [50] and theoretical data [52].
According to table 1, the calculation with CAS(6,6) and cc-pVTZ basis set shows a good agreement with the experimental data for both the equilibrium geometry and vibrational frequencies. This model also displays a sufficient representation of the dipole moment with a slightly larger value than found experimentally. Therefore, we concluded that the model of CAS(6,6) with cc-pVTZ basis set adequately represents the target molecule to be used in the following scattering calculation. The displacement matrix output by the selected model is then used to generate normal coordinate input for fixed-nuclei scattering calculations.  Comparison between the calculated equilibrium structure, vibrational frequencies denoted by ω 1 , ω 2 , ω 3 and ω 4 for N-H wagging, H-N-H scissoring, N-H symmetric stretch and N-H asymmetric stretch modes respectively, the dipole moment of NH 3 and experimental [50] and theoretical data [52]. The number combination such as 'CAS(8,7)' (elsewhere in the text below) indicates that eight electrons are freely distributed in seven active orbitals.

CAS
Basis set Dipole moment Exp. [ Table 3. Vertical excitation energies (in unit of eV) of NH 3 using different models at the equilibrium nuclear geometry.

CAS Basis set Vertical energy
State 6.38 [53]

Scattering model
In the R-matrix scattering calculations, we set a sphere of radius a = 12 a 0 for the inner region. The Gaussian-type orbitals are used for the expanded partial waves of the continuum up to l ≤ 4. Wavefunctions of the target's electronic states Φ N i are constructed using the CAS configuration interaction (CAS-CI) method using the CASSCF orbitals from MOLPRO. 17 generated target states below the cutoff energy 15 eV are included in the CC calculation. C s symmetry, a subgroup of the natural C 3v of NH 3 , was used since both the MOLPRO and R-matrix codes only accept the Abelian point group as valid. The correlation between these two point groups was given for convenience in table 2.
Vertical excitation energy is important to accurately represent the target in fixed-nuclei scattering calculations. Table 3 compares the quantities of NH 3 calculated using different models with the previously published data [53]. As illustrated in table 3, the CAS(6,6) with the cc-pVTZ basis set correctly yields comparable vertical transition energies to experimental values. It is thus appropriate to be used in the scattering calculation. The CAS model can be expressed as Table 4 compares the position and width of the two resonances calculated using the selected scattering model with the previously reported data. The two resonances located at approximately 5.5 and 10.5 eV are validated by experiments for DEA of NH 3 . They are formed by capturing the incident electron into 4a 1 orbital by the parent excited states of NH 3 with the configurations given in the table, and thus classified as Feshbach resonances of symmetries 2 A 1 and 2 E, respectively. As illustrated in the table, the calculated position of the lower resonance at approximately 5.5 eV agrees well with all the experimental values. The position of the higher resonance is 0.4 eV larger than the experimental observations. The currently calculated widths of the 5.5 eV and 10.5 eV resonances are in good agreement with the theoretical values reported by Rescigno et al [60]. By contrast, the position for the lower resonance determined by Rescigno et al [60] locates at 4.19 eV, showing a significant difference of over 1 eV to the present result. They stated that the resonance position was underestimated due to the use of the molecular orbitals (MOs) optimized for the parent excited states [60]. These comparable results with the available theoretical and experimental values indicate the accuracy of our model in describing the electron-NH 3 scattering process.
Note that the equilibrium nuclear geometry (q i = 0) was used to obtain all of the above discussed target properties. However, it is important to know for the scattering calculations how well the harmonic vibration of the target molecule could be reproduced by our model. Therefore, we plotted the potential energy curves (PECs) of NH 3 computed with the R-matrix code for each normal mode and compared with the harmonic potentials. Figures 2(a) and (c) show that even though our calculated harmonic frequencies of N-H wagging and symmetric stretch are larger than the experimental values, they agree satisfactorily well to support that our model is feasible to be chosen to represent the vibrating NH 3 . Figures 2(b) and (d) show very good agreements with harmonic potentials for other vibrational modes. NH 3 is a textbook example of an anharmonic system owing to the character of double minimum potential along the N-H wagging mode, i.e. the umbrella mode, as shown in figure 1 in [62]. The vibrational states of NH 3 in umbrella vibration are thus tunneling splitting with the forms of asymmetric and symmetric wavefunctions. The widths of the split ground (ν = 0) and first excited (ν = 1) vibrational states are only 0.8 and 35.8 cm −1 [62]. Therefore, we restricted the considered vibrational transition in the present calculation between ν = 0 to ν = 1 to reduce the effect induced by the tunneling splitting of the anharmonic system in the normal mode approximation. While it is currently difficult to improve on this approximation, the present study on the umbrella mode provides an approximate evaluation of its vibrational excitation cross sections.
To avoid the limitation of repeated scattering calculation for many normal coordinates, a Gauss-Legendre quadrature with ten points picked up in the PECs was applied [63]. This numerical technique was used to compute a definite integral through summing up the weighted function values over the specified point of nuclear geometry. We performed R-matrix scattering calculations for ten Gauss nodes to obtain the fixed-nuclei S-matrix elements. With the extensively tabulated weights, the vibrational S-matrix element, S ν ′ i l ′ λ ′ ,νilλ , defined by equation (2) can be computed numerically and the cross section for vibrational excitation is further obtained by equation (3).

Total elastic cross section
In figure 3, we present the total elastic cross sections calculated with Quantemol-EC and compare with the results of Gianturco [19] and Alle et al [16] recommended by Itikawa [40], as well as the recently reported theoretical calculations [17,25]. The obtained cross section presents a divergent characteristic at low scattering energies for a dipolar molecule. It drops rapidly to the minimum at approximately 2.5 eV followed by a broad resonant structure centering at around 10 eV, demonstrating a trend in accordance with the previously reported experimental observations and theoretical predictions. Below 2.5 eV, our result lies on the top of the  [16], red full circle; theoretical calculation by Gianturco [19], green dashed curve; Homem et al [17], blue long dashed curve; and Kaur et al [25], violet dotted-dash curve.
Gianturco [19] curve, but it is much higher lying above the measurements of Alle et al [16] due to the huge uncertainties in the experiment caused by the extrapolations of their low-energy DCSs to the forward and backward angles. For higher scattering energies up to 20 eV, our result agrees well with the experimental measurements, but it is underestimated compared with the Gianturco [19] curve and the other two theoretical results. This is likely owing to the fact that the inelastic channels are explicitly included in our CC model, thus lowering the elastic scattering cross sections at higher energies.

DCS
To achieve the convergence of DCS for the dipolar target molecule in a Quantemol-EC calculation, the Born closure procedure is used to account for the effects induced by the neglected higher partial waves, i.e. l > 4. Rotational motion is introduced to remedy the divergence of the DCS at small scattering angles caused by the dipole potential. The DCS is finally computed by summing the rotationally resolved (J = 0 → J ′ = 0 ∼ 5) cross sections. The obtained Born-corrected DCSs of NH 3 by electron impact at the energies of 2 eV, 5 eV, 7.5 eV, and 10 eV are shown in figure 4.
At 2 eV, all the theoretical DCSs are somewhat overestimated compared with the experimental measurements of Alle et al [16] and our DCS curve is highest lying above the other three theoretical results [17,24,64]. This is mainly due to the overestimated dipole moment of 0.679 with atomic unit compared to 0.611 of Munjal and Baluja [24], 0.656 of Brescansin et al [64] and 0.578 of Homem et al [17]. Our Born-corrected curve is forward-peaked at small scattering angles due to the long-range nature of the electron-NH 3 dipole interactions. However, this is not verified by the experimental DCS of Alle et al [16] which only covers angle from 20 • to 130 • . This is because of a well-known defect in experiment due to the low-angular discrimination against forward elastic scattering electrons. The issue has been intensively discussed in the elastic scattering electron with water, where large uncertainties are introduced to the small-angle DCS and thus the total elastic cross section for the dipolar molecule, especially at low scattering energies [65,66]. The problem was addressed in a recent study by Kadokura et al [67] wherein the obtained elastic DCS for tiny scattering angles agrees well with the corresponding R-matrix calculation with Born corrections.
At 5 eV and 7.5 eV, our results are in excellent agreement with the experimental and theoretical results. All of them demonstrate an approximate d-type angular distribution. The only difference is that the DCS at 5 eV above 130 • is lower lying to the other theoretical results. We assume that this maybe related to the detected 5.5 eV resonance as discussed above. At 10 eV, our DCS curve is overall comparable to the other theoretical results, except that of Akter et al [27] obtained by a semi-empirical IAM which works better at energies above 30 eV. The difference is that our curve is still forward-peaked at small scattering angles with a moderate dip at around 130 • compared with that of Yuan and Zhang [21] and Mahato et al [26]. With the increasing of the scattering energies, the long-range dipolar potential becomes less relevant. Our DCSs are thus in better agreement with the experimental and theoretical results at the higher energies.

MTCS
MTCS calculated in the energy range 0.01-10 eV is shown in figure 5 and compared with the results of Gianturco [19] recommended by Itikawa [40], and with other recent theoretical results [17,24,25]. The MTCS is derived from DCS and describes the backward scattering. As we stated that the overestimated dipole moment of NH 3 results in higher DCSs at lower scattering energies within the entire angle range. Our computed MTCS is thus larger than the results of Gianturco [19]. However, this could not instantly conclude that our MTCSs are overestimated. Because the MTCS computation only includes the rotational transitions over JK = 00 → 00, 10, 20 in the work of Gianturco [19], while the DCS used for integrating the MTCS in our calculation is over JK = 00 → J ′ K ′ = 0K ′ ∼ 5K ′ summed for all K ′ . Beyond 5 eV, the experimental results are becoming reliable and allow for comparison. One could see a good agreement with all the MTCSs reported previously.

Total ionization cross section
The present total ionization cross section is computed by the BEB method [36] inserted in the Quantemol-EC suite and is shown in figure 6. The method combines the binary encounter theory describing the collisions between two free electrons, and the dipolar interaction theory of Bethe for the interaction between the incident and target electrons at high incident energies. The maximum value at around 75 eV of our curve seems slightly overestimated compared with the theoretical   resonances are smeared out by the velocity distribution of the electrons.

Electronic excitation cross section
In figure 7, we present the cross sections for the electronic excitation from the ground state X 1 A 1 of NH 3 to the four lowest excited states a 3 A 1 , A 1 A 1 , b 3 E and B 1 E. The cross section curves for excitations to triplet and singlet states behavior as predicted by the optically-forbidden and -allowed transitions according to the theory of Inokuti [68]. As shown in figures 7(a) and (b), the cross section curve for excitation to triplet states rises rapidly from the threshold, and drop monotonically after attaining the maximum. In contrast, the excitation cross sections for the singlet states increase slowly from the threshold and then vary smoothly. All the peaks found in graphs around 10.5 eV is due to the presence of a resonance, which was not detected in Munjal's work [24].

Electron-impact dissociation cross sections
The electron-impact dissociation usually specifies the neutral dissociation through dissociative excitation, i.e. electronic excitation to excited states which are dissociative or curvecrossing with those dissociative states. Therefore, the cross sections for the process can be obtained by summing up the dissociative excitation cross sections [35]: Anti-and weak-bonding properties of the virtual MOs of NH 3 lead to the general dissociation of NH 3 in the electronically excited states [69][70][71]. We therefore assumed that the excited NH 3 will dissociate with near unit quantum efficiency. The present calculation was restricted to the lowest-lying electronic states including the X 1 A 1 ground state and a 3 A 1 , This is mainly attributed to three reasons: (1) high-lying electronic states are convoluted with each other. Dissociation pathways are thus not predictable, (2) the cross sections for the electronic excitation to those higher-lying excited states are typically more than one order of magnitude smaller than the aforementioned states. Truncating the number of the excited states is reasonable due to the fact of overestimating the neutral dissociation possibilities, and (3) previous studies on the electron-impact dissociation and photodissociation of NH 3 [72][73][74][75][76][77][78] limited their considerations within low-lying electronic states, such that scarce discussion is available for the higher states.
As the optimized geometries of the excited states (which could be regarded as Rydberg states attached to the ground state NH + 3 with a planar structure) are planar, the excited states are usually denoted as X 1 in the D 3h symmetry point group. According to the investigations on the electron-impact dissociation and photodissociation of NH 3 [72][73][74][75][76][77][78][79][80], we could roughly identify the associated dissociation pathways to NH 2 +H  (8) and to NH + H 2 fragments One needs to notice that (1) the global potential energy surface (PES) of the electronic states for polyatomic molecules can theoretically correlate with multiple dissociation limits depending on the symmetry restriction. Although conical interactions frequently arise between different electronic states, for instance the X 1 A ′ 1 , A 1 A ′ ′ 2 , and B 1 E ′ ′ along the NH 2 +H path and a 3 A ′ ′ 2 , b 3 E ′ ′ along the NH+H 2 path, the above correlations represent the most favorable dissociation pathways, (2) the individual association is carefully assigned based on the symmetry and spin conservation rules, and the spin selection rules in photodissociation process. Electronimpact excitation is not restricted by these rules due to the possible exchange with the scattering electron. However, the shapes of the corresponding cross sections are different for spin-allowed and -forbidden transitions [68], which also contributes to the assignment of the correlations, and (3) the path to NH + H + H that is energetically possible through the coupling of the 1 Σ + g ground state H 2 and the repulsive 3 Σ + u state is not considered for simplicity and owing to the inconclusive transfer efficiency. Additionally, the dissociation channels to N + H + H + H or N + H 2 + H are also ignored in the present study due to their access at relatively higher electron scattering energy and much smaller cross sections. The computed dissociation cross sections for NH + H 2 (upper panel) and NH 2 + H (bottom panel) paths are shown in figure 8. The obtained curves display the characteristic behavior of the optically-allowed singlet and -forbidden triplet excitations [68]. Molecules dissociate under electron scattering through other two mechanisms: dissociative ionization and DEA besides the neutral dissociation. Cross sections for dissociative ionization of NH 3 had been computed and reported by Hamilton et al [35] in 2017 using the same R-matrix frame. Computations for DEA cross sections of NH 3 are in progress and will be presented in a forthcoming publication.  [29], blue dotted-dash curve. Figure 9 displays the cross sections for vibrational excitations from the ground and the first vibrational excited states for each normal mode of NH 3 . We compared the present computed cross sections with the estimated results by Itikawa [40] using the Born method and unpublished experimental data of Hayashi [29]. The overall shape of vibrational excitation cross sections for mode 1 and mode 2 are in good agreement with the compared results as seen in figures 9(a) and (b), while the magnitude of the cross section is different. The discrepancies from the cross sections could be tentatively attributed to the inherent differences of the used methods. It is worth noting that both Itikawa's results and Hayashi's experimental data are indistinguishable due to the extremely close vibrational frequencies of N-H symmetric and asymmetric modes. In other words, the cross section for excitation v 3 = 0 → 1 includes the contribution from the v 4 = 0 → 1 transition. This is, however, not a problem in our theoretical approach where we can exactly define the vibrational excitation of each mode and compute the corresponding cross sections as given in figure 9(c). In figure 9(d), the summation of the calculated vibrational excitation cross sections for N-H symmetric and asymmetric modes were plotted. By comparing with the available results, we conclude that our calculations provide a more reasonable range of values for the vibrational excitation cross sections. Additionally, asymmetric modes are generally dominant in the vibrational excitation and thus resulting cross sections of higher values due to the much larger changes in the molecular dipole moment [81]. This is in accordance with our prediction for mode 4 of N-H asymmetric stretch mode which presents cross sections with nearly one order of magnitude higher than the mode 3 of N-H symmetric stretch mode as shown in figure 9(c). As there is no reliable experimental or advanced theoretical treatment on vibration excitations, the cross sections calculated in this study can constitute the necessary basic data for the modeling of the discharge processes.

Uncertainty estimation
Similar to the experimental studies, it is increasingly accepted that uncertainty estimation should be a routine part performed in the theoretical calculations of electron-molecule scattering [82]. The computed cross sections with uncertainties should be the primary source data for plasma modeling. However, uncertainty quantification of the final calculated cross sections from a given theoretical method is often impossible or very difficult due to the 'model uncertainties', the unavoidable 'numerical uncertainties' and the hard-toquantify uncertainty propagation through the various stages of a calculation, e.g. from target property to scattering. Although there is no well-defined general approach to assess the uncertainties associated with electron-molecule collisional computations at present, a coarse estimation of uncertainty is still strongly recommended. In this section, convergence studies are carried out to assess the uncertainty for all the cross sections calculated above through following the guidelines in the review work of Chung et al [83].
The uncertainties of the computed cross sections could be generally estimated by varying the key scattering parameters in the calculations. An equivalent way is theoretically analyzing the physically important intermediate quantities associated with the target properties and scattering processes, such as the dipole moment and eigenphase sum, etc. We first investigated the stability of the dipole moment of NH 3 at the equilibrium geometry for structural convergence tests. Two sets of R-matrix calculations (1) using the CAS(6,6) and increasing the size of basis sets and (2) increasing the CAS with cc-pVTZ basis set were performed. The obtained results are shown in figures 10(a) and (b), demonstrating that as more functions and active spaces are added to the basis set and CAS, the dipole moments show diminishing basis set and CAS dependence, respectively. The convergent behavior of this quantity indicates that the structure calculations achieve a sufficient level of stability and accuracy before providing input to the subsequent scattering calculations. Since the cross sections for total elastics, differential, momentum transfer, and electronic excitation scattering processes are expressed in terms of the S-matrix: σ ∼ |S| 2 /E el , the fixed-nuclei S-matrix enters as the most important source of uncertainty. As illustrated in figures 10(c) and (d), the S-matrices calculated with increasing the basis set and CAS yield an uncertainties about 2% and 4%, respectively. In an ideal scenario, the uncertainty of the S-matrix should be propagated toward the final results. Therefore, the uncertainties of the present calculated total elastics cross section, DCS, MTCS, and electronic excitation cross section are overally estimated herein to be 6%.
One source of uncertainty for the electron-impact dissociation cross sections is the same as the electronic excitation as 6% since they are obtained by summing the cross sections of electronic excitation to dissociative states. Another main identifiable source is assuming the excited NH 3 in a dissociative state by electron impact will definitely dissociate, i.e. the probability of dissociation is 100%. Corresponding uncertainty has been discussed by Yuen et al [84] to be under 20%. As we aforementioned that multiple dissociation pathways correlated with the global PES of a polyatomic molecule. Uncertainties may also arise from the ignored subordinate dissociation pathways, but they are partially remedied by the truncation of the higher-lying dissociative states, thus resulting in an uncertainty of roughly 10%. The total uncertainty entered for the calculated electron-impact dissociation cross sections is thus estimated to be within 36%.
The assessment for the uncertainties of the computed cross sections for vibrational excitations was described in our previous study [45]. The uncertainty of the final cross sections depends on the wave functions of the target and the scattering electron. In a sense, the uncertainty of the dipole moment is a measure of the overall uncertainty of the wave functions used in the R-matrix calculations and, correspondingly, the uncertainty of the final cross sections. With the assumption that the relative uncertainty of the dipole moment is of the same order of magnitude as the relative uncertainty of its derivative, and further assuming that the order of magnitude of the vibrational excitation cross section is determined by the square of the derivative of the permanent dipole moment and polarizability of the target versus the normal coordinates, we can estimate the uncertainty of the obtained cross section with respect to the wave function of the target. We estimated that the uncertainty in the dipole moment of NH 3 is less than 2% and the uncertainty in the resulting cross section is less than 4%. The energy choice of the S l ′ λ ′ ,lλ (q) in equation (2) generates another source of uncertainty in the final vibrational excitation cross section. The right two panels of figure 10 can provide an idea about the energy variation of S l ′ λ ′ ,lλ (q), which gives an uncertainty in the cross sections of about 16%. Neglecting other uncertainty sources, the overall uncertainty of the present vibrational excitation cross section is about 20%.
Regarding the total ionization cross section, the experimental benchmark data are available and can be used for essential validation. Considering the uncertainty of the experimental results of Rejoub et al [33] recommended by Itikawa et al [40] and the good agreement with the experimental results, the accuracy of the total ionization cross section calculated in the present study is estimated to be almost the same as the experiment within 8%.
For the convenience of plasma modeling, we compiled all the scattering cross section data calculated in the present study accompanied by uncertainty estimations, as shown in figure 11. We recommend plasma modelers trying to use this set of cross sections for low-energy electron scattering with NH 3 calculated in the unified R-matrix framework. Corresponding numerical data could be found in the supplementary.

Conclusions
In summary, we computed the electron-impact total elastic, DCSs, MTCSs, ionization and electronic excitation cross sections of NH 3 using the frame of the R-matrix method with Quantemol-EC suite. Scattering models with various combinations of CAS and basis sets were tested to yield comparable cross section sets with the available experimental and theoretical results. The electron-impact dissociation cross sections of NH 3 were estimated based on the idea of Hamilton et al [35] used for NF x (x = 1, 2, 3). In addition, vibrational excitation cross sections for each vibrational mode of NH 3 were first theoretically calculated by the approach that combines the fixed-nuclei R-matrix method, normal mode approximation and the vibrational frame transformation. The experimentally unrecognized cross sections for vibrational excitations of N-H symmetric and asymmetric stretch modes with close frequencies could be separately described in our theoretical approach. For the vibrational excitation cross section, the present calculation could be regarded as a predictive work and assumed to be more reasonable to be used for plasma modeling as extensive tests with different target molecules [45] and different scattering models had been performed to confirm that the uncertainty of the present calculation is acceptable. Together with the dissociative ionization cross sections obtained by Hamilton et al [35] and the DEA cross sections being theoretically treated in progress, a complete set of cross sections in an unified framework based on the R-matrix method will be provided for NH 3 -containing plasma modeling. All the cross sections reported in the present study are available in the supplementary files for potential use.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).