Calculated cross sections for electron collisions with NF₃, NF₂ and NF with applications to remote plasma sources

To model the plasma chemistry of gas mixtures including NF₃ the cross sections for the following process which occur in low to intermediate energy electron–molecule collisions were considered:

Elastic scattering:

$$\begin{eqnarray}&&{e}^{-}+{AB}\to {AB}+{e}^{-}.\end{eqnarray} \tag{ 1 }$$

Inelastic scattering:

$$\begin{eqnarray}&&{e}^{-}+{AB}\to {{AB}}^{* }+{e}^{-}.\end{eqnarray} \tag{ 2 }$$

Superelastic scattering (for NF only):

$$\begin{eqnarray}&&{e}^{-}+{{AB}}^{* }\to {AB}+{e}^{-}.\end{eqnarray} \tag{ 3 }$$

Electron impact dissociation:

$$\begin{eqnarray}&&{e}^{-}+{AB}\to A+B+{e}^{-}.\end{eqnarray} \tag{ 4 }$$

Dissociative attachment (DA):

$$\begin{eqnarray}&&{e}^{-}+{AB}\to A+{B}^{-}.\end{eqnarray} \tag{ 5 }$$

Metastable electron impact dissociation (for NF only):

$$\begin{eqnarray}&&{e}^{-}+{{AB}}^{* }\to A+B+{e}^{-}.\end{eqnarray} \tag{ 6 }$$

Electron impact ionisation:

$$\begin{eqnarray}&&{e}^{-}+{AB}\to {{AB}}^{+}+2{e}^{-}.\end{eqnarray} \tag{ 7 }$$

Electron impact dissociative ionisation:

$$\begin{eqnarray}&&{e}^{-}+{AB}\to A+{B}^{+}+2{e}^{-}.\end{eqnarray} \tag{ 8 }$$

The asterisk is used to denote an electronically excited state. Below we discuss the theoretical procedures used.

The R-matrix method treats electron scattering from molecules by dividing the space of the problem into two separate regions [22] comprising an inner region which contains within it the wavefunction of the molecular target, and an outer region in which only the incident, scattering electron is considered. An R-matrix calculation constructs and solves an energy-independent wave equation for the inner region whose solutions are then used to solve the much simpler, energy-dependent problem of the scattering electron in the outer region. By making the inner region of the problem energy independent and only the outer region energy dependent, we can resolve the outer region on a very fine energy grid which gives all the cross section features and structures.

All low-energy calculations reported here used the polyatomic implementation of the UK molecular R-matrix code UKRMol [23]. Most of the calculations were performed using the Quantemol-N expert system [24] which both runs UKRMol and provides the various high-energy approximations discussed below and the DA estimator [25]. A full review of the molecular R-matrix method has been given in [26].

Established electron scattering theory provides a range of models for treating the interaction of the incident scattering electron with the bound molecular electrons [26]. For this work three such methods were employed for calculation and validation of results.

The simplest model used was the static exchange (SE) method. In this method the target wavefunction is frozen in its Hartree–Fock (HF) ground state and is not allowed to relax in response to the scattering electron. This method therefore can only be used to calculate cross sections for elastic processes and can only detect shape resonances where the scattering electron is temporarily trapped behind a potential barrier created by the molecule. The SE method cannot detect Feshbach resonances which involve the excitation of bound electrons.

The second model employed is the static exchange plus polarisation (SEP) method in which the target is maintained in its HF ground state but energy exchange is allowed between the scattering electron and the bound molecular electrons. In this model a single target electron can be excited into one of the unoccupied orbitals of the target, known as virtual orbitals (VOs). This mode of excitation if found to give a particularly good representation of the polarisation of the molecule by the scattering electron; SEP techniques are often used to produce a reliable representation of shape resonances.

The third and most sophisticated model used is based on considering several target states through the use of a close-coupling (CC) expansion. Here the target states are represented using a complete active space (CAS) configuration interaction (CI) model [27] in which bound electrons from the highest (valence) occupied molecular orbitals are excited to the lowest unoccupied molecular orbitals. This model can calculate cross sections for electronically inelastic processes; it also gives reliable Feshbach resonances which are the temporary anion states where the scattering electron is trapped following excitation of the target.

None of the models considered above can treat electron-impact ionisation. Here the method used to calculate the total ionisation cross sections is the binary encounter Bethe (BEB) method of Kim and Rudd [19]. This method uses the orbital binding energy and electron kinetic energies of the bound orbitals to calculate the ionisation cross sections. The BEB method is an amalgamation of the Mott theory [28] which describes hard, close collisions with small impact parameters and work by Bethe [29] who showed soft collisions with large input parameters essentially take place as dipole interactions between the incident and ionised electron. The BEB method therefore accounts for both collision types which take place to calculate total ionisation cross sections for fast incident electrons. The BEB total ionisation cross section, ${\sigma }_{\mathrm{BEB}}$, is given by:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{BEB}}=\displaystyle \frac{S}{t+u+1}\left(\displaystyle \frac{1}{2}\left(1-\displaystyle \frac{1}{{t}^{2}}\right)\mathrm{ln}(t)+1-\displaystyle \frac{1}{t}-\displaystyle \frac{\mathrm{ln}(t)}{t+1}\right)\,,\end{eqnarray} \tag{ 9 }$$

where $t=\tfrac{T}{B}$, $u=\tfrac{U}{B}$ and $S=4\pi {a}_{0}^{2}N{\left(\tfrac{R}{B}\right)}^{2}$ and where T is the kinetic energy of the incident electron, B and U are the binding energy and the average kinetic energy respectively of the electrons in a sub shell, and N is the number of bound electrons with the constants the Rydberg constant, R, and Bohr radius, a₀.

The BEB method only yields total ionisation cross sections. To calculate the partial cross sections for the various ionisation and dissociative ionisation breakup cross sections involved it was assumed that the ratio of the charged fragments produced would be the same as that of the observed fragments in the mass spectrum of that molecule at the energy the mass spectrum was taken. Starting from this assumption partial cross sections are obtained by enforcing the correct thresholds for the various ionisation and dissociation processes while also ensuring the resulting cross sections are both continuous and sum to the calculated total cross section. With these constraints in place equation (9) can be rewritten in terms of ${\sigma }_{f}$, the cross sections for formation of each fragment:

$$\begin{eqnarray}&&{\sigma }_{f}=\displaystyle \frac{{{\rm{\Gamma }}}_{f}S}{{t}_{f}+u+1}\left(\displaystyle \frac{1}{2}\left(1-\displaystyle \frac{1}{{t}_{f}^{2}}\right)\mathrm{ln}({t}_{f})+1-\displaystyle \frac{1}{{t}_{f}}-\displaystyle \frac{\mathrm{ln}({t}_{f})}{{t}_{f}+1}\right),\end{eqnarray} \tag{ 10 }$$

where ${{\rm{\Gamma }}}_{f}$ is the branching ratio of the fragment and ${t}_{f}=\tfrac{T}{B-{D}_{f}}$ where D_f is the threshold to dissociation of fragment f.

This method was validated by calculating the dissociative ionisation cross sections of ammonia and making a comparison with published results. The mass spectrum of NH₃ at 100 eV was obtained from the NIST molecular database [30] and the ratios of the intensity of the spectra for NH₃⁺, NH₂⁺, NH⁺ and N⁺ were thus taken to be the ratios of the ionisation and dissociative ionisation cross sections of ammonia at 100 eV. Note that isotopically-substituted fragments appearing in the mass spectra were ignored and no fragment corresponding to H⁺ appears in the mass spectrum. The thresholds of the cross sections were taken from the experimental ionisation energies [31].

The ionisation and dissociative ionisation cross sections for NH₃ found using this method are compared to the measurements of Rao and Srivastava [32] in figure 1. Our predicted total ionisation cross section calculated using the BEB method is an average of 9% greater than the measured results between the NH₃ ionisation threshold and 1000 eV, albeit the differences are greater at lower energies and largely disappear at higher energies. The theoretical dissociative ionisation cross sections of the dominant fragments, NH₃⁺ and NH₂⁺, show excellent agreement with the previous results. For the minor fragments, NH⁺ and N⁺, there is a greater divergence between predictions and measurement at higher energies, possibly due to the absence of the H⁺ fragment in the theoretical calculation due to its absence in the mass spectrum used. This comparison nonetheless suggests that our theoretical model should be capable of giving reliable predictions of dissociative ionisation cross sections, especially when a complete mass spectrum fragmentation pattern is available.

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Dissociation occurs when molecules are excited to electronic states which are either unbound or have curve-crossings to unbound states. The total electron impact dissociation cross section can therefore be taken to be the sum of excitation cross sections to unbound states:

$$\begin{eqnarray}&&{\sigma }_{{\rm{eid}}}^{{\rm{tot}}}=\displaystyle \sum _{i=1}^{\infty }{\sigma }_{{\rm{ex}}}^{i}.\end{eqnarray} \tag{ 11 }$$

However to fully understand the dynamics of a plasma, electron-impact dissociation cross sections are required to specific products and states of products. To obtain these requires branching ratios for the dissociation cross sections, which is a difficult problem in itself. In this paper several methods are used to estimate the relevant branching ratios based on the asymptotes of potential energy surfaces where available and experimental photoionisation results. In the latter case we make use of the rules of spin conservation.

Electron impact excitation cross sections were scaled to high energies, summing over all symmetries, using the BEf procure of Kim [33]. BEf scaling is, in fact, two scalings: binary-encounter (BE) scaling and f scaling where qualitatively the BE scaling can be thought of as replacing the incident electron flux, usually defined in terms of the incident electron velocity at infinity, with an effective flux altered by the interaction of the incident electron with the target. f scaling is the empirical scaling factor for the electric dipole oscillator strengths which improves the accuracy of the target wavefunction.

The DA cross sections of the molecules were calculated up to 20 eV using Quantemol's DA estimator [25] which uses resonance parameters taken at equilibrium geometry and other appropriate data to provide estimated cross sections. Above 20 eV the DA cross sections are taken to be zero due to the lack of resonances at higher energies. This is consistent with the standard understanding of this process.

The remaining cross sections: elastic, momentum transfer and neutral dissociation were scaled to high energies by assuming these processes are dominated by dipole transitions with the effect of exchange and spin changing transitions going to zero. Dipole transitions scale as $\tfrac{\mathrm{ln}(E)}{E}$ where E is electron energy. Where the calculated cross sections showed non-physical structure at energies approaching 20 eV, this non-physical structure was assumed to be an artefact of the calculation for example due to using only single geometry, incomplete continuum orbital sets or pseudoresonances. The extrapolation to higher energies therefore smoothed over these structures in the cross sections approaching 20 eV.

The molecular geometries of NF₃, NF₂ and NF used are given in table 1 and are based on data obtained from the NIST CCCBDB website [34]. At these geometries, NF₃, NF₂ and NF have ${{C}}_{3v}$, ${{C}}_{2v}$ and C${}_{\infty v}$ symmetry, respectively.

After tests the Dunning's augmented Gaussian type orbital (GTO) aug-cc-pVTZ basis set was selected for the NF calculations. The use of the diffuse basis set improves the calculation of the diffuse excited states of the molecule subsequently improving the accuracy of the super elastic cross sections of the molecule and neutral dissociation of the metastable states of the molecule.

The cc-pVTZ GTO basis set was selected for the NF₃ and NF₂ calculations. The use of augmented basis sets for these targets interfered with the construction of complete continuum basis in scattering calculations on these two molecules and so were not used. These basis sets were used in all calculations.

The ground state of NF₃ is a closed shell. Iterative optimisation of NF₂ gave a X ²B₁ ground state with the configuration (1a₁² 1b₂² 2a₁² 3a₁² 2b₂² 4a₁² 3b₂² 5a₁² 1b₁² 6a₁² 1a₂² 4b₂² 2b₁¹) and NF has an X ${}^{3}{{\rm{\Sigma }}}^{-}$ ground state with the configuration (1${\sigma }^{2}$ $2{\sigma }^{2}$ $3{\sigma }^{2}$ $4{\sigma }^{2}$ $1{\pi }^{4}$ $5{\sigma }^{2}$ $2{\pi }^{2}$). These configurations can be written more concisely [1-6a₁, 1b₁, 1-4b₂, 1a₂]²⁴ [2b₁]¹ and [1-5σ, 1π]¹⁴ [2${\pi }^{}$]² respectively. This latter notation is used below. An ab initio dipole moment of 0.352 D was calculate for NF₃, 0.0536 D for NF₂ and 0.102 D for NF using the CAS-CI model. The published experimental dipole value for NF₃ is 0.23 D [35]. Experimental values of the NF₂ and NF dipole moments could not be obtained. Comparison when using experimental and theoretical dipole moments of NF₃ showed that use of the lower, measured, dipole lowered the value of the elastic cross section as shown in figure 2. This lowering is more pronounced at low energies.

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A Born correction was applied to the total cross sections calculated. The cross sections of species with dipoles converge slowly with partial waves and the Born correction makes up for the omitted partial wave states with $l\gt 4$. These corrections were calculated using the program BORNCROSS [36] which is regularly used in conjunction with the R-matrix suite. For NF₃ the experimental dipole moment was used when making the Born correction; for NF and NF₂ only theoretical dipole moment values are available. The cross sections presented in this work are rotationally unresolved and rotational excitation is treated as an elastic process. However, when making the Born correction to the momentum transfer cross sections, the various rotational contributions were computed using the code POLYDCS [37] and then summed over.

Elastic cross sections were calculated using CAS-CI calculations which roughly correspond to treating the various 1s and 2s orbitals as frozen and allowing the electrons to be distributed freely amongst the various 2p orbitals. For the NF₃ calculation. Orbitals [1-9a₁, 1-4a₂]²⁶ were frozen and a CAS of [10-15a₁, 5-8a₂]⁸ was used. For the NF₂ calculation orbitals [1-5a₁, 1b₁, 1-3b₂]¹⁸ were frozen and a CAS [6-9a₁, 2-3b₁, 4-6b₂, 1a₂]⁷ was used.

For the NF calculation orbitals [1-4σ]⁸ were frozen and a CAS of [5-8σ, 1-3π]⁸ was used. The momentum transfer cross sections are calculated using the same calculation model. The elastic and momentum transfer cross sections of NF₂ are compared in figure 3.

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Dissociative electron attachment or DA is the electron scattering process whereby the scattering electron attaches to the molecule in a resonance state which causes the molecule to fragment. DA is very important in technological plasmas as is can be the initial step at low energies which leads to the breakup of the feedstock gases and creation of negative ions; as such DA is of paramount interest in this investigation.

A semi-empirical treatment for calculating DA cross sections was used in this investigation [25] based upon the resonances found in the electron–molecule interaction detected by the program RESON [38]. Resonance calculations were made using the SE, SEP and CC methods, the most reliable results being found with the SEP calculations, as explained in the results section.

SEP calculations, with the target described in section 3.1, were found to converge NF₃ resonance parameters using 53 VOs, (12-44a₁, 7-28a₂). Similarly 77 VOs, (7-33a₁, 3-17b₁, 5-27b₂, 2-13a₂) converged the NF₂ calculation, and 41 VOs, (6-18 σ, 3-17 π, 1-13 δ), converged the NF calculation.

In addition to the cross section for capture into the resonant state calculated from the resonance positions and widths provided by RESON, the DA cross sections depend upon survival probabilities for each resonance which is the probability of the resonance state dissociating before it autoionizes. The survival probabilities are calculated from the electron affinity of the most likely anion fragment, the vibrational frequency of the bond broken to create this fragment and the dissociation energy of this bond. In the case of DA of NF₃, NF₂ and NF the most likely anion fragment was taken to be F⁻. F has electron affinity 3.401 eV [39]. The vibrational frequencies of the N–F bonds in the three molecules and the dissociating energies of these bonds are given in table 2.

No metastable states were detected for NF₃ or NF₂. The dissociation energy of the N–F in NF₃ bond is 2.52 eV [41, 42] and the first excited state of this molecule found in the CAS-CI calculation described in section 3.2 is the a ³E with energy 8.61 eV. The vertical excitation energy of this first triplet state was found experimentally to be 6.58 eV [45] and calculated by theory as 8.32 eV [13].

The dissociation energy of the N–F bond in NF₂ is 3.30 eV [41, 42] and the lowest excited state of this molecule found in the CAS-CI calculation is the A ²A₂ state which lies at 4.34 eV. For both NF₃ and NF₂ the first excitation threshold lies above the dissociation threshold. We therefore assume that excitation to this and higher states will lead to dissociation, and therefore no super-elastic collisions (collisions of the second kind) or impact dissociation from a meta-stable state will occur in electron impact processes with these molecules.

Metastable states were detected, however, in NF. The dissociation energy of the N–F bond in NF is 3.12 eV [41]. From the X ${}^{3}{{\rm{\Sigma }}}^{-}$ ground state of NF it was found that NF has two metastable, spin-singlet states below the dissociation energy, a ${}^{1}{\rm{\Delta }}$ with energy 1.34 eV and b ${}^{1}{{\rm{\Sigma }}}^{+}$ at 2.32 eV found in the CAS-CI calculation. These values are in good agreement with those found in previous studies. Our calculated vertical excitation energies for each molecule are shown in tables 3–5, where they are compared to values available in the literature.

Electron impact dissociation occurs via excitation to electronically excited states of the target which then dissociate [51]. For an accurate calculation of these processes a large number of electronically excited states need to be considered. Born corrections to the electron-impact excitation cross sections were used to account for long range dipole effects [36, 52].

Using the CAS-CI calculation models described in section 3.2 and the target described in section 3.1 results were converged with a treatment of NF₃ placing the eight highest electrons in 23 orbitals. These electrons were excited up to 39 excited states below 23 eV. The CAS-CI treatment of NF₂ placed the seven highest electrons in 19 orbitals; inclusion of 19 excited states with an energy cut off of 15 eV converged the NF₂ calculation.

The NF metastable states mean that superelastic collisions and electron impact dissociation from these states also need to be considered. The superelastic cross section is calculated as the sum of the cross sections from the metastable states to the ground state. Metastable dissociation is calculated as the sum of all excitation cross sections which start from a metastable state and excite a state above the N–F dissociation energy. This model treats the two metastables as a single entity and ignores collisions which interconverts between the two metastable states. A CAS-CI treatment of NF which placed the eight highest electrons in 11 orbitals was used for these calculations exciting the electrons up to 64 excited states with an energy cut off of 18 eV.

Total electron impact dissociation cross sections were thus obtained; from these branching ratios were estimated for dissociation to specific products and states. The recent NF potential energy curves computed by Wan et al [53] show there are two major dissociation products of NF:

$$\begin{eqnarray}&&\mathrm{NF}(A{}^{3}{\rm{\Pi }})+e\to {\rm{N}}{(}^{2}{D}_{u})+{\rm{F}}{(}^{2}{P}_{u})+e\end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}&&\mathrm{NF}(Y)+e\to {\rm{N}}{(}^{4}{S}_{u})+{\rm{F}}{(}^{2}{P}_{u})+e,\end{eqnarray} \tag{ 13 }$$

where Y in equation (13) represents all states of NF other than A ${}^{3}{\rm{\Pi }}$. While previous studies [49] predicted that excited states of NF above b ${}^{1}{{\rm{\Sigma }}}^{+}$ are repulsive, Wan et al [53] identified very shallow well in the excited states of NF capable of supporting bound states. Here, we approximate all states above b ${}^{1}{{\rm{\Sigma }}}^{+}$ as being repulsive. The shallowness of the wells identified by Wan et al and the vanishingly small Franck–Condon factors from the ground electronic state makes this a legitimate approximation. Although the a ${}^{1}{\rm{\Delta }}$ and b ${}^{1}{{\rm{\Sigma }}}^{+}$ curves are identified as crossing the A ${}^{3}{\rm{\Pi }}$ curve in this work we assume that predissociation of these states to produce ${\rm{N}}{(}^{4}{S}_{u})\,+{\rm{F}}{(}^{2}{P}_{u})$ fragments does not occur and that these states purely dissociate to ${\rm{N}}{(}^{2}{D}_{u})+{\rm{F}}{(}^{2}{P}_{u})$. The dissociation energy producing these latter fragments is 5.50 eV taking the dissociation energy to ${\rm{N}}{(}^{4}{S}_{u})+{\rm{F}}{(}^{2}{P}_{u})$ to be 3.12 eV [41] and the energy difference between the atomic state products to be 2.38 eV [53].

Potential energy surfaces for excited states of NF₂ are not available and therefore branching ratios were estimated by induction from photolysis experiments. Ground state NF₂ is a doublet so photodissociation only gives products which couple to doublet spin symmetry. Papakondylis and Mavridis [54] show explicitly:

$$\begin{eqnarray}&&{\mathrm{NF}}_{2}(A{}^{2}{A}_{1})\to \mathrm{NF}(X{}^{3}{{\rm{\Sigma }}}^{-})+{\rm{F}}{(}^{2}{P}_{u})\end{eqnarray} \tag{ 14 }$$

and

$$\begin{eqnarray}&&{\mathrm{NF}}_{2}(B{}^{2}{A}_{1})\to \mathrm{NF}(X{}^{3}{{\rm{\Sigma }}}^{-})+{\rm{F}}{(}^{2}{P}_{u})\end{eqnarray} \tag{ 15 }$$

and

$$\begin{eqnarray}&&{\mathrm{NF}}_{2}(C{}^{2}{B}_{2})\to \mathrm{NF}(a{}^{1}{\rm{\Delta }})+F{(}^{2}{P}_{u})\end{eqnarray} \tag{ 16 }$$

and Collins and Husain [55] suggest that:

$$\begin{eqnarray}&&{\mathrm{NF}}_{2}(D{}^{2}{A}_{2})\to \mathrm{NF}(b{}^{1}{{\rm{\Sigma }}}^{+})+\ {\rm{F}}{(}^{2}{P}_{u}).\end{eqnarray} \tag{ 17 }$$

Papakondylis and Mavridis [54] note that the two NF₂ ²A₁, A²A₁ and B²A₁, surfaces undergo an avoided crossing adding an additional energy barrier of 0.356 eV to their dissociation energy. They also discuss a crossing between the two ²A₁ states and those of C ${}^{2}{B}_{2}$ symmetries which can lead to production of NF $(a{}^{1}{\rm{\Delta }})$ from the two ²A₁ states. However, this crossing only arises from population of the antisymmetric b₂ vibrational levels which, as vibrational excitation is not discussed in this work, means this process is neglected. Additionally, Papakondylis and Mavridis rationalise the delayed appearance of NF(${}^{1}{\rm{\Delta }}$) as being due to the presence of an avoided crossing creating a barrier to the dissociation of the B ${}^{2}{B}_{2}$ state however this barrier is below the vertical excitation energy of the C ${}^{2}{B}_{2}$ state and consequently does not affect the threshold at which we calculate this dissociation occurring. Collins and Husain do not discuss the potential energy surface of the D ${}^{2}{A}_{2}$ state. Their suggestion that this state dissociates is assumed to be correct.

Photolysis cannot be used to show dissociation products for the quartet excited states of NF₂, however selection rules dictate the products of the dissociation of the NF₂ quartet states must be in a doublet state and a triplet state. Furthermore, as the fluorine atom produced by the dissociation is in a doublet state the NF product must, therefore, be in a triplet state. The most likely triplet state NF product is the ground X ${}^{3}{{\rm{\Sigma }}}^{-}$ state therefore:

$$\begin{eqnarray}&&{\mathrm{NF}}_{2}(a{}^{4}{A}_{2})\to \mathrm{NF}(X{}^{3}{{\rm{\Sigma }}}^{-})+{\rm{F}}{(}^{2}{P}_{u})\end{eqnarray} \tag{ 18 }$$

and

$$\begin{eqnarray}&&{\mathrm{NF}}_{2}(b{}^{4}{B}_{1})\to \mathrm{NF}(X{}^{3}{{\rm{\Sigma }}}^{-})+{\rm{F}}{(}^{2}{P}_{u}).\end{eqnarray} \tag{ 19 }$$

Potential surfaces of the a ${}^{4}{A}_{2}$ and b ${}^{4}{B}_{1}$ states of NF₂ are not available. We assume in this work that the quartet states are parallel to their doublet state equivalents and therefore assume these states are dissociating and that any additional dissociation barrier created by an avoided crossing is beneath the vertical excitation energy of the states. Furthermore we do not take predissociation into account for these quartet states.

As with NF₂, potential energy surfaces of NF₃ were not readily available and therefore, as for NF₂, results were inducted from photolysis results. NF₃ having a singlet ground state, photodissociation takes place on singlet excited states. The peak positions and thresholds in table 3 of Seccombe et al [56] compared with their NF₂ excited state energies given in their table 1 and cross referenced with the vertical excitation energies calculated in this paper and given in table 4, suggest the following impact breakups occur:

$$\begin{eqnarray}&&{\mathrm{NF}}_{3}(a{}^{1}E)\to \mathrm{NF}(b{}^{1}{{\rm{\Sigma }}}^{+})+{{\rm{F}}}_{2}\end{eqnarray} \tag{ 20 }$$

and

$$\begin{eqnarray}&&{\mathrm{NF}}_{3}(b{}^{1}{A}_{1})\to \ {\mathrm{NF}}_{2}(A{}^{2}{A}_{1})+{\rm{F}}{(}^{2}{P}_{u})\end{eqnarray} \tag{ 21 }$$

and

$$\begin{eqnarray}&&{\mathrm{NF}}_{3}(c{}^{1}{A}_{2})\to {\mathrm{NF}}_{2}(B{}^{2}{A}_{1})+{\rm{F}}{(}^{2}{P}_{u})\end{eqnarray} \tag{ 22 }$$

and

$$\begin{eqnarray}&&{\mathrm{NF}}_{3}(d{}^{1}E)\to {\mathrm{NF}}_{2}(C{}^{2}{B}_{2})+\ {\rm{F}}{(}^{2}{P}_{u}),\end{eqnarray} \tag{ 23 }$$

where it is assumed the dissociation to NF will also produce F₂, as opposed to 2F, as this product has a lower threshold and also because of the better match of this threshold to the threshold given in table 3 of Seccombe et al [56]. Photodissociation does not occur via NF₃ triplet states; however, we assume that the symmetry of the transitions will remain the same and therefore only spin transitions need to be taken into consideration. Spin-changing transitions are allowed for electron impact and triplet NF₃ excited states will dissociate to both doublet and quartet states of NF₂. Spin statistics suggest that quartet states will occur with twice the probability of the doublet states; while for triplet and singlet states of NF, triplet state products will have triple the probability of singlet state products. Branching ratios are thus obtained from these transition rules and probabilities and are associated with the following dissociations:

$$\begin{eqnarray}&&{\mathrm{NF}}_{3}(A{}^{3}E)\to \mathrm{NF}(b{}^{1}{{\rm{\Sigma }}}^{+})+\ {{\rm{F}}}_{2}\ \end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&\,\,\,\,\,\to \,\mathrm{NF}(B{}^{3}{{\rm{\Sigma }}}^{+})+\ {{\rm{F}}}_{2},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\,{\mathrm{NF}}_{3}(B{}^{3}{A}_{1})\to {\mathrm{NF}}_{2}(A{}^{2}{A}_{1})+{\rm{F}}{(}^{2}{P}_{u})\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&\,\,\,\,\,\,\to \,{\mathrm{NF}}_{2}(a{}^{4}{A}_{1})+{\rm{F}}{(}^{2}{P}_{u}),\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&{\mathrm{NF}}_{3}(C{}^{3}{A}_{2})\to {\mathrm{NF}}_{2}(B{}^{2}{A}_{1})+{\rm{F}}{(}^{2}{P}_{u})\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&\,\,\,\,\,\to \,{\mathrm{NF}}_{2}(b{}^{4}{A}_{1})+{\rm{F}}{(}^{2}{P}_{u}),\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&\,{\mathrm{NF}}_{3}(D{}^{3}E)\to {\mathrm{NF}}_{2}(C{}^{2}{B}_{2})+{\rm{F}}{(}^{2}{P}_{u})\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&\,\,\,\,\,\to \,{\mathrm{NF}}_{2}(c{}^{4}{B}_{2})+{\rm{F}}{(}^{2}{P}_{u}).\end{eqnarray} \tag{ 31 }$$

The nature of the NF₃ states is not discussed by Seccombe et al [56]. Their singlet states are implicitly identified as being dissociating states which we take to be the case. Predissociation was not taken into account. We assume in this work that the triplet states of NF₃ are parallel to their singlet equivalents and therefore are also dissociating states.

For total neutral dissociation cross sections of NF, NF₂ and NF₃ we expect the possible overestimate of these cross sections resulting from assumptions about the dissociative nature of the excited states to be ameliorated by the fact that in reality the number of these excited states is under-estimated in our calculations.

Koopman theorem's calculations give a simple, ab initio ionisation energy for molecules. For NF₃, NF₂ and NF the Koopman ionisation energies are 15.09 eV, 10.33 eV and 17.10 eV respectively, based on the target described in section 3.1. The experimental ionisation energy of NF₃, NF₂ and NF are given as 12.94 eV, 11.63 eV and 12.10 eV respectively [31]. Due to the large difference between the theoretical and experimental values, the experimental ionisation energies were used when calculating ionisation cross sections.

The mass spectrum of NF₃ at 100 eV was obtained from the NIST Mass Spectrometry Data Center [30]. This spectrum gives a fragmentation pattern for the full range of NF₃ fragments and therefore using it will not have the problem with missing fragments found in our model of NH₃ breakup. However, mass spectra are not available for NF₂ or NF and so the branching rations of the ionisation and dissociative ionisation breakups were estimated by truncating the fragmentation pattern obtained in the NF₃ mass spectrum.

DA cross sections were initially calculated using the SE and CAS-CI calculation models however the cross sections for NF₃ produced by these methods compared very poorly with experimental results of Nandi et al [10]. The SEP method is well-known to be most reliable for characterising low-lying shape resonances, upon which these DA cross sections rely. The model gave much more satisfactory results for DA in NF₃. It was therefore assumed that the SEP model would also be most reliable for NF₂ and NF, for which no experimental results are available for comparison.

The SEP calculations were converged by increasing the number of VOs and using the eigenphases as a diagnostic [60]. Increasing the number of VOs in the calculation had the effect of lowering the energy of the resonances. A SEP calculation involving 49 VOs converged the resonance parameters from which the NF₃ DA cross section was calculated. The results of this calculation compare well with the experimental results of Nandi et al [10] as shown in figure 10. The shape resonances detected in NF₃ by this SEP calculation are presented in table 6. The positions and widths of these resonances detected in the CC calculation described in section 3.2 are also given in this table. It is clear that the CC model places these resonances at much too high an energy which can be attributed to difficulties with converging polarisation effects in the CC procedure [61].

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For the NF₃ SEP calculation convergence was obtained when all VOs lying below an energy cut-off of 125 eV were retained. Based on these findings, the same energy cut-off was used in the NF and NF₂ SEP calculations leading to the inclusion of 74 virtual states in the NF₂ calculation and 54 virtual states in the NF calculation. The resonances detected by these calculations are shown in tables 7 and 8 for NF₂ and NF, respectively. Also shown are the positions and widths of these resonances calculated in the CC calculations described in section 3.2. The DA cross sections of NF₂ and NF based on the SEP resonances are shown in figure 10.

Table 8.  Resonances found in NF.

State	SEP		CC
	Position (eV)	Width (eV)	Position (eV)	Width (eV)
${}^{2}{\rm{\Pi }}$	1.697	1.579	2.389	0.234

Total ionisation cross sections for NF₃, NF₂ and NF were calculated up to 5000 eV and the mass spectrum of NF₃ was used to define the branching ratios of the fragments. This method gave continuous cross sections for each fragment with the correct thresholds although the differentials of the cross sections are not necessarily continuous. The cross sections for the total ionisation, ionisation and dissociative ionisation cross sections are shown in figures 11–13.

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Electron impact ionisation of NF₃ has actually been fairly well studied [11, 12, 16, 62, 63]. Our results give total cross sections somewhat larger than the earlier measurements of Tarnovsky et al [11] but are closer to the more recent experimental and theoretical study of Rahman et al [62]. Rahman et al's results were recommended by the recent review by Song et al [18] on a number of grounds, not least by comparison with the reliability of measurements using the same experiments for other systems such as CF₄.

The partial cross sections or branching ratios following electron impact ionisation are also not in complete agreement. Tables 9 and 10 show respectively relative intensities of ion yields in mass spectra and relative cross sections published by Tarnovsky et al [11] and Rahman et al [62]. The cross section values used in the ratios presented on table 10 are the values at the energies of the mass spectra on table 9. It can be seen that different measurements of relative ion yields are not in complete agreement. Furthermore there are other problems: for example Rahman et al [62] measure significant N⁺ at energies below its formation threshold. Our branching ratios are based on mass spectroscopy results as discussed above. Comparison of relative ion intensities to relevant cross section ratios give confidence in this method. The ordering of fragments by relative intensity published by Tarnovsky et al [11] and Rahman et al [62] is the same as their ordering of cross section magnitudes. Furthermore the ratios are comparable when uncertainties, as mentioned above, are taken into account.

Figures 14–16 give our predicted electron impact dissociation results obtained by summing the electron impact electronic excitation cross sections for all states not identified as being metastable for energies above the dissociation threshold. Fragmentation patterns were obtained on the basis of measured photodissociation data as described above.

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