Calculated cross sections for low energy electron collision with OH

The hydroxyl radical, OH, is an important component of many natural and technological plasmas but there is little available information on processes involving its collisions with low-energy electrons. Low-energy electron collisions with OH are studied in the framework of the R-matrix method. Potential energy curves of some of the low lying target states of doublet and quartet symmetry which go to the O(3P)+H(2S), O(1D)+H(2S) and O(1S)+H(2S) asymptotic limits are obtained for inter nuclear separations between 1 - 6 a0. Scattering calculations are performed at the OH equilibrium geometry Re = 1:8342 a0 to yield cross sections for elastic scattering, electronic excitations form the X2\Pi ground state to the three excited states of A2Sigma+, a 4Sigma, 1 2Sigma symmetry and for electron impact dissociation of OH. The positions and widths for negative ion resonances in the e{OH system are used estimate the cross section for dissociative electron attachment to OH which is found to be significant at electron energies about 1.5 eV.


Introduction
The hydroxyl radical, OH, is key component of many plasmas include atmospheric ones, particularly if the air is humid [1][2][3][4][5][6][7] or near liquid water [8][9][10], and plasmas formed during combustion [11,12]. Models of such plasmas require rates or cross sections for key processes occuring in the plasma but for OH these are largely unknown [13] and hence are absent from major data compilations [14,15]. Indeed the best source of electron-OH rates [5] appears to be the weighted total cross-section (WTCS) caclulations of Riahi et al. [16] which, for OH, are based on old and unproven electron collision cross sections [17].
Given the difficulty in measuring electron collision cross sections with open shell species such as OH, it would appear to be timely to use theoretical methods to establish a reliable dataset of cross sections and rates for electron collisions with OH is expected to govern its chemistry and hence it is necessary to have reliable cross sections for different electron induced processes in OH.
The starting point of our collision calculation requires accurate potential energy curves (PECs) for OH. Several studies on the PECs of OH exist. Calculations on the OH doublet and quartet states were performed as early as 1973 by Easson and Pryce [18]. Langhoff et al [19] studied the ground and some of the 2 Σ + states of OH to obtain their dipole moments and radiative lifetimes. A configuration interaction (CI) study on the 2 Σ − states of OH was performed by van Dishoeck et al [20]. Subsequently, they extended this work to a systematic study of the ground (X 2 Π) and several excited states of doublet and quartet symmetry [21]. There are also some recent and more sophisticated calculations using large GTO basis sets performed on some of the low lying OH states [22,23].
Compared to OH, its anionic states are much less studied even though these are known to play an important role in collision processes such as the dissociative electron attachment (DEA). Singlet and triplet OH − states of Σ + and Π symmetries were obtained by Sun and Freed [24] at the OH equilibrium bond length of R e = 1.8342 a 0 to calculate vertical excitation energies of the excited states of OH − . Several calculations on the PECs of these singlet and triplet states of OH − also exist [25][26][27][28]. More recently, calculations on the OH − anionic states were performed with large GTO bases by Srivastava and Satyamurty [22] and Vamhindi et al [29]. All of the studies indicate that the X 1 Σ + state of OH − is strongly bound where as the 1 Π and 3 Π excited states are only quasi-bound as these resonance states lie in the continuum of the OH neutral plus free electron system. Suprisingly few studies on electron plus OH collisions exist. A limited study of the vibrationally inelastic cross sections, both integrated and differential, for the excitation of the ν = 1 vibrational level of the electronic ground state was performed by Chen and Morgan [26] in the energy range 0 − 3 eV using the R-matrix method. A much more detailed calculation of the elastic differential, integral, momentum-transfer cross sections as well as grand-total (elastic + inelastic) and total absorption cross sections for electron-OH collisions were performed by Sobrinho et al [30] using the Schwinger variational method and the distorted-wave approximation. Since no other results were available on e-OH collisions, Sobrinho et al compared their cross sections with the corresponding cross sections for e-H 2 O collisions which they found to be remarkably similar. The ionisation potential for OH is 13 eV [31]. As mentioned above, cross section for electron impact ionisation and rate coefficients for OH(X 2 Π)→ OH(A 2 Σ + ) excitations were obtained by Riahi et al [16] using the WTCS theory which is essentially a model calculation.
In the present work, we present cross sections and associated rates for the OH molecule for elastic scattering, electron impact electronic excitation, and electron impact dissociation, We also study electronic states of the OH − negative ion and resonances in the e-OH system at the OH equilibrium bond length R e = 1.8342 a 0 . Finally, an estimate of the DEA cross section, is made using a model proposed by Munro et al [32]; we are not aware of the DEA of OH being included in models of OH-containing plasmas. The paper is organized as follow: Section 2 and Section 3 report the theoretical framework of R-matrix and the configurations for target and scattering calculations respectively. Section 4 shows the results for potential energy curves, couplings and cross sections. Finally, Section 5 reports our conclusions and perspectives.

The R-matrix Method
Our calculations are permormed using the R-matrix method the details of which can be found in reviews by Burke [33] and Tennyson [34]. The R-matrix method is based on division of the configuration space into an inner region, here a sphere of radius 11 a 0 centred at the molecular centre-of-mass, and an outer region exterior to this sphere. In the inner region, the wave function of the (N +1)-electron system (OH + e − ) is written as a close coupling (CC) expansion, where A is the antisymmetrisation operator, Φ i is the N-electron wave function of the i th target state, F i,j are continuum orbitals and χ i are two-centre L 2 functions constructed by making all (N + 1)-electrons occupy the target molecular orbitals (MOs), and takes into account the polarization of the N-electron target wave function in presence of the projectile electron.
An R-matrix is then built at the boundary of the R-matrix sphere using the inner region wave function. The R-matrix is then propagated to asymptotic distances and matched with known asymptotic functions [35]. The matching yields the K-matrix from which all scattering observables can be extracted.
As Slater type orbitals (STOs) are known to provide a better target representation for diatomic targets, we used the diatomic version of the UK molecular R-matrix codes [36] which uses STOs to represent the target wavefunction. The continuum was represented by numerical orbitals in a partial wave expansion about the molecular center of mass. Since the OH target is neutral, the numerical orbitals were chosen to be spherical Bessel functions. A Buttle correction [37] was also used to allow for the arbitrary fixed boundary conditions imposed on the continuum basis orbitals.

The OH target
A configuration interaction (CI) model was used to represent the OH target. As the choice of basis sets affect the quality of such calculation we tested several STO basis sets, namely those of Cade and Huo [38], Emma et al [39] and Langhoff et al [19]. Finally basis set II given in Table 1 of Langhoff et al was chosen as it gave excitation energies at the OH equilibrium bond length R e = 1.8342 a 0 that were in good agreement with other calculations [19,[21][22][23]. However, at larger bond lengths, the Langhoff et al basis set gave severe linear dependence in our calculation. To mitigate this, one 2s basis function centered on the H atom was deleted and the resulting 'trimmed' basis set was used in all subsequent calculations. This new basis set contained 10 s-type, 6 p-type, 2 d-type basis functions centered on the O atom and 3 s-type and 2 p-type basis functions centered on the H atom respectively.
The STOs were used to build a basis of 35 molecular orbitals consisting of 23σ, 10π and 2δ orbitals. An initial set of SCF calculation was made for the X 2 Π and B 2 Σ − states of OH. Two sets of natural orbitals (NOs) of 2 Π and 2 Σ − symmetry were then obtained by doing a complete active space + singles and doubles (CAS+SD) calculation using these SCF orbitals. The NOs were then used in a subsequent CI calculation.
Tests showed that the target excitation energies were sensitive to the choice of the natural orbitals included in the CAS CI calculations. A compromise set was therefore chosen in which the 3σ, 4σ and 5σ target orbitals were represented by 2 Σ − NOs and the remaining σ, π and δ target orbitals by 2 Π NOs.
We tested several target models. Of these, the model denoted (1σ) 2 (2σ − 6σ, 1π − 2π) 7 was selected; this model has the 1σ orbital frozen and the CAS was defined by (2σ − 6σ, 1π − 2π) 7 . This model gave the best target ground state and excitation energies. Figure 1 shows the behavior of eight lowest states of OH used in our calculation. The asymptotic limits of each of these curves are also shown in the figure. In common with previous electronic structure studies [23], we find that curves above the first excited states are repulsive.
The vertical excitation energies of some of the doublet and quartet states of OH at its equilibrium bond length, R e = 1.8342 a 0 , are shown in Table 1. These are compared with the calculations of van Dishoeck et al [21] (which are more comprehensive) and those in Refs. [19,22,23]. The excitation energies are in good agreement with those of van Dishoeck et al except for the a 4 Σ − and the b 4 Π states which are estimated form the corresponding PECs. Note also that van Dishoeck et al used a different value, R e = 1.85 a 0 , for their equilibrium bond length. The excitation energy for the D 2 Σ + state appears to be higher than that quoted in Qin and Zhang [23] as their value is adiabatic.
The dipole moment of the X 2 Π state was found to be 1.602 D and is in good agreement with the corresponding MCSCF value 1.612 D of Werner et al [40] and the measured value for the vibrational ground state of 1.655 D [41].

Scattering calculations
Our calculations are performed at a single geometry, namely the OH equilibrium geometry R e = 1.8342 a 0 . We have used 14 OH natural orbitals (8σ, 4π, 2δ) and a (2 − 6σ, 1 − 2π) 7 CAS target wave functions which allows for 2 virtual orbitals per symmetry. These were augmented by continuum orbitals F ij expressed as a truncated partial wave expansion about the center of mass retaining partial waves with l ≤ 6 and m ≤ 2 in the expansion. Since the target was neutral, the radial parts of the continuum function was chosen to be spherical Bessel functions and solutions below 109 eV were retained. A Buttle [37] correction was used to correct for the effect of fixed boundary condition used to generate the functions. This produced 59σ, 49π, 40δ continuum functions which were then Schmidt orthogonalized to the target NOs.
Scattering calculations were performed on the 1 Σ + , 1 Σ − , 1 Π, 1 ∆, 3 Σ + , 3 Σ − , 3 Π and 3 ∆ total symmetries of the e+OH system. The summary of the target states used for each symmetry in the CC expansion Eq. (5) is shown in Table 2. Since the contribution to the cross sections from calculations of ∆ symmetry were already found to be small, we did not consider higher symmetries in the calculations.

Results
In the following subsections, we present our results for the OH − bound states, resonance positions and widths at R e = 1.8342 a 0 , cross sections for elastic scattering, electronic excitations and estimate of the DEA cross sections. A more complete study of the DEA process would require detailed negative ion resonance curves and widths as a function of geometry and we propose to undertake this in a subsequent paper. To the best of our knowledge, results for the cross sections presented here have never been studied before. Potential energy (Hartree)

Bound and resonant states
The inner region solutions obtained were used to construct an R-matrix on the boundary. In the outer region, the potential was given by the diagonal and off-diagonal dipole Symmetry Number Target states coupled 1 Σ + 4 one 2 Π, two 2 Σ + and one 2 ∆ states 1 Σ − 3 one each of 2 Π, 2 Σ − , 2 ∆ states 1 Π 4 one each of 2 Π, 2 Σ + , 2 Σ − and 2 ∆ states 1 ∆ 4 one each of 2 Π, 2 Σ + , 2 Σ − and 2 ∆ states 3 Σ + 4 one each of 2 Π, 2 Σ + , 2 ∆ and 4 Π states 3 Σ − 5 one each of 2 Π, 2 Σ − , 2 ∆, 4 Π, and 4 Σ − states 3 Π 5 one each of 2 Π, 2 Σ + , 2 Σ − , 4 Π, and 4 Σ − states 3 ∆ 6 one each of 2 Π, 2 Σ + , 2 Σ − , 2 ∆, 4 Π, and 4 Σ − states moments of the OH target states. The R-matrices were propagated in this potential to 50 a 0 and then matched with exponentially decaying functions obtained from a Gailitis expansion [35]. To find the bound states, the searching algorithm of Sarpal et al [42] with the improved nonlinear, quantum defect based grid of Rabadán and Tennyson [43] was used. This method, originally developed by Seaton [44], searches for the zeros of an energy dependent determinant B(E) using either an energy or a quantum defect grid. The zeros of B(E) can be shown to correspond to the bound state energies. For resonance calculation, the R-matrix was propagated to 70 a 0 to obtain stable results. It was then matched with Coulomb functions using the Gailitis expansion procedure of Noble and Nesbet [35]. The recursive program RESON in the R-matrix code suit [45] was used to detect resonances and fit the resonances to a Breit-Wigner profile to obtain their energies (E) and width (Γ) with an initial energy grid 0.5 × 10 −3 Ryd.
The resonance positions and widths of some of the low lying resonances that are relevant for DEA are shown in Table 3. Interestingly, the 1 3 Σ + resonance is given by Vamhindi and Nsangou [29] but not by Srivastava and Satyamurthy [22], even though both these works study OH − resonant states in some detail.
A limited number of studies on the OH − states exist of which those of Refs. [22,24,26,28,29] are noteworthy. In particular, Chen and Morgan [26], Srivastava and Satyamurthy [22] and Vamhindi and Nsangou [29] obtained potential energy curves for some of the OH − states. These studies indicate that only the X 1 Σ + state of OH − is stable and bound. The a 3 Π, A 1 Π, 1 3 Σ + lie above the parent neutral X 2 Π state for R ≤ 3.0 a 0 and are of resonant character. However, for larger internuclear distances, these states become stable as they lie below the neutral X 2 Π state. The b 3 Π was obtained only by Srivastava and Satyamurthy and is of fully resonant character as it lies above the OH(X 2 Π) ground state for internuclear distances R.
For a comparison of the relative positions of the 1 Π and 3 Π resonant states whose  Table 4. Except for the corresponding results of Sun and Freed [24] which appear too high, all other results are in good agreement with each other. Table 5 compares the energy difference ∆E between the OH(X 2 Π) ground state and the X 1 Σ + , a 3 Π and A 1 Π states of OH. ∆E for the X 1 Σ + state is the electron affinity, E a , of OH. Our results are compared with those of Srivastava and Sharma [22], Chen and Morgan [26], Werner et al [40] and the experimental value of E a given by Schulz et al [46]. Our values of ∆E agree reasonably with those of Srivastava and Sharma. Our computed value of the electron affinity of OH is also in close agreement with all other results, and deviates by about 0.27 eV from the experiment value [46].  Figure 2 shows cross sections for elastic scattering from OH. A Born correction to the cross sections was done to include contribution from higher partial waves. As is usual for molecules with a   significant permanent dipole moment [47], the elastic cross section is strongly peaked at low energies due the strong forward scattering associated with the long-range potential due to the dipole moment. This effect is difficult to detect experimentally but has very recently been observed in low-energy collisions with water using specially designed apparatus sensitive to scattering angles of less than 3.5 • [48]. The elastic cross section also features sharp peaks around 2.5 eV due to capture into the OH − A 1 Π and a 3 Π resonant states. There is also a broad shape resonance like feature around 5 eV which may be due to temporary capture into resonant states with excited OH states as parent states. Figure 3 shows cross sections for excitation of the OH(X 2 Π) ground state to the A 2 Σ + , a 4 Σ − and the 1 2 Σ − states. The cross sections show sharp features due to negative ion resonances. Figure 4 shows the rate coefficients for the X 2 Π → A 2 Σ + . These are compared with the only available rate coefficients given by Riahi et al [16]. The rate coefficients of Riahi et al are obtained by fitting to an Arrhenius form k = a θ b exp (−c/θ). Though the shape of the rate coefficient curves agree well, the rates given by Riahi et al are much larger than ours, particularly at the higher end of the temperature range. In terms of electron energy, the rates we presented are in the range 0 − 3 eV. In this low energy regime, the R-matrix method is known to give good results and hence we believe our rates to be more reliable than those of Riahi et al . It is known that electron impact dissociation occurs via electronic excitation, particularly through excitation to dissociative states. OH has three repulsive dissociative states, namely the a 4 Σ − , 1 2 Σ − and the b 4 Π states, which go to the O( 3 P) + H( 2 S) asymptotic limit. Assuming that excitation to these states above the dissociation threshold (D 0 = 3.88 eV) results in dissociation, we present in Fig. 5 our estimate of the cross section for electron impact dissociation of OH. We predict that this process produces a significant quantity of excited, O( 1 D) atoms. There appears to be no experimental or theoretical data on this process to compare with.

Dissociative electron attachment
To the best of our knowledge, the dissociative electron attachment (DEA) to OH has never been studied before, theoretically or experimentally, though several works on DEA of H 2 O exist [49,50] (see also [51]). A detailed study of the DEA process would require full resonance curves and resonance widths across all internuclear distances considered and is not attempted here. However, we try to give an estimate of the DEA cross sections using an approximation developed by Munro et al [32] Since the details of the method can be found in Ref. [32], we only present the essentials. The inputs for the method are the resonance positions and widths for OH − resonances at a single geometry, here the equilibrium geometry R e = 1.8342 a 0 , the PEC of the target X 2 Π ground state, and the electron affinity of OH, which is taken to be 1.56 eV from column 2 of Table 5. The target PEC is chosen to be a Morse form with dissociation energy D e = 4.51 eV. The A 1 Π and a 3 Π resonance potentials were chosen to be of a Morse form while the 1 3 Σ + resonance potential was chosen to be of an exponentially decreasing form following the shapes given by Srivastava and Satyamurthy [22] and Vamhindi and Nsangou [29] for these resonant states. We mention however, that both these works [22,29] treat resonant part of the A 1 Π, a 3 Π and 1 3 Σ + potentials like bound states in their quantum chemistry calculation and hence must be treated with caution [52]. Moreover, because of their treatment of resonant states as bound, they are not able to provide the resonance widths.  Table 14 of Ref. [51]. Figure 6 shows our estimate of the DEA cross section. We assume that DEA produces only O − since the singlet resonances cannot correlate with the H − +O( 3 P) limit and all three resonance curves go asymptotically to O − +H( 2 S) [22]. Since this is an approximate calculation, we are not able to comment on the details. We find a sharp peak around 3 eV which is clearly due to the A 1 Π, a 3 Π resonances and a second broader peak around 6 eV which can be attributed to the 1 3 Σ + . Since no other results are available for comparison, we have shown the experimental values [50,51] for O − ion production from the dissociative electron attachment to H 2 O which also displays a similar two peak structure. We conclude therefore that the DEA process for the production of O − ions proceeds via the A 1 Π, a 3 Π resonances below 5 eV, but above 5 eV the 1 3 Σ + resonance mainly drives the DEA process. These cross sections are not insignificant and should be included in models of OH plasmas.

Conclusion
We study electron collision with OH using the R-matrix method. Scattering calculations are performed at a single geometry, namely the OH equilibrium geometry R e = 1.8342 a 0 to obtain cross sections for elastic scattering and electronic excitations to the lowest three excited states, namely the A 2 Σ + , a 4 Σ − and the 1 2 Σ − states. We also obtained an estimate of the electron impact dissociation cross section of OH on the assumption that electronic excitation to the states going to the O( 3 P)+H( 2 S) limit above the dissociation threshold leads to dissociation. The scattering calculations also yield OH − bound states and negative ion resonances in the e-OH system. The X 1 Σ + ground state of OH − was found to be bound and several resonances that are likely to be important for DEA were identified. Since a detailed study of the DEA process is beyond the scope of this paper, an estimate of the DEA cross section was obtained using the A 1 Π, a 3 Π and 1 3 Σ + resonances and their widths at a single geometry. To the best of our knowledge, these cross sections are being reported for the first time.
A spreadsheet containing our cross sections is provided as supplementary data to this article.