Rate constant calculations on the N( 4 S) + OH( 2 P) reaction

Transcription

1 Chemical Physics Letters 431 (2006) Rate constant calculations on the N( 4 S) + OH( 2 P) reaction David Edvardsson *, Christopher F. Williams, David C. Clary Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom Received 12 September 2006; in final form 29 September 2006 Available online 10 October 2006 Abstract Calculations of the rate constant for the reaction N( 4 S) + OH( 2 P) have been performed at several temperatures using the rotationally adiabatic capture centrifugal sudden approximation (ACCSA) in combination with ab initio electronic structure theory. The rate constants show good agreement with experimental data. The calculated temperature dependence of k(t) is predicted to have a maximum value of cm 3 molecule 1 s 1 at 66 K. Ó 2006 Elsevier B.V. All rights reserved. 1. Introduction It is well known that neutral neutral reactions involving free radicals play an important role in interstellar chemistry (see e.g. [1]). These reactions can proceed without a potential barrier for most angles of approach and thus have high rate constants (k cm 3 molecule 1 s 1 ) [2,3], even at temperatures as low as K. In fact, many of these reactions show an inverse temperature dependence with rate constants increasing as the temperature is lowered [2 4]. The inclusion of fast neutral neutral reactions into chemical models of dense interstellar clouds can have a profound effect on molecular abundances [5], which makes the determination of rate constants for this class of reactions very important. Theory has a special role to play in this field since experimental investigations of these highly reactive species are difficult to perform. The processes behind nitrogen initiated chemistry in the interstellar medium are of particular interest. The detection of nitric oxide (NO) in interstellar molecular clouds has been reported by several observers [6 8]. However, detection of molecular nitrogen (N 2 ) has proved far more difficult and the first observation of this molecule was reported only a few years ago [9]. It has been suggested * Corresponding author. address: david.edvardsson@chem.ox.ac.uk (D. Edvardsson). [10] that the formation of N 2 is controlled by the two reactions Nð 4 SÞþOH! NO þ H and Nð 4 SÞþNO! N 2 þ O: ð2þ Reactions (1) and (2) can explain how reservoirs of nitrogen molecules are created and also serve as precursor steps towards formation of more complex molecules e.g. ammonia. An accurate determination of the rate constants for these reactions is therefore of great importance. A comparatively small number of experimental studies on the title reaction have been reported in the literature. The only investigations which have focussed on the determination of the rate constant are the discharge-flow and flashphotolysis experiments as reported by Smith et al. [11 13]. These measurements have been performed at temperatures as low as 103 K, and show that the rate constant for reaction between N( 4 S) and OH radicals increases monotonically as the temperature is lowered. By extrapolation of the data, the authors in [13] suggest the use of k(t) =A Æ T B, with A = cm 3 molecule 1 s 1 and B = 0.17, in the modelling of interstellar cloud chemistry. Theoretical investigations, on the other hand, have been quite numerous. These can be divided into two categories: those dealing with the electronic structure problem of reaction (1), e.g. [14 18], and those treating the dynamical part ð1þ /$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi: /.cplett

2 262 D. Edvardsson et al. / Chemical Physics Letters 431 (2006) [19 21]. The electronic structure calculations predict a reaction pathway for addition of N to OH without a potential barrier [15,16]. The reaction proceeds on a 3 A 00 surface to form an intermediate NOH complex bound by 70 kcal mol 1. Calculations predict a barrier of about 8 kcal mol 1 to dissociation of NOH to form NO + H [15], but this is much below the reagent energy and do not impede on the reaction. The equilibrium geometries and relative stabilities of NOH, HNO and the HON - M HNO transition state have been well characterized using ab initio methods at the CCSD(T) level of theory by Lee [17] and more recently by Jursic [18]. The dynamics of reaction (1) were studied by Guadagnini et al. [19] using a standard quasiclassical traectory method (QCT) and the 3 A 00 potential energy surface calculated in [16]. The authors present thermal rate constant at temperatures between 300 and 3000 K, i.e. at much higher temperatures than those prevalent in interstellar clouds. Their results show rate constants essentially independent of the temperature and cross sections which are not strongly dependent on collision energy or initial rotational state. Using the same potential energy surface, Chen et al. [21] performed a time-dependent wavepacket calculation (TDWP) and report rate constants at a similar temperature interval. These authors calculate a larger reaction cross section than the QCT results in [19] and a rate constant with positive temperature dependence. The only study which explicitly considers temperatures of interstellar importance is the Statistical Adiabatic Channel Model (SACM) calculations by Cobos [20]. Cobos used a potential energy surface based mainly on ab initio data from Pauzat et al. [15] and predicts a positive dependence of the rate constants as temperature is increased. The present Letter is an investigation of reaction (1) utilising a combination of ab initio electronic structure calculations and the adiabatic capture centrifugal sudden approximation (ACCSA) method [22,23]. The ACCSA method is readily applied to barrierless reactions with a realistic description of the potential energy surface (e.g. [24,25]), such as the N + OH reaction. The remainder of this Letter is organized as follows: The second section gives a brief description of the ACCSA method. Next, we present the details of the electronic structure and the ACCSA calculations. The fourth section includes the discussion of the results and finally, conclusions are given. 2. Theory 2.1. The adiabatic capture approximation We consider the Hamiltonian for the entrance reaction channel which, in the body-fixed coordinate frame, can be written as bh ¼ h2 1 o 2 2l R or R þ B^ 2 þ b J ^ 2 þ V ðr; cþ: ð3þ 2 2lR 2 Here, l is the reduced mass of the collisional system, B is the rigid rotator rotational constant, ^ is the rotational angular momentum operator, b J is the total angular momentum operator and V(R, c) is the atom molecule interaction potential. The distance from the center of mass of the diatom to the atom is R and c is the Jacobi angle. The centrifugal sudden approximation (CSA) [26,27] is invoked by replacing b J ^ 2 by ½JðJ þ 1Þh 2 þ^ 2 2X 2 h 2 ], where X is the proection of the total angular momentum along the atom-diatom center of mass vector. For fixed values of R and J = 0, the Hamiltonian in Eq. (3), reduces to bh rot ¼ B^ 2 þ 1 2lR 2 ð^ 2 2X 2 h 2 ÞþV ðr; cþ: This Hamiltonian can easily be diagonalized in a basis of spherical harmonics, Xæ, to yield a set of rotationally adiabatic potential energy curves X (R). If a classical capture approximation is made, the reaction probability P JX becomes a step-function which takes on the values 1 if E c P V JX and 0 if E c < V JX, where V JX is the effective potential defined by V JX JðJ þ 1Þ ¼ X ðrþþ Bð þ 1Þ: ð5þ 2lR 2 To determine state-selected cross sections for a given collision energy E c, the maximum value of the total angular momentum, J = J max (,X,E c ), for which V JX for all R is less than E = E c + E must be determined. Together with the simple expression for P JX this allows the reaction cross section, as obtained from the partial-wave expansion, to be written as rð; E c Þ¼ p 2lE c X¼ X¼ ½J max ð; X; E c Þþ1Š 2 : Using Eq. (6), the initial selected rate constants k (T) can be obtained from a Maxwell Boltzmann average sffiffiffiffiffi Z 8 1 k ðt Þ¼ ðk B T Þ 3=2 de c rð; E c ÞE c exp E c ; pl 0 k B T ð7þ where k B is the Boltzmann constant. If the cross sections are fit to the functional form [22] rð; E c Þ¼A E B c ð8þ the integral evaluation in Eq. (7) can be performed analytically and the state selected rate constants written as sffiffiffiffiffi 8 k ðt Þ¼ A ðk B T Þ 0:5þB CðB þ 2Þ; ð9þ pl where C(x) is the Gamma function. The total rate constants k(t) can then be obtained by a Boltzmann average over the initial rotational states. ð4þ ð6þ

3 D. Edvardsson et al. / Chemical Physics Letters 431 (2006) Computational details 3.1. Ab initio calculations Electronic structure calculations have been carried out using the GAMESS-UK program package [28]. In capture theory, only the entrance channel of the reactions need to be considered and therefore the ab initio calculations could be restricted to treat the addition of N to OH. Further, since the radiation field in cold molecular clouds is weak both the nitrogen atom and the OH molecule were considered to be in their respective electronic ground state. Calculations of the 3 A 00 state of the NOH molecule was performed in C s symmetry at the complete active space SCF (CASSCF) level of theory and with the augmented cc-pvtz basis set [29]. The innermost core O1s and N1s orbitals were frozen and the active space was chosen to consist of the 9 valence orbitals {1a 0,2a 0,3a 0,1a 00,4a 0,5a 0,2a 00,6a 0,7a 0 }. The CASSCF space thus defined, together with 12 active electrons (CASSCF(12,9)), contains 1722 configuration state functions (CSF) for the particular state studied. A potential energy surface was calculated on a grid in R and c. The O-H distance was kept fixed at the geometry optimized value r OH = 0.96 Å. A total number of 300 grid points were calculated in the range 1 6 R 6 20 Å and 0 < c < 180. At all grid points, natural orbitals were calculated and the CASSCF wavefunctions were analyzed in terms of the most important CSFs. Subsequently, the CSFs with a weight larger than 0.05 were used to construct a reference space from which single- and double excitations were performed in a direct multi-reference configuration interaction (MRSD-CI) procedure, using the natural orbitals as starting orbitals. The internal space consisted of the active orbital set from the CASSCF calculations and the external space consisted of the complete virtual orbital space. The total number of states generated in this way was close to and the weight of the reference wavefunction was always close to A multi-reference analogue of the Davidson correction (MRSD-CI + Q) was performed to give the final energies to be used. Corrections for basis set superposition errors (BSSE) have not been carried out ACCSA calculations Adiabatic capture calculations have been performed using a code specifically designed for calculations of cross sections and rate constants using the Hamiltonian in Eq. (4). The present model uses a rigid rotor approximation and do not include any coupling between electronic and rotational angular momenta. These approximations have previously been shown to give reliable results when applied to fast low-temperature reactions [2,22]. The calculated ab initio potential energy surface was, for fixed values of the Jacobi angle, fitted to an even power series in the distance variable R according to V ðrþ ¼ A R B 6 R C 8 10 R ð10þ in a least-squares procedure. A smooth surface for arbitrary values of R and c was obtained by interpolation using two dimensional cubic splines. The Hamiltonian in Eq. (4) was diagonalized on a radial grid ranging from 1.5 to 16 Å with a grid spacing of 0.1. The interaction potential was for each value of R expanded in terms of the Legendre polynomials V ðr; cþ ¼ Xl max a l ðrþp l ðcos cþ: l¼0 ð11þ In this way, matrix elements of the potential, ÆXV(R,c) 0 Xæ, can be expressed analytically in terms of the Wigner 3 symbols. The calculated cross sections r(,e c ) were least-squares fit to the functional form of Eq. (8). State-selected rate constants k (T) were obtained from Eq. (9) and then Boltzmann averaged to obtain total rate constants k(t). Calculations were performed over a temperature range between 0 and 600 K. Good convergence of the results was obtained using 20 angular basis functions. The largest initial rotational quantum number ( max ) and total quantum number (J max ) were 40 and 200, respectively. The value of the rotational constant B was cm 1. The total rate constants have been multiplied by a temperature dependent factor F el ðt Þ¼ g el ð12þ Q el to take into account the thermal fine structure distribution of the reactants. In Eq. (12), g el is the degeneracy of the potential energy surface (a factor three in the present calculations) and Q el is the electronic partition function of the reagents. The latter is defined as Q el ¼ 4 ð2þ2expð 205=T ÞÞ; ð13þ where the first part is the contribution from the electronic sublevels in N and the second part arises from the 2 P 3/2 and 2 P 1/2 states in OH [30]. 4. Results The present ab initio calculations confirm and extend previous electronic structure calculations on the title system and predict a potential energy surface which is purely attractive for a large range of collision angles c = The minimum energy of the calculated potential occurs at R = 1.35 Å and c =108, which is in good agreement to the calculated minimum energy geometry of NOH [17]. The intermediate NOH complex is found to be strongly bound by 71 kcal mol 1 relative to the N + OH dissociation limit. The ACCSA state-selected reactive cross sections are shown in Fig. 1. As can be seen, the cross section decreases with increasing collisional energy. This is because a higher

4 264 D. Edvardsson et al. / Chemical Physics Letters 431 (2006) OH ( = 0) Reaction Cross Section (a 2 ) = 5 = Collision Energy (ev) Fig. 1. State-selected reactive cross sections for the reaction N + OH as a function of collision energy. initial translational energy makes it more difficult for the nitrogen atom to be captured in the potential well. A higher initial rotational quantum number also makes formation of an intermediate complex more difficult thus reducing the cross section. The calculated cross sections are slightly higher than values calculated by Guadagnini et al. [19] for which reported values are 37.8 and 33:0a 2 0, at a collision energy of 0.10 ev for = 0 and = 10, respectively. It should be noted that the electronic statistical factor F el is not included in the cross sections. The observed trends in the cross sections are directly reflected in the state-selected rate constants k (T). Fig OH ( = 0) k (10 10 cm 3 molecule 1 s 1 ) = 5 = T (K) Fig. 2. State-selected rate constants for the reaction N + OH as a function of temperature.

5 D. Edvardsson et al. / Chemical Physics Letters 431 (2006) shows the rotationally selected rate constants for a range of values and in the temperature interval K. The stateselected rate constants are quite sensitive to the initial quantum number and decrease as is increased. However, for a fixed value of the k (T) increase when the temperature is raised. This temperature dependence is reduced as the initial quantum number is increased, and for =20 the temperature dependence has essentially vanished. Calculation of the total rate constant k(t) allows for a comparison between the calculations and experimental results. The total ACCSA rate constant is shown in Fig. 3. In contrast to the state-selected rate constants, the overall rate constant k(t) shows a more complex behaviour with a positive temperature dependence up to ca. 70 K and a slight negative temperature dependence for higher temperatures. This is due to the Maxwell Boltzmann average of the different k (T) and the temperature dependent electronic statistical factor in Eq. (12). This factor asymptotically converges towards 3/8 at low temperatures and 3/16 at high temperatures. The decrease of F el (T) as the temperature increases directly carries over to the total rate constant. The high temperature limit of F el (T) reflects the fraction of the number of reactive and the total number of potential energy surfaces. At low temperatures, the splitting of the fine-structure levels due to spin-orbit coupling becomes more important and the high temperature degeneracy is lost. The agreement between the ACCSA rate constants and experimental data is very good, especially for T P 300 K. As can be inferred from Table 1, the largest deviation from experiment is only a factor of 1.6 at 103 K. In the interval between 100 and 300 K the theoretical rate constant curve Table 1 Theoretical (ACCSA) and experimental [11 13] rate constants a for N+OH T (K) ACCSA Experiment ± ± ± ± ± ± ± ± ± ± ± 0.37 a Rate constants are given in units of cm 3 molecule 1 s 1. does not exhibit the same steep gradient as the experimental data show. Further, the ACCSA calculations indicate that the maximum value of k(t) occurs around T =66K. This is an intriguing prediction for future low-temperature experiments. It is interesting to compare the present ACCSA results to other theoretical investigations. The ACCSA rate constants are close to the QCT results of Guadagnini et al. [19] at T = 300 K. These authors used a constant electronic factor 3/16 in the calculations of the rate constants. This is a good approximation for temperatures T > 1000 K, but at T = 300 K the high limit value of F el (T) has not been reached. As can be seen from the ACCSA calculations, F el (T) has a large influence on k(t) at low temperatures. Rate Constant (10 11 cm 3 molecule 1 s 1 ) Exp. Ref. [11] Exp. Ref. [12] Exp. Ref. [13] QCT Ref. [19] SACM Ref. [20] TDWP Ref. [21] Present study T (K) Fig. 3. Calculated and experimental total rate constants k(t) for the reaction N + OH as a function of temperature.

6 266 D. Edvardsson et al. / Chemical Physics Letters 431 (2006) The TDWP calculations by Chen et al. [21] predict a slight, but positive temperature dependence over the temperature interval studied and thus deviate from the present results. This can probably be explained by the difficulty in converging the wavepacket calculations at low collision energies. The use of a different potential energy surface and the use of the J-shifting approximation in the TDWP calculations, could also explain the different results obtained. In [20], Cobos employs the SACM method to obtain a temperature dependence µt +0.30, and calculates a rate constant at 10 K of cm 3 molecule 1 s 1. The present ACCSA calculations predict a higher value, cm 3 molecule 1 s 1. However, both values are in the range required for reaction (1) to fit observed relative abundances of NO in the cold molecular cloud L134N [7,20]. At this temperature, both the SACM and ACCSA values are considerably lower than that obtained by extrapolation of experimental data in [13] and new experiments at low temperatures are thus needed to verify the theoretical predictions. 5. Conclusions We have shown that by using a combination of ab initio electronic structure theory and the ACCSA method for calculating rate constants for the N + OH reaction, good agreement with experimental data in the temperature range K can be obtained. At temperatures below 100 K no experimental data is currently available which allows for a comparison. In this temperature regime we predict a slight increase of k(t) until it reaches a maximum value of cm 3 molecule 1 s 1 at 66 K. The study shows that the ACCSA method is a valuable tool in the prediction of rate constants for fast chemical reactions and we hope that the present results will spur future low temperature experiments on this reaction. Acknowledgements This proect was supported by the Molecular Universe Research Training Network FP6 No. MRTN-CT and the US Office of Naval Research Prediction and Control of Chemical Reactions, ONR Grant No. N One of the authors (C.F.W.), acknowledges funding from the Natural Environment Research Council (NERC). References [1] E. Herbst, Chem. Soc. Rev. 30 (2001) 168. [2] D.C. Clary, Ann. Rev. Phys. Chem. 41 (1990) 61. [3] I.W.M. Smith, Angew. Chem. Int. Edit. 45 (2006) [4] T. Stoecklin, C.E. Dateo, D.C. Clary, J. Chem. Soc. Faraday Trans. 87 (1991) [5] R.P.A. Bettens, H.-H. Lee, E. Herbst, Astrophys. J. 443 (1995) 664. [6] D. McGonagle, L.M. Ziurys, W.M. Irvine, Y.C. Minh, Astrophys. J. 359 (1990) 121. [7] M. Gerin, Y. Viala, F. Pauzat, Y. Ellinger, Astron. Astrophys. 266 (1992) 463. [8] M. Gerin, Y. Viala, F. Casoli, Astron. Astrophys. 268 (1993) 212. [9] D.C. Knauth, B.-G. Andersson, S.R. McCandliss, H.W. Moos, Nature 429 (2004) 636. [10] E. Herbst, W. Klemperer, Astrophys. J. 185 (1973) 505. [11] M.J. Howard, I.W.M. Smith, Chem. Phys. Lett. 69 (1980) 40. [12] M.J. Howard, I.W.M. Smith, J. Chem. Soc. Faraday Trans. 77 (1981) 997. [13] I.W.M. Smith, D.W.A. Stewart, J. Chem. Soc. Faraday Trans. 90 (1994) [14] P.J. Bruna, C.M. Marian, Chem. Phys. Lett. 67 (1979) 109. [15] F. Pauzat, Y. Ellinger, G. Berthier, M. Gérin, Y. Viala, Chem. Phys. 174 (1993) 71. [16] R. Guadagnini, G.C. Schatz, S.P. Walch, J. Chem. Phys. 102 (1995) 774. [17] T.J. Lee, Chem. Phys. Lett. 223 (1994) 431. [18] B.S. Jursic, J. Mol. Struct. (Theochem) 496 (2000) 207. [19] R. Guadagnini, G.C. Schatz, S.P. Walch, J. Chem. Phys. 102 (1995) 784. [20] C.J. Cobos, Int. J. Chem. Kinet. 27 (1995) 219. [21] M.-D. Chen, B.-Y. Tang, K.-L. Han, N.-Q. Lou, J.Z.H. Zhang, J. Chem. Phys. 118 (2003) [22] D.C. Clary, Mol. Phys. 53 (1984) 3. [23] D.C. Clary, Mol. Phys. 54 (1985) 605. [24] D. Reignier, T. Stoecklin, S.D. Le Picard, A. Canosa, B.R. Rowe, J. Chem. Soc. Faraday Trans. 94 (1998) [25] S.D. Le Picard, A. Canosa, D. Reignier, T. Stoecklin, Phys. Chem. Chem. Phys. 4 (2002) [26] R.T. Pack, J. Chem. Phys. 60 (1974) 633. [27] P. McGuire, D.J. Kouri, J. Chem. Phys. 60 (1974) [28] GAMESS-UK is a package of ab initio programs. Available from: < M.F. Guest, I.J. Bush, H.J.J. van Dam, P. Sherwood, J.M.H. Thomas, J.H. van Lenthe, R.W.A. Havenith, et al., Mol. Phys. 103 (6 8) (2005) 719. [29] J.T.H. Dunning, J. Chem. Phys. 90 (1989) [30] D.C. Clary, H.-J. Werner, Chem. Phys. Lett. 112 (1984) 346.