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Collision-theory calculations for some quadrupole-quadrupole reactions of O(1D)
Volume 169, number 3
CHEMICAL PHYSICS LETTERS
COLLISION-THEORY CALCULATIONS FOR SOME QUADRUPOLE-QUADRUPOLE
1 June 1990
REACTIONS OF O(‘D)
Leon F. PRILLIPS Chemistry Department, University of Canterbury, Christchurch, New Zealand
Received 13 October 1989; in final form 15 March 1990
Langevin-type capture rate constants have been obtained for collisions of 0( ‘D) with CO*, H1, N2 and 02, using approximate quasiclassical trajectory calculations with a combined quadrupole-quadrupole+ London t Morse potential, over the temperature range lo-600 K. The calculated capture rates depend about equally on the quadrupole+ptadrupole and London potentials and are independent of the Morse potential. Rate constants are obtained in good agreement with listed experimental values at 300 K when the calculated capture rates are multiplied by a factor of 0.2 for electronic degeneracyand by a factor of 1.O,0.545 or 0.455 for orientation. These orientation factors derive from the angular factorsof the quadrupole-quadrupole potential. For the reactions with O2 and CO, the orientation factor is 1.0 (no preferred orientation), for reaction with H2 it is 0.545 (end-on collisions favoured), and for reaction with N2 it is 0.455 (perpendicular collisions favoured).
1. Introduction For a bimolecular reaction with no activation energy, the rate of crossing the centrifugal barrier in the long-range intermolecular potential constitutes a reliable upper bound for the overall reaction rate. In many instances the calculated capture rate, i.e. the rate of crossing the centrifugal barrier, either is equal to the reaction rate or is equal to an easily predicted multiple of the reaction rate. In other cases more elaborate calculations are required to take account of additional barriers in the entrance or exit channels for formation and decay of the short-lived collision complex. Such barriers, which must lie below the en-
ergy of the separated reactants if they are not to give rise to an Arrhenius-type activation energy, reduce the overall reaction rate by introducing either back reflections, from the viewpoint of a trajectory calculation, or an entropy bottleneck from the viewpoint of a statistical calculation. The many existing calculations for such reactions vary greatly in sophistication, ranging from purely classical, Langevin-type calculations of capture rates on analytical potential curves to quantum-scattering calculations on high-quality ab initio surfaces. A number of different quantum-scattering approaches have been
compared by Clary and Henshaw [ 11 in a paper which forms part of a notable Faraday discussion on chemical reaction dynamics [ 21. An agreeable feature of these reactions, and one which is not widely appreciated, is that very good predictions of rate constants can often be made on the basis of relatively unsophisticated theories, in which category must be included the collision theory that is the basis of the present paper. The class of bimolecular reactions with no activation barrier includes many ion-molecule reactions and most radical-radical reactions. Such reactions are usually very fast, and many of them are important in systems such as planetary atmospheres and interstellar clouds, systems in which conditions are far removed from ordinary laboratory pressure and temperature. Fortunately, the calculations that can be made for such reactions work at least as well for the extremes of low pressure (where there are no complications arising from the need to distinguish between weak and strong collisions) and low temperature (where the range of rotational quantum numbers that must be included is small) as they do for more ordinary conditions. Consequently, useful predictions can often be made for rate constants which would be very difficult to determine experi-
0009-2614/90/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland )
253
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mentally. In addition, and probably more important, the calculations can provide insight into the factors which control and limit radical-radical reaction rates, into the effects of particular reactant energy distributions and, for the more sophisticated calculations, into the significance of particular product energy distributions, This paper gives the result of some calculations of reaction rates using Langevin-type collision theory, by which is meant an essentially classical calculation of barrier-crossing rates on an analytical potential surface. The main aim of the paper is to show that such simple calculations can furnish worthwhile insight as well as useful numbers. The calculations employ a recently developed computer program [ 31 to obtain capture rates over the centrifugal barrier in a long-range quadrupole-quadrupole + London potential, by means of approximate, quasiclassical trajectory calculations. The calculations are quasiclassical in the sense that rotation of reactants is quantised, but not the orbital angular momentum of collisions, and they are approximate to the extent that the program runs reasonabIy quickly without resorting to Monte Carlo methods, while still giving results which are in good agreement with accurate classical trajectory calculations. The program has been used previously to calculate dipole-dipole capture rates for the reaction of 3H with NO [ 41 and for a number of atmospheric radical-radical reactions [ 51, dipole-quadrupole capture rates for some reactions of 0( 3P) [ 61, and ion-dipole, ion-quadrupole, and ion-induced dipole capture rates for a number of reactions of CH: [7]. The analytical potential used here comprises the quadrupole-quadrupole potential I’oo = (3Qi QJ4r’)
( 1 - 5 cos’f3, - 5 cos2&
+ 17 cos20, cos2& + 2 sir?& sir?& cos*$~ +16sin8,
cos8,
sin&cost$cos~+...),
(1)
where the quadrupole moments Qi and Q2 and the angles are as defined by Buckingham [ 81, with a superimposed isotropic London potential V,=-3a,cu2hv,Y2/2(Y,+V2)r6,
(2)
where (pi and IY~are polarizabilities and u1and v2 are frequencies which correspond to the first ionization potentials of the interacting species. There is also the 254
1 June 1990
option of superseding the combined potential by a Morse potential at, distances less than the separation at which the Morse potential becomes more strongly attractive than the most negative sum of potentials (1) and (2). Expression (2) for the London potential is usually considered to underestimate the strength of the interaction by up to a factor of two for ground state species. For electronically excited species the interaction is expected to be weaker and in extreme cases may even become repubive. However, for the interaction of a ground-state molecule with a species such as 0( ‘D) in a low-lying metastable state, in the framework of the usual perturbation treatment, the effect of perturbations involving the ground-state of the metastable is likely to be greatly outweighed by perturbations involving continuum states, so that the London potential is essentially the same as that between two ground state species having the same polarizabilities and ionization potentials. For the interaction of a molecular quadrupole with an atomic quadrupole (species 2), the value of 9 in eq. ( 1) is set at zero and the value of 0, is chosen for each value of 8i, to give the largest absolute value of the angular factor in parentheses. Finally, the sign of the potential is adjusted, if necessary, to ensure that V,, is always negative. The computer program allows the sign of the atomic quadrupole moment to be specified and returns the fraction of positive angular factors in the resulting formula for Voo when 8i is varied in equal steps over the range 0-2x (with weighting proportional to sin 8, ) _The significance of this fraction (which, with proper allowance for the number of occurrences of equal and opposite maximum values of the angular factor, is either 0.545, for QiQ2 negative, or 0.455, for Q,Q2 positive) is as follows: For a molecule such as CO*, with a negative quadrupole moment, the most favourable direction of approach for an atomic quadrupole with a positive quadrupole moment is along the internuclear axis of the molecule. The fraction of angular factors which correspond to this approach, i.e. for which VW is automatically negative because of the sign of QlQ2, is 0.545. For the remaining 0.455 of the angular factors the sign of VW has to be adjusted by the program to make it correspond to an attractive interaction. Thus, assuming the distribution of collisions over 01 to be proportional to sin&, a fraction 0.545 of the total
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CHEMICAL PHYSICS LETTERS
capture rate must relate to “end-on” collisions and a fraction 0.455 to “perpendicular” collisions and, if reaction occurs for only one kind of collision, the calculated capture rate should be multiplied by the appropriate fraction. This general conclusion needs to be qualified by the observation that the angular dependence of the chemical forces which lead to a preference for one kind of collision orientation over another is not. likely to be identical with the angular dependence of the quadrupole-quadrupole potential, so that there is necessarily some uncertainty in the factors that are here taken as 0.545 and 0.455. Also, at intermediate angles the state of 0( ‘D) is best regarded as a quantum mechanical superposition of states corresponding to opposite signs of Q. Thus the conclusion is to some extent qualitative rather than quantitative. Nevertheless, the difference between 1.O and the factors 0.545 and 0.455 is large enough to be outside the limits of error of the experimental results with which the calculations are to be compared, and on this basis it may be possible, by comparing the calculated capture rate with the measured reaction rate, to conclude either that one particular orientation is required for reaction, or that there is no preferred orientation. For the reaction of 0( ‘D) with Hz, Fitzcharles and Schatz [9] found that quasiclassical trajectory calculations on two different potential surfaces led to opposite conclusions conceming the relative importance of end-on and perpendicular collisions, so such deductions must be treated with caution. Nevertheless, in that instance one of the surfaces was clearly preferable to the other on other criteria and, in cases where good agreement is obtained between the experimental rate constant and the calculated capture rate, such deductions deserve to be taken seriously. In addition to possible factors less than unity for orientation, the raw capture rate must also be multiplied by a factor for the fraction of collisions which occur on the lowest potential surface, collisions on other surfaces being assumed unable to lead to sufficiently close approach of the reactants. For 0( ID) in a system of C, symmetry the required factor is 0.2. More detailed discussion of this point is given elsewhere [5].
1 June 1990
2. Results and discussion For all of the reactions considered here the experimental outcome is physical quenching of 0 ( ‘D) to the 3P state. This is assumed to require a close approach of the two reactants on an attractive potential-energy surface. Calculated capture rates for the reaction of 0 ( ’ D ) with CO* are given in table 1. The quadrupole moment of 0( ID) was taken to be 1.07x 1O-26esu cm*, from the calculations of Gentry and Giese [ lo], with Buckingham’s definition of Q. The polarizability of 0( ‘D) is expected to be the same as that of 0 (3P), namely 0.802 x 1O-24 cm3, since the two states arise from the same electron configuration [ 111. Other data used in the calculation are listed below the table. The listed value of k3,,,,,the rate constant for this reaction at 300 K, is 1.1x lo-” cm3 molecule-’ s-‘, with an uncertainty amounting to a factor of 1.2 [ 121; kT is independent of temperature to within experimental error. The results in column 2 of table 1 were obtained for combined Morse, quadrupole-quadrupole and London potentials, the London potential being as given by eq. (2). Replacing the Morse potential by a hard-sphere potential led to results which differed from those in column 2 by not more than 1 or 2 in the fourth significant figure, so these results are not given in the table. The results in column 3 were obtained by omitting the quadrupole-quadrupole potential and those in column 4 by omitting the London potential. These two potentials appear to be of about equal importance in determining the rate constant of this reaG tion. Columns 5 and 6 show the effect of increasing the magnitude of the London potential by factors of 1.4 and 2.0, respectively. For comparison with the experimental rate constant, the capture rate has to be multiplied by a factor of 0.2, for electronic degeneracy, and possibly by a factor near 0.5 to take account of any preference for perpendicular or end-on collisions. For the reaction with C02, the present results imply that there is no preferred orientation, values of 8.50x10-“, 8.98~ 10-l’ and 9.60x lo-” cm3 molecule-’ s-’ being obtained for k,, with only the electronic degeneracy factor and with the London potential (2) multiplied by factors of 1.0, 1.4 and 2.0, respectively. Best agreement with experiment is obtained with the factor 2.0, and the last column of table 1 contains kT values calculated over 255
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Table 1 Results for reaction of O(‘D) with CO*. Units of k. 10-r’ cm3 molecule-’ s-‘. Factors used with London potential in parenthesis. k,,=cnpture rate; k_,=capture rate with London potential omitted; k_o=capture rate with quadrupole-quadrupole potential omitted, k,=calculated reaction rate at T, orientation factor= 1.O ‘) T(K)
kcapP(l.0)
k_,( 1.0)
k.dO.0)
%P( 1.4)
kuP(2.0)
kr(2.0)
10 20 30 50 75 100 150 200 250 300 350 400 500 600
31.7 34.1 35.7 37.8 39.5 40.7 42.2 42.7 42.7 42.5 42.0 41.5 40. I 38.6
20.5 22.7 24.1 25.9 27.4 28.6 30.1 31.3 32.0 32.5 32.7 32.9 33.2 33.5
28.8 30.4 31.5 32.8 33.8 34.2 34.3 33.8 33.1 32.4 31.7 30.9 29.6 28.5
32.8 35.4 37.2 39.5 41.4 42.1 44.1 44.8 45.0 44.9 44.5 44.2 42.9 41.5
34.2 37.0 38.9 41.5 43.7 45.2 46.9 47.8 48.0 48.0 47.8 47.3 46.1 44.1
6.84 7.40 7.78 8.30 8.74 9.04 9.38 9.56 9.60 9.60 9.56 9.46 9.22 8.94
N Data: Morse (r, (A), D. (kJ mol-I), o.(cm-‘))=(1.2,50,500); lO-26 esu cm;; &a&ability (CO, j =2,65 x lCt’24cm3.
the whole temperature range using this factor. The calculated results show only a small temperature dependence, in agreement with experiment. Similar conclusions to those in the last paragraph concerning the relative importance of Morse, London and quadrupole-quadrupole potentials can be drawn from calculations for the reactions of 0 ( ‘D ) with HZ,Nz and OZ.Because of the small quadrupole moment of 02, the London potential plays a correspondingly more significant role in this case. The calculated small temperature dependences for these reactions are also in accord with experiment. Table 2 shows results for the reaction with Hz with all potentials included, with the London potential (2) multiplied by factors of 1.0, 1.4 and 2.0, as before. The listed value of kaOeis 1.Ox 1O- ‘Ocm3 molecule- ’ s-l with an uncertainty factor of 1.2. The fao tors of 1.0, 1.4 and 2.0 for the London potential lead to boo values of 0.77x10-lo, 0.83~10-‘~ and 0.94x lo-“, respectively, when the corresponding capture rates are multiplied by factors of 0.2 for electronic degeneracy and 0.545 for orientation. All three results are consistent with the listed value but there is a clear preference for the factor 2.0. Use of the factor 0.545 for orientation implies that end-on collisions are required for reaction, in agreement with the results of Fitzcharles and Schatz for their best ab in256
Ionizationpotentials
(cm-‘)=94000,111000;
Q(COr)= -4.2~
Table 2 Results for reaction of O(‘D) with Hz. Units of k: IO-” cm3 moleculei’ s-‘. Factors used with London potential in parenthesis. Orientation factor=0.546 D) T(R)
k&1.0)
kW(l.4)
ka(2.0)
kT(2.0)
10 20 30 50 75 100 150 200 250 300 350 400 500 600
52.2 56.5 59.0 61.5 62.5 63.6 65.8 67.9 69.1 71.3 12.6 73.8 75.6 77.9
54.4 59.4 61.8 64.5
56.4 62.3 66.0 69.4 71.1 73.0 75.8 78.6 80.8 82.8 84.3 85.8 87.9 89.5
6.16 6.80 7.21 7.58 1.16 7.97 8.28 8.58 8.82 9.04 9.21 9.37 9.60 9.77
65.8 67.5 70.5 73.0 75.0 76.6 78.0 79.2 80.8 82.0
‘) Data:Mome (rmD,(kJmol-r),~(cm-‘))=(1.2, 720,360(l); Ionization potentials/cm-‘=94000, 124000; Q(H2)= t 0.651x10-26esucm2;polarizabilityH,=0.8l9~l0-24cm3.
itio surface, but the difference from the factor 0.455 for broadside collisions is not suffkient for an unequivocal distinction. The last column of table 2 gives kT values for the reaction with Hz over the whole temperature range, calculated using the factors 2.0
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for the London potential and 0.545 for orientation. Table 3 gives results for the reaction with N2 analogous to those for H2 in table 2. The listed value of ,&, for this reaction is 2.6X lo-” cm3 molecule-’ s-l, with an uncertainty factor of 1.2. Factors of 1.0, 1.4 and 2.0 for the London potential lead to values 3.34x lo-“, 3.62~ lo-” and 3.91 X lo-” cm3 molecule- l s- ’ for 500 when the capture rate is multiplied by factors of 0.2 for electronic degeneracy and 0.455 for orientation, The first value of /c300is slightly larger than the upper limit of the experimental value: in view of the uncertainty regarding the factor to be used for the preferred orientation, this degree of agreement is probably acceptable. Use of the factor 0.455 for orientation implies that perpendicular collisions are required for reaction, which is perhaps surprising in view of the anticipated tendency of this reaction system to form an intermediate NzO* complex. However, the reaction with Hz shows a preference for end-on collisions, even though it probably proceeds through a triangular H20* complex, so the conclusion is not unprecedented. One might speculate that an oblique approach is dictated by angular momentum requirements for barrier crossing, as appears to be the case of the reaction of O(3P) with OH [ 131. The alternative to assuming that there is Table 3 Results for reaction of 0( ‘D) with N,. Units of k IO-” cm3 molecule-r s-i. Factors used with London potential in parenthesis. Orientation factor=0.454 ‘) T(R)
kXP(1.0)
10 20 30 50 15 100 150 200 250 300 350 400 500 600
26.2 21.8 29.1 30.8 32.4 33.4 34.9 35.9 36.5 36.8 37.1 31.3 37.4 37.3
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CHEMICAL PHYSICS LETTERS
k(l.4) 27.7 29.5 31.0 ’ 32.9 34.7 35.9 31.7 38.7 39.5 39.9 40.3 40.4 40.5 40.4
k(2.0)
Wl.0)
29.4 31.6 33.3 35.6 37.6 39.0 40.9 41.9 42.6 43.1 43.3 43.5 43.6 43.5
2.38 2.52 2.64 2.80 2.94 3.04 3.17 3.26 3.31 3.34 3.37 3.39 3.40 3.39
‘) Data: Morse (re (A), De (kJ mol-‘), w3 (cm-‘)) = (1.2,350, 1285); Ionization potentials (cm-‘) =94000, 126000; Q(N2)=-l.5~10126 esu cm’ polarizability (N2)=l.76x 10mZ4cm3.
Table 4 Results for reaction of O(‘D) with OZ. Units of k lo-” cm3 molecule-’ s-i. Factors used with London potential in parenthesis. Orientation factor= 1.O&r T(R)
k-w(l.0)
kw( 1.4)
kP(2.0)
Wl.4)
10 20 30 50 75 100 150 200 250 300 350 400 500 600
18.7 20.6 21.7 22.7 23.1 23.1 22.7 21.8 20.7 19.6 18.5 17.5 15.7 14.3
20.4 22.7 24.0 25.2 25.7 25.7 25.2 24.2 23.0 21.7 20.5 19.4 17.5 15.7
22.1 25.2 26.7 28.1 28.6 28.7 28.2 27.1 25.1 24.3 23.0 21.8 19.6 17.8
4.08 4.54 4.80 5.04 5.18 5.18 5.04 4.84 4.60 4.34 4.10 3.88 3.50 3.14
a~Data:Morse(r,(A),D,(kJmol-‘),o~(cm-’))=(l.2,292, 1040); Ionization potentials (cm-‘) =94000, 97000; Q(Oa)= -0.4~ 10ez6 esu cm*; polarizability (O,)= 1.60~ 10wg4cm3.
preferred orientation is to suppose that a similar factor arises from the presence of a barrier in the entrance or exit channel for the quenching reaction, which seems less likely. Table 4 contains capture rates for the reaction with O2 similar to those given in table 2 and 3 for reactions with Hz and NZ. The listed value of k3m is 4.6 x 10-l’ cm3 molecule-’ s-‘, again with an uncertainty factor of 1.2. Use of the factors 1.O, 1.4 and 2.0 for the London potential, with a factor of 1 .Ofor orientation, gives k300 values of 3.92 X lo-“, 4.34~ lo-” and 4.86x lo-” cm3 molecule-’ s-l, respectively. All three values are in agreement with the listed value; the values of /cTin the last column of table 4 were calculated using a factor of 1.4 for the London potential. For this reaction the results imply that there is no preferred orientation, which again is counter to the intuitive expectation that the reaction should proceed through a triangular 0; complex. a
3. Conclusions For a quadrupole-quadrupole reaction between an atom and a molecule it is possible to make a plau257
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sible distinction among reactions requiring end-on or perpendicular collisions and reactions with no preferred collision orientation on the basis of the sign that the atomic quadrupole must take in order to produce an attractive interaction. The angular factors for this interaction are such that about 55% of collisions will have end-on and 45% will have perpendicular orientations. If one orientation or the other is preferred, the calculated capture rate, after allowance for electronic degeneracy, has to be multiplied by the appropriate factor for comparison with the experimental rate constant. The present approximate, quasiclassical trajectory calculations of Langevin-type capture rates lead to results in remarkably good agreement with experiment provided it is assumed that the reactions of 0( ‘D) with CO2.and O2 have no preferred collision orientation, while the reaction with Hz requires end-on collisions and the reaction with N2 requires perpendicular collisions. For best agreement the London potential given by eq. (2) needs to be multiplied by a factor of 2.0 (H,, COz), 1.4 (OI), or 1.0 (N,).
258
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Acknowledgement
I am grateful to a referee for pointing out the necessity of including a sin 13~weighting of the angular factors of the quadrupole-quadrupole potential. References [ I ] DC. Clary and J.P. Henshaw, Faraday Discussions Chem. Sot. 84 (1987) 333. [ 2 JFaraday Discussion No. 84, Dynamics of elementary gasphase reactions (Roy. Sot. Chem., London, 1987). [3] L.F. Phillips, J. Comput. Chem. 1I (1990) 88. [4] L.F. Phillips, Chem. Phys. Letters 168 (1990) 197. [ 51L.F. Phillips, J. Phys. Chem., submitted for publication. [6] L.F. Phillips, Chem. Phys. Letters 165 (1990) 545. [ 71 L.F. Phillips, J. Phys. C&n., in press. [8] A.D. Buckingham, Advan. Chem. Phys. 12 (1967) 107. [ 91 M.S. Fitzcharles and G.C. Schatz, J. Phys. Chem. 90 ( 1986) 3634. [ lo] W.R. Gentry and CF. Giese, J. Chem. Phys. 67 (1977) 2355. [ 111P.C. Schmidt, M.C. Boehm and A. Weiss, Ber. Bunsenges. Physik. Chem. 89 (1985) 1330. [ 121R. Atkinson, D.L. Baulch, R.A. Cox, R.F. Harnpson Jr., J.A. Kerr and J. Troe, J.,Phys. Chem. Ref. Data 18 (1989) 881. [ 131N. Markovic, G. Nyman and S. Nordholm, C&em. Phys. Letters 159 (1989) 435.
CHEMICAL PHYSICS LETTERS
COLLISION-THEORY CALCULATIONS FOR SOME QUADRUPOLE-QUADRUPOLE
1 June 1990
REACTIONS OF O(‘D)
Leon F. PRILLIPS Chemistry Department, University of Canterbury, Christchurch, New Zealand
Received 13 October 1989; in final form 15 March 1990
Langevin-type capture rate constants have been obtained for collisions of 0( ‘D) with CO*, H1, N2 and 02, using approximate quasiclassical trajectory calculations with a combined quadrupole-quadrupole+ London t Morse potential, over the temperature range lo-600 K. The calculated capture rates depend about equally on the quadrupole+ptadrupole and London potentials and are independent of the Morse potential. Rate constants are obtained in good agreement with listed experimental values at 300 K when the calculated capture rates are multiplied by a factor of 0.2 for electronic degeneracyand by a factor of 1.O,0.545 or 0.455 for orientation. These orientation factors derive from the angular factorsof the quadrupole-quadrupole potential. For the reactions with O2 and CO, the orientation factor is 1.0 (no preferred orientation), for reaction with H2 it is 0.545 (end-on collisions favoured), and for reaction with N2 it is 0.455 (perpendicular collisions favoured).
1. Introduction For a bimolecular reaction with no activation energy, the rate of crossing the centrifugal barrier in the long-range intermolecular potential constitutes a reliable upper bound for the overall reaction rate. In many instances the calculated capture rate, i.e. the rate of crossing the centrifugal barrier, either is equal to the reaction rate or is equal to an easily predicted multiple of the reaction rate. In other cases more elaborate calculations are required to take account of additional barriers in the entrance or exit channels for formation and decay of the short-lived collision complex. Such barriers, which must lie below the en-
ergy of the separated reactants if they are not to give rise to an Arrhenius-type activation energy, reduce the overall reaction rate by introducing either back reflections, from the viewpoint of a trajectory calculation, or an entropy bottleneck from the viewpoint of a statistical calculation. The many existing calculations for such reactions vary greatly in sophistication, ranging from purely classical, Langevin-type calculations of capture rates on analytical potential curves to quantum-scattering calculations on high-quality ab initio surfaces. A number of different quantum-scattering approaches have been
compared by Clary and Henshaw [ 11 in a paper which forms part of a notable Faraday discussion on chemical reaction dynamics [ 21. An agreeable feature of these reactions, and one which is not widely appreciated, is that very good predictions of rate constants can often be made on the basis of relatively unsophisticated theories, in which category must be included the collision theory that is the basis of the present paper. The class of bimolecular reactions with no activation barrier includes many ion-molecule reactions and most radical-radical reactions. Such reactions are usually very fast, and many of them are important in systems such as planetary atmospheres and interstellar clouds, systems in which conditions are far removed from ordinary laboratory pressure and temperature. Fortunately, the calculations that can be made for such reactions work at least as well for the extremes of low pressure (where there are no complications arising from the need to distinguish between weak and strong collisions) and low temperature (where the range of rotational quantum numbers that must be included is small) as they do for more ordinary conditions. Consequently, useful predictions can often be made for rate constants which would be very difficult to determine experi-
0009-2614/90/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland )
253
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mentally. In addition, and probably more important, the calculations can provide insight into the factors which control and limit radical-radical reaction rates, into the effects of particular reactant energy distributions and, for the more sophisticated calculations, into the significance of particular product energy distributions, This paper gives the result of some calculations of reaction rates using Langevin-type collision theory, by which is meant an essentially classical calculation of barrier-crossing rates on an analytical potential surface. The main aim of the paper is to show that such simple calculations can furnish worthwhile insight as well as useful numbers. The calculations employ a recently developed computer program [ 31 to obtain capture rates over the centrifugal barrier in a long-range quadrupole-quadrupole + London potential, by means of approximate, quasiclassical trajectory calculations. The calculations are quasiclassical in the sense that rotation of reactants is quantised, but not the orbital angular momentum of collisions, and they are approximate to the extent that the program runs reasonabIy quickly without resorting to Monte Carlo methods, while still giving results which are in good agreement with accurate classical trajectory calculations. The program has been used previously to calculate dipole-dipole capture rates for the reaction of 3H with NO [ 41 and for a number of atmospheric radical-radical reactions [ 51, dipole-quadrupole capture rates for some reactions of 0( 3P) [ 61, and ion-dipole, ion-quadrupole, and ion-induced dipole capture rates for a number of reactions of CH: [7]. The analytical potential used here comprises the quadrupole-quadrupole potential I’oo = (3Qi QJ4r’)
( 1 - 5 cos’f3, - 5 cos2&
+ 17 cos20, cos2& + 2 sir?& sir?& cos*$~ +16sin8,
cos8,
sin&cost$cos~+...),
(1)
where the quadrupole moments Qi and Q2 and the angles are as defined by Buckingham [ 81, with a superimposed isotropic London potential V,=-3a,cu2hv,Y2/2(Y,+V2)r6,
(2)
where (pi and IY~are polarizabilities and u1and v2 are frequencies which correspond to the first ionization potentials of the interacting species. There is also the 254
1 June 1990
option of superseding the combined potential by a Morse potential at, distances less than the separation at which the Morse potential becomes more strongly attractive than the most negative sum of potentials (1) and (2). Expression (2) for the London potential is usually considered to underestimate the strength of the interaction by up to a factor of two for ground state species. For electronically excited species the interaction is expected to be weaker and in extreme cases may even become repubive. However, for the interaction of a ground-state molecule with a species such as 0( ‘D) in a low-lying metastable state, in the framework of the usual perturbation treatment, the effect of perturbations involving the ground-state of the metastable is likely to be greatly outweighed by perturbations involving continuum states, so that the London potential is essentially the same as that between two ground state species having the same polarizabilities and ionization potentials. For the interaction of a molecular quadrupole with an atomic quadrupole (species 2), the value of 9 in eq. ( 1) is set at zero and the value of 0, is chosen for each value of 8i, to give the largest absolute value of the angular factor in parentheses. Finally, the sign of the potential is adjusted, if necessary, to ensure that V,, is always negative. The computer program allows the sign of the atomic quadrupole moment to be specified and returns the fraction of positive angular factors in the resulting formula for Voo when 8i is varied in equal steps over the range 0-2x (with weighting proportional to sin 8, ) _The significance of this fraction (which, with proper allowance for the number of occurrences of equal and opposite maximum values of the angular factor, is either 0.545, for QiQ2 negative, or 0.455, for Q,Q2 positive) is as follows: For a molecule such as CO*, with a negative quadrupole moment, the most favourable direction of approach for an atomic quadrupole with a positive quadrupole moment is along the internuclear axis of the molecule. The fraction of angular factors which correspond to this approach, i.e. for which VW is automatically negative because of the sign of QlQ2, is 0.545. For the remaining 0.455 of the angular factors the sign of VW has to be adjusted by the program to make it correspond to an attractive interaction. Thus, assuming the distribution of collisions over 01 to be proportional to sin&, a fraction 0.545 of the total
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capture rate must relate to “end-on” collisions and a fraction 0.455 to “perpendicular” collisions and, if reaction occurs for only one kind of collision, the calculated capture rate should be multiplied by the appropriate fraction. This general conclusion needs to be qualified by the observation that the angular dependence of the chemical forces which lead to a preference for one kind of collision orientation over another is not. likely to be identical with the angular dependence of the quadrupole-quadrupole potential, so that there is necessarily some uncertainty in the factors that are here taken as 0.545 and 0.455. Also, at intermediate angles the state of 0( ‘D) is best regarded as a quantum mechanical superposition of states corresponding to opposite signs of Q. Thus the conclusion is to some extent qualitative rather than quantitative. Nevertheless, the difference between 1.O and the factors 0.545 and 0.455 is large enough to be outside the limits of error of the experimental results with which the calculations are to be compared, and on this basis it may be possible, by comparing the calculated capture rate with the measured reaction rate, to conclude either that one particular orientation is required for reaction, or that there is no preferred orientation. For the reaction of 0( ‘D) with Hz, Fitzcharles and Schatz [9] found that quasiclassical trajectory calculations on two different potential surfaces led to opposite conclusions conceming the relative importance of end-on and perpendicular collisions, so such deductions must be treated with caution. Nevertheless, in that instance one of the surfaces was clearly preferable to the other on other criteria and, in cases where good agreement is obtained between the experimental rate constant and the calculated capture rate, such deductions deserve to be taken seriously. In addition to possible factors less than unity for orientation, the raw capture rate must also be multiplied by a factor for the fraction of collisions which occur on the lowest potential surface, collisions on other surfaces being assumed unable to lead to sufficiently close approach of the reactants. For 0( ID) in a system of C, symmetry the required factor is 0.2. More detailed discussion of this point is given elsewhere [5].
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2. Results and discussion For all of the reactions considered here the experimental outcome is physical quenching of 0 ( ‘D) to the 3P state. This is assumed to require a close approach of the two reactants on an attractive potential-energy surface. Calculated capture rates for the reaction of 0 ( ’ D ) with CO* are given in table 1. The quadrupole moment of 0( ID) was taken to be 1.07x 1O-26esu cm*, from the calculations of Gentry and Giese [ lo], with Buckingham’s definition of Q. The polarizability of 0( ‘D) is expected to be the same as that of 0 (3P), namely 0.802 x 1O-24 cm3, since the two states arise from the same electron configuration [ 111. Other data used in the calculation are listed below the table. The listed value of k3,,,,,the rate constant for this reaction at 300 K, is 1.1x lo-” cm3 molecule-’ s-‘, with an uncertainty amounting to a factor of 1.2 [ 121; kT is independent of temperature to within experimental error. The results in column 2 of table 1 were obtained for combined Morse, quadrupole-quadrupole and London potentials, the London potential being as given by eq. (2). Replacing the Morse potential by a hard-sphere potential led to results which differed from those in column 2 by not more than 1 or 2 in the fourth significant figure, so these results are not given in the table. The results in column 3 were obtained by omitting the quadrupole-quadrupole potential and those in column 4 by omitting the London potential. These two potentials appear to be of about equal importance in determining the rate constant of this reaG tion. Columns 5 and 6 show the effect of increasing the magnitude of the London potential by factors of 1.4 and 2.0, respectively. For comparison with the experimental rate constant, the capture rate has to be multiplied by a factor of 0.2, for electronic degeneracy, and possibly by a factor near 0.5 to take account of any preference for perpendicular or end-on collisions. For the reaction with C02, the present results imply that there is no preferred orientation, values of 8.50x10-“, 8.98~ 10-l’ and 9.60x lo-” cm3 molecule-’ s-’ being obtained for k,, with only the electronic degeneracy factor and with the London potential (2) multiplied by factors of 1.0, 1.4 and 2.0, respectively. Best agreement with experiment is obtained with the factor 2.0, and the last column of table 1 contains kT values calculated over 255
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Table 1 Results for reaction of O(‘D) with CO*. Units of k. 10-r’ cm3 molecule-’ s-‘. Factors used with London potential in parenthesis. k,,=cnpture rate; k_,=capture rate with London potential omitted; k_o=capture rate with quadrupole-quadrupole potential omitted, k,=calculated reaction rate at T, orientation factor= 1.O ‘) T(K)
kcapP(l.0)
k_,( 1.0)
k.dO.0)
%P( 1.4)
kuP(2.0)
kr(2.0)
10 20 30 50 75 100 150 200 250 300 350 400 500 600
31.7 34.1 35.7 37.8 39.5 40.7 42.2 42.7 42.7 42.5 42.0 41.5 40. I 38.6
20.5 22.7 24.1 25.9 27.4 28.6 30.1 31.3 32.0 32.5 32.7 32.9 33.2 33.5
28.8 30.4 31.5 32.8 33.8 34.2 34.3 33.8 33.1 32.4 31.7 30.9 29.6 28.5
32.8 35.4 37.2 39.5 41.4 42.1 44.1 44.8 45.0 44.9 44.5 44.2 42.9 41.5
34.2 37.0 38.9 41.5 43.7 45.2 46.9 47.8 48.0 48.0 47.8 47.3 46.1 44.1
6.84 7.40 7.78 8.30 8.74 9.04 9.38 9.56 9.60 9.60 9.56 9.46 9.22 8.94
N Data: Morse (r, (A), D. (kJ mol-I), o.(cm-‘))=(1.2,50,500); lO-26 esu cm;; &a&ability (CO, j =2,65 x lCt’24cm3.
the whole temperature range using this factor. The calculated results show only a small temperature dependence, in agreement with experiment. Similar conclusions to those in the last paragraph concerning the relative importance of Morse, London and quadrupole-quadrupole potentials can be drawn from calculations for the reactions of 0 ( ‘D ) with HZ,Nz and OZ.Because of the small quadrupole moment of 02, the London potential plays a correspondingly more significant role in this case. The calculated small temperature dependences for these reactions are also in accord with experiment. Table 2 shows results for the reaction with Hz with all potentials included, with the London potential (2) multiplied by factors of 1.0, 1.4 and 2.0, as before. The listed value of kaOeis 1.Ox 1O- ‘Ocm3 molecule- ’ s-l with an uncertainty factor of 1.2. The fao tors of 1.0, 1.4 and 2.0 for the London potential lead to boo values of 0.77x10-lo, 0.83~10-‘~ and 0.94x lo-“, respectively, when the corresponding capture rates are multiplied by factors of 0.2 for electronic degeneracy and 0.545 for orientation. All three results are consistent with the listed value but there is a clear preference for the factor 2.0. Use of the factor 0.545 for orientation implies that end-on collisions are required for reaction, in agreement with the results of Fitzcharles and Schatz for their best ab in256
Ionizationpotentials
(cm-‘)=94000,111000;
Q(COr)= -4.2~
Table 2 Results for reaction of O(‘D) with Hz. Units of k: IO-” cm3 moleculei’ s-‘. Factors used with London potential in parenthesis. Orientation factor=0.546 D) T(R)
k&1.0)
kW(l.4)
ka(2.0)
kT(2.0)
10 20 30 50 75 100 150 200 250 300 350 400 500 600
52.2 56.5 59.0 61.5 62.5 63.6 65.8 67.9 69.1 71.3 12.6 73.8 75.6 77.9
54.4 59.4 61.8 64.5
56.4 62.3 66.0 69.4 71.1 73.0 75.8 78.6 80.8 82.8 84.3 85.8 87.9 89.5
6.16 6.80 7.21 7.58 1.16 7.97 8.28 8.58 8.82 9.04 9.21 9.37 9.60 9.77
65.8 67.5 70.5 73.0 75.0 76.6 78.0 79.2 80.8 82.0
‘) Data:Mome (rmD,(kJmol-r),~(cm-‘))=(1.2, 720,360(l); Ionization potentials/cm-‘=94000, 124000; Q(H2)= t 0.651x10-26esucm2;polarizabilityH,=0.8l9~l0-24cm3.
itio surface, but the difference from the factor 0.455 for broadside collisions is not suffkient for an unequivocal distinction. The last column of table 2 gives kT values for the reaction with Hz over the whole temperature range, calculated using the factors 2.0
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for the London potential and 0.545 for orientation. Table 3 gives results for the reaction with N2 analogous to those for H2 in table 2. The listed value of ,&, for this reaction is 2.6X lo-” cm3 molecule-’ s-l, with an uncertainty factor of 1.2. Factors of 1.0, 1.4 and 2.0 for the London potential lead to values 3.34x lo-“, 3.62~ lo-” and 3.91 X lo-” cm3 molecule- l s- ’ for 500 when the capture rate is multiplied by factors of 0.2 for electronic degeneracy and 0.455 for orientation, The first value of /c300is slightly larger than the upper limit of the experimental value: in view of the uncertainty regarding the factor to be used for the preferred orientation, this degree of agreement is probably acceptable. Use of the factor 0.455 for orientation implies that perpendicular collisions are required for reaction, which is perhaps surprising in view of the anticipated tendency of this reaction system to form an intermediate NzO* complex. However, the reaction with Hz shows a preference for end-on collisions, even though it probably proceeds through a triangular H20* complex, so the conclusion is not unprecedented. One might speculate that an oblique approach is dictated by angular momentum requirements for barrier crossing, as appears to be the case of the reaction of O(3P) with OH [ 131. The alternative to assuming that there is Table 3 Results for reaction of 0( ‘D) with N,. Units of k IO-” cm3 molecule-r s-i. Factors used with London potential in parenthesis. Orientation factor=0.454 ‘) T(R)
kXP(1.0)
10 20 30 50 15 100 150 200 250 300 350 400 500 600
26.2 21.8 29.1 30.8 32.4 33.4 34.9 35.9 36.5 36.8 37.1 31.3 37.4 37.3
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k(l.4) 27.7 29.5 31.0 ’ 32.9 34.7 35.9 31.7 38.7 39.5 39.9 40.3 40.4 40.5 40.4
k(2.0)
Wl.0)
29.4 31.6 33.3 35.6 37.6 39.0 40.9 41.9 42.6 43.1 43.3 43.5 43.6 43.5
2.38 2.52 2.64 2.80 2.94 3.04 3.17 3.26 3.31 3.34 3.37 3.39 3.40 3.39
‘) Data: Morse (re (A), De (kJ mol-‘), w3 (cm-‘)) = (1.2,350, 1285); Ionization potentials (cm-‘) =94000, 126000; Q(N2)=-l.5~10126 esu cm’ polarizability (N2)=l.76x 10mZ4cm3.
Table 4 Results for reaction of O(‘D) with OZ. Units of k lo-” cm3 molecule-’ s-i. Factors used with London potential in parenthesis. Orientation factor= 1.O&r T(R)
k-w(l.0)
kw( 1.4)
kP(2.0)
Wl.4)
10 20 30 50 75 100 150 200 250 300 350 400 500 600
18.7 20.6 21.7 22.7 23.1 23.1 22.7 21.8 20.7 19.6 18.5 17.5 15.7 14.3
20.4 22.7 24.0 25.2 25.7 25.7 25.2 24.2 23.0 21.7 20.5 19.4 17.5 15.7
22.1 25.2 26.7 28.1 28.6 28.7 28.2 27.1 25.1 24.3 23.0 21.8 19.6 17.8
4.08 4.54 4.80 5.04 5.18 5.18 5.04 4.84 4.60 4.34 4.10 3.88 3.50 3.14
a~Data:Morse(r,(A),D,(kJmol-‘),o~(cm-’))=(l.2,292, 1040); Ionization potentials (cm-‘) =94000, 97000; Q(Oa)= -0.4~ 10ez6 esu cm*; polarizability (O,)= 1.60~ 10wg4cm3.
preferred orientation is to suppose that a similar factor arises from the presence of a barrier in the entrance or exit channel for the quenching reaction, which seems less likely. Table 4 contains capture rates for the reaction with O2 similar to those given in table 2 and 3 for reactions with Hz and NZ. The listed value of k3m is 4.6 x 10-l’ cm3 molecule-’ s-‘, again with an uncertainty factor of 1.2. Use of the factors 1.O, 1.4 and 2.0 for the London potential, with a factor of 1 .Ofor orientation, gives k300 values of 3.92 X lo-“, 4.34~ lo-” and 4.86x lo-” cm3 molecule-’ s-l, respectively. All three values are in agreement with the listed value; the values of /cTin the last column of table 4 were calculated using a factor of 1.4 for the London potential. For this reaction the results imply that there is no preferred orientation, which again is counter to the intuitive expectation that the reaction should proceed through a triangular 0; complex. a
3. Conclusions For a quadrupole-quadrupole reaction between an atom and a molecule it is possible to make a plau257
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sible distinction among reactions requiring end-on or perpendicular collisions and reactions with no preferred collision orientation on the basis of the sign that the atomic quadrupole must take in order to produce an attractive interaction. The angular factors for this interaction are such that about 55% of collisions will have end-on and 45% will have perpendicular orientations. If one orientation or the other is preferred, the calculated capture rate, after allowance for electronic degeneracy, has to be multiplied by the appropriate factor for comparison with the experimental rate constant. The present approximate, quasiclassical trajectory calculations of Langevin-type capture rates lead to results in remarkably good agreement with experiment provided it is assumed that the reactions of 0( ‘D) with CO2.and O2 have no preferred collision orientation, while the reaction with Hz requires end-on collisions and the reaction with N2 requires perpendicular collisions. For best agreement the London potential given by eq. (2) needs to be multiplied by a factor of 2.0 (H,, COz), 1.4 (OI), or 1.0 (N,).
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Acknowledgement
I am grateful to a referee for pointing out the necessity of including a sin 13~weighting of the angular factors of the quadrupole-quadrupole potential. References [ I ] DC. Clary and J.P. Henshaw, Faraday Discussions Chem. Sot. 84 (1987) 333. [ 2 JFaraday Discussion No. 84, Dynamics of elementary gasphase reactions (Roy. Sot. Chem., London, 1987). [3] L.F. Phillips, J. Comput. Chem. 1I (1990) 88. [4] L.F. Phillips, Chem. Phys. Letters 168 (1990) 197. [ 51L.F. Phillips, J. Phys. Chem., submitted for publication. [6] L.F. Phillips, Chem. Phys. Letters 165 (1990) 545. [ 71 L.F. Phillips, J. Phys. C&n., in press. [8] A.D. Buckingham, Advan. Chem. Phys. 12 (1967) 107. [ 91 M.S. Fitzcharles and G.C. Schatz, J. Phys. Chem. 90 ( 1986) 3634. [ lo] W.R. Gentry and CF. Giese, J. Chem. Phys. 67 (1977) 2355. [ 111P.C. Schmidt, M.C. Boehm and A. Weiss, Ber. Bunsenges. Physik. Chem. 89 (1985) 1330. [ 121R. Atkinson, D.L. Baulch, R.A. Cox, R.F. Harnpson Jr., J.A. Kerr and J. Troe, J.,Phys. Chem. Ref. Data 18 (1989) 881. [ 131N. Markovic, G. Nyman and S. Nordholm, C&em. Phys. Letters 159 (1989) 435.