J. Phys. Chem. 1993,97, 13629-13636
13629
A Classical Trajectory Study of Possible Symmetry Restrictions in the H2+/H2 Proton/Atom Transfer Reaction C . A. Picconattot Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, Indiana 46556 Gregory I. Gellene' Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409- 1061 Received: July 20, 1993; I n Final Form: September 28, 1993a
The suggestion that the title reaction may be subject to symmetry restrictions [J.Chem. Phys. 1992,96,4387] is investigated by using classical trajectory techniques to determine capture collision rates as a function of reactant isotopic composition, neutral rotational state, and translational temperature. A comparison of the theoretical results with available experimental reaction rates provides no evidence for a symmetry restriction at 300 K. It is suggested that the effect of a possible symmetry restriction may be best observed for reactions of J = 0 neutrals at T < 10 K where dynamical constraints restrict coupling of orbital and rotational angular momentum. The contribution of the ion-quadrupole potential to the capture collision rate is examined in detail and found to be significant only for J = 0 neutrals and to increase in importance with decreasing temperature, The implications of this result for low-temperature isotopic fractionation are addressed.
I. Introduction Over the last 10 years, experimental studies of various types have established the existence of kinetic isotope effects (KIE) which are dependent on nuclear symmetry considerations rather than the more familiar mass considerations. Work in this area was largely motivated by stratospheric ozone mea~urementsl-~ which indicated that the heavy isotopomers ( 4 9 0 3 and 5 0 0 3 ) were equivalently enhanced by up to40% above their natural abundance levels. Somewhat smaller enhancements ( 10%) have been observed in the laboratory+I0 for ozone formed by association reactions of 0 2 0. Additionally, similarly small enhancements have been reported for S ~ F Iand O C02 formed from association reactions of SF5 SFr; and CO 0 respectively.I1J2 Recently, we have observed non-mass-dependent KIE in two additional systemsI3J4with the magnitude of the effect considerably greater than those previously reported: ( I ) in the formation of 04+ by association of 0 2 + 0 2 , the heavy isotopes of oxygen were equivalently incorporated up to 100 times their natural abundance levels; (2) in the formation He2+ by association of He+ He, the isotopicallyheteronuclear 'Hez+ ion was produced at 3 times the rate of either homonuclear ion (6He2+or 8He2+). In the theory of non-mass-dependentKIE, it had been suggested that isotopic fractionation cannot ensue directly from symmetry alone and that the explanation of the effect can be formulated on the basis of incompleteenergy randomizationin the association complex.15 However, because of the large magnitude of the KIE and its Occurrence in an atomlatom association reaction where energy randomization is not possible, our results could not be rationalized by any existing theory and a new approach was adopted. The proposed explanation was based on a new symmetry correlation scheme16 which connected symmetry distinct rovibronic states of the reactants with distinct electronic symmetries of the system in the interaction region independent of the detailed geometry of the collision complex. When identical nuclei are present, the Pauli principle restricts the form of the allowed wave functions depending on the particular internal energy states of the reactants. Thus, when the system contains asymptotic BornN
+
+
+
+
+
+ Current address:
Department of Chemistry, Columbia University, New
York, N Y 10027. e Abstract
published in Aduonce ACS Abstracts. November 1, 1993.
0022-3654/93/2091- 13629$04.00/0
Oppenheimer electronic state degeneracies, the probability that the reactants interact on the particular Born-Oppenheimer potential energy surface (PES) relevant to the chemical process ofinterest can vary with the internal energy states of thereactants. It is important to emphasize that because the theory is based on the asymptotic states of the reactants, the predicted kinetic restrictions are by no means rigorous. The extent to which the macroscopic kinetics of a system containing identical nuclei will be affected by the restrictions can be thought of as depending on the extent to which the asymptotic description of the reactant wave functionsremains good as the system approaches a separation where the Born-Oppenheimer electronic state degeneracies are relaxed. Although all of the presently known systems exhibiting nonmass-dependent KIE are association type reactions, this is not required by the theory. An important question, therefore, is whether this effect could be observed in a more general reaction where the chemistry involved a substantialrearrangementof atoms in going from reactants to products. In our first paper on the subject,I6 we suggested that the H2+/H2 protonlatom transfer reaction:
+
H2+ H2
+
H3++ H
might be just such an example. In this case the asymptotic BornOppenheimer electronic degeneracy arises from the inability to specify which diatom is the ion and which is the neutral when the diatoms are identical. Application of the symmetry correlation scheme to reaction 1 (and its isotopic analogs) yielded the results summarized in Table I where N+ and N are the Hund's coupling case b rotational quantum numbers for the ion and the neutral respectively and 1 is the quantum number for the relative angular momentum of the two diatoms. Because even and odd 1 collisions occur with equal probability (except at very low temperature), Table I indicates that, in the absence of coupling between the electronicstates, only half of the collisionsof isotopically identical diatoms would access the ground state and undergo reaction. This led to the prediction that the relative rate of reaction 1 for identical vs nonidentical diatoms might be as much as a factor of 2 slower. Although reaction 1 has been the subject of several multiple PES dynamics calculation^,'^-^^ the absence of detailed nuclear symmetry considerations in the treatment of the PES 0 1993 American Chemical Society
13630 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993
Picconatto and Gellene Z
TABLE I: Correlation of Pauli Allowed Asymptotic Wave Functions for H2+/H2 (D2+/D# with the Electronic States (9elec)6 of H4+ (D4+) I N N+ parity coupling even even odd odd
even odd even odd
even odd odd even
+ +
}
even even odd odd
even odd even odd
odd even even odd
-
)
]
'*
41
l.(N++N)
'8s
)
'*
l.(N++N)
'8s
a The correlation for HD+/HD is given by interchanging t$* and rpgS. The ground and excited states are denoted by bgSand '*, respectively.
TABLE 11: Comparison of Experimental and Theoretical 300 K Thermal Rate Constantsa for the H2+/H2 Proton/Atom Transfer Reaction and Its Isotopic Varients kLC
kAWd
kc'
2.11
2.13
2.19 f 0.02
1.4,'1.44/ 1.44,m 1.60: 1.6 f 0.1; 1.54f 0.15k
1.49
1.51
1.58 A 0.02
HD+/HD
1.66/1.80,h 1.66,m 1.8f 0.1'
1.72
1.74
1.82 f 0.02
H2+/D2
3.2 f 0.6'
1.83
1.85
1.95 f 0.03
D2+/H2
3.0 f 0.6'
1.83
1.85
1.92f 0.02
reactant H2+/H2
kemtb 2.02,f1.85,g2.11,h 2.0 f 0.1,i 2.08 f 0.03; 2.12 f 0.14k
D2+/D2
Figure 1. Coordinate system used in the trajectory calculations.
This question is addressed in the present study by using classical trajectory techniques to determine collision rates. The purpose of the study is 2-fold: (1) to determine whether or not the rate data in Table I1can be taken as evidence for a non-mass-dependent KIE in reaction 1, and if not, (2) to determine conditions under which a non-mass-dependent KIE might be observed experimentally. 11. Theory
a Rate constants are in units of cm3mol-' s-*. Most experimental rates are for a near Franck-Condon distribution of vibrational levels of the ion. Reference 25. Reference 26. The values reported in ref 16 are in error because the wrong value of Q was used. e Present results. /Reuben, B. G.; Friedman, L. J. Chem.Phys. 1962,37,1636.g Warneck, P.J. Chem. Phys. 1967,46,502. Reference 42. Clow, R. T.;Futrell, J. H. In?. J. Mass Spectrom. Zon Phys. 1972, 8, 119.J Reference 40. Drewits, H. J. Znt. J. Mass Spectrom. Zon Phys. 1976, 19, 313. 'Stevenson, D. P.;Schissler, D. 0. J. Chem. Phys. 1955, 23, 1353. Huntress, W.T.,Jr.; Elleman, D. D.; Bowers, M. T. J. Chem. Phys. 1971,55,5413.
couplings in those studies makes it difficult to apply the results to the question of non-mass-dependent KIE. Experimentally, however,theredoes appear to be some evidence for this non-mass-dependent KIE. Table I1 lists experimentally determined rates for reaction 1 and its isotopic variants. Most significant in the present context is the increased rate reported for H2+/D2 and D2+/H2(i.e., nonidentical reactants) relative to that of H2+/H2, D2+/D2,and HD+/HD (i.e., identical reactants). However, there are reasons to be concerned that the reported rates for H2+/D2 and D2+/H2 may be high. Because reaction 1 is significantly exoergic (1.7 eV) and is believed to proceed by a direct mechanism without any observable barrier,22323the thermal rate of the reaction should be well approximated by the collision rate. Indeed, there is evidencethat changes in the internal energy of the ion by more than 1 eV causes only -20% change in the reaction rate.24 Columns 3 and 4 of Table I1 list classical collision rates calculated using the Langevin pure polarization model of Gioumousis and Stevenson25(kL) and the average quadrupole theory ( ~ A Q O )of Su and Bowers.26 Table I1 indicates that the predictions of these classical theories are generally in good agreement with the reported rates of reaction 1with identical diatoms. However, the reported rates for reaction 1 with nonidentical diatoms are approximately 70% larger than the classical theoretical predictions. Thus, either the contribution of the quadrupole of H2 to the collision rate is much larger than predicted by the AQO theory, or the reported rates for H2+/D2 and D2+/H2 reactants are too high.
A. The Hamiltonian. At thermal energies, the ion-molecule collision frequency is largely determined by the long range electrostatic potential. Therefore modeling the H2+ion as a point charge and the H2 molecule as a polarizable point quadrupole is justified in the classical trajectory calculations. In terms of the coordinatesof Figure 1, the potential energy of the model system (U) at an ion-molecule separation R and orientation angle y is given by27 W,y) =
qQ(3(cos2y) - 1) - q2[3a + (ail - a.)(3(C0S2 7 ) - 111 (2) 2 ~ 3 6R4 where q is the charge of the ion, Q is the quadrupole moment of H2 (0,128 esu A2),28cy11 and cyL are the components of the Hz polarizability tensor parallel and perpendicular to the diatomic axis respectively (cull - cyL = 0.3016 A2),29and cy = (all + 2a1)/3 is the isotropic polarizability of H2 (0.819 The second term in eq 2 is always attractive, whereas the first term can be attractive or repulsive depending on the value of y and the relative signs of q and Q. When the center of mass of the system is taken to be at rest, the kinetic energy of the model system (7) is the sum of the rotational kinetic energy of the neutral (Trot) and the relative ion-molecule translational kinetic energy (Ttr,,,). In terms of the coordinates in Figure 1 and their conjugate momenta (Pi), Trot and Ttrans are given by
(34
where p is the ion-molecule reduced mass, I is the moment of inertia of neutral, and the total squared angular momentum of the neutral rotor and the relative ion-molecule motion are denoted
The Hz+/H2 Proton/Atom Transfer Reaction
The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13631
byj2and L2,respectively. The Hamiltonian used in these studies is given by
H = Ttrans + Trot + u
(4)
and the equations of motion that follow from eq 4 are given in appendix A. The equations of motion were integrated using a variable step size sixth-order predictor-corrector integration alg~rithm.~'The computer codes were checked by back integrating several trajectories, and the maximum permissible step size was determined by requiring total energy and angular momentum to be conserved to 1 part in IO6. B. Initial Conditions. Five uniform deviates32in the range from 0 to 1, ei (i = 1-5), were generated at the start of each trajectory. The center of mass of the neutral molecule was taken to be at the origin of the (XYZ) system, and its initial state was randomly chosen according to33
smallest R for which the force in the R direction (dPR/dr) given by dPR/dt = L2/&
- dU/dR
(8) can be greater than zero. Setting eq 8 equal to zero, solving for L2, and substituting into eq 4 gives
q 2 P a + (a11 - a , ) ( 3 ( C O S 2 Y) - I)]
(9) 6R4 To find the smallest value of R which can satisfy eq 9, P Rand ~ j 2 are set to zero and y is chosen so as to maximize dPR/dt. In general, two cases need be considered. Case 1: qQ > 0. This is the case of present interest and dPR/ dt is maximized by y = 0 for 9QR > 4q(q - al)and y = r/2 for 9QR < 4q(all - al). Therefore, depending on the relative magnitude of all, al,and Q,ROis given as the smallest positive real root of either 2HR4
+ qQR - q2aI1= 0
for
y
=0
(1 Oa)
or 4HR4 - qQR - 2q2al = 0 for y = r / 2 where h is Planck's constant divided by 2 r and the choice of J = 0,1,2, ...initiated the molecule with a total angular momentum equal to that for the corresponding rigid rotor quantum state. The initial velocity (v) of the ion was taken to be in the Z-direction (i.e., Px = Py = 0 and Pz = pbl) and 1 . I chosen randomly from a Maxwell transmission distribution according to
sb'X'e
dX' = q
where X = pb12/2keT,k~ is the Boltzmann constant, and Tis the absolute temperature. The ion was chosen to have an initial position in the XZ-plane (i.e., Y = 0) with
2 = -2,
(5g)
where b,,, was an impact parameter above which the collision probability was negligible, and 20was chosen so that initially TtrPns > lo3 lq for 99.9% of the trajectories. With these choices for initial condition distributions, the collision rate as a function of translational temperature and quadrpole rotor state is given approximately by33
kc(J, r ) = (8k,T/*~) 1'2rbm,2(NJNtot) (6) where N,/Ntotis the ratio of the number of trajectories that result in a collision to the total number of trajectories run. The criteria by which a particular trajectory was judged to result in a collision is discussed in the next section. Finally, the collision rate as a function of temperature alone is obtained from the weighted sum:
where P(J,T) is the probability of a neutral rotor state at a given temperature. C. Trajectory Termination. Trajectorieswere terminated when either PR> 0 (a noncollision) or R I Ro (a collision) occurred. Ro is the distance at which any trajectory with PR< 0 can be guaranteed to reach R = 0 and was determined for each trajectory by the method of Bhowmik and Su34 modified for application to the present potential. Briefly, Ro is determined by finding the
(lob)
Equations 10a and 10b can be analytically solved by the so-called "solution by radicals" method.35 Case 2: qQ < 0. This case is included for completeness and it is found that y = 0 maximizes dR/dt for all values of R with Ro given by the smallest positive real root of eq loa. D. Tunneling Corrections. Because the systems of present interest involve relatively small reduced masses ( p = 1-2 amu), anattempt was made toestimatea correctionto k, due toquantum mechanical tunneling through the angular momentum barrier (L2/2pR2). At the point where a particular trajectory would be terminated and labeled as a noncollision, the possibility that tunneling would allow the trajectory to continue to smaller R was estimated by the semiclassicalapproach of Marki and Miller.36 In this approach tunneling is taken to have occurred if e I e-2u (1 1) where e is a newly generated uniform deviate and u is the classical action integral "through the barrier". For ease of application, the tunneling path was taken to be a straight line involving only radial motion (i.e., the angular coordinates: O,qj,[, q, and hence y are fixed) with Po, Po, Pt, and P,,independentlyconserved. This path necessarily conserved angular momentum, and conservation of energy was ensured by requiring that the potential where the barrier is "entered" (U(Rl,yl)) and "exited" (U(R2,rz)) be identical. To ensure that the trajectory would proceed to smaller Rafter tunneling, R1 was taken as the last calculated point before reaching the barrier where PR < 0, and R2 was determined by the solution of r2
=-+L
r2 L
U ( R , ~ ) - ~ - U ( R ~ , . =I )O (12) 2pR2 2kR, With this tunneling path, u is given by F(R)
u=
h-' J,,"'[2pF(R)]''*
dR
Because the equations of motion were integrated using Cartesian position and conjugate momentum coordinates, it was necessary to determine the corresponding spherical coordinates and conjugate momentum for theevaluation of u. The required equations are given in appendix B.
Picconatto and Gellene
13632 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993
TABLE III: Ion/Neutral Collision Rates' ion/ neutral
T
(K)*
0
initial neutral rotational level 1 2 3 4
H2+/H2
300 2.65(8) 200 2.68(8) 80 2.89(6) 10 3.35(5) 1 4.23(14)
H2+/HD
300 2.35(3) 1.91(5) 1.93(5) 1.916) 1.88(2) 200 2.43(6) 1.97(5) 1.88(5) 1.88(5) 1.96(2) 80 2.65(4) 1.88(2) 1.90(5) 10 3.08(6) 1 3.87(10)
5
2.13(2) 2.12(5) 2.21(2) 2.18(5) 2.09(5) 2.16(6) 2.15(3) 2.07(7)
H2+/D2
300 2.39(4) 115 2.50(3) 10 2.95(6)
1.89(6) 1.88(6) 1.83(6) 1.83(5) 1.82(8) 1.80(3) 1.79(3) 1.81(2) 1.79(8)
HD+/Hz
300 200 80 10 1
2.40(6) 2.39(3) 2.63(6) 3.07(5) 3.91(8)
1.91(4) 1.98(5) 1.94(2) 1.89(5) 1.91(7) 1.90(6) 1.88(3) 1.98(7)
HD+/HD
300
2.16(4)
D2+/Hz
300 2.23(4) 115 2.44(7) 10 2.89(4)
1.89(2) 1.77(5) 1.82(6) 1.82(3) 1.74(7)
Dr+/Dz
300 1.88(4) 115 2.00(4) 10 2.44(6)
1.60(5) 1.51(4) 1.48(3) 1.47(3) 1.44(4) 1.51(4) 1.51(6) 1.50(2) 1.49(8)
1.77(4) 1.73(4) 1.72(3) 1.70(4)
cm3 mol-' SKI;the value in Rate constants are in units of parentheses is one standard deviation in the units of the last digit reported. Translational temperature. a
111. Results
Ion/neutral collision rates for a particular isotopiccomposition, translational temperature, and initial neutral rotational state were determined using eq 6 with NtOtequal to 5000 (J = 0) or 2000 (J # 0). Statistical information on the reliability of the results wasobtainedbyrepeatingeachcalculation3-6 times (Le., 15 00030 000 ( J = 0) or 6000-12000 (J # 0) total trajectories). The translational temperatures and isotopic compositions investigated were chosen to allow comparison with the available 300 K experimental rate data and to address KIE temperature dependence under natural isotopic abundance conditions. The results obtained for the classical collision rates (Le., nontunneling) are listed in Table 111. In general, near Langevin rates are obtained for all collisions with J # 0, independent of temperature and isotopic composition. However, for collisions with J = 0, rates larger than the Langevin rate are calculated which increase significantly with decreasing temperature. The increase in the collision rate due to tunneling was generally small ranging from -0.5-1.595 at 300 K to 3.545% at 1 K.
IV. Discussion A. The Role of the Quadrupole. The principal effect of the quadrupole on the collision dynamics has its origin in the ionquadrupole interaction potential (first term in eq 2) which, in the present case becomes attractive for cos y < 1/& = 0.58. The importance of this interaction can be readily assessed by consideringthe distribution of cos y as a function of R. Because U(R,y) is an even function of cos y, the mean value of cos y( (cos y )) conveys no significant dynamicalinformationand the standard deviation of the distribution (ocosr= ((cos y - (cos y))Z)I/*) is examined instead. A plot of (cos y) f uccos-, us R for H2+/H2(J=O) collisions at translational temperatures of 300, 80, and 10 K is shown in Figure 2 where the dashed lines represent the result for a random distribution of quadrupole orientation (Le., ucOsr2 = 1/3). The results indicate substantial preferential alignment of the quadrupole in the attractive orientation even at R = 50 au, with the extent of alignmentincreasingwith decreasing separation
1 -_____________-___-----------------_---.
-1.0 -0'5 0
10
20
50
40
30
R (au) Figure 2. (cos y ) & ucOs-,as a function of R for H2+/H*(J=O)collisions at translational temperatures of 300,80, and 10 K. In each case (cos y ) = 0 independent of R as expected. The dashed lines represent the values of (cos y ) f um-, expected for a random distribution of HZ
orientation.
o
present results
-CeHi er ai.
1.e
--- Kosmas
J
.-.- Bates eta/.
5I: 1.01.2
0.81 , 0
100
1
I
200
300
Figure 3. Comparison of the trajectory calculation results for the H2+/ Hz(ful1y equilibrated) collision rate temperature dependence with the predictions of Celli et al. (ref 37), Kosmas (ref 38), and Bates et al. (ref 39). The "A"' denotes the prediction of AQO theory (ref 26).
and translational temperature. This ability to align in the field of the ion largely accounts for the enhanced collision rates of the J = 0 states. A plot analogous to Figure 2 for H2+/H2(J#O) indicates that the quadrupole orientation remains randomly distributed for R > 10 au, independent of translational temperature. Because COS y) - 1) = 0 for a random orientation distribution, U(R,y) = q2/2R4(the pure polarization potential) for J # 0 and near Langevin collision rates result. Theories of the temperature dependence of ion+padrupoie collision rates have been developed by Celli et al.," Kosmas?* and Bates and Menda8.39 Although the anisotropicpolarizability of the neutral was neglected in those studies, meaningful comparisonsto the present results can be made nevertheless. Figure 3 compares the predictions of these theories with the calculated H2+/H2 collision rate for a fully equilibrated H2 rotational state distribution. For T > 200, the classical adiabatic invariance approach of Kosmos3* and the average free energy approach of Celli et a1." are in good agreementwith the trajectory calculations, however, the agreement becomes increasingly worse with decreasing temperature. The low-temperature underestimation of the classical trajectory result may seem surprising as Celli et 41. argued that their approach should give an upper bound to the
The H2+/H2 Proton/Atom Transfer Reaction true classical collision rate. The resolution of this apparent conflict lies in the assumption of Celli et al. (and Kosmas) that the rotational constant ( B ) of the quadrupole was small compared to kBT when rotational state averaging was performed. Considering that B/kB = 85.5 K for H2, the discrepancy at low temperature is not surprising. The semiclassical adiabatic invariance approach of Bates and MendaH39provides the least satisfactory comparison in this case as it underestimates the collision rate and predicts a temperature dependence opposite to the present results. The validity of these conclusions might be questioned on the basis that an electron-transfer reaction would interchange the role of the ion and the neutral and the present treatment neglects rotational states of the ion. However, Eaker and Schatz’*have determined that the electron-transfer reaction is unimportant for R > 8 au which is comparable to Ro at 300 K and smaller than Ro at lower temperature. Thus the trajectories were judged to result in a collision (or not) before the scrambling of ion and neutral identities by electron transfer is expected to occur. B. Comparison with Experiment. The results listed in Table I11 can be used in eq 7 with T = 300 K to calculate thermal k, values which are listed in the last column of Table I1 for comparison with the available experimental data. The comparison indicates that the proton/atom transfer reaction for Hz+/H2, D2+/D2, and HD+/HD proceeds at essentially the classical collision rate, whereas the reported experimental rates for Hz+/D2 and D2+/H2 reactions seem to be in error as they are significantly larger than k,. It would appear, therefore, that the experimental results in Table I1 do not provide support for a symmetry-dependent KIE at 300 K. This point will be consideredfurther in the next section. Using the relative rate data of Chupka et ~ 1and. assuming ~ ~ a Franck-Condon distribution of H2+ vibrational states. Theard and Huntress40determined a rate of (2.48 f 0.13) X 10-9 cm3/ (molecules) for reaction 1 at 300 K with Hz+(u=O). This value, which is 13 f 6% larger than k,, suggests that the present neglect of “chemical” contributions to the potential in eq 2 may underestimate somewhat the attractiveness of the actual potentia1.22 The only available experimental information on the effect of the H2 rotational state on the reaction rate comes from the photoionizationstudies of Chupka et al.24using normal and para hydrogen where a 610% effect was found. In satisfactory agreement with this, a value of k, which is 9(f2)% larger than the result for normal hydrogen is calculated using the results in Table I11 and a 300 K para hydrogen rotational state distribution in eq 7. Unfortunately,the H2+/H2 proton/atom transfer reaction rate appears not to have been determined at any temperature below 300 K where the present calculations predict that the enhanced k, of the H2(J=O) should become significantly more pronounced. Such measurements would be highly desirable and may provide evidence for a significant role of the ion-quadrupole interaction in determining ion-molecule reaction rates, which has yet to be experimentally dem~nstrated.~’ C. Symmetry-Dependent HE. I Effect. The absence of convincing experimental evidence for a symmetry restriction in the H2+/H2 proton/atom transfer reaction at 300 K raises two questions: (1) If the symmetry restriction as outlined in the Introduction applies, how is it essentially completely overcome at 300 K? And (2) do conditions exist where the restriction would give rise to observable kinetic effects (e.g., symmetrydependent isotopic fractionation)? Table I indicates that the strength of the restriction depends on the “goodness” of the asymptotic quantum number I because mixing of the ground and excited H4+ electronic states at large H2+/H2 separation is mediated by coupling of the orbital angular momentum with the rotational angular momentum of either 8 2 + or H2 (Le., b(N + N+)). Although ion rotation was not considered in the present calculations, the classical orbital and neutral angular momenta
The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13633 J-0
0 -
10
20
5
z 1
.-,
0
30
J=1
40
50
-
-40 0
10
20
30
40
50
0
10
20
30
40
50
T=80K
,
,
10
20
, . , . , 30 40 50 XI
-*o.+.
Figure 4. ( at2) and ( AL2) f a ~asia function of R for H2+/H2(J=O,I) collisions at translational temperatures of 300, 80, and 10 K. In each case the bold curve represents (AL2) and the lighter curves represent (AL2)f U U Z . Note the different scales used in the six plots.
could (and did) couple as a trajectory proceeded to shorter R. Therefore, some insight into the ‘goodness” of I as a quantum number can be gained by examining the extent to which its classical analog is preserved. This point was addressed by determining the distribution of AL2 (defined as the change from the asymptotic value of L2) as a function of R for H2+/H2(J=O,1) trajectories at translational temperatures of 300,80, and 10 K. These AL2distributions are represented by plots of ( AL2) and ( AL2) f U A L as ~ a function of R in Figure 4 where some trends are evident. The ALz(J=l) distributions at intermediate value of R are broader than the corresponding AL2(J=O) distributions indicating that initial H2 rotation enhances the exchange of angular momentum. It is particularly relevant to the experimental comparisonsthat L2at 300 K is not preseved at separations as large as 50 au with u ~ p ( R = 50 au) being f2.5h2 and f1.7h2 for J = 0 and J = 1, respectively. The classical trajectory calculations thus indicate that the ionquadrupole potential may be sufficiently anisotropic to provide the required AI = f 1 and allow mixing of the ground and excited H4+ in the asymptotic region of these surfaces. If this is correct, then the results suggest that a facile breaking of the symmetry restriction through an LN interaction may account for the near equivalence of the experimental reaction and classical collision rates at 300 K. When it is assumed that the best conditions for observing the effects of the symmetry restriction are those for which L2 is largely preserved in the asymptotic region of the H2+/H2 potential, the results in Figure 4 indicate that the experimental determination of the H2+/H2(J=0) reaction rate at T I 10 K may provide direct information on the importance of symmetry restrictions in this reaction.
13634
Picconatto and Gellene
The Journal of Physical Chemistry, Vol. 97, No. 51, 1993
Before leaving the discussion of AL2 effects, it is interesting to note that (AL2(J=O)) < 0 while (AL2(J=1)) = 0 a t intermediate value of R for the three temperatures investigated. The decrease of ( L 2 )for H2(J=O) would contribute somewhat to the enhanced k, by giving rise (on average) to a reduced angular momentum barrier. This kinetic, rather than potential, energy contribution to thereaction rateappearsnot to have been identified previously and should apply to ion-dipole collisions as well. JEffects. The enhanced kcof Hz(J=O) together with the slow rate of ortho/para interconversion in the absence of a catalyst introduces the possibility of a symmetry-dependent KIE based on the rotational state distribution of the reactants. To illustrate this effect we consider the following kinetic scheme, appropriate when the fractional abundance of deuterium is low (e.g., natural abundance conditions), where the reaction rate is taken to be the
L
1
0
200
100
300
T (K) Figure 5. Predicted deuterium enhancement (defined by eq 20) as a function of temperature for normal and fully equilibrated H2 rotational
distributions. complete ortho/para relaxation would cause H2 to be dominantly in the J = 0 rotational level and some deuterium fractionation may occur by the 1-dependent KIE because restrictions would not apply to H2+/HD or HD+/H2 reactions. However, given the magnitude of the observed deuterium fractionation in some interstellar clouds,43these symmetry-dependent KIE should be considered only as a possible contributing factor.
Acknowledgment. Acknowledgement is made to the Donors of the Petroleum Research Fund administered by the American ChemicalSociety(PRF22721-AC5) and toNSF(Grant CHE9024091) for partial support of this work. In the rate expressions for reactions 15-18, /3 represents the fractional yield of deuterium in the ionic product. Defining predicted deuterium enhancement (Enh) in the ionic product as
Appendix A Equations of Motion Before specifying the equations of motion, it is convenient to define the following four auxiliary relationships:
R = (X2 cos y = ( X sin 5 cos q it is shown in appendix C that the enhancement for a low fractional abundanceof deuterium which is initially statistically distributed among the reactant molecules is given approximately by
Enh =
J
Presently, there is no direct information available on the magnitudes of and 1917 which enter eq 20 through k15 and kl7. Although there is evidence from measurements of the relative product yields of HD+/HD and H2+/D2 reactions42that deuterium is favored in the ionic product at 300 K, for the present purpose of emphasizing the effect of rotational state distributions on deuteriumenhancement,Bls and are taken to be thestatistical value of 0.75. Using the results in Table I1 and rotational distributions for normal and fully equilibrated Hz, deuterium enhacement factors were calculated as a function of temperature and are plotted in Figure 5. For fully equilibrated H2 the enhancement is -4.08 independent of temperature. Conversely, the enhancement for normal H2 increases with decreasing temperature from a value of -0.08 at 300 K to +0.19 at 1 K. The preceding discussion of symmetry-dependent KIE may have relevance for deuterium fractionationobserved in interstellar clouds.43 The question of ortho/para relaxation of H2 in interstellar clouds is a complex one which modeling indicates depends in a large part on the history and age of the If ortho/para relaxation is incomplete then some deuterium fractionation can occur by the J-dependent KIE. Alternatively,
(21)
+ Y sin $. cos q + 2 cos t ) / R
au = - 3qQ(3(cos2 Y) - 1) aR
+ Y2+ 22)'/2
2 ~ 4 2q2[3a
(22)
+
+ (all - a,)(3(cOs2 3R5
y ) - l)] (23)
In terms of eqs 21-24, the equations of motion which follow from the Hamiltonian given in eq 4 are
The H2+/H2 Proton/Atom Transfer Reaction
dPf -=”dt
The Journal of Physical Chemistry, Vol. 97,No. 51, 1993 13635
P’cos f‘ [sin3
E X cos [cos q
+ Ycos f‘ sin q - Z s i n E) (33) R
-X sin [ sin q
+ Y sin f‘ cos q )
R
(34)
Appendix B Tunneling Path Relationships
Because the trajectories were integrated using X,Y, Z, E, q, and their conjugate momentum as dynamical variables and, as described in section IID, the tunneling path was taken to involve only radial motion with 8,4, E, q, their conjugate momentum, and PRconserved, application of the tunneling correction required the determination of barrier “exit” values for the Cartesian position and momenta coordinates which satisfied conservation of energy and angular momentum in terms of the corresponding barrier “entry” values. Denoting the values of the nonconserved dynamical variables where the barrier is “entered” and “exited” by the subscripts 1 and 2 respectively, conservation of B and 4 (and hence cos B = Z/Rand tan 4 = Y/X) requires the following relationships:
k15pHD(J)I
(43)
where [H2] and [H2+] represent respectively the total concentration of hydrogen molecules and ions of any isotopic composition, and a is the fractional abundance of deuterium atoms (assumed to be small). Taking the predicted ion product ratio to be equal to the formation rate ratio and dividing by the statistical product ion ratio: [HZD+Istat=[H3+lstat
301
- CY
(44)
gives the result
3c{[(1
(45)
- a ) k 1 4 + 2ak181PH2(J) + 2ak16PHD(J)l
J
y2
= YI (X2/XI 1
(36)
With the value of R2 determined as described in section IID, Z2 is given immediately by eq 35. Substitution of eqs 35 and 36 into eq 21 allowed X2 to be determined by
x2= f
(37)
Substitution of eq 45 into eq 19 followed by expansion in powers of a and retention of only the constant term, gives eq 20. References and Notes (1) Mauersberger, K. Geophys. Res. Lett. 1981, 8, 935. (2) Mauersberger. K.Geoohvs. Res. Lett. 1987. 14. 80. (3) Abbas, M. k;Guo, J.;’Carli, B.; Mencaraglia, F.fCarlotti, M.; Nolt, I. G. J . Geophys. Res. 1987, 92, 13231. (4) Heidenreich, J. E.,111; Thiemens, M. H. J . Chem. Phys. 1983, 78,
___.
9513
where the sign of X2 is required to be that of XI.Substitution of eq 37 into eq 36 allows Y2 to be determined. Because P, and Pt are conserved along the tunneling path, conservation of angular momenta requires that PX,
= LY/Z2 + PZ2(X2/Z2)
(38)
where Li denotes a Cartesian component of the relative orbital angular momentum vector. Substitution of eqs 38 and 39 into the expression for PR
allows Pz, to be determined by
Finally, substitution of eq 41 into eqs 38 and 39 allow Px, and Py, to be determined. In an exact treatment, PR would be zero at the point where the barrier was “entered” and “exited”. However, as discussed in section IID, the necessity of using a finite step size in the trajectory calculations resulted in PR having a small magnitude, negative value at these points. Appendix C Deuterium Enhancement
From the reaction scheme detailed in eqs 14-1 8 the formation rate of H3+ and H2D+ can be written as
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