J. Phys. Chem. 1992, 96, 217-223 concentrations, and it was found that in all the solvents the decay rate was not affected by higher concentrations. In the presence of an excess of 02,the rate goes to the higher limit of 8 X lo7 s-I. In nonpolar solvents, the charge-transfer state is not stabilized and contact ion pairs (CIP) are formed. In polar solvents like dichloromethane and in acetonitrile, the solvents stabilize the CT state by solvation and they form solvent separated ion pairs ( S I P ) . The decay curve for the dichloromethane solution at 470 nm shows that after bleaching not all of them are recovered back but a part of them are lost due to the interactions with solvent molecules. In alcohols, the triplet is long lived. The interaction here is through solvation and H bonding which stabilizes the CT state. The triplet yield is very low in alcohols, as is the case with MK, which decreases with increasing polarity. The only competing reaction here seems to be quenching by 02.In deoxygenated solutions, the observed decay rate is of the order 104 s-', whereas in the O2saturated solution, the decay rate is lo9 M-' s-l. The ketyl radical formation is eliminated because of the reduced population in )nx* state. Conductivity Studies. The dc conductivity data were obtained using the 308-nm laser for MK and the 480-nm laser for DEAW. The signal observed for deoxygenated MK in cyclohexane showed that there was a fast geminate recombination (within a few nanoseconds after the pulse) of MK+' and e- after a biphotonic process, as shown by the square dependence of the signal amplitude with the laser intensity. This demonstrates the possibility of different photoprocesses with varying mechanisms as the intensity and energy of the laser increases. For this reason, we used a lower laser intensity and hence lower energy to avoid biphotonic processes. The sample saturated with SF, gave a similar signal with a reduction in signal amplitude and decay rate due to the sca-
217
venging of the electrons by SF,. This was the case with toluene also.
Conclusions The important finding in this study is the close resemblance of the photochemistries of DEAW and MK. The use of DEAW as an efficient sensitizer in the visible region in some solvents and its capability of being used as a laser dye in some other solvents are recognized. The dipole moment of the first excited singlet state of DEAW is estimated to be 14.0 D from the solvatochromic shifts in the absorption and fluorescence spectra. The quantum yield of fluorescence increases with increasing polarity of the solvent with the exception of alcohols, it varies with different excitation wavelengths. The quantum yield for triplet-triplet absorption increases with decreasing E: values. The first-order decay rate constant of the triplet state in alcohols is much smaller than that in nonpolar solvents, indicating the stabilization of the triplet state. In nonpolar solvents, the energy of the lowest triplet state is estimated to be -57 kcal mol-', and its fluorescence quantum yield is very low, with a high triplet yield which is essential for a molecule to act as a triplet sensitizer. In alcohols, the energy of the lowest triplet state is very low (-35 kcal mol-'), and low triplet yield with a relatively high fluorescence quantum yield makes it an efficient dye to be used in lasers. This dual role of DEAW makes it an excellent candidate for the use in both photoimaging and lasers.
Acknowledgment. We greatly appreciate the work of Dr. K. H. Schmidt in setting up the flash photolysis apparatus and for many helpful discussions.
State-testate Vibration-Translation and Vibration-Vlbration Rate Constants in H2-H2 and HD-HD Collisions Mario Cacciatore Centro di Studio per la Chimica del Plasmi. Istituto di Chimica Generale ed Inorganica, Universita di Bari, 70126 Bari, Italy
and Gert Due Billing* Chemistry Laboratory III, H . C. 0rsted Institute, University of Copenhagen, 2100 Copenhagen 0, Denmark (Received: May IS, 1991)
A semiclassical collision model and a potential energy surface fitted to ab initio data has been used to calculate V-V and V-T/R rate constants for vibrational transitions in H2-H2 and HD-HD collisions. The theoretical values are compared with recent experimental data.
Introduction Vibrational energy transfer in atom-diatom and diatom-diatom collisions plays an important role for the determination of nonthermal vibrational energy distributions in many physical important situations, as e.g., low-pressure plasmas' and chemical IR laser systems2 (1) Cacciatore, M.; Capitelli, M.; Benedictis, S.; Dilonardo, M.; Gorse,C.
Vibrational kinetics dissociation and ionization of diatomic molecules under non-equilibrium conditions. In Non-equilibrium Vibrational Kinetics; Capitelli, M., Ed.; Topics of Current Physics; Springer: Berlin, 1986. (2) Moore, C. B., Ed. Chemical and Biochemical Applications of Lasers; Academic Press: New York, 1974.
Especially for H2the vibrational energy transfer in collisions with vibrationally excited H2 molecules has within the last few years been of great interest, e.g., for the production of negative ions: a field related to the nuclear fusion technology." Also studies in the nonequilibrium vibrational kinetics in hydrogens as well (3) Gorse, C.; Capitelli, M.; Bacal, M.; Bretagne, J.; Lagana, A. Chem. Phys. 19%9,117, 177. (4) Hopmann, H.; van Amersfoort, W., Eds. Proceedings of the III European Workshop on Production and Applications of Light Negative Ions;
FOM Institute: Amsterdam, 1988. ( 5 ) Cacciatore, M.; Capitelli, M.; Dilonardo, M. Chem. Phys. 1978, 34,
193.
0022-3654/92/2096-217%03.00/0 0 1992 American Chemical Society
218
The Journal of Physical Chemistry, Vol. 96, No. 1. 1992
as other diatomic molecules] has revealed an urgent need for V-V and V-T rate constants between vibrationally excited molecules. However, most experimental determinations of V-V and V-T rate constants in diatomic molecules have been dealing only with the first few vibrational levels. From an experimental point of view the direct determination of state-to-state rates in hydrogen is in fact prevented due to the difficulty in pumping and probing selected vibrational transitions for molecules with no dipole moment. However, very recently nonlinear optical methods, specifically CARS probing, stimulated Raman excitation, and optoacoustic detection techniques have appeared.6x”-’3 In this way it has been possible to determine not only the state resolved V-V and V-T rates between the lower levels of hydrogen (and its isotopes), but also by the introduction of scaling laws to give an estimate of the rates from the upper 1evels.I’ Also the population switching technique by two laser near-resonant stimulated Raman scattering constitutes an important step forward in the creation of excited states and hence for the development of the understanding of excited-state dynamics.I6 The appearance of these new experimental informations has renewed our interest in further theoretical calculations on the vibrational energy exchanges in hydrogen. We have used a semiclassical modell’ and a potential which is a slightly modified version of one previously determined19 by fitting to a b initio CI calculations. This modification of the potential improved the agreement with experimentally determined rates for the vibrational relaxation of hydrogenz0 at low temperatures. The new potential differs significantly from the model potential used by Billing and Fisher24and as a consequence the V-V rates, which previously had only been calculated with this potential, are changed significantly. The reason for this is obviously that the V-V rates depend upon the second derivatives of the potential with respect to the bond length. Since information upon this dependence is scarce and eventual agreement with V-T relaxation data only requires a good first derivative we see that the agreement between theory and experiment for the V-V rates is much more difficult to obtain due to the lack of good potential hypersurfaces. In the present paper we further extend the calculations to include also the HD-HD system for which experimental information has recently become available. The semiclassical collision model which is used here is also improved as compared to the one used by Billing and Fisher, namely by expanding the total wave function in product states of anharmonic (Morse) functions for the isolated molecules, rather than using the energy-corrected harmonic oscillator However, the differences between the present and previous calculations are for transitions among the lower lying vibrational states almost entirely due to the changes made in the potential. However, the present collision model allows us to extend the calculations to vibrational states closer to the dissociation limit since it includes anharmonicity in the proper way. Hence the vibrational range covered can be extended so as to allow for a critical evaluation of vibrational scaling laws. The semiclassical model (ref 17) used in the present calculations has previously been tested against exact quantum calculations for collinear atom-diatom7 and diatom-diatom8 collisions. The model (6) Rohlfing, E. A.; Rabitz, H.; Gelfand, J.; Miles, R. B.; De Pristo, A. E. Chem. Phys. 1980, 51, 121. Rohlfing, E. A,; Gelfand, J.; Miles, R. B.; Rabitz, H. J . Chem. Phys. 1981, 75, 4893. (7) Billing, G. D. Chem. Phys. Letf. 1975,30, 391; J . Chem. Phys. 1976, 64, 908. (8) Billing, G. D.; Jolicard, G. Chem. Phys. 1982, 65, 323. (9) Price, R. J.; Clary, D. C.; Billing, G. D. Chem. Phys. Lett. 1983, 101, 269. Billing, G. D. Chem. Phys. 1986, 107, 39. (IO) Schwenke, D. W.; Truhlar, D. G. Theor. Chim. Acta 1986,69, 175. ( I 1) Rohlfing, E. A.; Rabitz, H.; Gelfand, J.; Miles, R. B. J . Chem. Phys. 1984, 81, 820. (12) Kreutz, T. G.; Gelfand, J.; Miles, R. B.; Rabitz, H. Chem. Phys. 1988, 124, 359. (13) Arnold, J.; Dreier, T.; Chandler, D. W. Chem. Phys. 1989, 133, 123. (14) Pirkle, R. J.; Cool, T. A. Chem. Phys. Lett. 1976, 42, 58. (15) Bott, J. F. J . Chem. Phys. 1976, 65, 3921. (16) Gaubatz, U.; Rudecki, P.; Becker, M.; Schiemann, S . ; Kii lz, M.; Bergmann, K. Chem. Phys. Lett. 1988, 149, 463.
Cacciatore and Billing TABLE I: Potential Parameters parameter ref 24 ref 19 C 2.782 0.920 a1 1.868 1.476 42 0.000 0.03528 2.4984 RO d 0.5000 0.3324 3.31 & I A 2.81 a 6.51 7.74 b 19.41 0.00
present
units
0.920 1.583 0.03528 2.4984 0.3324
hartree au-l
3.31
A A-1
2.81 7.74 0.00
~ I I - ~
au
eV eV
A6
AB
is extended to 3D by treating the rotational motion classically. This approach has been tested against quantum (coupled states) calculations for atom-diatom system^.^ For diatom-diatom systems the number of converged quantum calculations for vibrational excitations is scarce, since more than 1000 coupled channels will have integrated. Recent attempts have, however, been made by Schwenke and Truhlar’O for the HF-HF system. For the generation of a large number of rates in a large temperature range one has, however, to rely on approximate but well tested methods as the one used in the present paper. The PotentialThe potential for the Hz-Hz system has previously been fittedI9 to ab initio values for the i n t e r a ~ t i o n .Here ~ ~ we use a slightly modified version which gives good agreement (1-2% in the energy range relevant for the present calculations) with recent experimental beam measurements of the isotropic partz6 Thus we have
where the short-range part is
VsR = 4C exp(-a(R)R) X cosh (6a(R)rlcos rl)cosh (6a(R)rzcos y2) (2) and
+
a ( R ) = a] a2(R - Ro)Z
(3)
Here ri is the bond distance in molecule i (i = 1 , 2 ) , y i the angle between the center of mass vectors R and ri, and R the distance between the center of masses for each molecule. The quadrupole interaction is Vw =
3 -e(rl)
e(r2) x
16RS ( 1 - 5 COS’ yI
- 5 cos2 72 - 15 COS’ 71 cos2 yz + 2(sin y1 sin y2 cos($1 - $2) - 4 cos y1cos yz)2)(4)
where the angles $i together with yi specify the orientation of the diatomic molecule in a coordinate system having its z axis along the R axis. The quadrupole moment is
e ( r i ) = e(0.248 + 0.172(ri - e q ) / a o ) esu A2
(5)
(17) Billing, G. D. Comput. Phys. Rep. 1984, I, 237; Chem. Phys. Lett. 1983, 97, 188. Billing, G. D.; Cacciatore, M. Chem. Phys. 1983, 82, 1 . (18) Muckerman, J. T.; Kanfer, S.; Gilbert, R. D.; Billing, G. D. Classical Path and Quantum Trajectory Approaches to Inelastic Scattering. To be published. (19) Billing, G. D. Chem. Phys. 1977, 20, 35. (20) Cacciatore, M.; Capitelli, M.; Billing, G. D. Chem. Phys. Letf. 1989, 157, 305. (21) Cacciatore, M.; Billing, G . D. Chem. Phys. 1981, 58, 395. (22) Duff, J. W.; Blais, N . C.; Truhlar, D. G. J . Chem. Phys. 1979, 71, 4304. (23) Farrow, R. L.; Chandler, D. W. J . Chem. Phys. 1988, 89, 1994. (24) Billing, G. D.; Fisher, E. R. Chem. Phys. 1976, 18, 225. (25) Silver, D. M.; Stevens, R. M. J . Chem. Phys. 1973,59,337. Bender, C. F.; Schaefer 111, H. F. J . Chem. Phys. 1972, 57, 217. (26) Norman, M. J.; Watts, R. D.; Buck, U. J . Chem. Phys. 1984, 81, 3500.
The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 219
H2-H2 and HD-HD Collisions
I ‘
I
I
I
I
I
I
I
I
Theory
Ducuing
t
,,’
et,al.,‘
I
1000
I
1
I
1500
2000
2500
T(’K)
Figure 2. Comparison of experimental low-temperature’O and high-temp e r a t ~ r e ~rates ~ . ’ ~for the V-T relaxation rate and the theoretical values (dashed curve) obtained with the present potential.
is a matrix element responsible for the Coriolis (centrifugal stretch) coupling which couples the rotational (treated classically to obtain ji(r)) to the vibrational motion. and I$, are the eigenfunctions and the eigenvalues for the Morse oscillators. The Morse constants for hydrogen are taken from H e r ~ b e r gand ~ ~those for H D are obtained by mass scaling. For molecules with small moments of inertia as hydrogen this coupling is important.28 From the time-dependent Schrijdinger equation we obtain a set of coupled equations for the amplitudes ~ , , ~ , ~ ~These ( t ) . are solved simultaneously with 18 classical equations of motion for the translational and rotational motion of the molecules. Since the time-dependent Schrijdinger equation involves real and imaginary terms, we then obtain 18 + 2 X N coupled real equations, where N is the number of product vibrational states. Typically we use N = 50-60 in order to obtain convergence for the cross sections. From the solution of these equations we obtain the so-called average cross sections defined by”
*:,
2
3
L
5
6
7
0Rla,
Figure 1. Comparison of angle-averaged theoretical potential (solid line) with the experimentally determined (0) (ref 26) at low energies.
where e is the electron charge and finally the dispersion potential is approximated by
In Table I we have given the parameters used in the present calculations together with those used by Billing and Fisher24and Billing.I9 The Billing and Fisher potential gave a V-T rate constant at 300 K, which was about a factor of 2 too large compared with the experimental value, whereas the Billing potential which was a fit to ab initio calculations underestimated the V-T relaxation rate with about 20%. The Billing and Fisher potential included an R8term but no switching term in the dispersion potential. Previously only the more approximate Billing and Fisher potential had been used to calculate V-V rates.24 The present potential was modified as compared to the Billing potential just by changing the CY] parameter so as to obtain good agreement with the angle averaged potential of Buck et a1.26(see Figure 1). This modification also improved the agreement with the experimental value for the V-T relaxation making it almost perfect in the temperature range from 300 to 3000 K (see Figure 2). We therefore believe that it is justified to use this surface for extensive calculations of the V-V rates also. The new potential has a minimum for the isotropic part at 6.6 bohr of 11.3 X hartree. It is compared in Figure 1 with the potential obtained by Buck et a1.26 Also the quadrupole parameters are identical with those of ref 26. Method In the semiclassical method which has been used here the wave function for the quantum degrees of freedom (the vibrational motion) is expanded in rotationally distorted Morse vibrational product states, i.e.
where
1 (2j, + 1)(2j2 + 1)(21 + l)-CIanl,n2-nl’,n2~12 (10) Nt where Zi are moments of inertia of the two molecules and p is the reduced mass of the relative motion. For each trajectory we solve a set of time-dependent equations for the amplitudes obtained by inserting eq 7 in the time-dependent Schrtidiiger equation. The amplitudes are initially anlln2, = 6,,,,, and the quantity anln2-nl,ni in eq 10 denotes the amplitudes for a transition from state nlnz to n,’ni. We average over Nt trajectories with a given initial “classical” energy U,where
U = Ekin + E::) + E::)
(1 1)
The initial variables which are chosen randomly are 1 (orbital angular momentum) between [&Imax],j i (rotational angular momenta) between [O&max], ml (the projection of I) between -1 and +I, mj, (the projection of j i ) between -jiand j i and the corresponding angles q!, qji, P,, and 8, (i = 1, 2). These action-angle variables then specify the initial condition expressed in terms of Cartesian coordinates and momenta. Typically about 150-250 trajectories are needed in order to obtain convergence for the average cross sections to within 20-30%. The dynamical calculations have been performed on the CRAY/ 1 (27) Herzberg, G . Spectra of Diatomic Molecules; Van Nostrand: Princeton, NJ, 1950. (28) Billing, G. D. Chem. Phys. 1975, 9, 359.
220 The Journal of Physical Chemistry, Vol. 96, No. I, 1992 log K $
_ _ _--- -- - -&LING + FISHER (19761
I I
I ’
1
-13 1
Cacciatore and Billing In a first-order theory the amplitude for a V-V transition would be proportional to
where ho is the energy mismatch for the transition. Considering just the short-range part of the potential we get dt exp(iot)624R)2COS
-
li
exp KREUTZ et a1 119881
.I
I
I
I
700
500
I
I
1
I
K
-
Figure 3. Comparison of experimental” and theoretical values of the V-V rate constant for the process: H2(u=2)+ H2(u=0) H2(u=l)+ H2(u=1)as a function of temperature. Upper dashed curve is the old values from ref 24 and the lower solid curve is the present result obtained with a new improved potential energy surface. The lower dashed curve gives the rate constant obtained with the original Billing potential.lg
computer at Cineca/Bologna. Once the average cross sections are determined we can calculate the rate constants. They are obtained as
where /3 = l/kT, nl,nzthe initial, nl’, n2/ the final, 1.1 is the reduced mass and the energy U is now the symmetrized energy defined by17 1
+ (@I2
= u + -AE 2 16U where AE = EnIt+ E,; - E,, - E,,,, tmin= 0 for AE < 0 (endothermic) and tmin= AE for AE > 0 (exothermic) since Umin= t
cos yZ V ( R J ~ , ~ ~ ~ Y ~ ,(16) Y~)
(1/ 4 ) 1 4 . Thus each quantum transition has its own set of ”best” trajectories. The above expression (1 3) is the effective energy obtained by using the mean trajectory approach.17 For a 1D system with just translational and vibrational degrees of freedom it can be derived using a Gaussian wavepacket description of the translational motion followed by a projection onto plane incoming and outgoing waves.18 The expression used here is a 3D generalization of this result. In practical calculations one would find the average cross sections for a set of Uvalues (typically around 10) and would use eq 13 when calculating the integral (12).
Results In the present paper we report upon the calculation of some rates for V-V and V-T/R processes in H2-H2, H2-D,, and HDH D collisions. Figure 3 shows the rate constants k(0,112,1) for the process H~(u=O)+ H2(~=2) H,(u=l) + H 2 ( ~ = l ) (14)
-
in the endothermic direction. Comparison is made between our present calculation (using both the original and the slightly modified potential) and a previous one due to Billing and Fisher.24 Although the collision models used were not quite the same, the large difference found between the two sets of calculations is due to differences in the potential energy surface. The present PES has a much smaller vibrational coordinate dependence and the V-V rates are very sensitive to this particular feature of the PES.
Thus comparing the quantity CY)^ at a center of mass distance corresponding to the turning point at Ekin 500-1000 cm-I, we would get a factor of 3.2 difference between the Billing and Fisher and the present potential. For this reason alone the amplitude would in a first-order theory deviate by a factor of 3.2 and the probability by a factor of 10. Since the Billing and Fisher potential is steeper (see Figure 1 of ref 19) we would again based on first-order arguments obtain an additional factor of about 4 difference between the rates. This simple qualitative analysis therefore explains the large difference found also in the numerical calculations. Based upon the same arguments the Billing potential would only give V-V rates about 30% lower than the present ones. This ansatz is confirmed by the comparison made in Figure 3. The experimental data of Kreutz et a1.I2 at 298 K lie between the two calculated values. But a low-temperature value of lo-’’ cm3 at T = 82 K obtained by TeitelbaumB is smaller than the value we would obtain at this temperature by a factor of 8. However, due to the classical treatment of rotations the rates are only reported down to 300 K, although, due to the small energy mismatch, the V-V transitions are actually less sensitive to the quantum nature of the rotational motion. In any case a quantum treatment of the rotational motion would probably increase the rates rather than decrease it. Thus unless it appears that something is wrong with the present PES we conclude that the experimental value given by Teitelbaum is too low. The value obtained by Teitelbaum was extracted from the experimental data of Audibert et al.30assuming a simple linear scaling law for the V-T rates and a quadratic one for the V-V rates. However, our theoretical investigations do not support such simple scaling relations. On the contrary, our calculated V-T rates show (at small vibrational quantum numbers) an almost linear behavior in a log plot?O The experiment of Kreutz et al.12 determined both the V-T rate k(2,110,O) and the V-V rate k(2,110,1). If we instead use thevalue for the V-T rate, we get from Table I of ref 12 a V-V rate of cm3/s, Le., in much better agreement with our about 6-7 X value (seeFigure 3). On the other hand, the Raman experiment by Farrow and Chandler23gives a V-V rate for this transition which is a factor of two higher than the value determined by Kreutz et al. In any case the experimental determination of V-V transition rates provide a very sensitive probe of the potential. Comparison of the V-V rate for the process H 2 ( ~ = 1 )+ D ~ ( u = O ) H2(0=0) + D2(~=1) (17)
-
with experimental data (see Figure 4) shows essentially the same picture. The V-V rates obtained with the Billing-Fisher surface overestimate the rates and yield too steep a temperature dependznce, whereas the results obtained with new surface underestimates the rates but has about the correct temperature dependence. Figure 5 shows the single quantum V-T and V-V exothermic rates for the processes HZ(u-0) + H ~ ( u ) HZ(U=O) + Hz(u-1) (18)
+
H,(u=l)
+ H2(u)
H,(u=Oj
+ H2(u+1)
(19)
as a function of the vibrational quantum number u at T = 300 K. We notice that both rates increase with u up to about u = 4. The rates can at a given temperature increase for two reasons: (29) Teitelbaum, H . Chem. Phys. Lett. 1984, 106, 69. (30) Audibert, M. M.; Vilaseca, R.;Lukasik. J.; Ducuing, J. Chem. Phys. Lett. 1975, 31, 232; 1976, 37, 408.
The Journal of Physical Chemistry, Vol. 96, NO. 1, 1992 221
Hz-H2 and HD-HD Collisions ,
cmVsec
t
I
1
I
I
i
/"
t 10-'5'2h LOO 600 800 I000 1200 1LbO
'K
-
'
F i 4. Comparison of experimental and theoretical values for the V-V process: H2(u=l) + D2(u=0) Hz(u=O) + D2(u=1). The dashed line indicate values obtained with the Billing-Fisher potential and the solid line values obtained with the present potential. Experimental data from ref 14 are indicated by ( 0 )and from ref 15 by (m). The vertical lines indicate the experimental error bars. TIK)
- 12
-
+
--
-
Figure 6. Theoreticallydetermined rate constants as a function of mperature for the V-V processes: H2(u=0) + H2(u) Hz(u=l) Hz(u-l), where u = 3 (curve l), u = 5 (curve 2), v = 7 (curve 3), u = 9 (curve 4), u = 10 (curve 5 ) , and u = 12 (curve 6). Also the three double-quantum transitions H2(u=0) Hz(u) H,(v=l) + H2(c-2), where u = 7 ( 0 ) ,v = 9 (0),and u = 11 (A)are shown.
+
-13
YI
m5 6 -lL 5 c
2
c
-13
s 0)
c
e
-
-
m
CD
5 v)
E
-15
$
-8 I
1
2
-lL
I
3
5
7
9
v
Figure 5. Comparison of rate constants for V-V and V-T processes as a function of vibrational quantum number at 300 K. The V-V rates are shown in the exo- and the endothermic direction (see also text).
I
I
1
I
I
1
1
LOO
500
600
700
800
900
I
T (OK1
first because the energy mismatch decreases and second because an excited molecule is a dynamically more efficient quencher. For the V-T processes (18) the energy mismatch decreases as u increases so that both effects work in the same direction and the rate constant increases. For the V-V transfer processes (19) the energy factor works in the opposite direction of the dynamical coupling effect. At a given value of u (u 4) the energy mismatch has increased sufficiently such that this is the dominating factor and hence the V-V rates decrease. It has been discussed a t which vibrational level the V-T transitions are more efficient than the V-V transitions. In order to see that, it is necessary to compare the exothermic V-T rate k ( O , O l u , ~ l with ) the endothermic V-V rate k ( O , l l u , ~ l ) . For Hzat 300 K the crossing occurs at a value of u around u = 3 and for HD between u = 3 and u = 4. For HD Rohlfing et al.II find the crossing between u = 4 and u = 5 . As we shall see below, this is due to the much higher V-V rates at low u levels predicted by these experiments. Figures 6 and 7 show a number of V-V rates among higher u states as a function of temperature. It is worthwhile noticing
-
+
-
Figure 7. V-V rate constants for the processes: H2(u) H2(w) H,(u+l) H z ( w l ) with the same energy mismatch (242.6 cm-I) as a function of temperature. Curve 1 (u = 6, w = 8), curve 2 (u = 4, w = a), curve 3 ( u = 2, w = 4), and curve 4 (u = 1, w = 3).
+
that the energy mismatch for the transitions reported in Figure 7 is almost constant (around 242 cm-I). Therefore, the aforementioned increase of the rate constant with the vibrational quantum number must be ascribed to the dynamical coupling between the vibrational eigenstates of the two oscillators. This behavior is evident in the full range of temperature, and this proves that the energetic factor does not modulate the coupling integrals. A similar behavior has been pointed out for the V-T relaxation in H2.20 It has been discussed whether the vibrational multiquantum transitions can be important in diatomics, a question of relevant importance when the rate constants are indirectly determined from relaxation data. It is quite clear that multiquantum transitions contribute to the dynamical relaxation of the high-lying vibrational states in C0.31 Therefore, one would expect an increase in the
222
Cacciatore and Billing
The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 I
I
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-
TABLE III: V-T Rates k(v,v-llO,O) (in cm3 s-l) for the Processes HD(v) HD(v = 0) HD(v-I) + HD(v = 0) as a Function of Y at Three Temperatures
+
theoretical values
T = 300K U
(expt)
1
1.0-2.0 (-16)"" 0.87-2.5 (-16)32 2.5-7.0 (-16) 3.9-6.0 (-15) 0.5-1.0 (-14)
2 5 6 a
Means (1.0-2.0)
X
T = 300 K 5.66 (-17)
T = 500 K 3.90 (-16)
T = 700 K 1.89 (-15)
2.59 (-16) 6.17 (-15) 1.58 (-14)
1.54 (-15) 2.75 (-14) 1.00 (-13)
6.17 (-15) 1.03 (-13) 4.03 (-13)
10-l6.
Y
-15
1 u
5 t
bo0
500 600 700 800 900 T
[OK1
-
" /
t
F w e 8. V-T rate constants for the processes: H2(u) + H2(w) H2(u) + H2(w-l) as a function of temperature. Curve 1, u = 7, w = 8 (AE =
2459.1 cm-I); curve 2, u = 7, w = 7 (AE = 2701.6 cm-I); curve 3, u = 5, w = 5 (AE = 3187.1 cm-I); curve 4, u = 3, w = 4 (AE= 3429.7 cm-I); curve 5 , u = 3, w = 3 (AE = 3672.3 cm-I); curve 6, u = 2, w = 3 (AE = 3672.3 cm-I); and curve 7, u = 2, w = 2 (AE = 3914.9 cm-I).
-.
TABLE 11: V-V Rates Constants k(v,v-llO,l) (in cm3 s-I) for the Process HD(Y)+ HD(v = 0) HD(v-1) + HD(Y= 1) as a Function of Y at Three Temperatures
theoretical values
ref 11 u
T = 300 K
T = 300 K
T = 500 K
1 2 5 6
0.9-3.0 (-14) 1.0-3.0 (-15) 1.0 (-15)
2.60 (-15)" 2.44 (-15) 7.85 (-16) 3.11 (-16)
3.44 (-15) 4.10 (-15) 3.14 (-15) 1.95 (-15)
"Means 2.60
T = 700 K 3.79 (-15) 5.80 (-15) 6.02 (-15) 4.36 (-15)
X
relative importance between single and multiquantum transitions in molecules with large anharmonicity like H2. This is shown in Figure 6. Thus the double quantum transition H2(u=0)
+ H2(u=9)
-
H2(u=1)
+ Hz(u=7) + 518.3 cm-I (20)
is much more efficient than the corresponding one-quantum transition H2(u=0)
+ H2(u=9)
-
H2(u=l)
+ H 2 ( ~ = 8 )- 1940.9 cm-' (21)
In Figure 8 we have reported the exothermic rate constants for the V-T exchanges H2(u)
+ H2(u)
-
H ~ ( u - 1 )+ H2(u)
(22)
as a function of the temperature. At a given T the rate constant increases with u due to the energy mismatch. On the other hand, the effect of the vibrational coupling is evident when comparing the transition (3,2)-(3,3) with the transition (3,2)-(2,2) (curve 5 and 6, respectively). Obviously, this effect is more pronounced (31) Billing, G. D.; Cacciatore, M. A. Chem. Phys. Lett. 1983, 94, 219.
-1
v Figure 9. V-T and V-V rates at 300 K for HD(u) + HD(u=O) HD(lr1) + HD(u=O) or HD(u=l) as a function of u.
-
for Hz in higher u states. Thus the rate k(7,6)0,0) is 10 times smaller than the rate k(7,617,7) at T = 300 K. Tables I1 and 111 show the V-V and V-T rates for the processes
+ HD(u=O) HD(u) + HD(u=O) HD(u)
-+
+ HD(u=l) HD(u-I) + HD(u=O) HD(u-1)
(23)
(24)
and compare with recent experimental data. We notice that the agreement is good for the V-T rates but that the V-V rates for small u (u = 2) differ by a factor of 8. This in turn yields a V-V crossing of the V-T rates (Figure 9) for a u value between 3 and 4 as mentioned above. It should be noticed, however, that the experimentally determined rates were obtained indirectly by assuming a specific semiempirical scaling law for the u dependence of the rates. If the relaxation of HD and Hz is mainly a V-T relaxation, is., the rotational/vibrational quenching is not dominating, then we should expect a smaller relaxation rate for HD than for H, as found theoretically. It would therefore be interesting to fit the experimental data with a smaller k( lOlO0) rate and a sharper increase with vibrational level than the one used by Rohlfing et al. Also the V-V scaling law used appears not to be flexible enough because our data tend to follow their curve when a crossing a t u, = 4 is assumed but then to approach the u, = 5 curve for u 2 6. Simple scaling laws have previously been shown to be questionable, both in s e m k l a s s i ~ a l ~ ~ * ~ ' and quasiclassical trajectory studies.22 We therefore hope that continued progress in the experimental studies will eventually lead to the possibility for a direct determination of state resolved rates in hydrogen and other diatomic molecules. Conclusion
In the present paper we have reported upon V-V and V-T rates for energy transfer in the H2-H2, H2-D2, and HD-HD systems.
J . Phys. Chem. 1992, 96, 223-226 The rates are obtained without any fitting to experimental relaxation data whatsoever. The potential has been fitted to ab initio calculations, which consider the vibrational coordinate dependence of the potential. Only a slight change of one of the parameters of this fit, so as to obtain agreement with an independently determined angle averaged potential, has been introduced. It has been demonstrated that this change gives almost perfect agreement with experimental V-T relaxation rates. The V-V rates change by 3Q-50% by this modification. With the available amount of information on the vibrational coordinate dependence of the potential and considering the good agreement with experimental data for the V-T relaxation it is at present difficult to argue for the use of other potentials than the one used here. No attempt has been made to modify the potential to get agreement with experimental V-V data, since they are only available at a few temperatures and do not agree among each other. Improvement of the model so as to use Morse eigenstates in the expansion of the total wave function increases the computational effort as compared with the previously used energy-corrected harmonic oscillator model, but it enables us to calculate rates a t high vibrational levels and hence to be able to investigate the usefulness of semiempirical scaling laws for interpretation of experimental data or for simulation of bulk chemical systems. Our results indicate that the scaling laws which currently are used to extract state to state information from experimental data may be too
223
simple. Two different surfaces, both intended for the study of chemical reactions, are available in the literature. The potential of Varandas and MurrelP5 is a fit to the same ab initio points as those used in ref 28. Due to the different purpose of this fit it is a less accurate representation of the a b initio information than the one used here, where only the “inelastic” region is probed. After the present work was initiated a new potential energy surface appeared.36 The fit to the new a b initio points contains many more parameters than the present and would therefore be much more expensive to use than the present. It would, however, be interesting to test the sensitivity of both V-T and V-V rates to these other surfaces.
Acknowledgment. This research was supported by “Progetto Finalizzato Chimica Fine 11” of the Italian CNR and by the Danish Natural Science Research Council. Prof. M. Capitelli and Dr. Rosanna Caporusso are also acknowledged. Registry NO, Hz,1333-74-0; D, 7782-39-0. (32) Barroux, C.; Audibert, M. M. Chem. Phys. Lett. 1979, 66, 483. (33) Kiefer, J. H. J . Chem. Phys. 1968,48, 2332. (34) Dove, J. E.; Jones, D. G.; Teitelbaum, H. IVSymposium on Combustion, Uniuersity Park, PA, 1972; Combustion Institute: Pittsburgh, 1973. (35) Varandas, A. J. C.; Murrell, J. H. Discuss. Faraday Soc. 1975, 62, 92. (36) Schwenke, D. W. J . Chem. Phys. 1988,89, 2076.
Catalytic Synthesis of Ammonia Using Vibrationally Excited Nitrogen: Effect of Vibrational Relaxation Niels E. Henriksen,* Gert D. Billing, Department of Chemistry, H . C. 0rsted Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark
and Flemming Y. Hansen Fysisk-Kemisk Institut, The Technical University of Denmark, DTH 206, DK-2800 Lyngby, Denmark (Received: June 3, 1991)
In a previous study we have considered the catalytic synthesis of ammonia in the presence of vibrationally excited nitrogen. The distribution over vibrational states was assumed to be maintained during the reaction, and it was shown that the yield of ammonia increased considerably compared to that from conventional synthesis. In the present study the nitrogen molecules are only excited at the inlet of a plug flow reactor, and the importance of vibrational relaxation is investigated. We show that vibrational excitation can give an enhanced yield of ammonia also in the situation where vibrational relaxation is allowed.
1. Introduction The reaction mechanism of the catalytic synthesis of ammonia is believed to consist of seven elementary steps.’,2 The rate-determining step over iron catalysts is found to be the dissociative chemisorption of nitrogen. We have d ~ n eextensive ~ . ~ fundamental calculations of the probability for dissociation of N2on a catalytic active rhenium crystal, which has similar catalytic properties as iron. All six degrees of freedom of the N2 molecule were considered as well as the atomic motion of the metal atoms allowing for energy exchange between the molecule and the crystal. The interaction potential was adjusted such that available information on dissociation energies, vibrational frequencies, etc., is reproduced. The dynamics was described by a mixed quantum-classical ap(1) Ertl, G . Coral. Reu.-Sci. Eng. 1980, 21, 201. (2) Stoltze, P.; Norskov, J. K. Phys. Rev. Lett. 1985, 55, 2502. (3) Henriksen, N. E.;Billing, G. D.; Hansen, F. Y . Surf.Sci. 1990, 227,
224. __
proach and the calculations showed that translational as well as vibrational energy is effective in enhancing the dissociation probability. An increased temperature will therefore give a higher reaction rate-the obtainable yield at equilibrium will, however, decrease due to the well-known temperature dependence of the equilibrium constant for the ammonia synthesis. Excited vibrational states of the nitrogen molecule are not populated to any significant extent a t normal temperatures. Thermal excitation is not effective due to the large vibrational energy spacing of nitrogen, and the vibrational degree of freedom do not come into play under normal conditions. Suppose we have “vibrationally hot” nitrogen molecules available for the ammonia synthesis. An increased rate of reaction will follow, and the obtainable yield of ammonia will be enhanced, since the equilibrium constants for synthesis of ammonia from (pseudo-) nitrogen in specified vibrational states are predictedS to increase
..
(4) Billing, G. D.; Guldberg, A.; Henriksen, N. E.; Hansen, F. Y . Chem. Phys. 1990, 147, 1.
( 5 ) Hansen, F. Y . ; Henriksen, N . Sci., in press.
E.;Billing, G. D.; Guldberg, A. Surf.
0 1992 American Chemical Society
