2636
J . Phys. Chem. 1991, 95, 2636-2644
the simplest reasonable model that the predominant alkane loss channel is reaction with OH, the changing concentrations of these four species are used to infer the age and origin of low-lying air masses. However, predictions of the variations in relative alkane concentrations, based on previous rate constants and the recent field observations, conflict with the observed ratios. Accurate determinations of the rate constants for these reactions are therefore important to implicate either the model itself or the rate constants used in it'as the source of these discrepancies. The lower values of these rates found in the present study lead to increased disagreement between predictions and observations, indicating that the models may neglect important second-order effects and imply that assumptions regarding the spatial uniformity of the alkane tracer molecules or the effects of dilution on the traveling air masses may be incorrect.*' VI. Conclusions High-resolution infrared flash kinetic spectroscopy is used with a flow tube arrangement for time- and frequency-resolved studies of the O H radical. The OH is produced from photolysis by an excimer laser pulse and monitored via direct infrared absorption of light from a high-resolution color center laser which is tuned to the peak of or scanned over individual transitions from a specific rovibrational, spin-orbit and A-doublet quantum state. The collinear geometry of the excimer and color center lasers, along with the absolute infrared cross sections which have been measured
previously in this spectrometer, allows direct determination of OH number densities from the fractional IR absorption. The performance of this spectrometer is demonstrated by measurements of reaction rates of OH with ethane, propane, n-butane, and isobutane. These studies are facilitated by the high sensitivity and fast response of the real-time IR absorption signals. Complications due to wall reactions are eliminated, as the radicals are probed at the center of the flow cell. Extensive checks are performed to assure that the measurements are not complicated by competing radical-radical reactions, energy-transfer mechanisms, or temperature effects. Determinations of the four OH hydrocarbon rate constants are made which indicate that a significant number of previous studies have yielded excessively high values for the ethane, n-butane, and isobutane reactions. It is recommended that the rate constants currently used in atmospheric modeling be revised to reflect the lower values from the present study and that the assumptions underlying current atmospheric air flow models be reexamined in light of discrepancies between observations and predictions based on these rate constants.
+
Acknowledgment. This work has been supported by grants from the Air Force Office of Scientific Research. The authors thank Dr. A. R. Ravishankara and Dr. C. J. Howard for many valuable discussions. Registry No. HNO,,7697-37-2; OH, 3352-57-6; ethane, 74-84-0; propane, 74-98-6; butane, 106-97-8; isobutane, 75-28-5.
Structural Dependence of HF Vfbratlonal Red Shms in Ar,HF, n = 1-4, via Hlgh-Resolutfon Slit Jet Infrared Spectroscopy Andrew McIlroy, Robert Lascola, Christopher M. Lovejoy, and David J. Nesbitt*lt Joint institute for Laboratory Astrophysics, University of Colorado and National institute of Standards and Technology, and Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-440 (Received: May 10, 1990)
-
The rotationally resolved v = 1 0 HF stretching spectra of Ar,,HF, n = 1-4, have been observed by using a slit jet, difference frequency infrared laser spectrometer. The red shift of the HF vibrational frequency is seen to be sensitively dependent on the placement of the Ar with respect to the projection of the HF dipole moment; the largest incremental red shift is observed for the ArHF linear geometry. The n = 1-3 red shifts account for almost half (9.65-19.26 cm-') of the shift observed for H F in an Ar matrix (42.4 cm-I), suggesting that only the nearest neighbors contribute significantly to the perturbation of the H F vibrational frequency. The geometry of the experimentally observed isomer of Ar4HF places the fourth Ar in what would be the second coordination layer from the HF where it has little effect ( 20 portion of the Ar2HF P branch. The feature just to the blue of the Ar3HF Q branch may be due to a Q branch of a hot band transition, larger cluster, or a second isomer of Ar3HF tetramer. Ar,HF Q Branch Ar,HF P Branch 1
1
10 1
5 1
1
1
1
1
1
1
1
i
Ar,HF Q Branch T
v,
-
cm-'
3944.4756 3944.3963 3944.3167 3944.2365 3944.1575 3944.0781 3943.9976 3943.91 82 3943.8388 3943.7588 3943.6792 3943.5988 3943.5194 3943.4397 3943.3591 3943.2798 3943.2000 3943.1 200 3943.0401 3942.9599 3942.8798 3942.8001 3942.7205 3942.6409 3942.5610 3942.4809 3942.3218 3942.0853 3942.0052 3941.9264 3941.8473 3941.7682 3941.6897 3941.6099 3941.53 I6 3941.4534 3941.3748 3941.2969 3941.2185 3941.1406 3941.0629 3940.9852 3940.9080 3940.83 14 3940.7538 3940.6765 3940.5999 3940.5238 3940.4478 3940.3723 3940.2966 3940.2198 3940.1463 3940.0695
McIlroy et al. 0 Line Observed Positions
obs-calc:
x 103cm-l -0.2 0.0 0.0 -0.6 0.1 0.4 -0.3 0.1 0.5 0.3 0.6 0.1 0.5 0.7 0.0 0.6 0.7 0.6 0.5 0.2 0.0 0.0 0.2 0.2 0.0
-0.5 -0.5 0.4
0.0 0.4 0.3 0.0 0.4 -0.8 -0.5 -0.2 -0.5 -0.3 -0.6 -0.6 -0.6 -0.7 -0.5 0.0 -0.4 -0.9 -0.8 -0.5 -0.2 0.4 0.6 -0.6 1.3 0.2
oCalculated values from least-squares fit. See Table I1 and text. I
1.2
3941.4
3941.6
,
3941.8
I
3942.0
I
3942.2
u (cm-L) Figure 5. A detail of the Ar,HF and ArlHF spectrum taken in a Ne expansion. Note the Q branch of the Ar,HF HF u = 1 0 stretch spectrum between P(5) and P(6) of the Ar,HF spectrum. The increase in signal to noise from Figure 4 is most likely due to a combination of the colder expansion (5 K vs 10 K) and lower HF concentration (0.1% vs 1.0%) resulting in the formation of fewer multiple HF containing
-
clusters.
rotation and translation. This excess energy is only 3% lower than available to predissociating U H F = 1 ArHF complexes, in which extremely long lived (20.3ms) behavior is experimentally observed. (21) Maitland, G. C.; Smith, E. B. Mol. Phys. 1971, 22, 861.
C. Ar3HF. Gutowsky et al. have also observed pure rotational transitions in ground vibrational state Ar3HF in the microwave.' This tetramer is a symmetric top with the three Ar atoms forming a triangular base into which points the hydrogen of the HF (see Figure IC). If one makes the reasonable assumption that the transition moment for the HF stretch lies along the HF bond, then the Ar3HF HF stretching spectrum is expected to be a parallel band. This same assumption was implicit in our prediction of the band structure for Ar2HF, and for which only B type transitions were observed. Just to the red of the Ar2HF spectrum, the expected Ar3HF P, Q, and R branches of a parallel type transition of a symmetric top are observed (see Figures 4 and 5 ) . Ground-state energy differences predicted from the microwave work are again compared with the R(J'-l) - P(J'+I) infrared differences to confirm the assignment of these transitions (see Table 111) to Ar3HF. The Q branch's blue shading indicates that B and/or (C-B)must increase upon vibrational excitation. Again, this is qualitatively consistent with the contraction of the Ar-HF
HF Vibrational Red Shifts in Ar,HF TABLE IV: A r B F HF Stretch Y = 1
2F AC DJ DJK A DKc
-
0 Fit Resultsa u=l 3942.1636 (3) 0.039634496 0.0396736 (19) 4.537 (110) x 10-5 2.22836 X 2.262 (24) X -1.919 X -2.128 (42) X 3.12 (46) X IO“
‘All values in cm-I. bValuesfrom the microwave work of Gutowsky et al? Values in parentheses are 2u estimates of the uncertainty. cDue to the parallel ( A K = 0) nature of these transitions, only differences (upper - lower) of these quantities are determinable.
bond length observed for ArHF and Ar2HF. However, the Pand R-branch transitions show no splitting due to K structure, indicating a small A(C-B). Since no K splitting is observed in the P and R branches to within f0.003 cm-I, these transitions are used to obtain vo, B’, and Dj’from a least-squares fit to a centrifugally distorted rigid rotor, with B” and DJ” fixed at the microwave values.’ Some additional information is also available from the Q-branch band profile which cannot be fully described with only uo, AB, and AD) The Q branch is therefore modeled by a stick spectrum generated from the transitions between centrifugally distorted symmetric rigid rotors convoluted with Gaussian profiles for each transition. In a least-squares fit with uo, B’, and Dj’fixed at the values determined from the P and R branches and B”, Dj”, and DjK” fixed a t the observed ground-state values, this model is used to determine A(C-B), ADK,and DjK’. The Gaussian fwhm is fixed at 0.0020 cm-l (i-e., apparatus limit measured on ArHF transitions for these expansion conditions) and the temperature at T = 10.2 K, the value determined from a (HF), RQo ul spectrum recorded under identical jet conditions. Here we are relying on equilibration between rotational distributions, an assumption which has been tested in the slit jet geometry.’* To verify the assumption of rotational equilibration, we then float the rotational temperature in these fits and find T = 10.5 f 0.5 K with the other parameters unchanged, indicating rotational equilibration with (HF), within experimental error. Table IV lists all constants determined for Ar3HF. Note that the incremental red shift from the addition of the third Ar is 4.429 cm-I, again much smaller than the 9.654-cm-l shift caused by the first Ar, but only 14% less than the 5.173-cm-l incremental shift due the addition of the second. As with ArHF and Ar2HF, the rotational constants of Ar3HF increase upon vibrational excitation, indicating a contraction of the cluster. Analogous to the ArzHF behavior, the H F to Ar distance decreases by 0.0015 A, and the Ar to Ar, center of mass distance decreases by 0.0020 A. Again this apparent decrease in the Ar-Ar bond distance and increase in the centrifugal distortion parameters could be due to an increase in the vibrationally averaged tilt of the Ar, plane. Also as observed for the smaller clusters, the line widths are instrument limited, indicating a long vibrational lifetime (516 ns) for the H F stretch in the Ar,HF tetramer. A small Q-branch-like feature is observed just 0.13 cm-’ to the blue of the Ar3HF Q branch and is shown in Figure 4. This feature could be due to a larger cluster or even an isomer of Ar3HF. However, the proximity of the Q-branch structures suggests that it arises from a hot band of Ar3HF built on a low-frequency van der Waals vibration. This latter interpretation is corroborated by the slight blue shift with respect to the Ar3HF fundamental Q branch since van der Waals excitation would be expected to weaken the bond and therefore shift the HF frequency back toward the free H F monomer value. The intensity of this feature is found to change much like the Ar3HF Q branch as a function of distance downstream and total pressure in a 0.1% H F in Ar expansion. Furthermore, in a colder expansion (T = 4.5 f 0.4 K, determined from floating Tro,in the Q-branch fits) of HF:Ar:Ne:He = 0.001:0.33:0.50:0.17, this feature decreases in intensity with respect to the Ar3HF Q branch from 16% of the integrated intensity of the Ar3HF Q branch to 8%. A simple Boltzmann analysis of the relative intensities in the 5 and 10 K
The Journal of Physical Chemistry, Vol. 95, No. 7, 1991 2641 TABLE V: Ar.HF HF Stretch Y = 1
transition P(9) P(8) P(7) R(6) R(7) R(8) R(9) R(10)
v,
-
0 Observed Line Positions
cm-‘
obs-calc:
~ 1 0cm-I ’
3942.1016 3942.0601 3942.0186 3941.4355 3941.3941 3941.3518 3941.3115 3941.2706
0.0 0.3 0.6 0.0 0.0 -1 .o -0.1 0.3
“Calculated values from least-squares fit. See Table I1 and text. TABLE VI: Ar,HF HF Stretch v = 1 u = Ob
2ib E DJ
DJK
-
0 Fit Results” u=l 3941.7260 (14) 4.3 (12) x 10-5 0.020796 18 0.0208 12 ( 18) 3.543 x 10-8 3.543 x 10-86 2.04 X IO“ 2.04 X IOdb
“All values in cm-I. bValuesfrom the microwave work of Gutowsky et al.’ Values in parentheses are 2u estimates of the uncertainty. CDue to the parallel (AK = 0) nature of these transitions, only differences (upper - lower) of these quantities are determinable.
expansions indicates that this blue-shifted feature could come from a state -5-15 cm-l higher in energy than the ground state of Ar3HF. However, no associated P- or R-branch lines, which would provide a B constant for the complex, could be found to assist in identifying the lower state of these transitions. An unambiguous assignment of this feature to a specific cluster size and/or vibrational state must therefore await a further sensitivity enhancement. D. Ar4HF. Recently one of many possible isomers of Ar4HF has been observed in the microwave by Gutowsky et a1.” via pure rotational transitions in the ground vibrational state. In this isomer, the structure is a C,, symmetric top, with the fourth Ar added not directly adjoining the HF, but instead forming a distorted tetrahedron of Ar atoms with the H F pointing into one face (see Figure Id). Again, a parallel band is expected for the H F stretching motion. Since the additional Ar lies more than 5 A from the HF, we might expect it to change the red shift very little from the Ar3HF value. Indeed, in the cold ( T = 4.5 K) Ne/Ar expansion described above, a small Q branch is observed just 0.437 cm-’ to the red of the Ar3HF Q branch and several weak P- and R-branch lines are observed -0.4 cm-: above and below the Q branch (see Figure 5 ) . Comparison of ground-state combination differences for the observed R(6)-R(8) and P(8)-P(lO) lines with the values predicted from the microwave work confirms the assignment of these transitions (see Table V) to the prolate symmetric top structure of Ar4HF shown in Figure Id. Due to the additional Ar mass along the threefold symmetry axis, this axis switches from the C to the A axis between Ar3HF and Ar4HF. Fits of the Q-branch band contour and P- and R-branch-line positions following the methods described in section IIIc for Ar3HF verify the assignment of these transitions to the H F stretching motion of Ar4HF (see Table VI). In these fits, the ground-state constants are held fixed a t the values of Gutowsky et a1.* The upper-state centrifugal distortion terms are fixed at the groundstate values since they are not uniquely determinable with the data set available here. However, since only a limited range of J values is observed (6 I J I 11) and only small changes in the centrifugal distortion constants are observed for Ar,HF, this should be a reasonable approximation. Like Ar,HF, the Ar4HF Q branch is blue shaded due to the increased attraction between the vibrationally excited HF and the Ar atoms. It is also interesting to note that AC for Ar3HF and A4 for Ar4HF differ by less than 5%, indicating that the fourth Ar does not greatly perturb the interaction of the H F with the Ar, triangle. Though the P- and R-branch transitions are very weak, the spectral widths appear to be instrument limited as observed for the smaller complexes signifying a long lifetime for the H F vibration in this isomer of
2642 The Journal of Physical Chemistry, Vol. 95, No. 7, 1991
.
3945.0
$
TABLE VIII: Red Shift Model R d W AvA,-Hdpredicted) projb potd (proj)e (a>' (pot)# esc n Avo,,, 4.76 (9.6)' (9.6)* 9.4 (9.6)' 1 9.654 (9.6)' 7.44 17.5 15.4 2 14.827 16.1 11.0 15.9 10.64 25.3 14.3 22.6 23.0 3 19.257 22.6 10.99 26.2 23.2 14.3 23.0 4 19.697 22.4
.
E
& 3035.0
...............................matrix ........red .....shift ................................... I
0.0
1.0
2.0
I
3.0
4.0
I
5.0
I
6.0
I
7.0
Number of Ar atoms Figure 6. Plot of the Ar,HF HF stretch red shifts versus the number of Ar atoms, n. The red shift observed in an Ar matrix is marked by the dotted line. The fact that the cluster red shifts appear to roll off at only 1/2 of the matrix value is most likely a result of the Ar,,HF geometries. which 'solvate" only 1/2 of the HF molecule (see text for details). TABLE VII: ArJiF HF Stretch Vibrntiosrl Origins nlld Red Wfts (V" *- y o ) incremental n Y,. cm-I red shift, cm-l red shift, cm-l 0 3961.4229O 1 3951 .768gb 9.654 9.654 2 3946.5919 14.827 5.173 3 3942.1634 19.260 4.429 4 3941.7260 19.697 0.437 matrix 3919c 42.4 (I Value from Guelachvili.25 bValue from Lovejoy and Nesbitt.I6 CValuefrom Mason et aI.l4
the pentamer. This is also supported by the observed intensities, since significant broadening would render P/R-branch transitions unobservably small compared to the Q branch. The rotational temperature of this five-member complex is found to be 5.3 f 0.6 K from least-squaresfits to the Qbranch band contour. This result is in good agreement with the value found for Ar3HF under the same conditions and indicates that tetramer and pentamer are produced quite cold in the slit jet and appear to equilibrate to the same rotational temperature.
IV. Discussion A. Red-Shifr Models. The red-shifting of the H F stretch upon 'solvation" by Ar atoms is due to the H F stretch coordinate dependence of the van der Waals interaction potential. In the case of the Ar-HF system, these electronic interactions lower the effective vibrational force constant and frequency. As more Ar atoms are added, this perturbation should become larger until the successive Ar atoms are no longer close enough to perturb significantly the H F electronic structure. Since the addition of one Ar lowers the H F vibrational frequency only 0.24% and the complete encasement of H F in an Ar matrix lowers it only 1.196, these electronic perturbations are surely quite subtle and may depend sensitively on the specific properties of the interacting species.
The magnitude of the incremental red shifts observed for the sequential addition of Ar atoms is not linear, but drops by almost half with the addition of the second and third Ar atoms, and the fourth exhibits little additional red shift (see Figure 6 and Table VU). All of these data point to a strong dependence of the red shift on the vibrationally averaged geometry of the complex. However, the spectroscopic effects of rare gas solvation in all van der Waals systems are not always so sensitively dependent on geometry and number. For example, for the addition of up to six He atoms and/or Ne atoms to 12,Levy and co-workers observe a linear dependence of the electronic blue shifts.22 They propose (22) Kenny, J. E.; Johnson. K. E.; Sharfin, W.; Levy,D. H. J . Chsm. Phys. 1980, 72, 1109.
McIlroy et al.
"11 values in cm-l except n. bRed-shift model based on projection of the HF dipole moment on the HF center of mass to Ar bond using geometries of Gutowsky et al." CPredictedvalues using the electrostatic red-shift surface of Liu and Dykstra for ArHF to determine the contribution of each Ar in the vibrationally averaged geometry.u dPredicted red shift based on relative well depth on ArHF potential energy surface.' eThe dipole projection model averaged over approximate two-dimensional harmonic oscillator bend wave functions based on the average bend angles reported by Gutowsky et alew 'Predicted red shifts using a combination of the Liu and Dykstra red shift surface" and angular averaging over approximate two-dimensional harmonic oscillator bend wave functions determined from the average bend angles reported by Gutowsky et al." #Predicted red shift using relative ArHF well depth with angular averaging. *All values in column scaled to reproduce n = 1 shift.
that the linear behavior is due to the existence of six independent, equivalent binding sites on I2 whose effect on the electronic transition energy is identical. The nonlinear behavior of the Ar,,HF vibrational red shifts may simply indicate an inequivalence of binding sites and/or cooperative three-body effects which influence the red shift. (i) Rigid Red Shift Models. While all of the clusters discussed here are probably quite nonrigid, many of their physical properties including vibrational red shifts may be approximately described by their vibrationally averaged geometries. The potential energy surface for H F vibrationally excited ArHF of Nesbitt et a1.4 shows a significant minimum in the linear Ar-HF configuration which allows wide amplitude bending, but not free internal rotation of the HF.4 This leads to a vibrationally averaged bend angle in the dimer of 41 .6°.8 Surprisingly, Ar,HF magnetic dipole-dipole hyperfine splittings observed by Gutowsky et al. indicates only a -2% variation in the -40° average H F bend angle 6 for n = 1-4.6-s This would suggest that the H F bending potential in these complexes is not becoming significantly more isotropic as they grow larger. Thus it may be reasonable to consider the red shifts as a function of vibrationally averaged H F bending geometry or perhaps even equilibrium geometry. Since the largest red shift per Ar is observed for the addition of the first Ar in a linear, equilibrium geometry and smaller incremental red shifts are observed for Ar atoms positioned off the symmetry axis (seeTable VII), the magnitude of the red shift may scale with the projection of the H F dipole moment on the Ar to H F bond. For simplicity in this first treatment, we freeze the hydrogen at its probable equilibrium position in these complexes (Le., the H F bond coincident with the symmetry axis and the H atom pointing toward the Ar atoms). If one neglects R-dependent effects since the R's shown in Figure 1 are relatively constant, except for the fourth Ar atom in Ar4HF, a simple model for the red shifts, AYAR,HF, based on the sum of cos y projections of the dipole moment on each Ar-HF bond can be constructed as nk
AVAr,HF
=
cAvArHF i= 1
cos Ti
(1)
where nAris the number of equivalent Ar atoms, AVA~HF is the ArHF red shift, and y, is the angle between the symmetry axis and ArHF bond. Using the angles shown in Figure 1, we calculate the values listed in the column labeled "proj" in Table VIII. For n = 4, the Ar at 5.86 A is assumed to have no direct contribution to the red shift, although as shown in Figure 1, c and d, there are small differences between n = 3 and 4 in the vibrationally averaged positions of the three closest Ar atoms. This simple model gives reasonably good agreement with the observed experimental results; however, the n = 3, 4 shifts are overestimated by 17%.
The Journal of Physical Chemistry, Vol. 95, NO.7, 1991 2643
H F Vibrational Red Shifts in Ar,,HF Liu and Dykstra have developed a more rigorous and quantitative model for vibrational red shifts based on the electrostatic interactions between the subunits of the com~lex.2~In this theory the anharmonic HF stretching potential is modified by the addition of an approximately linear term resulting from the dependence of the electrostatic interactions on the HF stretching coordinate, r, the slope of which determines the red shift. For the ArHF system, Liu and Dykstra have calculated a red-shift surface as a function of radial and angular Jacobi coordinates (R, 8) which reproduces the ArHF red shift.% By interpolating on this surface one can predict the red-shift contribution for an off-axis Ar atom as a function of R. Again, we assume the H F to be frozen with the H F bond on the symmetry axis. We can then predict the red shifts for Ar,,HF, n = 1-4, complexes assuming that these shifts can then simply be added together and that the AI-Ar and higher order multibody interactions do not significantly change this red shift surface. The published surface does not extend to the position of the fourth Ar in Ar4HF, but the rapid falloff of the red shift with R suggests that the fourth Ar should have essentially no contribution. These electrostatic predictions (see the column labeled “es” in Table VIII) show the same qualitative pattern as the simple projection results, but now underestimate the n = 2 and 3 values by 26%. This may suggest that either the excellent red-shift agreement of Liu and Dykstra for ArHF may be fortuitous or that multibody effects are important. A further comparison of the observed and calculated red shifts for NeHF, KrHF, NeDF, and ArDF indicates that the 26% error found for the larger Ar,,HF, n = 2, 3, clusters may be more representative of the accuracy of this techniqueaZ4 It has been noted in the past that red shifts may correlate with van der Waals well depth,% in which larger shifts would correspond to larger well depths, and thus a stronger van der Waals interaction. If one assumes that the contribution to the red shift of a single Ar is proportional to the van der Waals well depth at its vibrationally averaged position, a third method of predicting red shifts becomes available through the use of an ArHF potential energy surface. We use here the HF u = 1 surface developed by Nesbitt et al. using rotational RKR method^.^ One can readily evaluate the value of the potential energy for a given Ar atom’s vibrationally averaged R and with the HF frozen with HF bond coincident with the symmetry axis and then predict the red shift from AVA~,HF =
V(Ri,yi) VArHF i-1
(2)
where VAIHF is the van der Waals well depth at the ArHF vibrationally averaged geometry, nAris the number of Ar atoms, and V(Ri,Ti) is the value of the potential at the ith Ar atom’s vibrationally averaged position. Note that this method is explicitly scaled to reproduce the ArHF red shift exactly. Column “pot.” of Table VIII lists these predictions which again qualitatively reproduce the observed trend quite well. This model predicts red shifts of 15.9, 22.6, and 23.0 cm-’ for n = 2-4, respectively, compared to experimental shifts of 14.827, 19.257, and 19.697 cm-I. Note that no special assumptions are made regarding the fourth Ar in this model, and its incremental red shift is indeed predicted to be -0.4 cm-’in excellent agreement with the observed value of 0.437 cm-I. Though off by as much as 17% for Ar3HF, this method provides the most accurate estimate of the three techniques discussed here. (ii) Effects of Bend Averaging on Red-Shut Models. As noted above, these complexes are by no means rigidly fixed even at their vibrationally averaged geometries and in particular the HF almost certainly executes wide amplitude bending motion due to the small mass of the hydrogen. Certainly the above models have completely neglected the extensive H F bend vibrational averaging that must take place. Here we will investigate briefly the effects of such angular averaging on the models described above. An approximate ground-state bending wave function can be determined from the (23) S. Liu, L.; Dyhtra, C. E.J. Phys. Chem. 1986, 90,3097. (24)S.Liu. L.;Dykstra,C. E. Chcm. Phys. Lrrr. 1987. 136.22.
average bend angles provided by Gutowsky et and the a p proximation of the bends as harmonic modes. We further assume that the two HF bending modes are degenerate for all Ar,HF complexes, an approximation that should be worst for Ar2HF where the out-of-plane bend, q5 = 90°, is very different from the in-plane bend, q5 = Oo. These wave functions are then used to find the expectation value of the red shift from (3)
*
where &shift is the red shift oppator and is the harmonic oscillator bend wave function. Oldshift for the dipole moment projection model is given by eq 1 where the projection angle, the angle between the H F bond and the Ar-HF bond to the ith Ar atom, is no longer fixed at y as it is for the rigid 8 = Oo, q5 = Oo geometries assumed above. For the electrostatic model?4 a slice of the red shift surface at the appropriate R value foz each cluster is fit to a Gaussian as a function of 8 to form Ordshift. The expectation values are calculated numerically with R fixed at the vibrationally averaged value for the electrostatic and van der Waals potential well models. The averaged values are shown in the last three columns, “(proj)”, “(es)”, and “(pot.)”, of Table VIII. In general, the effect of vibrational averaging does not change the qualitative trend in the red shifts. Note that averaging the electrostatic n = 1 value yields a very low result (AY,,~ = 4.76 cm-I compared to Avexpt = 9.65 cm-I) even for ArHF for which the surface was developed. In part, this may be due to the neglect of radial averaging which would sample regions of higher predicted red shifts. It should be noted that the behavior observed for the lowering of the HF stretch vibrational frequency is well reproduced by these simple models based only on two-body effects. This suggests that inequivalent binding sites, not three-body effects, may account for most of the nonlinear red shift behavior. Thus the linear dependence observed for the electronic blue shifts of I2 clusters could be largely due to the high symmetry of the I2 and the corresponding availability of up to six equivalent binding sites.22 If this is indeed the case one might expect that a more linear vibrational red shift would result for clusters built on a symmetric infrared chromophore such as C 0 2 where near T shaped vibrationally averaged geometries for the A r m 0 dimer are 0bserved.2~ B. Comparison to Ar Matrix Red Shift. The HF vibrational frequency observed in an Ar matrix is 3919 cm-1,13*14 and with our data we may consider how many atoms need to surround the H F before this red shift is obtained. Figure 6 displays the H F vibrational frequency as a function of number of Ar atoms attached to the cluster, and the Ar matrix value is marked by the dashed line. Since the total red shift observed in the matrix is only 42.4 cm-l and a red shift of 19.3 cm-’ is obtained with only three Ar atoms, it seems plausible that only the nearest neighbors may contribute significantly to the red shift. Matrix studies indicate that in general the H F lies in an octahedral site (Le., 12 nearest neighbors) in which it executes near free rotation.” This would require an average shift per Ar of -3.5 cm-’ for the entire 42 cm-I to be accounted for by the first shell alone. For Ar3HF, there is a shift of -6.4 cm-’/Ar which is close to but already a somewhat larger shift per atom than required to explain the bulk values. Of course, any angular dependence of these shifts such as suggested by Liu and D y k ~ t r awould ~ ~ tend to reduce this discrepancy, since only some portion of the Ar sites would contribute significantly. The addition of the fourth Ar in the geometry seen in Figure Id, essentially in the second layer away from the HF, is observed to increase the red shift by only -2%. This supports the hypothesis that nearest neighbors are most important in determining solvent red shifts of vibrations. An examination of the Ar potential energy surface shows this should not be too surprising since at R = 5.86 A, the separation of the fourth Ar and H F center of mass, the van der Waals interaction energy has (25)Guelachvili, G.Opr. Commun. 1974,19, 150. (26)Stced, J. M.;Dixon, T. A.; Klemperer, W. J . Chem. Phys. 1979, 70. 4095;1981, 75, 5977(E).
2644
J. Phys. Chem. 1991, 95. 2644-2652
dropped to 56% of its maximum value. As noted earlier, the Liu and Dykstra red-shift predictions also appear to go to zero for R of this magnitude.24 Thus these small clusters indicate that the Ar matrix red shift may be accounted for by the nearest neighbors alone, with little or no contribution from atoms past the first coordination shell. V. Conclusions The first rotationally resolved, high-resolution infrared spectra of Ar,HF, Ar3HF, and Ar4HF have been recorded in the H F stretching region near 3950 cm-I. Combined with the earlier observation of ArHF, these data allow us to examine the red shifting of the H F stretching motion as a function of complex size and geometry. By n = 3, the red shift of the Ar,HF is half way to the matrix value of 3919 cm-’ from the uncomplexed value of 3961 cm-l. We find that the magnitude of the red shift depends sensitively on the position of the Ar atoms with respect to the HF dipole moment. Hence the largest effect occurs for the addition of the first Ar atom in a linear ArHF geometry. With the addition of a fourth Ar in a position equivalent to that of an atom in a second solvation shell, the red shift increases only 2.2% beyond the value for Ar3HF. This radial dependence to the red shifts suggests that only the nearest neighbors of the HF contribute significantly to the vibrational red shift of the H F stretch and thus some inert gas condensed-phase properties may be well approximated in relatively small clusters which close the first solvation shell. No line broadening beyond the instrumental limit is observed for any of the spectra, indicating a long lifetime (516 ns) for the H F vibration in these clusters. The relatively high signal to noise ratios observed for these large clusters offer the opportunity to extend the considerable high-
resolution work on two-particle van der Waals complexes into a regime where the transition to condensed-phase properties may be investigated. These single-mode IR laser experiments have the advantage of rotational resolution which allows the determination of cluster geometries and provides a probe of the temperature of the expansion. This last point may prove particularly important since previous slit jet experiments have shown essentially complete equilibration of internal and rotational degrees of freedom. This equilibration indicates that the rotational temperature is likely a good probe of the internal “temperature”, an important parameter in many molecular dynamics simulations of clusters. To extend this work, we are now investigating the applicability of the H F v = 1 ArHF pair potential of Nesbitt et al. to these larger clusters. Combined with the observation of combination bands of the H F stretch plus intermolecular vibrations, these calculations should provide insight into the importance of three-body forces and additivity of two-body potentials. A clear focus of these computations will be to attempt to reproduce the free-rotor behavior observed in the octahedral matrix site. To complement this last calculation, we are also pursuing the spectroscopy of still larger clusters in order to explore the transition from the large-amplitude bender of ArHF to the free-rotor behavior proposed in the matrix work.
Acknowledgment. Support from this research by National Science Foundation Grant No. CHE86-05970 through the University of Colorado as well as grants from the Henry and Camille Dreyfus Foundation and the Sloan Foundation is gratefully acknowledged. Registry No. Ar, 7440-37-1; HF, 7664-39-3.
Ultraviolet-Visible and Raman Spectroscopy of Diatomic Manganese Isolated in Rare-Gas Matrices A. D.Kirkwood: K. D. Bier, J. K. Thompson,*T. L. Haslett, A. S. Huber, and M. Moskovits* Department of Chemistry and the Ontario Laser and Lightwave Research Centre, University of Toronto, Toronto, Ontario MSS 1 A l , Canada (Received: June 4, 1990)
Numerous absorption bands in the UV-vis spectrum of matrix-isolated Mnz are assigned to transitions between singlet, triplet, quintet, and septet electronic spin states of the diatomic. These assignments are based on the temperature dependence of the intensities of these absorptions resulting from population redistribution among the low-lying spin states of the Mn2molecule. The exchange energy (J) obtained from the observed temperature dependence ranges from -8 to -1 1 c d . Fluorescence spectra of Mnz in krypton matrices are also reported. The resonance Raman spectrum of diatomic manganese isolated in solid xenon matrices was observed with IZg+ ground-state vibrational constants: w,” = 68.1 cm-’ and w/x,” = 1.05 cm-l.
Introduction
The characterization of small metal clusters is an important link in understanding the chemical and physical properties of the respective bulk materials. Transition-metal clusters are particularly interesting because of their enhanced reactivity due to the presence of d-orbital electrons. Diatomic manganese is unique in that it is a weakly bound van der Waals molecule similar to the group 1IA metal dimers rather than its neighboring first-row transition-metal diatomics. In this paper more of the spectroscopy of dimanganese will be examined. Nesbetl was the first to propose that Mn2 was an antiferromagnetic diatomic with a lZg+ground state based on an a b initio ‘Resent address: Schlumberger Well Services, 5000 Gulf Freeway,P.O. Box 2175, Houston, TX 77252-2175. *Resent addms: Dow Chemical, Patents Department, 1776 Bldg., Midland, MI 48674.
Hartree-Fock calculation with exchange interaction. The Heisenberg exchange Hamiltonian describing such systems can be written as
H = -.IS,&, where Sa and S, are the electron spin operators of the two manganese atoms and J is the exchange energy. The eigenenergies of this Hamiltonian are given to first order by an equation that is similar to the Land&interval expression E(S) = -(J/Z)[S(S + 1 ) - 2s(s + l ) ] Here S represents the total spin of the molecule and s is the atomic spin. Assuming that the bonding in manganese dimer results solely (1)
Nesbet, R. K. Phys. Rev. 1964, A135,460.
0022-3654191 12095-2644%02.50/0 0 1991 American Chemical Society
