J . Phys. Chem. 1991, 95, 9638-9647
9638
the stearic acid. This was probably because the weaker alignment of the methyl groups of the hemicyanine were easily disturbed by the interaction between the deposited hemicyanine film and the water or the stearic acid during the deposition processes of the stearic acid overlayer in steps B4 and B5. Therefore, a well-constructed 2-type multilayer is supposed to be produced with the hemicyanine molecule which is modified with a more hydrophobic moiety than methyl groups. Those measurements such as X-ray diffraction method13 and Stark effect spectral0 are known to be employed to characterize the structure of LB films. However, they can be used only to distinguish Y-type structure from those of X- or Z-type, and it seems rather impossible to determine the pointing direction of the specific molecules in a submerged layer by this method. Contact angle measurement has also been utilized to determine whether the outermost layer of a given multilayered LB sample is hy( I 3) Matsuda, A.; Sugi, M.; Fukui, T.: Iizima, S.;Miyahara, M.: Otsubo, Y . J . Appl. Phys. 1977, 48, 771.
drophilic or hydrophobic, and undoubtedly this method cannot be utilized to analyze the inner structure of a multilayer. On the other hand, the SHG-CI method is applicable to the layer of optically nonlinear heteromolecules which is sandwiched among other nonlinearly inactive molecules as demonstrated in this report, because this method is quite sensitive to the polar structure of a given interface. The present study is based on the phase inversion of SH waves and this property is characteristic to an even-ordered nonlinear optical phenomenon such as SHG. Therefore, the phase inversion in SH radiation induced by a certain molecular configurational change at a given interface should be emphasized as another advantage of the S H G method besides its surface specificity and simplicity. In conclusion, the SHG-CI method was applied to analyze the molecular configuration of hemicyanine layer installed in multilayered LB films by the Y - and the Z-type deposition. It was demonstrated for the first time to clarify which direction the axis of the hemicyanine molecule pointed in a given layer sandwiched among other nonlinearly inactive layers.
FEATURE ARTICLE Scaling Laws for Inelastlc Collision Processes In Diatomic Molecules J. I. Steinfeld,* Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
P. Ruttenberg, Atomic Collisions Data Center, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colorado 80309
G. Millot, G . Fanjoux, and B. Lavorel Laboratoire de Spectronomie Moldculaire et Instrumentation Laser, Universite de Bourgogne, U.R.A. CNRS No. 777, 6 Bd Gabriel, 21000 Dijon, France (Received: May 16, 1991; In Final Form: August 23, 1991)
A variety of fitting and scaling laws have been developed for the purpose of modeling rotational energy transfer (RET) in diatomic molecules. These include exponential energy gap (EGL), statistical power gap (SPG), and dynamically based angular-momentum scaling laws (e.g., the energy-corrected sudden approximation, ECS). These scaling laws are tested against state-to-state energy-transfer data for diatomic halogens, and stimulated Raman Q-branch band shapes in nitrogen. For state-testate RET in halogens, an ECS scaling law, modified to account for restrictions on angular-momentum transfer, is found to be superior to the EGL. When all available data on Raman band shapes in N2, particularly including the collision-induced Raman line shifts, are taken into account, the angular-momentum-based ECS-EP scaling law again provides the best representation of the data. We conclude that dynamically based scaling laws are to be preferred for modeling rotational energy transfer in diatomic molecules. Several unresolved questions and possible future directions for energy-transfer scaling laws and fitting procedures are discussed, including extension to polyatomic systems, possible contributions to the line width from elastic dephasing processes, and the development of global fitting procedures which will simultaneously account for line shape, line shift, and (when available) state-to-state RET measurements on molecular systems.
I . Introduction The development of highly specific and selective methods for molecular quantum state preparation and detection, such as laser-induced fluorescence, optical double resonance, and molecular beam techniques, has made available a large body of data on state-testate inelastic processes in diatomic and small polyatomic molecules. Indeed, a recent survey' of literature on the diatomic
halogens retrieved over 2000 measurements of rates and/or cross sections for vibrational, rotational, and electronic relaxation processes in these systems alone. ( I ) (a) Steinfeld, J. I . Rate Data for Inelastic Collision Processes in the Diatomic Halogen Molecules. J . Phys. Chem. Rex Dora 1984, 13, 445. (b) Steinfeld, J. I . Supplement to Rate Data for Inelastic Collision Processes in the Diatomic Halogen Molecules. J . Phys. Chem. Re/ Data 1987, /6. 903.
0022-365419 1 12095-9638%02.50/0 0 1991 American Chemical Society
Feature Article It is generally recognized that quantum-state-resolved data are not all independent; on the contrary, rates for processes which differ only slightly in initial and/or final state are closely correlated. In principle, such correlations may be derived ab initio from formal scattering theory. While straightforward, such calculations can be quite tedious, especially for heavy systems (such as the diatomic halogens) and large quantum numbers. Classical trajectory calculations have been performed for inelastic dihalogen-atom collisions2 but such calculations require sampling over a wide range of initial conditions to be accurate. Despite the availability of large-scale computers, quantum close-coupled calculations remain feasible only for low rotational quantum numbers, due to the rapidly expanding size of the matrices involved. In either case, an accurate intermolecular potential function is required if the calculations are to yield meaningful results; such potentials are by no means well-known, particularly if one of the collision partners is in an electronically excited state. For these reasons, a number of attempts have been made during the past several years to find semiempirical systematizations or condensations of such data, in which a small number of measured data would suffice to determine an entire state-to-state array. Expressions of this type are known asfitting laws or scaling laws. Properly speaking, fitting laws are empirical expressions with freely adjustable parameters, such as a power series in which the coefficients are fit by a least-squares method to a set of data, while the functional form and some of the parameters of scaling laws may be determined from a theoretical model. The terms tend to be used somewhat interchangeably in the literature, however. Having recognized that the number of independent observables is, in general, much smaller than the number of data for inelastic processes obeying scaling laws, we seek an approach which will identify those observables in the most economical and efficient manner. One such approach is that of information theory, in which the entropy content of an array of data (for example, the set of state-to-state kinetic parameters) is maximized subject to one or more constraints which incorporate available information about the ~ y s t e m . ~ The - ~ simplest prescription, which is a single constraint specifying the average amount of energy transferred during relaxation, leads to an exponential energy-gap law,3 or EGL. Previous analyses6 of rotation-vibration energy-transfer data in 12*appeared to indicate that an EGL scaling law could give a satisfactory representation of these data. There have been numerous suggestions, however,'+ that rotational-energy-transfer rates are better represented by an angular-momentum-based scaling law, rather than an energy-based one. The form of this law most commonly encountered is the energy-corrected sudden (ECS) scaling law, discussed by De Pristo et a1.I0 A particularly important application of scaling laws for rotational energy transfer is to the modeling of band shapes for stimulated Raman scattering (SRS) of simple diatomic molecules, such as N2,I1-l6C0,16J7and O2,I8at moderate to high densities. (2) See,for example: Rubinson, M.;Garetz, B. A.; Steinfeld, J. 1. J. Chem. Phys. 1974,60, 3082. (3) Procaccia, 1.; Levine, R. D. J. Chem. Phys. 1975.63, 4261 and references therein. (4) Levine, R. D. Annu. Reo. Phys. Chem. 1978, 29, 59. ( 5 ) Levine, R. D.; Kinsey, J. L. In Atom-Molecule Collision Theory: A Guide for the Experimentalist; Bernstein, R. B., Ed.; Plenum Press: New York, 1979; pp 693-750. (6) Rubinson, M.; Steinfeld, J. 1. Chem. Phys. 1975, 4, 167. (7) Barnes, J. A.; Keil, M.; Kutina, R. E.; Polanyi, J. C. J. Chem. Phys. 1980, 72, 6306; 1982, 76. 913. (8) Brunner, T. A.; Smith, N.; Karp, A. W.; Pritchard, D. E. J. Chem. Phys. 1981, 74, 3524. (9) Brunner, T. A.; Pritchard, D. E. In Dynamics of the Excited State; Lawley, K. P., Ed.; Wiley: New York, 1982; pp 589-641. (IO) De Pristo, A. E.; Augustin, A. D.; Ramaswamy, R.; Rabitz, H. J. Chem. Phys. 1979,71,850. De Pristo, A. E. J. Chem. Phys. 1981, 74,5037. (11) Rahn, L. A.; Palmer, R. E. J . Opt. Soc. Am. 1986, 8 3 , 1164. (12) (a) Greenhalgh, D. A.; Porter, F. M.; Barton, S. A. J. Quant. Spectrosc. Radiat. Transfer 1985, 34, 95. (b) Rahn, L. A.; Palmer, R. E.; Koszykowski, M. L.; Greenhalgh, D. A. Chem. Phys. Lett. 1987. 133, 513. (13) Lavorel, B.; Millot, G.; Bonamy, J.; Robert, D. Chem. Phys. 1987, 115, 69.
The Journal of Physical Chemistry, Vol. 95, No. 24, 1991 9639
In such analyses individual state-to-state rate constants, represented by a scaling law, are summed or averaged to predict either line widths of resolved Raman features, or the band shape of a collapsed feature such as the Raman Q-branch. The scaling or fitting laws employed in such analyses tend to be more elaborate than a simple EGL or statistical power-gap (SPG) law, with additional adjustable parameters available for fitting to the spectra. It has been noted" that, since individual state-to-state rates are not determined from this procedure, quite different scaling laws can give equivalent fits, if different physical models for the underlying rate matrix are postulated. Indeed, various versions of the EGL, SPG, and ECS scaling laws have all been claimed to give superior representations of the available data."-I8 Reliable scaling laws are also needed for infrared absorption line widths, particularly when modeling atmospheric transmission, where tens of thousands of individual features may need to be included. Attempts have been made, for example, to model theoretical predictions"" of air-broadened ozone absorption line widths by a simple polynomial e x p r e ~ s i o n . ~In~ ,this ~ ~ case too, an accurate scaling law, based on theoretical models and validated by appropriate experimental measurements, would be extremely valuable. During the course of the halogen data survey,' a comparative analysis was made25of the applicability of various scaling laws to the critically reviewed data on rotational- and vibrational-energy-transfer rates. Following a summary of the various scaling laws in section 11, we present the main conclusions of this analysis in section 111. The application of these scaling laws to modeling the recent N2 S R S Q-branch data is discussed in section IV. In section V, we summarize the conclusions of this study and indicate possible future directions for work in this area, including a brief description of recent experimental results on rotationally inelastic processes in small polyatomic molecules, and the use of additional types of spectroscopies to establish the applicability of various scaling laws to these processes. 11. Energy and Angular-Momentum Scaling Laws for Rotationally Inelastic Collision Processes Scaling laws for inelastic processes have been reviewed by several a ~ t h o r s , ~but ~ , ~the ' notation varies considerably from one treatment to another. In this section, we summarize the various forms that have been proposed, using the notation most commonly employed in treatments of Raman line widths and band shapes, but expressing the formulas in terms of state-testate rate constants k(ji-+jr), where jiand j r denote initial and final rotational states, respectively. (14) Koszykowski, M. L.; Rahn, L. A.; Palmer, R. E.; Coltrin, M. E. J. Phys. Chem. 1987, 91,41. (15) Bonamy, L.; Bonamy, J.; Robert, D.; Lavorel, B.; Saint-Loup, R.; Chaux, R.; Santos, J.; Berger, H. J . Chem. Phys. 1988, 89, 5568. 1161 Millot. G. J. Chem. Phvs. 1990. 93. 8001. (17) Looney, J. P.; Rosasco,'G. J.; Rahn; L. A.; Hurst, W. S.; Hahn, J. W . Chem. Phys. Lett. 1989, 161, 232. (18) Millot, G.; Saint-Loup, R.; Santos, J.; Chaux, R.; Berger, H.; Bonamy, J., to be published. (19) Gamache, R. R.; Rothman, L. S.Appl. Opt. 1985, 24, 1651. (20) Gamache, R. R.; Davies, R. W. J . Mol. Spectrosc. 1985, 109, 283. (21) Gamache, R. R. J . Mol. Spectrosc. 1985, 114, 31. (22) Hartmann, J. M.; Carny-Peyret, C.; Raud, J. M.; Bonamy, J.; Robert, D.J . Quanr. Spectrosc. Radiat. Transfer 1988, 40, 489. (23) Flaud, J. M.; Camy-Peyret, C.; Rinsland, C. P.; Devi, V. M.; Smith, M. A. H.; Goldman, A. Appl. Opt. 1990, 29, 3667. (24) Smith, M. A. H.; Rinsland, C. P.; Devi, V. M. J. Mol. Spectrosc. 1991, 147, 142.
(25) Steinfeld, J. I.; Ruttenberg, P. "Tests of Scaling laws for Inelastic Collision Processes in Diatomic Halogen Molecules"; Report No. 23, Atomic Collisions Data Center, Joint Institute for Laboratory Astrophysics, Boulder, CO, 1983. (26) McCaffery, A. J.; Proctor, M. J.; Whitaker, B. J. Annu. Reo. Phys. Chem. 1986, 37, 223. (27) (a) Robert, D. In Vibrational Spectra and Structure; Bist, H. D., Durig, J. R., Sullivan, J. F., Eds.; Elsevier: Amsterdam, 1989; Vol. 178; pp 57-82. (b) Levy, A.; Lacome, N.; Chackerian, C., Jr. In Spectroscopy of the Earth's Atmosphere and Interstellar Molecules; Rao, K. N., Weber, A., Eds.; Academic Press: New York, in press
9640
The Journal of Physical Chemistry, Vol. 95, No. 24, 1991
The transformation of units between rate parameters and line-broadening parameters WJJfmay be. found in Bonamy et al.,Is viz.
(I)
where k is expressed in cm3 molecule-’ s-I, D is the number density of colliding species at standard conditions ( T = 273 K, P = 1 atm). (Re is the broadening coefficient in cm-I atm-I, and c is the speed of light (2.997 92458 X 1O’O cm s-I). It should be noted. however, that the relationship between state-to-state rate constants and observed line widths may be far more complex than the simple units conversion expressed by eq I , and indeed will depend on the type of spectroscopy being considered.I6 (a) Exponential-GapLaws. The simplest representation of the dependence of inelastic collision rate on initial and final states is one in which the rate decreases exponentially with the amount of energy transferred in the collision. Such an “exponential energy gap law” (EGL) may be derived from the entropy maximization principle of information t h e ~ r y . ~The , ~ inelastic collision rate connecting an initial vibration-rotation state (D,, j , ) with final state (of, if) may be expressed as k(uj,-+uJf;T) = A( T)ko(uj,-wdf;T) exp(-XIE,,, - E,,,J/kT\ (2) where the “prior” or statistical rate is given by
and X is a fitting constant, sometimes called the ”surprisal ~ a r a m e t e r ” . In ~ eq 3, A = (Ei - Ef)/2kT (the energies are calculated from spectroscopic term values), and K , ( A ) is the modified Bessel function of the first order and second kind.28 For vibrationally elastic collisions (pure rotational energy transfer), eq 2 may be reduced to a simple two-parameter formZ5
Steinfeld et al. et aI.,l7introduces an additional factor of this type and a further parameter 6’,to yield a doubly modified EGL (M2EG): kM’EG +j,;T) = 1 + AEf/kT6’ ‘ ( T ) [ l + .Er/,,)
1
1
1
+ AE,/kTd +
AEi/kT
1
expl-P(Ef (7)
Equation 7 appears to offer little improvement over the MEG scaling lawI7 and has not been widely used. (b) Statistical Power-Gap Laws (SPG). Pritchard and cow o r k e r ~suggested ~.~ a power law in AE as an empirical alternative to the EGL for fitting their experimental dataZ9on 12*-He and 12*-Xe collisions. This SPG scaling law has the form
where Bo is the molecular rotational constant. The principal differences between EGL and SPG are (i) fitting is with a dimensionless parameter a , which can be obtained from the EGL by applying entropy maximization to In IAEl; (ii) the possibility of incomplete reorientation of angular momentum is allowed for by the factor N,,. If there is no restriction on MI following a collision, then N Ais just the statistical degeneracy 2jf 1 used in eqs 3 and 4; but if reorientation of M j is restricted in a collision, then a correspondingly smaller factor is a p p r ~ p r i a t e .As ~ ~before, an SPG law for pure VET processes could be obtained by summing over j r
+
(9)
Several modifications and extensions have been proposed to increase the flexibility of the SPG. Looney et aI.I7 formulated a three-parameter hybrid statistical power-exponential gap (SPEC) scaling law of the form
kEGL(ji-jf;T) = a(t)(2jf + 1) exp(-P(Ef - E i ) / k T )= a ( n ( 2 j f+ I)e-aAEIkT(4) which is often used to characterize final-state distributions. In eq 4, /3 is the exponential scaling parameter, and a( r ) is used here, and in all subsequent expressions, to represent a temperaturedependent parameter which expresses both the overall magnitude of the rate constant and the units in which it is expressed, e.g., rate coefficient (cm3 molecule-’ SKI),relaxation time (s-l Torr-I), cross section (cm2 molecule-I), line width (cm-’ atm-I), etc. This scaling law can also be applied to pure vibrational-energy-transfer (VET) processes by summing the prior rates over j , This gives3 kEGL(ui+-of;T)= a(T) expl-XJE, - E,.,l/kT]A2eAK2(A)
(5)
where K 2 ( A ) is the modified Bessel function of the second order and second kind.28 Several modifications to the EGL have been proposed. Rahn and ~ o - w o r k e r s ’have l ~ ~ ~described a modified EGL, or MEG. of the form
(6)
The MEG introduces an additional factor which roughly accounts for finite collision duration (a more exact form is included in the ECS scaling law, discussed below in section c), with an additional fitting parameter 6 . The quantity A, which may be estimated from interaction lengths and closest approach distances, typically has a value of 1.5 or 2. A further modification, proposed by Looney (28) Abramovitz, M.; Stegun, 1. A. Handbook of Mathematical Functionr: National Bureau of Standards Monograph 5 5 ; US.Government Printing Office: Washington, DC. 1964: pp 374-379 and 417-429.
Greenhalgh et al.,12 on the other hand, chose to expand the SPG into a power series, which they termed the polynomial inverseenergy gap (PIG) model. This has the form “I
kPIG(j,+jf;T) = a T A g ( j l )
1x0
c,
IEf - Eil‘
( 1 1)
where gGr) is the statistical degeneracy factor for the final rotational state j,. The SPG (eq 8) is equivalent to a polynomial energy-gap law having a single term with a noninteger exponent 1 = CY. The PIG law (eq 1 1 ) requires I,, 3 adjustable parameters; in fitting N 2 Raman Q-branch data, Greenhalgh and = 5, or eight paco-workersi2 took six terms in the sum (I,, rameters) to fit 19 measured line widths. They concluded that the MEG law gave a better representation of the data. (c) Angular-Momentum-BasedScaling Laws. In contrast to the preceding scaling laws, which are statistical in nature and based on the amount of energy transferred, are the expressions based on the sudden approximation in inelastic scattering theory.I0 These scaling laws have been used successfully to analyze data for Na2*-M collisions8 and many other systems including Raman band shape^.'^^'^^'^ The energy-corrected sudden (ECS) scaling law for a transition from initial rotational statejt to final state ji may be expressed as
+
(29) Dexheimer, S. L.: Durand. M . : Brunner, T. A,: Pritchard, D. E. J. Chem. Phys. 1982, 76, 4996.
The Journal of Physical Chemistry, Vol. 95, No. 24, 1991 9641
Feature Article
C
The symbols in eq 12 have the following meanings: j, is the larger of Uiv jr),30
is a 3-j
and R= 1
+
1 -~?/6 I
1
+ w;, - A ~ 2 / 2 4
is an adiabaticity correction which accounts for the finite duration of a collision. A stipulates any restriction on angular momentum transfer due to nuclear-spin statistics (A = 1 for a heteronuclear diatomic but A = 2 for a homonuclear diatomic molecule such as N2 or I,; A(&) = f 3 for ammonia, etc.), and T, is an effective collision duration given by T, = I,/&,,, where 1, is a characteristic distance in the molecule and B,h is the mean thermal velocity, B,h = ( b k T / ~ p ) I / ~I,. is generally the parameter which is fit in applying the ECS law; by taking the limit T~ 0, one recovers the infinite order sudden (10.5)approximation, with flj = RL = 1, which may be applicable at high temperatures. Several forms of the ECS have been proposed, differing priThe standard power-law marily in choice of the basis rates WoL. (ECS-P) expression is
-
IO-''
-
IIIIIIIII IIIIIIIII IIIIIIIII
-20
-
0
.
.
20
IIIIII
40
60
It - l i
Figure 1. Rotational energy transfer (RET) data from ref 34 for 12* + 4He (vi = 25, ji = 34 Or = 25, jr) with EGL scaling law (eq 4).
+
since EL/Bo = L(L 1). A final, working expression for the ECS-EP (ECS-j*) scaling law is thus kEcs-9ji+jf; T) =
with a( 7') = a( To)(T/To)-Nand a fitting parameter a. Since L(L 1) is just a rotational energy divided by Bo, the direct relationship between the ECS-P law and the SPG law, eq 8, is apparent. It may also be noted that the temperature-dependent a( r ) factor may be taken outside the summation in eq 12, thereby giving a clear basis for the ( T/To)-N dependence suggested by GamachemJ1 and others22for pressure-broadened line widths. For collisions between heavy diatomic molecules (e.g., 12*) and light collision partners (e.g., He) it may be necessary to restrict the amount of angular momentum which can be transferred. This may be done by modifying the basis rates WoLwith an exponential attenuation term
+
where j * represents some upper limit on the number of angular momentum quanta which can be transferred in a collision. Incorporating I@ given by eq 14 with the ECS scaling law, eq 12, gives the ECS-EP (or ECS-j*) law, which generally gives the best representation of experimental data.I6J8 While j * is considered as an adjustable parameter in the fit, its magnitude should be comparable with the orbital angular momentum available in the collision. This quantity can be estimated as
7
Fab/h
(15)
where p is the collision-reduced mass, the mean thermal velocity has been defined above, and b is a 'typical" impact parameter in an inelastic collision. This implies that j * == I = [(8c(/ ~ ) ' / ~ b / h ] ( k T )so ~ /it~ is , reasonable16 to rewrite eq 14 as
B
(30) Note that in eq 12 detailed balance is automatically ensured by the factor (2jl+l) exd[E(ji)-E(j,)]kfl, as it is in eqs 5 and'9 by A2&K2 (A). In several of the other scaling laws described (e& MEG (eq 6) or SPEG (eq IO)), detailed balance must be introduced in an ad hoc way by including the additional prescription (2jr+l)k(ji-=jr) = (2ji+l)k(jpji) exp(AEif/kfl. (31 ) Rose, M. E. Elementary Theory of Angular Momentum; Wiley: New York, 1957. (32) Rotenberg, M.; Bivins, R.; Metropolis, N.; Wooten, J. K. The 3-j and 6-j symbols; M.I.T. Press: Cambridge, MA, 1959.
ji jr
(0
O O
2L * ) [L(L + I)]" +
The sum in eq 17 may be taken over sufficient accuracy.
exp(-PEL/kT) (17)
vi - jflIL 5 vi + jd with
111. State-to-State Rotational Energy Transfer in Diatomic Halogens Two data sets for state-testate rotational energy transfer (RET) in electronically excited iodine (I2*, B3110u+)have been tested against the various scaling laws described in the preceding section. The first was the data for 12* (ui = 13, ji = 41, 81, 91, or 113) + M (=Xe, He), which had been reported and analysed by Dexheimer et aI.% Comparison with the various scaling and fitting laws is generally done in terms of a goodness-of-fit statistic x 2 / v . When x 2 / v = 1, this means that the data are being fit, on the average, to within the reported experimental standard deviation a; when x 2 / v is significantly greater than 1, the expression used is inadequate to fit the data. Results obtained in ref 25 essentially duplicated those of Dexheimer et al.29933 The EGL fit (eq 4, with parameter kT/P = 62 cm-') gave a statistical goodness-of-fit to the data of x2 = 10.4, indicating significant deviations beyond experimental error. The SPG [eq 8, a = 0.8661" or ECS-P [eqs 12 and 13, with a = 0.91 and 1, = 3.2 A] scaling laws gave essentially perfect fits to the data, with x2 = 1.0-1.2. An 10s limit, obtained by setting I, = 0, gave a noticeably poorer fit with parameter a = 0.87 ( x 2 = 2.5). An earlier but more extensive set of data34on RET in 12*(ui = 25, ji = 34) + M(=3He, 4He, Kr, and 12) at T N 350 K was then analyzed, since these data had been previously employed in a "surprisal analysis"6 to validate the EGL for RET. Closer inspection of the fits in ref 6, however, revealed small, significant, and consistent deviations from the EGL fits. Comparisons were made, using the x2 criterion, among EGL (eq 4), ECS-P (eqs 12 and 13), and ECS-EP (expressions 12 and 14) scaling laws. The (33) The only exception is the value for the parameter a in the SPG fit reported in ref 29, which is in error by a factor of 23 due to a programming error (Pritchard, D., private communication). (34) Steinfeld, J. I.; Klemperer, W. J . Chem. Phys. 1965, 42, 3475.
9642
The Journal of Physical Chemistry, Vol. 95, No. 24, 1991
TABLE I: Comparative Fits for Iz*(vi = 25, j = 34) + M
-
scaling law (eq)
Steinfeld et al.
12*(vf = 25, jr)t M
parameters"
x 2 /V
Figure 1
M = 4He EGL (4) 10s IOS-EP ECS (12, 13) ECS-EP (1 2, 14)
a = 0.001 1 a = 0.9802 a = 0.2648 a = 0.403 a = 0.3379
k T / @ = 16 cm-' a = 1.48 a = 0.969 a = 1.129 a = 0.977
EGL (4) 10s-EP ECS-EP (12, 14)
a = 0.00145 a = 0.0488 a = 0.0619
M = )He k T / p = 14 cm-I a = 0.533 a = 0.513
EGL (4) ECS (12, 13)
a = 0.0024 a = 0.3206 a = 0.3046
M = k T / @ = 14.8 cm-' a = 0.981 a = 0.979
a = 0.00144 a = 0.622 a = 0.524
M = Kr k T / P = 22.1 cm-' a = 1.06 a = 1.05
10s EGL (4) ECS (12, 13)
10s
I, = -14 8, I, = 37 A
j'
= 34.2
2.4 4.6 0.99 1.98 0.89
I, = 47 A
j * = 19.9 j * = 20.7
3.15 2.86 2.24
j * = 31.7
2
3 4 5
12
1,=3A
1.66 2.6 2.5
I, = 7.5 A
1.37 0.55 0.55
6
"The parameter a is in units of cross section (A2);to obtain rate coefficients in cm3 s-', multiply by 12.7 X lo-'*.
-20
0
.
If
.
20
40
60
- 11
Figure 2. Data as in Figure 1 with 10s and [OS-EP (IOS-j*) scaling
results are summarized in Table I and Figures 1-7. It is clear that, in almost every case, angular-momentum-based (ECS or 10s) scaling laws provide a better representation of the data than does the EGL. Including the angular momentum restriction (eq 14) provides a significant improvement for the lighter systems (M = 3He, 4He), but there appears to be relatively little difference between the 10S(Ic = 0) and ECS forms of the scaling law. We conclude that angular-momentum-based scaling laws are indeed the correct ones to use for RET and should be employed whenever such data are to be described. Several interesting observations may be made about the magnitudes of some of the parameters appearing in Table I. The first concerns the adiabaticity correction Qj = (1 + ~:/6)-' which appears in eq 12. Consider 12*(B) in high-o, low-j ( u = 43,; = 12) and low-c, high-j ( u = 14, j = 100) states, with
W,J-2
( u = 43), 0.0267
20
0
40
60
-J, Figure 3. Data as in Figure 1 with ECS and ECS-EP (ECS-j*) scaling laws (eqs 12 and 13 or 17). If
laws.
E, = 0.022 cm-'
-20
cm-I ( u = 14)
= 4TCB"O' + y2) = 1 X 10" radians s-' ( c = 43,; = 12) 1 X I O i 2 radians s - I ( r = 14,; = 100)
Thus, the angular velocity of a rotating 1, will be in the range 10"-1012radians/s. For T~ = lc/O, take IC = 1 A = cm. For 12*-He, L' = 1.27 X IO5 cm s-l, T~ = 8 X s. For I,*-Xe.
PI1
The Journal of Physical Chemistry, Vol. 95, No. 24, 1991 9643
Feature Article
-= 10s-jn ECS-jr
---
t
a
-20
0
20
40
h-ji Figure 7. Data as in Figure 6 for 12*
+ Kr.
+
+
150, and a reduction factor of exp[-1(1 l)/j*(j* l ) ] would not be noticeable. We may also note that this relationship had been suggested in the early work34 on RET in 12*.
-= E G L --- ' ECS
--- 10s
IV. Scaling Laws for Rotational Energy Transfer in NZ (a) Determination of the SPEC Parameters. The usual procedure15-16for obtaining the state-to-state rates for rotational energy transfer from Raman spectroscopic data is to invert the measured line-broadening coefficients using the various fitting or scaling laws described previously. The parameters are obtained with the aid of the customary sum rule, - R e x W f j= r j (18)
a
j'# j
a
\
t I0-.k
t
,,,,,,,,l,,,,,,,,,l,,,,I,,,0,1,,,,,,,5,1,
I/
\ -20
0
.
It
.
20
40
- Ji
Figure 6. RET data from ref 34 for 12* + I, (ui = 25, ji = 34 25, j r ) with EGL, ECS and 10s scaling laws.
-
uf =
0 = 2.7 X 104 cm s-l, T~ = 3.7 X s. Let us construct a table covering the possible range of the factor 1 + 1:/6:
I,*-He 12*-Xe
oj= 10" s-l (low j ) 1.00001 1.00002 ,
oj= 10l2s-' (high j ) 1.001 1.023
We see that this factor is very close to 1, and the 10s should, in fact, be a good approximation for most 12*-M collisions. The worst case is for high-j states with heavy collision partners (I2*-Xe, 12*-Iz), and even then the correction term is only a few percent. A comment may also be made about the parameter j * , appearing in eq 14, which represents a limit on the number of angular momentum quanta which can be transferred in a collision. Using eq 15, we may estimatej* for several cases. For 12*-4He, the reduced mass p = 3.95 atomic units ( m H= 1.008), 0 = 12.7 X IO4 cm-l at 298 K, and we can take an average impact parameter b = 4 X 1 0-8 cm. These values, inserted in eq 15, give I = 31.6; the closeness of this value to the 10s-j* fitted parameter j * = 31.7 is, of course, fortuitous, but the magnitude is physically correct. For 12*-3He collisions, I should be reduced by a factor (3/4)Il2 to give 26. and the parapeter fit j * = 20 is consistent with this. For collisions with Xe, I or j * would be approximately
and the broadening due to pure vibrational dephasing is suffkiently small that it may be neglected.35 The collisional broadening coefficients for the N2 Q-branch used in this analysis are those measured by stimulated Raman spectroscopy (SRS)"*36 and extended to higher rotational quantum numbers." This set of broadening coefficients covers a wide temperature range (295-1500 K) and a large number of rotational levels (up to j = 40 a t the higher temperatures). This set of data had been previously analyzed,13 with the result that an EGL scaling law appeared to be able to reproduce the experimental spectra up to the highest pressure employed. In the reanalysis of these data presented here, several factors enter into consideration when choosing a suitable scaling law. While the PIG model1&(eq 11) is able to reproduce the observed line widths (as, indeed, do almost all the models), it is not able to reproduce the band shape if the collisional shift is properly taken into account; equivalently, it is unable to reproduce accurately the collision-induced line shifts." The PIG model is also not able to model recently reported experimental data38on state-to-state rotational energy-transfer rates in N2(u=1). For these reasons, we do not consider the PIG model further, but instead begin by testing the temperature dependence of the SPEG parameters A( T), /3, and 6 [eq 101, which have been determined from the set of Raman line-broadening coefficients at each temperature. The procedure used is the nonlinear weighted least-squares method described in ref 16. The results are presented in Table 11, where the uncertainties are 1 standard deviation. The parameters /Iand 6 do not exhibit a significant variation as a function of temperature, ( 3 5 ) A value of y, = (2.8 f 1.6) X cm-I amagat-' has recently been obtained (ref 41). (36) Lavorel, B.;Millot, G.; Saint-Loup, R.; Wenger, C.; Berger, H.; Sala, J . P.; Bonamy, J.; Robert, D. J . Phys. (Paris)1986, 47, 417. (37) The precise values for the high-j levels do not make a large contribution, but must be included for an accurate fit. (38) Sitz, G . 0.;Farrow, R. L. J . Chem. Phys. 1990, 93, 7883.
9644
The Journal of Physical Chemistry, Vol. 95, No. 24, 1991
TABLE 11: Fit of
N, SRS Self-Broadening Data to SPEC Law a ( 7'). mK/atm 1.871 i 0.088 38.27 f 5.41 53.36 f 3.04 1.632 f 0.042 19.47 f 1.67 1.854 f 0.042 2.121 f 0.056 11.65 f 0.75 1.666 f 0.059 16.01 f 1.48 1.846 f 0.102 10.78 f 0.86 1.919 f 0.071 8.39 f 0.82 1.759 f 0.079 6.16 f 0.67 4.02 f 0.34 2.324 f 0.087
a
T, K 295 298 500 600 7 30 940 1000 1310 I 500
a, mK/atm
6
0.1 12 f 0.044 0.235 f 0.019 0.130 f 0.023 0.307 f 0.019 0.204 f 0.025 0.157 i 0.024 0.128 f 0.025 0.1 30 f 0.027 0.044 f 0.022
6,: 1.844 f 0.031
weighted average
Steinfeld et al.
0.1 13 0.166 0.528 0.461 0.501 0.586 0.610 0.495 0.345
(49.281 f 0.343)(T/295)-'
TABLE 111: Parameter Values for MEC, SPEG, ECS-P, and ECS-EP Scaling Laws a(To=295 K), scaling law cm-l atm-I N R 6
SPEG ECS-P ECS-EP
O
E-c
-0.5
26.44 f 0.26 49.28 f 0.34 38.57 f 0.27 18.92 f 0.62
1.365 f 0.005 1.256 f 0.013 0.915 f 0.017 1 . 1 12 f 0.015
0.367 0.831 1.068 0.804 0.617 0.709 0.676 0.568
6,: 0.176 f 0.028
power law fit:
M EG
46.86 f 0.25 24.12 f 0.14 19.85 f 0.12 15.25 f 0.10 11.73 f 0.04 10.34 f 0.05 7.97 f 0.06 6.73 f 0.03
1.890 f 0.029 1.844 f 0.031
256*0.0'3
I,,
(Y
a
1.174 f 0.018 0.176 f 0.028
0.1309 f 0.0062
1.245 f 0.066 0.742 f 0.012
0.748 f 0.047 3.50 f 0.37
Q
ref
1.02 1,13 1.14 0.97
13 present work 15 16
SPEC
T
1
-1.5
-
0
-2.0 0.0
I
I
I
0.5
1 .o Log (TITO)
1.5
2.0
Figure 8. log [a(T)/a(To)] vs log (T/T,) using 8, and I , in SPEG scaling
law. but the observed scattering is probably due to experimental uncertainties. Similar variations in fl and 6 are observed when the MEG model is used, which is consistent with attributing the variation to experimental uncertainty. We therefore consider the parameters p and 6 to be independent of temperature, and we use the weighted average values (pa and 6,) of these parameters to determine a consistent set of a ( T ) (next-to-last column of Table 11). The standard deviation u obtained at each temperature (last column) may be compared with the one obtained when the parameters are free to vary (first column of Table 11). We observe a small degradation of the quality of the fit when the parameters (3, and 6, are used, at each temperature, but the values of u remain very close to 1 ( < I for most of the cases), indicating that the fit remains very good. The amplitude parameter a( T ) obtained with fl, and 6, is reproduced very well by the power law a ( T ) = u ( T , ) ( T / T , ) - ~ (see Figure 8 ) . The values of the parameters a( To)and N are given in Table 111. Raman line-broadening coefficients calculated with the best fit parameter are shown in Figure 9, in which only five temperatures are displayed to clarify the figure. The agreement is relatively good and quite similar to the one previously obtained using the MEG or ECS scaling laws. Our best fit SPEG parameters are also given in Table 111, along with the parameters previously obtained for the MEG,I3 ECS-P,IS and ECS-EPi6 models. The standard deviation for the whole set of broadening coefficients shown in the next-to-last column does not depend greatly on the choice of scaling law, indicating that any of the models is adequate to reproduce both the temperature and rotational quantum number dependences. Let us remark that our parameters for the MEG and SPEG models are somewhat dif-
0
r o t a t i o n a l quontm rider ( J )
35
Figure 9. Calculated Raman line-broadening coefficients for N, at selected temperatures using SPEG scaling law.
ferent from those used by Sitz and Farrow38 to fit their direct measurements of state-to-state rotational-energy-transfer rates. These differences arise from the method used to determine the parameters. For the MEG model, the parameters used by Sitz and Farrow have been obtained by Koszykowski et aI.l4 using a I-atm Raman spectrum of the Q-branch recorded at 295 K, and for the SPEG model, we are not certain how their parameters have been obtained. In Figure IO, we compare the experimental values of stateto-state rates from Sitz and Farrow for Aj = 2 and A j = 4. We see that the ECS-P does not reproduce the experimental data very well. On the other hand, the three other models (SPEG, MEG, and ECS-EP) reproduce these data relatively well. We consider this question further at the conclusion of this section. (b) Simulation of SRS Band Shapes in Nitrogen. A critical test of the validity of the various fitting and scaling laws is to compare the simulation of the S R S Q-branch at high density with experimental spectra. Let us note that the relaxation matrix W allows us to simulate the collapse of the Q-branch at high density through the following expression: 1
FRS(w)= -(olfilI)* Re 7r
CPjr(G-l(w))jub ldi
(19)
where ( G ( U ) ) , ~=, i(w - wl,,,o,r - nAj .Jhjd, + nWjd,with n the number density of perturbers, wIj,,o, fhe &jr) line position, and Ajdr the collisional line-shifting coekicient of the QGr) line. We usually assume15that Ajdrdoes not depend on the rotational quantum numberif so that Ajdracts as an overall frequency shift A. One example of comparison between experimental and theoretical collapsed Q-branch is given in Figure 1 1 at 70.2 amagat and 295 K. This comparison is made by fitting only an overall scale factor and an overall frequency shift A. It clearly appears
The Journal of Physical Chemistry, Vol. 95, No. 24, 1991 9645
Feature Article
Density (amagat)
lo] A
0 experimental
0
I
100
150
h
+
ECSEP ECS-P
r
o
m
4
w
50
"s
T
w
0 r
400-
Y
1
l a '
0
2
4
6
8
1
0
1
2
Initial J I
5 1 0 experimental 4-
0
meg
A
ea-p ecs-ep SPeg
0
X
3-
P
2-
A
f
1-
fA
I b l
0
1JE.
1
Ram Sill (
,
0.1
1
)
1
1
1
1
1
1
1
1
1
1
IUI.
Figure 11. Comparison between experimental N2 Raman Q-branch spectra and scaling-law predictions. The residuals of the fit are shown below each spectrum.
from this figure that the best band shape is obtained with the ECS-EP model; a small deterioration is observed with the SPEG
Figure 12. Raman line shifts vs density obtained with various scaling-law models.
and MEG models, whereas a larger discrepancy appears with the ECS-P model. At this point, the comparison concerns only the band shape. A very important test of the ability of a rate law to model the Wfjelements is the value of the collisional shift A extracted from the model. The derived shift is then compared with measurements on isolated lines at low density, as shown in Figure 12 for the N2 Raman Q-branch data and the scaling laws we have discussed here. In the figure, the solid line represents a linear extrapolation of the shift measured a t low densities. It is essential that the low-density measurement be accurate and reliable for this comparison to be valid. We have used the j-averaged value39 (-3.6 X cm-' amagat-' at 295 K, which is consistent with the value (-3.5 f 0.5) X cm-' amagat-' ~ b t a i n e d 'by ~ differential inverse Raman spectroscopy, and an upper and lower bound (-3 X cm-' atm-' < A < -4 X lo-' cm-' atm-I) given by Koszykowski et aI.l4," Since other measurements from very high density spectra give shift values A = -3.2 X lW3 cm-' amagat-' and -3.25 X 10" cm-' amagat-',42,43we are confident that this value is the correct one for N2-N,. The value of A = -6.1 X IO-' cm-' reported earlier by Wang and Wright,44which was obtained simply by measuring the shift of the peak frequency, does not take into account the contribution from motional narrowing and is therefore significantly in error. With this value for A, it is clear from Figure 12 that the ECS-EP is the only model which gives the necessary agreement with the low-density measurements. For the other scaling laws (MEG, and particularly SPEG and ECS-P) to be acceptable, it would be necessary to have either a pronounced nonlinearity in A in the low-density (0-25) range or a nonzero shift at zero pressure, both of which are completely unphysical hypotheses. The reason that a given model does not give the exact line shift is certainly due to imperfections in the calculation of the W matrix. Indeed, changes in off-diagonal W matrix elements have the effect of shifting the Q-branch, appearing as an overall frequency shift. So the determination of the collisional shift from comparison with the S R S band shape at high density constitutes a critical test of fitting/scaling laws, and the ability of a relaxation model to give the exact Q-branch shape does not necessarily mean that it can (39) Gonze, M. L.; Saint-Loup, R.; Santos, J.; Lavorel, B.; Chaux, R.; Millot, G.;Berger, H.; Bonamy, L.; Bonamy, J.; Robert, D. Chem. Phys. 1990, 148, 417. (40) The density in amagat units is the ratio of the molar volume of gas at STP to the actual volume at a given temperature and pressure. For N1, I atm at 295 K corresponds to 0.926 amagat. (41) Lavorel, B.; Oksengorn, B.; Fabre, D.; Saint-Loup, R.; Berger, H.. to be published. (42) Osin, M. N.; Smirnov, V. V.; Fabelinskii, V. 1. Report No. 108 of the General Physics Institute: Moscow, 1984; see also ref 43. (43) Lavorel. B.; Chaux, R.; Saint-Low, R.; Beraer, - H. Opt. Commun. 1987, 62, 25.
(44) Wang, C. H.; Wright, R. B. J . Chem. Phys. 1973, 59, 1706.
9646 The Journal of Physical Chemistry, Vol. 95, No. 24, 1991
also give the exact absolute collisional shift, with a linear density dependence in the binary collision regime. Thus we conclude that for modeling both the broadening and the shift of the N2 Raman Q-branch spectra, the dynamically based ECS-EP scaling law is the only acceptable model. Very recently, Bonamy et al.45have reconsidered the ECS-EP scaling law in light of the state-to-state rate mea~urements'~ recently reported for N2. They suggest replacing the adiabaticity factor flJ2 = ( I
+ wij.A21,2/24v2)-2
in eq 12 by the alternate expression fly = (1
+ Wij.&21,2/
12vy
in order to eliminate unphysical behavior of the relaxation matrix W which appears in the modeling of the rotational angular momentum relaxation correlation function for high j levels. Such unphysical behavior at high j (or large A j ) , which corresponds to a divergence of the correlation function at long times, does not appear in the present study, in which we have modeled linebroadening coefficients, state-to-state rates, and narrowed Qbranch band shapes. This is particularly important for large I,, as in the ECS-EP model for N2. While the modified adiabaticity factor Qjt, used in the ECS-EP scaling law, gives a good representation of the state-to-state rates, and is able to account for the Raman band shape, it appears to be unable to reproduce the line shift for the collision-narrowed N2 Raman Q-branch. Indeed, fitting an ECS-EP scaling law to the data, using fly in place of fit, essentially reproduces the results of using a statistically based fitting law such as MEG or SPEG (cf. Figure 12), which we had concluded was unsatisfactory in that it predicted either a nonlinearity in A in the low-density regime or a nonzero pressure shift at zero density, both of which are themselves unphysical. We address this point further in the following section. V. Discussion and Conclusions The analyses of both microscopic state-to-state RET rate data for halogens, and of more highly averaged Raman band shapes and shifts in nitrogen, lead to essentially the same conclusion: an angular-momentum-based scaling law derived from a dynamical theory, and specifically the ECS-EP formulation, is superior to either exponential energy-gap scaling laws or statistical fitting laws for representing these data in a compact form. Additional, more recent measurements of RET in diatomic halogensspecifically, the heteronuclear IF( B3110+)~pecies~~-corroborate this result. The RET data for IF*-He and IF*-Xe collisions show pronounced departure from EGL scaling. Comparisons of these data with several of the RET scaling laws had been promised in ref 46, but these analyses have not yet appeared. For certain experimental systems, an EGL scaling law may be acceptable. For example, VET data, such as that reported recently by Parmenter and co-workers" for 12*-He at high kinetic energies, appear to follow an EGL very well. An EGL may also be appropriate for RET, when the data are sufficiently sparse that no meaningful distinction can be made among the various scaling and fitting laws. For example, the development of the time-resolved infrared diode-laser double-resonance (TRIRDLDR) technique has made it possible to obtain state-to-state RET data for small polyatomic molecules with precision approaching that of laser-induced fluorescence measurements in diatomic mole~ ~ l e s . Such * ~ measurements ~ ~ . ~ ~ ~ have ~ ~ been carried out, for example, on methane,48 silane,49and ammonia.S0 Typically, a scaling law is used to represent RET rates in conjunction with
(45) Bonamy, L.;Thuet, J. M.; Bonamy, J.; Robert, D. J. Chem. Phys. 1991, 95. 3361. (46) Davis, S. J.; Holtzclaw, K. W. J. Chem. Phys. 1990, 92, 1661. (47) Krajnovich, D. J.; Butz, K. W.; Du, H.; Parmenter, C. S. J. Chem. Phys. 1989, 91, 7705. (48) Foy, B.; Hetzler. J.; Millot, G.; Steinfeld, J. I. J. Chem. Phys. 1988, 88, 6838. (49) Hetzler, J. R.; Steinfeld, J. I . J. Chem. Phys. 1990, 92, 7135. (50) Abel. B.; Coy, S. L.; Klaassen. J. J.; Steinfeld, J. I., in preparation.
Steinfeld et al. a kinetic master-equation model,5' which is iterated until agreement with experimental level population vs time signals is achieved. Since the data are relatively sparse in these systems-only a small number of rotational states may be accessible from the initially populated level, although a complex pattern of fine-structure selectivity may be superimposed on the basic scaling l a ~ ~ ~ . ~ ~ - i t is in general not possible to distinguish between exponential-gap or power-law behavior. Typically, an EGL, such as eq 4, can be used to represent the data. It may be instructive to note the results of such an analysis for the three systems mentioned above. For methane48self-collisions, an EGL parameter p = 0.8 has been obtained, while in silane the distribution appears much broader, with 0 = 0.05 (for F and E symmetry levels) and p = 0.075 (for A symmetry levels). An analysis of the results for ammoniaMgives 0 = 2.1 f 0.1, reflecting the much narrower final-state distribution resulting from collisions between two polar molecules. Further measurements on these systems, and especially a comparison of scaling laws and parameters deduced from TRIRDLDR and SRS experiments on the same molecules (as has been discussed above for the N2-N2 system) would be highly desirable; the latter experiments have recently been carried out in D i j ~ n . ~ * Several comments concerning remaining uncertainties and possible future directions, particularly in the area of Raman bandshape analysis, may be appropriate at this point. 1. The assumption of a constant, j-independent collisional shift A needs to be carefully assessed. A variation of A withj is possible (but would be difficult to observe in practice) and could modify the band shape in the intermediate-density (ca. 10 amagat) regime. 2 . The models used in the scaling laws only take account of R-T transfers. Near-resonant R-R transfer has been implicated in both methane RET4* and Raman self-broadening in H2.53 Possible contributions from R-R transfer should be estimated, or if possible measured, and introduced explicitly into scaling-law expressions for line widths and line shifts. 3. Angular momentum reorientation during RET will also affect the form of the scaling laws and thus the computed band shapes. As noted above, complete reorientation of MJduring a collision will introduce a statistical-weight factor N , = 2jf+ 1. Recent measurement^,^^ however, suggest that reorientation of MJmay be severely limited in systems such as N2-N2,and indeed that alignment may be preserved in inelastic collisions. 4. More accurate determination of the temperature coefficient N would be desirable, but is often difficult in practice. Since theoretical values2"-**of N typically lie between 0.7 and 1.0, it is an important but not extremely sensitive parameter for scaling broadening coefficients. 5 . A particularly important area of investigation is the measurement and intercomparison of time-resolved relaxation rates and frequency-domain measurements of line widths (and line shifts, if possible) for the same sets of transitions and physical systems. In addition to providing a more stringent test of the scaling laws, as noted below, such measurements can help to elucidate the contribution of elastic dephasing terms to line broadening. Neglect of the elastic contributions is a standard and nearly universally made a p p r o x i m a t i ~ n ,and ' ~ ~ is~ ~implicit in the definition of eq 1. Recent measurements on RET and infrared line broadening in some small polyatomic molecules, such as ozone56and ammonia,50suggest, however, that the rotationally inelastic processes may account for only a fraction of the total (51) Steinfeld, J. 1.; Francisco, J. S . ; Hase, W. L. Chemical Kinetics and Dynamics; Prentice-Hall: Englewood Cliffs, NJ, 1989; Chapter 14. (52) Millot, G.; Lavorel, B.; Steinfeld, J. I. J. Quant. Spectrosc. Radial. Transfer, in press. ( 5 3 ) Rahn, L. A.; Farrow, R. L.; Rosasco, G . J. Phys. Reo. A , in press. (54) Farrow, R. L.; Sitz, G . 0. Abstracts, International Symposium on Coherent Raman Spectroscopy, Samarkand (U.S.S.R.), 1990. (55) DePristo, A. E.; Rabitz, H. J. Quant. Spectrosc. Radiat. Transfer 1979, 22, 65. (56) Flannery, C.; Mizugai, Y.; Steinfeld, J. I.; Spencer, M. N. J. Chem. Phys. 1990, 92, 5164. Flannery, C . ; Klaassen, J. J.; Steinfeld, J. 1.; Spencer, M. N.; Chackerian, C.. J r . J . Quam. Spectrosc. Radiaf. Transfer 1991, 46,
73.
9647
J. Phys. Chem. 1991, 95,9641-9653 line width, and that contributions from elastic processes also need to be considered. In summary, neither Raman line widths and band shapes by themselves, nor direct measurement of state-testate rates (because of the limited experimental accuracy of the latter), appear to be sufficient for a clear discrimination between the various fitting and scaling laws. The combination of several types of measurements, particularly including Raman Q-branch narrowing at high densities, and accurate and reliable measurement of collisional shifts, is necessary for a definitive test of the rate laws. The optimum procedure for determining rate laws and parameters would be a simultaneous, properly weighted, global fit of Raman linewidths, state-to-state rates (when available), and collisionnarrowed S R S Q-branch bandshapes over a range of densities, taking proper account of collisional shifts as well. The development of such a procedure is currently in progres~.~' Furthermore, preliminary results a t the Laboratoire d e Spectronomie MolCculaire et Instrumentation Laser in Dijons8 seem to indicate that high-resolution Coherent Anti-Stokes Raman Spectroscopy (57) Roche. C.. unpublished results. (58) Fanjoux, G.,unpublished results.
(CARS) is a little more sensitive than S R S to the type of model used to calculate the collision-narrowed Q-branch. It is likely that inclusion of high-resolution collision-narrowed CARS Q-branch data could further improve the suggested procedure. Acknowledgment. Support for this work was provided in part by the Office of Standard Reference Data and Quantum Physics Division of the National Bureau of Standards (now NIST), and by the Advanced Laser Technology Division of the U.S. Air Force Weapons Laboratory. J.I.S. thanks Jean Gallagher, Steve Leone, and the staff of JILA and the Atomic Collisions Data Center for their assistance, hospitality, and enthusiasm during the initial stages of this work; the members of the Laboratoire de Spectronomie Mol&culaireet Instrumentation Laser (SMIL) for their hospitality during a recent sabbatical visit; and the NASA Office of Space Science and Applications, Upper Atmosphere Research Program and Planetary Atmospheres Program, for their continuing support of our recent research. The Centre National de la Recherche Scientifique, Paris, and the Conseil RCgional de Bourgogne are gratefully acknowledged for their financial support of the work at SMIL. Registry No. N2, 7727-37-9.
ARTICLES Role of Charge Transfer for the Vibrational-Mode-Specific Chemical Reaction of NH,+(v) and NH, Akitomo Tachibana,**tSusumu Kawauchi,t***5 Yuzuru Kurosaki,t Naoto Yoshida,t Takea Ogihara, and Tokio Yamabet.* Department of Hydrocarbon Chemistry, Faculty of Engineering, Kyoto University, Kyoto 606, Japan (Received: December 5, 1990: In Final Form: July 18, 1991) New reaction pathways involving (NH3.NH3)+ charge-transfer (CT) complex formation have been proposed for the vibrational-mode-specificreaction of NH3+(v)with NH3 using the concept of a meta-IRC (intrinsic reaction coordinate). It is known experimentally for the reaction that there are three elementary processes, electron transfer (ET), proton transfer (PT), and hydrogen abstraction (HA). The CT complex formation plays a key role in our proposed reaction pathways for both ET and HA processes: in ET the reactants form the CT complex, which then dissociates to NH3+ and NH3 with an electron transferred from one to the other, while in HA the CT complex does not dissociate but undergoes a rearrangement to form the (NH4.NH2)+,complex, which dissociates to NH4+ and NH2. The overall HA process has been predicted to have no energy barrier and to be exothermic by 15.8 kcal/mol at the HF/4-31G level of calculation. We have suggested that three characteristic transition points (TPs) exist for the reaction pathway of HA: (1) the "primary" TP of the CT process prerequisite for the HA to occur, (2) the geographical TP of the potential surface, Le., the conventional transition state (TS) for the HA process, and (3) the "secondary" TP of the geometrical change for the HA process. For the ordinary reaction coordinate, the conventional TS merges the TP of CT and the geometrical TP, but in this case the character of the conventional TS splits into the three TPs. In particular, we have found that the CT process around the primary T P is most significant for the vibrational-mode specificity of the ET and HA processes. Moreover, our mechanism elucidates the non-state-selectedexperimental results: the reaction proceeds via PT (85%) rather than HA (15%) and ET is negligible.
Introduction Three elementary processes, proton transfer (PT),hydrogen abstraction (HA), and electron transfer (ET),' in the reaction of neutral ammonia N H 3 and vibrationally excited ammonia cation NH3+(o) have been studied state selectively.* It has been found 'Additional addras: Division of Molecular Engineering, Graduate School of Engineering, Kyoto University, Kyoto 606,Japan. 'Additional address: Institute for Fundamental Chemistry, 34-4 Nishihiraki-cho, Takano, Sakyo-ku, Kyoto 606. Japan. 'On leave from Kawasaki Plastics Laboratory, Showa Denko K. K., Kawasaki 210, Japan.
TABLE I: Total Energies0 (au) species
enerw
species
NH3 NH3+ NHd+ NH2
-56.106692 -55.801670
(NHyNH3)' (NH,.NHJ+
-56.458884 -55.474727
TS
energy -1 11.952851 -1 I 1.977768 -1 I 1.944909
"At the HF/4-3IG level.
that PT is suppressed while HA and ET are enhanced by the excitation of v2 umbrella-bending vibrational mode of the NH3+."
0022-3654/91/2095-9647%02.50/0 0 199 1 American Chemical Society
