J . Phys. Chem. 1993,97, 3151-3156
3151
Vibrational States of ArC02: Analysis of an Internal Dynamical Transition Using Self-Consistent Field Techniques T. R. Horn and R. B. Gerber Fritz Haber Institute and Department of Physical Chemistry, Hebrew University, Jerusalem, Israel
Mark A. Ratner’ Department of Chemistry, Northwestern University, Evanston, Illinois 60208 Received: October 27, 1992
The van der Waals cluster molecule A r c 0 2 is studied computationally by using the vibrational self-consistent field (SCF) approximation, with an approximate but reasonable potential function. Calculations are carried out both for the full six-dimensional motion and for a reduced two-dimensional problem in which the C 0 2 is held rigid. An interesting dynamical transition is found in the motion of the Ar atom. Its equilibrium geometry is a symmetric T-shape, and for low excitations both the radial and the angular motions in the CO1 plane resemble the states of anharmonic oscillators (smaller intervals with higher excitations). Above the sixth state of the bend in the angle 6, however, the bend spectrum changes to that of a rigid rotor, with spacings of 2Bne for quantum number ne. The one-dimensional effective S C F potentials along the 8 coordinate and plots of the wave function both show a dynamical transition, in which, above ne = 6, the motion of the Ar in the C02 plane is essentially that of a rigid rotor in the 6 coordinate. Calculations of the principal moments of inertia support this interpretation.
I. Introduction van der Waals molecules offer significant insights into the interaction potentials and the dynamics of weakly bonded systems.’ The strongly anharmonicforce fieldscharacteristicof such systems give rise to an unusual type of motion and often to spectroscopic patterns reflecting a dynamical behavior very different from that of strongly bound species. Understanding the dynamics of van der Waals vibrations is essential for using spectroscopy of such systems to learn about the interactions involved. Recent spectroscopicwork has provided accurate and elegant data sets to compare with theoretical characterization of the structure and dynamics of van der Waals molecules.’ These molecules are distinguished by high-amplitude vibrations; often these motions exhibit, even in low-lying states, extensive delocalization, in which several different geometries are sampled in a manner similar in spirit if not in degree to electronic motion in atoms. Due to the much heavier mass of the atom compared to the electron, however, the localization that characterizes ordinary chemical bonds is also seen in most van der Waals species. A fascinating question then becomes the extent to which the competing force of delocalization (due to kinetic energy) in soft weakly-structured potentials competes with the localization due to minima in the potential surface and to the relatively large atomic masses. Advances in experiment permit measurements of sufficient accuracy to search for structural changes in van der Waals molecules due to changes in the extent of localization in different energy eigenstates. In this sense, van der Waals molecules act as a testing ground for the structural and dynamic effects of long-range, noncovalent forces. Within van der Waals species, preferred geometric and energetic motifs can result in very different properties, best described by using differing, appropriate coordinates. Examples can be found even in triatomic species. For example, the “three ball” structures such as Ar, or XeHe2 are best characterized in hyperspherical coordinates,2 since only the distances are reasonably sharplydefined, even in theground state. These molecules are extensively “floppy”, in that their geometries can vary from equilateral to nearly linear, assisted only by ground-state and
zero-point vibrational energy. Very different behavior is found in such ‘ball and stick” species as IzNe, in which the T-shaped geometry is dominant both in the ground state and for fairly substantial excitations away from the ground-state geometry.3.4 The inert-gas complexes with C02 comprise a well-studied case of atomftriatom van der Waals complexes; accurate spectroscopic characterizations of the Arc02 cluster are a~ailable.~-* Additionally, realistic potential surfaces have been developed for this ~ y s t e m . ~ - In l ~ the present contribution we analyze the dynamical behavior of these complexes, using a reasonable potential surface and full three-dimensionality and employing the self-consistent field approximation24J6-21 for vibrational dynamics. The self-consistent field model permits analysis of structural and spectroscopicchanges by describing the states of the system in terms of the best approximate separable model. In so doing, it permits study of effective potentials in reduced dimensionality and of energy levels as a function of quantum numbers in the approximately separable coordinates. Utilizing this analysis, we observe a fascinating change in the ArC02 molecule. As the bending excitation of the argon in its T-shaped equilibrium geometry is increased, the amplitude for argon motion becomes larger and larger until eventually it undergoes a dynamical transition; for excitation levels above this dynamical transition, it behaves like an internal rotor with spacings increasing by a constant (0.6 cm-I), correspondingto twice an effective rotational constant. Below the threshold, excitation in this bending motion occurs in an anharmonic well, with spacings typical of an anharmonic oscillator. This dynamical transition differs substantially from that of other van der Waals molecules (Ar3, He12) that have been studied by SCF techniques and demonstrates a fascinating limit in which the potential minimum arising from van der Waals interaction is strong enough to localize at low excitation energies but not as excitation energy is increased. In section 11, the self-consistent field model is presented, and equations are given describing its application to the four-atom ArC02 species. Calculations are presented in section 111, both with full three-dimensionality and in a specific subcase in which
0022-3654/93/2091-3151$04.00/0 0 1993 American Chemical Society
3152 The Journal of Physical Chemistry, Vol. 97, No. 13. 1993
Horn et al. of eq 1, we obtain the two separable single-mode SCF equations
i 0
e>
IO
IO
*
6
2
2
fl
2
4
6
8
0
x(bohr)
Figure 1. Ar-CO2 potential (for C02 fixed at its equilibrium geometry) as a function of Ar position. The contour lines shown vary from -8.00( - ~ ) E to H -3.00(-4)E~,with spacings of 5 . 0 0 ( - 5 ) E ~between them. The minimum of the potential is located at R = 6.6 bohr and 0 = r/2 radians.
These equations must be solved iteratively for the wave functions in the 0 and R coordinate; the notation ( F ) e indicates that the averaging is done only over the angular coordinate. A more general case involves the four internal degrees of freedom of the C02, as well as the motions of the argon atom. This leads to the Hamiltonian operator
motions of the C 0 2are frozen. Finally, section IV presents some of the salient features observed in the calculation, and discusses them in terms of the dynamical transition. 11. Self-Consistent Field Model
The full dynamics of the four-atom system, assuming J = 0 and separating overall center of mass motion, is a six-dimensional dynamical problem. A full analysis of the six-dimensional problem is beyond current computational methodology, and, in addition, the aim of computations such as those undertaken here is not only semiquantitative analysis of the energy levels but also an understanding of the dynamical behavior of the molecule in different energy regimes. For such interpretation, the selfconsistent field (SCF) approximation is both efficient and convenient. SCF approximations have been shown useful in analysis of vibrational dynamics of several van der Waals clusters.24*18-21 Comparison with configuration interaction calculations, which are essentially exact on a given potential surface, have shown that S C F approximations, carried out in reasonable coordinates, yield quite accurately the energy levels and wave functions for van der Waals species.2J,20We will therefore complete this study using the SCF approximation, in normal mode coordinates for the C02 species and angular and radial distortion coordinates for the argon against the C02. The potential surface used was adapted from those developed by Dykstra, Billing, Pack, et aL9-I3 The intramolecular C02 potential, based on an inversion of spectroscopic data, is from Romanowski et al.22 Figure 1 shows a cut of the potential for the argon moving in the plane of (rigid for illustration) C02. Notice the minima at the T-shaped geometry, with larger extensions in the angular than the radial coordinate. This shows both that the equilibrium geometry is T-shaped and that facile excitations along the 0 distortion coordinate are to be expected. The Hamiltonian for argon interacting with a rigid CO? molecule could be written, for J = 0, as
V R , @ (1) Here is the reduced mass of Ar and the C02 molecule, a is the distance between the end oxygens of C 0 2 , and p2 is the corresponding reduced mass. In the S C F approximation, we assume that the wave function for theargon motioncan be written as a simple product. Inserting this into the Schradinger equation with the Hamiltonian
where we have denoted the motions of the C02 molecule as Ql (symmetric stretch), Q3 (asymmetric stretch), Qza,and Q2b(the degenerate bend motions). In the presence of the argon, the two components of the bend are no longer strictly degenerate. In this case, there are six SCF equations, as follows:
Simultaneous solution of the SCF equations (6)-( 11) (for flexible C02) or (3) and (4) (for rigid C02) yield the SCF wavefunctions. The final SCF equations are characterized by quantum numbers in each mode. The potential used in the four-atom dynamics is taken from a combination of the CO2 potential derived by Romanowski et
Vibrational States of Arc02
The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 3153
TABLE I: Bound-State Energies for ArC02' nRIn8
0
0
-157.00 -120.32 -89.37 -63.80 43.13
1
2 3 4
2
1
-143.52 -130.96 -107.56 -95.71 -77.49 -66.55 -52.96 4 3 . 1 0 -33.25 -24.05
3
4
-119.47 -109.24 -84.95 -75.47 -56.75 4 8 . 3 4 -34.39 -26.95 -15.42 -7.27
5
6
-100.58 -67.70 42.50 -20.83
-95.37 -63.86 -39.09 -16.84
The SCF bound-state energies of Arc02 (C02 frozen) identified by their quantum numbers nR and n,; all energies are in cm-I and the zero origin is defined as the infinite separation of the reagents Ar and C02. The potential minimum is -184.9 cm-I.
TABLE 111: Ground and First Pure Excitation States of A r c 0 2 (in cm-1 Relative to the Zero-Point Energy) in Combination with the Ar Stretching Mode R nR (nRo000) (nR1000) (nR0100) (nR0010) (nR001'0) (nRW1) 0 0 13.47 1358.34 667.65 667.43 2347.85 1 36.71 49.40 1395.04 704.29 704.14 2384.55 2 67.61 79.48 1426.01 735.19 735.04 2415.47 1451.53 760.71 3 93.19 104.01 760.62 2441.03
TABLE IV Ground and First Pure Excitation States of ArC02 Tabulated (in cm-I Relative to the Zero-Point Energy) in Combination with the Ar Bendinn Mode B ~
TABLE 11: Stepwise Bending Excitation Frequencies' nk
- .:I
0-
1
1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9 - IO
-
IO-. 11 12 12- 13 13- 14 11
nR
0
1
2
13.5 12.6 11.5 10.2 8.7 5.2 3.4 4.4 4.9 5.5 6.1 6.7 7.3 7.9
12.8 11.8 10.8 9.5 7.8 3.8 3.6 4.3 4.9 5.5 6.1 6.7 7.3 7.8
11.9 10.9 9.8 8.4 5.8 3.4 3.7 4.3 4.9 5.5 6.0 6.6 7.2
The separation between consecutive bending states (i.e., e,nH.n,,+l) ( n =~0, I , and 2). All energies are in cm-I, and the C02 molecule is frozen. ecn,.n,,)) for fixed values of n R
al. by generalized SCF inversion22 and an Ar-COZ interaction developed by Bi1ling.l' It is the Billing potential that is shown in Figure 1. Both parts of this potential should be reasonably accurate: the C02 part gives vibrational levels accurate to better than OS%, and the Ar-C02 part was fit to accurate scattering and electronic structure results. 111. Calculations and Results
Two classesof calculations were done; in the first, the C02 was frozen at its equilibrium geometry, and a two-dimensionalSCF problem was solved. The energies obtained, for J = 0, are given in Table I. The zero energy for the potential surface in these calculations is taken to the separated Ar atom and C02 molecule. The energy at the minimum is -184.94 cm-I, and the zero-point energy is -157.00 cm-I. The first and second pure stretching excited states are at -120.32 cm-I (1,O) and at -89.37 cm-I (2, 0) (here theindices indicatethequantumnumbers ofthestretching and bending modes, nR and ne). The intervals among the first three stretch states are then 36.7 and 31 .O wavenumbers, with a decreasing size indicative of the strong cubic anharmonicity in the stretching mode potential. The bending excitations calculated for the different stretching modes are also given in the table. In the first few bending excitations, theintervals again get progressively smaller as the anharmonicityof the bending potentialis explored. The tabulated intervals from Table I are shown in Table 11; note that the successive excitations of the bending frequency become smaller until the 6 7 transition, after which they become larger again. Thisis true for thelowest threevaluesoftheradialstretchquantum number n ~ although , the actual spacings get smaller as n~ increases. This last effect is, again, a demonstration of the anharmonicity of the potential surface-one sees by studying Figure 1 that, as the radial excitation is excited, the potential becomes flatter in the 8 direction, as is reflected in the decreased excitation frequencies. Interesting comparisons can be drawn between these results and those obtained for the full six-dimensionalproblem by using
-
no
(oneoooj
(lnfio0)
(On,l00)
0
0 13.47 25.99 37.49 47.73 56.33 61.56 64.99
36.71 49.40 61.22 72.01 81.42 89.23 93.14 96.65
1358.34 1371.81 1384.38 1395.85 1406.13 1414.80 1420.02 1423.43
1
2 3 4 5 6 7
~
~
~~~~~
(Ond10) (O~ROI'O) (Onm1)
667.65 681.05 693.54 704.94 715.13 723.76 728.92 732.33
667.43 680.90 693.43 704.93 715.16 723.76 728.99 732.42
2347.85 2361.34 2373.91 2385.32 2395.56 2404.25 2409.43 2412.87
SCF. Tables 111 and IV show the computed frequencies for different combination excitations of C02 modes with argon stretching and bending, for motion on the full potential surface. Both stretching and bending excitations of COZ couple only very weakly (shifts of less than 0.3 cm-1) with Ar motion.
IV. Discussion There are two sets of observations from these computations that might be of some significance. Changes in the C 0 2 frequenciesare extremely small, with no computed shift exceeding 0.3 cm-I. Experimentally, the asymmetric stretch is observed to red-shift very slightly.6 In sharp contrast to the relatively uninteresting behavior of the COZvibrations, higher excitation in the 8 and R coordinates yields fascinating dynamical changes. In both the full results and the results with frozen C02, the intervals in the excitationfrequencywith increasingne first becomesmaller and then, above ne = 7,become larger again (Tables I1 and IV). The second differences among the 0 excitations, as deduced for instance from Table 11, show that, above the 7 8 transition, the second difference is 0.6 cm-l for all succeeding transitions. This behavior in the spectrum can be explained by analysis of the Ar wave functions. In Figures 2-6, we illustrate the effect of SCF bending and stretching motion for selected states of our 'frozen" C02 computations. Figure 2 depicts the effective SCF potentials V , and V Bfor the ground state (the curved solid lines). The straight solid lines denote Q and to, the single-mode stretching and bending energies. The dashed lines show the single-mode wave functions +R and $8. The potential VR has its minimum at roughly 6.64 bohr, corresponding to the T-shaped minimum in the potential surface of Figure 1. The classical turning points for the 0 motion, in the ground state, occur at roughly u / 1 6 radians on either side of the minimum. Correspondingly,$0 has a narrow peak centered at */2. As the number of excitationsin the bending coordinate 0 is increased, the effective 8 potential becomes very much broader and more asymmetric (Figures 3-6). As this happens, the minimum in the R coordinate moves to larger values of R, as bending excitation gives a centrifugal effect and expands the length of the van der Waals bond. With increasing excitation in the 8 coordinate, the bending potential becomes wider and samples considerably wider bending angles, though it remains in the half-plane of the full molecularplane. (In fact, motion around the C 0 2axis is essentially free rotorlike, so that such localization would not be observed experimentally. This free rotorlike motion has been ignored in our calculations, and, as we have seen for the 12NeSCF cal~ulation,3~*~ it simply adds rotorlike excitations to the spectrum for the 8 and R modes.) Upon further increase of the 8 excitation energy, the effective barriers become lower. For the ( 0 , 6 ) mode shown in Figure 5 ,
-
3154 The Journal of Physical Chemistry, Vol. 97, No. 13, 1993
Horn et al.
I
5
6
7
8 R(bohr)
9
1
0 'I 5
0
w2
,''
6
1
"
8 R(bohr)
9
0
e (radians) Figure 4. Single-mode potentials for (0,5).
I
\
I
0
'
5
u)
1
lrJ2
0
7l
e (radians) Figure 2. Single mode potentials are shown for the ground state (0,O) of Arc02 for frozen C02. In the upper graph, the single-mode stretching potential VR is denoted by a solid curve line; the dashed line represents the wavefunction,+,+ Thestraight solid lineindicates the relative p i t i o n of the stretching mode energy LX, drawn between its classical turning points. The lower figure shows the single mode bending potential, VR (solid curve), with the dashed line indicating its corresponding eigenfunction,.#)I and the solid straight line denoting the position of (0.
,
,
',--,
R
O'Ool
0
\
/ "
'
6
.
A
7
\
'
'
8 R(bohr)
'
9
I
-
1
0
0.0005 m
I
c
0
/
I
R(bohr)
0
ul
x
0 Iradians)
Figure 5. Single-mode potentials for (0,6).
t ----- \ 1-J
--------:
0 0
x12
II
€I( radians)
Figure 3. Single-mode potentials for (0.1).
the barriers at the linear geometry are just slightly higher than that at the T-shape, and on adding only one more excitation (Figure 6), the barriers become q u a l , and the wave function is effectively delocalized. This can perhaps be seen more clearly in Figures 7-10, which show the computed wave function in the molecular plane for the frozen situation with motion allowed only along B and R. The molecule explores larger and larger regions of radial space, until at the (0,6)state the entire radial region is sampled. In a certain sense, this behavior corresponds to an intramolecular melting,23with the molecule exploring new regions of space with increased excitation energy.
The plots both of the SCF wave function in the 0 coordinate and of the two-dimensional function for ne exceeding 6 seem to indicate essentially free.-rotor motion. Such free-rotor motion would indeed be expected to show spacings going like (21 1)B. where 1 is a rotational quantum number and B a rotational constant. The second differences, then, would be constant at a value of 28. Table I1 shows precisely this behavior, with B = 0.3 cm-I. One way to understand this behavior centers on the influence of the kinetic energy on the B coordinate. This term introduces a repulsive component proportional to R 2 into the effective stretching potentials, VR.This repulsion increases with increasing excitation in the B coordinate and leads to an increase in &in. This in turn enhances the bending motion of the argon atom, allowing it to move further along the potentialvalley for increasing bending excitationsuntil it reachesthe threshold where full planar motion is permitted. Another manifestation of this dynamical transition, one that might be experimentally observable, involves the moments of inertia. Table V shows the calculated moments of inertia ( x is
+
Vibrational States of ArCOz
The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 3155 10
5
6
1
8 R(bohr)
9
1
1
0
F
I
-10 1 -10
0
10
x(bohr)
Figure 9. SCF probability density of the (0,6) bending excitation of ArC02.
0' 0
'
c9 I I(
x/2 (radians)
e Figure 6. Single-mode potentials for (0,7). Note that the is now fully delocalized.
Q
0 +g
function
c4 L
0
0 Q
0
@
h L
r
10 I -10
0 0 e >.
I
I
0 x(bohr)
10
Figure 10. SCF probability density for the (0,7) bending excitation. Note that,as in Figure6, theangular motion isfullydelocalized (rotorlike). -10 I -10
,
I
0
10
x( bohr)
Figure 7. SCF probability density (I+(R,0)12 sin 0) of the ground state of Arc02 as a function of Ar's position relative to C02 (see Figure I); it was obtained by squaring the modulus of the SCF wavefunction and is normalized over the volume element RZdR de. The contour lines vary from 0.02 bohr-' to 0.10 bohr-' with a spacing of 0.02 bohr-I.
TABLE V
Computed Moments of Inertia for Differing Levels of Excitation' ground
R excitation Ql excitation Q2. excitation Q2h excitation QJ
excitation
ng= 1 ng = 2
i./
-01-
ng = 3 ng = 4 ng ng ng
5
=6
=7
ng
8
ng
=9
1 716 850 1 847 680 1 716 920 1 717 940 1 716 910 1 716 920 1 692 440 1 661 680 I 620 930 1 564 360 1474680 1321 100 1 206 020 1 215 840 1 194920
496 720 50 090 496 430 496 580 496 440 496 430 580 320 675 470 787 900 926 780 1 121 380 1471 110 1 600 800 1 591 070 1612040
6.393(-02) -1.129 -0.809 -0.473 -5.84(-02) 0.654 -2.006 -0.617 0.648 -6.1 15(-02) -1.078 0.177 -2.535 -0.215 0.700
Note substantial changes near the'dynamical melting" transition at no = 7. Theeffectivemomentsof inertiaoftheArC02clusterdetermined
0 10 x(bohr) Figure 8. SCF probability density of the (0,l) bending excitation of ArCOz as a function of Ar's position relative to C02.
by integration over system wavefunctions. The x axis refers to the C02 molecular axis with y pertaining to the axis perpendicular to the CO2 molecular axis coplanar with the argon. All moments are expressed in atomic units.
the C02 axis). Note that there are only small differences in the moments of inertia upon excitation of one quantum either in the COZmotions or the van der Waals excitations. On the other hand, substantial excitation in the ne coordinate sharply reduces R,, and, concomitantly, increases R,,-this is precisely what would be expected as the argon atom leaves the y axis on its excursions, having a smaller projection on they axis and a large moment of inertia around it. The change in moment of inertia maximizes, again, above ne = 6 for both the decrease in R,, and the increase in Ryy.
The SCF calculations show, in this molecule, a dynamical transition that corresponds to a complete change in the nature of the excitation, as the barrier for effective one-dimensional motion along the 0 coordinate disappears and motion that was an anharmonic vibration at low excitation numbers becomes essentially an intramolecular rotation for higher quanta. In this system, complementary behavior is observed to the 'three ball" system2J.20 of both Ar3 (RRKM statistical limit, facile energy exchange, extreme excursions around an averaged equilibrium structure, even in zero point) and I2Ne (localization of the van
-10
3156 The Journal of Physical Chemistry, Vol. 97, No. 13, 1993
der Waals atom near the T-shaped geometry even at high excitation).2.3.20 An interesting comparison can be drawn between the present study, in which increase of total energy results in an internal dynamical transition to a delocalized state, and a previous study of a model potential problem for quantum l o c a l i z a t i ~ n .In~ ~the previous work, with coupled quartic potentials, increase in total energy led to a dynamical trapping, with the emergence of a trapped trajectory in the classical trajectory simulation and of a dynamical barrier in the single-mode SCF potential. It thus appears that, depending on the nature of the potential surface,distributions can become more delocalized (melt) or more localized with increase in total energy. Similar behavior is observed in a number2sofother physical systems, from asymmetric rotors to Trojan asteroids, and one might well anticipate many other examples in highly-excited molecular species.
Acknowledgment. The Fritz Haber Institute is supported by the Minerva Gesellschaft,Miinchen. We thank Curt Wittig and George Schatz for helpful remarks and the Chemistry Division of the NSF for support. References and Notes ( I ) For recent review, cf., e.g.: Miller, R. E. Science 1988, 240, 447. Nesbitt, D. J. Chem. Reo. 1988, 88, 843. (2) Horn, T. R.; Gerber, R. B.; Ratner, M. A. J . Chem. Phys. 1989, 91, 1813. Gerber, R. B.; Horn,T. R.; Williams,C. J.;Ratner, M.A. In Dynamics oJPolymericvander Waals Complexes; Halberstadt, N., Janda, Eds.; Plenum Press: New York, 1990; pp 343-354. (3) Horn, T. R.;Gerber, R. B.; Valentini, J. J.; Ratner, M. A. J. Chem. Phys. 1991, 94, 6728.
Horn et al. (4) Bacic. Z.; Light, J. C. Annu. Reo. Phys. Chem. 1989, 40, 469. ( 5 ) Steed, J. M.; Dixon, T. A.; Klempcrer, W. J . Chem. Phys. 1979,70, 4095. (6) Sharpe, S. W.; Reifschneider, D.; Wittig, C; Beaudet. R. A. J . Chem. Phys. 1991, 94, 253. (7) Sharpe, S . W.; Sheeks. R.; Wittig, C.; Beaudet, R. A. Chem. Phys. Lett. 1988. 151. 267. (8) Fraser, G . T.; Pines, A. S.; Suenram, R. D. J . Chem. Phys. 1988,88, 6157. (9) Pack, R. T. J . Chem. Phys. 1976, 64, 1659. (IO) Parker, G.A.; Snow, R. L.; Pack, R. T. J. Chem. Phys. 1976,64, 1668. ( I I ) Billing, G. D. Chem. Phys. 1984, 91, 327. (12) Dykstra, C. E. J. Am. Chem.Soc. 1989, I l l , 6168. ( I 3) Dykstra, C. E. J. Comp. Chem. 1988, 9, 476. (14) Hough, A. M.; Howard, B. J. J . Chem. Soc., Faraday Trans. 2 1987, 83, 191. (IS) Rotzoll, G ;Lubbert, A. J . Chem. Phys. 1979, 71, 2275. (16) Bowman, J. M. J. Chem. Phys. 1986, 68, 608. Acc. Chem. Res. 1986, 19, 202. (17) Gerber, R. B.; Ratner, M. A. Chem. Phys. Lerrr. 1979, 68, 198. (18) Gerber, R. B.; Ratner, M. A. Adu. Chem. Phys. 1988,70,97-132. (19) Ratner, M. A.; Gerber, R. B. J . Phys. Chem. 1986, 90, 20. (20) Horn,T. R.;Gerber, R. B.; Wil1iams.C. J.;Ratner, M.A. In Advances in Molecular Vibrations and Collision Dynamics; Bowman,J. M.; Ratner, M. A., Eds.; JAI Press, Inc.: 1991; Vol. IA, pp 215-253. (21) Bacic, Z.; Gerber, R. B.; Ratner, M. A. J . Phys. Chem. 1986, 90, 3606. (22) Romanowski, H.;Gerber, R. B.; Ratner, M.A.J. Chem. Phys. 1988, 88, 6757. (23) Beck, T. L.;Berry, R. S . J . Chem. Phys. 1988,88, 3910. Beck, T. L.; Doll, J. D.; Freeman, D. L. /bid. 1989, 90,5651. (24) Barboy, B. M.; Schatz, G.C.; Gerber, R. B.; Ratner, M. A. Mol. Phys. 1983, 50, 353. (25) Rau, A. R. P. Rev. Mod. Phys. 1992. 64, 623.
