J. Phys. Chem. 1982, 86, 1107-1111
1107
Quantum Effects of Vibrational Excltation in an Idealized Three-Body Reactive Collision M. S. Bowers and K. 1. Tang' Department of Physlcs, Pacmc Lutheran Unlverslty, Tacoma, Washington 98447 (Received: Ju/y 7, 7981)
The H + Hzreactive collision with the Hzin the vibrational excited state is simulated by considering an idealized potential energy surface which is a two-dimensional duct with a 60° bend. The activation energy is simulated by a step function barrier in the corner region. The exact transition probabilities and reaction rates are obtained from both quantum and classical mechanics. These results are compared with the corresponding ones of the ground-state target molecules. They show (1)vibrational enhancement is much larger in quantum mechanics than in classical mechanics; (2) quantum mechanical tunneling is more pronounced in the case that the target molecule is in the excited state; (3) the quantum theory predicts that the vibrational adiabatic reaction channel (v, = 1to ug = 1)is strongly preferred to the nonadiabatic channel outside the tunneling region. These results suggest that the discrepancies between experiments with excited target molecules and classical trajectory calculations are due to quantum effects.
Introduction The study of the H + H2 chemical reaction plays a special role in chemical kinetics because it is the simplest and most fundamental exchange reaction between an atom and a molecule. Consequently, the study of this reaction has received much attention. In recent years, the development and perfection of infrared lasers, which can selectively prepare and detect vibrationally excited molecules, have made possible experimental studies of state-selected chemical reactions. The rate constants for the H + Hz reaction have been measured by Gordon et al.' with the target molecule vibrating in the first vibrationally excited state. It was found that the vibrationally excited states are strongly preferred to the vibrational ground state by the product molecules. Also, experiments by Heidner and Kasper, showed that vibrational excitation greatly enhances the reaction rate of the reaction. These results seem to be confirmed by experiments by Kneba, Welhausen, and W ~ l f r u m . ~ On the other hand, the classical trajectory calculations,4 which have previously been demonstrated to be adequate in explaining elementary reactions when the target molecule is in the ground vibrational state, gave reaction rates which are two orders of magnitude smaller than the measured ones when the initial molecule is in the vibrationally excited state. Also classical results showed that product molecules have no preference in the distribution of the final vibrational states, which seem to be in contradiction with experiment. Since the experiments are rather involved, there are questions regarding the origin of these discrepancies. Although three-dimensional approximate quantum calcul a t i o n ~suggest ~ that they are quantum effects, there is at present no rigorous three-dimensional quantum treatment with the target molecule in the vibrationally excited state to settle these questions. In this paper, we simulate this reaction with an idealized model which was first introduced by Hirschfelder, Wigner, and Hulburt? It is exactly solvable in both quantum' and (1)E.B. Gordon, B. I. Ivanov, A. P. Perminov, V. E. Balalaev, A. N. Ponomarev, and V. V. Filatov, Chem. Phys. Lett., 68,425 (1978). (2)R.F. Heidner, III,and J. V. V. Kasper, Chern. Phys. Lett., 15,179 (1972). (3)M. Kneba, U.Wellhausen, and J. Wolfrum, Ber. Bunsenges. Phys. Chem., 83,940 (1979). (4)H. R.Mayne, Chem. Phys. Lett., 66,487(1979);H. R. Mayne and J. P.Toennies, J. Chem. Phys., 70,5314 (1979). (5)J. C. Sun,B. H. Choi, R. T. Poe, and K. T. Tang, Phys. Reo. Lett., 44,1211(1980). (6)J. 0. Hirschfelder and E. Wigner, J. Chern. Phys., 7,616 (1939); H. M.Hurlburt and J. 0. Hirschfelder, ibid., 11, 276 (1943). 0022-3654/82/2006-1107$01.25/0
classical mechanicsSs Previous results of this model are confined to the case in which the target molecule is in the ground vibrational state. In this paper, we report both quantum and classical results for the case in which the target molecule is in a vibrationally excited state. We compare them with the corresponding results of the ground-state target molecules. The purpose of this study is to see if this simple model can shed some light on whether the large discrepancies between experiment and classical theory are indeed quantum effects.
The Model The model consists of a linear system of three particles as shown in Figure 1. Initially, particle A impinges on the bound system of particles B and C; after collision, either particle A is reflected or a rearrangement occurs yielding a new bound system of particles A and B with the now free particle C moving away. If the three particles have finite masses the kinetic energy terms are not separable when expressed in terms of the symmetric interparticle coordinates (rl,rz). However, these terms are separable in the coordinates (Rl,r2)or (rlJlz),where R1(R,) is the distance from particle A (C) to the center of mass of the bound system of particles B and C (A and B). In order to make the equations of motion symmetric, additional scaling factors are introduced: x = [MA(MB + Mc)2/MBMc(MA + MB + Mc)]'/2R1 (I) Y = r2
u = [Mc(MA + MB)2/MAMB(MA+ MB + Mc)]'/2R2 (2) v = PI where MA, MB,and MCare the masses of particles A, B, C, respectively. In both quantum and classical mechanics, the motion of the center of mass of the whole system can be separated out. To describe the internal motion quantum mechanically, we have the Schrodinger equation written in terms of x and y I-h2/2~)[(aZ/ax2)+ (a2/ay2)1 + Vb,y)I$= E$ (3)
or in terms of u and u ( - h z / 2 ~ ' [ ( a z / d ~ 2+) (d2/av2)]+ V(U,V))$ = EJ. (4) (7)K. T. Tang, B. Kleinman, and M. Karplus, J. Chem. Phys., 50, 1119 (1969). (8)D.W. Jepsen and J. 0. Hirschfelder, J. Chem. Phys., 30, 1032 (1959).
0 1982 American Chemical Society
1108
Bowers and Tang
The Journal of Physical Chemistry, Vol. 86.No. 7, 1982
In both the classical and quantum mechanical cases, we assume that particles B and C are initially bound together as a molecule. If it is in the ground vibrational state, it has a zero-point energy of E,* = 1; if it is in the first excited state, its vibrational energy is E,* = 4. To simulate the H H, reaction, we set 8 = 60° and VIII* = 3.3, corresponding to a barrier height of 0.424 eV used in the previous study.'O
+
Methodology Quantum-Mechanical Solution. If initially BC is vibrating in its first excited state, the wave function #, which is the solution of the Schrodinger equation, subject to one unit of incident flux, has the form in region I 2a
J/l
Figure 1. Reaction coordinates and idealized potential energy surface for atom-diatomic molecule collision.
where p = M$/lc(MB + M J ' , p' = MAMB(MA + M J ' , V(x,y)and V(u,u)are the potential energy function in terms of (x,y) and (u,u), respectively, and E is the total energy in the center-of-mass system. In classical mechanics, the kinetic energy is also diagonalized in terms of these coordinate^.^ The Lagrangian L of the system can be expressed either in terms of x and Y
L = ' / ~ [ ( d x / d t )+ ~ (dy/dtl21 - V(X,Y)
(5)
or in terms of u and u
L = '/p'[(du/dt)2
+ (du/dt),]
- V(U,U)
To complete the specification of the problem an idealized potential energy function is introduced. The reaction path is divided into three parts (Figure 1). Region I represents the particle system A + BC, region I1 represents the rearranged system AB + C, and region I11 contains the interaction of the three particles. The potential is defined as follows: region I: v=o y < l - m y > l whenx>S region 11:
-
v =- o
- a
ul
whenu>S
(8)
region 111: when x < S and u < S V = VI,, where the quantity S and 1 are defined in Figure 1. It is convenient to scale the energy units as follows: E * = (8p12/ h2)E V* = (8p12/h2)V (9)H. Eyring and M. Polanyi, 2.Phys. Chem., B12,279(1931);J. 0. Hirschfelder, Int. J. Quantum Chem., 111 S , 17 (1969).
(9)
and in region I1 m
na
qI1= nC= l T, exp(ik,u) sin -u1
(10)
where
kn2 + (na/l), = (2p/h2)(E -
V)
To relate the R,'s and T,'s to the incident flux, we have to know the solution in region 111. It is convenient to use polar coordinates in that region. Let the point (x,y) = (0,O) be the origin, and let the x axis be the reference line for the polar angle 4. The wave function in region I11 can then be expressed as
(6)
The three-body collision dynamics is then described by the solution of the Lagrangian equations generated either from eq 5 or from eq 6 . It is convenient to use the (x,y) coordinates prior to collision, and the (up)coordinates after the rearrangement. These coordinate systems are related by a linear transformation which causes the reaction path coordinates ( x and u) to intersect at an angle 6 which depends solely on the masses of the particlesg (Figure 1): tan2 8 = MB(MA + MB + Mc)MA-lMc-l (7)
+ n=l R , exp(ik,x) sin E1 y
= exp(-ik2x) sin -y 1
m
$111
=
C (AnFn+ BnGJ
nil
(11)
with
J',(R,4) = Jmn(aR) sin (mn4)
(12)
Gn(R,4) = Jm(n-l/Z)(aR)sin[m(n - f/2)41 + [Jrn(n-l/z)((~R~)/Jm(n+l/~)((~R~)l X
Jm(n+l/Z)(aR)sin [m(n + '/)@I (13)
+
where m = 2 ~ 1 8R, z = x 2 y2, 4 = tan-' ( y / x ) , R: = l2 + S2,(Y = [2p(E - V I , I ) / ~ ~and ] ' / J~k,is the Bessel function of order k. At the two interfaces where the different regions meet, the wave function and its first derivative must be continuous. These conditions determine the coefficients A,,, B,, R,, and T, uniquely. It is clear, from eq 10, that the probability, ITn*I2,for a rearrangement to take place with the new molecule vibrating in the nth state is (k,/kz)1T,12. Following the procedure of Tang, Kleinman, and Karplus,' we obtained our quantum probabilities of reaction. Classical Solution. The Lagrangian equation generated from eq 5 is identical with that of a particle of mass p moving on a two-dimensional plane under the influence of the potential V(x,y). Therefore, the interconversions of the kinetic and potential energies in the course of the three-body collision can be considered as the corresponding interconversion between the kinetic and potential energies of a single particle moving on a potential surface. Thus, we can trace the trajectory of the representative particle to determine whether the rearrangement has taken place. If the particle is initially in region I, but ends up in region 11, then the system has been rearranged. With the potential defined in eq 8, any possible motion of the system is a connected sequence of straight-line (10)K. T.Tang and P. B. Liebelt, Chem. Phys. Lett., 17,614(1972).
Quantum Effects of Vibrational Excitation
The Journal of Physical ChemMy, Vol. 86, No. 7, 7982
1.0
1109
7 t
r
L
1 .a c I
0
0
L k
-+
2
6
4
8
REDUCED TRANSLATIONAL ElvERGY
c
Figure 3. Reaction probabilitiesas functions of translation energy from quantum mechanical calculations with target molecules in the first vibrationally excited state.
Y
Flgure 2. Classical trajectories on the idealized potential surface.
segments reflected or refracted at the discontinuities of the potential (Figure 2). The angle of incidence y on the first potential interface is determined by the ratio of the x and y components of the momentum, which are the square roots of the translational and vibrational energies, respectively. We mign the same vibrational energy to the initial system in both the classical and quantum mechanical calculations. Thus the angle y is fixed by the total energy of the system. The particular point at which a representative particle strikes the interface (line PQ of Figure 2) corresponds to the phase of the relative motion of the three-body system. If the laws of refraction and reflection are correctly represented in Figure 2, then it is evident from the geometry that for a given total energy all trajectories striking the interface at points on the line PQ between P and M will be reflected back into region I, and those striking at points between M and Q will be transmitted to region 11. Thus the probability, P, of a rearrangement occurring is P = d / l , where d is the distance MQ and 1 is the channel width PQ. This process of following the trajectory through the reactive region has been greatly simplified by Jepsen and Hirschfelder8 with the principle of the kaleidoscope. Using their method, we obtained our classical result. Reaction Rate. Assuming a thermal distribution of relative translational energies (Et) we can obtain the rate constant K ( T ) from the reaction-probability P(E,)
K(T) = ( 2 7 r ~ ~ k T ) - ~ / ~exp(-E,/lzT) ~ ~ P ( E ~ dEt )
(14)
where k is Boltzmann's constant, T is the absolute temperature, and pa is the reduced mass of the A + BC system."
Results Reaction probabilities as a function of reduced translation energy (E,*= E* - E,*) from quantum mechanical calculations with target molecules in the first vibrational state are shown in Figure 3. In general, most of the product molecules are populated in the highest vibrational state allowed. If the energy is such that the second vibrational channel has not yet opened, then, except at very low energies, the adiabatic transition u, = 1to u5 = 1 (u, ~
~
~~
(11) M. A. Eliason and J. 0. Hirschfelder, J. Chem. Phys., 30, 1426 (1959).
' .a .Oi A n m
$
" 4
L Y
.4t 0
2
4
6
8
REDUCED TRANSLATIONAL ENERGY
Flgure 4. Comparison between quantum and classical reaction probabllities with target molecules in the first vibrationally excited state.
is the vibrational state of the target molecule and uB the vibrational state of the product molecule) is more strongly preferred than the nonadiabatic transition u, = 1to ug = 0. Comparisons between quantum and classical reaction probabilities with the initial molecule in the first excited state are shown in Figure 4. For the quantum result, we have summed all possible transitions. Since in classical mechanics the vibrational energies of the final system will be continuously distributed, we assume that the system in in the highest state possible whenever the vibrational energy of the rearranged system exceeds the threshold of that state; that is, if the energy is less than that required by the first excited state, the system is considered to vibrate in the ground state, if it is enough for the first excited state but less than the second excited state, the system is considered to be in the first excited state, etc. The energy that corresponds to the threshold of the first excited state of the product molecule is indicated by the single arrow and that of the second excited is indicated by the double arrow on the classical curve in the figure. Compared with Figure 3, we see that the final state distributions according to quantum mechanics and to classical mechanics are completely different. For example, the quantum probability predicts that most of the product moIecules will be in the first excited state for Et* 5 5, whereas the classical mechanics shows that they will be entirely in the ground state. The other prominent feature in Figure 4 is that the quantum mechanical "tunneling" is considerable. This is
1110
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982
Bowers and Tang
V
1.0 k
A
T
f
_..I__~-
CONSTANT K
y.
2
I
.4 r
I
‘,\
i
\
3
1
0
REDUCED
Flgure 5. Comparison between quantum and classlcal reaction probabllitles with target molecules in the ground state.
because the vibrational energy of the initial molecule is available for the system to overcome the activation barrier in quantum mechanics, but not in classical mechanics. Other than that, the overall agreement between quantum and classical results is reasonably good. This makes one to expect that, at very high temperatures, the classical and quantum rate constants will be similar to each other. The quantum and classical reaction probabilities with the initial molecules in the ground state were compared in the work of Kleinman and Tang.12 In Figure 5, we reproduce their results with VI,,* adjusted to 0.33. The quantum mechanical tunneling, which is not negligible even in this case, is, however, much smaller as compared with those shown in Figure 4,where the initial molecule is in the excited state. Again, it is seen that the zero-point energy of the initial ground-state molecule, which is incorporated into both the quantum and classical calculations, is available for overcoming the activation energy only in the quantum description. These reaction probabilities are used to calculate the reaction rates according to eq 14. The results are shown in Figure 6. The quantum rate constants Qo and the classical rate constants CL,with the initial molecule in the ground state are essentially the same as those reported by Tang and Liebelt.lo Here we compare them with the quantum Q1and the classical CL1 rate constants with the initial molecule in the first excited state. With one additional vibrational quanta in the initial molecule, the rate constants are enhanced (increased) in both classical and quantum results. However, the enhancement is much larger in the quantum mechanical results. For example, at 500 K the quantum enhancement is close to two orders of magnitude, while the classical enhancement is relatively small. It is interesting to note that the classical rate constants for both ground-state and excited-state initial molecule fall in straight lines in the Arrhenius plot. The curvatures in the quantum results are commonly attributed to the “tunneling” which are explicitly demonstrated in Figures 4 and 5. At high temperatures (greater than 3000 K) the classical rate constants for both cases approach the corresponding quantum rate constants as expected. Discussion Given the model, both the quantum and classical calculations are exact. There is no ambiguity in the results. The main conclusion of this study is that there is a large quantum effect in the reaction probability when the target (12) B. Kleinman and K. T. Tang, J. Chem. Phys., 51, 4587 (1969).
2
3
1 GOO/T
T R A ~ S L A T I O N A ENERGY ~
Figure 8. Quantum and classical reactlon rate constants of ground state and first excited state target molecules. The temperature Tis in K, and the rate constant K is in cm/(molecule s).
molecule is in the vibrational excited state. To draw inference from these results to the H + H2 reaction, one must ask the question as to what extent this simple model represents reality. The two most drastic simplifications of this model are the collinear restriction and the step function type potential surface. Ever since the inception of the modern approach to the chemical dynamics, one-dimensional calculations of reactive collisions have played very important roles in the theoretical development. Since the interaction potential is minimum when three hydrogen atoms are on a line in the region where the reaction takes place, it is not unreasonable to expect that the reaction probability obtained from the onedimensional calculation bears some resemblence to the real three-dimensional process. As for potential surfaces, there are some tests available. Diestler13 studied the collinear reaction with a “realistic” Wall-Porter-type potential. He used a close-coupling technique to obtain numerically exact reaction probability curves. He found a large number of features which are qualitatively similar to the results of Tang et al.,’ who used this simple model. This led him to conjecture that “there is regular correspondence between the structure of the probability curves for potential energy surfaces which have globally different character”. A similar conclusion was arrived by Truhlar and Kuppermann’* in their exhaustive study of the collinear H H2 reaction with an ab initio potential surface. Using a finite difference boundary value method, they obtained exact reaction probability curves. Upon comparing their results with that of the corresponding case of this simple model, they found that the qualitative agreements are “surprisingly good”. These similarities in reaction probabilities are also translated into reaction rates. Tang and Liebeltloaveraged the reaction probabilities from this simple model and found the activation energy from the Arrhenius plot to be 0.30 eV which is almost identical with the value 0.299 eV found by Truhlar and Kuppermann on a much more sophisticated surface. While all these tests are for the case where the target molecules are in the vibrational ground state, there is no reason to think that the conclusion will be different for the vibrationally excited target molecules. Clearly, the quantitative results of this simple model cannot be expected to represent reality, nevertheless the qualitative features may still have some validity in the real
+
(13) D. J. Diestler, J. Chem. Phys., 50, 4746 (1969). (14) D. G. Truhlar and A. Kuppermann, J. Chem. Phys., 56, 2232 (1972).
J. Phys. Chem. 1982, 86, 1111-1116
three-dimensional reactions.15 Indeed, the conclusions of this study are in good qualitative agreement with the recent three-dimensional approximate quantum calculation^.^ This study not only gives the added evidence that the large discrepancies between experiment and classical theory when vibrational excitation is involved are true quantum effects, it also shows that the reason for these discrepancies is the fact that the vibrational energy is much more effective in promoting the reaction in quantum mechanics, especially in the tunneling region. This large quantum effect might have been anticipated from the fact that the patterns of the quantum stream lines (15) For recent discuesions on the relationshipbetween one and three dimensions, see J. C. Sun,B. H. Choi, R. T. Poe, F d K. T. Tang, Chem. Phys. Lett., 82, 255 (1981),and references therein.
1111
involving excited states are fundamentally different from those of the ground state.16 Professor Hirschfelder stated in one of his papers16that he learned long ago that there is much to learn from a thorough study of simple systems. In honoring him at his seventieth birthday, we find that this simple system he devised 40 years ago still yields information of current interest. Since vibrational enhancements may have farreaching consequences in the practical question of population inversion, this simple model, like many of his other works, may yet speak to the future.
Acknowledgment. This research is supported by the National Science Foundation Grant CHE-7809808 and PRM-7921430, and by a grant from Research Corporation. (16) J. 0. Hirschfelder and K. T. Tang, J. Chem. Phys., 64,760 (1976).
Mass Polarization and Breit-Pauli Corrections for the Polarizability of 4He F. Welnhold Thewetlcal Chemistry Instnote and 08partment of Chemistry, Unlverslty of Wlsconsln-Madison, Madlson, Wisconsin 53706 (Recelved: July 7, 1981; In Flnal Form: July 28, 1981)
We have evaluated the leading relativistic and nuclear motion corrections for the static dipole polarizability a. of the 'He ground state, using accurate Hylleraas-type variational wave functions. The net correction is calculated to be 1.031 X au, even though the contributions of some of the individual terms are more than 20 times this size. With this correction the theoretical polarizability,including all known effects but the Lamb shift, stands now at C Y ~ ~ ~ 1%) ' ( ' H=~1.383199 au compared with the experimental value measured recently by Gugan and Michel, %@(4He 1%) = 1.383223 f 0.000067 au. The a priori theoretical value may be instrumental in establishing - an absolute temperature scale in the 4-21-K region, and in refining the value of the gas constant R.
Introduction The effects of special relativity and of nuclear motion are usually ignored in theoretical treatments of polarizabilities and related properties of light atoms and molecules. Indeed, the modification of Schrodinger's nonrelativistic equation to properly incorporate these effects leads to well-known formal difficulties (particularly when electromagnetic fields are present) that are the subject of active current research.l Recent advances in the precision of experimental measurements have impelled renewed theoretical interest in these small effects, which may lie within the limits of present or foreseeable experimental determination. In the present paper we have considered the leading relativistic and nuclear motion corrections to the static electric dipole polarizability a. of the 1%ground state of the helium atom, which represents a particularly favorable testing ground for precision comparisons of theory and experiment. The polarizability a. is the basic parameter describing the electronic distortion in the presence of an external electric field? and hence enters the interpretation of many types of experiments, including adsorption, dielectric (1) K.-H. Yang and J. 0. Hmhfelder, J. Chem. Phys., 72,5863 (1980); K.-H. Yang,J. 0. Hirschfelder, and B. R. Johnson, in press, and references therein. (2) For theoretical reviews, see A. Dalgamo, Adu. Phys., 11,281 (1962); R. R. Teachout and R. T. Pack, At. Data, 3, 195 (1971). 0022-3654182l2086-1111$01.25/0
constant, atomic molecular beam, optical interferometry, field emission microscopy, ion mobility, index of refraction, and Stark effect measurement^.^ However, the earlier experimental values of a. were subject to large uncertainties and disturbing mutual inconsistencies, which precluded comparisons with accurate theoretical value^.^ The experimental situation has recently been altered dramatically by the work of Gugan and Michel; who employed high-precision capacitance measurements in conjunction with techniques of low-temperature gas thermometry ("dielectric constant gas thermometry") to obtain a value of the 4He polarizability C X ~ ~ ~ P= ~1.383223 ( ~ H ~ f) 0.000067 au
(1)
with an experimental accuracy nearly two decades beyond that previously available? The quoted uncertainty is only ~
~~~~~~~
(3) For a review of experimentalmethods, see P. J. Leonard, At. Data Nucl. Data Tables, 14, 22 (1974). Earlier experimental values for the polarizability of helium are summarized in ref 8. (4) D.Gugan and G. W. Michel, Metrobgia, 16,149 (1980);Mol. Phys., 39,783 (1980). (5) This value wan obtained from the exp6rimental molar polarizability A, = 4/s~Nfi&0 = 0.517257 f O.oooO25 cm3/molwith the appropriate Bohr radius ao('He) = 0.5292496 X 10" cm for the 'He isotope. (Numerical conversion factors used in the present work are given in Appendix B.) Note that the apparent "discrepancy"between theory and experiment (ref 4) resulted from incorrect uae of the infinite-maasBohr radius a o ( m ) in this conversion factor.
0 1982 American Chemical Society
