J. Phys. Chem. 1986, 90. 3599-3603 centrifugal barrier when J 1: 1 # j because sin (x/2) = 0. Conversely sin (x/2) = 1 for J 1: 1 - j and the decreasing steps in j are accompanied by simultaneous decreases in 1 that allow progressively more rapid tunneling. These patterns offer some encouragement for an eventual interpretation of a t least the gross features of the experimental spectrum.
6. Discussion An efficient scheme for the rapid estimation of the positions and lifetimes of even very narrow multichannel resonances has been outlined. As a measure of the time involved, the current four-channel version requires 10-15 s per resonance on a Hewlett-Packard desk-top computer, and extension to more channels is merely a matter of generalizing eq 22 and 26. The scheme is in principle exact, subject to the restriction of a short-range coupling potential, apart from the use of the JWKB connection formulas (eq 5 and 6 ) which are known to be extremely reliable. The major approximation, as implemented, is the use of a unitarized distorted-wave internal S matrix, but this could even be replaced by a close-coupled equivalent because the method would still aid the location of very narrow resonances, which would be computationally prohibitive by an energy scan using closem. The use of asymptotic Percicoupled integration to R Val-Seaton coefficients is a second inessential approximation, but
-
3599
it was shown by test calculations to be accurate to well within 10% and the same is probably true of the distorted-wave approximation. The results quoted in secton 5 are therefore believed to be accurate to within an order of magnitude for both level shifts and widths. The main physical conclusions in relation to H3+ are as follows: (1) Line widths in the experimentally observed range' can be accounted for only if the predissociation is inhibited by several centrifugal barriers. The number of such long-lived ( 10-7-10-9 s) non RRKM states is, however, only a fraction of the total, and lifetimes of the order of 10 ps, as expected for unimolecular decomposition, appear to be more typical. (2) High orbital angular momenta required to produce barriers of the requisite magnitude give rise to rotational energy separations approaching the energy of a COz laser photon, and the system H+/Hz has an appropriate dipole moment to couple with the electromagnetic field. (3) Lifetimes of the calculated long-lived states show a systematic trend according to the magnitude of the resultant J arising from the relative orientations of the orbital and rotational angular momenta. There is a good prospect that these preliminary observations will ultimately lead to an interpretation of at least the gross features of the observed spectrum. Registry No. H3+, 28132-48-1.
An Improved Bessel Uniform Approximation of Noninteger Order for Use in Semiclassical Collision Theory: Application to Vibrationally Inelastic Molecular Collisions J. N. L. Connor* and W. J. E. Southallt Department of Chemistry, University of Manchester, Manchester M13 9PL, England (Received: January 21, 1986)
A new choice for the order of the semiclassical noninteger Bessel uniform approximation is proposed and used to calculate transition probabilities for collinear atom-oscillator collisions. Calculations have been performed for two collision systems: a Morse oscillator interacting via an exponential potential and a harmonic oscillator interacting via a Lennard-Jones potential. Results are reported for both classically allowed and classically forbidden transitions. The new noninteger Bessel approximation remains accurate in situations where earlier uniform approximations break down.
I. Introduction Semiclassical techniques are widely used in the theory of molecular collisions and chemical reactions. A semiclassical approach provides an excellent physical understanding of the scattering, as well as often being very accurate. The most successful and complete application of semiclassical methods so far has been to the elastic scattering of atoms.' Semiclassical theories for inelastic and reactive collisions were developed about 15 years ago by Marcus,2 and independently by Miller.3 These dynamically exact theories construct an approximate solution to the Schrodinger equation, using real or complex valued solutions of Hamilton's classical equations (for reviews, see ref 4-6). There are also many dynamically approximate theories that employ semiclassical techniques.' The semiclassical theory of inelastic molecular collisions has been most highly developed and tested for the canonical problems of vibrationally inelastic scattering in collinear atom-molecule In particular, various uniform approximations for the S matrix involving Airy, Bessel, and Laguerre functions have been derived and shown to be accurate under certain conditions. 'Present address: IBM(UK) Ltd., 126 Washway Road, Sale, Cheshire M33 lDB, England.
0022-3654/86/2090-3599$01.50/0
The uniform Airy formula2' is an asymptotic approximation for an oscillating integral with two coalescing real or complex (1) Connor, J. N. L. In Semiclassical Methods in Molecular Scattering and Spectroscopy, Proceedings of the NATO Advanced Study Institute held in Cambridge, England, Sept, 1979, Child, M. S., Ed.; Reidel: Dordrecht, The Netherlands, 1980; pp 45-107. (2) Marcus, R. A. Chem. Phys. Lett. 1970, 7, 525. J . Chem. Phys. 1971, 54, 3965. (3) Miller, W. H. J . Chem. Phys. 1970, 53, 1949. (4) Miller, W. H. Adu. Chem. Phys. 1974, 25, 69. 1975, 30, 77. (5) Connor, J. N. L. Chem. SOC.Reu. 1976, 5 , 125. (6) Child, M. S. Adu. Atom. Mol. Phys. 1978, 14, 225. In Ref 1, pp 155-177. (7) Clark, A. P.; Dickinson, A. S.;Richards, D. Adu. Chem. Phys. 1977, 36, 63. Dickinson, A. S.; Richards, D. Adu. Atom. Mol. Phys. 1982, 18, 165. Billing, G. D. Comput. Phys. Rep. 1984, I , 237. (8) Connor, J. N. L. Annu. Rep. Chem. Soc., Sect. A : Phys. Inorg. Chem. 1973, 70, 5. (9) Miller, W. H. J . Chem. Phys. 1970,53,3578. Chem. Phys. Lett. 1970, 7,431. J . Chem. Phys. 1971, 54, 5386. (10) Wong, W. H.; Marcus, R. A. J . Chem. Phys. 1971, 55, 5663. (11) Stine, J.; Marcus, R. A. Chem. Phys. Lett. 1972, 15, 536. (12) Miller, W. H.; George, T. F. J . Chem. Phys. 1972,56,5668. George, T. F.; Franchino, H. D. Phys. Rev. A 1973, 8, 180. Lin, Y-W.; George, T. F. J . Chem. Phys. 1974,61, 5396. (13) Stine, J. R.; Marcus, R. A. J . Chem. Phys. 1973, 59, 5145. (14) Eastes, W.; Doll, J. D. J . Chem. Phys. 1974, 60, 297.
0 1986 American Chemical Society
3600 The Joiirnal of Physical Chemistry. Vol. 90, No. 16, 1986 stationary phase (saddle) points. Physically, the two stationary phase points correspond to trajectories in which the oscillator makes a transition from an initial state n = 0, 1, 2, ... to a final state m = 0, 1, 2. .... However, for a near elastic n n transition, the uniform Airy transition probability becomes inaccurate. To deal with this situation, Stine and MarcusI3 introduced a uniform Bessel approximation of integer order. Although this formula is reliable for elastic transitions at low collision energies, it becomes inaccurate as the collision energy increase^;^^^*^.*^ and for classically forbidden elastic transitions it cannot be applied at all.20.24I n a similar way, the uniform Laguerre approximation" also loses accuracy.3" To overcome the breakdown of the integer Bessel approximation at high collision energies, one of the authors and M a ~ n derived e~~ a uniform Bessel approximation of noninteger order. The order of the Bessel functions was chosen by a physical argument.25using information contained in the final-action initial-angle plot. However, a defect of the results reported in ref 25 is that transition es of greater than unity can occur for elastic transitions at low collision energies, and microscopic reversibility is not always satisfied for inelastic transitions, although the deviations are small. We can summarize the situation described above by saying that none of the uniform semiclassical approximations currently in use are reliable for all values of the collision energy. The purpose of the present paper is to report results for a new choice of the order of the noninteger Bessel approximation, one which removes the defects mentioned above. We apply the new formula to two atom-oscillator collision systems: a Morse oscillator with an exponential interaction (the MOEXP system) and a harmonic oscillator with a Lennard-Jones interaction (the HOLJ system). We have used these collision systems for the following reasons: (i) Extensive exact quantum mechanical transition probab are available for comparison. In particular, Clark and Dickin(see also ref 33) have performed quantum calculations for the MOEXP system, while Gutschick et al.34have reported results for the HOLJ case. (ii) Uniform semiclassical calculations have previously been carried out for the same systems.'S,'R-2','' (iii) Other approximation^^^^^^ for vibrational energy transfer can be compared with the semiclassical and quantum transition
-
( 1 5 ) Connor. J. N. L. Mol. Phvs. 1974. 28. 1569. (16) Sutherlky, T. A,; Wakeham, W . A. J . Phys. B 1975,8, 2538. Mol.
Phys. 1976, 32. 579. (17) Fraser, S. J.: Gottdiener, L.; Murrell, J. N.Mol. Phys. 1975,29, 415, 1585. Gottdiener. L. [bid. 1975. 29. 1585: J . Chem. Phvs. 1975, 63. 2433. (18) Duff, J. W.; Truhlar, D. G. Chem. Phys. 1975: 9, 243. J . Chem. Phys. 1975, 63, 4418. Chem. Phys. 1976, 17. 249. (19) Schinke. R.; Toennies. J. P. J Chem. Phys. 1975, 62, 4871. (20) Connor, J. N. L.; Mayne, H. R. Mol. Phys. 1976, 32, 1123. (21) Child, M. S.: Hunt, P. M. Mol. Phys. 1977, 34, 261. (22) Rusinek, I.; Roberts, R. E. J . C h e m Phys. 1976, 65, 872. 1978, 68, 1147. (23) Child, M. S. Mol. Phys. 1978, 35, 759. 1979, 38, 1023. Hunt, P. M.; Child, M. S. J . Phys. Chem. 1982, 86, 1116. (24) Connor, J. N. L.; Mavne, H . R. Mol. Phys. 1979, 37, 1 . In eq 3.9 for 3/2(Imf)'/3 read (3/21mf)2:i, (25) Connor, J. N. L.; Mayne, H. R. Mol. Phys. 1979, 37. 15. (26) Collins, M. A,; Gilbert, R. G. Mol. Phys. 1977, 34, 1407. Augustin, S. D. J . Chem. Phys. 1982. 77, 3953. 1983. 78, 206. Herman, M. F.: Currier, R. Mol. Phys. 1985, 56, 525. (27) Ya Aksenov, S.; Potapov, V . S. Teor. Mat. Fiz. 1984,59,400 [English translation: Theo. Math. Phys. 1984, 59, 5811; Khim. Fiz. 1985, 4, 356. (28) Connor, J . N. L.; Marcus, R. A. J . Chem. Phys. 1971, 55, 5636. (29) Pechukas, P.; Child, M. S. Mol. Phys. 1976, 31, 973. (30) Connor, J. N. L.; Lagan& A., unpublished calculations. (31) Clark, A. P.; Dickinson. A. S. J . Phys. B 1973, 6, 164. (32) Clark, A. P.; Dickinson, A. S., private communication. (33) Scherzinger, A. L.; Secrest, D. J . Chem. Phys. 1980, 73, 1706. (34) Gutschick, V. P.: McKoy. V.; Diestler. D. J. J. Chem. Phys. 1970, 52, 4807. (35) Skodje, R. T.; Truhlar, D. G. J . Chem. Phys. 1984,80, 3123. Skodje, R. T. Chem. Phys. Lett. 1984, 109, 227. (36) DePristo, A. E.; Augustin, S . D.: Ramaswamy, R.: Rabitz, H. J . Chem. Phys. 1979, 71, 850. DePristo, A. E.; Ramaswamy, R. Chem. Phys. 1981,57. 129. Ramaswamy, R. Pramana 1981,16, 139. DePristo, A. E. J . Phys. Chem. 1982,86, 1334. Vasan, V. S.; Cross, R. J. J . Chem. Phys. 1982, 77,4507. Ree, T.; Kim. Y . H.: Shin, H. K . Chem. Phys. Letr. 1983,103. 149.
Connor and Southall probabilities. There are many other approximate theories of collinear atom oscillator scattering in the l i t e r a t ~ r e . ~ .When ~ , ~ ~ assessing the utility and accuracy of these collinear theories, it is helpful to keep the following criteria in mind: (i) The approximate theory should be applied to collisions involving both harmonic and anharmonic oscillators. This is because some theories are much more accurate for harmonic than for anharmonic oscillators. (ii) Transition probabilities should be calculated for many different initial and final states, not just from a single initial state, e.g. the ground state. (iii) The theory should be capable of extension into three dimensions. The semiclassical calculations reported below satisfy all these criteria. This paper is arranged as follows. In section I1 we describe the noninteger Bessel uniform approximation. We discuss the choice of the order of the Bessel functions in section 111. The classical Hamiltonians for the MOEXP and HOLJ collision systems are given in section IV. We describe and discuss our results in section V. Conclusions are in section VI. Our research is based on the important work of Marcus on the semiclassical theory of molecular We are therefore very pleased to contribute this paper to the R. A. Marcus Commemorative Issue of The Journal of Physical Chemistry. 11. Bessel Uniform Approximation In this section, we give the explicit form of the semiclassical Bessel uniform approximationz4for the transition probability of m. Since the order of the Bessel a vibrational transition n functions is in general nonintegral, we will denote the resulting semiclassical transition probability by P;Fn (where NIB noninteger Bessel). It is convenient to consider separately classically allowed and classically forbidden transitions. For a classically allowed transition, P;Yn is given by24
-
PEE,, = 1 / 2 ~ [ ( p I ' / * + p21/2)2({2- u ~ ) ' / ~ J ~ ~+( { )
-
p2'/')*({' - u~)-'/*{~J,'~(()] (1)
In eq 1, Jv(() is a regular Bessel function of argument l a n d order u , while .Iv'({) denotes the derivative dJ,(()/d{. In addition, p1 and p2 are classical transition probabilities p, = Idfi/dwolwo=w,o-l i = 1, 2
(2)
where e(&) is the "quantum-number variable" for a collision with initial quantum number n = 0, 1, 2, ..., and wo is the initial angle variable of the oscillator. The subscripts i = 1, 2 label the two coalescing real stationary phase trajectories. The argument l of the Bessel functions is calculated from ((2
-
U*)1/2
- u cos-] ( u / { ) =
7214 -
A,l
(3)
where A(?) is the classical phase A ( t ) = - 2 ~ S ' w ( t ) i ( t ) dt - l r R ( t )PR(t) dt 10
+ !12r (4)
' 0
In this equation, R is the separation distance of the atom from the oscillator and pRis the conjugate momentum (see section IV for the classical Hamiltonians). The integration in eq 4 is along a classical trajectory from the initial time to to the final time t at the end of the collision. For a classically forbidden transition, the expression for PEE,, is24 (37) Gentry, W.R. In Atom-Molecule Collision Theory. A Guide for the Experimentalist, Bernstein, R. B., Ed.; Plenum: New York, 1979; Chapter 12. (38) Marcus, R. A. J . Chem. Phys. 1972,56,311,3548; 1972,57,4903. Faraday Discuss. Chem. SOC.1973,55, 34. J . Chem. Phys. 1973,59,5135. (39) Fitz, D. E.; Marcus, R. A. J. Chem. Phys. 1973, 59, 4380. Kreek, H.; Marcus, R. A. Ibid. 1974, 61, 3308. Kreek, H.; Ellis, R. L.; Marcus, R. A. Ibid. 1974, 61, 4540. 1975, 62, 913. Fitz, D. E.; Marcus, R. A. Ibid. 1975, 62, 3788. Liu, W-K.; Marcus, R. A. Ibid. 1975, 63, 272, 290. Turfa, A. F.; Fitz, D. E.; Marcus, R. A. Ibid. 1977, 67, 4463. Turfa, A. F.; Liu, W-K.; Marcus, R. A. Ibid. 1977, 67, 4468.
The Journal of Physical Chemistry, Vol. 90, No. 16, 1986 3601
Vibrationally Inelastic Molecular Collisions
P:!& = xp[(l
+ sin p ) ( v 2 - [2)'/2~,2({) + (1 - sin p) x (v2 - [2)-'/2{2J,'2([)]( 5 )
where the amplitude p and phase p are defined by
and the argument of the Bessel functions is found from v cosh-' (v/{) - ( v 2 - l2)'l2= IIm All
(7)
So that eq 1-7 can be applied, it is also necessary to define v, the order of the Bessel functions. This is considered in the next section. 111. Choice of v
For the integer Bessel approximation, the order is given byI3 v =k
) m - nl
(8)
As mentioned in the Introduction, the integer Bessel approximation becomes inaccurate for elastic transitions (k 0) as the energy increase^^^^^^^^^ and is inapplicable for classically forbidden elastic transition^.^^,^^ Transition probabilities calculated from the integer Bessel formula will be denoted by PLYnin the following. The rwninteger Bessel approximation was applied in ref 25 with the order v determined by v = Im - Aavl
(9)
where fiav
= 1/z(fimax+ Amm)
(10)
-
with LX and fimin the maximum and minimum values of the A(@) plot, respectively. For the elastic 0 0 transition in the HOLJ system, v changesZSfrom v = 0 at low collision energies to v = 1.3 at high energies, where the transition is classically forbidden. This increase in v with energy avoids the breakdown that occurs for the integer Bessel approximation. In particular for v >, 0.5, P g agrees with the uniform Airy result to within 5%.2s The reason for the increase in v is that, as a function of energy, fi,,, first decreases to -0.5 and then increases, crossing the m = 0 line when the transition becomes classically forbidden (see the figure in ref 20). However, the prescription (9) has two w e a k n e s ~ e s :(i) ~ ~at low energies, the elastic transition probabilities can be greater than unity, which implies that v is not tending to zero sufficiently fast as the energy decreases, (ii) microscopic reversibility (Le., the is not rigorously satisfied for inelastic property that PEEn = collisions. To overcome these difficulties, we propose a new choice for v in this paper, namely v = Im - nl
+ I(m - mav)(n- fiav)l
-
(11)
where mavis the same as in eq 10, but with the oscillator initially n transition, eq 11 in state m rather than n. For an elastic n simplifies to v = (n -
(12) It is evident that microscopic reversibility is rigorously obeyed for eq 11 since it is symmetric in n and m. Also for an elastic transition, eq 12 shows that v will approach zero faster as the collision energy decreases than will eq 9. Transition probabilities calculated from the asymmetric formula 9 will be denoted Pz!!;, while those computed from the symmetric formula 1 1 will be written Pf;l!!i. We have also investigated some other choices for v; these will be briefly discussed in section V. fia,)2
IV. Collinear Atom-Oscillator Hamiltonian The Hamiltonian for the MOEXP collision system (in reduced units) is's931 H = (2m)-'pRZ
+ €(A) + exp[-a(R = E
- q)]
(13)
-
TABLE I: Comparison of Quantum Results with Semiclassical Bessel Approximations, for the 0 0 Transition of the HOW Collision System' NIBS p pNlBa INT p E n m Pw-n Pm-n/ m-n m-xIPm-n Pm-nJ m-n Classically Allowed Transitions 1.55 1.65 1.85 2.05 2.25 2.45 2.55 2.65 2.85 3.05 3.45 3.80 4.00 4.20 4.50 4.60
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4.70 4.80 5.20
0 0 0
------------
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.9999 0.9992 0.9946 0.9835 0.964 0.936 0.918 0.898 0.852 0.799 0.674 0.555 6 0.420 6 6
1.001 1.005 1.009 1.016 1.026 1.040 1.049 1.060 1.042 0.997 0.933 0.910 [0.444]' 0.912 [0.302Ic [0.277Ic
1.045 1.055 1.030 1.016 1.043 1.064 1.073 1.082 1.068 1.031 0.975 0.944 [0.455]' 0.929 [0.303]' [0.277]'
1.0008 1.003 1.009 1.016 1.025 1.038 1.047 1.056 1.035 0.979 0.859 0.741 [0.318Ic 0.538 [0.0912]c [0.0457Ic
Classically Forbidden Transitions 0 0 0
6 0.245 0.157
[0.259]' 0.947 1.045
[0.259Ic 0.943 1.032
d d d
"The quantum transition probabilities are from Table I1 of GutSchick et al. (ref 34), who label the oscillator states n = 1, 2, 3, ..., rather than n = 0, 1, 2, .... *Exact quantum result not available. 'Values in square brackets are absolute values, not ratios. dThe integer Bessel approximation is not applicable to classically forbidden elastic transitions. where E is the total energy and the Morse oscillator energies are given by t(fi)
=A
+ '/z - ( 4 D ) 4 ( f i + 7*)2
(14)
and q = (2D)'/2(1n [ I
+ (t/D)'/* cos (2nw)l - In [ I - (C/O)]) (15)
In our calculations we used the same reduced parameters as in the exact quantum calculation^,^'^^^ namely (m,a,D) = (0.667, 0.314, 9.3) which represents approximately He colliding with H,. The Hamiltonian for the HOLJ system isZ0334
-.
r
H = (2m)-'pR2+ fi
+ y2 + 4e =E
(16)
with q = (2fi
+ I)'/*
sin (2nw)
(17) As before,20~2s~34 the parameters (m, e, 0)are given the values (0.5, 0.005707, 46.71) which approximately represents an H2 molecule colliding with another H2 molecule which has its vibrational degree of freedom frozen. The calculation of the classical probabilities and phases for the MOEXP and HOLJ system were carried out in the same way as in our earlier work.~s~Zo~2s
V. Results and Discussion In this section, we report our results obtained from the noninteger Bessel approximation with the order v determined by eq 11. We pay particular attention to elastic n n transitions as it is for these transitions that Pi!: and PrLF can become inaccurate. Table I reports 0 0 semiclassical transition probabilities for the HOLJ system with the total energy E in the range 1.55 to 5.20. Note that for E Z 4.65 the 0 0 transition is classically forbidden. In order to see the trends more clearly, the semiclassical results have been expressed as ratios of the exact quantum transition pr~babilities~~ (which are denoted P-J. An exception is for energies close to the classically allowed/forbidden boundary
-
-
-
3602 The Journal of Physical Chemistry, Vol. 90, No. 16, 1986 TABLE 11: Same as Table I, Except for Nonground-State Elastic Transitions of the HOW Collision Svstem"
E 1.65 2.05 2.45 2.55 2.85 3.45 4.20 4.80 5.20 5.80 6.20 3.05 3.80 4.20 4.80 5.20 5.80 6.20 3.80 4.20 4.80 5.80 6.20 4.80 5.80 6.20 5.80 6.20
n
I 1 1
I I 1 1
I 1 1 1 2 2 2 2
2 2 2 3 3 3 3 3 4 4 4 5 5
---+
------------4
m 1
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4
5 5
Pmc, 0.9992 0.9835 0.936 0.918 0.843
0.580 0.179 0.0104 0.0222 0.175 0.299 0.969 0.755 0.536 0.177 0.03 13 0.0387 0.134 0.989 0.921 0.676 0.0805 0.00364 0.986 0.612 0.338 0.9835 0.882
pNlBs
1Pm-n 0.989 0.999 1.013 1.015 1.018 1.017 1.067 1.337 0.734 1.069 1.ooo 0.999 1.029 1.017 1.011 1.032 0.948 0.970 1.004 1.013 1.034 0.981 0.984
m-n
1.005 1.035 1.004 1.015 1.037
"All transitions are classically allowed
NlBa
Pm-nlpm-n 1.007 1.176 1.123 1.105 1.062 1.024 1.061 1.308 0.757 1.063
1.000 1.201 1.060 1.022 1.OD0 1.006 0.964 0.978 1.143 1.173 1.049 0.976 0.980 1.157 1.043 1.003 1.173 1.187
INT
p
Pm-nl m-n 0.989 0.996 0.999 1.000 1.004 1.014 1.067 1.337 0.725 1.086 1.003 0.993 I .003 1.007 1.023 1.064 0.938 0.970 1.004 0.992 1.002 1.022
1.025 1.004 1.004 1.008 1.015 0.992
Connor and Southall TABLE 111 Comparison of Quantum Results with Semiclassical Bessel Approximations for Elastic Transitions of the MOEXP Collision System" E n-m P,, pm-n/f'm-n NIBS Pm-nIPm-n NlBa Pm-nIPm-n INT
---
2 2 3
0
3 3 4 4 4 4 5 5 5
1 1 2 2 0-0 1 1 2 2 3 3 0- 0 1 1 2 -2 3 3 4 -4
5 5 5 6 6 6 6 6 6 6 8 8 8 8 8 8 8
1
0 1
0 -0
---
-
5 5 0-0
--
1 1 2 2 3 3 4-4 5 5 6-6 0-0 1 I 2 2 3-3 4-4 5 5 6 -6
-
--
+
0.9998 0.9998 0.993 0.991 0.999 0.968 0.946 0.972 0.995 0.924 0.843 0.862 0.925 0.980 0.999 0.864 0.695 0.663 0.716 0.818 0.925 0.987 0.715 0.364 0.222 0.173 0.175 0.216 0.300
1.000 1.000 1.003 1.001 1.000 1.010 1.001 1.001 1.000 1.025 1.004 1.001 1.002 1.001 1.000 1.045 1.009 1.002 1.000 1.002 1.005 1.001 1.098 1.019 1.005 0.988 0.977 0.98 1 0.990
1.010 1.047 1.035 1.066 1.054 1.062 1.013 1.054 1.090 1.079 1.009 1.015 1.045 1.083
1.050 1.088 1.017 1.005 1.011 1.027 1.064 1.116 1.115 1.019 1.000 0.988 0.977 0.977 0.987
1.000
1.000 1.002 1.001 1.000 1.008 1 .oo 1 1.000 0.999 1.018 1.004 1.000 1.000 0.999 1.000 1.032 1.007 1.002 0.999 1.000 1.000 0.999 1.078 1.019 1.005 0.988 0.977 0.981 0.990
(namely, for E = 4.0, 4.5,4.6,4.7) where no exact quantum results "The quantum results are from Clark and Dickinson (ref 32) of which values for E = 8 are 18 state calculations. All transitions are are available. classically allowed. We do not report results using the uniform Airy approximation, as they have been extensively discussed in previous work.'5.20.25 TABLE IV: Same as Table 111 Except for Inelastic Transitions of the Also, we shall not consider the primitive semiclassical approxiMOEXP Collision SystemQ mation or the classical semiclassical approximation for the E n m P,,,-,, PE!$/P,-, P~!!~/Pmcn Pm-n/Pm-n INT transition probability, since these approximations are generally unreliable.- Note that the classically allowed (forbidden) results Classically Allowed Transitions in Tables I-V are estimated to be accurate to f l ( f 2 ) in the third 8 0 - 1 0.247 0.976 0.951 [0.980] 0.972 significant figure. 8 1-2 0.321 0.997 0.991 [1.000] 0.997 8 2-3 0.339 1.000 1.000 [1.003] 1.000 Table I clearly demonstrates the breakdown in PA!!$ for E 3 8 3-4 0.341 1.000 1.000 [1.000] 1.000 3.80, an energy range where both PNTBa and PNIBsremain accurate. For E C 3.05, we have P F z = Po, % 6 P 2 y w i t h P F z rising 8 4-5 0.335 1.003 1.003 [1.003] 1.003 8 5-6 0.319 1.006 1.006 [l.OOO] 1.003 above unity at very low collision energies. This also occurs for the nonground-state elastic collisions of the HOLJ system reported Classically Forbidden Transitions in Table 11, all of which are classicall allowed. For exam le, 2 0 1 0.246(-3)b 0.955 0.955 [0.955] 0.955 3 0 1 0.731(-2) 0.959 ?.!? = 1.16 whereas & ?!. = 0.980 and P{$ = 0.958 [0.958] 0.959 at E = 2.05, P 4 0 1 0.315(-1) 0.959 0.956 [0.956] 0.959 0.983. An interesting feature in Table I1 is that some very small 0.984 0.984 [0.980] 0.984 4 I 3 0.307(-4) transition probabilities arise from a destructive interference in the 5 0 1 0.739(-1) 0.959 0.952 [0.961] 0.959 semiclassical calculations. For these cases, the relative error can 5 2 5 0.365(-7) 0.819 0.819 [0.819] 0.819 be large, although the absolute error is not more than f0.006. 6 0 - 1 0.129 0.961 0.953 [0.969] 0.961 Elastic transition probabilities for the MOEXP system are given 6 0 2 0.665(-2) 0.950 0.947 [0.950] 0.950 in Table 111. Again PY.!: is accurate whereas P;!: can arise 6 I 3 0.841(-2) 0.993 0.992 [0.992] 0.993 above unity. The breakdown in PA% that occurs for the HOLJ 6 3 5 0.597(-1) 1002 1.007 [0.995] 1.000 system is not evident in Table I11 because even at the highest 8 0 2 0.353(-I) 0.958 0.949 [0.958] 0.958 8 1 4 0.561(-2) 0.988 0.986 [0.988] 0.988 energy of E = 8.0, Po,o has only dropped to 0.715. Semiclassical inelastic transition probabilities for the MOEXP "The quantum results are from Table 1 of Clark and Dickinson (ref and HOLJ systems are reported in Tables IV and V, respectively. 31), except for E = 8, which are 18 state calculations from ref 32. Results for both classically allowed and classically forbidden Values in square brackets are for P ~ ~ l P , , - , .6Numbers in partransitions have been calculated. Since the noninteger Bessel entheses are powers of ten by which the entry must be multiplied. approximation using the prescription (9) does not satisfy microand PNIBa Tables IV and V. A much more extensive set of inelastic uniform scopic reversibility, transition probabilities for both PNIBa iG Airy results can be found in ref 15, 20, and 2 5 . are given. It can be seen that PZ!!; = PLYn=;?:P = P,, ,,,.r This result is expected on general grounds, because it has been All the calculations for PZ?; discussed above have used eq 11 shown that, for u large, the noninteger Bessel approximation to define u. We have also investigated more general formulae, becomes equivalent to the uniform Airy a p p r o x i m a t i ~ n , ' ~ , ~ ~in~ particular ~~ which is known to be accurate for inelastic collisions. In practice, (18) Y = )m - nJ + I(m - ?$?,")(?I - fi,")IP a value of Y k 0.6 seems to be s ~ f f i c i e n t .For ~ ~ this reason, only a few inelastic semiclassical transition probabilities are shown in with @ = 1, 3 / 2 , 2 , 5/2. For inelastic collisions, the noninteger Bessel transition probabilities are almost independent of the value of w. However, the elastic transition probabilities are more sen(40) Connor, J. N . L. Chem. Phvs. Lerr. 1974, 25, 611
-
-----
-+
-+
-+
J. Phys. Chem. 1986, 90, 3603-3606
TABLE V Same as Tables I and I1 Except for Inelastic Transitions of the H O U Collision System” E n-m .,P P!~YP,,,P~;/P,.P!,Y,,/P,, Classically Allowed Transitions 4.2 4.8 5.2 5.8 6.2
1 2 0 1 0
3.8 4.2 5.8 5.8 6.2
2 2 2 0 1
-----
2 3 2 3 3
0.313 0.238 0.300 0.258 0.183
0.997 1.000 0.967 0.996 0.945
3 3 4 5 4
0.982(-2)* 0.698(-1) 0.878(-1) 0.350(-5) 0.644(-1)
0.997 1.004 0.940 0.992 0.923
tO.9781 [0.971] tO.9601 [0.988] tO.9401
0.994 0.992 0.963 0.996 0.945
Classically Forbidden Transitions 0.986 0.984 0.993 0.914 0.984
1.013 1.004 0.994 0.903 0.977
[0.985] [0.974] [0.983] [0.914] tO.9771
0.985 0.981 0.991 0.914 0.983
’Values in square brackets are for Pf;jlSmP,,-,,,. bNumbers in parentheses are powers of ten by which the entry must be multiplied.
-
sitive, with the p > 1 0 0 transitions in the HOLJ system being a little less accurate at high energies than those which have p = 1. Note that for an elastic collision, p = is the same as Overall, the simplest and most convenient choice for the parameter p seems to be p = 1. We have also investigated different ways of calculating A, and fiav.The prescription (10) uses the values of A,,, and Amin and can be considered as a two point average of the information contained in the A ( @ ) curve. Often the A(@) curve is generated on a grid of points equally spaced in Awo. Then another way of averaging the information in the A(wo) curve is
Pzy.
N
A,, =
E
h=O
A(hAwo)/(N + 1)
(19)
3603
where N = l/A@. For example, if A d = 0.1 (or 0.05) we have an 11 (or 21) point average. We used an 11 and 21 point average in some of our calculations, but the results were generally similar to those obtained from the two point average (10).
VI. Conclusions The uniform semiclassical formulae currently in use for collinear vibrationally inelastic scattering are not reliable for all values of the collision energy. We have shown in this paper how a new choice for the order of the noninteger Bessel approximation removes the defects encountered previously. In particular, the new formula is accurate over a wide range of collision energy for 0 0 transitions in the HOLJ system and microscopic reversibility is obeyed for inelastic transitions. The order v has been chosen on physical grounds by using information contained in the A(wo) plot. Although reasonable, our prescription for v should nevertheless be regarded as an empirical one. It would be interesting to see if our choice can be justified on more rigorous grounds, or if better prescriptions for v can be found. Finally, we emphasize that the present treatment can be extended to more realistic collisions, involving additional degrees of freedom, for which a semiclassical S matrix element is represented by a multidimensional integral. This is achieved by using a generalization of the Morse lemma for the mapping of the exponent of the multidimensional integral and has been described in section 4 of ref 24.
-
Acknowledgment. Support of this research by the Science and Engineering Research Council is gratefully acknowledged. The numerical calculations were carried out on the CDC 7600 computers of the University of Manchester Regional Computer Centre.
“Quantum Integrable” Systems Break the Noncrossing Rule Philip Pechukas Department of Chemistry, Columbia University, New York, New York 10027 (Received: January 29, 1986; In Final Form: April 1 1 , 1986)
“Primitive” semiclassical theory predicts that classically integrable systems violate the quantum noncrossing rule. The theory ignores tunneling, which is the mechanism by which such systems avoid crossing; Uzer, Noid, and Marcus have recently shown how to calculate tunneling splittings at avoided crossings. Actually, the splittings vanish for a subclass of integrable systems, those that are “quantum integrable”. These systems behave just as “primitive” semiclassical theory predicts: they break the noncrossing rule.
Introduction “Primitive” semiclassical theory of bound states often predicts an energy spectrum that is simpler-more degenerate-than the true spectrum, because the theory misses splittings of energy levels due to tunneling.’ The standard example is one-dimensional motion in a symmetric double well; according to semiclassical theory, the levels are doubly degenerate, while in fact they are split by barrier penetration. Another example: the noncrossing rule is regularly broken in semiclassical theory. Vary a parameter in the Hamiltonian and the semiclassical energy levels cross with impunity, because the classical objects associated with these levels are distinct tori in phase space having no points of contact-no interaction-with one another. It is the nonclassical “tails” of the wave functions, extending beyond the classical tori, that overlap and interact and enforce the noncrossing rule.
Still, the “primitive” theory is almost right. In the semiclassical regime of small h , tunneling splittings are very small compared to the typical spacings between energy levels: a tunneling splitting is expected to be exponentially small with h, O(exp[-()/h]], while the typical spacing between levels is O(hN),where N i s the number of degrees of freedom. Therefore, avoided crossings are expected to be isolated events, involving only two levels and occurring over a range of parameters that is small--O(exp[-( )/ h])-with respect to the @(tiN)range between successive “collisions” of a given level with its neighbors. To calculate these small level splittings in the vicinity of an avoided crossing requires a “uniform” improvement on “primitive” semiclassical theory, and Marcus has shown how to do it.* The (2) Uzer, T.; Noid, D. W.; Marcus, R. A. J . Chem. Phys. 1983, 79,4412.
See also: Wilkinson, M. Proceedings of the Second International Conference (1) For an extended discussion see: Pechukas,P. J. Chem. Phys. 1983,78,
3999.
0022-3654/86/2090-3603$01.50/0
on Quantum Chaos, to appear, for a general theory of tunneling splittings; Hose, G.; Taylor, H. S.; Richards, D. J . Phys. B 1985, 18, 51, for a discussion of avoided crossings in the quadratic Zeeman problem.
0 1986 American Chemical Society
