786
J. Phys. Chem. 1981, 85, 786-791
Vibrational Energy Transfer and an Improved Information-Theoretic Moment Method. Comparison of the Accuracy of Several Methods for Determining State-to-State Transition Probabilities from Quasiclassical Trajectories Donald G. Truhlar,” Brian P. Reid,+ Dale E. Zurawskl,’ and Jon1 C. Gray Depatfment of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received: October 22, 1980)
We have applied several methods, both old and new, to predict state-to-state vibrational transition probabilities from quasiclassicaltrajectory calculations. The methods are applied to 106 cases where accurate quantal resulta are available and in each case the rms error in the probabilities is computed. The average rms error for the standard histogram method is 0.055. Quadratic smooth sampling, a new method introduced here, reduces this to 0.051; and improved histogram method I1 of Bowman and Leasure reduces it further to 0.049. Even better accuracy can be obtained by using information theoretic moment methods. Minimizing the information-theory entropy subject to the constraint that the quantized final-state distribution has the same first two moments of final vibrational quantum number variable, i.e., ( n2)and (n:), as calculated from the trajectory end conditions gives an average rms error of 0.036. Continuing to add moments until no feasible solution exists raises this to 0.045, but using two moments when the initial vibrational quantum number nl is zero and three or four moments when it is not reduces this to 0.033. Using the moments ( n2- nl) and ( (nz- n J 2 )yields an average rms error of 0.031. Finally, using these two moments when nl = 0 and augmenting them with ((n2- n J 3 ) or ( (nz - n1)3)and ( (n2- nJ4) when nl # 0 reduces the average rrns error to 0.028.
I. Introduction Classical trajectory calculations are the most practical method for simulating molecular collision processes except possibly for some atom-diatom collisions involving light atoms at low energy where accurate quantal calculations may be feasible.’“ Quasiclassical trajectory calculations, in which the internal states of molecular colliders are assigned quantized energies or action variables as initial conditions and the classical equations of motion are integrated with no further quantization, are particularly popular. At the end of such trajectories, the molecular action variables are no longer quantized and the question arises: How can one best extract state-to-state transition probabilities from the end conditions of the trajectories? This question arises for both rotational and vibrational transitions. Rotational transitions are often classically allowed and histogram (bin) techniques have been reasonably successfuk2 The present contribution is concerned explicitly only with vibrational transitions, for which histogram methods have generally been less successful.~Q We consider four systemsl“-ll of collinear atom-diatom collisions (He-H2 with harmonic and Morse oscillator potentials and 2He+ Br2and H + Hz with Morse oscillator potentials) with many total energies (in the range 2 h . q to 25.08hwJ and several initial states (with vibrational quantum numbers 0-5) for a total of 106 cases. For each case we have computed the state-to-state transition probabilities from quasiclassical t r a j e c t ~ r i e sby ~ ?several ~ methods, and we compare the results to accurate quantalQJOresults. The methods are judged by computing root-mean-square deviations of the quasiclassicaltransition probabilities from the quantal ones. We examine several categories of methods for extracting the state-to-state transition probabilities from the trajectory end conditions, namely, the histogram method,12 improved histogram methods,13smooth sampling method^,'^'^-^^ and informa‘Land0 Undergraduate Research Fellow, summer 1980. Marietta College, Marietta, OH 45750. National Science Foundation Undergraduate Research Participant, summer, 1977. Materials Research Center, Coxe Laboratory 32, Bethlehem, PA 18015.
*
0022-3654/81/2085-0786$01.25/0
tion theoretic moment methods.21-28 Sections I1 and I11 give information about the systems studied and the methods used. The results are presented and discussed in section IV. 11. Systems The first three systems we consider are model systems (1) R. N. Porter and L. M. Raff in “Dynamicsof Molecular Collisions”, Part B, W. H. Miller, Ed., Plenum, New York, 1976, p 1. (2) M. D. Pattengill in “Atom-Molecule Collision Theory”, R. B. Bernstein, Ed., Plenum, New York, 1979, p 359. (3) D. G. Truhlar and J. T. Muckerman in “Atom-MoleculeCollision Theory”, R. B. Bernstein, Ed., Plenum, New York, 1979, p 505. (4) J. N. L. Connor, Computer Phys. Commun., 17, 117 (1979). (5) G. C. Schatz in “Potential Energy Surfaces and Dynamics of Calculations”,D. G. Truhlar, Ed., Plenum, New York, p 287; D. A. Micha, ibid.,p 685. (6) J. W. Duff and D. G. Truhlar, Chem. Phys., 9,243 (1975). (7) D. G. Truhlar, Znt. J . Quantum Chem. Symp., 10, 239 (1976). (8) J. W. Duff and D. G. Truhlar, Chem. Phys., 17, 249 (1976). (9) J. C. Gray, G. A. Fraser, D. G. Truhlar, and K. C. Kulander, J. Chem. Phys., 73, 5726 (1980). (10) A. P. Clark and A. S. Dickinson, J. Phvs. B,6, 164 (1973). (11) K. C. Kulander, J. Chem. Phys.; 69, 5064 (1978). (12) The original applications are M. Karplus and L. M. Raff, J. Chem. Phys., 41, 1267 (1965);K. G. Anlauf, J. C. Polanyi, W. H. Wong, and K. B. Woodall, ibid., 49,5189 (1968). References to later work are given in ref 7. See also ref 1-6. (13) J. M. Bowman and S. C. Leasure, J. Chem. Phys., 66,1756 (1977); 67,4784(E) (1977). (14) R. G. Gordon, J. Chem. Phys., 44, 3083 (1966). (15) R. G. Gordon and R. P. McGinnis, J. Chem. Phys., 55, 4898 (1971). (16) P. Brumer, Chem. Phys. Lett., 28, 345 (1974). (17) N. C. Blais and D. G. Truhlar, J. Chem. Phys., 66,5335 (1976). (18) N. C. Blais and D. G. Truhlar in “State-to-State Chemistry”, P. R. Brooks and E. F. Hayes, Ed., American Chemical Society, Washington, DC, 1977, p 243. (19) J. W. Duff, N. C. Blais, and D. G. Truhlar, J. Chem. Phys., 71, 4304 (1979). (20) J. N. L. Connor and A. Lagan& Computer Phys. Commun., 17, 145 (1979). (21) J. W. Duff and D. G. Truhlar, Chem. Phys. Lett., 36,551 (1975). (22) H. Kaplan, R. D. Levine, and J. M a , Mol. Phys., 31,1765 (1976). (23) U. Halavee and R. D. Levine, Chem. Phys. Lett., 46, 35 (1977). (24) R. J. Gordon, J. Chem. Phys., 65,4945 (1976). (25) R. J. Gordon, J. Chem. Phys., 67, 5923 (1977). (26) R. J. Gordon, J. Chem. Phys., 71, 4720 (1979). (27) S. Chapman and S. Green, J. Chem. Phys., 67, 2317 (1977). (28) See also I. Procaccia and R. D. Levine, J. Chem. Phys., 63,4261 (1975); 64, 808 (1976).
0 1981 American Chemical Society
The Journal of Physical Chemistry, Vol. 85, No. 7, 198 1 787
State-to-State Vibrational Energy Transfer Probabilities
parametrized by Clark and Dickinson;lo they have also been studied by later w o r k e r ~ . ~ The - ~ ~Hamiltonians ~~ consist of a harmonic or a Morse oscillator for the diatomic and an exponential nearest-neighbor repulsion between the atom and the diatom. We study the He H2 systems1° with both harmonic and Morse oscillator potentials and the 2He Brz systemlowith a Morse oscillator potential. Here 2He is a structureless particle with a mass like H2. For the two Morse oscillator systems, we consider only total energies below the dissociation energy. The fourth system is a model for H + H2with the LEPS interaction potential of Ku1ander.l'~~~ This system has also been restudied in later workagIt treats H2 as a Morse oscillator. This system allows for reactive collisions, and we do consider energies above the reaction threshold and even above the dissociation energy of the Morse oscillators. In such cases we renormalized the state-to-state inelastic transition probabilities Pllnzwhere nl and n2are initial and final vibrational quantum numbers, respectively. The total probability for other types of events is PT1 = Pfl, +P$, (1) where is the probability of dissociation, and Elis the probability of reaction. The renormalized state-to-state transition probabilities for the inelastic events are then defined by9
+
+
el
TABLE I: Test Casesa system
E*
n1
He t H, (HO)
0
1-2 3-5 He + H, (MO) 0 1-5 "e t Br, 0-1 2-3 4 5 H H, 0
+
1
4
2,3,4, 5,6, 8 6. 8 5; 6, 8 2, 3 , 4 , 6 6. 8 2; 5, 6, 1 0 5, 6, 1 0 6,lO 6 6.82, 7.34, 8.27, 8.72, 9.18, 10.10, 11.02, 11.94, 12.61, 12.85, 13.77, 14.27, 14.69, 15.61, 16.52, 16.98, 17.44, 17.91, 18.36, 19.28,' 20.21,' 21.12,b 22.03,b 22.53b 10.10, 11.02, 11.94, 12.85, 14.27, 14.69, 16.52, 16.98, 17.85,' 19.28' 20.21,' 21.12,' 22.03,' 22.53b 11.94, 12.85, 14.27, 14.69, 15.61,b 16.52,' 16.98,' 17.44,' 17.91,' 18.36,' 19.28,' 20.21,'21.12,c 22.03 ' 22.53,' 23.09,' 23.89,' 25.Oid
a Unless indicated otherwise the quasiclassical Pzl = 0 and the quantal Pzl < 0.045, where Pzl is defined in eq 1. For these cases the quasiclassical Pr1= 0 but the quantal PT, > 0.045. For these cases the quasiclassical P z > 0.1 but the quasiclassical P,", = 0. For this case the quasiclassical Pg = 0.50 and the quasiclassical P,",= 0.02.
'
so that
CPftrn2= 1
(3)
nz
In the rest of the article all inelastic probabilities are normalized as in (2) and (3) so we omit the superscript N. At most of the initial relative translational energies and vibrational states involved in this study the probability of reaction and dissociation in a quasiclassical trajectory are both zero. The 106 test cases considered here are listed in Table I. We use the notation that the reduced total energy is E* = E/hwe (4) where E is the total energy and we is the vibrational frequency. 111. Methods
The trajectory program31 and the trajectory calculat i o n ~ are ~ - described ~ elsewhere. At the beginning of each trajectory the diatomic molecule has its energy fixed at Evib = (nl + '/z)hwe- (nl + f / 2 ) 2 h W e X e (5) where w p x ,is the Morse anharmonicity constant. For each case (initial vibrational quantum number nl and total energy E),we calculated 36-120 trajectories, evenly spaced in initial vibrational phase ~ h i f t ~ql. ' , ~For ~ each trajectory we calculated the final vibrational radial action variable33 Ji(ql).From this we compute the more convenient final vibrational quantum number ~ a r i a b l e ~by l%~ nz(q1) = [J,'(q,)/hI - Y2 (6) Next we wish to use the results of the trajectories to ap(29) J. N. L. Connor, Mol. Phys., 28, 1569 (1974). (30) For a discussion of the Kulander potential see also the footnote in J. C. Gray, G. A. Fraser, and D. G. Truhlar, Chem. Phys. Lett., 68,359 (1979). (31) J. W. Duff and D. G. Truhlar, Chem. Phys., 4, 1 (1974). (32) W. H. Miller, J. Chem. Phys., 53, 3578 (1970). (33) H. Goldstein,"Classical Mechanics",Addison-Wesley, Reading, MA, 1950, p 291. Goldstein calls the action variable Ji.
proximate the quantum mechanical probability Pnln2(E) that an inelastic collision, having begun in state nl,ends in state n2. The only property of the trajectories that we use for this purpose is the function n2(q1). For some methods we require the full distribution of calculated values of n2(q1). For other methods we require only certain integrals over this distribution. We call these integrals moments, and we introduce the notation, for any function f[n2(q1)]of n2(q1)
where S(ql) is unity if the trajectory is an elastic or inelastic collision and zero if it is dissociative or reactive. As an example ( (n2- n1)2)denotes the mean-square deviation of nz(ql) from the initial vibrational quantum number. Most moments were computed by simply averaging over the trajectories actually computed. To ensure that we had enough trajectories to well converge the moment, we checked its value against the value computed using only every second trajectory. In the cases with the largest deviations we recomputed the moment by a more careful procedure, Le., we interpolated f[n2(ql)] and integrated it accurately by repeated Gauss-Legendre i n t e g r a t i ~ n . ~ A. Histogram Method. In the histogram method, nz(ql) is simply rounded to the nearest integer. To increase our precision for a given number of trajectories, we did not simply bin the trajectories and compute the fraction in each bin; rather we used quadratic interpolation of the trajectory function n2(q1)to estimate the limits of the q1 intervals that correspond to each bin. The resulting probabilities are called PFln B. Improved Histogram hiethod. Bowman and Leasure13suggested two "improved histogram" methods. For both methods the first step is to compute the histogram probabilities pf;Inz as in section A and to compute the first moment ( n 2 ) .i n improved histogram method I, the next step is to readjust the probabilities to obtain a set p'yizof
.
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Truhlar et al.
The Journal of Physical Chemlstty, Vol. 85,No. 7, 198 1
probabilities that have the least-squares minimum deviation from P:n, consistent with the constraint
( n 2 )= Cn2PL7!2
(8)
02
TABLE 11: Ordered Lists of Moments name of list
momentsa
.
123 ( n J , (n,?, (n,3), (n24), (n,5), , . (n,),( n z z ) ,(nZ4),( n Z 6 )(n,*), , . .. 124 213 (A2),( A ) , ( A 3 ) , (A4),( A 5 ) , . . . 214 (A’), (A), (A4), (A6),( A 8 ) , . . . a A 5 n,-n See example of (A 2, explained in second paragraph of section 111.
and the constraint that the probabilities sum to unity. Improved histogram method I sometimes leads to negative probabilities. To avoid these we here introduce the renormalized improved histo ram method I. In the new method we first compute B then we set any negative biased distribution (in an information theoretic ~ e n s e ~ ~ B ~ ) values equal to zero, and we renormalize the remaining that is consistent with values to unity. The results are called Bowman and Leasure introduced a second method to avoid negative CPnlnz = 1 (17) nz probabilities; they called it the improved histogram method 11. This method is like method I except the least-squares and with N moment constraints of the form step is performed only on nonzero values of Pf;tnz and it (fi) = Cfi(n2)Pnln, i = 1, 2, ..., N (18) involves the sum of the squares of the relative deviations na rather than the absolute deviations. In this method, if ElnZ Two decisions are required here: (i) what to use for N , and = 0, then PkHiz= 0, and if > 0, then P!,yiz > 0. (ii) what to use for the functions fi. We will consider these C. Smooth Sampling Methods. It is useful to restate in turn. the histogram method more formally. In the histogram (i) There are at least two philosophies for choosing N . method each trajectory is assigned a set of probability One23is to increase N until the results are independent functions P:: of increasing it further. A second philo~ophy,39~*~~ the one P3n,(q1)1 = 1 n2 = we prefer, is to try to develop a scheme, based on the accuracy of the quasiclassical moments (fi), for choosing =0 n2 # n&) (9) N so that the resulting probabilities agree as well as poswhere n&) is the nearest integer to n2(q1).Then sible with the accurate quantal results. One of the main objectives of the present study is to systematically test p;nz = (PF,) (10) various such schemes. Our philosophy here is that, because Using this notation we can define the smooth sampling the quasiclassical moments have inaccuracies (Le., they do methods very concisely. The original smooth sampling not agree perfectly with the accurate quantal moments), m e t h ~ d ~ Jwill ” ~be called linear smooth sampling (LSS). the results may sometimes get worse as one adds additional In this method the probability functions are34 quasiclassical moments as constraints. One wants to know how many moments to use to get the best possible prePZ;,SS[n2(q1)I= 1 - ln2(q1)- n,(ql)l if n2 = n&) dictions of the quantal results from the trajectories. = 1 - In&) - ncc(ql)l if n2 = ncc(41) (ii) The second decision is closely related to the first. = 0 otherwise Again we want to know which choice of quasiclassical (11) moments allows us to predict the most accurate state-tostate transition probabilities from the results of the trawhere n,&) is the second closest integer to n2(Q1) and jectories. where we observe the exception We will examine several schemes. Each scheme consists P!fs[n2(ql)l= Jon2 if n2(q1)< 0 (12) of choosing a particular ordered list of moments and a number of N of moments from the ordered list. The four The final probabilities are ordered lists considered in this article are explained in Table 11. A scheme is identified by giving the name of Pfil2 = (P31,SS) (13) the ordered list and either N or a rule for determining N . In the present work we introduce a variant of this meFor example scheme 123:6 consists of using the first six thod; we call it quadratic smooth sampling (QSS). In this moments from ordered list 123. For a more complicated method we have example, scheme 213:(4 - 2Jn10)means using the first four moments from ordered list 213 when nl is not zero and E f S [ n d q l ) l= 1- In2(~1) - nC(q1)l2if n2 = n,(qJ using the first two moments from this list when nl is zero. = Ind~1) - nC(q1)l2if n2 = ncc(ql) Sometimes two or more possible schemes yield identical = 0 otherwise results. For example, it is obvious from Table I1 that (14) schemes 213:2 and 214:2 are identical. Following up on a suggestion made earlier,7 we tested except that many schemes in which the number N of moments to be E:S[n2(~1)1 = 6OnZ if nZ(a1) < 0 (15) used is an increasing function of the number of states for which P:nz # 0. This latter number is called H. As an The final probabilities are example, scheme 214:min (H,2) consists of using the first two moments from ordered list 214 when H I 2 and using E$: = ( E 3 (16) only the first moment ( N = 1) from this list when H = 1. D. Information Theoretic Moment Methods (Classical The determination of the least-biased distribution conP Matrix Theory). In the information theoretic moment sistent with a set of moments requires, at least implicitly, the probabilities are assigned as the least-
e::’.
en,
(34) Here we point out that eq 10 of ref 7 is misprinted;r and (1 - x ) should be interchanged. The idea is that the probability is apportioned to the two nearest integers in inverse proportion to their distance from nz(Q1).
(35) E.T.Jaynes in “Statistical Physics”, Vol. 3, “1962 Brandeis Institute Lectures in Theoretical Physics”, W. A. Benjamin, New York, 1963, p 81. (36) R. Baierlein, “Atoms and Information Theory”, Freeman, San Francisco, 1971, Chapter 3.
State-to-State Vibrational Energy Transfer Probabilities
that one specify a prior expectation or a set of a priori probabilities. As b e f ~ r e , we ~ ~use , ~ the prior expectation that all final states are equally probable. Other possible choices are discussed el~ewhere.~ One stumbling block in applying information theoretic analyses in the past has been the difficulty of actually solving the constrained minimization problem. In our own past we accomplished this by solving N simultaneous transcendental equations for N unknown Lagrange multipliers using iterative techniques for solving nonlinear sets of equations. One difficulty with this method is that sometimes the iterative process does not converge; then one does not know whether (i) no solution exists or (ii) a solution exists but it has not been found. Alhassid, Agmon, and L e ~ i n e re-formulated ~~b~ the computational problem as the minimization of a concave function F and developed necessary and sufficient inequalities which must be satisfied for the existence of a feasible solution, Le., a solution nonnegative. Given that a feasible of eq 15 with all PnInz solution exists, the iterative minimization of F is theoretically guaranteed to converge to a unique solution because of the concavity of F. Agmon et developed a computer program LP incorporating these procedures. We have used this program, with minor modifications and a new driver program, for the present work, and it is a pleasure to report that we found their procedures to be very convenient and completely satisfactory in all respects. The check for the existence of a feasible solution allows us to apply any “scheme”, as defined above, to any case. To do this we require one more rule. If no feasible solution exists with the desired N , decrease N successivelyuntil a feasible solution exists. For example, when we say that we have applied scheme 213:6 to all 106 cases we mean we actually used six moments whenever a feasible solution exists for six moments, but for other cases we used the maximum number of moments for which a feasible solution exists. Thus scheme 213:6 differs from scheme 2135 only for those cases where a feasible solution does exist for six moments. Stated another way, 213:6 is really a short notation for 213:min (M,6) where M is the maximum value of N for which a feasible solution exists in a given case. For the 106 test cases considered here, the highest value of M we encountered was 12.
IV. Results For each of the 106 cases, we applied many methods to predict the transition probabilities. For each method applied to each case, we computed the rms deviation of the approximate transition probabilities from the accurate quantal ones for all the energeticallyallowed states. Then we took a simple average of these rms errors over all the cases. The average rms errors for a large number of information theoretic schemes are presented in Table 111. In Table IV we compare the average rms errors of several of the simpler and the more successful information theoretic schemes to the average rms errors of the other methods. V. Discussion A . Histogrammic Methods. Table IV shows that the average rms error for the histogram method is 0.055. This is improved only slightly by the IH1 and RIHl procedures, but improved histogram method I1 does reduce the average rms error to 0.049. Examination of the various columns (37) Y. Alhassid, N. Agmon, and R. D. Levine, Chern. Phys. Lett., 53, 22 (1978). (38) N. Agmon, Y. Alhassid, and R. D. Levine, J.Cornput. Phys., 30, 250 (1979).
The Journal of Physical Chemistty, Vol. 85, No. 7, 1981 789
TABLE 111: Average rms Errors of Several Information Theoretic Schemes Applied to All 106 Cases
N
123:N
124:N
213:N
214:N
1 2 3 4 5 6 3 - 6nla 4 - 26nlo max(H- 2,2) max( H - 6,2) max(H- 11,2) min(H, 2) min(H, 3 - 6nl,,) min(H + 1,3 - 6 n l o ) min(H, 4 - 2€jn1,) min[(H t 2)1’2, ( 4 - 26n,o)I M best possibleb
0.084 0.036 0.036 0.039 0.041 0.042 0.033a 0.033 0.039 0.036 0.035 0.039 0.036 0.033’ 0.036 0.037
0.084 0.036 0.039 0.041 0.042 0.042 0.033’ 0.034 0.037 0.035 0.035 0.039 0.036 0.033a 0.037 0.037
0.039 0.031 0.031 0.033 0.034 0.035 0.0278 0.0280 0.034 0.032 0.030 0.031 0.0279 0.0278 0.0277a 0.029
0.039 0.031 0.036 0.041 0.041 0.041 0.030‘ 0.034 0.034 0.036 0.030 0.031 0.030 0.030a 0.034 0.039
0.045 0.030
0.042 0.030
0.039 0.0249
0.042 0.0259
a Best method in column. b This row is for cheaters. It gives the average rms error if one uses for each case the value of N that yields the most accurate probabilities.
of Table IV shows that the increased accuracy of improved histogram method I1 is quite uniform as a function of nl and E*. Thus we must judge this method as an unequivocably successful improvement over the conventional histogram method. To our knowledge the only previous tests of the improved histogram methods were the two cases in the original paper.13 B. Smooth Sampling. The motivation of various workers for using the smooth sampling method has presumably been that it gives faster convergence of Monte Carlo trajectory calculations. However, it also causes the calculation to converge to different answers. In the present study all results are well converged with respect to sampling density, and we are concerned with the question of the accuracy of the converged results with respect to quantum mechanics. Table IV shows that the usual smooth sampling method, LSS, is significantly worse than the histogram method for nl = 0 for E* 5 10, but better for nl = 0 for E* > 10, and also slightly better for nl # 0. Therefore, there is not much difference overall (LSS actually has a slightly higher average rms error, 0.0551 vs. 0.0546). In contrast, Table IV shows that the new smooth sampling method proposed here, QSS, is almost always better than both conventional histogramming and LSS. It reduces the average rms error to 0.0514. QSS does not do as well, 0.0492, as improved histogram method 11. This is primarily because it does much worse for nl = 0 and only slightly better for nl # 0. C. Information Theoretic Moment Methods. Next we consider the information theory methods. The procedure for all moment calculationsin the literature so far has been to use the ordered list 123. There were two considerations that made US reexamine this starting point. First of all we compared the moments obtained from the quasiclassical trajectories to the accurate quantum mechanical moments for all 106 cases. The first two moments, (n,) and ( n2,),are generally reasonably accurate, but the accuracy of the third moment, (nZ3),is generally poor. For example, it is often the wrong order of magnitude or it may have the wrong sign. The fourth moment, in conjunction with the second, carries information about whether the distribution is highly peaked or flat or even bimodal. I t is certainly conceivable that quasiclassical trajectories would carry more information about this property than
790
The Journal of Physical Chemistry, Vol. 85, No. 7, 1981
TABLE IV: Average rms Errors (Times
Truhlar et al.
l o z )for Several Methods Applied to Various Groups of Cases
n, E* no. of cases
0
55
0 >10 19
histogram improved histogram I renormalized IH1 improved histogram I1 linear smooth sampling quadratic smooth sampling
4.0 3.8 3.9 1.6 6.0 3.8
8.6 8.6 8.5 8.4 8.6 7.8
6.9 6.8 6.8 6.0 7.7 6.4
4.3 4.2 4.2 2.1 5.7 4.2
7.3 7.3 7.3 7.3 7.1 6.6
6.2 6.2 6.2 5.4 6.6 5.7
123:l 123:2 123:3 123:4 123:5
2.0 1.4 1.4 1.4 1.4
20.4 5.2 5.3 5.3 5.3
13.7 3.8 3.9 3.9 3.9
2.4 1.5 1.6 1.6 1.6
17.0 5.3 5.2 5.4 5.8
123:(3 - 6 n , 0 ) b 123:(4 - 2Sn10) 123:min(H- 1,4 - 26,1,)c 123:min(H,4 - 26,,,,) 124:3 124:max(H- 2,2) 124:min(H t 1,3 - t i n l o )
1.4 1.4 1.4 1.4a 1.4 1.4 1.4
5.3 5.3 6.7 6.7 5.0 5.2 5.0
3.9 3.9 4.8 4.8 3.7 3.8 3.7
1.5 1.5 1.5 1.5a 1.6 1.5 1.5
213:l 213:2 213:3 213:4 213:5 213:6
2.1 1.8 1.8 1.8 1.8 1.8
2.7 2.2" 2.3 2.3 2.3 2.3
2.5 2.1a 2.1 2.1 2.1 2.1
213:max(H- 2,2) 213:(3 - 6 n 1 0 ) 213:(4 - 2Fjnl0) 213:min(H- 1 , 4 - 2ti,I,)c 213:min(H, 4 - 26,10) 213:M 214:3 214:max(H- 2,2)
1.8 1.8 1.8 2.0 1.8 1.8 1.8 1.8
2. 2a 2.3 2.3 2.3 2.3 2.3 2.3 2.2"
2.1a 2.1 2.1 2.2 2.1 2.1 2.1 2.1"
Best method in column.
1-5 ~ 6 . 5 46.5 12 21
all 0 1-5 ~ 6 . 5 ~ 1 0410 33 20 35
5.1 5.1 5.1 4.4 4.6 4.8
1-4 >10 32 4.4 4.4 4.4 4.4 4.3 4.3
all >10 51 4.7 4.7 4.7 4.4 4.4 4.5
0 all 39 4.7 4.6 4.6 3.2 5.1 4.5
1-5 all 67 5.9 5.9 5.9 5.9 5.7 5.5
11.7 3.9 3.9 4.0 4.3
4.1 l.la 2.7 4.0 4.0
5.4 4.6 3.8 3.6 3.8
4.9 3.3 3.4 3.7 3.9
3.2 1.3 2.1 2.8 2.7
11.4 5.0 4.5 4.5 4.9
8.4 3.6 3.6 3.9 4.1
5.2 5.4 5.9 6.2 5.0 4.8 5.0
3.8 4.0 4.3 4.5 3.8 3.6 3.7
l.la 1.1" 1.2 l.la 4.0 3.1 1.1"
3.8 3.6 3.6" 3.6 4.0 4.2 4.0
2.8 2.7" 2.7 2.7" 4.0 3.8 2.9
1.3 1.3 1.4 1.3a 2.8 2.3 1.3
4.5 4.5 4.7 4.9 4.5 4.5 4.5
3.3 3.3 3.5 3.6 3.9 3.7 3.3
2.6 1.7 1.8 1.8 1.8 1.8
4.0 3.5 3.3 3.6 3.7 3.7
3.5 2.9 2.8 3.0 3.0 3.0
2.6 1.1" 2.7 4.0 4.0 4.2
5.3 4.6 3.8 3.6 3.8 4.1
4.3 3.3 3.4 3.7 3.9 4.1
2.6 1.4 2.3 2.9 2.9 3.0
4.6 4.0 3.6 3.6 3.8 3.9
3.9 3.1 3.1 3.3 3.4 3.5
1.7 1.7 1.7 2.1 1.7
3.1" 3.3 3.6 3.3 3.5 3.7 3.9 3.3
2.6' 2.7 2.9 2.9 2.9 3.0 3.2 2.7
3.0 l.la 1.1" 1.4 1.1" 4.2 4.0 3.1
5.0 3.8 3.6 3.6a 3.6 5.2 4.0 4.7
4.3 2.8 2.7 2.8 2.7" 4.8 4.0 4.1
2.3 1.4 1.4
4.0 3.6 3.6 3.4a 3.6 4.9 4.0 4.0
3.4 2.78 2.80 2.82 2.77" 3.9 3.6 3.4
1.8 1.9 1.7
410
1.8 1.4 3.0 2.9 2.4
all all 106 5.5 5.4 5.4 4.9 5.5 5.1
min(H t 1,3 - t i n l o ) gives same results as ( 3 - 6 n l o ) . c If H - 1is 0, set N = 1.
about the skewedness of the final-state distribution. Hence we wished to try using (n,4)without using ( n 2 3 ) .One reasonable way to do this is simply to delete all odd moments except the first, leading to ordered list 124. Table I11 shows, however, that scheme 124:3 does not do as well as scheme 123:3. With N = 2 the two ordered lists are identical, and they lead to an average rms error of 0.0362. Adding (n?) yields scheme 123:3 with an average rms error of 0.0365. In comparison, scheme 124:3 has an average rms error of 0.0386. Adding both ( nZ3)and ( n t )yields scheme 123:4 with an even worse average rms error, 0.0389. Table IV shows that the largest difference between the methods occurs for nl = 0 at E* > 10. For those cases N = 2 provides far better accuracy than any of the choices with N I 3. For nl # 0 and E* > 10, where N = 3 or 4 is significantly better than N = 2, schemes 123:3, 1243, and 123:4 have average rms errors 0.0383,0.0397, and 0.0359, respectively. Thus ( nZ3)is still more helpful than ( nZ4), and using both is best of all. Trying higher moments for these cases (nl # 0 and E* > 10) does not improve the accuracy, e.g., schemes 1244 and 123:5 lead to average rms errors of 0.0408 and 0.0385, respectively. Combining the above observations, plus the fact that scheme 123:2 is uniformly better than scheme 123:1, leads one to a very simple combination of the methods, namely, to use scheme 123:2 for nl = 0 and scheme 123:4 or scheme 123:3 for nl # 0. These combinations are called schemes 123:(4- 26 ) and 123:(3 - 6nlo), and they yield average rms errors for cases of 0.0335 and 0.0333, respectively. One can actually do slightly better with 124:(3 - 6,10), which yields 0.0332.
8
all
TABLE V: Approximate and Quantal Transition Probabilities for He + H, (HO) with n, = 3, E* = 5 n2
0 1 2 3 4 rms error
accurate
123:l
0.0002 0.0054 0.0770 0.915 0.0028
0.0574 0.0950 0.157 0.260 0.430 0.355
213:l 3x 3x 0.0377 0.925 0.0377 0.0 24
The three schemes just mentioned all have a serious deficiency though for cases with nl # 0. In many such cases ( n z )= nl and M = 1. Then scheme 123:N or 124:N for any N uses only the first moment ( n z )as explained at the end of section 111; but the least-biased distribution consistent with ( nz)is a very poor approximation to the true quantal distribution of final states. A typical case is illustrated in Table V. It is clear that the trajectories are indicating a very elastic situation but that this information is not contained in ( nz).Using (n?) instead offers no great improvement. Information about the width of the finalstate distribution can be obtained from the pair of values ( n z )and (n;), which contain information equivalent to (nz)and ((nz- (nZ)P),but, as already mentioned, one typically finds in such cases that no feasible solution exists ) used. Using ((nz.-nJ2) when both ( n 2 )and ( n Z 2are provides a way to incorporate the width information in a single moment, and this is the motivation for scheme 213. Table N shows that schemes based on the ordered list 213 provide a dramatic improvement over schemes based on the ordered list 123 for n, # 0 and E* 5 10. A t higher
The Journal of Physical Chemlstry, Vol. 85, No, 7, 7981 791
State-to-State Vibrational Energy Transfer Probabilities TABLE VI: Average rms Errors (Times lo2)for Several Methods Applied to Various Groups of Cases n1 0 0 0 1-5 1-5 1-4 E* 0-6.5 6.5-15 15-23 0-6.5 6.5-15 15-26 no. ofcases 12 16 11 21 24 22 5.3 8.6 5.4 4.0 histogram 4.0 4.7 6.4 8.4 5.5 4.0 improved 1.6 3.0 histogram I1 4.7 4.7 8.6 4.9 3.8 linear 6.0 smooth sampling quadratic 3.8 4.6 5.1 7.8 5.0 3.9 smooth sampling 123:l 123:2 123:3 123:4 123:M
2.0 1.4 1.4 1.4 1.4
2.7 1.1 1.2 1.2 1.2
5.3 1.5 4.3 6.5 6.8
20.4 5.2 5.3 5.3 5.3
10.0 5.2 4.6 4.8 5.9
4.5 4.5 3.7 3.6 5.2
124:3 124:4 124:M
1.4 1.4 1.4
2.5 2.5 2.5
4.6 6.3 6.4
5.0 5.0 5.0
4.7 5.0 5.2
3.8 3.9 4.0
213:l 213:2 213 :3 213:4 213:M
2.1 1.8 1.8 1.8 1.8
2.5 1.1 1.2 1.2 1.2
3.3 1.5 4.3 6.5 6.8
2.7 2.2 2.3 2,3 2.3
5.6 5.2 4.6 4.8 5.5
5.4 4.5 3.7 3.6 5.2
214:3 214:4 214:M
1.8 1.8 1.8
2.5 2.5 2.5
4.6 6.3 6.4
2.3 2.3 2.3
5.5 6.9 6.9
3.9 4.1 4.6
energies, one does not encounter the nearly elastic situations that cause large errors for scheme 123. Using scheme 213:(4 - 26, o) instead of 123:(4 - 26,,& lowers the average rms error hom 0.0335 to 0.0280. With the ordered list 213, one can achieve slightly better accuracy by using the value of H in the prescription for N . Thus schemes 213:min(H + 1,3 - anlo) and 213:min(H,4 2an10)yield 0.0278 and 0.0277, respectively. However, it may be inconvenient to use the value of H in the prescription for more complicated problems. Examination of Table IV shows that scheme 213:(4 2anP) is always close to the optimum choice. Its biggest deficiency occurs for nl # 0 for E* I10. Here one could obtain better accuracy by using N = 3 than N = 4. But for nl # 0 for E* > 10, N = 4 is more accurate than N = 3. Thus we recommend schemes 213:(3 - 6, o) at low and intermediate energy and scheme 21 3:(4 - 26,d at very high energy. Table I11 shows that schemes 213:N are preferred over schemes 214:N for any N except 1 or 2, for which the two schemes are identical. One could attempt to find more complicated schemes for ordering the moments or picking N. The limit of accuracy that one can attain by adjusting N is indicated by the last row of Table I11 where we show the accuracy one would obtain if one always knew the optimum values of N for a given ordered list. We see that schemes 213:(3 anlo) and 213:(4 - 2an10)yield 0.0278 and 0.0280 whereas optimum choices of N yield 0.0249. The disadvantage of
more complicated rules for choosing N is that, although they might work well for the set of cases used to optimize them, their validity is probably less general than the simple schemes 213:(3 - 6, o) and 213:(4 - 213,~~). Tables I11 and I s also show the accuracy one would obtain by using in each case the greatest number of moments for which a feasible solution exists. The accuracy of this procedure is worse than using any fixed N in the range 2-6. Thus we do not recommend this procedure. Table VI compares the results for E* < 15 where P:l is zero or close to zero to those for E* > 15 where it is not. As explained in sections I1 and 111, in the latter cases the inelastic probabilities are estimated by using information from only the inelastic trajectories and the errors might be compounded by the errors in Pzl. The table shows that for nl = 0, the accuracy of all methods except linear smooth sampling (which is not very accurate at low and medium energies) decreases as E* is increased beyond 15. For nl # 0, though, the accuracy generally improves when E* is increased past 15. Notice, however, that schemes based on the moment lists 213 and 214 are particularly accurate at the energies below E* = 6.5. Finally we compare to previous work. We have already mentioned that the most commonly used method, the histogram method, yields an average error of 0.055. In contrast, scheme 123:2, as employed by yields 0.036, and scheme 123:M, as recommended by Levine and C O - W O ~ ~ yields ~ ~ S 0.045. , ~ ~ ~Using ~ ~ two moments for nl = 0 and three moments otherwise gives scheme 123:(3 - anlo), which reduces the average rms error to 0.033. Then, using the ((nz- n#) moments with i = 2, 1, and 3, we have reduced the average rrns error to 0.028, a factor of 2.0 better than the histogram method. The results show that we have been successful in our original g0a13*7921 of finding a simple rule for which moments to use in order to obtain an accurate semiclassical correspondence. Although we have considered three mass combinations, we have not done enough work to be able to comment on trends in the results due to mass combination. The agreement or disagreement of the methods with quantal results as a function of mass combination would be an interesting subject for future work.
VI. Conclusion Using the moments ( n z- nl) and ((nz- n1)') with nl = 0 and augmenting them with ( ( n z- n1I3)when nl # 0 yields an average rms error for the 106 test cases of 0.028. This compares favorably with the average rms error of 0.055 for the standard histogram method or the average rms error of 0.033-0.036 for various versions of the information theoretic method we introduced previously. Acknowledgment. We are grateful to Gerald A. Fraser and James W. Duff for collaboration on the trajectory calculations and other assistance. We thank R. D. Levine for supplying a listing of the computer program LP by Agmon and Levine. This work was supported in part by the National Science Foundation through grants CHE7506416, CHE77-27415, and SM176-02077.
