628
J. Phys. Chem. 1984, 88, 628-636
Reaction-Path Hamiltonian Model of Partial Widths for Vibrationally Elastic and Inelastic Decay of Adiabatically Trapped Reactive Resonances Rex T. Skodje, David W. Schwenke, Donald G. Truhlar,* Department of Chemistry and Chemical Physics Program, University of Minnesota, Minneapolis, Minnesota 55455 and Bruce
C. Garrett
Chemical Dynamics Corporation, Columbus, Ohio 43220 (Received: May 25, 1983) We present a model to obtain the parameters of collisional resonances (quasibound states) in collinear A + BC reactions. The resonances are interpreted as quasibound states trapped in wells of adiabatic potential curves as functions of the minimum-energy-path coordinate. The adiabatic partial widths are obtained from semiclassical tunneling calculations. The nonadiabatic partial widths are found by using a Feshbach golden-ruleformula with the decay mediated by an internal centrifugal interaction proportional to the curvature of the reaction path. Exact two-dimensional scattering calculations are then carried out and fit to the isolated-narrow-resonance formula to obtain the accurate partial widths. The model results agree well with the accurate values.
I. Introduction The adiabatic approximation, in one or another coordinate system, with an appropriate coordinate or coordinates singled out as “slow”, provides a useful starting point for the description of many reactive resonances. (For recent examples and references to earlier work, see ref 1-5.) In systems for which the adiabatic approximation is good, the existence of resonances can be inferred from the presence of bound or quasibound states supported by the adiabatic potential curves. These adiabatically trapped states are usually easy to find because the adiabatic approximation yields an effective one-dimensional Hamiltonian. In a recent paper6 on adiabatic models for reactive tunneling, we showed how to use Feshbach resonance theory7-10with a reaction-path Hamiltonian” to compute the virtual coupling between states that correspond to different adiabatic quantum numbers. Here we extend this method to calculate partial widths for resonances in reactive scattering. In order to test the accuracy of the new approximation scheme, we present calculations for collinear atom-diatom reactions for which accurate quantal results are obtainable. We calculate the accurate resonance energies, widths, and partial widths by converged quantum mechanical coupled-channel calculations for these collinear reactions, and we compare these to the results of our Feshbach-theory resonance calculations.
system of ref 12: Let ( x , y ) be mass-scaled coordinates that diagonalize the kinetic energy with reduced mass I.L in each mode, and let the reaction path be the union of the paths of steepest descents in (x,y) space from the saddle point (s = 0) to reactants (s = -m) and to products (s = a). The local vibrational coordinate u is a straight line perpendicular to the reaction path and is defined as positive on the dissociation side. The Hamiltonian is
and K ( S ) is the curvature of the reaction path. Define an adiabatic basis &(u;s) for this Hamiltonian as the solution of
11. Theory
+
Consider a collinear collision A BC, which may be elastic, inelastic, or reactive to produce AB + C. We use the coordinate (1) A. Kuppermann in “Potential Energy Surfaces and Dynamics Calculations”, D. G. Truhlar, Ed., Plenum, Press, New York, 1981, p 375. (2) B. C. Garrett, D. G. Truhlar, R. S . Grev, G. C. Schatz, and R. B. Walker, J . Chem. Phys., 85, 3806 (1981). (3) B. C. Garrett and D. G. Truhlar, J . Phys. Chem., 86, 1136 (1982). (4) J. M. Launay and M. Le Dourneuf, J . Phys. B, 15, L455 (1982). (5) E. Pollak and R. E. Wyatt, J . Chem. Phys., 77, 2689 (1982). (6) R. T. Skodje, D. G. Truhlar, and B. C. Garrett, J . Chem. Phys., 77, 5955 (1982). (7) H. Feshbach, Ann. Phys. (N.Y.),5, 357 (1958), 19, 287 (1962). (8) W. H. Miller, Phys. Rev., 152, 70 (1966); W. H. Miller, Chem. Phys. Lett., 4, 627 (1970). (9) L. Fonda in “Scattering Theory: New Methods and Problems in
Atomic, Nuclear, and Particle Physics”, A. 0.Barut, Ed.; Gordon and Breach, New York, 1969, p 129. (10) C. L. Shoemaker and R. E. Wyatt, Adv. Quantum Chem., 14, 169 (1981); J . Chem. Phys., 77, 4994 (1982). (11) G. L. Hofacker, Z . Naturforsch. A , 18, 607 (1963); R. A. Marcus, J . Chem. Phys., 45,4493,4500 (1966); Discuss. Faraday SOC.,44, 7 (1968); J . Chem. Phys., 49,2610, 2617 (1968); S . F. Fisher, G. L. Hofacker, and R. Seiler, ibid., 51, 3951 (1969); M. V. Basilevsky, Chem. Phys., 24, 81 (1977); W. H. Miller, N. C. Handy, and J. E. Adams, J . Chem. Phys., 72, 99 (1980).
0022-3654/84/2088-0628$01.50/0
+ [V(s,u)- V(s,u=O)]- t,(s) where a curvature zero point term6 is dropped going from ( 3 ) to (5). A solution to the Schrodinger equation
H*(s,u) = E*(s,u)
(6) where E is the total energy [we define the zero of energy as V(s=-m,O)], can be written in this basis as ( ~ , u l * ) = *(s,u) = U i ( s ) 4 i ( u ; s ) i
(7)
A projection operator Q, can be defined which projects out of an arbitrary wave function the adiabatic component with vibrational quantum number n, i.e. (s3ulQ,,l*) = Q,,*(s,u)= f,(sM,(u;s)
(8)
Explicitly (12) D. G. Truhlar, J . Phys. Chem., 83, 188 (1979); B. C. Garrett and D. G. Truhlar, ibid., 83, 1052, 3058 (E)(1979); 87, 4553 (E) (1983).
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 629
Collisional Resonances in Collinear Reaction Q,*(s,u) = ~ , ( u ; sJdu’ ) J ( u’;s)~,*(u’;s)\k(s,u’) =
4fl(w)(4fll*) (9) where the Jacobian is J(u;s) = 9
( 1Oa)
quantum number is greater than 0 and the resonance energy G m is below the adiabatic threshold, i.e. where Va(s=--m,n)
and where we use round bra-kets for the u space and angular bra-kets for the (s,u) space, such that
(41142) = JdU J(u;s)41*(u;s)42(u;s) =
J Jdu
ds J(u;s)*l*(s,u)*2(s,u)
(lob) (1Oc)
(s,441)(4~1*3) = 41(u;s)I d ” ’ J ( u ’;S)~Z*(U’;S)*~(~,U’) (1 0 4
Define another projector as Pn= 1 - Q,
(11)
~ n = min t ~ ~
Va(s= m ,n)
(21)
Such a resonance is core-excited type 1 in the notation of Taylor et al.I4 In such a case adiabatic tunneling makes no contribution to the resonance widths, and the resonance becomes a true bound state in the adiabatic approximation. As is well-known, the resonance state corresponds to an unnormalized eigenstate of the Schrdinger equation with a complex eigenenergy. The position of the resonance is given by the real part of the eigenenergy, while the width (full width at halfmaximum) is twice the imaginary part. Thus we will have solved the problem when this eigenvalue is found. Take the complex resonance energy to be
Using the orthonormality relation
(4nldd) = J ~ uJ(U;S)4,*(U;s)~n~(U;s)=
6nnl
(12)
these definitions yield QnQnf QPnf
=
(13)
6nrfQn
= (1 - 6nd)Qn
PPn, = Pn - QA1
- and)
(14) (15)
The vibrationally adiabatic potential curves are defined as
Va(s,n)= V(s,u=O) + en($)
(16)
In the calculations presented here E,($) in eq 16 is always calculated by a Morse potential fit to the u dependence of V(s,u). It is expected that the present formalism will be most valid for systems of low-to-moderate reaction path curvature. When the reaction path curvature becomes too high, the reference coordinate system (s,u) no longer yields a satisfactory adiabatic separation. The first approximation we make is to set all derivative-ofcurvature terms, dK/ds and d2K/ds2,equal to zero. This procedure is motivated and discussed at length in ref 6; a brief recap of the argument is that these terms are very sensitive to small changes in the reaction path that should not have a large physical effect on the dynamics, and we would expect the effect of these terms to largely cancel in a calculation without other approximations. As a result of this approximation, (1 /q)(a/as)(l/q)(a/as) becomes (i/12)(a2/as2) in eq 1. The resonances we consider here are interpieted as systems trapped in wells in the vibrationally adiabatic potential curves. For example, consider the adiabatic approximation for reactioncoordinate motion when the bound degree of freedom is in vibrational state n:6
If this equation has a resonance solution with reaction-coordinate quantum number u, we denote the resonance as (n,u),the resoThe nance solution of (17) as f K(s), and the state vector as I*,). wave functionfK(3) is expected to be well localized near s = 0 between two maxima or local maxima in Va(s,n). The vibrationally adiabatic approximation for the resonance-state wave function is (s,ul*fl,)
=
*fl”(S,U)
=f W # h ( u ; s )
(18)
Thus
QnPnU) = PnU)
To find it we proceed as in Feshbach t h e ~ r y . ~In, ~the original Feshbach formalism,’ the resonance-state vector is obtained by a diagonalization in the closed-channel subspace. Here, as in Miller’s applications of the Feshbach formalism to electronic structure problems,s we replace the closed-channel projector of the original formalism by a projector onto an approximate resonance state that is an eigenstate of a zero-order Hamiltonian. Rather than obtain the approximate resonance state by an S C F or configuration-interaction c a l c ~ l a t i o n as , ~ ~suggested by Miller,8 in this article we make the adiabatic approximation and use Qn as the resonance projector, thereby avoiding the need for diagonalization. According to Feshbach’s original formalism the width is given by a “golden-rule” formula involving closed- and openchannel projectors; in the present case these are replaced by Q, and P,, respectively, to yield
r,,~= 2~l(*fl,lQ,HPfll~,,,j~12 = 2nlJ Jds du J(u;s)*n”*(s,u)Q,HP,(s,ulG,,,J)I*
(23b) (23~)
In these equations rmij is the partial width for channelj and the channel states I&, j) satisfy (PnHPn - ~ m ) I G n u ~ = j) 0
(24)
The sum over j in (23a) is over all scattering states which solve (24) and are normalized as (G’,j’lG,j) = a(& - G’,6jj,
(25)
The resonance wave function solves7
We will now adopt the so-called “sharp resonance” approxim a t i ~ n .In~ this approximation the width is assumed small and the nonadiabatic coupling is dropped in (24) and (26). Then (24) becomes m
~[~’I4i)(4iI~I4i)(4~II - EnulIGnuj) = 0 i=O
(27)
where the prime denotes that the i = n term is omitted from the sum and the sum limit m is the highest open vibrational state. Since there is no nonadiabatic coupling the solutions of (27) are
(19)
A . Case 1. First we consider the case where the resonance
(13) T. C. Thompson and D. G. Truhlar, J . Chem. Phys., 76, 1790 (1982); 77, 3777 (E) (1983); Chem. Phys. Lett., 92, 71 (1982).
630 The Journal of Physical Chemistry, Vol. 88, No. 3, 1984
(~J4&fI"J) = fp ? g n U 7 j ) 4 / p ; 4
(28)
where 1, denotes the particular value of the vibrational quantum number for solution j . There are two linearly independent sets of scattering boundary conditions for each 1, and since 1, takes on the values from 0 to m,we see that in case 1 the upper limit in (23) is J = 2(m 1 ) (29a)
+
if all these channels are open, and J=2m+1
Skodje et al.
(29b)
if the highest channel is open for reactants or products but not both. We will consider only these cases. The translational wave functions in (28) satisfy
[(4/pl4l/)- G " l f p , ~ n u J )= 0
(30)
where s< and s> are the classical turning points in the well. The partial widths are given by the Gamow formula16-1s
where ~ ( 6 , )is the vibrational frequency for s-coordinate motion and PIOj(&,)] is the primitive of an oscillator with total energy &lw, semiclassical tunneling probability through either the reactant or product side barrier, depending on j , and given by The barrier-penetration integral Oj(gm)is given in the smallcurvature (SC) approximation6J9as
In the sharp-resonance approximation (26) becomes
(QnHQn- Gnu)lqnu)= 0
(31)
where
and eq 23 becomes J
r,, = 2 7 jc = I$ 1
Jds du J(u;slf~(s)(4,,lH14,~~~,~,,u,j)12 (32)
B. Case 2. When the resonance energy 6, is greater than the channel threshold E',h', it is possible for the particle to tunnel through the adiabatic barrier to decay. We now consider this case, which includes shape resonances in the ground state as well as core-excited type III4 resonances. In this case there will be an adiabatic contribution to the width due to tunneling through the barrier@) in V,(s,n). To handle this conveniently within the framework of the Feshbach formalism the projection operator Q,, is split into two parts; one part, Qi is the resonance projector, while the other, Q", is a continuum projector. The roles played in case 1 by Qn and P,, are now to be assigned to Ql, and (Q", Pn), respectively. This makes two changes from case 1. First, we must replace eq 23 by
+
rnu
=
J
2m
27
is the effective mass, which accounts for the effective collapse of the wave function around a near-optimal tunneling path through the two-dimensional coordinate space, d' and d, are the classical turning points of barrier j , and u,>(s) is the outer classical turning point of the u coordinate when the local vibrational energy at s is cn(s). For comparison with the above method, which is used for the final results in this article, we also performed some calculations in which pe&) is replaced by the original p (this is called ~ ~ , ~ ~in ) which uniform semithe MEP a p p r o x i m a t i ~ n ~and/or classical f o r m ~ l a e l ~are * ~used ~ * in ~ ~place of (37) and (38). Thus we will not need Q",, but we still must identify @, for use in (35). We adopt the following reasonable definition for this projector
C1 I ( * ~ l Q P P n I ~ n u , ~+) I2z7 1=2m+l
I(*nulQ3Q3&lw,j)12
J=
(33) where in this case m = n. The two continuum states with 12m+l = n and 12m+2 = n are determined by
(P3a- gnu)l&nu,j)
=0
(34)
-
&n")lqnu)
=0
(35)
instead of (31). For case 2, the total width is written as the sum of two terms
rnU= r; + r i 2m
=
=
+m
so < s
< s1
(43)
so > s or sI < s
(36a)
(s=sl,ul*n") = (s=so,ul*nu) = 0
(44)
In practice we used eq 43 and 44 for all resonances, not just those above the threshold. So the resonance wave functions for both cases 1 and 2 are given by
J
r?IU,J
J=1
vfi(s,n) = V,(s,n)
and so and s1 are the positions of the left and right (local) maxima of V,(s,n). So
Second, the resonance state now will satisfy
(Q%Q:,
where
+
c
/=2m+l
r?I",J
(36b)
where I?; is the nonadiabatic partially summed width which comes from the first 2m terms in (33) and I?; is the adiabatic partially summed width which is the contribution from the last sum. In C. Hamiltonian Matrix Elements. The quantities necessary for use in the golden-rule formula, eq 32 for case 1 or eq 33 for practice, instead of the golden-rule formula for the adiabatic partial widths we use a simpler treatment based on semiclassical ap(16) W. H. Miller, J. Phys. Chem., 83, 960 (1979). proximations. In the semiclassical adiabatic (SA) treatment the (17) J. N. L. Connor, in "Semiclassical Methods in Molecular Scattering energies &LA(which differ slightly from those calculated for Spectroscopy", M. S. Child, Ed.; Reidel, Dordrecht, Holland, 1980, p 45. treating the nonadiabatic terms) are obtained by ~ o l v i n g ~ , ~ * ~and ~ (18) J. N. L. Connor and A. D. Smith, Mol. Phys., 43, 397 (1981). (14) H. S. Taylor, G.V. Nazaroff, and A. Golebiewski, J. Chem. Phys., 45, 2822 (1966). (15) L. D. Landau and E. M. Lifshitz, "Quantum Mechanics", 2nd ed, Pergamon, Oxford, 1965, p 163.
(19) R. T. Skodje, D. G. Truhlar, and B. C. Garrett, J. Phys. Chem., 85, 3019 (1981).
(20) B. C. Garrett, D. G.Truhlar, R. S. Grev, and A. W.Magnuson, J . Phys. Chem., 84, 1730 (1980). (21) J. N. L. Connor, Mol. Phys., 15, 37 (1968).
Collisional Resonances in Collinear Reaction
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 631
case 2, are not yet clearly defined. At issue are the matrix elements of the Hamiltonian between the adiabatic basis vectors, Le. ffi/(s) E
(4ilHI4,)
These quantities are needed for i = 1 to calculate the continuum wave functions via eq 30 and to calculate the resonance wave function via eq 46. The off-diagonal elements are needed to evaluate the interaction matrix elements in ( 3 2 ) or (33). The problem with these matrix elements is that the integral in eq 47 is divergent because of the singularity of v-' at u = 1 / K ( S ) . However, the divergence is due to the breakdown of natural collision coordinates rather than to a singularity in the physics, and it is possible to obtain meaningful asymptotic expansions in terms of coupling operators involving positive powers of the reaction-path curvature.6g22 First consider the diagonal matrix elements ( ~ n ~ l / ~ 2As~ ~ , discussed above these are needed for the calculation of vibrationally adiabatic scattering states in classically allowed regions of s and for the calculation of vibrationally adiabatic resonance states, which are well localized in classically allowed regions of s. In ref 3 it was shown that reaction-path curvature apparently has only a small effect on vibrationally adiabatic states in classically allowed regions for systems like those studied in the present article. Thus we neglect curvature in these elements, Le., we set 17 = 1 for these elements. Furthermore, as justified for diagonal elements in ref 6, we replace (4ild2/ds214i)by ( I # J ~ I ~ J ( ~ ~ / ~ S ~ )where , (4il~i) is unity by eq 12. Thus eq 30 and 46 reduce to
and where the subscripts R and L denote that the differential operators act to the right and left, respectively. The expectation values in (52) were evaluated analytically by using the harmonic approximation for the wave functions dn and 4,. Using (52), we calculated the nonadiabatic partial widths from
I't,
The total width is then calculated from (36a) where is zero for case 1 and is estimated by the semiclassical method for case 2. The decomposition of the width into partial widths is considered further in section D. D. Partial Widths and Intensities. The actual width of the resonance as observed in the energy dependence of the scattering amplitude in any of its decay channels is given by eq 23 or 32 in case 1 and eq 33 in case 2. This total width is obtained by summing the partial widths for all open decay channels9as defined in eq 23a and 23b. Although the energy dependence of the is determined by rnv and is the same in all channels, ) resonance . its contribution to the scattering amplitude in any channel is determined by the partial width. For collinear collisions the state-to-state transition probabilities are given by Pjy = ITjY12
where Tjjlis a scattering matrix element. Then, for a sharp resonance9
where lTJI2=
cJJ
respectively. Equation 49 must be solved for G, and f i ( r ) . For the cases considered here, for which m = n or n - 1 , eq 48 must be solved 2n times for the f f;(s,Gnu,j)with lj = 0, 1, ..., 2n - 1. It was also necessary to treat the diagonal matrix elements in the classically forbidden region for the adiabatic partial widths. Implicit in the semiclassical treatment of eq 37-41 is the approximation (only valid for the classically forbidden regions)
(55)
(56)
rflL,J
and is the background, or "direct", contribution, which is assumed to be slowly varying over the width of the resonance. In the absence of a resonance the transition probability connecting channelsj and j'would be l ~ J t 1 2 . Thus the resonance contribution to the transition probability is
1qJ,12
AP:JS(E) = ITJJ~(E)12 -
(57)
If is assumed truly constant over the width of the resonance, then as E varies from much less than to much more than G,,, A$J?(E) can vary between the limits A- IAPJJ?(E)IA+
(58)
+
(59)
where Finally consider (4n1(1/112)(a2/as2)14,1)for n # 1. This matrix element accounts for the nonadiabatic partial widths and so it cannot be neglected. Instead we approximate it by making the expansion 1/q2
'v
1
+
2K(S)U
A+ =
2M?l,3
JJ'
2Mfl,,
JJ""1
(51)
Then we make H,/(s) Hermitian by taking an arithmetic average with its Hermitian conjugate. These two steps yield
M?W;JJ'
=
rllViJrltU,J'/rn?
(61)
Clearly M,,,jJ. is a measure of the "strength" of the resonance for a given transition. 111. Model Calculations
+
We consider collinear A BC collisions on three potential energy surfaces: Porter and Karplus' surface no. 2 for H3 (PK2),23 the Truhlar-Kuppermann surface for H3 (TK),24 and Stern, Persky, and Klein's GSW surface for ClH,.25 The quantities Va(s,l) and K ( S ) needed for eq 48, 49, and 52 are computed on a grid of about 400 s values by the methods
(22) G. L. Hofacker and R. D. Levine, Chem. Phys. Lett., 9, 617 (1971); R. D. Levine, ibid., 10, 510 (1971); M. V . Basilevsky, Mol. Phys., 26, 765 (1973); 28, 617 (1974).
(23) R. N. Porter and M. Karplus, J. Chem. Phys., 40, 1105 (1964). (24) D. G. Truhlar and A. Kuppermann, J . Chem. Phys., 52,3841 (1970); 56, 2232 (1972). (25) M. J. Stern, A. Persky, and F. S. Klein, J . Chem. Phys., 58, 5697 (1973).
632 The Journal of Physical Chemistry, Vol. 88, No. 3, 1984
Skodje et al.
20 15
where kj' = [ 2 p ( E - Va(s=km,lj))/h2]1/2
(70)
The normalization of these solutions is
-3
-2
I
0
2
3
s (ao)
-
Figure 1. Vibrationally adiabatic potential curves for the two lowestD H D on the energy vibrational states of the reaction D H D Porter-Karplus no. 2 potential energy surface are shown as solid curves. The dashed horizontal line represents eigenvalue resonance energy calculated as the eigenvalue of eq 23. The long tick marks on the left and right ordinates indicate the energy of the n = 1 threshold.
+
+
developed for generalized-transition-state-theory calculation^.^^*^^ Va(s,l) includes anharmonicity by Morse approximation I.26 For the rest of the calculations, Va(s,l)and K(S) are evaluated at required s values by numerical interpolation (and, if necessary for convergence in asymptotic regions, by extrapolation) of values on this grid. Figure 1 illustrates the functions V(s,u=O),'V,(s,O), and Va(s,l) for D H D on the P K 2 surface. Equation 49 was solved by the bound-state finite-difference boundary-value method (FDBVM)28,29with zero boundary values at the two global maxima s = s< and s, of Va(s,I). The eigenvalues of the finite-difference Hamiltonian matrix were found with EISPACK subprograms. Grid sizes of 600, 800, and 1000 points were used, and the eigenvalues were extrapolated to zero stepsize by the h4 extrapolation method of Richardson.28 In practice the eigenvalues for the 600-point grid agreed with the extrapolated values to five significant figures. Equation 48 was solved by the scattering finite-difference boundary-value Three grids were used for each case, with approximately 10, 15, and 25 points per shortest deBroglie wavelength. The finite difference equations were first solved with arbitrary linearly independent boundary conditions:
+
This normalization is consistent with the normalization specified by (12), (25), and (28) for the solutionsff;(s,Gn,,j) that give the reative and nonreactive partial widths. These solutions have outgoing components that are entirely reactive or nonreactive; they are given by
To calculate the partial width integrals, eq 47, we quadratically interpolated the continuum solutions, eq 7 2 and 73, and the interaction matrix element, eq 42, to the finest bound-state grid, and the quadrature was carried out by the trapezoidal rule. The partial widths did not fit a simple polynomial form in the grid spacing because of the interpolation, but the results were converged to within 1% with respect to expanding the grid size. The primitive semiclassical calculations for resonance energies3 and phase integrals for tunnelingIg were carried out with methods described elsewhere. For the uniform semiclassical calculations the resonance energy was calculated by finding the maximum of the probability of reaction as a function of E as calculated by the uniform semiclassical formula of Connor.21 In this formula we used p:$(s) for the barrier-penetration phase integrals but the original p for the phase integral in the well. The uniform semiclassical width was calculated by the analytical formula of Connor and Smithlg with the same phase integrals. In summary the uniform semiclassical results are based on eq 24 of ref 21 and 3.6, 3.7, and 4.5 of ref 18 with +(e,) of ref 18 replaced by +(e1) + +(e2) of ref 2 1.
IV. Exact Calculations
where s i and sCp are the end points of the grid which are arbitrary except that they lie in the asymptotic region. The + ; ( ~ , € , , ~are ,j) real and are obtained by solving finite-difference equations with a LINPACK subprogram. The usual scattering solutions gT;(s,GnU,j) we will present the are linear combinations of the $i(s,6nu,j); equations only for the case (29a5, in which case they satisfy
To gauge the accuracy of our model calculations and to provide a standard set of results for reactive resonance partial widths, we have performed exact two-dimensional reactive scattering calculations. Although reliable calculations already exist for the systems we c o n ~ i d e r , I * ~visual ~ , ~ ' -estimates ~~ of the resonance parameters from published figures were undependable. Therefore we repeated the calculations for a fine grid of energies for nine cases (listed in Table I). The accurate quantal scattering calculations were performed by the R matrix propagation method of Light and (27) B. C. Garrett, D. G. Truhlar, R. S.Grev, and A. W. Magnuson, J . Phys. Chem., 84, 1730 (1980); 87,4554 (E) (1983). (28) D. G. Truhlar, J . Comp. Phys., 10, 123 (1972). (29) D. G. Truhlar, Chem. Phys. Lett., 15, 483 (1972). (30) D. G. Truhlar and A. Kuppermann, J . Am. Chem. Sor., 93, 1840 (197 1).
(26) B. C. Garrett and D. G. Truhlar, J . Phys. Chem., 83, 1079 (1979); 84, 682 (E) (1980); 87, 4553 (E) (1983).
(31) J. Manz, E. Pollak, and J. Romelt, Chem. Phys. Lett., 86, 26 (1982). (32) G. C. Schatz and A. Kuppermann, Phys. Rev. Lett., 35, 1266 (1975). (33) J. P. Dwyer, Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1978. (34) G. C. Schatz and A. Kuppermann, J . Chem. Phys., 59,964 (1973). (35) M. Baer, U. Halavee, and A. Persky, J . Chem. Phys., 61, 5122 (1974).
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 633
Collisional Resonances in Collinear Reaction
TABLE I: Characterization of the n := 0 and 1 Vibrationally Adiabatic Potential Curvesa,b n=O n=l A
Va(-,O) 3 P'a(-,O) 3 kcal/mol kcal/mol
+ BC
VaC-, 11, kcal/mol
Va(-, 1 ), kcal/mol
so, a,
Va(So, kcal/mol
s,in,
a,
va(smin, 11, kcal/mol
s,, a,,
Va(s-191 1, kcal/mol
0.51 0.49 0.42 0.42 0.36
21.86 19.80 19.78 17.29 19.04
H + H, H + DH D + HD D + D, T + HT
6.29 5.46 5.46 4.47 5.15
6.29 5.46 5.46 4.47 5.15
18.32 15.96 15.96 13.13 15.09
18.32 15.96 15.96 13.13 15.09
PK2 Surface -0.51 -0.49 -0.42 -0.42 -0.36
21.86 19.80 19.78 17.29 19.04
0.00 0.00 0.00
18.28 18.28 15.65 11.73 14.47
+ H,
6.29
6.29
18.32
18.32
TK Surface -0.64
22.92
0.00
18.28
0.64
22.92
C l + H, C1 D, CI + T,
6.19 4.40 3.60
1.24 6.04 5.53
18.03 12.92 10.62
15.44 12.01 10.49
SPK Surface -0.68 -0.44 -0.34
18.98 14.60 12.81
-0.06 --0.08 -0.10
13.32 11.70 10.95
0.44 0.32 0.28
16.67 13.62 12.35
H
+
O.OOc 0.00
a Zero of energy is A infinitely far from BC with BC at classical equilibrium. Origin of s is the saddle point; s, and s, are reactant-side and product-side local maxima in Va(s,l); and smill is the local minimum of Va(s,l) between so and s , . This system actually shows twin minima on either side of this location with energies lower than the given value by
