3398
J. Phys. Chem. 1994,98, 3398-3406
Transition State Resonances by Complex Scaling: H
+ Hz and H + MuH
Naomi Rom and Nimrod Moiseyev’ Department of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel Received: October 18, 1993; In Final Form: December 27, 1993’
A novel application of the complex coordinate method to reactive scattering systems is performed. Transition state resonance energies and lifetimes as well as the wave functions of the collinear H3 and H M u H transition complexes on two different potential energy surfaces (PK2 and LSTH) were calculated by using two complex scaling methods: conventional and exterior scaling. We report, partially for the first time, the exact complex energies of the four lowest resonances of H3 and the five lowest resonances of HMuH. The eigenfunctions of these resonances are plotted, and hyperspherical modes quantum numbers are assigned to each one of them. A natural expansion analysis computation is carried out, revealing the separable nature of each resonance. The results are in good agreement with previous ones, when available. The complex scaled eigenfunctions are shown to be more compact and smooth than the unscaled or exterior-scaled corresponding eigenfunctions and, consequently, can be expanded by a minimal number of grid points. I. Introduction Over the past 20 years numerous attempts have been made at calculating the transition state (“dynamical”) resonance energies and lifetimes in A + BC AB C type reactions.’“ The phenomenon of resonances appearing in reactive collisions, expressed as sharp peaks in reaction probability vs energy diagrams, is well understood from the plots of the diagonal corrected vibrational adiabatic hyperspherical model potentials derived by Romelt.3 The resonances are the quasibound states which are supported by the effective potential curves (see Figures 2 and 4 in ref 3). The use of hyperspherical coordinates (or polar, in the case of a collinear reaction) in the construction of the effective potential curves is a crucial point in Romelt’s3 and Kuppermann’sz analyses. These resonances can be shape, Feshbach, or both shapeand Feshbach type resonances, depending on the system and on the energy. Some of the methods which have been applied to the calculation of transition state resonances in exchange reactions are direct search of S-matrix complex energies,da stabilization procedures,‘ye time-independent scattering calculations,lb~2~4b,cJ-h and very recently sensitivity analysis5 and time-dependent spectral intensity calculations.6 However, although such considerable effort has been put into the investigation of the prototype exchange reaction H + H2 H2 + H, detailed information on the transition state resonances on various available potential energy surfaces (PES) is far from being complete. In this publication we report accurate resonance complex energies and functions for the four lowest-lying resonances in the collinear H H2 reaction and for the five lowest-lying resonances in the collinear H MuH reaction, on two commonly used PESs: PK2’ and LSTH.* The procedure that we used to obtain them is the complex coordinate method9 (CCM), which for the first time is applied to full-collision reactive processes. The only application to date of the CCM to a real three-atomic physical system was made by Lipkin, Moiseyev, and Leforestier’o to the half-collision process of vibrational predissociation of the NeIC1 van der Waals complex. The notion of the applicability of the CCM to reactive systems comes from the main assumption of the transition state theory: An activated complex with a finite lifetime is created as an intermediate species in the A + BC reactive collision and dissociates to give the products. Since the CCM has been developed to treat problems consisting of metastable states, such as the transition complex, it seems natural
-
+
-
+
a
+
Abstract published in Advance ACS Abstracts, March 15, 1994.
0022-3654/94/2098-3398$04.50/~
to implement it in systems supporting dynamical resonances. The CCM produces (on the basis of the Balslev-Combes theoremga) a single, square integrable (provided that the scaling parameter is large enough) complex wave function for each resonance state, as distinct from a combination of continuum wave functions which is necessary to represent each resonance when other scattering methods based on the solution of the Lippmann Schwinger equation are used, and gives the resonance complex energies of which the real part is the resonance position and the imaginary part is the resonance width (which is inversely proportional to the lifetime). The fact that the resonance wave function is square integrable enables the use of computational procedures which originally were developed for bound systems. It might have been expected that other absorbing boundary conditions methods, such as the optical potential method, could be used to obtain resonance complex energies in reactive scattering problems since they were successful in calculating state-to-state transition probabilities and cumulative reaction probabilitieslla* in these systems. However, to the best of our knowledge, resonance energies and widths could not have been computed accurately by using the optical potential method.lld The outline of the paper is as follows: In section I1 we give a brief description of the reactive scattering Hamiltonian of the studied systems and show the application of the CCM toa collinear exchange reaction by using two different complex scaling schemes: (a) conventional and (b) exterior complex scaling.12 In section 111our results for the transition state resonance energies are presented, and in section IV the resonance eigenfunctions are shown and analyzed as to their separability. Section V concludes. 11. Resonances in Reactive Scattering Collisions
There are typically three types of coordinate sets which are most commonly used for A + BC type reactions: (a) Jacobi coordinates, which provide a different set of vibrational and translational coordinates for each rearrangement channel, (b) hyperspherical coordinates, which describe the whole region of coordinate space with only one set of coordinates (in the case of a collinear reaction these are simply polar coordinates), and (c) natural reaction coordinates, which as in (b) describe the whole coordinate space with only one set of coordinates, being in this case the reaction coordinate and the modes perpendicular to it. We chose to implement the CCM in the studied ABC systems using the hyperspherical coordinates since, as we shall show, the application of the CCM to it is most straightforward (in the case of Jacobi coordinates, for example, a separate analytic contin0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3399
Transition State Resonances by Complex Scaling
uation for each of the two sets of Jacobi coordinates will have to be made, and the solutions will have to be matched in the interaction region. This difficulty is avoided when using either hyperspherical or natural reaction coordinates), and all types of A BC reactions can be treated with them (including heavylight-heavy reactions which cannot be treated at all with natural reaction coordinates). For the sake of simplicity we shall deal with collinear reactions, and in order to reduce the computing time, we shall use A BA AB A symmetric reactions. As is well-known, the latter is helpful since in symmetric reactions the potential energy function, V ( p , a),is symmetric with respect to half of the skew angle amax (to be defined below), and hence the solution have either odd or even symmetry with respect to am,,/2 and can be handled separately. Moreover, we shall be interested only in the even symmetry solutions since all the resonances in symmetricreactionsbelong to this kind of s0lution2~~ (as discussed in ref 3, only the effective potentials with even symmetry support resonances). Usually, when the CCM is used, an analytical continuation of the reaction coordinate is performed, leaving the other set of coordinates of the system unscaled. When polar coordinates are used, neither the radial coordinate, p, nor the angular coordinate, a,can represent the reaction coordinate throughout the whole coordinate space, but instead a function of both is required. However, p seems as good a candidate for analytic continuation since in the asymptotic limit (in both the reactant and the product valleys) it coincides for certain angles with the true reaction coordinate. Another point which supports the idea that p can be treated as the reaction coordinateis that the adiabatic potentials,2.3 which in the most primitive adiabatic scheme are the solutions of the angular Schrainger equation when p is taken as a parameter,2 plotted as a function of p support quasibound states which explainthe existenceof the dynamical resonances. Indeed, our calculations with complex scaled p give the correct results for the studied systems. Let us briefly summarize the relevant formulas which are needed when the problem of collinear A BC reactions is solved. The polar coordinatesp and a are defined in terms of the standard mass-weighted Jacobi coordinates R (which is proportional to the distance between A and the center of mass of the BC molecule, RA,BC)and r (which is proportional to the vibrational coordinate of BC, rgc):
+
+
-
+
+
with the skew angle a,,, = arctan(mgM/mAmc). mA, mg, and mc are the masses of the atoms A, B, and C, accordingly, and Mis the total mass of the ABC complex. p describesthe reactants for a < a,/2 and the products for a > amaX/2. The Hamiltonian of the ABC system in polar coordinates is given byI3
fr = --ti2 (2 a2 + 7 I + 2~
ap
4p
:-)aa2a2 1
+ 3 ( p , a)
(4)
m.n
where
po
Figure 1. Two complex contours of integration used in this paper: conventional (CCM; see eq 10) (full line) and exterior scaling (PIES;see eq 13) (broken line). 0 is the rotation angle, po is the smallest value that p gets in the computation, and p* is the point from which exterior scaling is started (taken in the asymptotic region of the potentia1 surface).
size of the box in the radial direction is given by Ap = pmx - po where pmaxis some large value of p ensuring convergenceof the results. Similarly,
(2m - l ) m
xm(a)= -sin amax
amax
m = 1,2, ... (6)
Note that only the even sine functions with respect to amax/2 are used in the calculation, as explainedabove. The normalization condition for the eigenfunctions of H is
(7) Two types of analytic continuation are used in this work (see Figure 1 for a schematic representation): Conventional and exterior complex scaling. In the former approach the scaled coordinate is rotated into the complex plane by an angle 6 at any given point in the coordinatespace, whereas in the latter approach the coordinate is kept on the real axis where the potential is varying (i.e., in the interaction region) and is rotated by the angle 6 into the complex plane only in the asymptotic region (i.e., in the product and reactant valleys). In both schemes the coordinate which is scaled is the radial coordinate, p. The criterion to choose an optimal scaling angle, eOpt, which gives the most accurate result in all complex scaling methods is a variational one:
This requirement also implies the two following conditions:
(3)
with p = dmAmgmc/M. The eigenfunctions 4(p, a) of are spanned as a linear combination of particle in a box functions in both the radial and angular coordinates: @(P, a) = CCm,. x m ( a ) un(P)
R c t p ) (bohr)
is taken to be in the inner classically forbidden region. The
In practice, calculations with various values of 6 are carried out and in each one the resonance energies E, = E - 1(I'/2) are obtained. Then, a &trajectory plot is made (i.e., JE,, is plotted vs WE,, as a function of e) in which a cusp appears.14 The rotational angle in which this occurs is eopt. We shall describe the computational algorithm below. (a) Reactive Scattering by Conventional Complex Scaling. In the conventional complex scaling, the scaled coordinate, p, is rotated into the complex plane by an angle 6. The complex radial coordinate, per is given by (see Figure 1) Po = ( P
PO^ + PO
(10)
3400 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
Rom and Moiseyev
where 7 = exp(i8). Since both eqs 9a and 9b need to be simultaneously satisfied, two independent real parameters are used for the optimization of the complex energy: The rotation angle 8 and the radial box size Ap. In our calculations po was fixed and pmaxvaried. The scaled Hamiltonian in the conventional CCM framework is
\
and
and its matrix elements are
P
a = -(a AP
The matrix elements of the scaled potential V(p8,a) and of l/pZe can be evaluated by Filon’s integration method’s or by the Harris, Engerholm, and Gwinn (HEG) method,16which is equivalent to the Gaussian quadrature procedure. (b) Reactive Scattering by Exterior Complex Scaling. In the exterior complex scaling (ECS) method, the scaled coordinate (p) remains real in the interaction region, Le. when p < p*, and rotates into thecomplexplanewhenp 1p*. Thecomplexcontour in the ECS approach is given by pe as follows (see Figure 1): Po
=(’(p-p*) exp(i8) + p*
ifp
