J. Phys. Chem. 1995, 99, 15624-15626
15624
Coordinate Transformations Relevant to Fragmentation U. Fano Physics Department & James Franck Institute, University of Chicago, Chicago, Illinois 60637 Received: June 27, 1999
Viewing molecules as aggregates of nuclei and electrons, jointly represented in hyperspherical coordinates, affords describing their dissociation or ionization as a reorganization of these coordinates.
I. Introduction Chemical reactions-and analogous processes-involve the coalescing of two (or more) reactants into a single “complex”, followed by the complex fragmenting into two (or more) products. Coalescing and fragmenting are mutually reciprocal processes, represented mathematically by complex-conjugate “Jost” matrices.’ Constructing either one of these matrices, by means of wave functions of the complex evolving from their most compact to a fully fragmented form, provides thus an adequate description of reactiom2 This paper focuses on a single, static, step of this description, namely, on the coordinate transformation that factors a complex’s wave function into functions of the separating fragments. This step has been formulated previously through ad hoc wave-function schematizations; a generic procedure to perform it is presented here. An important aspect of fragmentation lies in allotting the initial energy, parity, and angular momentum among the fragments and their respective motions about the center of mass. The (apparent) earlier failures to quantify this process reflect presumably an insufficient parameterization of fragmentation models. Even more important may prove parametrizing adequately the inverse formation of a complex, whose internal distribution of energy and angular momentum may influence a collision’s outcome critically. Hyperspherical coordinates, consisting of a hyperradius of inertia R and of a large set of angular (Le., dimensionless) variables represented collectively by a unit vcctor R, lead us to indicate the relevant wave function as Y ( R ; R ) . This notation implies mapping the evylution of a complex’s structure onto a progressive rotation of R, paced by the growth of R from 0 to 00 and governed by the complex’s Schrodinger e q ~ a t i o n .Our ~ present task deals with changing the set of R coAmponents,at a constant value of R where a portion of Y ( R ; R )has become localized along a fragmentation axis. A specific feature of hyperspherical dynamics helps implement the present approach: The R variable commutes with the kinetic energy operator of R’s Hamiltonian, R’s evolution being thus dubbed a “kinematic r ~ t a t i o n ” .The ~ same feature also removes a longknown difficulty of the Born-Oppenheimer approach to dis~ociation.~ This paper aims primarily at identifying parameters that quantify the allotment of energy and angular momentum, through eqs 10, 1 1 , 13, and 14. These equations still fail to describe the energy allotment fully, as this process stretches to nearly infinite separation through the remaining action of van der Waals or analogous interactions between the fragments. The origin of the hyperspherical coordinates, i.e., the “zero” of the system’s “configuration vector” R, is conveniently laid @
Abstract published in Aduance ACS Abstracts, October 1, 1995.
at the complex’s center of mass. Contributions to by several particles at positions 7i are also conveniently weighted by the square root of their masses, R itself being thus defined by
h,
N
N
i= 1
i= 1
Setting the origin at the center of mass thus implies the condition N
C m i F i= o i= I
Fragmentation of a complex amounts, in this framework, to resolving each of the expressions 1 and 2 into two-actually three-terms, corresponding to each of two fragments (with separate centers of mass) and to their distances astride their common center. Each of the fragments, together with their separation, will contribute to R, to M, and to mL7j. Achieving this goal only requires simple algebra and trigonometry, as well as appropriate notations, described in section 11. An additional task, outlined in section III, concerns-the corresponding allotment of the total angular momentum J of the complex among three terms, *
J =7,
+ 7, + 7,
(3)
representing the angular momenta of the two fragments and of their joint relative motion about the initial center of mass. The procedure outlined here appears readily amenable to a recursive description of multifragmentation events.
11. Vector Algebra of Coordinates The position coordinates of single particles Fi, with i = 1 , 2,
..., N , as well as the particle masses mi, shall now be cast into two groups with labels: i = 1 , 2 ,..., n, j = n + l , n + 2 ,..., N n
(4)
N
The new parameter’s value, 0 5 p 5 d 2 , is a fragmentation constant. The centers of mass of the two fragments are separated by a vector $ directed from the center of fragment 1 to that of fragment 2. The two fragments’ centers of mass lie-with respect to their joint center of mass-at the positions
0022-3654/95/2099-15624$09.00/0 0 1995 American Chemical Society
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J. Phys. Chem., Vol. 99,No. 42, 1995 15625
Coordinate Transformations Relevant to Fragmentation -
=-p
mjTj
sin2 p,
= jj c o s 2 p (6)
M sin2 p
z
The interfragment vector rotates freely, of course, within constraints to be described in section III; its magnitude will also vary, govemed by its relevant Schrodinger equation with a potential reflecting the fragments’ residual interaction (Coulomb, van der Waals, or other), which lies beyond our present scope. The central task of the present section consists of partitioning the hyperradius R into the fragments’ hyperradii (R1, R2) and the interfragments’ distance p , at a constant value of R itself. To this end we introduce intrafragment (non-coplanar) radial distances (Figure 1)
-. ti= T~+ jj sin2p, zj = T~- > cos p 2
(7)
n
i= I
- sin2p12 =
E milZil2-
-2 C m i t i * psin2p + M ~ [ Psin4I p~= i
COS~
1
+ p2 sin4p ) (8a)
M cos2 N i=n+ I
i 2 E mjxj.p cos2p + M sin2
,HI;~~C O S p ~
j
M sin2p ( R ;
=
+ p2 cos4p ) (8b)
The middle terms of the central expressions in both equations vanish, owing to the analogue of eq 2 for either fragment. The entries (R12,Rz2) on the right of each equation correspond to the definition in eq 1 of the hyperradius for either fragment and of the fragment’s fractional mass (eq 5). The combined allotment of masses and hyperradii emerges now by adding eqs 8a and 8b to yield
Rl=R
sin ?j cos a sin q sin a 2 cos , R2=R , p=R2 cos 11 sin 11 sin 211
111. Angular Momenta Labels IJM> in a k$s state representation, or appended to a wave function Y(R;_R),correspond to the invariant squared angulg momentum IJI2 = J(J l)h2 of an isolated complex and to S s component ,Malong a refeqnce axis 2. The complex’s configuration vector R correlates to J and thereby to 2, though often indirectly, through a “body frame” that coincides here with the complex’s inertial axes. This particular connection proves relevant to our process, whose fragmentation axis 2, introduced in section II, coincides with the axis of Zowest inertia at sufficiently large values of p . Sets of Euler angles (q, 8, v ) familiarly describe the connection between 2 and a body frame, q representing the frame’s azimuth about 2, 8 the angle between 2 and a body axis (tbe lowest inertia axis here), and ly the body’s azimuth about p. The wave function Y of complexes with cylindrical symmetry may depend on the Euler angles through a single standard factor dM,,,(q,8,v), but its expansion into such factors is generally required. Concerning the relations among the magnitudes and orientations of the angular momenta on the right of eq 3, recall that quantum physics replaces consideration of the angles between vector pairs with- that of fjquantized) values of the sums of altemative pairs, Ji J k = Jik. In the present context attention might focus on the partition
+
6
J’
+ 7, + 5,= + 7, = 7
(13) because the angular momentum J, of the fragment’s rotation
N
MRe ~ Emir: = i= I
+
M [ C O Sp~R , ~ sin2p ~
, +2 sin2p cos2p p2]
(9)
The coefficients of R22, p2) in eq 9 are proportional to the respective masses of the two fragments and to their reduced mass pertaining to the joint motion about the common center of mass. The distribution of MR2 among the three terms of eq 9 generates two new parameters, considering the square-root-mass weighting in eq 1 and its analogues, namely: (a) the ratio between the mass-weighted sum of the two fragments’ sizes and their equally weighted distance, represented by the angular parameter ?j
generally in different planes.
+
and evaluate the expressions
E
Figure 1. Vector diagram for eqs 7 and 8. The i and j triangles lie
=tan-‘
+
cos p R , sin p R2 sin p cos p p
(10)
which vanishes in the limit of full fragmentation, and (b) the ratio of RI and R2, represented by the angular parameter
a = tan-L(R2/Rl)
(11)
Inverting these equations yields the coordinate transformations:
about their center of mass may remain nearly constant after fragmentation. In generd, however, one considers the set ofi three altemative couplings represented, in terms of integer (or half-integer) quantum numbers, by ket symbols
Each step of the summation of two (or more) angular momenta is worked out quantitatively with “Wigner” (or “Clebsch-Gordan”) coefficients. The probability amplitude of any of the addition “schemes” (eq 14) tuming into another one is govemed by “angular momentum algebra” and represented through 6j coefficients.6 Transformations of the wave function of a complex, Y(R;k), by the coordinate or angular momentum changes of section 11, yield generally a superposition of states with altemative sets of J quantum numbers in altemative combinations (eq 14). Further analysis of this aspect will hinge on the results of prototype studies.
Acknowledgment. It is a privilege to dedicate this paper to Zdenek Herman, of whose science and art I have been a major
15626 J. Phys. Chem., Vol. 99, No. 42, 1995 beneficiary. Partial help in its preparation has been provided by NSF Grant PHY-9217874.
References and Notes (1) Newton, R. G. Scattering Theory of Waves and Particles; McGrawHill: New York, 1966.
Fano (2) Fano, U. Phys. Rev. A 1981, 24,2402. (3) Smith, F. T. Phys. Rev. 1960, 120, 1058. (4) Macek, J.; et al. Phys. Rev. A 1987, 35,3940. (5) Bohn, J. L.;Fano, U. Phys. Rev. A 1994, 50,2893. (6) Edmonds, A. R. Angular Momentum in Quantum Mechanics; Princeton University Press: Princeton, N.J., 1960.
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