Department of Physics and Astronomy, University of Tennessee, Knoxville, ... and Oak Ridge National Iaborutory, P.O. Box 2009, Oak Ridge, Tennessee $7881.
JUNE 1994
VOLUME 49, NUMBER 6
PHYSICAL REVIEW A
Harmonic-oscillator
structure of classically unstable motion
J. H. Macek
and S. Yu. Ovchinnikov'
Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee $7996 1-501 and Oak Ridge National Iaborutory, P. O. Box 2009, Oak Ridge, Tennessee $7881 (Received 12 October 1993; revised manuscript received 1 March 1994) A quantal description of classcially unstable motion which employs the integration of the Schrodinger equation along a path in the complex-coordinate plane is described. It is shown that a region in the complex-coordinate plane exists where the Schrodinger wave function is well described in a harmonic-oscillator approximation. The breakup of the hydrogen molecule ion H~ is used as a prototype to illustrate the method.
PACS number(s): 34.90.+q, 34.20. —b, 31.10.+z
The motion of microscopic systems along classically unstable trajectories plays a key, but controversial, role in quantal descriptions of model problems [1]. In particular, the breakup of three charged particles is thought to occur via motion on potential saddles where classical orbits are unstable [2,3], whereas other theories [4] consider that these motions are highly unlikely, since clear mechanisms for classical systems to remain on the top of potential barriers are not apparent. In this Rapid Communication we show that the localization of wave functions on the top of the barrier easily emerges Rom considerations of the propagation of systems along paths where reaction coordinates R, which measure sizes of systems, are taken to be complex. This bypasses the need to find a mechanism for quantal systems to remain localized in regions of classical instability in order to compute the propagation of wave functions through such regions. The breakup of H2+ is used as a prototype to illustrate the essential features of this new picture. Atomic units are used throughout. To set complex path integration in its most general context, consider a Schrodinger equation of the form [
—(1/2M)(d /dR
)
+'R(R;z)]@ = E4',
where B is a reaction coordinate, z represents a set of dimensionless coordinates orthogonal to R [5], 'R is the Hamiltonian for the system minus the kinetic-energy operator corresponding to the reaction coordinate R, and M is a parameter with dimensions of mass. The exact form of W for H2+ is well known and need not be given here [5]. Only one general property of 'R is pertinent to the discussion presented here, namely that the potential energy V(R, z) of the three interacting particles contains a barrier region where the potential has locally an inverted harmonic-oscillator structure —2 k(R) z with k 0 in at least one of the coordinates x. Then the system is classically unstable at the point x = 0 and the corresponding Schrodinger wave is locally unbounded. DifFerential equations such as Eq. (1) may be inte-
(
'Permanent address: Ioffe Physical-Technical Petersburg, Russia. 1050-2947/94/49(6)/4273(4)/$06. 00
Institute, St.
49
grated along any path in the complex R plane to connect solutions at some initial R = R, to solutions at a final B = Ry. This Bexibility in the choice of R is used to select some path in the complex plane along which good approximations to the exact 4' are evident. The structure of Eq. (1) suggests that an appropriate path can be found by computing the eigenvalues of R(R;z). These eigenvalues e„(R)are interpreted in terms of a single function e(R) which is single valued on a multisheeted Riemann surface. Conventional energy eigenvalues e„(R)represent the values of the function e(R) on the real axis for different sheets n. The energy eigenvalues for H2+ are computed using the program of Ovchinnikov and Solov'ev [5]. Figure 1 shows a Riemann surface for the ug eigenfunctions of H2+ constructed by plotting Re(R) vs R. Different sheets join at the branch points. The connection between the branch points of e(R) and classically unstable orbits has been discussed in Ref. [6]. In this Rapid Communication we show that when the instability corresponds to motion on a potential saddle, the eigenvalues e(R) have a harmonic-oscillator structure in a region adjacent to the branch points. The branch points of interest here are called T-series branch points in Ref. [5] and are exhibited more clearly by plotting the real part of the effective principle quantum number n(R) = 1/g — e(R) vs ~R as in Fig. 2. Figure 2(b) shows the surface from the "back" side with the ImR & 0 region foremost. Notice that the complex structure on the real axis merges at the branch points with a remarkably Bat region that extends an infinite distance. An infinite number of such Bat sheets connect with the sheets corresponding to conventional potential-energy curves on the real axis. Figure 1(b) shows, for example, a front view of the energy surface. Here one sees how the 3dcr sheet joins with the 180 sheet at a branch point Bq, 3g that marks the border of the flat surface seen in Fig. 2(b). Part of the 5go surface passes through the Hat portion and joins the iso sheet at a branch point Bzs~5g~ with Baser, 3cgcr + R1so, 5g . The new branch point marks the border of another Bat surface which lies below the surface shown in Fig. 2(b). These interleaving surfaces have been cut away in Fig. 2(b) to show the first Hat surface clearly. The wave functions to these flat regions of e(R) are &p (R; r) corresponding represented by harmonic-oscillator functions in the coordinate r equal to the distance from the center of mass R4273
1994
The American Physical Society
R4274
J. H
MACEK AND S. YU OVCHINNIKOV '
divided b y R. Along the direction parallel to the nuclear axis the potential is locally that of an inverted oscillator with spring constant k(R) = — 32Z/ Z is the nuclear charge. With the scaled coordinates
=
mo,
w ere mo is the
reduced mass of the electron relative to the two pro' a factor tons. The eigenfunctions of 'R(R ; x~reincorporate ) ' Qk(R)m(r)x'] with Qk(R)m(R) = —i4/2ZR, exp[ ——,
so that
p(R; x)
oc
exp[i2V2ZRe~Bx
] exp[
—4/2Zlm~Rx2 (2)
)
Since Im g O0 the wave function y(R ;x~ x is ocaized ' on the toop o of thee inverted oscillator near z = 0. The locus of branch points se arates t gion om the region where e(R) has mainly thee —Z 2 n racteristic of electrons in th e 6e ld so point
e ~(R)
e n(R)
ia) 59
0
Re s(R)
-0.5
2
(b)
Imv R
FIG. 1. Plot of the real part of the ener ei l R Th e p loot represents a Riemannn surface surfa on
a single-valued
e n(R)
which
e(R) is
function. Th e values along the real axis are the th energy eigenvalues e (R) of o H 2 +.. (a) e axles foremost ( is a view with the ReR 's and (b) has the ImR 3scr surface passes through the leer surface and a rane point Rq,
FIG. 2. Plot of n(R) = 1/ ~e(R) vs ~R. (a) "Front" view of the s . surface with the real axis foremost.
b
"Bac
view of the sur ace
STRUCTURE OF CLASSICALLY. . .
HARMONIC-OSCILLATOR
49
charges. The harmonic-oscillator region occurs only for complex R, as illustrated schematically in Fig. 3. In this region the energy eigenvalues are given by
e(R)
= —[(4Z —1)/R] —(nz + z)i[(k(R)/m(R)] ~ = —[(4Z —1)/R] —(nz + z)i4(2Z/mo) ~ (1/R
),
~
(8) where n2 is zero or a positive integer. The corresponding expression for n(R) is
n(R)
—i(n2+ 2)2/2Z/mo(4Z —1)
(4)
Equation (4) shows that Re(R) is a linear function of Re~R and independent of Im~R for all nz, thus the harmonic-oscillator region corresponds to the Bat portion of the surface seen in Fig. 2 for n2 —— 0. Figure 3 shows a path through the harmonic-oscillator region along which the complete Schrodinger equation (1) is integrated. The path passes through the harmonicoscillator region which connects the point Ro, where all three particles are close together with the asymptotic region where the &agments are &ee. In the classicaltrajectory approximation with dt = MdR/K(R), where K(R) = /2M[E+ (4Z —1)/R], the system propagates, to good approximation, according to the Schrodinger equation for a time-dependent harmonic oscillator [7—9]. For real R the propagator is known exactly from the work of Peterkop [10] and others [11,12] and is readily used for complex R to obtain, for example, the Wannier threshold law for the ionization of atomic hydrogen by proton or electron impact. As emphasized by Rau, only the zeroenergy solutions are needed to obtain the threshold law. Rau's method for extracting the threshold behavior from the zero-energy solutions has been criticized by Peterkop [13]; thus it is instructive to see how the threshold law emerges &om the construction presented here. The probability P(E) for ionization by the top-of-thebarrier mechanism is given in the adiabatic approximation by the expression
K(R)
4
3—
0
10
To obtain the correct threshold law the diabatic value of e(R) must be used. This quantity can be obtained from Ref. [7], but it was obtained much earlier in Ref. [11], where the diabatic energy eigenvalue was identified as a complex screening potential. In the notation used here the diabatic energy eigenvalue e"(R) obtained from Ref.
[11] is
e"(R)
= —[(4Z —1)/R] —i(~/[(4Z —1)/2M](1/R)
~
(6) where [11]
= ~R(4Z —1)-'~'
65
R4275
15
20
25
Re&R (a.u. )
FIG. 3. Schematic illustration of the complex R plane, showing how the T series of branch points separates the region where e(R) has the familiar Hp+ Rydberg electron-energy-level structure [e(R) —Z /2n near the real axis from the region where it has the harmonic-oscillator ] structure of Eq. (3). Also shown is a path through the latter region along which the Schrodinger equation is integrated to obtain the threshold law for ionization.
= —-+ ([16Z/(4Z —l)](M/
)
+ ( —)} ~,
(7)
and where the mass ratio M/mo equals 916 or 4 for the ionization of atomic hydrogen by proton or electron impact, respectively. In both cases Z = 1. The diabatic energy eigenvalue is substituted into Eq. (8), the integral over R is carried out, and the result expanded in powers of the total energy E and lnE to obtain the total ionization cross section: der
= const x E~~
(8)
Equation (8) with (7) represents the standard result obtained by integrating along the real axis, with the additional assumption that fragmentation pertains only to a part of a wave function that remains near the top of the barrier as the particles separate to infinite distance; i.e. , perfectly absorbing boundary conditions are implicitly assumed. In addition, it is necessary to determine the energy and angular distribution of the ejected electrons in order to compute the total cross sections. These distributions are ambiguous, with diH'erent procedures giving diferent results. Furthermore, solutions with energies E & 0 must be matched at large but finite R to &ee particle solutions. In contrast, all of these issues are avoided here since the total cross section is determined directly. When the integration is taken along an appropriate path in the complex plane the component of the wave function localized near the top of the barrier at Ro remains localized in that region as the system propagates through the harmonic-oscillator region to the fragmentation region. The total ionization cross section is then given directly by Eq. (5). The angular and energy distributions are not needed to extract the value of the Wannier index. Equation (8) is presented as an example of integrating along a path in the complex R plane to treat the top-of-the-barrier classically unstable motion quantally. The key to the new perspective that emerges &om this extension of standard methods is the identification of a harmonic-oscillator region illustrated quantitatively for the H2+ system. It must be emphasized that such regions exist for a wide variety of classical motion near a point of unstable equilibrium in the potential. It is only necessary that Imk(R) 0 for a region to exist where the eigenvalue e(R) has a spectrum characteristic of a harmonic oscillator with a corresponding wave function y(R; x) localized on the top of a potential barrier. The role that such regions play in dynamical processes depends upon the details of how systems get into and out of
)
J. H. MACEK
R4276
AND S. YU. OVCHINNIKOV
such regions. In the particular example presented here, one border of the re~ion is close to the origin, so that propagation of the wave function from the origin to Ro can be computed essi. ntially exactly, using, for example, the R-matrix methocl. Also, the harmonic-oscillator region extends to infinite R; thus the oscillator solutions connect the small-R region with the fragmentation region. The Wannier threshold law provides a simple consequence of this connection, as we have seen. Since propagation along classically unstable trajectories is most relevant to systems where the semiclassical approximation holds, it is desirable to compute the eigenIn this connecvalue e(R) in the WKB approximation. tion, note that the Bohr-Sommerfeld quantization conditions do not adequately represent the eigenvalues for complex R since they cannot describe branch points. To obtain semiclassical equations for complex R, consider a generic potential with a barrier at fixed R with turning points z;, i = 0, 1, 2, 3 and a top-of-the-barrier position zT. For simplicity, we consider potentials symmetric about the origin at z = zT —0. The eigenvalues of the Schrodinger equation for this potential are computed approximately by joining %KB solutions in the potential-well region with parabolic cylinder functions in the top-of-the-barrier region. This gives this semiclassical quantization condition
P(R) + g(+)(R) = 7r[n'+ (4 p 1)/8],
(9)
where the minus sign refers to even functions (n' even) and the plus to odd functions (n' odd), and where
P(R)
=
q(R; z)dz, Z2
QIi+
(R)
= arg{I'(2 + 1)/2 + ia(R)]} +a(R) (1 —in[a(R)/2)/2,
with
q(R z)
= Q —2m[e(R) —V(R, z)]
and
a(R)
4i =— x~
q(R; z)dz
= [m/ —V" (R, zT')]
~
[V (R, zT )
—e(R)],
where the primes denote derivatives with respect to z and the approximate expression for a(R) holds when the turning points z2 and z3 are near the top of the barrier. The branch of the square root in the de6nition of q(R; z) is taken such that the imaginary part of q(R, z) [1] K. Richter et al. , J. Phys. B. 225, 3929 (1992). [2] G. H. Wannier, Phys. Rev. 90, 817 (1953). [3] U. Fano, Rep. Prog. Phys. 46, 97 (1983). [4] A. Temkin, Phys. Rev. Lett. 49, 365 (1982). [5] S. Y. Ovchinnikov and E. A. Solov'ev, Zh. Eksp. Teor. Fiz. 90, 921 (1986) [Sov. Phys. JETP 63, 538 (1986)]; Comments At. Mol. Phys. 22, 69 (1988). [6] D. 1. Abramov et aL, Phys. Rev. A 42, 6366 (1990). and J. Macek, J. Phys. A 22 [7] D. Jakubassa-Amundsen
49
is negative. Note that P(R) is the usual WKB phase and Po+ (R) is an additional phase coming from the top-ofthe-barrier region described by the parameter a(R). Along the real axis for e(R) V(R;zT), the phase Po(+) (R) varies between — and vr/4 vr/4. This limited variation of the term that comes from the top-of-the-barrier region precludes any marked oscillator structure on the real axis for fixed R. Alternatively, for complex R, Po varies rapidly near a(R) = i(nz + 2), so that solutions of Eq. (9) with 0 & nz oo are always possible for some values of complex R. The energy eigenvalues are those of a harmonic oscillator with a complex spring constant as seen in Eq. (3) for the specific example of H2+. The harmonic-oscillator region is separated from the region near the real axis by the T series of branch points. In contrast to the Bohr-Sommerfeld quantization conditions, Eq. (9) describes branch points in energy eigenvalues as functions of complex R, owing to the Po+(R) term. These are just the T series of branch points that form the boundary of the harmonic-oscillator region illustrated in
)
(
Fig. 3. The harmonic-oscillator
structure for some region of complex R emerges from the generalized semiclassical quantization conditions of Eq. (9). Branch points separate the harmonic-oscillator region from regions where the energy eigenvalues are given by the Bohr-Sommerfeld formula. As articulated in Ref. [5] these branch points play a key role in describing transitions in systems where the parameter R varies with time. Since the harmonicoscillator region provides a quantal description of classical motion near a point of unstable equilibrium, we see that there is a class of transitions that relates to such classically unstable motion. Ionization of atoms by charged-particle impact near the threshold for ionization represents one well-known example of this connection. We have seen, for example, how the Wannier threshold law derives quantally from propagation through the harmonic-oscillator region, while Wannier obtained the law classically by considering motion along unstable trajectories. By integrating in the complex plane, many controversial issues related to the matching of wave functions at boundaries are avoided.
This research has been sponsored by the Division of Chemical Sciences, U. S. Department of Energy, under Contract No. DE-AC05-84OR21400 managed by Martin Marietta Energy Systems, Inc. Support for collaboration with the IoKe Physical Technical Institute, St. Petersburg, Russia, is provided by the National Science Foundation under Grant No. PHY-9213953. 4151 (1989). G. J. Popadopoulos, Phys. Rev. D ll, 2870 (1975). [9] A. K. Kazansky and V. N. Ostrovsky, J. Phys. B 25, 2121 (1992). [10] R. Peterkop, J. Phys. B 4, 513 (1971). [11] A. R. P. Rau, Phys. Rev. A 4, 207 (1971). [12] James M. Feagin, J. Phys. B 17, 2433 (1984). [13] R. Peterkop, J. Phys. B 16, L587 (1983). [8]
