Fachbereich Physik, Universitat Kaiserslautern, 6750 Kaiserslautern, Federal Republic of GermanyReceived November 24, 1986; accepted January 13, 1987
Exact rigid-body solutions for two-electron atoms have been derived within classical mechanics. These solutionsdescribe equilibrium configurations of the electrons, i.e., the electrons do not experience any force. Exactly twonondegenerate equilibria exist with singly periodic rotation. Linear stability of these classical solutions emergesfrom the structure of highly nonlinear, coupled equations of motion rather than from geometrical potential barriers.This new type of strong correlation effect generates an atomic rotation spectrum of resonances converging to thedouble-escape threshold. These resonances should be observable in multiphoton absorption spectra. The bindingmechanism of these resonances is provided by trapping the correlated electron pair onto an attractive phase-spacedomain. The motion within this basin of attraction consists of periodic, quasi-periodic, and chaotic components.
INTRODUCTION
Atomic spectra are often analyzed within the framework ofthe independent-electron model. In this model each elec-tron is assumed to move in a nonlocal potential generated bythe other electrons. In critical situations of strong or evendominant correlation, however, the independent-electronmodel breaks down. Collective electron motions ratherthan approximate independent electron motions then takeplace. Numerous examples of strong correlation effects areknown, particularly in cases in which an electron pair movesin an unfilled shell. For the simplest case of two-electronatoms such as He and H- but also for alkali earth atoms andnegative alkali ions, correlation phenomena have been stud-ied theoretically in great detail.'
A prototype for a strongly correlated motion of a highlyexcited electron pair was reported long ago by Wannier.2He considered the classical equations of motion for a nonro-tating He-like atom and derived an equilibrium configura-tion for the electron pair. The equilibrium occurs if thethree bodies (nucleus plus two electrons) form a collinearconfiguration that has equal electron-nucleus distances (ri= r), with the nucleus in the center corresponding to theangle 12 = COS-'(Pl P2) = r. The motion itself is symmetri-cally stretching, i.e., the electron-nucleus distances aretime-dependent functions, r(t) = r 2(t). Below the double-escape threshold, this motion is bounded by two turningpoints; above threshold, this motion is unbounded and leadsto double escape. The potential energy surface of a two-electron atom shows saddle structure in the neighborhood ofthe equilibrium; the angular correlation is stable, whereasthe radial correlation is unstable. This instability of theWannier mode implies a finite lifetime of Wannier ridgeresonances3 -5 below threshold and a competition betweensingle and double escape above threshold. We remark thata classical description of the Wannier mode is a suitableapproximation because the motion takes place along anequilibrium domain, i.e., where wave-number variations aresmall.
In this paper we report on an entirely new type of correlat-ed motion different from the Wannier mode. We look forclassical orbits of a two-electron atom that describe a rotat-ing rigid body, i.e., we seek exact solutions of the equations of
motion such that the two electron-nucleus distances, r1 andr2, and the electron-electron separation, r12 = I - r2l, areconstant in time:
The only degree of freedom is then an overall rotation of theatom. The existence of rigid-body solutions for an atom isfar from obvious because the potential energy surface has nominimum. However, we will see below that rigid-body solu-tions indeed exist for a two-electron atom. The electronsare then in an equilibrium such that they do not experienceany net force. We presume that analogous solutions existfor many-electron atoms and quite generally for many-bodyCoulomb systems including molecules. The classical rota-tion of a rigid body (top) may be singly periodic or doublyperiodic in time. The latter case occurs only for the asym-metric top. The rigid atom orbits presented below are sin-gly periodic. The projection of the angular momentum ontoa body-fixed axis is then conserved. It can indeed be shownthat doubly periodic equilibrium atoms do not exist.
Rigorous stability of rotating equilibrium atoms cannot beexpected because the potential energy surface has no mini-mum. Within the framework of a linear stability analysis,however, we find a confinement for the atom, i.e., orbits aretrapped onto an attractive domain of the phase space. Thisattractor is provided by the structure of the nonlinear andcoupled equations of motion. From a quantum-mechanicalviewpoint this classical attractor provides the binding mech-anism for a new type of resonance. Level positions of theseresonances are obtained semiclassically.
The lifetimes are at present unknown; the structure ofclassical orbits, however, indicates long-lived states. Ex-perimental data for comparison do not exist; atomic spec-troscopy below the double-ionization threshold is highly de-sirable.
CLASSICAL EQUATIONS OF MOTIONHyperspherical coordinates have proved to be particularlyconvenient for the analysis of two-electron correlations.1' 6' 7
We use here a parameterization of position coordinates thatincludes Euler angles because we want to separate overall
rotations from internal motions. We parameterize labora-tory-fixed electron position vectors ri (i = 1, 2) in terms ofbody-fixed vectors pi as follows:
cos a -sina 01 1 0 0
ri = sin a cosa 0 (0 cos1 -sinBL 0 = 1 L0 sinB cos 3
cos y -siny 01X sin-y cos 0 Pi, (2)
L 0 0 0]
where a, 1, and y are Euler angles. We now describe the
body-fixed vectors pi in terms of three hyperspherical coor-dinates r, 4, and <,:
cos 4'
P1,2 =sin4'
cos 1/2( + 2
(I 2 ) ,sin _/2(P 3
0]
(3)
where the plus (minus) belongs to the electron labeled by thesubscript 1 (2). These coordinates have the following mean-ing. The hyperradius r is given by
r = rl2 + r22, (4)
r1 and r2 being the electron-nucleus distances. The pseudo-angle 4 (O _ 4' 7r/4) parameterizes the inertia I in itsdiagonal form
I, = r2 sin2 4,
I2 = r2 cos2 4,
I3 = I 2 = r2, (5)
i.e.,
cos 24' = 211 (6)I3
is the asymmetry of the atom. The subscripts 1, 2, and 3 inEqs. (3) and (5) label the principal axes. Different particleconstellations with the same principal moment of inertia aredistinguished by the kinematic rotation angle <, (O so _ 27r).
It is now straightforward to calculate the kinetic energy(in atomic units)
Vol. 4, No. 5/May 1987/J. Opt. Soc. Am. B 789
ric top, the moment of inertia given by Eq. (5). The Coriolisterm finally reads as
Tcor = 12(a cos 1 + j),r 2 sin 24.
The potential energy
V=-Z + 1 + 1r2 JIrl-r2I
has the form
V = -C(4' f),r
where the function C is given by7
C(4', o) =-Z21 12 [(1 + cos 24 sin p)-1/2
+ (1 - cos 24' sin f)-1/2]
+ (1 - cos24 cos )-1/2
(11)
(12)
(13)
(14)
The Lagrange equations of motion can now be written. Thevariation with respect to the Euler angles a, , and -y leads tothe equations
r2 [sin 2(3(sin
2 4 sin 2 -y + COS2 4' COS2 y) + COS2 3]
+ 'r2 cos 3 -1/21r2 sin ,B cos 24 sin 2-y
+ 1/2 r 2 cos 13 sin 2' =L
dtd [Ar2 (sin 2 4' cos2 -y + cos2 4' sin2 -y)
-/ 2ar2 sin sin 2-y cos 24']
= /2 ar sin 2/3(sin2 4' sin2 -y + cos2 4 cos2 Y-1)
-1/2a/3r2 cos 13 sin 2-y cos 24 - r2 sin
-L / 2a 4 r sin 1 sin 24,
dt ('yr+ ar 2 cos 1 + '/2 1r2 sin 24')
= - /2 a2 r2 sin2 13 sin 2-y cos 24
(15)
(16)
+ /2,32r2 sin 2'y cos 24 - 1r 2 sin 13 cos 2-y cos 24'. (17)
In Eq. (15) we have introduced the laboratory-fixed z com-ponent of the angular momentum:
(18)
L being the Lagrange function. From the variation withrespect to the internal coordinates r, 4', and <,, we find that
T = /2(ft2 +kr22) (7)
in terms of the new coordinates. One finds, as usual, threecontributions,
T = Tint + Trot + Tcor, (8)
where
Tint = 1/2 + 2r + 41,f) (9)
is the internal kinetic energy and
Trot = /2c 2 r2 sin2 1(sin 2 4 sin 2 y + cos2 4' cos2 -y) + cos2 1]
X We search now for exact solutions of the equations of motion0 such that we may describe a rotating rigid atom. "Rigid".c here means that the mutual distances between the particless are constant in time; see Eq. (1). In terms of our coordi-.+ nates, Eq. (1) reads as
O r = f = f = 0. (24)
It can be shown analytically that the coupling between over-all rotations and internal motions forbids a doubly periodicrotation for rigid-body solutions. 8 Also, for the singly peri-odic rotation, the Euler angles 13 and y are constant:
=Y1 = . (25)
The angular momentum then shows in the laboratory-fixedz direction, L = L, and its body-fixed projection is alsoconserved. The solution of this algebraic problem was de-rived elsewhere.8 9 Here we give only the result. Exactlytwo solutions are found, an equilibrium rotor and an equilib-rium top (see Fig. 1), which are referred to in what follows ase-rotor and e-top for brevity. The constant coordinate val-ues are the following:
(i) for the e-rotor
r = L 2/CO, o0= (o = ,
Bo = arbitrary, 'o0 = 0.
(a)
r,
zN
-4 r2/_ _ _
Fig. 1. Equilibrium configurations for a two-electron atom: (a) e-rotor, (b) e-top. The symbols Z and e denote the positions of thenucleus and the electrons, respectively. L is the angular momen-tum. In (a) the electrons move on one circular orbit, 1800 out ofphase. In (b) each electron performs a circular motion in its ownorbit. These orbits are parallel to each other, and these circularmotions are in phase.
(26)
(ii) for the e-top
L 2CO= CO'os2 °
(o = O. 0 = 7r/2,
= sin-' (4Z)- 1/3,
The geometrical shape of the e-rotor is coincidentally thesame as that of the Wannier configuration. The three parti-cles are in a collinear configuration, the nucleus in the centerand the electrons at the ends. The electron-nucleus dis-tances are equal; spo = r corresponds to r = r2. The angle O= 0 corresponds to a solid angle 012 = CoS1( 1
* 2) = .However, in contrast to the Wannier mode, the length of thee-rotor is constant such that the sum of the centrifugalrepulsion and the electron-electron repulsion compensatesfor the electron-nucleus attraction. The constant C in Eqs.(26) is given by
Co = C(' 0, 'P0rotor = 21/2 (4Z - 1),Z being the nuclear charge.
The shape of the e-top is an isosceles triangle, nucleus andone electron, respectively, at the corners. The electron-nucleus distances are equal; o = 0 corresponds to r = r2 if012 < /2. The solid angle 012 is given by 012 = 2 o; its valuefor a nuclear charge of Z = 1, for instance, is 012 780. Theoverall size of the triangle is again given by a total compensa-tion of all acting forces. The constant Co' in Eqs. (27) hasthe value
Co' = -C(4' 0 , O) top = 23/2 Z - 21/6 Z'1 /3 . (29)
The rotation axis is parallel to the interelectronic separationr12 = r- r2. This is the principal axis with an intermediatemoment of inertia; the e-top is an asymmetric top.
A remarkable result is the expression for the energy forboth the e-rotor and the e-top. For a rigid body one expectsa classical rotation energy proportional to the squared angu-lar momentum,
EL = BL 2.
The rotation constant B is given by B = 1/21, I being the(b) moment of intertia. In our situation, however, the momentof inertia depends on L. The equilibrium hyperradii r for
the e-rotor and the e-top are proportional to L2 [see Eqs. (26)and (27)], the moment of inertia is proportional to r2 [see Eq.(5)], and the rotation constant B is therefore proportional toL-4 .
The rotation energy thus reads as
EL = - bL2
where the constant b is given by
(30)
b f/2 C02 for the e-rotor (31)
1/2(Co' cos 4'o)2 for the e-top
Quantizing the rotation, we must replace the classical quan-tity L2 by L(L + 1), where L is now a positive integer. 10
From Eq. (30) we then obtain two series of atomic rotationlevels converging to the double-escape threshold
EL = E _ bEwhere tn L(L + 1)'
where the new rotation constant b is given by Eq. (31).
(32)
-
t
TYo = . (27)
(28)
e+
Vol. 4, No. 5/May 1987/J. Opt. Soc. Am. B 791
The unexpected result [Eq. (32)] emerges from the highnonrigidity of the atom. For a small centrifugal stretchingthe traditional rotation energy expression is"l
EL - BL(L + 1)-DL2 (L + 1)2+ ..., (33)valid, of course, for
DL(L + 1) << B. (34)
The softness of the Coulomb forces, however, allows for avery large centrifugal stretching such that Eq. (33) does notconverge. Equation (32) is an exact result.
M= - Fi (40)
The stability of atrajectory x0 can then be obtained from thespectrum of Liapunov characteristic exponents (LCE's).'2
In our situation the LCE's are simply equal to the eigenva-lues of the matrix M because the equilibrium trajectories x0are constant.
We now report on the LCE spectra for the e-rotor and thee-top; details of the calculation may be found elsewhere.13'14
The e-rotor LCE spectrum can be calculated analytically;the result reads as follows:
= io2 -3,STABILITY
The investigation of stability of a mode of motion is a stan-dard problem of mathematical physics. The most conve-nient way to proceed is as follows. We employ Eq. (15) toeliminate the angular velocity a. Equations (16), (17) and(19)-(21) then represent a coupled system of second-orderdifferential equations for the five generalized coordinates q= (, -y, r, 4, v). We now introduce the momenta conjugateto qi,
Pi =-'
to rewrite the equations in Hamiltonian form:
=Hqi = -
aHPi -aqi
(35)
These are 10 first-order equations replacing the 5 second-order equations. The symbol H in Eqs. (35) is the Hamilto-nian restricted to a fixed value of L,. The 10 equations [Eq.(35)] are coupled and nonlinear; their solution [q(t), p(t)] =
x(t) describes the motion of the electron pair in a 10-dimen-sional phase space. Formally, Eqs. (35) have the structureof a general dynamical systemi , Fi(xl, ...m, X1r), n10or, in matrix notations
x = F(x). (36)
Our equilibrium solutions given in Eqs. (26) and (27) arefixed points (also called critical points) of Eq. (36). For anequilibrium the coordinates qi are constant, and the momen-ta are equal to zero; an equilibrium solution is thereforedescribed by a constant phase-space vector x0:
F(xo) = 0. (37)
Now consider trajectories in the neighborhood of an equilib-rium trajectory x0:
x=x 0 +y. (38)
For small departures from the equilibrium, we linearize Eq.(36) and obtain
y = 9)ly, (39)
where the 10 X 10 matrix 9) is given by its elements
3,4 = ,
X56= iCL0 1'7,8 - (-s + s2+ t)12L-3,
Xs,1 = +i(s + S2 + t) 1 2L-3, (41)
with
s = Co4(1 + 3 cos2 13 + C1- _C)
t = Co7C2 (sin2 10 +
The angle 0o describes the orientation of the rotor; see Eqs.(26). Figure 1(a) assumes that 1o = 0. The constants C0, Cr,and C2 are the Taylor series coefficients of the function C(i,,) [see Eq. (14)] at the saddle point
C(A )=-C + /C12 - 1/8C2(#- 7)2 + , (42)
and the numerical values of the coefficients are given by Eq.(28) and by
C1 -1/2
C2 = 2-1/2(12Z - 1). (43)
For the e-top the calculation must be done numerically.The result for a nuclear charge of Z = 1, corresponding toH-, is
X1,2 = A1.2658iL-3 ,
X34 = +3.2747iL3 ,
X56 = +22.134L-3 ,
X7,8 = 055896[exp(+1.16717i)]L 3 ,
9,10 = 0.55896[exp(+4.30876i)]L- 3 , (44)
valid for any value of the angular momentum L > 0. Onecan easily verify that the sum of all X is equal to zero, as itmust be for a conservative system.
The essential point is the following: In both cases thematrix 9R is block diagonal such that subsets of the LCE'sdescribe motions in subspaces of the phase space. Thisobservation also confirms that hyperspherical coordinatesare appropriate for three-body Coulomb systems. Table 1summarizes the correspondence between LCE subsets andphase subspaces. Two results are noteworthy: First, theLCE's describing the hyperradial motion are purely imagi-nary for both the e-rotor and the e-top; these motions are
C
>4
0
CD
0
c:
Mo
Xo
Hubert Klar
792 J. Opt. Soc. Am. B/Vol. 4, No. 5/May 1987
Table 1. Correspondence of the LiapunovCharacteristic Exponentsa
a See text. The LCE's for the e-rotor are given in Eqs. (41) and those for thee-top in Eqs. (44). The momenta conjugate to coordinates qi = (r, fl, y, P, so)are denoted by pq,-
therefore bounded. Since the hyperradius is the only coor-dinate that permits us to describe' a fragmentation of theatom (r - a), we conclude that there is linear stabilityagainst fragmentation for both the e-rotor and the e-top.For the e-rotor the hyperadial motion is periodic, whereasfor the e-top this motion is quasi-periodic because the fre-quencies IX1,21 and 1X3,41 are incommensurable.1 4
The second remarkable result is that the hyperangularmotion in both cases has a real-positive LCE. This gener-ates strong sensitivity on initial conditions and leads, ingeneral, to a chaotic time evolution in compact domains withdimension >3. For the e-rotor the positive LCE (X7) de-scribes a motion in the four-dimensional subspace (, Mo, ppp,) of the radial and angular correlations of the electron pair,pa and p being the momenta conjugate to 4' and o. Forsmall departures from the equilibrium the angle 4' describesangular correlation
012 cos1(iQ1 * 2) - r-242,
and the angle so describes radial correlation
r2 -r2 2r12+ r2 -
The overall rotation is stable; the orientation 13 of the rotor isconserved in linear approximation, and the motion in the (y,py) phase plane is periodic. For the e-top, relations (45) and(46) must be replaced by
012 24' (47)
and
r 12 - r2
2
2 + 2 2 X COS 2 i.(48)
For small departures from equilibrium we find a quasi-peri-odic time evolution for the angular correlation as we did forthe hyperradius. The radial correlation up [see relation (48)]and the rotation of the e-top described by the angles 13 and yshow a chaotic time evolution in the corresponding six-di-mensional phase subspace.
The result of this stability analysis may be summarized asfollows. The rotating equilibrium atom is linearly stableagainst decay. Rigorous stability can be neither expectednor concluded from this analysis. Nonperiodic irregularmotions within a finite volume are identified by positiveLCE's.
OUTLOOK
The above results may be employed to construct a quantum-mechanical wave function in the neighborhood of the equi-librium. We remark that, near equilibrium, the motion isessentially classical because local wave-number variationsthere are small. Away from turning surfaces, a wave func-tion has then a Wentzel-Kramers-Brillouin structure
4(rl, r2) = A exp(iS),
where S is the classical action and A is an approximatelyconstant amplitude.
For the purpose of illustration, we restrict ourselves hereto the e-rotor and determine the action S in terms of a powerseries
S = aL + '/2(r - ro)2Srr + -YST + WY2S,,,
+ 1/-Y2s, + V /2(So - 7r)2S VV + .... 5The coefficients of this expansion are obtained by the re-quirement that the gradient of S be equal to the momentumof the electrons. Within the linear approximation, thesecoefficients can be expressed by the LCE's with the follow-ing result:
Srr = + iC02L-3,
S, = L COS 13o,Sly = iCo-L 3 sin2 10,So = XCo-2L4,S4,a = (4X)-1C0C2L-2, (51)
where X is one of the LCE's 7, ... , Xo; see Eqs. (41). Wereject the negative imaginary coefficient Srr because thisyields an exponentially diverging wave function. We thenfind a wave function showing the structure locally:
where the function is bounded on the hypersphere withradius r. A confinement of the atom is described by aGaussian centered at r = L 2/CO. Equation (52) thereforedescribes a resonance; for the e-top one finds a similar re-sult. 4
The binding mechanism for this new type of resonance isprovided by trapping the electron pair as a whole onto anattracting equilibrium domain of the phase space, a mecha-nism substantially different from the traditional picture of a"bound state in a continuum." Wannier ridge resonances3 -5
may also be considered charge distributions trapped into anequilibrium. There are, however, two important differencesbetween a Wannier ridge resonance and the new type ofresonance described in this paper. First, the collective mo-tion of the two-electron charge density is a breathing motionfor the Wannier ridge resonance, whereas for our resonancethis is a rotation. Second, within a Wannier ridge resonanceindividual orbital angular mementa up to imax n 0.5nl/2 mixup.'5 In our case the equilibrium electrons are n circularorbits corresponding to max = n - 1. For large principalquantum numbers, n >> 1, this difference should be observ-able. A substantial mixing in the excitation spectrum of the6sl2s - 10sl3s two-photon transition in a with the 6hnh
0
+a
0-4
0
.)C)
-4
Hubert Mlar
(49)
(50)
Vol. 4, No. 5/May i987/J. Opt. Soc. Am. B 793
Rydberg series was observed recently.'6 We attribute thisrather complex excitation profile at least partly to the for-mation of an equilibrium configuration. Features typicalfor an equilibrium, however, should manifest themselvesmore clearly in high intrashell states with nonzero totalangular momentum. Such experimental data do not seemto exist at present. Direct excitation probabilities for thesestates are expected to be small if the initial state is a Rydbergstate, because the irregular nonperiodic motion within anequilibrium state has only little overlap with the highlyperiodic motions within a Rydberg state. On the otherhand, the smallness of this overlap provides a long lifetimefor the resonance.
ACKNOWLEDGMENT
Financial support by Deutsche Forschungsgemeinschaft(SFB 91) is gratefully acknowledged.
* Present address, Fakultdt fir Physik, Albert-Ludwigs-Universitat, 7800 Freiburg, Federal Republic of Germany.
REFERENCES AND NOTES
1. U. Fano, Rep. Prog, Phys. 46, 97 (1983), and references therein.2.'G. Wannier, Phys. Rev. 90, 817 (1953).3. U. Fano, J. Phys. B 13, L519 (1980).4. S. J. Buckman, F. H. Read, and G. C. King, J. Phys. B 16, 4036
(1983).5. A. R. P. Rau, J. Phys. B 16, L699 (1983).6. H. Klar, J. Math. Phys. 26, 162i (1985), and references therein.7. H. Klar and M. Klar, J. Phys. B 13, 1057 (1980).8. H. Klar, Phys. Rev. Lett. 57, 66 (1986).9. H. Klar, Z. Phys. D 3, 353 (1986).
10. The value L = 0 must be excluded because the equilibria need afinite rotation.
11. G. Herzberg, Molecular Spectra and Molecular Structure (VanNostrand, New York, 1978).
12. H. G. Schuster, Deterministic Chaos (Physik Verlag, Wein-heim, 1985).
13. H. Klar, Few Body Syst. 1, i23 (1986).14. H. Klar, Z. Phys. D (to be published).15. A. R; P. Rau, J. Phys. B 17, L75 (1984).16. W. E. Cooke, L. A. Bloomfield, R. R. Freeman, and J. Bokor, in
Fundamentals of Laser Interactions, F. Ehlotzky, ed., Vol. 229of Lecture Notes in Physics (Springer-Verlag, New York, 1985),p. 187.
