Helium Atom 707

CHAPTER 37. HELIUM ATOM 707

e e

θ r1 r2 Figure 37.1: Coordinates for the helium three body problem in the plane. ++ Chapter 37 He ++ --He e e Figure 37.2: Collinear helium, with the two electrons on opposite sides of the nucleus. r r Helium atom 1 2

trajectories in sect. 6.3. Here we will find periodic orbits and determine the relevant eigenvalues of the Jacobian matrix in sect. 37.1. We will explain in sect. 37.5 “But,” Bohr protested, “nobody will believe me unless I why a quantization of the collinear dynamics in helium will enable us to find parts can explain every atom and every molecule.” Rutherford of the full helium spectrum; we then set up the semiclassical spectral determinant was quick to reply, “Bohr, you explain hydrogen and you and evaluate its cycle expansion. A full quantum justification of this treatment of explain helium and everybody will believe the rest.” helium is briefly discussed in sect. 37.5.1. —John Archibald Wheeler (1986)

(G. Tanner) 37.1 Classical dynamics of collinear helium

o far much has been said about 1-dimensional maps, game of pinball and Recapitulating briefly what we learned in sect. 6.3: the collinear helium system other curious but rather idealized dynamical systems. If you have become consists of two electrons of mass me and charge −e moving on a line with respect S impatient and started wondering what good are the methods learned so far to a fixed positively charged nucleus of charge +2e, as in figure 37.2. in solving real physical problems, we have good news for you. We will show in this chapter that the concepts of symbolic dynamics, unstable periodic orbits, The Hamiltonian can be brought to a non–dimensionalized form and cycle expansions are essential tools to understand and calculate classical and quantum mechanical properties of nothing less than the helium, a dreaded three- p2 p2 2 2 1 body Coulomb problem. H = 1 + 2 − − + = −1 . (37.1) 2 2 r1 r2 r1 + r2 This sounds almost like one step too much at a time; we all know how rich and complicated the dynamics of the three-body problem is – can we really jump from The case of negative energies chosen here is the most interesting one for us. It three static disks directly to three charged particles moving under the influence of exhibits chaos, unstable periodic orbits and is responsible for the bound states and their mutually attracting or repelling forces? It turns out, we can, but we have to resonances of the quantum problem treated in sect. 37.5. do it with care. The full problem is indeed not accessible in all its detail, but we are able to analyze a somewhat simpler subsystem – collinear helium. This system There is another classical quantity important for a semiclassical treatment of plays an important role in the classical dynamics of the full three-body problem quantum mechanics, and which will also feature prominently in the discussion in and its quantum spectrum. the next section; this is the classical action (33.15) which scales with energy as

The main work in reducing the quantum mechanics of helium to a semiclassi- e2m1/2 S (E) = dq(E) p(E) = e S, (37.2) cal treatment of collinear helium lies in understanding why we are allowed to do (−E)1/2 so. We will not worry about this too much in the beginning; after all, 80 years and many failed attempts separate Heisenberg, Bohr and others in the 1920ties from with S being the action obtained from (37.1) for E = −1, and coordinates q = the insights we have today on the role chaos plays for helium and its quantum (r1, r2), p = (p1, p2). For the Hamiltonian (37.1), the period of a cycle and its spectrum. We have introduced collinear helium and learned how to integrate its 1 action are related by (33.17), Tp = 2 S p.

706 helium - 27dec2004 ChaosBook.org version14, Dec 31 2012 7 6 as 5 709 2 r 0, the 4 = ctron is 1 rom the = r 1 r 2 3 p he dynamics 2 cleus nearly si- 2 at s which we now nner particle hits rguments one de- depends now sen- n be reconstructed ugh to kick one of 1 ≤ rization of spectral ntial along the line en by the fugative. lectrons escapes for 2 electron (say electron n–electron exchange; r scape originates from plained in chapter 21, tal energy in the range ChaosBook.org version14, Dec 31 2012 0 s approach the nucleus .3 (b). We plot here the 7 6 5 4 3 2 1 0 ely, these nearly missed 0, ajectory is plotted in fig- motions dominate when- 2 r > 1 p , and pass close to the triple collision 2 r = 1 as the relevant axis. The dynamics can also r and treat a crossing of the diagonal 2 r 2 r ≥ and 1 1 r 0. As the unstructured gray region of the Poincarésec- = 2 r The cycle 011 in the fundamental domain or 1 r illustrates, the dynamics is chaotic whenever the outer ele 37.4: 1 (full line) and in the full domain (dashed line). r 2 r 37. HELIUM ATOM ≥ 1 Figure r ]. There is no energy barrier which would separate the bound f , figure 37.4. As a consequence, we can restrict ourselves to t fundamental domain r 2 r −∞ The collinear helium dynamics has some important propertie Things change drastically if both electrons approach the nu , = 1 1 − the dynamics in the fundamental domain is the key to the facto helium - 27dec2004 a hard wall reflection. Theby dynamics unfolding in the the full trajectory domain through can back-reflections. the As ex can the momentum transfer betweenthe the particles electrons out be completely. large eno In other words, the electron e the near triple collisions. this symmetry corresponds to ther mirror symmetry of the pote list. 37.2.1 Reflection symmetry The Hamiltonian (6.10) is invariant with respect to electro turning point of the inner electron.almost Only symmetrically if along the the two line electron in the unbound regions of the phaseduces space. that the From outer general electron kinematic a will not return when multaneously. The momentum transfersitively between on the how electrons the particlestriple approach collisions the render origin. the dynamics Intuitiv chaotic.ure A 37.3 (a) typical where tr we used tion for small The remaining electron is trapped in[ a Kepler ellipse with to 1) can escape, with an arbitrary amount of kinetic energy tak close to the origin during aever collision. the Conversely, regular outer electron is faralmost from any the starting nucleus. condition, the As system one is of unbounded: the one e CHAPTER be visualized in a Poincarésurface of section,coordinate see and figure momentum 37 of thethe nucleus, outer i.e., electron whenever the i 10 9 8 7 1 6 r 5 4 3 exercise 37.1 2 1 0 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 708 1 p (37.3) (37.4) b) force on 0 no longer = ul when calcu- , 2 2 10 2 Q 2 . ron has a maximal ions (6.19) are sin- Q 2 2 2 1 P The electrons are at- = 8 2 l dynamics is regular Q Q to the old coordinates negligible if the outer ChaosBook.org version14, Dec 31 2012 2 1 2 r on momentum is trans- = P P 2 6 1 1 4 4 r, less intuitive as a visual- p 1 r 0 or = = 1 2 4 = ˙ ˙ Q Q 2 1 , 1 Q 1 , the equations of motion take form 2 Q 2 P = r 2 1 1 r ; ; r = 0 / 8 6 4 2 0             10 1             dt 0, i.e., when both electrons hit the nucleus p 2 2 2 2 1 r 4 12 4 12 = = Q Q a) R R τ 12 d + + R 1 , – 1 1 r 2 2             1 r 2 2 2 1 Q Q Q = axis; (b) − − 2 2 r r 2 2 2 1 8 8 P P − − , 2 2             2 1 1 2 Q near the nucleus. Q Q ect on the charge of the nucleus, diminishing the attractive 1 37. HELIUM ATOM 2 2 r ff = = = 1 2 0) for collinear helium. Strong 1 ˙ ˙ P P r = when discussing the dynamics and the periodic orbits. 2 r 2 After a Kustaanheimo–Stiefel transformation The new coordinates and the Hamiltonian (6.18) are very usef (a) A typical trajectory in the r , 1 (6.19) and reparametrization of time by Individual electron–nucleus collisions at pose a problem to agular numerical only at integration the routine. triple The collision equat 37.2 Chaos, symbolic dynamics and periodic orbits Let us have a closer look at the dynamics in collinear helium. tracted by the nucleus. Duringferred an between electron–nucleus the collisi innerscreening and e outer electron. The inner elect lating trajectories for collinear helium; theyization are, howeve of the three-body dynamics.r We will therefore refer at the same time. CHAPTER the outer electron. Thiselectron is electron far – from electron thelike interaction nucleus in is at the a 1-dimensional Kepler collision problem. and the overal helium - 27dec2004 37.3: plane; the trajectory enters here along the 2 axis and escapes to infinity along the Poincarémap ( chaos prevails for small r Figure 001 0011 000011 0010110 0110111 01 0001 000001 711 by such meth- try left for the idence indicates the nucleus. The y; a time reversal mbersome to pro- ing. All cycles up to 0 cycle would corre- with zero kinetic en- nfinitely far from the ith symmetries, which ChaosBook.org version14, Dec 31 2012 t all periodic orbits are 6 1 he dynamics and is thus ed to intermittent behav- 011 0111 that we have already noted 001011 011111 4 1 r 2 r 0 2 6 4 2 0 1 r 2 = 0. r 1 ≥ 0 cycle is not in this list. The ≥ i − r 1 r domain, the 2 r – 1 r

37. HELIUM ATOM ) and the condition 2 r + 1 r ( Some of the shortest cycles in / 1 Note that the fixed point Search algorithm for an arbitrary periodic orbit is quite cu + 2 r orbit is the limiting caseergy. of The an orbit is electron in escapingmarginally the to stable. regular infinity The (i.e., existence separable) of limitior this of generating orbit t the is quasi–regular also dynamics relat for large nucleus while the inner electron bounces back and forth into that the dynamics of collinear heliumunstable. is hyperbolic, and tha spond to the situation where the outer electron sits at rest i based on the binary symbolic dynamicssymbol phase length space 16 partition (some 8000ods, prime with cycles) some have examples been shown computed in figure 37.5. All numerical ev helium - 27dec2004 can be easily founddynamics by in a the one-parameter fundamental search. domain is The time only reversal symme symmetr gram. There is, however, a class of periodic orbits, orbits w CHAPTER in figure 37.3 (b). / 37.5: 2 − 1 r / 2 fundamental domain. The last figure,00101100110111, the is 14-cycle an example of awith no typical symmetry. cycle The orbits are shown in the full itineraries refers to the dynamics in the Figure collinear helium. The classicalmotion collinear is electron bounded by the potential barrier − 710 0 and = amental 2 , leaves us r 2 r rémap in the ount the num- = analysis of the . One can show 1 m with the same . This suggests 2 r 15.7.2. However, 2 r dic orbits in multi- r tories and calculate polation algorithms epend sensitively on der the collinear he- = he similarity between in depth: sect. 21.6, p. 434 ne two-body collision = n be achieved only af- ChaosBook.org version14, Dec 31 2012 pondence between bi- 4, and the fundamental 0 or between two collisions damental domain, with 1 r 1 2 r = r riction of the periodic orbit 2 = between two collisions with r 1 2 r r = 1 r ce to calculate semiclassical spectral ffi 0; = 2 r 0. = 2 r 0 cycle. 0 before coming back to the diagonal ; the symbolic dynamics is linked to the Poincarémap 2 = r

37. HELIUM ATOM 2 ≥ r symbolic dynamics corresponding to the dynamics in the fund 1 r the nucleus with the nucleus 0 or Empirically, the symbolic dynamics is complete for a Poinca The potential in (37.1) forms a ridge along the line = 1: if a trajectory is reflected from the line 0: if the trajectory is not reflected from the line binary 1 fundamental domain, i.e., therenary exists symbol a sequences and one-to-one collinear corres trajectoriesexception in of the the fun determinants, to be implemented here inthe (37.15). fundamental Note domain of also the t collinear potentialdomain figure figure 37. 12.12 (b) inbinary symbolic the dynamics. 3–disk system, a simpler proble in general with a 2-dimensionaldimensional search. spaces have Methods been described to in find chaptergood perio 13. starting They guesses. d A systematicter search combining for multi-dimensional all Newton orbits methods ca with inter helium - 27dec2004 the symbols 0 and 1 are defined as domain 37.2.3 Periodic orbits The existence of aber binary of symbolic periodic dynamics orbitsmere makes in existence it of the easy these fundamental cycles to domain, does c not as su in sect. that a trajectory passing the ridger must go through at least o determinants. We need totheir determine periods, their topological indices phase and space stabilities.search trajec A to rest a suitable Poincarésurface of section, e.g. 37.2.2 Symbolic dynamics We have already made thelium claim that fully the chaotic. triple collisionssymbolic ren We dynamics lends have further no credence to proof the of claim. the assertion, but the CHAPTER a exercise 37.6 713 (37.6) (37.5) then has the O are given by itable orthogonal the direction of the llinear helium. The m he transformation to e energy manifold is m by ChaosBook.org version14, Dec 31 2012 ) along the two vectors t ( , rdinates (6.17), and state n of energy and displace- x T ) 2] Jacobian matrix. 2 × Q              H 1 2 − 2 1 Q Q , P P 1 H H Q − − ), which provides a transformation to mO H 2 T 2 P − , O R H R H R R H . 2 / / / / 1,2. The vector perpendicular to a trajec- P 2 1 1 2 = T + ) P P Q Q H 2 = 1 0, so the two vectors are orthogonal. , )) and to the energy manifold is given by the H H P 2 M P H H i 1 t = ( P H , H , ; the ‘trivial’ directions correspond to unit eigen- 2 i , 1 2 H m H P 1              P H 2 + 1 ( x m P Q ∂ ∂ P ∂ , P ∂ Q 2 ) 0 0 1 H H = n t H 2 Q ) H x ( = , − − v ∂ ∂ 1 n 2 H i , H x ∗ ∗ P Q P ∂ R + mn ∂ , R ) H R H are phase space vectors perpendicular to the trajectory and H 1 t R H ω R / , / ( mn 2 / 2 Q 2 / 22 12 1 2 2] matrix , , γ/ 2 1 1 = P 1 Q Q 2 ω H m m P Q × , Q γ ( H H ) H m , = , γ H H t ( ) γ − ( − 1 = t , and ) i ( m 21 11 t γ = ∗ ∗ ∗ 0 0 10 ( 2 m 1              ( H v Q | ) m m ∂ ∂ m t Q H ˙ ( H

37. HELIUM ATOM x ( l              ∇ = = = |∇ = = = = i = ) = m Q t ˙ m γ v m O ( H R x ). The vectors The vector locally parallel to the trajectory is pointing in The equations of motion for the reduced Jacobian matrix Next, we consider the orthogonal matrix v γ, two trivial eigenvectors corresponding to thements conservatio along a trajectorycoordinates can, transformations, however, see be appendix B. projectedlocal coordinates out We explicitly, will by here give su for t the the resulting regularized equations coo of motion for the reduced [2 By symmetry gradient of the Hamiltonian (6.18) phase space velocity (7.3) tory form with helium - 27dec2004 values on the diagonal in the 3rd and 4th column and row. The linearized motion perpendiculardescribed to by the the [2 trajectory on th with to the energy manifold in theJacobian 4-dimensional matrix phase (4.6) space of rotated co to the local coordinate syste local phase space coordinates centered on the trajectory CHAPTER ( exercise 37.5 712 kily, 1. By − ) plane. The = ng the orbits 2 r t start perpen- 2 , r ′ elf-retracing cy- 1 + nvariant and has 1 ace. r 1 r e Jacobian matrix ry as the only pa- formation; we can racing or not? All ions for the matrix l find time reversal 4-dimensional; the n either of the axis e same trace in the on, with the details p to length 5. There e ( o calculate stability se space is mapped ear helium with the o cycles are mapped + hat an orbit starts out the first two non-time mmetry. s up to length 5 fulfill ChaosBook.org version14, Dec 31 2012 2 2 011 in figure 37.5. r cycle is self-retracing if e., read backwards) and − 1 2 r 001101 in figure 37.5. Their − 0011 in figure 37.5; 001 in figure 37.5; ), by reversing the direction of the 2 p − , 1 p − 001011 and ( → ) 2 p if the itinerary is time reversal invariant and has an , 1 2 p r = 1 0 if the itinerary is time reversal invariant and has an even r = , or on the potential boundary 2 2 r r 001 in figure 37.5 are examples of self-retracing cycles. Luc = trajectories perpendicular to the boundaries and monitori

37. HELIUM ATOM 1 r ff 01 and 1, to the diagonal odd number of 1’s; an example is the cycle an odd number of symbols; an example is the cycle to the axis to the potential boundary if the itinerary is time reversal i number of symbols; an example is the cycle 0 or plane, but they trace out distinct cycles in the full phase sp 2 We are ready to integrate trajectories for classical collin Why is this observation helpful? A self-retracing cycle mus • • • r = – 1 2 cycles momentum of the orbit. Suchcle, an an orbit orbit must that be runs a back “libration” and forth along or the same s path in th symmetric periodic orbit isonto an itself when orbit changing whose ( trajectory in pha All cycles up to symbol lengthreversal 5 symmetric are time cycles reversal are invariant, determination cycles would require a two-parameteronto search. each The tw other byr time reversal symmetry, i.e., they have th shooting o help of the equations of motionsis (6.19) only and one to thing find not allelements yet cycles in u of place; the we Jacobian needindices. the matrix governing along We equat a will provide trajectoryof the in the main order derivation equations relegated t in to the the appendix next B.5. secti 37.3 Local coordinates, Jacobian matrix In this section, wealong will a derive collinear the helium equations trajectory. of motion The for Jacobianhelium - th matrix 27dec2004 is returning to the boundary withsymmetric the cycles right by symbol length varying werameter. the wil But starting how point can onthe we the relevant tell bounda information whether is a contained givenits in cycle itinerary the is is itineraries; self-ret invariant a undera time suitable reversal number symmetry of (i. cyclicthis permutations. condition. All The binary symbolic string tell dynamics at contains which boundary even the more totalperpendicular in reflection occurs. One finds t the shortest cycles that we desire most ardently have this sy CHAPTER dicular to the boundary ofr the fundamental domain, that is, o topological p m . for the motion in the collinear p p T / | σ p Λ | | p Λ | ln π 2 / p S ), Lyapunov exponent π 1.82900 0.6012 0.5393 2 3.61825 1.86225.32615 1.0918 3.42875.39451 1.6402 1.86036.96677 4 1.6117 4.43787.04134 6 2.1710 2.34177.25849 6 2.1327 3.11248.56618 8 2.1705 5.11008.64306 8 2.6919 2.72078.93700 8 2.6478 5.15628.94619 10 2.7291 4.59329.02689 10 2.7173 4.17659.07179 10 2.7140 3.3424 10 2.6989 10 10 10.13872 5.604710.21673 3.2073 3.032310.57067 3.1594 6.139310.57628 12 3.2591 5.676610.70698 12 3.2495 5.325110.70698 12 3.2519 5.325110.74303 12 3.2519 4.331710.87855 12 3.2332 5.000210.91015 12 3.2626 4.2408 12 3.2467 12 12 p 1 01 for the motion perpendicular to the collinear plane, and the 001 011 p 0001 0011 0111 σ 00001 00011 00101 00111 01011 01111 (in units of 2 000001 000011 000101 000111 001011 001101 001111 010111 011111 p S Action for all fundamental domain cycles up to topological length 6 p 37.1: m Table index plane, winding number 714 )] ) 2 2 P P 2 2 trix can P P . p H H )] and how to T 2 with respect + P + 2 2 e the Jacobian 1 1 P H = rized dynamics P Q 1 1 H p P coordinates. That Q ion can be imple- ntributions. When S + lculate trajectories. H t also allow for dis- H 1 elium is an invariant ted to displacements restrict the dynamics )( ivious to the other di- )( P atrix for dynamics for 2 1 1 ChaosBook.org version14, Dec 31 2012 2 Q 2 P P determinant (34.12). By ) H H H 8 P + + − 1 H 2 2 1 2 Q 2 Q Q Q 2 H H Q ( H ( H − + + ) (37.7) ) − 1 ) 2 2 1 ) P 1 P 2 Q P Q 2 1 P H H 1 2 P Q 2] matrices, with trivial eigen-directions 1 H Q Q Q 1 H H are given by × H P H H + + )( i j H )( l 2 2 1 1 + P Q P − Q 1 2 2 1 H P 2 Q Q Q 2 Q H Q H H H 2 , H depends on the trajectory in phase space and has the H Q )( 1 2 are the second partial derivatives of )(              + l 2 2 , Q H 2 2 P − 2 P 1 ( 0 0 0 P H , evaluated at the phase space coordinate of the classical 1 2 H 1 ( H i P = Q 2 Q P 1 + j + Q ∗ ∗ is proportional to the period of the orbit, H , Q 1 H 1 1 1 i , p Q . i 2 2 P Q Q H Q S , [2( H 22 12 j H H H 11 [2 l l 2 l ( ( 2 ( P 1 i 1 R R − − + − + P . The matrix 1 H ∗ ∗ ∗ 0 0 00 21 11 l 37. HELIUM ATOM l , = = = = the collinear helium subspace decouple, and the Jacobian ma j =              Q i ff = Q (0) 11 22 21 12 l l l l l H m 0 angular momentum is 6 dimensional. Fortunately, the linea There is, however, still a slight complication. Collinear h = L in and o where the relevant matrix elements with form be written in terms of two distinct [2 trajectory. 37.4 Getting ready Now everything is in place: the regularized equations of mot mented in a Runge–Kutta or anyWe other have integration scheme a to symbolic ca dynamicsfind and them know (at least how up many tomatrix cycles symbol whose there length eigenvalues enter are 5). the We semiclassical know spectral (33.17) how the to action comput to the coordinates Here 4-dimensional subspace of the full helium phaseto space. angular momentum If equal we zero, weis are not left a with problem 6 when phase computing space periodicmensions. orbits, they However, are the obl Jacobian matrixwe does calculate pick the up Jacobian extra matrix co placements for out the of full the problem, collinear we plane, mus so the full Jacobian m providing the remaining two dimensions. The submatrix rela CHAPTER helium - 27dec2004 remark 37.4 0. e- se 717 = 2 . That l n , share the = N 0 spectrum N 1 l r We therefore 1. How does = elium gen atom that rresponding to = ttribute angular − st semiclassical gure 37.6. The erent series can 1 series consists L f the full helium l erate; states with ff , that is, there are he classical phase N might not come as in figure 37.6. We = 2 ), corresponding to di z e of the spectrum. ith l n rth along a common inear helium. In the t always escape after N ... , at the collinear model ates which decay into l determinant in terms − N ChaosBook.org version14, Dec 31 2012 us section in mind: we 1 N , as the principal quantum = 0 for the helium three-body 1 N z L say. In order to obtain total l = 2 l 2 lie above the first ionization l and ≥ . . l and N N span 1 = l N ≪ 2 l l with 0 for fixed quantum numbers = n , Rydberg series corresponding to the same = 1 N l N L E ect which can only be understood by quantum ff as long as erent l , are removed by the electron-electron interaction. erent Rydberg series converging to each ionization ff , an e N . We will therefore refer to ff n + 1 bound state series, including even the ground state and show that the collinear model describes states for di 0 we need l . This is indeed the case and the He = N 2 = N L / N 2 1. As soon as we switch on electron-electron interaction the − = 0; we thus expect that we can only quantize helium states corr = N =

37. HELIUM ATOM N E L ect the helium case? Total angular momentum erent states corresponding to ff ff In the next step, we may even speculate which parts of the The hydrogen-like quantum energies (37.8) are highly degen erent angular momentum but the same principal quantum numbe di ff N moderate angular momentum of helium! We willthe also angular find momentum a semiclassical quantum number co the Rydberg series on thewill outermost see in left the side next of section that the this spectrum is indeed the case and th holds down to the 37.5.2 Semiclassical spectral determinant for collinear h Nothing but lassitude caneigenvalues. The stop only thing us left now to do from is to calculating set up our the fir spectra helium - 27dec2004 These series are the energetically lowest states for fixed ( can be reproduced by the semiclassical quantization ofaxis coll through the nucleus,expect that so collinear each helium has describes zero the angular Rydberg series momentum. w collinear helium, both classical electrons move back and fo of true bound states for all states of the helium ion We thus already have a rather good idea of the coarse structur principal quantum number arguments. number. We also see that all states threshold means that we expect a big surprise if we have the classical analysis of the previo states are no longer bounda states; bound they state turn of into the resonant helium ion st and a freealready outer found electron. that This one of the classical electrons will almos some finite time. More remarkable is the fact that the first, be identified also indegeneracies the between exact the helium di quantum spectrum, see fi problem is conserved. The collinear helium is a subspace of t space for threshold for angular momentum momenta to the two independent electrons, spectrum. Going back to our crude estimate (37.9) we may now a sponding to the total angular momentum zero, a subspectrum o di that a same energy. We recall from basic quantum mechanics of hydro CHAPTER the possible angular momenta for a given 716 n of ounts ective (37.9) (37.8) ff ating the two ogen like one- ial with e bility angles, and tum energy levels 1. p the discussion as accessible with the degrees of freedom ormation we need to = t) ionizations thresh- spects of our calcula- mics in the additional um. This will give us ChaosBook.org version14, Dec 31 2012 ~ of the nucleus neglect- he next section, an also then bound in hydrogen = ct. 37.5. ble 37.1. These numbers, e m 2, screening at the same time = = e , yielding the ground and excited Z is the mass of the electron. Through e . →∞ m N n > n with -coordinate in figure 37.1. It turns out that the linearized , 2 Θ n 1 2 2 2 N Z 2 − coordinate is stable, corresponding to a bending type motio e 2 2 m 2 Θ are obtained in the limit 1. In this way obtain a first estimate for the total energy N ~ 4 2 e − 2 = N − 37. HELIUM ATOM er here only a rough overview. For a guide to more detailed acc 1 = − ff = is the charge of the nucleus and n − , = N N Z Ze N E E E The numerical values of the actions, Floquet exponents, sta The simplest model for the helium spectrum is obtained by tre the linear configuration characterizes the linearized dyna ff dynamics in the degree of freedom, the topological indices for the shortest cyclesneeded are for listed the semiclassical in quantization ta implementedbe in helpful t in checking your own calculations. 37.5 Semiclassical quantization of collinear helium Before we get down tolet a us serious have calculation a of brief thea look helium preliminary quan at feel the for overall which structurehelp parts of of of our the collinear the spectr model helium – spectrumsimple and are as which possible are not. and to Intions order concentrate we to on kee o the semiclassical a see remark 37.4. 37.5.1 Structure of helium spectrum We start by recallingelectron atoms. Bohr’s The formula eigenenergies form for a Rydberg the series spectrum of hydr the two electrons. We willin need evaluating the the Floquet semiclassical exponents spectral for determinant all in se o helium - 27dec2004 charge This double Rydberg formula contains alreadyunderstand most of the the basic inf structure ofolds the spectrum. The (correc where the rest of this chapter we adopt the atomic units electrons as independent particles movinging in the the electron–electron potential interaction.like Both states; electrons the are inner electron will see a charge CHAPTER the nucleus, the outer electron will move in a Coulomb potent 719 ), re- . The σ σ (37.11) (37.13) (37.12) i π 2 perpendic- ± e minus sign sical spectral roduct of two e which equals iclassical spectral a functions, as in present in (34.12) ing number ., the eigenenergies 2 s of the eigenvalues ) the trace formula in nting displacements 18) we may proceed / nt directions perpen- E 1 form exp( ple repetitions of this ( ics along the collinear ChaosBook.org version14, Dec 31 2012 this degree of freedom ¯ − , N es of the shortest cycles ) π ) i ) e σ m , ir k p ( corresponds to the unstable π t 2 p − − e m ) not defined. In order to obtain − (1 the collinear space and E p ( , )(1 ¯ in ) N σ 0 2 p inverse hyperbolic periodic orbits with / = ir ∞ / σ 1 π m − 2 2) . / 0 e is the full winding number along the orbit | 1 ) = σ ∞ 1 and real. As in (19.9) and (34.18) we may + k r − σ ℓ 2) ( / > M π 1 = | (1 4 | + ) − ℓ − Λ 1 m ( r | , 2 π . Using the relation (37.12) we see that the sum − k ir − p ζ Λ − m Λ 0 e − = p ∞ with − rk det (1 m | } sS ) Λ 0 ( i r 2 ⊥ Λ ). The topological index = )(1 ∞ / e k 1 / r ⊥ 1 M k | 1 Λ r , = M Λ − Λ 2 − Λ | / sc { 1 − 1 ) 0 | (1 = E ∞ Λ − ,ℓ | k det (1 −

37. HELIUM ATOM − = ˆ = = H ) m , negative eigenvalues is the topological index for the motion in the collinear plan / k p ( in (37.10) is the expansion of the logarithm, so the semiclas sign corresponds to the hyperbolic t det ( p r to it makes it possible to write the denominator in terms of a p ± m erent ways, reflected by the absence of the modulus sign and th ff in collinear helium are listed in table 37.1. helium - 27dec2004 values for the actions, winding numbers and stability indic submatrices corresponding to the dynamics twice the topological length of thedicular cycle. to the The collinear two axis independe leadwhich to accounts a twofold for degeneracy an in additional factor 2 in front of the wind determinants. Stable and unstabledi degrees of freedom enter and dynamics in the collinear plane. Note that the factor The in front of det (1 ular thus write are resonances corresponding todeterminant the complex and zeros the mean of energy the staircase sem is absent in (37.10). Collinear helium is an open system, i.e flecting stable linearized dynamics. over positive in the stable degree of freedom, multiplicative underaxis multi are paired as orbit .The eigenvalues corresponding to the unstable dynam a spectral determinant as anas infinite in product (19.9) of by the expanding formof the (34. the determinants in corresponding (37.10) Jacobian inperpendicular matrices. term to the The collinear matrix space represe has eigenvalues of the where the cycle weights are given by CHAPTER determinant can be rewritten(19.9): as a product over dynamical zet N=8 N=8 N=7 N=7 N=6 N=6 N=5 N=5 2] 718 (37.10) × , -0.1 -0.04 -0.06 -0.08 -0.12 -0.14 -0.16 -0.18         N=4 2

/ 1 | ) p r M N=3 − s. The semiclassic- . As explained in ribed in sect. 6.3.1 ) 2 over all cycles of the π e ﬁrst few terms of its N=2 p ChaosBook.org version14, Dec 31 2012 um enters the classical det (1 determinant in the form | m 2 − / p 1 sS )) ( ⊥ ir N=1 p r e M − 0 -1 -2 -3 -0.5 -1.5 -2.5

det (1 E [au] E − ( r 1 1 4] Jacobian matrix decouples into two [2 = ∞ r × p −         exp , = e E sc m − ) E − 37. HELIUM ATOM 2 ˆ ~ e H = s det ( The exact quantum helium spectrum helium - 27dec2004 cycle expansion with the help ofal the spectral binary determinant symbolic (34.12) dynamic hasclassical been systems. written The as energy product dependencedynamics in collinear only heli through simplewhich makes scaling it possible transformations to write desc the semiclassical spectral of the periodic orbits of collinear helium and to write out th sect. 37.3, the fact that the [4 obtained by using the scaling relation (37.2) for the action with the energy dependence absorbed into the variable CHAPTER 0. The energy levels denoted by bars 37.6: = L have been obtained from fulltum 3-dimensional calculations quan- [37.3]. Figure for remark 37.5 . site 721 → ∞ (37.16) 2 ons, the r , 1 r ; the quantum namics with an havior of cycles he limit of large directly from its ... . arious ionization eacquainted with ical spectral det- odd number of 1’s t reproduces states − in the fundamental ﬀ ermine the zeros of these are the lowest ] ng on the symmetry indices and winding ) e the cycle expansion of the series after the hange. We may now ℓ re brackets. The cycle s intermittency in our ChaosBook.org version14, Dec 31 2012 ( 1 n channel t ) blem: there is no cycle 0 ℓ ( 011 t ] ) − ℓ ( 1 erent from those staying in the t ) ﬀ ) ℓ ℓ ( 01 ( 0111 t , the inner electron bouncing back t − ∞ 0 corresponds to the limit of an outer + ) ) = ℓ ℓ good quantum numbers ( 1 ( 011 t t 1 symmetry and enters the spectral deter- is very di r ) 1 corresponds to states being symmetric ℓ + 2 ( 001 ± t 0 cycle as C ) ℓ ( 001 − t →∞ [ 2 ) − r ℓ ( 0011 ) t ℓ or ( 01 t + 1 r − ) ) correspond to eigenfunctions where the wave function ℓ ℓ ( 0001 ( 1 t 2 . t [ 1 N t − − / 2 − − 1 1 are given in (37.12), with contributions of orbits and compo = ≈ ) ℓ = p ( ) t s N )

37. HELIUM ATOM ( s has been identified with an excitation of the bending vibrati ) E ( ℓ ( ℓ m /ζ /ζ 1 1 . This turns out to be equivalent to switching to a symbolic dy 2 r , 1 infinite alphabet. With this observation in mind, we may writ r To include those states, we have to deal with the dynamics in t (....) for a binary alphabet without the orbits of the same total symbolexpansion length depends collected within only squa onnumbers, the given for classical orbits actions, upthe stability cycle to expansion length formula 6 (37.16), infirst consider term table a 37.1. truncation To get r helium - 27dec2004 of the outer electron is stretched far out into the ionizatio states in each Rydbergthresholds series. The states converging to the v near core region. Awhere cycle both expansion electrons using are the localized binary in alphabe the near core regions: in the collinear helium. The symbol sequence spectral determinant. There is, however, still another pro or antisymmetric with respectstart to writing the down electron-electron the binary exc cycle expansion (20.7) and det system, a problem discussedgoing in far chapter out in 24. the channel We note that the be and forth into the singularity at the origin. This introduce exchange symmetry quantum number The weights two dynamical zeta functions defineddomain. as products The over symmetry orbits properties of an orbit can be read o electron fixed with zero kinetic energy at symbol sequence, as explained inin sect. the 37.2. itinerary An is orbit self-dual with under an the number CHAPTER minant in (37.15) with asubspace negative under or consideration. a positive sign, dependi 37.5.3 Cycle expansion results So far we haveerminant established and a have thereby factorized picked form up of two the semiclass 2 2 r en C 720 ≤ twice 1 2 (37.15) (37.14) 0 terms r r 12), i.e., etry. As = , with the fixed. The = ) ℓ k t. 37.5.1 be ( ℓ − 1 r in sect. 37.2. /ζ , with to factorize the 1 − − main, and the cy- ζ al domain cycle contributions les, i.e., cycles that ibrational modes in 1 formation. The sec- efore be interpreted ly for large negative ained by writing the keeping tential, unstable mo- resonances of the in- vicinity of the trajec- − own with increasing + ℓ ζ ChaosBook.org version14, Dec 31 2012 ss the axis e flow along a cycle cor- /ζ terms in an expansion of onding to the states sym- = k corresponds to ‘excitations’ /ζ which contributes as a phase k ℓ cient to take only the ffi . Higher erent roles in the semiclassical spec- k ff representing the single particle angular 2 , , . The index l ) ) ℓ ˜ ˜ . s s = t t ) ˜ s 2 1 − t + l − = (1 (1 l play very di ˜ ˜ s s (1 symmetry. Such cycles come in pairs, as two equiv- k symmetry as it is invariant under reflection across ˜ s 2 ) ) value corresponds to a full spectrum which we obtain a a 2 C t t 2 ℓ and C ) − − a ℓ t , antisymmetric resonances by the zeros of 1 (1 (1 ) − ℓ ( + a a (1 /ζ a in (37.14) are the weights of fundamental domain cycles. The = = ˜ s 0 in what follows. The symmetric subspace resonances are giv = values. As we are interested only in the leading zeros of (37. line corresponding to the electron-electron exchange symm 37. HELIUM ATOM t ) ) ) s 2 ℓ ℓ ℓ = ( ( ( + − r k /ζ /ζ /ζ = coordinate and can in our simplified picture developed in sec 1 1 1 decrease exponentially with increasing 1 ) Θ The integer indices Next, let us have a look at the discrete symmetries discussed r -line excitations and thus increasing m , k ff p ( to the cycle weights inas the a dynamical harmonic zeta oscillator quantum functionsthe number; can it ther corresponds to v tral determinant (37.12). A linearized approximationresponds of to th a harmonic approximation oftory. the potential Stable in motion the correspondstion to to a an harmonic inverted oscillator harmonic po oscillator. The index by the zeros of 1 already at half thecle period weights compared to the orbit in the full do along the unstable direction andverted can harmonic be oscillator identified with centered local ont the given orbit. The harmonic oscillator approximation willo eventually break d setting related to the quantum number symmetry now leads to the factorization of (37.14) 1 the zeros closest to the real energy axis, it is su at a right angle. The self-dual cycles close in the fundament Here, the first product isare taken not self-dual over under all the asymmetric prime cyc the determinant give corrections whichimaginary become important on alent orbits are mapped into each other by the symmetry trans ond product runs over all self-dual cycles; these orbits cro from the zeros of the semiclassical spectral determinant 1 momenta. Every distinct the helium - 27dec2004 explained in sects. 21.1.1semiclassical and spectral determinant. 21.5, The we spectrum corresp maymetric or use antisymmetric this with symmetry respectdynamical to zeta reflection functions can in be the symmetry obt factorized form CHAPTER into account. Collinear helium has a e 723 of the . We thus ℓ ects is given in ff structure ball all the way emiclassical he- er. victory and fold ff nd the semiclassi- d to the dynamics the collinear plane antum problem. of the classical prop- ter precision.” While acy. A sceptic might . They appear in various to o ant part of the helium traditional semiclassi- ChaosBook.org version14, Dec 31 2012 n H. Hopf [37.14] used a pectrum of helium. As tum number ? A straightforward nu- [37.12] who realized that hink about - what about us mass e table, that they are uniquely elium cycles have been found lly stable solutions for plane- ustaanheimo–Stiefel transfor- are root removal” trick (6.15), auli matrix two–body collision No stable periodic orbit and no . The KS transformation for the s. The technique was originally is the same as Dirac’s trick of getting , raction, tunnelling, ...? As we shall 2 normed algebras | ff Q | ers new insights into the = ff define r 2 | Q ects as di | ff = erent Rydberg series depicted in figure 37.6, a fact | ff Q Q quantization will not work and that we needed the full The full 3-dimensional Hamiltonian after elimination of th | with property 2 fermions by means of spinors. Vector spaces equipped with a / Q WKB Sources. Complete binary symbolic dynamics.

37. HELIUM ATOM ´ e 37.1 37.2 Having traversed the long road from the classical game of pin ´ esum R We have covered a lot of ground starting with considerations lium spectrum. We saw that the three-body problem restricte spectrum. We discovered thatwas the stable, dynamics giving rise perpendicular to to an additional (approximate) quan erties of a three-body Coulomb problem, and endingon with a collinear the appears s tocal be methods fully such chaotic; as this implies that a piece of unexpected luck the symbolic dynamics isspectrum, simple, including a the groundsay: state, “Why to bother a with all reasonable themerical accur semiclassical quantum considerations calculation achieves the same goal with bet periodic orbit theory to obtain leads to the semiclassicalcal s quantization of the collinear dynamics yields an import this is true, the semiclassical analysis o center of mass coordinates, and an account of the finite nucle now see, the periodic orbit theory has still much of interest Commentary Remark understood the origin of the di which is not at all obvious from a numerical solution of the qu to a credible heliumdown this spectrum enterprise. computation, Nevertheless, there wesuch is could still quintessentially much declare quantum to e t physical applications - as quaternions, octonions, spinor tary motions. The basic ideaKS was transformation in in place order as to early illustrate ascollinear a 1931, helium Hopf’s was whe invariant introduced in ref. [37.2]. Remark developed in celestial mechanics [37.6] to obtain numerica product and a known satisfy linear equation for spin 1 exception to the binary symbolic dynamics ofin the numerical collinear h investigations. A proof that all cycles arehelium - uns 27dec2004 CHAPTER ref. [37.2]. The general two–bodymation collision [37.5], regularizing a K generalization ofregularization Levi-Civita’s for [37.13] P motion in athe correct plane, higher-dimensional is generalization due ofby to the introducing Kustaanheimo a “squ vector exercise 37.7 in 722 N (37.17) 1 and the = qm nclude more j E − tates by calculat- . This is illustrated s into account, we nder ! Results obtained in sect. 37.5.1. The can be described as each Rydberg series m hase space dynamics 16 ances obtained by a cycle d from truncations of the energetically low- 1 or higher in (37.15) ChaosBook.org version14, Dec 31 2012 ,... , = 2 = those centered closest to , j ℓ 1 , 0 dash as an entry indicates a missing = 12 N = 5393 for the winding number, see , . j 0 m -series for fixed = N 1 8 σ , = 0 leads to 2 j ] = 1 ) σ s ) ( ) 1 2 4 . The exact quantum energies [37.3] are in the last column. ℓ j ( 2 ) + = π /ζ j . N 2 / + 2( 1 /ζ S + ( orbit, see figure 37.5. The additional quantum number . Repeating the calculation with 1 values), in figure 37.6. The simple formula (37.17) gives ) 2 1 ℓ ( − = N + /ζ m [ and 1.8290 for the action and − m

37. HELIUM ATOM = = Collinear helium, real part of the symmetric subspace reson π N 1 12 3.0970 22 2.9692 0.8044 32 2.9001 0.7714 43 — 2.9390 0.7744 33 — 2.9248 0.7730 0.3622 43 0.5698 2.9037 0.7727 0.3472 53 0.5906 — 0.7779 0.3543 64 — 0.5916 — 0.3535 44 0.5902 — 0.3503 0.2050 54 — 0.5899 — 0.3535 0.1962 64 — — 0.1980 7 0.2812 — 0.5383 — 0.2004 0.2808 0.2550 0.1655 0.5429 — 0.2012 0.2808 0.2561 0.1650 0.5449 — 0.2010 0.2811 0.2559 — 0.1654 0.2560 — 0.1657 0.2416 0.1508 0.1657 0.2433 0.1505 0.1413 0.2438 0.1507 0.1426 0.1508 0.1426 0.1426 , N n j 2 m / 1 E 37.2: S In order to obtain higher excited quantum states, we need to i Results of the same quality are obtained for antisymmetric s asymmetric stretch orbits in the cycle expansion (37.16),further covering away more from of the the p indeed center. reveal more Taking and longer more and states in longer each cycle comparison with the full quantum calculations. by the data listed inthe table cycle 37.2 expansion of for 1 symmetric states obtaine (37.17) corresponds to the principalstates quantum described number by defined the quantization conditionthe (37.17) are nucleus and correspond therefore(for to a the fixed lowest statesalready in a rather goodfrom estimate (37.17) for are the tabulated ground in state table of 37.2, helium see the 3rd column u Table ing the zeros of 1 reveals states in the Rydberg seriesest which series are in to figure the 37.6. right of helium - 27dec2004 table 37.1, the 1 cyclethe in the fundamental domain. This cycle expansion (37.16) up to cycleThe length states are labeled byzero their at principal that quantum level of numbers. approximation. A with CHAPTER The quantization condition 1 ) ff 2 r , 1 r axis, and 1 r the diagonal. Compute the the ff diagonal. Mod- ); The unit eigen- , in this case the 2 ff p r Λ = / 1 1 4 Jacobian matrix, see , r p × Λ The motion in the ( , the diagonal. Just as in the 1 given in local coordinates by , ff l axis or reflecting o a mirror. Miraculously, the symbolic axis and the i r fundamental domain 1 ff r 725 2 matrix using equations (37.6) together with ) where n × 0 a 0’s in a row. n Now either Useself-retracing special to symmetries find of alldimensional orbits Newton orbits search. such up as to length 5Collinear by helium cycle a stabilities. 1- 1 symbol for a bounce o 3-disk game of pinball, weto thus be know computed what for cycles the need cycle expansionGuess (37.16). some short cycles bythey requiring correspond that to topologically sequencesing of bounces to either the return- same Collinear helium cycles. ify your integration routine so thethe trajectory diagonal bounces as o o dynamics for the survivors againwith turns 0 out symbol to signifying be binary, a bounce o region between plane is topologically similar to3-disk the system, except pinball that motion the inpotential. motion a is in the Coulomb Just as in theif 3-disk viewed system in the dynamics the is simplified values are due todisplacements the along the usual orbit periodicto as orbit the well invariances; as energyprovides perpendicular manifold a are check of conserved;Compare the with the accuracy table 37.1; of latter you your should one get computation. the actions and eigenvalues for the cycles you founddescribed in in exercise sect. 37.5, 37.3. as Youduced may 2 either integrate the re- the generating function (37.7) or integrate the full 4 sect. K.1.eigenvalues of the form (1 Integration in 4 dimensions should give axis. In the chaotic 2 r 37.6. 37.5. um, i.e., the succession ChaosBook.org version14, Dec 31 2012 or 1 us ionization thresholds, see r electron atoms and semiclas- eed to take into account the lsequencesof form( is a succession of n Construct Check the 0 quadrant, or, are the cycle weights in (37.12), and cy- 0, show that the , see figure 6.1. p t Θ = ≥ i L r ), 2 Starting with the helium r , where , n 1 0 r a t 0 Do some trial integrations of the = ) are independent. Alternatively, ∞ n 2 P , 1 P , 2 Q , ´ e section for collinear Helium. 1 and the inter-electron angle , and angular momentum Q 2 ∞ r = and he 1 three bodyproblemcan be written inpendent terms coordinates of only, three the inde- electron-nucleus distances r Helium trajectories. a Poincarésection of figurelium 37.3b flow to that a reduces map.respond the Try he- to to delineate finite regions symboltions which sequences, cor- that i.e. follow theure initial same 37.3a condi- topological space itinerary forrough in a partition fig- finite can numberNewton-Raphson be of method bounces. used searches to Such exercise for initiate 37.5. 2–dimensional helium cycles, Kustaanheimo–Stiefel regularization forlium; collinear derive he- thehelium Hamiltonian equations (6.18) of motion and (6.19). the collinear Helium in theHamiltonian plane. in the infinite nucleusm mass approximation collinear helium equations of motionenergy (6.19). conservation, only Due three to of the thedinates phase ( space coor- you can integrate inconservation as 4 a dimensions check andtor. on use the the quality energy of yourThe integra- dynamics can be visualizedinal as a configuration motion in space the ( orig- better still, byPoincarésection, an exercise appropriately 37.4. chosenrun 2-dimensional Most away, do trajectories not be will surprised -lium the is classical unbound. collinear he- Try toest cycle guess of approximately figure the 37.4. short- A Poincar Kustaanheimo–Stiefel transformation. In order to obtain the Rydberg series structure of the spectr stands for an arbitrary binary symbol sequence and 0 37.3. 37.2. 37.4. 37.1. a exerHelium - 16apr2002 of states converging todynamics various of ionization orbits thresholds, which we make n large excursions along the Exercises collinear subspace these orbits are characterized by symbo cle expansion of indeed yield allref. Rydberg [37.4]. states For a up comprehensive the overview vario onsical spectra treatments of ref. two- [37.1]. EXERCISES A summation of the form 724 ectron ectron , see fig- N lectrons on 0 dynamics figuration are Θ= ation of the inter The semiclassical oth electrons are on the ChaosBook.org version14, Dec 31 2012 tates should also belong classical ates can be obtained with 37.5.1, prevailed until the elium resonances with ex- odern interpretation of the electron-electron exchange ics is complete is, however, m). These states form the al wave function, including n the position variables must erent Rydberg series sharing scussed in refs. [37.7,what 37.8, has been discussed in s and theoretical predictions e space is given by the triple quantum number lly stable both in the collinear ff tates is lifted by the spin-orbit e antiparallel and the total spin i In our presentation of collinear ring the collinear configuration ic spin wave function with total t in or end at the singular point main of this book. trons are fermions and that deter- ects due to the spin of the particles ff 0). However, the single electron picture , The classification of the helium states in (0 subspectrum in figure 37.6, and contain the quantum numbers, see also remark 37.5. A = 0 where electron-electron interaction becomes ) n N 2 l , , side of the nucleus. As both electrons again have N 1 l Θ = same . The fact that something is slightly wrong with the single el 2 l , 1 , see figure 37.1, and can not be described in terms of single el l Θ ect. The quantum states corresponding to this classical con erent from those obtained from the collinear dynamics with e Beyond the unstable collinear helium subspace. Spin and particle exchange symmetry. Helium quantum numbers. ff ff

37. HELIUM ATOM -quantum number correspond, roughly speaking, to a quantiz 0. See also ref. [37.2]. 37.5 37.3 37.4 N = 2 r A classical study of the dynamics of collinear helium where b erent sides of the nucleus. The Rydberg series related to the = ff 1 di picture laid out in sect. 37.5.1where is highlighted both when electrons conside are on the detailed account of the historicalspectrum development can as be found well in as ref. a [37.1]. m Remark are on the outermost rigth side in each breaks down completely in the limit quantization of the chaotic37.9]. collinear helium Classical subspacesect. and is 37.5 semiclassical follow di several considerations other directions, beyond all outside the same side of the nucleus revealsplane that and this configuration perpendicular is to fu the it. help of The an corresponding approximatetremely EBK-quantization quantum long which st lifetimes reveals h (quasienergetically - highest bound Rydberg series states for in a the given continuu principal angular momentum equal to zero, the corresponding quantum s to single electron quantum numbers ( energetically highest states for given ure 37.6. Details can be found in refs. [37.10, 37.11]. helium - 27dec2004 quantum numbers distinctively di the dominant e a given electronic angle terms of single electron1960’s; quantum a numbers, growing sketched discrepancy in betweenmade sect. experimental it result necessary to refine this picture. In particular, the d Remark labeled by the binary symbolic dynamcis,still and missing. that this The dynam conjectured Markovcollision partition manifold, of i.e., the by phas r those trajectories which star involved, such as the electronic spin-orbit coupling.mines Elec the symmetry properties of thethe quantum spin states. degrees of The freedom, tot musttransformation. be antisymmetric That under means the that a quantumhave state an symmetric antisymmetric i spin wave function, i.e.,is the zero spins (singletstate). ar Antisymmetric states havespin symmetr 1 (tripletstates). The threefoldcoupling. degeneracy of spin 1 s Remark CHAPTER helium we have completely ignored all dynamical e 726 , L413 24 s. B , 204 (1965). the zeta functionfor as the function minima. of Assmall real the imaginary eigenenergies part, energy in a and general contourcan plot have look such a yield as figure informed 20.1, find guesses. the zeros Betterclose by way are Newton would you method, be tosults sect. to the listed 20.3.1. cycle in table expansion How data 37.2? and in ref. quantum You [37.3]. re- can find more quantum , 261 (1994). , eds. D. H. Feng , 497 (2000). 29 , 4182 (1993). 218 72 48 , 3163 (1995). ChaosBook.org version14, Dec 31 2012 28 , 19 (1992). (1964). 2 2928 (1995). 73 75 , L565 (1991). (1956). 24 2 Linear and regular celestial mechanics J. Reine Angew. Math. Directions in Chaos Vol 4 (1931). Compute the lowest eigenen- 104 Ann. Univ. Turku, Ser. AI., Opere mathematische Math. Ann. ergies of singlet andtuting triplet cycle states data of intothe helium symmetric by the and antisymmetric substi- cycle zeta functions expansionProbably (37.15). the (37.16) quickest for way is to plot the magnitude of Lyapunov exponents right, butstability topological angles indices we take and on faith. Helium eigenenergies. and J.-M. Yuan (World Scientific, Hongkong), 245 (1992). (Springer, New York 1971). (1991). 37.7. [37.6] E.L. Steifel and G. Scheifele, [37.10] K. Richter and D. Wintgen, J. Phys. B [37.9] R. and Blümel W. P. Reinhardt, [37.7] G.S. Ezra, K. Richter, G. Tanner and D. Wintgen, J. Phy [37.11] D. Wintgen and K. Richter, Comments At. Mol. Phys. [37.8] D. Wintgen, K. Richter and G. Tanner, CHAOS [37.5] P. Kustaanheimo and E. Stiefel, refsHelium - 26mar2002 [37.4] G. Tanner and D. Wintgen Phys. Rev. Lett. [37.13] T. Levi-Civita, References [37.1] G. Tanner, J-M. Rost and K. Richter, Rev. Mod. Phys. [37.12] P. Kustaanheimo, [37.2] K. Richter, G. Tanner, and D. Wintgen, Phys. Rev. A [37.14] H. Hopf, REFERENCES [37.3] A., Bürgers Wintgen D. and Rost J. M., J. Phys. B