View source for Semiclassical theory of helium atom

''In memory of our teacher and friend, Dieter Wintgen, who died on the 16th August 1994 at the age of 37 years on the descent from the Weisshorn (4505 m).''


<strong>Semiclassical theory of helium atom</strong> refers to a description of the quantum spectrum of helium in terms of the underlying classical dynamics of the strongly chaotic three-body Coulomb system formed by the nucleus and the two electrons. 


[[Category:Quantum Chaos]]

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[[Category:Quantum Chaos]]


== Helium and its role for the development of quantum mechanics ==

=== Helium: an atomic three-body problem ===

The <strong> semiclassical theory of the helium atom </strong> (or other two-electron atoms)
follows the idea of computing and understanding the quantum energy levels starting from 
trajectories of the underlying classical system. In helium, the classical dynamics is given 
by the pair of interacting electrons moving in the field of the (heavy) nucleus. 
Two-electron atoms represent a paradigmatic system for the successful application of concepts of
[[quantum chaos|quantum chaos theory]] and in particular the
[[Gutzwiller trace formula|Gutzwiller trace formula]].

[[Image:Tanner-Richter-helium-coord.png|thumb|upright=1.2|The helium atom composed of two
electrons and a nucleus of charge Z=2 (from Tanner et al. 2000)]]

Helium, as the prototype of a two-electron atom, is composed of the
nucleus with charge Z=2 and two electrons, see  <figref>Tanner-Richter-helium-coord.png</figref>.
The interplay between the attractive Coulomb interaction between the nucleus
and the electrons and the Coulomb repulsion between the electrons
gives rise to exceedingly complicated spectral features, despite the
seemingly simple form of the underlying quantum Hamiltonian. Correspondingly,
orbits of the two interacting electrons, when considered as classical
particles, are predominantly characterized by chaotic dynamics and
cannot be calculated analytically. Hence, helium as a microscopic three-body
Coulomb system has much in common with its celestial analogue, the gravitational
[[three body problem|three-body problem]].

=== The failure of the "old quantum theory" ===

Modern semiclassical theory of the helium atom has its roots in the early
days of quantum theory: The observation that atomic spectra consist of discrete lines called
for a then novel theoretical approach, a quantum theory for atoms.
Bohr's early attempts were formulated in terms of quantum postulates and successfully reproduced
the energy levels of hydrogen by requiring periodic (elliptic)
Kepler electron motion with quantized radii, respectively momenta p,

:<math>\label{eq:Bohr}
\oint p d q = n h
</math>

[[Image:Tanner-Richter-helium-orbit-models.png|thumb|upright=1.0|He-orb|Periodic orbit configurations
of the helium electron pair that served as quasi-classical models for the ground
state (from Tanner et al. 2000)]]

(where n is an integer and h Planck's constant). It was natural to try
this approach also for helium, the simplest atom with more than one electron.
By applying Bohr's ad hoc quantization rule \eqref{eq:Bohr} to various 
periodic orbit configurations of the electron pair motion in helium
(see  <figref>Tanner-Richter-helium-orbit-models.png</figref>),
a number of leading physicists of that time, including Bohr, Born, Kramers,
Landé, Sommerfeld and van Vleck, tried to compute the ground state energy
of helium. However, without success: all models gave unsatisfactory results.

[[Image:Tanner-Richter-helium-Heisenberg-orbit.png|thumb|upright=1.6|H-orb| Heisenberg's proposal for
  Kepler-type electron pair motion in helium (from Tanner et al. 2000)]]

Heisenberg, then a student of Sommerfeld, devised a different trajectory
configuration with the electrons moving on perturbed Kepler ellipses on different
sides of the helium nucleus; in  <figref>Tanner-Richter-helium-Heisenberg-orbit.png</figref> Heisenberg's sketch
of this configuration posted in a letter to Sommerfeld in 1922 is shown. Assuming half-integer quantum numbers
in his letter, Heisenberg arrived at a helium ionization potential of 24.6 V very close
to the observed value of 24.5 V. However, discouraged by Bohr who did
not accept such half-integer orbital quantum numbers, Heisenberg never published his
results. Though the good agreement must be considered as accidental,
the Heisenberg model came closest to an adequate semiclassical description
of the helium ground state. Modern semiclassical theory reveals that
the association of energy levels with individual periodic orbits in the old
quantum theory was too simple-minded. Indeed, for chaotic systems such as the [[three body problem|three-body problem]] helium, it is the entirety of all periodic orbits which conspire to form the energy levels such as beautifully shown in [[Gutzwiller trace formula|Gutzwiller's trace formula]]. For a comprehensive account of the developments of the semiclassical theory for helium up to the year 2000, see Tanner et al. 2000.


The problems and failure of (most of) the attempts to quantize the
electron pair motion in helium marked the end of the "old quantum theory"
which was subsequently replaced by the "new quantum theory": quantum (wave) mechanics
which has proven very successful to this day.

== Spectral properties and quantum-mechanical concepts ==

By now considerable parts of the rich energy spectrum of the helium atom
have been computed quantum mechanically by numerically solving the
Schr&ouml;dinger equation for the two-electron Hamiltonian of helium. To that end,
besides the orbital dynamics, the spin degree of freedom of the two electrons has to be 
considered. The electron spins can be paired antiparallel or parallel leading to the distinction of
singlet states (total spin $S = 0$) and triplet states ($S=1$) often referred
to as parahelium and orthohelium, respectively.

<figref>Tanner-Richter-num-spectrum.png</figref> depicts, as a representative case, the level 
diagram of parahelium. The helium states and energy levels can be classified
as follows: (i) the ground state and bound singly excited states, (ii)
doubly excited resonant states, and (iii) unbound continuum states at
energies above the two particle fragmentation threshold that are not considered here.
States of category (i) are composed of one electron in a hydrogen-type
ground state with quantum number $N=1$ and the second electron being excited
with energy levels (labeled by $n=1,2,3...$) forming a Rydberg series
(see  <figref>Tanner-Richter-num-spectrum.png</figref>) converging to the first ionization threshold at an energy of
$-Z^2/2$ (in atomic units). In energy region (ii) the doubly excited states
have a finite lifetime; they can decay, owing to the mutual repulsive
interaction between the electrons, by autoionization where one electron
leaves the system while the second one remains bounded to the nucleus.
These doubly excited states are organized in doubly infinite level sequences
with quantum numbers N and n. As visible in <figref>Tanner-Richter-num-spectrum.png</figref>, 
they apparently form individual Rydberg series labeled by the index $N$, the hydrogen-like
principle quantum number of the energetically lower electron. However, closer inspection
of the energy region approaching complete fragmentation (i.e. the border to
regime (iii)) shows that neighboring Rydberg series perturb each other
more and more.
[[Image:Tanner-Richter-num-spectrum.png|thumb|upright=1.5|He-energy| Helium energy level diagram
(from Tanner et al. 2000) ]]
With further increasing energy, these states eventually form
a rather dense set of energy levels with seemingly irregular spacings, and
the specification of the two-electron states in terms of the quantum numbers (N,n)
looses its meaning at such high excitations: At these energies electron-electron
interaction gets increasingly important, and hence the concept of quantum numbers
(N,n) labeling independent electron states breaks down. The labels (N,n) can be
partly replaced by new, though approximate, quantum numbers representing the 
collective dynamics of the electron pair. However, due to the non-integrability
of the three-body Coulomb problem, a clear-cut classification is no longer possible
(see Tanner et al. 2000).
 

The increasing complexity of the energy spectrum close to the helium double
ionisation threshold can be experimentally revealed in photo-ionisation measurements.
The single photo-ionisation cross section is proportional to the
probability of ionising a helium atom by a photon at a given
frequency <math>\omega\ .</math> It can be compared directly to experimental data
measuring the electron flux obtained from shining a laser (at sufficiently
weak intensity to avoid effects due to multi-photon ionisation) onto
a helium target; a typical photoionisation signal for highly doubly excited helium states is shown in 
<figref>TPICS-puettner.png</figref> (Jiang et al. 2008), exhibiting irregular sequences
of peaks from overlapping resonances.

[[Image:TPICS-puettner.png|thumb|upright=2.2|TPICS| Total photoionisation cross section of helium; Ix refers to the ionisation threshold for the xth Rydberg series (from Jiang et al. 2008).]] 


== The helium atom - a semiclassical approach ==

The three-body Coulomb system helium is one of the most complex
systems which has been treated fully semiclassically using 
Gutzwiller's trace formula (Wintgen et al. 1992). The challenge is to describe quantum 
spectra or photoionisation cross sections of this few particle system
in terms of classical trajectories of the nucleus and the two
electrons alone. It turns out that the structure of the spectrum
is closely linked to features of the underlying classical few-body dynamics such as 
invariant subspaces in phase space, chaotic or nearly integrable
behaviour and the influence of collision events.

The bound and resonance spectrum as depicted in  <figref>Tanner-Richter-num-spectrum.png</figref> is linked
via
[[Gutzwiller trace formula|Gutzwiller's trace formula]] to the set of all periodic orbits 
of the system. Furthermore, it can be shown that photoionisation or absorption
spectra in atoms are related to a set of returning trajectories, that is, trajectories 
which start and end at the origin (Du et al. 1988). Note that these orbits are in 
general only closed in position space and thus not periodic. Interestingly,
in helium these are triple-collision orbits, that is, orbits for which both electrons hit the nucleus
simultaneously.

A good knowledge of the phase space dynamics is necessary to classify
and determine these sets of trajectories.

=== Classical dynamics ===

The classical three body system can be reduced to four degrees of freedom (dof)
after eliminating the centre of mass motion and incorporating the 
conservation of the total angular momentum.  As the nucleus
is about 1800 times heavier than an electron, one can work in the infinite nucleus mass 
approximation without loosing any essential features. After rescaling and
making all quantities dimensionless, one can write the classical Hamiltonian in the form

:<math>\label{He_scal}
H =  \frac{{\mathbf p}_1^2}{2}  + \frac{{\mathbf p}_2^2}{2}
- \frac{Z}{r_{1}} - \frac{Z}{r_{2}} + \frac{1}{r_{12}} =
                             \left\{\begin{array}{rcl} +1 & : & E > 0 \\
                                                        0 & : & E = 0 \\
                                                       -1 & : & E < 0
                                  \end{array} \right.
</math>

with nucleus charge <math>Z = 2</math> for helium (Richter et al. 1993).
The phase space in Eq. \eqref{He_scal} has 6 dof, the dynamics for fixed angular momentum takes 
place on 4 dof.
The <math>H = +1</math> regime corresponds to the region of positive energy where
double ionisation is possible. There exist no periodic orbits of the electron
pair and one does not find quantum resonance states in this energy regime, see
<figref>Tanner-Richter-num-spectrum.png</figref>.
It is the classical dynamics for negative energies, that is H= -1, which shows complex
behaviour, chaos, unstable periodic orbits and is linked to the bound and resonance
spectrum of helium in <figref>Tanner-Richter-num-spectrum.png</figref>. Only one electron can escape classically in this energy regime and it will
do so for most initial conditions.

==== Symmetries and invariant subspaces ====
[[Image:eZe-space.png|thumb|upright=1.0|eZe-spa|The collinear eZe configuration]]
The equations of motion derived from the Hamiltonian \eqref{He_scal} are invariant under
the transformation
<math>({\mathbf r}_1, {\mathbf r}_2) \rightarrow (-{\mathbf r}_1, -{\mathbf r}_2)</math>, as well
as <math>({\mathbf r}_1, {\mathbf r}_2) \rightarrow ({\mathbf r}_2, {\mathbf r}_1)\ .</math>
The symmetries give rise to invariant
subspaces in the full phase space. Trajectories which start in such a
subspace will remain there for all times thus reducing the relevant
degrees of freedom of the dynamics. Invariant subspaces are thus an extremely useful
tool to study classical dynamics in a high dimensional phase space. The subspace
most important for a semiclassical treatment is the collinear
eZe space where the electrons move along a common axis at different sides of the nucleus, see 
<figref>EZe-space.png</figref>. The dynamics in this space describes the spectrum
near the ground state as well as some of the Rydberg series in the energy spectrum, 
<figref>Tanner-Richter-num-spectrum.png</figref>. Furthermore, the photoionisation spectrum is dominated by the collinear dynamics. 
Heisenberg's early success is indeed related to the similarity of
his 'periodic orbit' in  <figref>Tanner-Richter-helium-Heisenberg-orbit.png</figref> with the shortest periodic orbit in the eZe space. We will discuss the most important properties of the dynamics in this
subspace in more detail below. Other subspaces are, for example, the collinear dynamics of both electrons on the same side of the nucleus giving rise to 'frozen planet states' (Richter et al. 1992) and the so-called Wannier ridge
space with <math>{\mathbf r}_1= {\mathbf r}_2; {\mathbf p}_1 = {\mathbf p}_2 </math>, which is, however, unstable 
with respect to perturbations away from the subspace and thus less relevant for the spectrum. It plays an important role as a gate-way for ionisation processes, see Lee et al. 2005, Byun et al. 2007.
For a more detailed description of the dynamics in other invariant subspaces, see Tanner et al. 2000 and references therein. 
[[Image:eZe-orbit.png|thumb|upright=2.0|eZe-orb|a) A typical orbit in the eZe - space; b)
trajectory in the Poincaré  surface of section <math> r_2 = 0 </math> (from Tanner et al. 2000)]]

=== Symbolic dynamics in the eZe collinear space ===

The dynamics in the eZe collinear space turns out to be fully chaotic with
a binary [[symbolic dynamics|symbolic dynamics]]. The two degrees of freedom are the distances
<math> r_i, i=1,2 </math> of electron <math> i </math> from the nucleus -
a typical trajectory is shown in  <figref>EZe-orbit.png</figref>. 
Note that the axis <math>r_i = 0</math> corresponds to binary collisions, 
that is, the electron "i" collides with the nucleus - see also the next section for
a discussion of collision events. One electron can escape
(ionise) to infinity leaving the other electron in a regular Kepler ellipse around
the nucleus. Interestingly, escape can only occur after both electron come close
to the nucleus simultaneously to allow for momentum transfer between the 
light particles. The triple collision (discussed below) serves thus
as the gateway to electron ionisation. The dynamics is nearly regular 
having a small, but positive [[Lyapunov exponent| Lyapunov exponent]],
if the electrons are far apart (that is, <math>r_1 \gg r_2</math> or 
vice versa), see the Poincaré  surface of section in <figref>EZe-orbit.png</figref>b).

The symbolic dynamics for the chaotic eZe - configuration maps each 
trajectory one-to-one onto a binary symbol string. The symbols are defined through
binary collisions, that is, 
* 1 if a trajectory crosses the line <math>r_1=r_2</math> between two collisions with the nucleus, (i.e. <math>r_1 = 0</math> or <math>r_2 = 0</math>);
* 0 otherwise.
[[Image:eZe-po.png|thumb|upright=2.0|eZe-po|Representative periodic orbits of the helium electron 
pair in the eZe - space (from Wintgen et al. 1992)]]
Note that the symbolic dynamics is closely related to the triple collision, that
is, the boundaries of the partition are given by trajectories starting in or 
ending at the singular point <math>r_1 = r_2 = 0</math> (triple collision manifolds).

The symbolic dynamics fully describes the topological properties of the
phase space; periodic orbits, for example, can be characterised by a
periodic symbol string 
<math>\overline{a} = \ldots aaaa\ldots</math> where <math>a</math>
is a finite binary symbol string.
There are infinitely many periodic orbits and they are all unstable 
with respect to the dynamics "in" the collinear plane. Some examples are shown in  <figref>EZe-po.png</figref>. The number of periodic orbits increases exponentially with 
the code length and thus with the period of the orbits. The 'asymmetric stretch' orbit 
<math>\overline{1}</math> is the shortest orbit in this subspace. The asymptotic periodic
orbit <math>r_1 \equiv\infty\ ,</math> <math>p_1\equiv 0</math> corresponds
to the notation <math>\overline{0}</math> in the binary code.
  
=== Collisions, regularisation and the triple collision ===

Collisions are an important feature in few-body dynamics as described above. There is
in particular a fundamental difference between two-body (or binary) collisions 
and many-body collisions where more than two particles collide simultaneously. Binary
collisions can be regularised, that is, the dynamics can be continued through the
singularity after a suitable transformation of the time and space variables. A
popular regularisation scheme is the Kustaanheimo-Stiefel transformation which
preserves the Hamiltonian structure of the equations. Binary collisions do not
add instability to the classical dynamics.

This is in contrast to triple collisions where both electrons hit
the nucleus simultaneously. The triple collision is a non-regularisable singularity,
that is, there is no unique way to determine the fate of a trajectory after it
has encountered a triple collision. The manifold of all orbits coming out of or
going into a triple collisions - the so-called triple collision manifold (Waldvogel 2002) - plays an 
important role in tessellating the full phase space and provides the symbolic
dynamics in the eZe space. Triple collision orbits always move along the so-called
Wannier orbit <math> r_1 = r_2 </math> when encountering the singularity. The triple collision singularity
thus acts as an infinitely unstable fixed point; a closer analysis shows that
the singularity itself has a non-trivial structure and topology which can be
illuminated using McGehee transformation techniques. 

For a discussion of the Kustaanheimo-Stiefel and McGehee transformations in the context of
three body Coulomb problems, see Richter et al. 1993 and Lee et al. 2005, respectively.

=== Semiclassical periodic orbit quantisation ===
[[Image:He-FT.png|thumb|upright=2.5|He-FT|The Fourier transformed part of the spectrum associated with the
eZe space (here denoted <math> K_{max} </math>) - the binary code (+,-) refers to the
code (0,1) introduced above (from Qiu et al. 1996)]]

The [[Gutzwiller trace formula|Gutzwiller trace formula]] marked a milestone in the 
development of semiclassical theories. It relates the spectrum of a quantum system
to the set of all periodic orbits of the corresponding classical system in terms of 
a Fourier-type relation where the eigenenergies and the actions of the classical 
periodic orbits act as Fourier-pairs. 
The classical dynamics of the eZe collinear configuration can be used for a quantisation of an 
important part of the helium spectrum due to a 'lucky' coincidence: It turns out that the electron motion
in the vicinity of the collinear space is stable in all degrees of freedom perpendicular
to the eZe space. The electrons carry out a regular bending-type vibration while performing
chaotic motion in the collinear degrees of freedom. This makes it possible to use the periodic orbits of the 
eZe configuration for a semiclassical description of parts of the spectrum for angular momentum
L=0 including the ground state. The existence of this connection can be shown by
Fourier methods. By inverting the [[Gutzwiller trace formula|Gutzwiller trace formula]] using Fourier transformation, one obtains an ''action spectrum'' related to the full quantum energy spectrum as shown in <figref>He-FT.png</figref> (Qui et al. 1996). The energy scaling relation for the classical actions
:<math>\label{action}
S_{po} = \frac{1}{\sqrt{|E|}} \tilde{S}_{po},
</math>
has been used here,
where <math> \tilde{S}_{po} </math> is the action of a periodic orbit (po) at fixed energy <math> E=-1</math>, see \eqref{He_scal}. The quantum spectrum used in <figref>He-FT.png</figref>  has been obtained from full 3D numerical calculations (B&uuml;rgers et al. 1995) and semi-empirical formulas based on approximate quantum numbers. (For more details on approximate quantum numbers, see Tanner et al. 2000).

[[Image:table-cycleexp.png|thumb|upright=2.5|tab-cycle| Quantum eigenvalues obtained from cycle expansion
techniques using periodic orbits up to length ''j''; the exact quantum results
are given in the last column (in atomic units), from Wintgen et al. (1992).]]
Each of the peaks in <figref>He-FT.png</figref> can be identified with a periodic orbit of the classical two-electron dynamics; furthermore all these periodic orbits lie in the eZe space confirming the statement that large parts of the quantum spectrum are determined by this invariant lower dimensional subspace - a truly amazing result. At last the
periodic orbits Niels Bohr was looking for have been found and they are quite
close to the solution proposed by Heisenberg which he himself did not dare to publish!
For a full blown semiclassical quantisation, one needs information of as many periodic
orbits as possible - these can be obtained systematically using the symbolic dynamics in the eZe space.
The most extensive semiclassical calculations so far made use of all periodic orbits up to
length 16 (<math> 2^{16} = 65536</math> orbits ) together with [[cycle expansion|cycle expansion]]
techniques to obtain energies as listed in Tab 9 (Wintgen et al. 1992). Pushing the semiclassical calculation 
to even higher energies is hampered by the exponential increase of the number of periodic orbits with increasing (symbol) length in chaotic systems - a general obstacle for semiclassical quantisation techniques.

=== Photoionisation cross sections ===

Information about atomic spectra is often experimentally obtained through measurements of the
photo excitation or ionisation, see <figref>TPICS-puettner.png</figref> for helium. 
An expression for the photo-ionisation cross section can be 
written in terms of the retarded Green function ''G(E)'' of the full
three particle problem, that is,
:<math>\label{cs-G}
\sigma(E) = -\frac{4 \pi}{c} \, \omega \,
  \Im \langle D \phi_{0}| G(E) |D \phi_{0}\rangle

</math>
where ''c'' is the speed of light, <math>\phi_{0} </math> is the initial state
wave function and <math> D = {\mathbf \Pi} \cdot ({\mathbf r}_1 + {\mathbf r}_2)</math>
is the dipole operator with <math>\mathbf \Pi</math>, the polarization of the incoming photon. Using again Gutzwiller's expression for the Green function in terms
of classical trajectories, one can relate the cross section to classical trajectories of the three-body dynamics.

Semiclassical methods are particularly useful when considering
the cross section in the limit <math> E \to 0\ ,</math> that is, at the double
ionisation threshold. Especially the regime just below the threshold with <math> E<0 </math>
is not accessible both to experiments and to fully numerical calculations
due to the large density of resonances. Using a semiclassical closed
orbit theory together with a semiclassical treatment of triple collision orbits, one
can make detailed predictions here; in particular, the cross section can be written in the form
(Byun et al. 2007, Lee et al. 2010)
:<math>\label{s_fl_2}
\sigma(E) \approx \sigma_0 + \frac{8 \pi^2 \omega}{c}\; |E|^\mu \;
\Re \left[2 \pi i\sum_{{\rm CTCO}_\gamma} a_\gamma
e^{i \tilde{S}_\gamma/\sqrt{E} - i \pi \nu_\gamma/2}\right] \, ,
</math>

where <math> \sigma_0 </math> gives a smooth background contribution and the sum is
taken over all ''closed triple collision orbits'' (CTCO), that is, trajectories
which start and end in the triple collision. It can be shown that CTCOs are
part of the eZe sub-space. Furthermore, <math>\tilde S</math>
is the classical action at energy <math>E = -1</math> as given in \eqref{action} and
<math>a_\gamma</math> is an energy independent coefficient related to the stability of a given
CTCO away from the triple collision. Most remarkably is the energy scaling
due to the exponent <math> \mu </math>, (for details see Lee et al. 2010),

:<math>\label{mu_t}
\mu= \mu_{eZe} + 2 \mu_{wr} =
\frac{1}{4}\left[\sqrt{\frac{100 Z-9}{4Z-1}} +
2\sqrt{\frac{4 Z - 9}{4Z -1}}\right],
</math>
[[Image:TPICSFT.png|thumb|upright=1.8|TPICSFT| Fourier transform of cross section data; the
peaks can be related to the CTCOs depicted in the insets (from Byun et al. 2007).]]
which can be obtained through a stability analysis of the triple collision itself. Here, <math>wr</math> relates to
a contribution from the so-called ''Wannier Ridge'' dynamics, an invariant subspace of the full dynamics where the two electrons are always at the same distance from the nucleus. The
exponents are related to Siegel exponents (see Waldvogel 2002) or Wannier exponents (Wannier 1953). The energy scaling describes the decay of the fluctuations in the photoionisation cross section towards the threshold as can be seen in 
<figref>TPICS-puettner.png</figref>. 

The CTCOs can in fact be seen in cross section data using a Fourier transformation of Eqn.
\eqref{s_fl_2}. The data shown in  <figref>TPICSFT.png</figref> are obtained from a 1D eZe cross section calculations (Byun et al. 2007)
and show a nice one-to-one correspondence between peaks and triple collision trajectories. 

Experimental and numerical studies confirm that the dominant contribution to the cross section signal is given by the
collinear eZe dynamics (Jiang et al. 2008) as predicted by the semiclassical analysis.

== Recent developments and open questions ==

=== Exploring the full phase space - approximate symmetries and global structures ===
Helium has provided a prime example where experimental and numerical results of the ''quantum'' 3-body problem give clear hints about interesting structures in the phase space of the classical dynamics. However, the story is not finished yet - at the time of writing (2013), large areas of the full 7 dimensional classical phase space are unexplored and the connection between ''approximate'' quantum numbers (Herrick's quantum numbers - see Lee et al. 2005, Sano 2010) is still unclear. This also opens up interesting links to celestial mechanics and triple collision encounters in three-body gravitational problems as discussed at the workshops on [http://www.pks.mpg.de/~fbd10/ Few Body Dynamics in Atoms, Molecules and Planetary Systems] in Dresden in 2010 and [http://www.pims.math.ca/scientific-event/130729-cmad2 Celestial, Molecular, and Atomic Dynamics (CEMAD)] in Victoria in 2013.


=== Highly doubly excited states - recent advances ===
The world record of experimentally accessing and numerically calculating highly doubly excited states in helium is currently held (in 2013) by Jiang et al. 2008 for total cross sections reaching helium resonances up to the ionisation thresholds N=17 and Czasch et al. 2005 for partial cross sections reaching N=13. Going even higher in the spectrum or considering helium under electromagnetic driving (Madronero et al. (2008)) is a formidable challenge asking for new numerical techniques to deal with the large basis sets necessary and experimental techniques to reach the resolutions required. Unusually for atomic physicists, the rewards may lie in looking at the Fourier transforms of their data.

=== Double ionisation of helium for strong laser fields and ultra-short pulses - probing correlated electron-electron dynamics ===
Studying double ionisation (DI) of helium by looking at the classical dynamics of the two electrons as they escape form the nucleus has a long history: Already in 1953, Wannier predicted an unexpected energy scaling of the DI cross section near the threshold governed by exponents similar to those found in Eq.\eqref{mu_t}. Interesting recent effects being considered are electron-electron correlation effects in strong laser fields and in attosecond pulses. In the strong field case, rescattering can lead to a large contribution to the DI cross section from ionisation events where both electrons escape from the nucleus along the same direction (Prauzner-Bechcicki et al. 2007). Two-photon DI in ultra-short pulses, on the other hand, shows a preference for back-to-back electron escape due to electron-electron repulsion (Feist et al. 2009). These and many other scenarios can be studied using classical electron dynamics.        

=== Semiclassics for many-body problems ===

While helium represents a prime example for the success of semiclassics for an interacting
few body system, generalizations to other many-body problems remain as a future challenge.

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* Y Qiu, J M&uuml;ller, and J Burgd&ouml;rfer, ''Periodic-orbit spectra of hydrogen and helium'', Phys Rev A 54:1922 (1996).

* K. Richter, J. S. Briggs, D. Wintgen, and E. A. Solov'ev, J. Phys. B 25, 3929 (1992).

* K Richter, G Tanner, and D Wintgen, ''Classical mechanics of two electron atoms'', Phys Rev A 48:4182 (1993).

* M M Sano, ''Semiclassical Interpretation of Electron Correlation in Helium'', J Phys Soc Japan 79:034003 (2010).

* J Waldvogel, ''Triple Collisions and Close Triple Encounters'', in ''Singularities in Gravitational Systems'', Lecture Notes in Physics, 590:81 (2002).

* G H Wannier, ''The Threshold Law for Single Ionization of Atoms or Ions by Electrons'', Phys Rev 90:817 (1953).

* D Wintgen, K Richter, and G Tanner, ''The semiclassical helium atom'', CHAOS 2:19 (1992);

== Recommended reading ==

* G Tanner, K Richter, and J-M Rost, ''The theory of two electron atoms: Between ground state and complete fragmentation'', Rev Mod Phys 72:497 (2000).
* P Cvitanović, R Artuso, R Mainieri, G Tanner, G Vattay, N Whelan and A Wirzba, '' Chaos: Classical and Quantum'', http://chaosbook.org/; see in particular http://chaosbook.org/chapters/helium.pdf
* M C Gutzwiller, ''Chaos in Classical and Quantum Mechanics'', Springer-Verlag, New York (1990).

== See also ==

* James Meiss (2007) [[Hamiltonian systems]]. Scholarpedia, 2(8):1943.

* Martin Gutzwiller (2007) [[Quantum chaos]]. Scholarpedia, 2(12):3146.

* Alain Chenciner (2007) [[Three body problem]] Scholarpedia, 2(10):2111.

[[Category:Quantum_Chaos]]