3-body Coulomb dynamics
We have to introduce
some basics before study helium atom.
See [1-3] for details or go directly to the "Collinear helium dynamics"
at the bottom
to skip the math.
Classical hydrogen atom
Consider at first classical
dynamics of the hydrogen atom. It is well
known that a bounded electron moves along a Kepler's elliptic orbit with
nucleus
in one of its focal points.
If the electron
has momentum pointing directly to the nucleus then the
ellipse degenerates in a line segment and electron oscillates along
the
line. In this case we get a system of one degree of freedom with the Hamiltonian
H = p2/2m
e2/r = E
where E is the total energy. We put m = e = 1 further.
Scaling
Kepler dynamics remains invariant under a change of energy up to a simple
scaling transformation;
a solution of the equations of motion at an arbitrary
energy E < 0 can be transformed into a solution at a fixed energy
Eo
= -1 by scaling the coordinates as
r(E) = r / (-E),
p(E) = (-E)1/2 p
together
with a time transformation
t(E) = t / (-E)-3/2 .
Regularization
H(p,r)
is singular at r = 0 therefore whenever the electron comes
close to the nucleus accelerations become infinitely
large. A regularization
of the two-body collinear collisions is achieved [1] by means of the Levi-Civita
transformation,
which consists of a coordinate dependent time transformation,
which stretches the time scale near the origin, and a canonical
transformation
of the phase space coordinates.
A time transformation dt = f(p,q)ds for a system described by
H(q,p)
= E leaves the Hamiltonian structure of the
equations of motion unaltered, if the Hamiltonian itself is transformed
into
H' = f(q,p)[H(p,q) - E]. We choose dt = r ds which lifts the
|r| -> 0 singularity and leads to a new
Hamiltonian
H' = rp2/2 - 1 - Er = 0 .
Then a canonical transformation of form
r = Q2, p = P/2Q
maps the Kepler problem into that of a harmonic oscillator with
Hamiltonian
H(Q,P) = P2/8 - EQ2 = 1 .
Collinear helium
Collinear
eZe helium with the two electrons situated along a line
on opposite sides of the nucleus [1-3] is a system of two degrees
of freedom
with the Hamiltonian
H = p12/2 +
p22/2
2/r1 - 2/r2 +
1/(r1 + r2) = E
We put
r1
= Q12,
r2 = Q22,
p1 = P1/2Q1,
p2 = P2/2Q2,
R122 = Q12
+
Q22,
ds = dt/r1r2 .
Then the new Hamiltonian
is
H = (Q22P12 +
Q12P22)/8
-
2R122 +
Q12Q22(
1/R122
E) = 0
and equations of motion are
P1' = 2Q1[2 -
P22/8
+ Q22(E -
Q22/R124)]
P2'
= 2Q2[2 -
P12/8 + Q12(E -
Q12/R124)]
Q1' = P1Q22/4
Q2' = P2Q12/4
where
prime denotes derivative with respect to fictitious time s.
Scaling transformations are now
Qi(E) = Qi / (-E)1/2 ,
Pi(E) = Pi ,
s(E) = (-E)1/2 s .
Note, that Pi does not depend on E and
|Pi|
-> 4 when Qi -> 0 .
Leapfrog algorithm
For the first-order
equations
r' = v , v' = F(v)
the simple leapfrog algorithm
is
r1/2 = r0 +
v0 dt/2
v1 = v0 +
F(r1/2) dt
r1 = r1/2 +
v1 dt/2
where subscripts 0
and 1 denote the values at the beginning
and end of a step respectively.
Collinear
helium dynamics
The overall dynamics depends critically on whether E > 0 or E < 0 .
If the energy
is positive both electrons can escape to infinity. More
interestingly, a single electron can still escape even if E
is negative,
carrying away an unlimited amount of kinetic energy, as the total energy of the
remaining inner electron has
no lower bound. Not only that, but one electron
will escape eventually for almost all starting conditions!
To the left
below you can see dynamics of the two electrons on the
phase plane
x = r1/2 = |Q| ,
y = 2pr1/2 = P sign(Q) .
To the right the same orbit is ploted on the
(r11/2,
r21/2) plane.
An animation stops when one of |Pi| is more then 7 .
Click
mouse to set new coordinates of electrons. Press "Enter" to set new
ds or "Delay".
[1] P.Cvitanovic et al.
Classical and Quantum
Chaos Chap.24: Helium atom
[2] Gregor Tanner, Klaus Richter and Jan-Michael Rost
"The
theory of two-electron atoms: between ground state and complete
fragmentation"
Rev.Mod.Phys. 72, 497 (2000)
[3]
Klaus Richter, Gregor Tanner and Dieter Wintgen
"Classical mechanics of two-electron atoms"
Phys.Rev.A
48, 4182 (1993)
Contents
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chaos
Next: Poincare maps
updated 9 Apr 2003