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.


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 .


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)

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updated 9 Apr 2003