9887

Three-Parameter Variational Wavefunctions for the Helium Isoelectronic Series

An optimized three-parameter variational wavefunction for the helium isoelectronic series , , , …, can give quite accurate values for the ground-state energies. This computation is based on a simplified version of Hylleraas' classic work on two-electron atoms [1]. The Schrödinger equation for a two-electron atom with nuclear charge can be written as
,
with the Hamiltonian
expressed in atomic units . For S-states, including the ground state, the wavefunction depends on just three variables: , , and . Hylleraas used these alternative variables: , , and . Consider simple trial functions of the form . We need the combination in order that is an even function of , thus a symmetric function with respect to the interchange of and .
According to the variational principle, the ground-state energy can be approximated by , where
.
The parameters , , and are then selected such that takes a minimum value. This is always an upper bound on the exact ground-state energy . To run this Demonstration, choose a value of corresponding to one of the two-electron atoms or ions. Then, vary the sliders determining various combinations of α, β, and γ. The ionization potential () is the energy to remove one electron, which leaves the atom in the one-electron energy state . The is expressed in (). As the blue and red areas become larger, exact values of the energy and ionization potential are being approached more closely. Also shown is the ratio of expectation values of potential and kinetic energies, . By the virial theorem for Coulombic systems, this ratio should equal . This value would be obtained by the exact solution of the Schrödinger equation, but also for an approximate solution with optimally chosen variational parameters.
If you want to look at the answers, the optimal values of , , and are tabulated in the Details section.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

The calculated optimal values are given in the following table.
Reference
[1] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Berlin: Springer–Verlag, 1957 pp. 146–154.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+