The 1958 Pekeris-Accad-WEIZAC Ground-Breaking Collaboration that computed Ground-States of Two-Electron Atoms (and its 2010 Redux)

By Christoph KOUTSCHAN Doron ZEILBERGER

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Appeared in the Mathematical Intelligencer Volume 33, Issue 2 (2011), 52-57
Written: May 26, 2010.
It is amazing what the eminent applied mathematician, Chaim Leib Pekeris, in collaboration with machine-language-programming-whiz Yigal Accad, could squeeze out of the "primitive" (by today's standards) computer WEIZAC. It is also amazing how Pekeris had the patience to derive, purely by hand, a monstrous (33-term!) (see Eq. (22) of Pekeris' 1958 paper) partial linear recurrence equation (with (fairly complicated) polynomial coefficients) that today can be derived by Maple or Mathematica in a fraction of a second (after a couple of hours of programming).
Note: This will form the subject of Doron Zeilbeger's 17th Chaim Leib Pekeris' Memorial Lecture to be delivered on June 30, 2010, 4:00-5:00pm, local time.

# Mathematica Package and Matlab code:

See Christoph Koutschan Mathematica/Matlab webpage .

# Sample Input and Output for the Maple package PEKERIS:

• If you want to see a whole computer-generated paper, done completely automatically (both the Pekeris part and the Accad part) reproducing, ab initio sections 1-4 of Pekeris' seminal paper, including Table III, using the largest ω as 11 (as in that paper)
the input gives the output.

• The above output file only reproduced the values in Pekeris' paper, for Z from 1 to 10. If you want an extended table for Z from 1 to 100 (in addition to doing all the rest)
the input gives the output.

(Note that the dependence on Z, not surprisingly, is almost linear)

• If you want to see a whole computer-generated paper, done completely automatically (both the Pekeris part and the Accad part) reproducing, ab initio sections 1-4 of Pekeris' seminal paper, including Table III, but with the largest ω equal to 16, getting a higher accuracy
the input gives the output.

• If you want to see a whole computer-generated paper, done completely automatically (both the Pekeris part and the Accad part) reproducing, ab initio sections 1-4 of Pekeris' seminal paper, including Table III, but with the largest ω equal to 21, getting even a higher accuracy
the input gives the output.

• If you want to see a whole computer-generated paper, done completely automatically (both the Pekeris part and the Accad part) reproducing, ab initio the energy calculation in C. L. Pekeris' follow-up paper (1 1S and 2 3S States of Helium, Phys. Rev. v.115(5), 1216-1221), for the ortho state (sec. 3) but not just for Helium (Z=2) but for all Z from 1 to 10,
the input gives the output.

• If you want to see the above but with Z from 1 to 62
the input gives the output.

Doron Zeilberger's List of Papers