I ended up writing a paper about what I figured out for helium. The paper starts off following pretty much the analysis of the helium atom that you'll find in the Wikipedia article. (There's actually a better version of this analysis on a University of Texas website http://farside.ph.utexas.edu/teaching/qmech/lectures/node128.html
but I'm just going to refer to Wikipedia for convenience.) There are basically three stages in the refinement of the analysis:
1. First case, you treat it as a hydrogen atom with two electrons in the same ground orbital. Scaling for the charge and reduced dimension, you get a ground state energy or -108.8 eV, or exactly -8 Rydbergs (where hydrogen is -1 Ry).
2. First improvement: you keep the same wave function as in case (1), but you calculate the repulsion energy of the two electrons. It turns out to be an exact integral, and it comes to 2.5 Ry. (It's not hard to do numberically either because of the spherical symmetry.) So the refined estimate of the energy is -5.5 Ry, or -74.8 eV. It's actually pretty close to the experimental value of -79.0 eV. I call this the "naive model".
3. Second improvement. You tweak the wave function by putting a variable parameter in front of the exponential decay. The nice thing is because you're still basically using the same wave function except for a scaling factor, you don't have to redo any of the hard 3-dimensional integrals. You just scale the ones you already have.
When you do case (3), you get an optimization parameter that lets you bring the energy down to -77.5 eV. Now you're within 1.5 eV of the true value. The funny thing is, why didn't you get even closer? The "naive model" was already within 4 eV of the true value, so this optimization trick has really only closed the gap by 60%.
That's why I find it odd what the guy at U of Texas says at the end of his article. He says: "Obviously, we could get even closer to the correct value of the helium ground-state energy by using a more complicated trial wave-function with more adjustable parameters. ". It's a funny statement because how obvious is it? Based on the numbers so far it doesn't appear to me that we're closing in fast enough.
In hindsight, I know the reason. You can make the wave function as complicated as you want and it will only get you so close and no closer. To get the correct ground state energy you have to do something quite tricky and unexpected, at least based on the analysis so far I'd have to say it's unexpected. But I'll get to that a little later.
What I did in my analysis was to generalize the equations in the Wikipedia article so they work on any size nucleus, as long as it has two electrons. So it works for the hydrogen negative ion (H-); it works for neutral helium; it works for singly ionized Lithium (Li+); and it works for doubly ionized beryllium (Be++). It works for all the atoms of course, but the first four are the ones I was able to look up and check the actual energies.
But before we get to my results, I'm going to brag about how clever I was in deriving these equations. I did a bunch of stuff using basic scaling principles that let me go from A to B in a single line when Wikipedia (and Texas) use a whole page of hard math to do the same thing. I think I'll tell you about that in my next post.