Extended quote from original article... "When Hagen started solving the problem himself, he immediately noticed a trend. The error of the variational approach

Extended quote from original article... "When Hagen started solving the problem himself, he immediately noticed a trend. The error of the variational approach was about 15 percent for the ground state of hydrogen, 10 percent for the first excited state, and kept getting smaller as the excited states grew larger. This was unusual, since the variational approach normally only gives good approximations for the lowest energy levels." Rings bells from celestial mechanics. You can end up with things like a series solution converging on an elliptical or circular orbit when you apply series methods to simple (or 'simple' in this case - it is quantum mechanics!) systems. I'll see if I can hack my way though their paper (linked to OA, downloadable in full text).