Wheeler (1986):
"But", Bohr protested, "nobody will believe me unless I can explain every atom and every molecule."; Rutherford was quick to reply, "Bohr, you explain hydrogen and you explain helium and everybody will believe the rest."
Bob S, I'd have to disagree with the statement it can "only be calculated approximately". It cannot be calculated
analytically by any
known method. But it
can be calculated to arbitrary precision.
There probably does exist an exact mathematical solution, at least for the nonrelativistic equation. They did find one for the classical three-body problem. It's unlikely though, that the mathematically exact solution is something which is itself easier to calculate than a numerical method.
The nonrelativistic ground state energy is -2.903724 a.u. Relativistic: -2.903855 (a difference by ~0.3 J/mol.. not much by any standard)
Just to compare some methods, since I have the numbers handy:
Hartree-Fock: -2.86
Thomas-Fermi model: -2.19
1st order perturbation theory: -2.75
The simplest, most exact method was the one Hylleraas used as early as 1929; a direct variational-principle approach, which is completely exact (nonrelativistic) for a given basis.
A single parameter (analytically solvable): -2.85
3 parameters: -2.847 (Hylleraas 1929)
6 parameters: -2.902 (Hylleraas 1929)
14 parameters: -2.90370 (Chandrasekhar 1955)
10257 parameters: -2.90372437703411959831115924519440444 (Schwartz, 2002)
Which is absurd, really, since relativistic effects come into play. Also, Helium is essentially a bit of a special case; Hylleraas method doesn't scale well to systems of more than two electrons. In other cases you'd probably use full-CI.
