The most precise analytic solution of the hydrogen energy levels Consider the problem where we want to calculate the energy levels of an electron in a hydrogen atom and we want to find the most precise analytic (i.e. a closed-form solution) solution that matches the most accurate experimental data. In this problem, there are no external electric or magnetic fields.
For this, I think we need:
The Dirac equation + the hyperfine structure + the Lamb shift
Is this all that we need to consider? If not, what are the other important effects that we need to consider?
I know that we need to consider the Zeeman (Stark) effect when we apply an external magnetic (electric) field.
Also, can someone please provide any references for the details of this calculation?
 A: Unfortunately it's not so simple. The Dirac equation has the Coulomb potential as an external potential, which means that it does not take account of the motion of the nucleus. So the biggest single correction isn't there. You can't correctly replace the electron mass by the reduced mass in the Dirac Equation, as you can in the Schrodinger Equation, because the removal of the centre-of-mass motion that is based on is strictly non-realativistic. I think you will need the Bethe Salpeter equation, and the reference is B&S Quantum Mechanics of One and two electron atoms (Springer, 1977).
This is why all accurate calculations of energy levels, even in hydrogen, are given in the form of a perturbation series of terms.
See also the answers to this question.
A: A comment too long for a comment:

*

*both are true, but there is no real  relationship between the two paragraphs of @CWPP's answer.

*At present time the motion of nucleus is well understood and taken into account.

*The long standing problem is the effect of quantum electrodynamics, that introduces the so called radiative correction, among which the Lamd shift is the most 'obvious'.
These corrections cannot be determined as a whole in closed analytics form.To match the accuracy of experiments one need to refine their calculation, which in a perturbation way appears as an expansion in powers of the fine structure constant alpha. (The series suggested by CWPP's final sentence). Curently at 4 or 6 order.

*Finally  the biggest problem now comes from the radius of the proton, which shifts the energy levels as the electron spendd some time inside it. This value is measured in various ways which give incompatible results !

