The variables in the Hartree-Fock equations unfortunately depend on
themselves, so they must be solved in an iterative manner. You will see
these iterations in the outputs you run in the lab. In typical cases, the
Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky
cases, convergence may be improved by changing the form of the initial
guess. Since the equations are solved self-consistently, Hartree-Fock is an
example of a *self-consistent field* (SCF) method.

Unfortunately, you are probably under the illusion that molecular orbitals
are somehow ``real'' or ``true.'' Except for the special case of the
hydrogen atom, this is completely false. Molecular orbitals are the
product of Hartree-Fock theory, and Hartree-Fock is *not*
an exact theory: it is an approximation to the electronic Schrödinger
equation. The approximation is that we pretend that each electron feels
only the *average* Coulomb repulsion of all the other electrons. This
approximation makes Hartree-Fock theory *much* simpler than the real
problem, which is an *N*-body problem. Unfortunately, in many cases this
approximation is rather serious and can give bad answers. It can be
corrected by explicitly accounting for electron correlation by density
functional theory (DFT), many-body perturbation theory (MBPT), configuration
interaction (CI), and other means.