  We present a method for solving the first-order field equations in a
post-Newtonian (PN) expansion. Our calculations generalize work of Bini and
Damour and subsequently Kavanagh et al., to consider eccentric orbits on a
Schwarzschild background. We derive expressions for the retarded metric
perturbation at the location of the particle for all $\ell$-modes. We find
that, despite first appearances, the Regge-Wheeler gauge metric perturbation is
$C^0$ at the particle for all $\ell$. As a first use of our solutions, we
compute the gauge-invariant quantity $\langle U \rangle$ through 4PN while
simultaneously expanding in eccentricity through $e^{10}$. By anticipating the
$e\to 1$ singular behavior at each PN order, we greatly improve the accuracy of
our results for large $e$. We use $\langle U \rangle$ to find 4PN contributions
to the effective one body potential $\hat Q$ through $e^{10}$ and at linear
order in the mass-ratio.
