  Quark orbital angular momentum in the proton is evaluated via a Lattice QCD
calculation of the second Mellin moment of the twist-3 generalized parton
distribution $\widetilde{E}_{2T} $ in the forward limit. The connection between
this approach to quark orbital angular momentum and approaches previously
utilized in Lattice QCD calculations, via generalized transverse
momentum-dependent parton distributions and via Ji's sum rule, is reviewed.
This connection can be given in terms of Lorentz invariance and equation of
motion relations. The calculation of the second Mellin moment of
$\widetilde{E}_{2T} $ proceeds via a finite-momentum proton matrix element of a
quark bilocal operator with a straight-line gauge connection and separation in
both the longitudinal and transverse directions. The dependence on the former
component serves to extract the second Mellin moment, whereas the dependence on
the latter component provides a transverse momentum cutoff for the matrix
element. Furthermore, a derivative of the matrix element with respect to
momentum transfer in the forward limit is required, which is obtained using a
direct derivative method. The calculation utilizes a clover fermion ensemble at
pion mass 317 MeV. The resulting quark orbital angular momentum is consistent
with previous evaluations through alternative approaches, albeit with greater
statistical uncertainty using a comparable number of samples.
