  Theoretical models of the (d,p) reaction are exploited for both nuclear
astrophysics and spectroscopic studies in nuclear physics. Usually, these
reaction models use local optical model potentials to describe the nucleon- and
deuteron-target interactions. Within such a framework the importance of the
deuteron D-state in low-energy reactions is normally associated with spin
observables and tensor polarization effects - with very minimal influence on
differential cross sections. In contrast, recent work that includes the
inherent nonlocality of the nucleon optical model potentials in the
Johnson-Tandy adiabatic-model description of the (d,p) transition amplitude,
which accounts for deuteron break-up effects, shows sensitivity of the reaction
to the large n-p relative momentum content of the deuteron wave function. The
dominance of the deuteron D-state component at such high momenta leads to
significant sensitivity of calculated (d,p) cross sections and deduced
spectroscopic factors to the choice of deuteron wave function [Phys. Rev. Lett.
{\bf 117}, 162502 (2016)]. We present details of the Johnson-Tandy adiabatic
model of the (d,p) transfer reaction generalized to include the deuteron
D-state in the presence of nonlocal nucleon-target interactions. We present
exact calculations in this model and compare these to approximate
(leading-order) solutions. The latter, approximate solutions can be interpreted
in terms of local optical potentials, but evaluated at a shifted value of the
energy in the nucleon-target system. This energy shift is increased when
including the D-state contribution. We also study the expected dependence of
the D-state effects on the separation energy and orbital angular momentum of
the transferred nucleon. Their influence on the spectroscopic information
extracted from (d,p) reactions is quantified for a particular case of
astrophysical significance.
