  The diagonal Born-Oppenheimer correction (DBOC) stems from the diagonal
second derivative coupling term in the adiabatic representation, and it can
have an arbitrary large magnitude when a gap between neighbouring
Born-Oppenheimer (BO) potential energy surfaces (PESs) is closing.
Nevertheless, DBOC is typically neglected in mixed quantum-classical methods of
simulating nonadiabatic dynamics (e.g., fewest-switch surface hopping (FSSH)
method). A straightforward addition of DBOC to BO PESs in the FSSH method,
FSSH+D, has been shown to lead to numerically much inferior results for models
containing conical intersections. More sophisticated variation of the DBOC
inclusion, phase-space surface-hopping (PSSH) was more successful than FSSH+D
but on model problems without conical intersections. This work comprehensively
assesses the role of DBOC in nonadiabatic dynamics of two electronic state
problems and the performance of FSSH, FSSH+D, and PSSH methods in variety of
one- and two-dimensional models. Our results show that the inclusion of DBOC
can enhance the accuracy of surface hopping simulations when two conditions are
simultaneously satisfied: 1) nuclei have kinetic energy lower than DBOC and 2)
PESs are not strongly nonadiabatically coupled. The inclusion of DBOC is
detrimental in situations where its energy scale becomes very high or even
diverges, because in these regions PESs are also very strongly coupled. In this
case, the true quantum formalism heavily relies on an interplay between
diagonal and off-diagonal nonadiabatic couplings while surface hopping
approaches treat diagonal terms as PESs and off-diagonal ones stochastically.
