  For a system without spin-orbit coupling, the (i) nuclear plus electronic
linear momentum and (ii) nuclear plus orbital electronic angular momentum are
good quantum numbers. Thus, when a molecular system undergoes a nonadiabatic
transition, there should be no change in the total linear or angular momentum.
Now, the standard surface hopping algorithm ignores the electronic momentum and
indirectly equates the momentum of the nuclear degrees of freedom to the total
momentum. However, even with this simplification, the algorithm still does not
conserve either the nuclear linear or the nuclear angular momenta. Here, we
show that one way to address these failures is to dress the derivative
couplings (i.e. the hopping directions) in two ways: (i) we disallow changes in
the nuclear linear momentum by working in a translating basis (which is well
known and leads to electron translation factors [ETFs]); and (ii) we disallow
changes in the nuclear angular momentum by working in a basis that rotates
around the center of mass (which is not well-known and leads to a novel,
rotationally removable component of the derivative coupling that we will call
electron rotation factors [ERFs] below, cf. Eq. 96). The present findings
should be helpful in the short term as far as interpreting surface hopping
calculations for singlet systems (without spin) and then developing new surface
hopping algorithm in the long term for systems where one cannot ignore the
electronic orbital and/or spin angular momentum.
