  This is the final paper in a series that introduces geodesic molecular
dynamics at constant potential energy. This dynamics is entitled NVU dynamics
in analogy to standard energy-conserving Newtonian NVE dynamics. In the first
two papers [Ingebrigtsen et al., J. Chem. Phys. 135, 104101 (2011); ibid,
104102 (2011)], a numerical algorithm for simulating geodesic motion of atomic
systems was developed and tested against standard algorithms. The conclusion
was that the NVU algorithm has the same desirable properties as the Verlet
algorithm for Newtonian NVE dynamics, i.e., it is time-reversible and
symplectic. Additionally, it was concluded that NVU dynamics becomes equivalent
to NVE dynamics in the thermodynamic limit. In this paper, the NVU algorithm
for atomic systems is extended to be able to simulate geodesic motion of
molecules at constant potential energy. We derive an algorithm for simulating
rigid bonds and test this algorithm on three different systems: an asymmetric
dumbbell model, Lewis-Wahnstrom OTP, and rigid SPC/E water. The rigid bonds
introduce additional constraints beyond that of constant potential energy for
atomic systems. The rigid-bond NVU algorithm conserves potential energy, bond
lengths, and step length for indefinitely long runs. The quantities probed in
simulations give results identical to those of Nose-Hoover NVT dynamics. Since
Nose-Hoover NVT dynamics is known to give results equivalent to those of NVE
dynamics, the latter results show that NVU dynamics becomes equivalent to NVE
dynamics in the thermodynamic limit also for molecular systems.
