  Path-integral molecular dynamics (PIMD) simulations have been carried out to
study the influence of quantum dynamics of carbon atoms on the properties of a
single graphene layer. Finite-temperature properties were analyzed in the range
from 12 to 2000~K, by using the LCBOPII effective potential. To assess the
magnitude of quantum effects in structural and thermodynamic properties of
graphene, classical molecular dynamics simulations have been also performed.
Particular emphasis has been laid on the atomic vibrations along the
out-of-plane direction. Even though quantum effects are present in these
vibrational modes, we show that at any finite temperature classical-like motion
dominates over quantum delocalization, provided that the system size is large
enough. Vibrational modes display an appreciable anharmonicity, as derived from
a comparison between kinetic and potential energy of the carbon atoms. Nuclear
quantum effects are found to be appreciable in the interatomic distance and
layer area at finite temperatures. The thermal expansion coefficient resulting
from PIMD simulations vanishes in the zero-temperature limit, in agreement with
the third law of thermodynamics.
