  A comprehensive analysis of the elementary processes behind the scanning
tunneling microscope controlled rotation of C$_2$H$_2$ and C$_2$D$_2$,
isotopologues of a single acetylene molecule adsorbed on the Cu(001) surface is
given, with a focus on the isotope effects. With the help of density-functional
theory we calculate the vibrational modes of C$_2$H$_2$ and C$_2$D$_2$ on
Cu(001) and estimate the anharmonic couplings between them, using a simple
strings-on-rods model. The probability of the elementary processes --
non-linear and combination band -- are estimated using the Keldysh diagram
technique. This allows us to clarify the main peculiarities and the isotope
effects of the C$_2$H$_2$ and C$_2$D$_2$ on Cu(001) rotation, discovered in the
pioneering work [Stipe et al., Phys. Rev. Lett. 81, 1263 (1998)], which have
not been previously understood.
