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A comprehensive analysis of the elementary processes behind the scanningtunneling 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 isgiven, with a focus on the isotope effects. With the help of density-functionaltheory we calculate the vibrational modes of C$_2$H$_2$ and C$_2$D$_2$ onCu(001) and estimate the anharmonic couplings between them, using a simplestrings-on-rods model. The probability of the elementary processes --non-linear and combination band -- are estimated using the Keldysh diagramtechnique. This allows us to clarify the main peculiarities and the isotopeeffects of the C$_2$H$_2$ and C$_2$D$_2$ on Cu(001) rotation, discovered in thepioneering work [Stipe et al., Phys. Rev. Lett. 81, 1263 (1998)], which havenot been previously understood.

Isotope effect in acetylene C$_2$H$_2$ and C$_2$D$_2$ rotations on Cu(001)