Isotope effect in acetyleneC2H2andC2D2rotations on Cu(001)

A comprehensive analysis of the elementary processes behind the scanning tunneling microscope controlled rotation of ${\mathrm{C}}_{2}{\mathrm{H}}_{2}$ and ${\mathrm{C}}_{2}{\mathrm{D}}_{2}$, 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 ${\mathrm{C}}_{2}{\mathrm{H}}_{2}$ and ${\mathrm{C}}_{2}{\mathrm{D}}_{2}$ on Cu(001) and estimate the anharmonic couplings between them, using a simple strings-on-rods model. The probability of the elementary processes, nonlinear and combination band, is estimated using the Keldysh diagram technique. This allows us to clarify the main peculiarities and the isotope effects of the ${\mathrm{C}}_{2}{\mathrm{H}}_{2}$ and ${\mathrm{C}}_{2}{\mathrm{D}}_{2}$ on Cu(001) rotation, discovered in the pioneering work [B. C. Stipe et al., Phys. Rev. Lett. 81, 1263 (1998)], which have not been previously understood.