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

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.