Vibrational frequencies in Car–Parrinello molecular dynamics

Car–Parrinello molecular dynamics (CPMD) are widely used to investigate the dynamical properties of molecular systems. An important issue in such applications is the dependence of dynamical quantities such as molecular vibrational frequencies upon the fictitious orbital mass μ. Although it is known that the correct Born–Oppenheimer dynamics are recovered at zero μ, it is not clear how these dynamical quantities are to be rigorously extracted from CPMD calculations. Our work addresses this issue for vibrational frequencies. We show that when the system is sufficiently close to the ground state the calculated ionic vibrational frequencies are ωM = ω0M[1 − C(μ/M)] for small μ/M, where ω0M is the Born–Oppenheimer ionic frequency, M the ionic mass, and C a constant that depends upon the ion–orbital coupling force constants. Our analysis also provides a quantitative understanding of the orbital oscillation amplitudes, leading to a relationship between the adiabaticity of a system and the ion–orbital coupling constants. In particular, we show that there is a significant systematic dependence of calculated vibrational frequencies upon how close the CPMD trajectory is to the Born–Oppenheimer surface. We verify our analytical results with numerical simulations for N2, Sn2, and H/Si(100)-(2×1).