How to Control Adiabaticity ?

Under which circumstances can the adiabatic separation be achieved, and how can it be controlled? A simple harmonic analysis of the frequency spectrum of the orbital classical fields close to the minimum defining the ground state yields [39]

$$\omega_{ij} = \left( {{ 2( \epsilon_i - \epsilon_j) } \over \mu }\right)^{1/2} \enspace ,$$ (42)

where $\epsilon_j$ and $\epsilon_i$ are the eigenvalues of occupied and unoccupied orbitals, respectively. This is in particular true for the lowest frequency $\omega_{\rm e}^{\rm min}$ , and an analytic estimate for the lowest possible electronic frequency

$$\omega_{\rm e}^{\rm min} \propto \left( {E_{\rm gap} \over \mu }\right)^{1/2} \enspace ,$$ (43)

shows that this frequency increases like the square root of the electronic energy difference $E_{\rm gap}$ between the lowest unoccupied and the highest occupied orbital. On the other hand it increases similarly for a decreasing fictitious mass parameter $\mu$ .

In order to guarantee the adiabatic separation, the frequency difference $\omega_{\rm e}^{\rm min} - \omega_{\rm n}^{\rm max}$ should be large. But both the highest phonon frequency $\omega_{\rm n}^{\rm max}$ and the energy gap $E_{\rm gap}$ are quantities that are dictated by the physics of the system. Therefore, the only parameter to control adiabatic separation is the fictitious mass. However, decreasing $\mu$ not only shifts the electronic spectrum upwards on the frequency scale, but also stretches the entire frequency spectrum according to Eq. (42). This leads to an increase of the maximum frequency according to

$$\omega_{\rm e}^{\rm max} \propto \left( {E_{\rm cut} \over \mu }\right)^{1/2} \enspace ,$$ (44)

where $E_{\rm cut}$ is the largest kinetic energy in an expansion of the wavefunction in terms of a plane wave basis set. At this place a limitation to decrease $\mu$ arbitrarily kicks in due to the maximum length of the molecular dynamics time step $\Delta t^{\rm max}$ that can be used. The time step is inversely proportional to the highest frequency in the system, which is $\omega_{\rm e}^{\rm max}$ and thus the relation

$$\Delta t^{\rm max} \propto \left( {\mu \over {E_{\rm cut}}} \right)^{1/2}$$ (45)

governs the largest time step that is possible. As a consequence, Car-Parrinello simulators have to make a compromise on the control parameter $\mu$ ; typical values for large-gap systems are $\mu$  = 300-1500 a.u. together with a time step of about 2-10 a.u. (0.12-0.24 fs). With the increase of computational power, longer trajectories with better statistics are possible which make errors from larger fictitious masses more evident and as a result there is a trend to stay away from aggressively large ficitious masses and timesteps and use more conservative parameters. Note that a poor man's way to keep the time step large and still increase $\mu$ in order to satisfy adiabaticity is to choose heavier nuclear masses. That depresses the largest phonon or vibrational frequency $\omega_{\rm n}^{\rm max}$ of the nuclei (at the cost of renormalizing all dynamical quantities in the sense of classical isotope effects). Other advanced techniques are discussed in the literature [33].