Ab initio molecular dynamics: Theory and Implementation

where ɛ j and ɛ i are the eigenvalues of occupied and unoccupied orbitals, respectively;see Eq. (26) in Ref. 467 for the case where both orbitals are occupied ones.It can be seen from Fig. 2 that the harmonic approximation works faithfully ascompared to the exact spectrum; see Ref. 471 and Sect. IV.A in Ref. 472 for a moregeneral analysis of the associated equations of motion. Since this is in particulartrue for the lowest frequency ωemin , the handy analytic estimate for the lowestpossible electronic frequencyω mine∝(Egapµ) 1/2, (53)shows that this frequency increases like the square root of the electronic energydifference E gap between the lowest unoccupied and the highest occupied orbital.On the other hand it increases similarly for a decreasing fictitious mass parameterµ.ω maxnIn order to guarantee the adiabatic separation, the frequency difference ωemin −should be large, see Sect. 3.3 in Ref. 513 for a similar argument. But boththe highest phonon frequency ωnmax and the energy gap E gap are quantities that adictated by the physics of the system. Whence, the only parameter in our handsto control adiabatic separation is the fictitious mass, which is therefore also called“adiabaticity parameter”. However, decreasing µ not only shifts the electronicspectrum upwards on the frequency scale, but also stretches the entire frequencyspectrum according to Eq. (52). This leads to an increase of the maximum frequencyaccording toω maxe∝(Ecutµ) 1/2, (54)where E cut is the largest kinetic energy in an expansion of the wavefunction interms of a plane wave basis set, see Sect. 3.1.3.At this place a limitation to decrease µ arbitrarily kicks in due to the maximumlength of the molecular dynamics time step ∆t max that can be used. The time stepis inversely proportional to the highest frequency in the system, which is ωemax andthus the relation( ) 1/2 µ∆t max ∝(55)E cutgoverns the largest time step that is possible. As a consequence, Car–Parrinellosimulators have to find their way between Scylla and Charybdis and have to makea compromise on the control parameter µ; typical values for large–gap systems areµ = 500–1500 a.u. together with a time step of about 5–10 a.u. (0.12–0.24 fs).Recently, an algorithm was devised that optimizes µ during a particular simulationgiven a fixed accuracy criterion 87 . Note that a poor man’s way to keep the timestep large and still increase µ in order to satisfy adiabaticity is to choose heaviernuclear masses. That depresses the largest phonon or vibrational frequency ωnmaxof the nuclei (at the cost of renormalizing all dynamical quantities in the sense ofclassical isotope effects).20