Ab initio molecular dynamics: Theory and Implementation

1e−050e+00−1e−051e−04E cons(a.u.)0e+00−1e−045e−030e+00−5e−030 1 2 3 4 5 6 7 8t (ps)Figure 5. Conserved energy E cons defined in Eq. (48) from Car–Parrinello (CP) and Born–Oppenheimer (BO) molecular dynamics simulations of a model system for various time stepsand convergence criteria using the CPMD package 142 ; see text for further details and Table 1 forthe corresponding timings. Top: solid line: CP, 5 a.u.; open circles: CP, 10 a.u.; filled squares:BO, 10 a.u., 10 −6 . Middle: open circles: CP, 10 a.u.; filled squares: BO, 10 a.u., 10 −6 ; filledtriangles: BO, 100 a.u., 10 −6 ; open diamonds: BO, 100 a.u., 10 −5 . Bottom: open circles: CP,10 a.u.; open diamonds: BO, 100 a.u., 10 −5 ; dashed line: BO, 100 a.u., 10 −4 .largest element of the wavefunction gradient which was required to be smaller than10 −6 , 10 −5 or 10 −4 a.u.; note that the resulting energy convergence shows roughlya quadratic dependence on this criterion.The outcome of this comparison is shown in Fig. 5 in terms of the time evolutionof the conserved energy E cons Eq. (48) on energy scales that cover more than threeorders of magnitude in absolute accuracy. Within the present comparison ultimateenergy stability was obtained using Car–Parrinello molecular dynamics with theshortest time step of 5 a.u., which conserves the energy of the total system toabout 6×10 −8 a.u. per picosecond, see solid line in Fig. 5(top). Increasing the30