Ab initio molecular dynamics: Theory and Implementation

Table 1. Timings in cpu seconds and energy conservation in a.u. / ps for Car–Parrinello (CP) andBorn–Oppenheimer (BO) molecular dynamics simulations of a model system for 1 ps of trajectoryon an IBM RS6000 / model 390 (Power2) workstation using the CPMD package 142 ; see Fig. 5 forcorresponding energy plots.Method Time step (a.u.) Convergence (a.u.) Conservation (a.u./ps) Time (s)CP 5 — 6×10 −8 3230CP 7 — 1×10 −7 2310CP 10 — 3×10 −7 1610BO 10 10 −6 1×10 −6 16590BO 50 10 −6 1×10 −6 4130BO 100 10 −6 6×10 −6 2250BO 100 10 −5 1×10 −5 1660BO 100 10 −4 1×10 −3 1060time step to 10 a.u. leads to an energy conservation of about 3×10 −7 a.u./ps andmuch larger energy fluctuations, see open circles in Fig. 5(top). The computer timeneeded in order to generate one picosecond of Car–Parrinello trajectory increases –to a good approximation – linearly with the increasing time step, see Table 1. Themost stable Born–Oppenheimer run was performed with a time step of 10 a.u. and aconvergence of 10 −6 . This leads to an energy conservation of about 1×10 −6 a.u./ps,see filled squares in Fig. 5(top).As the maximum time step in Born–Oppenheimer dynamics is only relatedto the time scale associated to nuclear motion it could be increased from 10 to100 a.u. while keeping the convergence at the same tight limit of 10 −6 . Thisworsens the energy conservation slightly (to about 6×10 −6 a.u./ps), whereas theenergy fluctuations increase dramatically, see filled triangles in Fig. 5(middle) andnote the change of scale compared to Fig. 5(top). The overall gain is an accelerationof the Born–Oppenheimer simulation by a factor of about seven to eight, see Table 1.In the Born–Oppenheimer scheme, the computer time needed for a fixed amount ofsimulated physical time decreases only sublinearly with increasing time step sincethe initial guess for the iterative minimization degrades in quality as the time step ismade larger. Further savings of computer time can be easily achieved by decreasingthe quality of the wavefunction convergence from 10 −6 to 10 −5 and finally to 10 −4 ,see Table 1. This is unfortunately tied to a significant decrease of the energyconservation from 6×10 −6 a.u./ps at 10 −6 (filled triangles) to about 1×10 −3 a.u./psat 10 −4 (dashed line) using the same 100 a.u. time step, see Fig. 5(bottom) butnote the change of scale compared to Fig. 5(middle).In conclusion, Born–Oppenheimer molecular dynamics can be made as fastas (or even faster than) Car–Parrinello molecular dynamics (as measured by theamount of cpu time spent per picosecond) at the expense of sacrificing accuracyin terms of energy conservation. In the “classical molecular dynamics community”there is a general consensus that this conservation law should be taken seriouslybeing a measure of the numerical quality of the simulation. In the “quantum chemistryand total energy communities” this issue is typically of less concern. There, itis rather the quality of the convergence of the wavefunction or energy (as achievedin every individual molecular dynamics step) that is believed to be crucial in orderto gauge the quality of a particular simulation.31