Ab initio molecular dynamics: Theory and Implementation

thonormality constraint can be written asC † (t + δt)C(t + δt) − I = 0 (234)] † [˜C + XC[˜C + XC]− I = 0 (235)˜C † ˜C + X˜C † C + C † ˜CX † + XX † − I = 0 (236)XX † + XB + B † X † = I − A , (237)where the new matrices A ij = ˜c † i (t + δt)˜c j(t + δt) and B ij = c † i (t)˜c j(t + δt) havebeen introduced in Eq. (237). The unit matrix is denoted by the symbol I. Bynoting that A = I + O(δt 2 ) and B = I + O(δt), Eq. (237) can be solved iterativelyusingX (n+1) = 1 [I − A + X (n) (I − B)2( ]+ (I − B)X (n) − X (n)) 2(238)and starting from the initial guessX (0) = 1 (I − A) . (239)2In Eq. (238) it has been made use of the fact that the matrices X and B are realand symmetric, which follows directly from their definitions. Eq. (238) can usuallybe iterated to a tolerance of 10 −6 within a few iterations.The rotation matrix Y is calculated from the orthogonality condition on theorbital velocitiesċ † i (t + δt)c j(t + δt) + c † i (t + δt)ċ j(t + δt) = 0. (240)Applying Eq. (240) to the trial states Ċ′ + YC yields a simple equation for YY = − 1 2 (Q + Q† ), (241)where Q ij = c † i (t+δt) ˙ c ′† i(t+δt). The fact that Y can be obtained without iterationmeans that the velocity constraint condition Eq. (240) is satisfied exactly at eachtime step.3.7.2 Advanced TechniquesOne advantage of the velocity Verlet integrator is that it can be easily combinedwith multiple time scale algorithms 636,639 and still results in reversible dynamics.The most successful implementation of a multiple time scale scheme in connectionwith the plane wave–pseudopotential method is the harmonic reference systemidea 471,639 . The high frequency motion of the plane waves with large kinetic energyis used as a reference system for the integration. The dynamics of this referencesystem is harmonic and can be integrated analytically. In addition, this can be combinedwith the basic notion of a preconditioner already introduced in the sectionon optimizations. The electronic mass used in the Car–Parrinello scheme is a fictitiousconstruct (see Sect. 2.4, Eq. (45)) and it is allowed to generalize the idea by72