Ab initio molecular dynamics: Theory and Implementation

introducing different masses for different ”classical” degrees of freedom 473,610,639 .In agreement with the preconditioner introduced in the optimization section, thenew plane wave dependent masses are{ µ H(G,G) ≤ αµ(G) =(µ/α) ( 1 2 G2 + V(G,G)) H(G,G) ≥ α , (242)where H and V are the matrix elements of the Kohn–Sham matrix and the potentialrespectively. The reference electron mass is µ and the parameter α has beenintroduced before in Eq. (208) as H Gc,G c. With the preconditioned masses andthe harmonic reference system, the equations of motion of the system areµ(G)¨c i (G) = −λ(G)c i (G) + δΦ i (G) + ∑ jΛ ij c j (G) . (243)where δΦ i (G) is the force on orbital i minus −λ(G). From Eq. (243) it is easyto see that the frequencies ω(G) = √ λ(G)/µ(G) are independent of G and thatthere is only one harmonic frequency equal to √ α/µ. The revised formulas forthe integration of the equations of motion for the velocity Verlet algorithm can befound in the literature 639 .The implications of the G vector dependent masses can be seen by revisitingthe formulas for the characteristic frequencies of the electronic system Eqs. (52),(53), and (54). The masses µ are chosen such that all frequencies ω ij are approximatelythe same, thus optimizing both, adiabaticity and maximal time step. Thedisadvantage of this method is that the average electron mass seen by the nuclei isdrastically enhanced, leading to renormalization corrections 75 on the masses M Ithat are significantly higher than in the standard approach and not as simple toestimate by an analytical expression.3.7.3 Geometrical ConstraintsGeometrical constraints are used in classical simulations to freeze fast degrees offreedom in order to allow for larger time steps. Mainly distance constraints areused for instance to fix intramolecular covalent bonds. These type of applicationsof constraints is of lesser importance in ab initio molecular dynamics. However, inthe simulation of rare events such as many reactions, constraints play an importantrole together with the method of thermodynamic integration 217 . The ”blue–moon”ensemble method 115,589 enables one to compute the potential of mean force. Thispotential can be obtained directly from the average force of constraint and a geometriccorrection term during a molecular dynamics simulation as follows:F(ξ 2 ) − F(ξ 1 ) =∫ ξ2ξ 1dξ ′ 〈 ∂H∂ξ〉 cond.ξ ′ , (244)where F is the free energy and ξ(r) a one–dimensional reaction coordinate, H theHamiltonian of the system and 〈· · · 〉 cond.ξthe conditioned average in the constraint′ensemble 589 . By way of the blue moon ensemble, the statistical average is replacedby a time average over a constrained trajectory with the reaction coordinate fixed73