Ab initio molecular dynamics: Theory and Implementation

The effective classical partition function Eq. (315) with a fixed discretization Pis isomorphic to that for N polymers each comprised by P monomers 233,126,120 .Each quantum degree of freedom is found to be represented by a ring polymer ornecklace. The intrapolymeric interactions stem from the kinetic energy T(Ṙ) andconsist of harmonic nearest–neighbor couplings ∝ ω P along the closed chain. Theinterpolymeric interaction is given by the scaled potential E (s)0 /P which is onlyevaluated for configurations {R I } (s) at the same imaginary time slice s.In order to evaluate operators based on an expression like Eq. (315) most numericalpath integral schemes utilize Metropolis Monte Carlo sampling with theeffective potentialV eff ={P∑ ∑ Ns=1I=11(2 M IωP2 R (s)I) 2− R (s+1) 1(I +P E 0 {R I } (s))} (317)of the isomorphic classical system 233,126,542,120,124,646,407 . Molecular dynamicstechniques were also proposed in order to sample configuration space, seeRefs. 99,490,462,501,273 for pioneering work and Ref. 646 for an authoritative review.Formally a Lagrangian can be obtained from the expression Eq. (317) by extendingit{P∑ ∑ N ( 1L PIMD =− 1 ( ) ) 22 M IωP2 R (s)I− R (s+1)I− 1 (P E 0 {R I } (s))}s=1I=12M ′ IP (s)I(318)with N×P fictitious momenta P (s)Iand corresponding (unphysical) fictitious massesM I ′ . At this stage the time dependence of positions and momenta and thus the timeevolution in phase space as generated by Eq. (318) has no physical meaning. Thesole use of “time” is to parameterize the deterministic dynamical exploration ofconfiguration space. The trajectories of the positions in configuration space, can,however, be analyzed similar to the ones obtained from the stochastic dynamicsthat underlies the Monte Carlo method.The crucial ingredient in ab initio 395,399,644,404 as opposed to standard233,126,542,120,124,646,407 path integral simulations consists in computing theinteractions E 0 “on–the–fly” like in ab initio molecular dynamics. In analogy to thiscase both the Car–Parrinello and Born–Oppenheimer approaches from Sects. 2.4and 2.3, respectively, can be combined with any electronic structure method. Thefirst implementation 395 was based on the Car–Parrinello / density functional combinationfrom Sect. 2.4 which leads to the following extended LagrangianL AIPI = 1 P+ ∑ ij+P∑s=1{∑iP∑Λ (s)ijs=1{∑Iµ(〈〈 ˙φ(s) iφ (s)i12 M ′ I∣∣∣φ (s)j〉(s) ˙φ [ i − E KS {φ i } (s) , {R I } (s)]〉− δ ij) }(Ṙ(s)) 2 ∑N 1(I −2 M IωP2I=1R (s)I}) 2− R (s+1)I, (319)111