Ab initio molecular dynamics: Theory and Implementation

structure calculations, see e.g. Refs. 494,49,496 and references therein. Rather itturns out that in the case of Car–Parrinello calculations using a plane wave basisthe resulting relation for the force, namely Eq. (64), looks like the one obtained bysimply invoking the Hellmann–Feynman theorem at the outset.It is interesting to recall that the Hellmann–Feynman theorem as applied to anon–eigenfunction of a Hamiltonian yields only a first–order perturbative estimateof the exact force 295,368 . The same argument applies to ab initio molecular dynamicscalculations where possible force corrections according to Eqs. (67) and (68)are neglected without justification. Furthermore, such simulations can of course notstrictly conserve the total Hamiltonian E cons Eq. (48). Finally, it should be stressedthat possible contributions to the force in the nuclear equation of motion Eq. (44)due to position–dependent wavefunction constraints have to be evaluated followingthe same procedure. This leads to similar “correction terms” to the force, see e.g.Ref. 351 for such a case.2.6 Which Method to Choose ?Presumably the most important question for practical applications is which ab initiomolecular dynamics method is the most efficient in terms of computer time given aspecific problem. An a priori advantage of both the Ehrenfest and Car–Parrinelloschemes over Born–Oppenheimer molecular dynamics is that no diagonalizationof the Hamiltonian (or the equivalent minimization of an energy functional) isnecessary, except at the very first step in order to obtain the initial wavefunction.The difference is, however, that the Ehrenfest time–evolution according tothe time–dependent Schrödinger equation Eq. (26) conforms to a unitary propagation341,366,342 Ψ(t 0 + ∆t) = exp [−iH e (t 0 )∆t/ ]Ψ(t 0 ) (70)Ψ(t 0 + m ∆t) = exp [−iH e (t 0 + (m − 1)∆t) ∆t/ ]× · · ·Ψ(t 0 + t max ) ∆t→0= T exp× exp [−iH e (t 0 + 2∆t) ∆t/ ]× exp [−iH e (t 0 + ∆t) ∆t/ ]× exp [−iH e (t 0 ) ∆t/ ]Ψ(t 0 ) (71)[− i ∫ ]t 0+t maxt 0dt H e (t)Ψ(t 0 ) (72)for infinitesimally short times given by the time step ∆t = t max /m; here T is thetime–ordering operator and H e (t) is the Hamiltonian (which is implicitly time–dependent via the positions {R I (t)}) evaluated at time t using e.g. split operatortechniques 183 . Thus, the wavefunction Ψ will conserve its norm and in particularorbitals used to expand it will stay orthonormal, see e.g. Ref. 617 . In Car–Parrinellomolecular dynamics, on the contrary, the orthonormality has to be imposed bruteforce by Lagrange multipliers, which amounts to an additional orthogonalizationat each molecular dynamics step. If this is not properly done, the orbitals willbecome non–orthogonal and the wavefunction unnormalized, see e.g. Sect. III.C.1in Ref. 472 .27