Ab initio molecular dynamics: Theory and Implementation

Ehrenfest molecular dynamics is certainly the oldest approach to “on–the–fly”molecular dynamics and is typically used for collision– and scattering–type problems154,649,426,532 . However, it was never in widespread use for systems with manyactive degrees of freedom typical for condensed matter problems for reasons thatwill be outlined in Sec. 2.6 (although a few exceptions exist 553,34,203,617 but herethe number of explicitly treated electrons is fairly limited with the exception ofRef. 617 ).2.3 Born–Oppenheimer Molecular DynamicsAn alternative approach to include the electronic structure in molecular dynamicssimulations consists in straightforwardly solving the static electronic structure problemin each molecular dynamics step given the set of fixed nuclear positions at thatinstance of time. Thus, the electronic structure part is reduced to solving a time–independent quantum problem, e.g. by solving the time–independent Schrödingerequation, concurrently to propagating the nuclei via classical molecular dynamics.Thus, the time–dependence of the electronic structure is a consequence of nuclearmotion, and not intrinsic as in Ehrenfest molecular dynamics. The resulting Born–Oppenheimer molecular dynamics method is defined byM I ¨R I (t) = −∇ I min {〈Ψ 0 |H e | Ψ 0 〉}Ψ 0(32)E 0 Ψ 0 = H e Ψ 0 (33)for the electronic ground state. A deep difference with respect to Ehrenfest dynamicsconcerning the nuclear equation of motion is that the minimum of 〈H e 〉has to be reached in each Born–Oppenheimer molecular dynamics step accordingto Eq. (32). In Ehrenfest dynamics, on the other hand, a wavefunction that minimized〈H e 〉 initially will also stay in its respective minimum as the nuclei moveaccording to Eq. (30)!A natural and straightforward extension 281 of ground–state Born–Oppenheimerdynamics is to apply the same scheme to any excited electronic state Ψ k withoutconsidering any interferences. In particular, this means that also the “diagonalcorrection terms” 340∫DI kk ({R I(t)}) = − dr Ψ ⋆ k ∇2 I Ψ k (34)are always neglected; the inclusion of such terms is discussed for instance inRefs. 650,651 . These terms renormalize the Born–Oppenheimer or “clamped nuclei”potential energy surface E k of a given state Ψ k (which might also be theground state Ψ 0 ) and lead to the so–called “adiabatic potential energy surface”of that state 340 . Whence, Born–Oppenheimer molecular dynamics should not becalled “adiabatic molecular dynamics”, as is sometime done.It is useful for the sake of later reference to formulate the Born–Oppenheimerequations of motion for the special case of effective one–particle Hamiltonians. Thismight be the Hartree–Fock approximation defined to be the variational minimumof the energy expectation value 〈Ψ 0 |H e | Ψ 0 〉 given a single Slater determinant Ψ 0 =det{ψ i } subject to the constraint that the one–particle orbitals ψ i are orthonormal12