Ab initio molecular dynamics: Theory and Implementation

The next step in the derivation of classical molecular dynamics is the task toapproximate the nuclei as classical point particles. How can this be achieved in theframework of the TDSCF approach, given one quantum–mechanical wave equationdescribing all nuclei? A well–known route to extract classical mechanics fromquantum mechanics in general starts with rewriting the corresponding wavefunctionχ({R I }; t) = A({R I }; t) exp [iS({R I }; t)/ ] (8)in terms of an amplitude factor A and a phase S which are both considered to bereal and A > 0 in this polar representation, see for instance Refs. 163,425,535 . Aftertransforming the nuclear wavefunction in Eq. (7) accordingly and after separatingthe real and imaginary parts, the TDSCF equation for the nuclei∂S∂t + ∑ I∂A∂t + ∑ I∫1(∇ I S) 2 +2M I1M I(∇ I A) (∇ I S) + ∑ Idr Ψ ⋆ H e Ψ = 2 ∑ I1 ∇ 2 I A2M I A(9)12M IA ( ∇ 2 I S) = 0 (10)is (exactly) re–expressed in terms of the new variables A and S. This so–called“quantum fluid dynamical representation” Eqs. (9)–(10) can actually be used tosolve the time–dependent Schrödinger equation 160 . The relation for A, Eq. (10),can be rewritten as a continuity equation 163,425,535 with the help of the identificationof the nuclear density |χ| 2 ≡ A 2 as directly obtained from the definitionEq. (8). This continuity equation is independent of and ensures locally the conservationof the particle probability |χ| 2 associated to the nuclei in the presence ofa flux.More important for the present purpose is a more detailed discussion of therelation for S, Eq. (9). This equation contains one term that depends on , acontribution that vanishes if the classical limit∂S∂t + ∑ I12M I(∇ I S) 2 +∫dr Ψ ⋆ H e Ψ = 0 (11)is taken as → 0; an expansion in terms of would lead to a hierarchy of semiclassicalmethods 425,259 . The resulting equation is now isomorphic to equations ofmotion in the Hamilton–Jacobi formulation 244,540∂S∂t + H ({R I}, {∇ I S}) = 0 (12)of classical mechanics with the classical Hamilton functionH({R I }, {P I }) = T({P I }) + V ({R I }) (13)defined in terms of (generalized) coordinates {R I } and their conjugate momenta{P I }. With the help of the connecting transformationP I ≡ ∇ I S (14)7