Passage from Quantum to Classical Molecular Dynamics in the Presence of Coulomb Interactions

Abstract

We present a rigorous derivation of classical molecular dynamics (MD) from quantum molecular dynamics (QMD) that applies to the standard Hamiltonians of molecular physics with Coulomb interactions. The derivation is valid away from possible electronic eigenvalue crossings.

On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H–J Equation

“…To begin, let us recall that in the framework of the usual Weyl quantization on R n it can be considered the following space of test functions (see for example [3,22])…”

“…The well known arguments within the framework of the Weyl quantization on R n (see for example [3,4,17,22]) must be adapted for the Weyl quantization on T n . The first result reads as follows Proposition 3.1 Let ψh be the solution of (3.1), t > 0 and f ∈ C ∞ c ((0, t) × T n × R n ; R) such that ∀s ∈ (0, t) it holds f (s, · ) ∈ A as in Definition 2.5.…”

“…On the other hand, since H is smooth, it is easily seen that equation This can be proved thanks to the phase space representation (2.12) for (4.14), recalling that the phase space transform φ is supposed to be compactly supported and by the equation for the Wigner transform (3.18) with test functions f = X (t)φ(x). Then, by following the same arguments in Lemma 3.2 of [3], it follows that dw ∈ C weak ([−T, +T ]; M + (T n × R n )) and that for any −T ≤ t ≤ T it holds the weak limit Whψh(t) dw t with test functions φ ∈ A ⊂ C b (T n × R n ), namely as in Definition 2.6.…”

“…About the above result, we recall Lemma 3.2 of[3], and we focus the attention on the additional continuous regularity of the limits{dw t } t∈[−T,T ] of Whψh(t) in L ∞ ([−T, +T ];A ) passing through sequences ash → 0. In fact, it can be easily proved that for our test functions φ ∈ A, the related functions h,φ (t) := η∈h 2 Z n T n φ(x, η)Whψh(t, x, η)dx (4.14)…”

Almost everywhere well-posedness of continuity equations with measure initial data

“…Assume, besides (a), (b) above, that (3) has uniqueness in L ∞ + ([0, T ]; L 1 ∩ L ∞ (R d )). Then the ν-RLF μ(t, μ) relative to b exists, is unique (by Theorem 2.2) and…”