  In the classical work by Irving and Zwanzig [Irving J.H. and Zwanzig R.W., J.
Chem. Phys. 19 (1951), 1173-1180 ] it has been shown that quantum observables
for macroscopic density, momentum and energy satisfy the conservation laws of
fluid dynamics. This work derives the corresponding classical molecular
dynamics limit by extending Irving and Zwanzig's result to matrix-valued
potentials for a general quantum particle system. The matrix formulation
provides the semi-classical limit of the quantum observables in the
conservation laws, also in the case where the temperature is large compared to
the electron eigenvalue gaps. The classical limit of the quantum observables in
the conservation laws is useful in order to determine the constitutive
relations for the stress tensor and the heat flux by molecular dynamics
simulations. The main new steps to obtain the molecular dynamics limit is to:
(i) approximate the dynamics of quantum observables accurately by classical
dynamics, by diagonalizing the Hamiltonian using a non linear eigenvalue
problem, (ii) define the local energy density by partitioning a general
potential, applying perturbation analysis of the electron eigenvalue problem,
(iii) determine the molecular dynamics stress tensor and heat flux in the case
of several excited electron states, and (iv) construct the initial particle
phase-space density as a local grand canonical quantum ensemble determined by
the initial conservation variables.
