On some 'local' force densities and stress tensors in molecular quantum mechanics

Within the density-functional theory of Hohenberg, Kohn and Sham, a comprehensive stress tensor is defined for a many-electron polyatomic system. This consists of contributions from kinetic energy, Coulomb energy and exchange correlation energy. The covariant derivative of the stress tensor yields the 'local' force density in three-dimensional space, which vanishes for static stationary states due to the balance between classical and quantum force densities. Some interesting implications of the zero force density are discussed. In particular, this turns out to be a special case of the Euler equation or the Navier-Stokes equation in fluid dynamics. The fluid-dynamical interpretation of the stress tensor is discussed and the viscosity coefficients are derived. The present work is likely to provide a basis for 'classical' interpretations of many molecular and solid-state phenomena.