The in situ pressure acting on the surface of an open system at the atomic level is defined and determined by the virial theorem for a proper open system, one whose spatial boundary and equations of motion are determined by the principle of stationary action. The quantum pressure is determined by the virial of the force resulting from the electronic momentum flux through the surface of the open system. A scaling procedure is used to demonstrate that the expectation value of the pressure-volume product of a proper open system is proportional to its surface virial. Previous work, in analogy with the classical virial theorem for a contained system, incorrectly relates the pressure to the external forces of constraint acting on a closed system. A neon vise consisting of a chain of three, four or five hydrogen molecules compressed between two neon atoms is used to introduce the quantum definition of pressure and study its effect on the mechanical properties of an atom and on the topology of the electron density. Pressures approaching 160 GPa have been calculated for the vise. The topology of the electron density and the homeomorphism it exhibits with the virial field are found to be invariant to an increase in pressure, the electron density accumulating to an ever increasing extent between all pairs of nuclei which serve as the sole attractors. The virial of the Ehrenfest force acting on the surface of a compressed molecule provides a measure of the increase in the electronic kinetic energy resulting from the applied pressure. The effects of pressure on the intra- and intermolecular bonding are discussed in terms of pressure-induced changes in the electron density and in the mechanical properties of the atoms. (C) 1997 American Institute of Physics.