Surface flux virial

Equation (18) is identical in form and content to the atomic statement of the virial theorem Equation (7) - the virial theorem for a proper open system - with the petit virial Zp being the analogue of the basin virial Vb and the surface flux virial Zs/ the analogue of the surface virial vs, the virial of the Ehrenfest forces acting on the surface of the open system. 

Schweitz s representation of the quantum stress tensor a (r) in terms of the flux density operator acting on the momentum in Equation (19) makes clear its interpretation as a momentum flux density. Schweitz does not, however, consider how the surface flux virial in the quantum case, Zs or i ,s, may be related to the pv product. This, as demonstrated in the following section, has been accomplished using the atomic statement of the virial theorem [12]. 

Pressure and the surface flux virial construction of the pv operator... 

Equation (22) properly relates the pressure to the virial of the force arising from the surface flux in the momentum density... 

Chemists have long been intrigued by the question, Does an atom in a molecule somehow preserve its identity An answer to this question comes from studies on the topological properties of p(r) and grad p(r). It has been shown that the entire space of a molecule can be partitioned into atomic subspaces by following the trajectories of grad p(r) in 3D space. These subspaces themselves extend to infinity and obey a subspace virial theorem (2 (7) + (V) = 0). The subspaces are bounded by surfaces of zero flux in the gradient vectors of p(r), i.e., for all points on such a surface,... 

Since the surface is not crossed by any gradient lines, it is referred to as the surface of zero flux. As further discussed below, the virial theorem is satisfied for each of the regions of space satisfying the zero-flux boundary condition. 

The non-vanishing of the flux of a quantum mechanical current in the absence of a magnetic field is what distinguishes the mechanics of a subsystem from that of the total system in a stationary state. The flux in the current density will vanish through any surface on which i// satisfies the natural boundary condition, Vi/ n = 0 (eqn (5.62)), a condition which is satisfied by a system with boundaries at infinity. Thus, for a total system the energy is stationary in the usual sense, 5 [i/ ] = 0, and the usual form of the hyper-virial theorem is obtained with the vanishing of the commutator average. 

Since the atom Q is bounded by a surface of zero flux, Z,(Q) = 0 and one obtains the atomic statement of the virial theorem,... 

The second equality given in eqn (6.70) follows from the definition of X(r) in eqn (5.49). The integration of this energy density over a region of space bounded by a surface of zero flux in Vp yields an energy e( ) which will satisfy the various statements of the atomic virial theorem,... 

In addition to the basin virial i b(n) arising from the internal forces, which is identical to that obtained in the field-free case (eqn (8.192)), there are further contributions to the virials of the forces acting on the electrons in the basin of the atoms resulting from their interaction with the external fields. The surface virial reduces to the flux in the virial of the mechanical stresses... 

The opening discussion will demonstrate that the definition of pressure is a problem that requires the physics of an open system, classical or quantum. This is an understandable result since the pressure acting on a system is the force exerted per unit area of the surface enclosing the system, the flux in the momentum density per unit area per unit time of the bounding surface. This understanding calls into question the use of the result obtained from the classical virial theorem for an ideal gas to define the pressure acting on a quantum system. 

The bond path is always found to be accompanied by a shadow graph, the virial path, first discovered by Keith, Bader, and Aray [17]. The virial path is a line of maximally-negative potential energy density in three-dimensional space that links the same pair of atoms that share a bond path and an interatomic surface of zero-flux. No theoretical basis has ever been provided that requires the presence of a virial path as a doppelganger of every bond path that links two chemically bonded atoms, however, there is no known computational violation of this observation to date known to the authors. The presence of the virial path links the concept of chemical bonding directly with the concept of energetic stability as amply discussed in literature on QTAIM. 

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