Atomic virial definition

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,... 

Integration of eqn (8.201) over an atomic volume for which the integral of V p(r) vanishes yields, term for term, the atomic virial theorem for a time-dep>endent system (eqn (8.193)) or for a stationary state (eqn (6.23)). Thus, eqn (8.201) is, in terms of its derivation and its integrated form, a local expression of the virial theorem. The atomic virial theorem provides the basis for the definition of the average energy of an atom, as discussed in Chapter 6. 

From eqn (6.30) it is clear that the virial of the electronic forces, which is the electronic potential energy, is totally determined by the stress tensor a and hence by the one-electron density matrix. The atomic statement of the virial theorem provides the basis for the definition of the energy of an atom in a molecule, as is discussed in the sections following Section 6.2.2. 

The atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

The definition of the energy of an atom in a molecule requires detailed consideration from a number of points of view and the following section is devoted to that task. The definition is shown to follow directly from the atomic statement of the virial theorem and, once having established this fact, the underlying equations are readily put down. We give the final equations here. The energy of an atom in a molecule, is purely electronic in origin... 

While the individual contributions to the virial tC ( 2) can be given physical interpretations, care must be exercised in this regard. The value of the total virial tC( 2) is independent of the choice of origin used in the definition of the virial operator, but this is not the case for the basin and surface terms treated separately. The origin-dependent terms appearing in the total virial for an atom are... 

The interaction between bonded atoms is characterized by the values of p(r), V-p(r), G r) and V(r) at the bond critical point. G(r) is the positive definite kinetic energy density and V(r) is the potential energy density. At a bond critical point, the kinetic and potential energy densities are related to the Laplacian by the local form of the virial relation ... 

The definition of the atomic contribution to the virial of the external forces of constraint is nontrivial. Keith has developed the procedure for the atomic partitioning of null properties properties such as the sum of the Feynman forces on the nuclei, that sum to zero for the entire molecule [51]. Thus in analogy with the expression for a total system, the energy of atom A is given by... 

It follows that Hartree-Fock atomic wavefunctions must satisfy the virial relation. Such solutions are, by definition, the best (lowest energy) attainable in a single deter-minantal form. Best includes all conceivable variation, linear or nonlinear, so all improvements achievable by scale factor variation are already present at the Hartree-Fock level, and r) = 1. 

QTAIM charges and classification of interactions at the BCP. The QTAIM theory predicts that several local indicators calculated at the BCP are closely related to the nature of the interactions between atoms [18, 19, 53, 70]. This prediction relies notably on the local definition of the virial theorem ... 

---