Atomic virial theorem

The postulation of an atomic virial theorem for the topologically defined atoms leads to a numtjer of important conclusions (Bader and Beddall 1972). 

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,... 

Because of the atomic virial theorem, eqn (6.24), the atomic energy E Q)... 

Because L(Ci) vanishes for an atom, integration of eqn (7.4) over the basin of an atom yields the atomic virial theorem... 

It was pointed out in the discussion of the atomic virial theorem for a stationary state (eqn (6.23)) that, while the values of the individual contributions to the virial of the subsystem are dependent upon the choice of origin for the vector r, their sum, which determines the total subsystem virial, is independent of the choice of origin. It is clear from eqn (8.193) that the same... 

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. 

As illustrated earlier, setting the generator equal to e-p defines the atomic force and the variational principle leads to the integral atomic force law, or the equation of motion for an atom in a molecule. Finally, it was shown that, when F = — sr-p, the commutator defines the electronic kinetic energy and virial for an atom, and the variational principle yields the relationship between these quantities, the atomic virial theorem. These three relationships—the equation of continuity, the equation of motion, and the virial theorem—form the basis for the understanding of the mechanics of an atom in a molecule. 

The atomic virial theorem for a system in the presence of an electromagnetic field is obtained by setting the generator F equal to f The commutators required for the evaluation of eqn (8.225), are readily evaluated giving... 

Since the integral of the Laplacian vanishes over fl, the integral of the local statement of the virial theorem (Eq. 6) over the volume of an atom O in a molecule yields the atomic virial theorem ... 

These equations resemble those for atoms [Eqs. (14.17) and (14.18)]. At / = 00 we have the separated atoms, and the atomic virial theorem gives... 

Equation (59) is useful in simulations where periodic boundary equations are employed. In the presence of periodic boundary conditions Eq. (57) should not be used [41]. The product of the force acting on a particle times its position vector is called virial, so Eqs. (57)-(59) are forms of the (atomic) virial theorem for the pressure. 

The sum of the atomic basin and surface virials equals the total virial V(i ) for the atom and in a stationary state one obtains the atomic virial theorem... 

A topological energy partitioning, independent of the atomic virial theorem (see Sect. 7), was proposed [48] in 2001. This iimovation led to the development of a segment of QCT called interacting quantum atoms (IQA) [28]. Since its implementation in the computer program AIMALL [71], IQA has become an increasingly popular tool in the armoury of interpretative quanmm chemical tools. 

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