Virial theorem, definition

Wigner s formula is open to criticism also on another point, since he assumes the existence of a stationary electron state where the density is so low that the kinetic energy may be neglected. This is in contradiction to the virial theorem (Eq. 11.15), which tells us that the kinetic energy can never be neglected in comparison to the potential energy and that the latter quantity is compensated by the former to fifty per cent. A reexamination of the low density case would hence definitely be a problem of essential interest. 

Use of Equation (1) in numerical work requires a means of generating x(r, r i(o) as well as the average charge density. Direct variational methods are not applicable to the expression for E itself, due to use of the virial theorem. However, both pc(r) and x(r, r ico) (39-42, 109-112) are computable with density-functional methods, thus permitting individual computations of E from Eq. (1) and investigations of the effects of various approximations for x(r, r ico). Within coupled-cluster theory, x(r, r ico) can be generated directly (53) from the definition in Eq. (3) then Eq. (1) yields the coupled-cluster energy in a new form, as an expectation value. 

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... 

The subtraction of the kinetic energy density G(r) from each side of the statement of the local virial theorem given in eqn (6.31) yields a definition of an energy density E,(r) as... 

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. 

The virial theorem plays a dominant role in the definition of pressure, in both classical and quantum mechanics. The following section demonstrates that the pv product for a proper open system is proportional to the surface virial, Equation (9), the virial of the Ehrenfest forces exerted by the surroundings on the open system [9,12],... 

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. 

While Pendas makes no attempt to disprove the quantum definition of pressure obtained through the scaling procedure of Marc and McMillan [22] as presented here, he does state that "the use of electron-only scaling to study stressed situations, where the virials due to the nuclear system cannot be neglected, is not a very consistent procedure." In reality, the physics of an open system does not neglect the virials due to the nuclear system, they are included in the total virial that is defined by taking the virial of the Ehrenfest force, see for example Equation (13). The virial theorem illustrates... 

The path whereby the maps C and D are each established is well defined. One solves the Schrodinger equation for each local potential v(r) to determine F, and then obtains the density p(r) from T via its definition. On the other hand, although the inverse maps C and D are known to exist, the specific paths establishing these maps are thus far unknown. However, the differential form of the virial theorem of Eq. (58) defines the path whereby the external potential v(r) is determined from the ground-state wavefunction T. The potential v(r) is the work done to bring an electron from infinity to its position at r against the field F(r) ... 

That is, the difference between the force equation and the differential virial theorem lies only in the definition of the stress tensor. The relationship between them is... 

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 ... 

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