Tensor virial theorem

The final conclusion is that the expression for the stress tensor developed in Sect. 7 is consistent with the tensor virial theorem. Furthermore, it is not permitted to add a divergenceless term to the stress tensor, and the partitioning of the external source term in Eq. (7.9) into an external force term G and the divergence of in Eq. (7.10) is correct. 

In a simulation [19] the pressure tensor is obtained from the virial theorem [78]... 

A differential virial theorem represents an exact, local (at space point r) relation involving the external potential u(r), the (ee) interaction potential u r,r ), the diagonal elements of the 1st and 2nd order DMs, n(r) and n2(r,r ), and the 1st order DM p(ri r2) close to diagonal , for a particular system. As it will be shown, it is a very useful tool for establishing various exact relations for a many electron systems. The mentioned dependence on p may be written in terms of the kinetic energy density tensor, defined as... 

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. 

In addition, we used Bader s atoms-in-molecules (AIM) theory [56,57] to help analyze some of the results. For convenience, we give here a very brief overview of this approach. According to the AIM theory, every chemical bond has a bond critical point at which the first derivative of the charge density, p(r), is zero. The (> r) topology is described by a real, symmetric, second-rank Hessian-of-/3(r) tensor, and the tensor trace is related to the bond interaction energy by a local expression of the virial theorem ... 

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]. 

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... 

A generalization of the virial theorem to all components of the stress tensor has been given in Refs. 11 and 53 and termed the "stress... 

A different type of structural parameter was considered recently by the present authors, namely X representing a homogeneous macroscopic strain defined as the linear scaling of all particle positions as x- 1+e)x. The e is a constant 3x3 strain tensor, and e=0 corresponds to some reference configuration. The conjugate force is in this case defined as the macroscopic stress a, and an explicit general expression denoted the "stress theorem" is derived by Nielsen and Martin (1983). The result is a generalization of the quantum virial theorem (Born et al., 1926),... 

FiG. 8 Evolution of the component a x of the stress tensor with time t for the C24 system. The results at every time t have been obtained either by applying the virial theorem and averaging over all dynamical trajectories (broken line) or by using a... 

In this arena, energetic properties are usually derived by integrating densities over real space domains, and not by examining appropriate scalar or vector fields. Exceptions to this rule exist the localized orbital locator (LOL) focuses on the topological properties of a kinetic energy density [34], and the QTAIM virial (t ) and energy density (J ) fields are commonly examined at critical points (CPs) of the density. The latter are however computed from the density and its derivatives through the QTAIM s local virial theorem [1], that depends on an arbitrary choice of the kinetic stress tensor [9]. 

Recall that the stress tensor is defined as the change in force on the surface of a volume element from a differential change in the area of that element (2). On the other hand, (12) and (13) indicate that the stress tensor is linked not only to the electronic potential energy (which defines the force on the electrons), but also to the kinetic energy. The stress tensor is thus revealed as the key component in the differential virial theorem [30, 35, 56, 57] ... 

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