Local virial theorem

If the second derivative, and hence the curvature of/, is negative at x, then / at x will be larger than the average of / at all neighbouring points, i.e. / concentrates at point x73. Therefore - V2p(r), which is the second derivative of a function depending on three coordinates x, y and z, has been called the Laplace concentration of the electron density distribution. Furthermore, the Laplacian of pir) provides the link between electron density p(r) and energy density Hir) via a local virial theorem (equation 8)67,... 

Thus, if it is assumed that the local virial theorem is valid for the model electron densities fitted to the experimental structure factors, the kinetic, g(r), and potential, v(r), energy densities may be mapped, as well as the energy characteristics of the (3,-1) bond critical points evaluated [38]. 

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... 

Electrostatic potential maps have been used to make predictions similar to these (Scrocco and Tomasi 1978). Such maps, however, do not in general reveal the location of the sites of nucleophilic attack (Politzer et al. 1982), as the maps are determined by only the classical part of the potential. The local virial theorem, eqn (7.4), determines the sign of the Laplacian of the charge density. The potential energy density -f (r) (eqn (6.30)) appearing in eqn (7.4) involves the full quantum potential. It contains the virial of the Ehrenfest force (eqn (6.29)), the force exerted on the electronic charge at a point in space (eqns (6.16) and (6.17)). The classical electrostatic force is one component of this total force. 

The application of the local virial theorem at the BCP implies that interactions for which < 0 are dominated by a local lowering of the potential energy, while those for which V p > 0 are dominated by a local excess in the kinetic energy (since G(r) > 0 and (r) < 0, always). 

The Laplacian is related, through the local virial theorem [32, 33], to the electronic kinetic (G(r) > 0) and potential energy (V(r)) densities... 

Local Virial Theorem for Ensembles of Excited States ... 

Virial theorem has an important role in quantum mechanics. It proved to be also very useful in density functional theory (see e.g. Parr and Yang, 1989). Several forms of virial theorem have been proposed in ground-state density functional theory, for example, local virial theorem (Nagy and Parr 1990), differential virial theorem (Holas and March 1995), regional virial theorem (Nagy 1992), and spin virial theorem (Nagy 1994b). In this chapter, local and differential virial theorems are extended to excited states in the frame of density functional theory. 

The virial theorem was also derived for ensanbles of excited states (Nagy 2002a). In the ground-state theory, several forms of the virial theoran were derived. The local and differential forms proved to be especially useful. In this chapter, the local virial theorem is derived for ensembles of excited states. In Section 7.2, the ensemble theory of excited states is summarized. The ensemble local virial theoran is derived in Section 7.3. Extension of the differential virial theorem of Holas and March (1995) to ensembles is presented in Section 7.4. Finally, Section 7.5 is devoted to discussion. 

LOCAL VIRIAL THEOREM FOR ENSEMBLES OF EXCITED STATES... 

Here, real orbitals are considered following Deb and Ghosh (1979) in their derivation of the local virial theorem. Divide first the Kohn-Sham eqnations by the orbital M , take the gradient of the equations, multiply with the orbitals and occupation numbers, and sum for all i ... 

It has the same form as the ground-state local virial theorem (Nagy and Parr 1990). 

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]. 

Holas A, March NH (1995) Exact exchange-correlation potential and approximate exchange potential in terms of density matrices. Phys Rev A 51 2040-2048 Nagy A, March NH (1997) Differential and local virial theorem. Mol Phys 91 597-602 Ehrenfest P (1927) Bemerkung fiber die angenaherte Gfiltigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Z Phys A 45 455-457... 

Laplacian of the electron density, W p(r) scalar derivative of the gradient vector field of p(r). It determines where electronic charge is locally concentrated, < 0, and depleted, > 0. The Laplacian can be partitioned into energy densities. It can be demonstrated that there is a local virial theorem... 

The Laplacian is especially telling quantity [17,55], since it is connected to the kinetic and potential energy densities at BCP, G(Fc) and y(Fc), respectively, by the following local-virial theorem expression ... 

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