Differential virial theorem

Differential Virial Theorem and Exchange-Correlation Potential.99... 

Differential Virial Theorem and Hartree-Fock Theory. 100... 

DIFFERENTIAL VIRIAL THEOREM AND EXCHANGE-CORRELATION POTENTIAL... 

The reason why the relationship in Equation 7.41 is called the differential virial theorem is because if we take the dot product of both sides with vector r, multiply both sides by pir), and then integrate over the entire volume, it gives... 

Our aim here is to apply the differential virial theorem to get an expression for the Kohn-Sham XC potential. To this end, we assume that a noninteracting system giving the same density as that of the interacting system exists. This system satisfies Equation 7.4, i.e., the Kohn-Sham equation. Since the total potential term of Kohn-Sham equation is the external potential for the noninteracting system, application of the differential virial relationship of Equation 7.41 to this system gives... 

Now we discuss the differential virial theorem for HF theory and the corresponding Kohn-Sham system. The Kohn-Sham system in this case is constructed [41] to... 

There are other noteworthy single excited-state theories. Gorling developed a stationary principle for excited states in density functional theory [41]. A formalism based on the integral and differential virial theorems of quantum mechanics was proposed by Sahni and coworkers for excited state densities [42], The local scaling approach of Ludena and Kryachko has also been generalized to excited states [43]. 

Exchange identities utilizing the principle of adiabatic connection and coordinate scaling and a generalized Koopmans theorem were derived and the excited-state effective potential was constructed [65]. The differential virial theorem was also derived for a single excited state [66]. 

Differential Virial Theorem for the Hartree-Fock Solution. 103... 

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... 

The differential virial theorem, obtained by us in [30], has the form of the following identity... 

From the differential virial theorem (165) for interacting electron systems one can obtain immediately an analogous theorem for noninteracting systems, just by putting m = 0 and replacing the external potential i (r) with Vs(r) ... 

The differential virial theorem (169) for noninteracting systems can alternatively be obtained [31], [32] by summing (with the weights fj ) similar relations obtained for separate eigenfunctions 4>ja(r) of the one-electron Schrodinger equation (40) [in particular the KS equation (50)]. Just in that way one can obtain, from the one-electron HF equations (33), the differential virial theorem for the HF (approximate) solution of the GS problem, as is shown in Appendix B, Eq. (302), in a form ... 

Many interesting integral relations may be deduced from the differential virial theorem, allowing us to check the accuracy of various characteristics and functionals concerning a particular system (for noninteracting systems see e.g. in [31] and [32]). As an example, let us derive here the global virial theorem. Applying the operation Jd rY,r, to Eq. (165), we obtain... 

Thus, finally Eq. (299) can be rewritten as the following differential virial theorem for the HF solution of the GS problem... 

To motivate the approach of Holas and March [52], let us briefly set out the early work of March and Young [53], who set up the so-called differential virial theorem for independent fermions moving in 1 dimension (x) only in a common potential energy V(x). If t(x) denotes the kinetic energy per unit length at position x, their result was... 

The first achievement of the study of Holas and March [99] is to establish the differential form of the above virial theorem [their Eq. (2.15)]. Again, as in the zero field case treated above, this differential virial theorem is interpreted as a force-balance equation. The well-known Lorentz force of electromagnetism then appears quite naturally in this equation. 

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. 

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... 

The ensemble differential virial theorem for spherically symmetric systems (Nagy 2011) has been recently derived ... 

The ensemble differential virial theorem for spherically symmetric systems (Nagy 2011) has recently turned out to be very useful in the solution of the orbital-free problem. A first-order differential equation has been derived for the functional derivative of the ensemble noninteracting kinetic energy functional ... 

THEORY FOR A SINGLE EXCITED STATE DIFFERENTIAL VIRIAL THEOREM... 

Recently, a new approach of treating a single excited state has been presented It is based on Kato s theorem and is valid for Coulomb external potential (i.e. free atoms, molecules and solids). It has the advantage that one can treat a single excited state. In this paper this theory is reviewed and the differential virial theorem is derived. Excitation energies are presented for the Li, Na and K atoms and inner-shell transition energies are shown for the Be atom. 

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