Virial theorem derivation

The ensemble virial theorems derived here have the same form as the ground-state virial theorems. The only difference is that they include quantities defined for the ensembles rather than for ground-state quantities. The reason for the similarity is that the ensemble Kohn-Sham equations have the same form as the ground-state equations. 

The virial theorems derived above are exact (i.e., exact functionals satisfy these equations). However, these equations do not generally hold for approximate functionals. Therefore, they can be used to judge the quality of the approximation. Exact theorems or equations can be very useful in constructing approximate functionals, too. 

The local form of the virial theorem derived from the subsystem hypervirial theorem is as follows [1] ... 

The pressure is usually calculated in a computer simulation via the virial theorem ol Clausius. The virial is defined as the expectation value of the sum of the products of the coordinates of the particles and the forces acting on them. This is usually written iV = X] Pxi where x, is a coordinate (e.g. the x ox y coordinate of an atom) and p. is the first derivative of the momentum along that coordinate pi is the force, by Newton s second law). The virial theorem states that the virial is equal to —3Nk T. 

In spite of its simplicity and the visual similarity of this equation to Eq. (7), we would like to note that Eq. (11) leads to a nontrivial thermodynamics of a partially quenched system in terms of correlation functions, see, e.g.. Ref. 25 for detailed discussion. Evidently, the principal route for and to the virial theorem is to exploit the thermodynamics of the replicated system. However, special care must be taken then, because the V and s derivatives do not commute. Moreover, the presence of two different temperatures, Pq and P, requires attention in taking temperature derivatives, setting those temperatures equal, if appropriate, only at the end of the calculations. 

In the derivation above, we have included the kinetic energy of the nuclei in the Hamiltonian and considered a stationary state. In Eq. II.3, this term has been neglected, and we have instead assumed that the nuclei have given fixed positions. It has been pointed out by Slater34 that, if the nuclei are not situated in the proper equilibrium positions, the virial theorem will appear in a slightly different form. (A variational derivation has been given by Hirschfelder and Kincaid.11)... 

In previous work, we have mainly used the DPM model to investigate the effects of the coefficient of normal restitution and the drag force on the formation of bubbles in fluidized beds (Hoomans et al., 1996 Li and Kuipers, 2003, 2005 Bokkers et al., 2004 Van der Floef et al., 2004), and not so much to obtain information on the constitutive relations that are used in the TFMs. In this section, however, we want to present some recent results from the DPM model on the excess compressibility of the solids phase, which is a key quantity in the constitutive equations as derived from the KTGF (see Section IV.D.). The excess compressibility y can be obtained from the simulation by use of the virial theorem (Allen and Tildesley, 1990). 

This equation may be used to derive the quantum mechanical virial theorem. For this purpose it is necessary to define the kinetic operator... 

The relationship between the exchange potential of DFT and the corresponding energy functional is established through the virial theorem. The two are related via the following relationship derived by Levy and Perdew [23]... 

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]. 

In this work, the electronic kinetic energy is expressed in terms of the potential energy and derivatives of the potential energy with respect to nuclear coordinates, by use of the virial theorem (5-5). Thus, the results are valid for ail bound electronic states. However, the functional derived for E does not obey a variational principle with respect to (Pg ( )), even though in... 

A virial theorem (5-8) applied to the electronic coordinates gives the kinetic energy (T) in terms of (V) and expectation values of the first derivative of V with respect to nuclear coordinates (66-72). For bound electronic states f)... 

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... 

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,... 

However, in the second set of data, reporting scans of the PES for a limited set of small molecules, it appears that the geometries obtained are satisfactory. Moreover, the nature of the technique used for the determination of Exc, namely the use of a "senior" Exe functional, or the use of the virial theorem, as well as the use of a line integration (not reported here), leads to quite similar geometries. This point is in accord with a similar conclusion obtained by van Gisbergen et ol. in their frequency-dependent polarizabilities [75] they choose to use a "mixed scheme" where a different approximation for fxc and Vxc were used, whereas fxc is the functional derivative of the exchange-correlation potential Vxc, with respect to the time-dependent density. 

Attempts to improve molecular wavefunctions so as to be able to calculate properties more accurately continue to be made, particularly via the constrained variational procedure. Two-particle hypervirial constraints were considered by Bjoma within the SCF formation,282 and he presented a perturbational approach to their solution.233 Using Scherr s wavefunction, and constraining p to satisfy the molecular virial theorem, a calculation on N2 led to rapid convergence.234-235 The constrained SCF orbitals are believed to be a closer approximation to the true tfi nearer the nucleus than further out. A later paper discussed the electron-density maps in comparison to the SCF derived maps, which confirm the conclusion that the wavefunction near the nucleus is improved.236... 

In the original derivation of the classical virial theorem given by Clausius, an expression corresponding to eqn (5.30) is also obtained. In the classical case one argues that the time average of d(f p)/dt vanishes over a sufficiently long period of time or that the motion is periodic to obtain the equivalent of eqn (5.31). ... 

Alternatively, the virial theorem may be derived directly from eqn (5.29), which expresses the forces in terms of the time derivative of the current density. Dotting the vector r into eqn (5.29) to obtain the virial of these forces gives... 

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. 

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