While (29) has the form of the usual virial-theorem equation of state, its derivation from the virial theorem appears to fail for periodic boundary conditions. Consider, for example, the virial in the infinite checkerboard, [Pg.11]

The starting point of the derivation of the virial theorem is the Schrodinger equation for the ith excited state. [Pg.250]

For a particular choice of the generator F, equations 14 and 16 both yield statements of the virial theorem. The virial theorem for an atom in a molecule obtained from equation 16 states that twice the average kinetic energy of the electrons in atom Q, T(Q), equals the negative of the virial of the forces exerted on the electrons, F (Q) [Pg.43]

The virial theorem was first stated by Clausius in 1870 for the expectation value of the product between the position of each particle and the force acting on it. Indeed, substituting in Eq. 8.15 for the position of a particle, and using Hamilton s equation of motion, yields [Pg.141]

Equation (14.7) is the quantum-mechanical virial theorem. Note that its validity is restricted to bound stationary states. [Tlie word vires is Latin for forces in classical mechanics, the derivatives of the potential energy give the negatives of the force components.] [Pg.460]

We have thus arrived at the fundamental result that the virial theorem together with the equations of state divides the multitude of stars into two classes those which contract and heat forever , and those which cool to zero temperature and to a final finite density. Real stars which evolve according to the cooling branch are the white dwarfs, and [Pg.37]

The volume contribution by the metallic 5f-5f and the covalent cation 5f-N2p bonds to a virial-theorem formulation of the equations of state of a series of light actinide nitrides was calculated in the self-consistent linear muffin tin orbital (LMTO), relativistic LMTO, and spin-polarized LMTO approximations [46]. The results for ThN give the same lattice spacing in all three approximations higher by ca. 3% than the experimental value, which discrepancy is attributed to the assumed frozen core ions [47]. [Pg.31]

A recently proposed theory for a single excited state based on Kato s theorem is reviewed. This theory is valid for Coulomb systems. The concept of adiabatic connection leads to Kohn-Sham equations. Differenticil virial theorem is derived. Excitation energies and inner-shell transition energies are presented. [Pg.247]

This is a remarkable equation that gives a general expression for the equation of state for any fluid system. The first term on the right-hand side is the well-known ideal gas contribution to the pressure. The second term encompasses the influence of all non-idealities on the pressure and the equation of state of the system. Notably, starting with the ensemble average in the virial theorem, we obtain the thermodynamic behavior of any fluid. [Pg.144]

Schweitz proceeded to show that for a gas with an isotropic and uniform distribution of momentum, Zs reduces to —3pv, and in the absence of internal forces, one obtains 2T = 3pv, a statement of the ideal gas equation. One notes that the derivation is based on the exchange of momentum through imaginary walls, while in the case of a petit ensemble, the derivation is based on the action of forces exerted on the particles by impenetrable walls. Schweitz concludes with a discussion of the use by Slater [14], March [20] and Ross [21] of the virial theorem for a petit ensemble stated in the form [Pg.291]

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