Energy quantum virial theorem

Show explicitly for a hydrogen atom in the Is state that the total energy is equal to one-half the expectation value of the potential energy of interaction between the electron and the nucleus. This result is an example of the quantum-mechanical virial theorem. 

The effect of pressure on the ground-state electronic and structural properties of atoms and molecules have been widely studied through quantum confinement models [53,69,70] whereby an atom (molecule) is enclosed within, e.g., a spherical cage of radius R with infinitely hard walls. In this class of models, the ground-state energy evolution as a function of confinement radius renders the pressure exerted by the electronic density on the wall as —dEldV. For atoms confined within hard walls, as in this case, pressure may also be obtained through the Virial theorem [69] ... 

A prime example is the so-called quantum-mechanical virial theorem that appears in countless chemistry textbooks. The theorem is purported to state that the relationship between the expectation values of kinetic and potential energies... 

The full usefulness of the classification using V Pb must await the development of the quantum mechanical aspects of the theory. The Laplacian of the charge density appears in the local expression of the virial theorem and it is shown that its sign determines the relative importance of the local contributions of the potential and kinetic energies to the total energy of the system, A full discussion of this topic is given in Section 7.4. 

The non-vanishing of the flux of a quantum mechanical current in the absence of a magnetic field is what distinguishes the mechanics of a subsystem from that of the total system in a stationary state. The flux in the current density will vanish through any surface on which i// satisfies the natural boundary condition, Vi/ n = 0 (eqn (5.62)), a condition which is satisfied by a system with boundaries at infinity. Thus, for a total system the energy is stationary in the usual sense, 5 [i/ ] = 0, and the usual form of the hyper-virial theorem is obtained with the vanishing of the commutator average. 

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. 

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.]... 

For a bound stationary state, the quantum-mechanical virial theorem states that 2(r) = Hi qiidV/dqi)), where the sum is over the Cartesian coordinates of all the particles. If F is a homogeneous function of degree n, then 2(7 ) = n(V). For a diatomic molecule, the virial theorem becomes = - u - R dU/dR) and (V) = 2U + R dU/dR), where U R) is the potential-energy function for nuclear motion. The virial theorem shows that at 7 , ( F) of a diatomic molecule is less than the total ( V) of the separated atoms, and ( T i) is greater than the total (Tei) of the separated atoms. 

Eq. (A.l 1) constitutes the quantum mechanical virial theorem for molecular solutes described within the PCM model, which involves, on the left side, the kinetic and the total potential energies for exact state-wavefunctions. The terms on right side of Eq.(A.ll) have a physical meaning which can be clarified with the aid of the Hellmann-Feynman theorem discussed in Chap. 2 (see Eq. 2.1 ). 

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