Quantum mechanics virial theorem

The relationship E = —T = V /2) an example of the quantum-mechanical 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. 

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

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... 

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. 

For a bound stationary state, the quantum-mechanical virial theorem states that 2 (r) = 2 i ( , (3 )), where the sum is over the Cartesian coordinates of all the particles. 

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 ). 

The Quantum-Mechanical Virial Theorem. In classical mechanics, the virial theorem states that... 

Van der Waals Forces, 351. The Quantum-Mechanical Virial Theorem, 355. The Restricted Rotator, 358. 

Lowdin, P.-O., Scaling problem, virial theorem and connected relations in quantum mechanics."... 

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]. 

There are several possible ways of introducing the Born-Oppenheimer model " and here the most descriptive way has been chosen. It is worth mentioning, however, that the justification for the validity of the Bom-Oppenheimer approximation, based on the smallness of the ratio of the electronic and nuclear masses used in its original formulation, has been found irrelevant. Actually, Essen started his analysis of the approximate separation of electronic and nuclear motions with the virial theorem for the Coulombic forces among all particles of molecules (nuclei and electrons) treated in the same quantum mechanical way. In general, quantum chemistry is dominated by the Bom-Oppenheimer model of the theoretical description of molecules. However, there is a vivid discussion in the literature which is devoted to problems characterized by, for example, Monkhorst s article of 1987, Chemical Physics without the Bom-Oppenheimer Approximation... ... 

P.O. Lowdin, Scaling Problem, Virial Theorem, and Connected Relations in Quantum Mechanics, J. Mol. Spectros. 3 (1959) 46. 

Whereas the quantum-mechanical molecular Hamiltonian is indeed spherically symmetrical, a simplified virial theorem should apply at the molecular level. However, when applied under the Born-Oppenheimer approximation, which assumes a rigid non-spherical nuclear framework, the virial theorem has no validity at all. No amount of correction factors can overcome this problem. All efforts to analyze the stability of classically structured molecules in terms of cleverly modified virial schemes are a waste of time. This stipulation embraces the bulk of modern bonding theories. 

Since difference electron densities, deformation densities or valence electron densities are not observable quantities, and since the Hohenberg-Kohn theorem applies only to the total electron density, much work has concentrated on the analysis of p(r). The accepted analysis method today is the virial partitioning method by Bader and coworkers which is based on a quantum mechanically well-founded partitioning of the molecular... 

The inherent valne of the topological method is that these atomic basins are defined by the electron density distribution of the molecule. No arbitrary assumptions are required. The atomic basins are quantum mechanically well-defined spaces, individnally satisfying the virial theorem. Properties of an atom defined by its atomic basin can be obtained by integration of the appropriate operator within the atomic basin. The molecular property is then simply the sum of the atomic properties. 

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. 

A theory is only justified by its ability to account for observed behaviour. It is important, therefore, to note that the theory of atoms in molecules is a result of observations made on the properties of the charge density. These observations give rise to the realization that a quantum mechanical description of the properties of the topological atom is not only possible but is also necessary, for the observations are explicable only if the virial theorem applies to an atom in a molecule. The original observations are among the most important of the properties exhibited by the atoms of theory (Bader and Beddall 1972). For this reason and for the purpose of emphasizing the observational basis of the theory, these original observations are now summarized. They provide an introduction to the consequences of a quantum mechanical description of an atom in a molecule. 

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. 

That this is indeed the differential form of the customary virial theorem is readily seen by multiplying Eq. (26) throughout by x and then integrating over all x from —oo to +00. Some elementary integrations by parts recovers the usual (integral) virial theorem of Clausius, in, of course, now fully quantum-mechanical form [54]. 

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