Virial Theorem condition

If the basic set xpk is chosen complete, the virial theorem will be automatically fulfilled and no scaling is necessary. In such a case, the wave function under consideration may certainly be expressed in the form of Eq. III. 18, but, if the basis is chosen without particular reference to the physical conditions of the problem, the series of determinants may be extremely slowly convergent with a corresponding difficulty in interpreting the results. It therefore seems tempting to ask whether there exists any basic set of spin orbitals. which leads to a most "rapid convergency in the expansion, Eq. III. 18, of the wave function for a specific state (Slater 1951). 

Since the surface is not crossed by any gradient lines, it is referred to as the surface of zero flux. As further discussed below, the virial theorem is satisfied for each of the regions of space satisfying the zero-flux boundary condition. 

We have previously demonstrated, using the virial theorem, that the field cannot be equal to zero over the surface S if this field is a force-free field. As a consequence, the condition 8S = 0 is verified only if the magnetic field is normal to the surface S. Less restrictive and more physical conditions can be chosen if the Eo field exhibits a nonnull irrotational component verifying the condition VAEoii = 0. In this case, the magnetic field is not force-free anymore ... 

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. 

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. 

Equating the results given in eqns (8.190) and (8.191), one obtains an expression for the virial theorem of an open system which satisfies the zero-flux boundary condition (8.109)... 

F.M. Fernandez, E.A. Castro, Virial theorem and boundary conditions for approximate wave functions, Int. J. Quant. Chem. 21 (4) (1982) 741-751. 

Figure 4.8 Calculation of the hydrogen 2,1 energy using the Slater 2s orbital rendered orthogonal to the hydrogenic Is orbital with the best choice for the Slater exponent subject to two conditions. In the first diagram, the SOLVER based solution is determined by the requirement that the Virial theorem coefficient be 1.00. In the second diagram, the best values for the kinetic and potential energy terms have been determined for the choice of the Slater 2s exponent. Note, the substantial cancellation of mismatches of the r grad transformed Slater function compared to the variation of the transformed exact function.

It has been customary to base the dynamical pressure on the virial theorem. However, the usual derivation, as given for example by Hirschfelder et is based on rigid-wall boundary conditions, rather than the periodic boundary conditions usually employed in the numerical calculations. As we shall show below, for periodic boundary conditions the virial theorem actually leads to a somewhat unusual result. Thus we derive a dynamical pressure based on the momentum flux across an element of area within the fluid, as discussed for example by Chapman and Cowling. As we shall see, the resulting expression for the pressure is identical to the one usually attributed to the virial theorem. 

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,... 

Equation (59) is useful in simulations where periodic boundary equations are employed. In the presence of periodic boundary conditions Eq. (57) should not be used [41]. The product of the force acting on a particle times its position vector is called virial, so Eqs. (57)-(59) are forms of the (atomic) virial theorem for the pressure. 

Use an approach similar to that in Problem 11.1 to discuss the effect of scaling in which all the interparticle distances in the wavefunction are multiplied by a parameter p. Hence establish the virial theorem For a system of particles with inverse distance interactions, (V) = —2(7 ) and (E) = V), either for a variationally optimized value of p or for an exact wavefunction. Then extend the theorem to admit external forces applied to the nuclei. [Hint Show, by change of variables in the integrations, how the expectation values depend on p. Use a stationary-value condition and suppose that p = l for the exact wavefunction. Note that the nuclei may be held fixed by applying forces opposite to the (Hellmann-Feynman) forces exerted by the electrons. These forces must be included in forming the expectation value of the classical virial.]... 

We consider in this section the variation principle in molecular electronic-structure theory. Having established the particular relationship between the Schrddinger equation and the variational condition that constitutes the variation principle, we proceed to examine the variation method as a computational tool in quantum chemistry, paying special attention to the application of the variation method to linearly expanded wave functions. Next, we examine two important theorems of quantum chemistry - the Hellmann-Feynman theorem and the molecular virial theorem - both of which are closely associated with the variational condition for exact and approximate wave functions. We conclude this section by presenting a mathematical device for recasting any electronic energy function in a variational form so as to benefit to the greatest extent possible from the simplifications associated with the fulfilment of the variational condition. 

CTombining (4.2.72) with (4.2.66), we may now turn the molecular electronic virial theorem around and regard it as setting up a condition on the Cartesian forces acting on the nuclei ... 

There are other conditions on these forces, related to the translational and rotational invariance of the electronic energy (which are always exactly fulfilled also for approximate wave functions). In particular, in a diatomic molecule, there are three translational conditions - two rotational conditions and one condition provided by the electronic virial theorem. Taken together, these conditions determine the diatomic nuclear force field completely. The only nonvanishing force acts along the molecular axis and may be obtained directly from (he kinetic and potential enei ies if the wave function is fully variational with respect to a scaling of both the electronic and the nuclear coordinates. 

The HF theorem is satisfied for a given diatomic molecule at a given internuclear distance, as proved by Hirschfelder and Coulson (1962), if the set of so-called hypervirial relations corresponding to all the parameters AJ occurring in H, of the type ([H, W,]> = 0 with Hermitian operators WJ, holds. These hypervirial relations are an alternative expression of the floating or stable condition, which will be discussed later. As expected from this viewpoint, the virial and HF theorems have an intimate relation (Frost and Lykos, 1956). 

The last restriction in this equation is needed to satisfy the conditions of the theorem which allows the topological reduction. In this series, there is one term with no yo 8f bonds and only one term with one yo 8f bond. This latter term has only two field points and just one bond. There are an infinite number of graphs with two yo 8f bonds, and even the series for these terms looks like a low-density virial series which might not be expected to be convergent or meaningful at high densities. 

The large maxima of the electron density are expected and are found at the nuclear positions Ra. These points are m-limits for the trajectories of Vp(r), in this sense they are attractors of the gradient field although they are not critical points for the exact density because the nuclear cusp condition makes Vp(Ra) not defined. The stable manifold of the nuclear attractors are the atomic basins. The non-nuclear attractors occur in metal clusters [59-62], bulk metals [63] and between homonu-clear groups at intemuclear distances far away from the equilibrium geometry [64]. In the Quantum Theory of Atoms in Molecules (QTAIM) an atom is defined as the union of a nucleus and of the electron density of its atomic basin. It is an open quantum system for which a Lagrangian formulation of quantum mechanics [65-70] enables the derivation of many theorems such as the virial and hypervirial theorems [71]. As the QTAIM atoms are not overlapping, they cannot share electron pairs and therefore the Lewis s model is not consistent with the description of the matter provided by QTAIM. 

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