The molecular electronic virial theorem

The exact molecular electronic energy is variational with respect to an arbitrary change in the wave function. In particular, therefore, it is variational with respect to the following uniform scaling of the electron coordinates  

Here a is the scaling factor and N the number of electrons in the system. The overall scaling factor has been incorporated so as to preserve normalization of the scaled wave function (see Exercise 4.3)  

Note that the undisturbed system is represented here by a = 1 rather than a = 0 as in the discussion of the Hellmann-Feynman theorem. The Hamiltonian (4.2.58) depends parametrically on the nuclear positions Rj. collectively denoted by R. Before evaluating the variational condition (4.2.57). we note the following identities (see Exercise 4.3) 

The molecular electronic virial theorem is usually written in a slightly different fashion. According to the Hellmann-Feynman theorem, we have the following relationship between the derivative of the total electronic energy and the derivative of the potential-energy operator  

Combining the virial theorem at equilibrium with the expression for the total electronic energy 

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Equation (14.24) is an example of the Hellmann-Feynman theorem.] Using these last two equations in the molecular electronic virial theorem (14.23), we get... 

If we had omitted from V, we would have obtained Eq. (14.24), which was used in deriving the molecular electronic virial theorem.] We have... 

The H-F theorem (2.1) is also involved in formulation of the molecular electronic virial theorem for the PCM model, as shown in Appendix A.2. 

A theorem that relates the kinetic energy T and potential energy F of a system in its stationary states. The molecular electronic virial theorem is formulated as follows ... 

The above expressions for the molecular electronic virial theorem have been given in terms of the force with respect to a uniform scaling of the nuclear iiamework. This scaling force may easily be related to the classical Cartesian forces on the nuclei... 

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 ... 

This simplification occurs since, in this case, the potential-energy operator scales in a simple fashion characteristic of the Coulomb potential when the coordinates of all particles (electrons and nuclei) are uniformly scaled. It is the failure of the electron-nuclear potential-energy operator (4.2.62) to scale in a simple fashion in response to the electron coordinate scaling that introduces the scaling-force term in the molecular electronic virial theorem. 

A.1 Molecular Electronic Virial Theorem for the Polarizable Continuum Model... 

In contrast to the symmetry requirements, the virial theorem is a dynamical requirement and, with the exception of atoms, can only be tested once the solution of the variational problem has been carried through. Or, to be a little more cautious, the imposition of the virial theorem on the form of a model of the molecular electronic structure is not easy. (It should be said at this point that the simple form,... 

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 energy obtained from a calculation using ECP basis sets is termed valence energy. Also, the virial theorem no longer applies to the calculation. Some molecular properties may no longer be computed accurately if they are dependent on the electron density near the nucleus. 

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... 

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 electron-nuclear interaction energy is the only attractive interaction in a molecular system and it is the decrease in the potential energy resulting from this interaction that is responsible for the formation of a bound molecular state from the separated atoms. Because of the virial theorem and in the absence of external forces acting on the nuclei, the total energy E equals i V (eqn (6.64)) where V = is the total potential energy. 

We are using here the fully relaxed molecular density function and the corresponding solvent effects on the average kinetic energy of the electrons, may be justified by invoking the virial theorem. 

In benchmark calculations [48, 120], however, one needs to perform atomic integrations of energy densities obtained from systems which satisfy the molecular virial theorem exactly (see discussion in Ref. [120]). Dr. Keith has written a link [121] for the Gaussian program [122] that implements Lowdin s scaling of the electronic coordinates [123, 124], the so-called self-consistent... 

This chapter discusses theorems that are used in molecular quantum mechanics. Section 14.1 expresses the electron probability density in terms of the wave function. Section 14.2 shows how the dipole moment of a molecule is calculated from the wave function. Section 14.3 gives the procedure for calculating the Hartree-Fock wave function of a molecule. Sections 14.4 to 14.7 discuss the virial theorem and the Hellmann-Feynman theorem, which are helpful in understanding chemical bonding. 

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