Virial theorem for atoms

At zero intemuclear distance we find that a = 2, which we recognize as the virial theorem for atoms, Eq. (18). Empirically, we find that... 

Several forms of wf have already been used within the field of MQS. These methods include the Hirshfeld partitioning [30], Bader s partitioning based on the virial theorem within atomic domains in a molecule [64], and the Mulliken approach [65]. For more information on all the three methods, refer to Chapter 15. 

This is the form of the virial theorem for a force law with / dependence. Note that the energy of a bound atom is negative, since it is lower than the energy of the separated cleetron and proton, which is taken to be zero. 

The postulation of an atomic virial theorem for the topologically defined atoms leads to a numtjer of important conclusions (Bader and Beddall 1972). 

The general time-dependent virial theorem for an atom in a molecule is derived from the atomic variational principle. We shall find a close connection between the expressions so obtained for the virial and those derived in the previous section for the force. In particular, the differential force law leads directly to a corresponding local expression for the virial theorem. 

It was pointed out in the discussion of the atomic virial theorem for a stationary state (eqn (6.23)) that, while the values of the individual contributions to the virial of the subsystem are dependent upon the choice of origin for the vector r, their sum, which determines the total subsystem virial, is independent of the choice of origin. It is clear from eqn (8.193) that the same... 

Integration of eqn (8.201) over an atomic volume for which the integral of V p(r) vanishes yields, term for term, the atomic virial theorem for a time-dep>endent system (eqn (8.193)) or for a stationary state (eqn (6.23)). Thus, eqn (8.201) is, in terms of its derivation and its integrated form, a local expression of the virial theorem. The atomic virial theorem provides the basis for the definition of the average energy of an atom, as discussed in Chapter 6. 

The atomic virial theorem for a system in the presence of an electromagnetic field is obtained by setting the generator F equal to f The commutators required for the evaluation of eqn (8.225), are readily evaluated giving... 

The expectation value of the commutator for the virial operator G(r) = r p yields 2T(Q) + vb( 2), twice the atom s electronic kinetic energy, T(Q), together with the virial of the Ehrenfest force exerted over the basin of the atom, vfc( 2) [4], In a stationary state these contributions are balanced by v,(S2), the virial of the Ehrenfest force acting over the surface of the atom. Expressing by v( 2), the total virial for atom 2, the virial theorem for a stationary state may be stated as [4]... 

Equation (18) is identical in form and content to the atomic statement of the virial theorem Equation (7) - the virial theorem for a proper open system - with the petit virial Zp being the analogue of the basin virial Vb and the surface flux virial Zs/ the analogue of the surface virial vs, the virial of the Ehrenfest forces acting on the surface of the open system. 

Thus the virial theorem for the average electronic kinetic and potential energies of a molecule will not have the simple form (14.17), which holds for atoms. We can, however, view Vg] as a function of both the electronic and the nuclear Cartesian coordinates. From this viewpoint V isa homogeneous function of degree -1, since... 

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) ... 

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. 

Using now the virial theorems for the atom and molecule, namely ... 

The wavefimction for the H atom Is state must obey the virial theorem for the Coulomb central potential, i.e. we must have T) = — U). Show that only = 1 gives a Is function obeying the virial theorem. 

The energy of a Is-electron in a hydrogen-like system (one nucleus and one electron) is —Z /2, and classically this is equal to minus the kinetic energy, 1/2 mv, due to the virial theorem E — —T = 1/2 V). In atomic units the classical velocity of a Is-electron is thus Z m= 1). The speed of light in these units is 137.036, and it is clear that relativistic effects cannot be neglected for the core electrons in heavy nuclei. For nuclei with large Z, the Is-electrons are relativistic and thus heavier, which has the effect that the 1 s-orbital shrinks in size, by the same factor by which the mass increases (eq. (8.2)). 

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. 

Chemists have long been intrigued by the question, Does an atom in a molecule somehow preserve its identity An answer to this question comes from studies on the topological properties of p(r) and grad p(r). It has been shown that the entire space of a molecule can be partitioned into atomic subspaces by following the trajectories of grad p(r) in 3D space. These subspaces themselves extend to infinity and obey a subspace virial theorem (2 (7) + (V) = 0). The subspaces are bounded by surfaces of zero flux in the gradient vectors of p(r), i.e., for all points on such a surface,... 

The conclusion that the local hardness is given entirely by the variable parts of the kinetic energy is very logical. It is the kinetic energy increase which limits the distribution of electron density in all systems with fixed nuclei. Since the equilibrium state of atoms and molecules is characterized by minimum energy, they will also be marked by maximum kinetic energy because of the virial theorem. This will put them in agreement with the principles of maximum hardness, for which much evidence exists. 

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] ... 

By the Hellmann-Feynman theorem, the expectation value < f -dV/dRua I f) is the force on nucleus N in the a direction. The force on each nucleus vanishes for a molecule in its equilibrium nuclear configuration the force also vanishes for an isolated atom. In these cases the virial theorem becomes (T) = -other cases, however, the second term on the right in Eq. (17) is non-vanishing. 

The topological analysis of the total density, developed by Bader and coworkers, leads to a scheme of natural partitioning into atomic basins which each obey the virial theorem. The sum of the energies of the individual atoms defined in this way equals the total energy of the system. While the Bader partitioning was initially developed for the analysis of theoretical densities, it is equally applicable to model densities based on the experimental data. The density obtained from the Fourier transform of the structure factors is generally not suitable for this purpose, because of experimental noise, truncation effects, and thermal smearing. 

The y parameters of the isolated atoms are readily obtained from Eqs. (3.20) or (3.21). Selected SCF results" are indicated in Table 10.2. For hydrogen, of course, y = 2 because of the virial theorem. 

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