Radial expectation value

Nevertheless, Mossbauer spectroscopy is sensitive to the core region of the absorber and therefore, it is of interest to investigate how the radial expectation values are influenced by relativity. This will be further elaborated here. 

By means of a modification of the TFD method in the near nnclear region for the electron and energy densities, which introduces exact asymptotic properties, radial expectation values and the atomic density at the nucleus are evaluated, comparing fairly closely to the HF results, with a large improvement of the TF estimates. In addition to this, momentum expectation values can be estimated from semiclassical relations. 

Among the average properties which play a special role in the study of quantum fermionic systems are the radial expectation value (r ), the momentum expectation value (j9 ) and the atomic density at the nucleus p(0) = <5(r)>. These density-dependent quantities are defined by... 

As we said previously, the radial expectation values constitute a test of the description of the density in different regions depending on a. For a smaller than -1 the most important region is the near nuclear one. Now we show in the Table 3 the values from a = -2 to a = 2 obtained with the present method and with HF. In Figures 2 and 3 are illustrated a = -2 and a = - 1 compared with the //F values. 

Table3 Some radial expectation values (r ) evaluated in the present work (PW) compared to the HFones for some neutral atoms.

J. B. Mann, Atomic Structure Calculations, Los Alamos Scientific Laboratory, Univ. California, Los Alamos, NM, Part I Hartree—Fock Energy Results for the Elements Hydrogen to Lawrencium, 1967 Part II Hartree-Fock Wave Functions and Radial Expectation Values, 1968. 

A radial expectation value of the normalized charge density distribution function for an arbitrary function of the radius, f r), is obtainable from the general formula... 

Frequently occurring radial expectation values are those for integral powers of r,... 

It may be asked how accurate energy-consistent pseudopotentials will reproduce the shape of the valence orbitals/spinors and their energies. Often radial expectation values < r > are used as a convenient measure for the radial shape of orbitals/spinors. Due to the pseudo-valence orbital transformation and the simplified nodal structure it is clear that values n < 0 are not suitable, since the resulting operator samples the orbitals mainly in the core region. Table 2 lists orbital energies, < r > and < > expectation values for the Db [Rn] 5f 6d ... 

Fig. 2. Time dependence of a hydrogen wave-packet with averaged principal quantum number h = 80 and FWHM An = 4. On top we show the autocorrelation junction l(tp(t)l ip(0)[) and on bottom the radial expectation value. Both values document the occurrence of periodic structures as well as revival and partial revival structures as indicated by vertical dotted lines.

However, the application of semiclassical models derived from the Thomas-Fermi approach has some drawbacks, including an overestimation of the electron density near the nucleus, which affects the dependence of low-order radial expectation values with Z, and the impossibility of describing atomic anions [1]. 

In the previous expression the radial expectation value (r appears. The contribution to this quantity from this region TZ is given by... 

Eqs. (5) and (7) can be thought of as changes in the distance and energy units, respectively. In these new, dimension-scaled, units the radial expectation value, (r) = has a much tamer dimension dependence, and the new groimd-state energy, E = — is completely independent of D. Thus, the one-dimension limit of the scaled Schrodinger equation, Eq. (6), is not such a bad model for the physical, three-dimensional, problem. 

The radial expectation value appearing in (78) can be estimated in two ways. The first way is by calculating it exactly using the D-dimensional ground state wave function (30) thereby obtaining... 

Figure 1.1 Radial expectation values for the valence s- and p-orbitals in periods 2 and 3 of the periodic table (approximate numerical Dirac-Hartree-Fock values from Ref. [14]). Figure adapted from Ref. [13].

Unlike the ligand dependence, the Ln dependences of X2(dc) and X2(ab) are different especially between Eu and Tb for all LnXa series. The value of X2(ab) decreases within the early Ln series and also within the late Ln series, but increases between Eu and Tb, while X2(dc) decreases monotonically. The origin of Ln dependence of X2(dc) is explained only from the squared radial expectation values of 4f orbital of Ln, whose value decreases monotonically... 

---