Radial expectation values for

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].

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. 

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 ... 

The nuclear quadrupole moment is an expectation value with respect to the nuclear wave function. For the nuclear ground state the nuclear wave function depends upon a radial parameter a, the nuclear spin quantum number / and its projection Mj so that the corresponding ket-vector is denoted as a, I, Mj). The properties of the nuclear spin (in general, an angular momentum) are well known and they can be fully exploited in expressing such an expectation value. For this purpose let us rewrite the electrostatic interaction energy, making use of the expansion in terms of the spherical harmonic functions... 

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. 

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... 

Orbital energies e (a.u.) and radial expectation values (r) (a,u.) for the valence shells of Ce and Lu from multi-conflguration Dirac-Hartree-Fock calculations for the average of the 4f 5d 6s and 4f 5d 6s configurations, respectively. The ratio of relativistic and corresponding nonrelativistic values is given in parentheses, Data taken... 

Fig. 1. Radial expectation values, , of the lanthanide for the wave functions at the Fermi level. (For Yb and Lu, we used the 4f wave functions at the center of gravity of the 4f density of states). The values are normalized by their Wigner— Seitz radii.

So the expectation value for the radial coordinate of the electron in this case is... 

Problem A 10.1 shows that the expectation values for the kinetic and potential energy components for any trial wavefimction are dependent on the exponential decay factor f. The decay factor controls how the electron is distributed along the radial coordinate and so affects the averages taken to produce the expectation values. 

In this calculation, the calculated maxima r are scaled against the radial expectation values of Mann. The multiplet level structure of the HF analysis is reproduced in detail. The fundamental assumption underlying this simple simulation is the indistinguishability of individual electrons in a collective, as emphasized by Schrodinger [18], Madelung [19], and Pauli [20]. This calculation has not been done for other elements, but once outer-level radial expectation values had been obtained, the procedure of Table 2 applies. 

The rationale behind this identification lies therein that the energy simulations assume uniform one-electron density within the characteristic volume, whereas an electron, decoupled from the nucleus by hydrostatic compression, is likewise confined to a sphere of radius tq at constant density. By exploiting this property, ionization radii were also calculated from the maxima of HFS wave functions normalized over spheres of constant density [24]. The same procedure now suggests itself for the calculation of such radii, directly from the calculated charge densities (p) and radial expectation values r, in Fig. 7. 

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