Orbital radial expectation values

Figure 3.3 Orbital radial expectation values < r > (in ao) of (left) the 4f elements cerium through lutetium and (right) the 5f elements thorium through lawrencium from 4-component relativistic Elartree-Fock calculations averaging over the n—2)f n—])d ns (/ = 1,14) valence configuration of the neutral atom...

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 ... 

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... 

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... 

Negative valence orbital energies and radial expectation values of the atoms Au, Hg (n=5),, E and njE (n=6) and the corresponding pseudo-atoms without the 4f or 5f shell and with a nuclear charge diminished by 14... 

Fig. 12. The radial expectation values, , of the wave functions of each orbital on the Fermi surface. Each value is normalized to the respective Wigner-Seitz atomic radius.

The steady decrease of 4f radial expectation values along the lanthanide series is often associated with the lanthanide contraction. However, it is a perfectly normal trend that atoms become smaller along a row in the periodic table. As pointed out by Lloyd [1], the relative contraction of ionic radii of+3 cations is larger for the 3d elements (Sc + -Ga +) than the 4f ones (La + - Lu +). Also, what is clearly seen in Figure 3.4 is that the size of the - -3 cations is dictated by the size of 5s and 5p orbitals rather than 4f. When the pioneer geochemist... 

To illustrate the construction of such a basis set, we consider the representation of the occupied carbon Hartree-Fock orbitals by an STO expansion that is minimal in the sense that it contains only one STO for each occupied Hartree-Fock orbital. The expectation values of r for the Is, 2s and 2p numerical Hartree-Fock orbitals are 0.2690ao l-589ao and 1.715ao, respectively. As basis functions, we choose STOs with expectation values (6.5.30) that match those of the Hartree-Fock orbitals. This prescription leads to an STO basis with the following radial fomos ... 

The expectation value of r is given by the radial part R(r) of the orbitals,... 

In Fig. 4.3 we plot the density dependence of the resulting exchange potentials. The relevant range of density values for electronic structure calculations is indicated by the j8-values at the origin and the expectation value of the radial coordinate of the lSl/2-orbital for the Kr and Hg atoms. One finds that relativistic effects are somewhat more pronounced for than for and are definitely relevant for inner shell features of high Z-atoms. 

As described by eq.(3), APb is connected with relativistic change in atomic radial wavefunctions. It is, however, difficult to examine the atomic-number dependence of the magnitude of the ratio l(( ) -(t)"0/

radial wavefunctions, respectively, for each AO. Then we investigated the atomic-number dependence of A/ (A= - "0 for each atom in the priodic table (H to Pu), where " and denote the expectation values of each atomic orbital for Hartree-Fock and Dirac-Fock... 

J1 t/w II / ) is the matrix element of the transition between the ground and excited state of the sensitizer and activator respectively (the calculation of these matrix elements in the intermediate-coupling scheme is now a well known procedure and may be found in references (9) and (79)) S is the overlap integral / is the interionic distance is a numerical factor that depends on the orientation of the coordinate axis and (Af rA 4/>2 is the expectation value of the radial integral of the 4/orbital. 

Ionic radii can be defined either by the maximum of the radial charge density, r, or the expectation value, (r , of an outer valence orbital. The DF... 

Figure 18 Qualitative picture of change in radial extent of metal d-orbitals in cfs° (right), cT s (middle) and cT (left) state. Abscissa is in arbitrary units of length the radical expectation value is given on the ordinate.

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