Iso-electronic sequences and orbital collapse

In an iso-electronic sequence, the only parameter that changes is the nuclear charge . Such calculations provide useful trends, particularly when plotted as a function of 1/Z, the perturbation parameter from Z-dependent perturbation theory, to be discussed in the next chapter.

An interesting phenomenon that can occur is a very rapid contraction of an orbital which is called orbital collapse. This could be an LS-dependent effect, but it can also occur along on isoelectronic sequence of values. This effect is most noticeable in the high-l orbitals. In hydrogen, it can be shown that the mean radius of an orbital is $\langle r\rangle = (1/2)[3n^2 -l(l+1)]$. Thus the the higher-l orbitals are more contracted. But in neutral systems, the high-l orbitals have a higher energy and are more diffuse. This is due, in part, to the ``angular momentum barrier'', the l(l+1)/r² term that appears in the definition of the ${\cal L}$ operator. In the Hartree model, where the radial equation has the form

$$\left(\frac{d^2}{dr^2} +\frac{2}{r}[Z-Y(a;r)] -\frac{l_a(l_a+1)}{r^2} -\varepsilon_{aa}\right)\P{a}{} = 0$$ (43)

it is possible for $- \frac{2}{r}[Z-Y(a;r)] +\frac{l_a(l_a+1)}{r^2}$ to have two wells, an inner well and an outer shallow well. When the lowest eigenfunction changes rapidly as a function of the nuclear charge from the outer well to the inner well, orbital collapse is said to occur.

2001-01-09