4.3 The central-field approximation 67Fig. 4.5The total potential in the
central-field approximation including
the term that is proportional tol(l+
1)/r^2 drawn here forl=2andthe
same approximate electrostaticVCF(r)
as shown in Fig. 4.4. The angular
momentum leads to a ‘centrifugal bar-
rier’ that tends to keep the wavefunc-
tions of electrons withl>0awayfrom
r= 0 where the central-field potential
is deepest.the fact that the central field itself depends on the configuration of the
electrons in the atom. For a more accurate description we must take
into account the effect of the outer electron on the other electrons, and
hence on the central field. The energy of the whole atom is the sum of
the energies of the individual electrons (in eqn 4.6), e.g. a sodium atom in
the 3s configuration has energyE
(
1s^2 2s^2 2p^6 3s)
=2E1s+2E2s+6E2p+
E3s=Ecore+E3s. This is the energy of the neutral atom relative to the
bare nucleus (Na11+).^10 It is more useful to measure the binding energy^10 This is a crude approximation, espe-
relative to the singly-charged ion (Na+)withenergyE(1s^2 2s^2 2p^6 )= cially for inner electrons.
2 E1s′+2E′2s+6E2p′ =Ecore′. The dashes are significant—the ten electrons
in the ion and the ten electrons in the core of the atom have slightly
different binding energies because the central field is not the same in
the two cases. The ionization energy is IE =Eatom−Eion=(Ecore−
Ecore′ )+E3s. From the viewpoint of valence electrons, the difference
inEcore between the neutral atom and the ion can be attributed to
core polarization, i.e. a change in the distribution of charge in the core
produced by the valence electron.^11 To calculate the energy of multi-
(^11) This effect is small in the alkalis and
it is reasonable to use thefrozen core
approximation that assumesEcore
E′core. This approximation becomes
more accurate for a valence electron in
higher levels where the influence on the
core becomes smaller.