Russel-Saunders - vs. jj-Coupling

Now the knowledge of electron characteristics in hydrogen atom let us to begin study of many-electron atoms.

The whole system function since internal electrons making up the closed shell give zero total kinetic moment and zero total spin. The action of these closed shell together with the nuclear potential on the external valency electrons can be expressed by the central potential V(r). The Hamiltonian operator for the valency electrons is then as follows (we're neglecting the nuclear spin here):

H = T + V(r) + | Hee + | Hss + | Hsl
kinetic + potential energy | electrostatic interaction of the external electrons | spin - spin interaction | spin - orbital interaction

Depending on strength of the last three terms there are two boundary cases:

1.)

Hsl << Hee , Hsl << Hss

The orbital kinetic moments and spins are considered independently. The orbital moments i of single electrons are then summed up into the total kinetic moment:

= Σi i

and also the individual spins are summed up giving the total spin:

= Σi i

After that we have the weak spin-orbital interaction (Hsl) and the moments and are summed up giving the total kinetic moment:

= +

This type of coupling is also referred to as LS-Coupling or Russel-Saunders-Coupling (1925) and it is true almost for all atoms (all light atoms).

The quantum numbers of Russel-Saunders-Coupling are orbital moment L, apin S, kinetic moment J (nomenclature: 2S+1LJ) and the magnetic quantum number M isn't clearly seen in this nomenclature.

2.)

Hsl >> Hee , Hsl >> Hss

In this type of spin-orbital coupling the interaction between spin of a single electron and spin i + orbital moment i of each other electron dominates giving the following total kinetic moment i of i-electron:

i = i + i

The total kinetic moment of the atomic state is then as follows:

= Σi i

This type of coupling is called the jj-Coupling and it can be met only in heavy atoms since because of the great vast of electronic shell the valency electrons are situated on great distances from each other and therefore the interaction between valency electrons is weak comparing with the spin-orbital interaction of a single electron.

The quantum numbers are ji (nomenclature: (j1, j2, j3, ...)J)

To prevent from being misunderstood: the pure mathematical addition of vectors i and i gives the same result ; but only addition of Si and Si and then + ; and also i + i and then Si, correspondingly should show us the dominating interaction. The total kinetic moment is the conservative quantity for both LS- and jj-Coupling.

jj - Coupling | Russel-Saunders- (LS-) Coupling | LS - jj Correlation for elements of the 4th main group