The effect of screening not only breaks the degeneracy between states with the samenbut different
ℓ, it even moves the 6s state, for example, to have lower energy than the 4f or 5d states. The 4s and
3d states have about the same energy in atoms because of screening.
26.2 The Hartree Equations
The Hartree method allows us to to change the 3Zdimensional Schr ̈odinger equation (Zelectrons
in 3 dimensions) into a 3 dimensionalequation for each electron. This equation depends on the
wavefunctions of the other electrons but can be solved in a self consistent way using the variational
principle and iterating.
ψ=φ 1 (~r 1 )φ 2 (~r 2 )...φZ(~rZ)
− ̄h
2
2 m
∇^2 i−Ze^2
ri
+e^2∑
j 6 =i∫
d^3 rj|φj(~rj)|^2
|~ri−~rj|
φi(~ri) =εiφi(~ri)In the Hartree equation above,εirepresents the energy contribution of electroni. The term
e^2
∑
j 6 =i∫
d^3 rj|φj(~rj)|2
|~ri−~rj| represents the potential due to the other electrons in which electronimoves.In this equation we can formally see the effect of screening by the other electrons. The equation is
derived (see Gasiorowicz pp 309-311) from the Schr ̈odinger equation usingψ=φ 1 φ 2 ...φZ. Since
we will not apply these equations to solve problems, we will not go into the derivation, however, it
is useful to know how one might proceed to solve more difficult problems.
An improved formalism known as the Hartree-Fock equations, accounts for the required antisym-
metry and gives slightly different results.
26.3 Hund’s Rules
A set of guidelines, known as Hund’s rules, help us determine the quantum numbers for the ground
states of atoms. The hydrogenic shells fill up giving well definedj= 0 states for the closed shells.
As we addvalence electronswe follow Hund’s rules to determine the ground state. We get a great
simplification by treating nearly closed shells as a closed shell plus positively charged, spin^12 holes.
For example, if an atom is two electrons short of a closed shell, we treat it as a closed shell plus two
positive holes.)
- Couple the valence electrons (or holes) to givemaximum total spin.
- Now choose the state of maximumℓ(subject to the Pauli principle. The Pauli principle rather
than the rule, often determines everything here.) - If the shell is more than half full, pick the highest total angular momentum statej=ℓ+s
otherwise pick the lowestj=|ℓ−s|.
This method of adding up all the spins and all the Ls, is called LS orRussel-Saunders coupling.
This method and these rule are quite good until the electrons become relativistic in heavy atoms
and spin-orbit effects become comparable to the electron repulsion(arond Z=40). We choose the