Hund's rules describe the ground state of multielectron atoms and ions. Rule 1 states that the atomic state with the highest total spin S will have the lowest energy. Rule 2 states that for a given spin multiplicity, the term with the largest orbital angular momentum L has the lowest energy. Rule 3 considers energy shifts due to spin-orbit coupling, where the splitting is determined by the spin-orbit coupling constant λ and the quantum numbers J, L, and S. Examples are provided to illustrate the application of Hund's rules to determine the ground states of silicon and titanium.

1. HUND’ S RULE

2. RULE 1
The lowest energy atomic state is the
one that maximizes the sum of the S
for all the electrons in the open
subshelll.

3. EXAMPLE
Consider the ground state of silicon.The
electronic configuration of Si is 1s²2s²2p⁶3s²3p²
(see spectroscopic notation). We need to
consider only the outer 3p² electrons, for which it
can be shown (see term symbols) that the
possible terms allowed by the Pauli exclusion
principle are 1D, 3P, and 1S. Hund’s first rule now
states that the ground state term is зp, which
has S=1.The superscript 3 is the value of the
multiplicity = 2S+1=3.The diagram shows the
state of this term with ML=1 and Ms=1.

4. RULE2
For a given multiplicity, the term with
the largest value of the total orbital
angular momentum quantum
number L has the lowest energy.

5. EXAMPLE
(Ti, Z=22)
Electron configuration 1s²2s²2p⁶3s²3p⁶3d²4s²
Open shell 3d² and the allowed terms include
three singlets (1S, 1D, and 1G) and two triplets
(3P and 3F). We deduce from Hund’s rule that the
ground state is one of the two triplets, and from
the Hund’s second rule that the ground state is 3F
(with L=3) rather than 3P 9with L=1). There is no
3G term since its (ML=4, Ms=1) state would
require two electrons each with (ML=2, Ms= +½),
in violation of the Pauli principle.

6. RULE3
This rule consider the energy shifts due to spin-
orbit coupling. In this case where the spin-
orbit coupling is weak compared to the
residual electrostatic interaction, L and S are
still good quantum numbers and the
splitting is given by:
∆E= ∫(L,S){L.S}
= (½)∫(L,S){J(J+1)-L(L+1-S(S+1)}

7. EXAMPLE
The 3P lowest energy term is Si consist of three
levels, J=2,1,0.With only two of six possible
electrons in the shell, it is less than half-full
and thus 3P₀ is the ground state.
For sulfur (S) the lowest energy term is again 3P
with spin-orbit levels J=2,1,0, but now there
are four of six possible electrons in the shell
so the ground states is 3P₂.

8. REFERENCES
 ^ G.L. Miessler and D.A.Tarr, Inorganic Chemistry
(Prentice-Hall, 2nd edn 1999)
[ISBN 0138418918], pp. 358-360

9. THE END