hund's rule

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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 з p, 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 M L=1 and Ms =1.

4. RULE 2 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. RULE 3 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, 2 nd edn 1999) [ISBN 0138418918], pp. 358-360

9. THE END