List of Hund's rules

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In atomic physics, Hund's rules refer to a simple set of rules used to determine which is the term symbol that corresponds to the ground state of a multi-electron atom. They were proposed by Friedrich Hund. In chemistry, rule one is especially important and is often referred to as simply Hund's Rule.

The three rules are:

  1. For a given electron configuration, the term with maximum multiplicity (maximum S \, ) has the lowest energy.
  2. For a given multiplicity, the term with the largest value of L \, has the lowest energy.
  3. For a given term, in an atom with outermost subshell half-filled or less, the level with the lowest value of J \, lies lowest in energy. If the outermost shell is more than half-filled, the level with highest value of J \, is lowest in energy.

These rules specify in a simple way how the usual energy interactions dictate the ground state term. The rules assume that the repulsion between the outer electrons is very much greater than the spin-orbit interaction which is in turn stronger than any other remaining interactions. This is referred to as the LS coupling regime.

Full shells and subshells do not contribute to the quantum numbers for total S, the total spin angular momentum and for L, the total orbital angular momentum. It can be shown that for full orbitals and suborbitals both the residual electrostatic term (repulsion between electrons) and the spin-orbit interaction can only shift all the energy levels together. Thus when determining the ordering of energy levels in general only the outer valence electrons need to be considered.

Contents

[edit] Rule #1

Main article: Hund's rule of maximum multiplicity

Due to the Pauli exclusion principle, two electrons cannot share the same set of quantum numbers within the same system. Therefore, there is room for only two electrons in each spatial orbital. One of these electrons must have (for some chosen direction z), S_Z = 1/2 \, , and the other must have S_Z = -1/2 \, . Hund's first rule states that the lowest energy atomic state is the one which maximizes the sum of the S \, values for all of the electrons in the open subshell. The orbitals of the subshell are each occupied singly with electrons of parallel spin before double occupation occurs. (This is occasionally called the "bus seat rule" since it is analogous to the behaviour of bus passengers who tend to occupy all double seats singly before double occupation occurs.)

It is often stated that this is the lowest energy atomic state because it forces the unpaired electrons to reside in different spatial orbitals, and this results in a larger average distance between the two electrons, reducing electron-electron repulsion energy. But, in fact, careful calculations have shown that this explanation can be wrong, at least for light systems.

[edit] Example

Hund's rules applied to Si. The up arrows signify electrons with up-spin. The boxes represent different magnetic quantum numbers
Hund's rules applied to Si. The up arrows signify electrons with up-spin. The boxes represent different magnetic quantum numbers

As an example, consider the ground state of silicon. The electronic configuration of Si is 1s^2, 2s^2, 2p^6, 3s^2, 3p^2 \, . We need consider only the outer 3p^2 \, electrons, for which it can be shown (see Term symbols) that the possible terms allowed by the Pauli exclusion principle are {}^1\!D ,{}^3\!P ,{}^1\!S . Hund's first rule now states that the ground state term is {}^3\!P \, which has S = 1 \, . The diagram shows the state of this term with ML = 1 and MS = 1.

[edit] Rule #2

This rule deals again with reducing the repulsion between electrons. It can be understood from the classical picture that if all electrons are orbiting in the same direction (higher orbital angular momentum) they meet less often than if some of them orbit in opposite directions. In that last case the repulsive force increases, which separates electrons. This adds potential energy to them, so their energy level is higher.

[edit] Example

For silicon there is no choice of triplet (S = 1) \, states, so the second rule is not required. The lightest atom which requires the second rule to determine the ground state is titanium (Ti, Z = 22) with electron configuration 1s2 2s2 2p6 3s2 3p6 4s2 3d2. Following the same method as for Si, the allowed terms include three singlets (1S, 1D, and 1G) and two triplets (3P and 3F). We deduce from Hund's first rule that the ground state is one of the two triplets, and from Hund's second rule that the ground state is 3F (with L = 3) rather than 3P (with L = 1).

[edit] Rule #3

This rule considers the energy shifts due to spin-orbit coupling. In the 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:

\begin{matrix} \Delta E & = & \zeta (L,S) \{ \mathbf{L}\cdot\mathbf{S} \} \\ \ & = & \ (1/2) \zeta (L,S) \{ J(J+1)-L(L+1)-S(S+1) \} \end{matrix}

The value of \zeta (L,S)\, changes from plus to minus for shells greater than half full. This term gives the dependence of the ground state energy on the magnitude of J \, .

[edit] Examples

The {}^3\!P \, lowest energy term of Si consists 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 {}^3\!P_0 \, is the ground state.

For sulfur (S) the lowest energy term is again {}^3\!P \, with spin-orbit levels J = 2,1,0 \, , but now there are four of six possible electrons in the shell so the ground state is {}^3\!P_2 \, .

[edit] Notes

Hund's rules are often used to order the excited levels. This is a common misapplication of the rules and is generally incorrect.

[edit] See also

For the notation of orbital angular momentum L see: Spectroscopic notation

[edit] References

  • Elementary Atomic Structure, physics, by G.K. Woodgate (McGraw-Hill, 1970) [ISBN 978-0198511564]
  • G.L. Miessler and D.A. Tarr, Inorganic Chemistry (Prentice-Hall, 2nd edn 1999) [ISBN 0138418918], pp.358-360
  • T. Engel and P. Reid, Physical Chemistry (Pearson Benjamin-Cummings, 2006) [ISBN 080533842X], pp.477-479

[edit] External links