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Hund's Rule


n atomic chemistry, Hund's rules formulated by German physicist Friedrich Hund around 1927, refer to a 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 has the lowest energy. Since multiplicity is equal to  2S + 1 \ , this is also the term with maximum  S \, . S is the spin angular momentum.
  2. For a given multiplicity, the term with the largest value of  L \,  has the lowest energy, where L is the orbital angular momentum.
  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. J is the total angular momentum, J = S + L.
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.

Rule 1


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.)
Two physical explanations have been given for the increased stability of high multiplicity states. In the early days of quantum mechanics, it was proposed that electrons in different orbitals are further apart, so that electron-electron repulsion energy is reduced. However this explanation is now obsolete. Accurate quantum-mechanical calculations (starting in the 1970s) have shown that the true reason is that the electrons in singly occupied orbitals are less effectively screened or shielded from the nucleus, so that such orbitals contract and electron-nuclear attraction energy becomes greater in magnitude (or decreases algebraically).
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 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.

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 the latter case the repulsive force increases, which separates electrons. This adds potential energy to them, so their energy level is higher.
For silicon there is no choice of triplet states  (S = 1) \, , 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).


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 \, .

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 \, .
If the shell is half-filled then  L = 0 \, , and hence there is only one value of  J \,  (equal to  S \, ) which will be the lowest energy state. For example in phosphorus the lowest energy state has  S = 3/2, L = 0 \,  for three unpaired electrons in three 2p orbitals. Therefore  J = S = 3/2 \,  and the ground state is 4S3/2.

Excited states


Hund's rules work best for the determination of the ground state of an atom or molecule.
They are also fairly reliable (with occasional failures) for the determination of the lowest state of a given excited electronic configuration. Thus in the helium atom, the 1s2s triplet state (3S) is correctly predicted by Hund's first rule to be lower than the 1s2s singlet (1S). Similarly for organic molecules, the same rule predicts that the first triplet state (denoted by T1 in photochemistry) is lower than the first excited singlet state (S1), which is generally correct.
However Hund's rules should not be used to order states other than the lowest for a given configuration. For example, the titanium atom ground state configuration is ...3d2 for which a naïve application of Hund's rules would suggest the ordering 3F < 3P < 1G < 1D < 1S. In reality, however, 1D lies below 1G.