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| In [[atomic physics]], '''Hund's rules''' refers to a set of rules that German physicist [[Friedrich Hund]] formulated around 1927, which are used to determine the [[term symbol]] that corresponds to the ground state of a multi-[[electron]] [[atom]]. The first rule is especially important in chemistry, where it is often referred to as, simply, [[Hund's rule of maximum multiplicity|Hund's Rule]].
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| The three rules are:<ref>G.L. Miessler and D.A. Tarr, Inorganic Chemistry (Prentice-Hall, 2nd edn 1999) [ISBN 0138418918], pp. 358–360
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| </ref><ref>T. Engel and P. Reid, Physical Chemistry (Pearson Benjamin-Cummings, 2006) [ISBN 080533842X], pp. 477–479</ref><ref>G. Herzberg, Atomic Spectra and Atomic Structure (Dover Publications, 1944) [ISBN 0486601153], p. 135 (Although, Herzberg states these as being two rules rather than three.)</ref>
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| # For a given [[electron configuration]], the term with maximum [[multiplicity (chemistry)|multiplicity]] has the lowest energy. The multiplicity is equal to <math> 2S + 1 \ </math>, where <math>S</math> is the total [[spin angular momentum]] for all electrons. The term with lowest energy is also the term with maximum <math> S \, </math>.
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| # For a given multiplicity, the term with the largest value of the total [[Azimuthal quantum number|orbital angular momentum quantum number]] <math> L \, </math> has the lowest energy.
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| # For a given term, in an atom with outermost subshell half-filled or less, the level with the lowest value of the [[total angular momentum quantum number]] <math> J \, </math> (for the operator <math>\boldsymbol{J}=\boldsymbol{L}+\boldsymbol{S}</math>) lies lowest in energy. If the outermost shell is more than half-filled, the level with the highest value of <math> J \, </math> is lowest in energy.
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| These rules specify in a simple way how usual energy interactions dictate the ground state term. The rules assume that the repulsion between the outer electrons is much greater than the spin–orbit interaction, which is in turn stronger than any other remaining interactions. This is referred to as the [[angular momentum coupling#LS coupling|LS coupling]] regime.
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| Full shells and subshells do not contribute to the quantum numbers for total {{mvar|S}}, the total spin angular momentum and for {{mvar|L}}, the total orbital angular momentum. It can be shown that for full orbitals and suborbitals both the residual [[electrostatic force|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 must be considered.
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| ==Rule 1==
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| {{Main|Hund's rule of maximum multiplicity}}
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| 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''<sub>''z''</sub> = ½, and the other must have ''S''<sub>''z''</sub> = −½. Hund's first rule states that the lowest energy atomic state is '''the one that 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.)
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| Two different physical explanations have been given<ref name=Levine>I.N. Levine, Quantum Chemistry (Prentice-Hall, 4th edn 1991) [ISBN 0205127703], pp. 303–304</ref> 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, accurate quantum-mechanical calculations (starting in the 1970s) have shown that the reason is that the electrons in singly occupied orbitals are less effectively screened or [[effective nuclear charge|shielded]] from the nucleus, so that such orbitals contract and electron–nucleus attraction energy becomes greater in magnitude (or decreases algebraically).
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| ===Example===
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| [[Image:Si Hund.png|thumb|250px|Hund's rules applied to Si. The up arrows signify [[electron]]s with up-[[Spin-½|spin]]. The boxes represent different [[magnetic quantum number]]s ]]
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| As an example, consider the ground state of [[silicon]]. The electronic configuration of Si is {{math|1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>6</sup> 3s<sup>2</sup> 3p<sup>2</sup>}} (see [[spectroscopic notation]]). We need to consider only the outer 3p<sup>2</sup> electrons, for which it can be shown (see [[Term symbol#Term symbols for an electron configuration|term symbols]]) that the possible terms allowed by the [[Pauli exclusion principle]] are <sup>1</sup>''D'' , <sup>3</sup>''P'' , and <sup>1</sup>''S''. Hund's first rule now states that the ground state term is <sup>3</sup>''P'', which has ''S'' = 1. The superscript 3 is the value of the multiplicity = 2''S'' + 1 = 3. The diagram shows the state of this term with ''M''<sub>''L''</sub> = 1 and ''M''<sub>''S''</sub> = 1.
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| ==Rule 2==
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| This rule deals 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.
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| ===Example===
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| For silicon there is no choice of [[triplet state]]s <math> (S = 1)</math>, so the second rule is not required. The lightest atom that requires the second rule to determine the ground state is [[titanium]] (Ti, [[atomic number|{{mvar|Z}}]] = 22) with electron configuration {{math|1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>6</sup> 3s<sup>2</sup> 3p<sup>6</sup> 3d<sup>2</sup> 4s<sup>2</sup>}}. In this case the open shell is {{math|3d<sup>2</sup>}} and the allowed terms include three singlets (<sup>1</sup>S, <sup>1</sup>D, and <sup>1</sup>G)
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| and two triplets (<sup>3</sup>P and <sup>3</sup>F). 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 <sup>3</sup>F (with <math>L = 3</math>) rather than <sup>3</sup>P (with <math>L = 1</math>). There is no
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| <sup>3</sup>G term since its <math>(M_L = 4, M_S = 1)</math> state would require two electrons each with <math>(M_L = 2, M_S = +1/2)</math>, in violation of the Pauli principle.
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| ==Rule 3==
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| 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, <math> L \, </math> and <math> S \, </math> are still good quantum numbers and the splitting is given by:
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| :<math> \begin{align} \Delta E & = \zeta (L,S) \{ \mathbf{L}\cdot\mathbf{S} \} \\
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| \ & = \ (1/2) \zeta (L,S) \{ J(J+1)-L(L+1)-S(S+1) \} \end{align} </math>
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| The value of <math> \zeta (L,S)\, </math> 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 <math> J \, </math>.
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| ===Examples===
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| The <math> {}^3\!P \, </math> lowest energy term of Si consists of three levels, <math> J = 2,1,0 \, </math>. With only two of six possible electrons in the shell, it is less than half-full and thus <math> {}^3\!P_0 \, </math> is the ground state.
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| For [[sulfur]] (S) the lowest energy term is again <math> {}^3\!P \, </math> with spin–orbit levels <math> J = 2,1,0 \, </math>, but now there are four of six possible electrons in the shell so the ground state is <math> {}^3\!P_2 \, </math>.
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| If the shell is half-filled then <math> L = 0 \, </math>, and hence there is only one value of <math> J \, </math> (equal to <math> S \, </math>), which is the lowest energy state. For example in [[phosphorus]] the lowest energy state has <math> S = 3/2,\ L = 0 </math> for three unpaired electrons in three 2p orbitals. Therefore <math> J = S = 3/2 </math> and the ground state is <sup>4</sup>S<sub>3/2</sub>.
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| ==Excited states==
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| Hund's rules work best for the determination of the [[ground state]] of an atom or molecule.
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| They are also fairly reliable (with occasional failures) for the determination of the lowest state of a given [[electron configuration#Energy — ground state and excited states|excited electronic configuration]]. Thus, in the helium atom, Hund's first rule correctly predicts that the 1s2s [[triplet state]] (<sup>3</sup>S) is lower than the 1s2s singlet (<sup>1</sup>S). Similarly for organic molecules, the same rule predicts that the first triplet state (denoted by T<sub>1</sub> in [[photochemistry]]) is lower than the first excited singlet state (S<sub>1</sub>), which is generally correct.
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| However Hund's rules should not be used to order states other than the lowest for a given configuration.<ref name=Levine/> For example, the [[titanium]] atom ground state configuration is ...3d<sup>2</sup> for which a naïve application of Hund's rules would suggest the ordering <sup>3</sup>F < <sup>3</sup>P < <sup>1</sup>G < <sup>1</sup>D < <sup>1</sup>S. In reality, however, <sup>1</sup>D lies below <sup>1</sup>G.
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| ==References==
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| <references />
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| ==External links==
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| *[http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/hund.html#c1 Hund's rules on HyperPhysics]
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| * [http://www.chem.purdue.edu/gchelp/gloss/hundsrule.html A glossary entry hosted on the web site of the Chemistry Department of Purdue University]
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| * [http://physicsweb.org/article/news/8/2/5 A PhysicsWeb article]
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| * [http://www.cond-mat.de/events/correl12/manuscripts/eder.pdf Multiplets in Transition Metal ions] in E. Pavarini, E. Koch, F. Anders, and M. Jarrell: Correlated Electrons: From Models to Materials, Jülich 2012, ISBN 978-3-89336-796-2
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| [[Category:Atomic physics]]
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| [[Category:Quantum chemistry]]
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| [[Category:Spectroscopy]]
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| [[Category:Rules]]
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