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In [[atomic spectroscopy]] '''Hund's rules''' predict which atomic energy level with quantum numbers ''L'', ''S'' and ''J'' is lowest. The rules are called after [[Friedrich Hund]] who formulated them in  1925.<ref>F. Hund, ''Zur Deutung verwickelter Spektren, insbesondere der Elemente Scandium bis Nickel.'' [On the interpretation of complicated spectra, in particular scandium through nickel]. Zeitschrift f. Physik, vol. '''33''', pp. 345-371 (1925).</ref> A group of atomic energy levels, obtained in the [[Russell-Saunders coupling]],  is concisely indicated by a [[term symbol]]. A term (also known as multiplet) is a set of simultaneous eigenfunctions of '''L'''<sup>2</sup> (total orbital angular momentum squared) and '''S'''<sup>2</sup> (total spin angular momentum squared) with given quantum numbers ''L'' and ''S'', respectively.
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If there is no spin-orbit coupling, the functions of one term are degenerate (have the same energy).


Hund's rules are:
In [[atomic spectroscopy]], '''Hund's rules''' predict the order of atomic energy levels with quantum numbers ''L'' (orbital), ''S'' (spin) and ''J'' (orbital plus spin). The rules are called after [[Friedrich Hund]] who formulated them in  1925.<ref>F. Hund, ''Zur Deutung verwickelter Spektren, insbesondere der Elemente Scandium bis Nickel.'' [On the interpretation of complicated spectra, in particular the elements scandium through nickel]. Zeitschrift für Physik, vol. '''33''', pp. 345-371 (1925).</ref>


# Of the Russell-Saunders states arising from a given [[electronic configuration]] those with the largest spin quantum number ''S'' lie lowest, those with the next largest next, and so on; in other words, the states with largest spin multiplicity are the most stable.
==LS coupling==
A group of atomic energy levels, obtained by [[Russell-Saunders coupling]],  is concisely indicated by a [[term symbol]]. As discussed in the article [[Russell-Saunders coupling]], closed shells and closed subshells have ''L'' = ''S'' = 0 and hence can be ignored in the coupling. A ''term'' (also known as ''multiplet'') is a set of simultaneous eigenfunctions of '''L'''<sup>2</sup> (total orbital angular momentum squared) and '''S'''<sup>2</sup> (total spin angular momentum squared) with given quantum numbers ''L'' and ''S'', respectively. That is, the respective eigenvalues are ''L(L+1)''ℏ<sup>2</sup>
and ''S(S+1)''ℏ<sup>2</sup>.
 
If there is no spin-orbit coupling, the functions of one term (fixed ''L'' and ''S'')  are (2''L''+1)×(2''S''+1)-fold degenerate (have the same energy). If there is weak spin-orbit coupling it is useful to diagonalize the matrix of the corresponding spin-orbit operator within the ''L-S'' basis [consisting of the (2''L''+1)×(2''S''+1) functions of the term] in the spirit of first-order [[perturbation theory]]. This lifts partially the degeneracy and introduces the new conserved quantum number ''J'', with |''L''-''S''| &le; ''J'' &le; ''L''+''S'', that labels a (2''J''+1)-fold degenerate energy level.
[[Image:Carbon levels.png|right|thumb|350px|Level scheme of the [[carbon]] atom <math>\scriptstyle (1s)^2(2s)^2(2p)^2</math>. Drawing is not on scale. On the left the energy
without any two-particle interaction. Then three-fold energy splitting after switching on electrostatic electron-electron interaction (''L'' and ''S'' good quantum numbers). Then splittings after switching on first-order spin-orbit coupling (''J'' good quantum number). Finally on the right [[Zeeman]] splittings in an external magnetic field.]]
 
==Formulation of the rules==
Hund's rules are:<ref>L. Pauling, ''The Nature of the Chemical Bond'', Cornell University Press, Ithaca, 3rd edition (1960)</ref>
 
# Of the Russell-Saunders states arising from a given [[electron configuration]] those with the largest spin quantum number ''S'' lie lowest, those with the next largest next, and so on; in other words, the states with largest spin multiplicity are the most stable.
# Of the group of terms  with a given value of ''S'', that with the largest value of ''L'' lies lowest.
# Of the group of terms  with a given value of ''S'', that with the largest value of ''L'' lies lowest.
# Of the states with given values of ''S'' and ''L'' in an electronic configuration consisting of less than half the electrons in a closed subshell, the state with the smallest value of ''J'' is usually the most stable, and for a configuration consisting of more than half the electrons in a closed subshell the state with largest ''J'' is the most stable.
# Of the states with given values of ''S'' and ''L'' in an electronic configuration consisting of less than half the electrons in a closed subshell, the state with the smallest value of ''J'' is usually the most stable, and for a configuration consisting of more than half the electrons in a closed subshell the state with largest ''J'' is the most stable.


The levels of the second sort, largest ''J'' most stable, can be seen as arising from holes in the closed subshell.
The levels of the second sort, largest ''J'' most stable, can be seen as arising from holes in a closed subshell.
 
==Examples==
 
* The ground state carbon atom, (1''s'')<sup>2</sup>(2''s'')<sup>2</sup>(2''p'')<sup>2</sup>, gives by [[Russell-Saunders coupling]] a set of energy levels labeled by [[term symbol]]s. Hund's rules predict the following order of the energies:
::<math>
^3P_{0} < ^3P_{1} < ^3P_{2} < ^1D_{2} < ^1S_{0}.
</math>
* The ground state oxygen atom, (1''s'')<sup>2</sup>(2''s'')<sup>2</sup>(2''p'')<sup>4</sup>, (a two-hole state) gives by Russell-Saunders coupling a set of energy levels labeled by term symbols. Hund's rules predict the following order of the energies:
::<math>
^3P_{2} < ^3P_{1} < ^3P_{0} < ^1D_{2} < ^1S_{0}.
</math>
 


==References==
==References==
<references />
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In atomic spectroscopy, Hund's rules predict the order of atomic energy levels with quantum numbers L (orbital), S (spin) and J (orbital plus spin). The rules are called after Friedrich Hund who formulated them in 1925.[1]

LS coupling

A group of atomic energy levels, obtained by Russell-Saunders coupling, is concisely indicated by a term symbol. As discussed in the article Russell-Saunders coupling, closed shells and closed subshells have L = S = 0 and hence can be ignored in the coupling. A term (also known as multiplet) is a set of simultaneous eigenfunctions of L2 (total orbital angular momentum squared) and S2 (total spin angular momentum squared) with given quantum numbers L and S, respectively. That is, the respective eigenvalues are L(L+1)2 and S(S+1)2.

If there is no spin-orbit coupling, the functions of one term (fixed L and S) are (2L+1)×(2S+1)-fold degenerate (have the same energy). If there is weak spin-orbit coupling it is useful to diagonalize the matrix of the corresponding spin-orbit operator within the L-S basis [consisting of the (2L+1)×(2S+1) functions of the term] in the spirit of first-order perturbation theory. This lifts partially the degeneracy and introduces the new conserved quantum number J, with |L-S| ≤ JL+S, that labels a (2J+1)-fold degenerate energy level.

Level scheme of the carbon atom . Drawing is not on scale. On the left the energy without any two-particle interaction. Then three-fold energy splitting after switching on electrostatic electron-electron interaction (L and S good quantum numbers). Then splittings after switching on first-order spin-orbit coupling (J good quantum number). Finally on the right Zeeman splittings in an external magnetic field.

Formulation of the rules

Hund's rules are:[2]

  1. Of the Russell-Saunders states arising from a given electron configuration those with the largest spin quantum number S lie lowest, those with the next largest next, and so on; in other words, the states with largest spin multiplicity are the most stable.
  2. Of the group of terms with a given value of S, that with the largest value of L lies lowest.
  3. Of the states with given values of S and L in an electronic configuration consisting of less than half the electrons in a closed subshell, the state with the smallest value of J is usually the most stable, and for a configuration consisting of more than half the electrons in a closed subshell the state with largest J is the most stable.

The levels of the second sort, largest J most stable, can be seen as arising from holes in a closed subshell.

Examples

  • The ground state carbon atom, (1s)2(2s)2(2p)2, gives by Russell-Saunders coupling a set of energy levels labeled by term symbols. Hund's rules predict the following order of the energies:
  • The ground state oxygen atom, (1s)2(2s)2(2p)4, (a two-hole state) gives by Russell-Saunders coupling a set of energy levels labeled by term symbols. Hund's rules predict the following order of the energies:


References

  1. F. Hund, Zur Deutung verwickelter Spektren, insbesondere der Elemente Scandium bis Nickel. [On the interpretation of complicated spectra, in particular the elements scandium through nickel]. Zeitschrift für Physik, vol. 33, pp. 345-371 (1925).
  2. L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaca, 3rd edition (1960)