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In
atomic physics Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
and
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
, Hund's rules refers to a set of rules that German physicist
Friedrich Hund Friedrich Hermann Hund (4 February 1896 – 31 March 1997) was a German physicist from Karlsruhe known for his work on atoms and molecules. He is known for the Hund's rules to predict the electron configuration of chemical elements. His work on H ...
formulated around 1925, which are used to determine the
term symbol In atomic physics, a term symbol is an abbreviated description of the total spin and orbital angular momentum quantum numbers of the electrons in a multi-electron atom. So while the word ''symbol'' suggests otherwise, it represents an actual ''valu ...
that corresponds to the
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
of a multi-electron atom. The first rule is especially important in
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
, where it is often referred to simply as Hund's Rule. The three rules are: # For a given
electron configuration In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon ato ...
, the term with maximum multiplicity has the lowest energy. The multiplicity is equal to 2S + 1 \ , where S is the total spin angular momentum for all electrons. The multiplicity is also equal to the number of unpaired electrons plus one. Therefore, the term with lowest energy is also the term with maximum S \, and maximum number of unpaired electrons with equal spin angular momentum (either +1/2 or -1/2). # For a given multiplicity, the term with the largest value of the total orbital angular momentum quantum number  L \, has the lowest energy. # 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  J \, (for the operator \boldsymbol=\boldsymbol+\boldsymbol) lies lowest in energy. If the outermost shell is more than half-filled, the level with the highest value of  J \, is lowest in energy. These rules specify in a simple way how usual energy interactions determine which term includes the ground state. 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 LS coupling regime. Closed shells and subshells do not contribute to the quantum numbers for total , the total spin angular momentum and for , the total orbital angular momentum. It can be shown that for full orbitals and suborbitals both the residual
electrostatic energy Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An ''object'' may be sa ...
(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.


Rule 1

Due to the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle (German: Pauli-Ausschlussprinzip) states that two or more identical particles with half-integer spins (i.e. fermions) cannot simultaneously occupy the same quantum state within a system that o ...
, 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'') ''m''''s'' = , and the other must have ''m''''s'' = −. Hund's first rule states that the lowest energy atomic state is the one that maximizes the total spin quantum number for 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 different physical explanations have been givenI.N. Levine, Quantum Chemistry (Prentice-Hall, 4th edn 1991) , pp. 303–304 for the increased stability of high multiplicity states. In the early days of
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, 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 shielded from the nucleus, so that such orbitals contract and electron–nucleus attraction energy becomes greater in magnitude (or decreases algebraically).


Example

As an example, consider the ground state of
silicon Silicon is a chemical element; it has symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic lustre, and is a tetravalent metalloid (sometimes considered a non-metal) and semiconductor. It is a membe ...
. The
electron configuration In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon ato ...
of Si is (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 In quantum mechanics, the Pauli exclusion principle (German: Pauli-Ausschlussprinzip) states that two or more identical particles with half-integer spins (i.e. fermions) cannot simultaneously occupy the same quantum state within a system that o ...
are 1''D'' , 3''P'' , and 1''S''. Hund's first rule now states that the ground state term is 3''P'' (triplet 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''''L'' = 1 and ''M''''S'' = 1.


Rule 2

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.


Example

For silicon there is only one triplet term, so the second rule is not required. The lightest atom that requires the second rule to determine the ground state term is
titanium Titanium is a chemical element; it has symbol Ti and atomic number 22. Found in nature only as an oxide, it can be reduced to produce a lustrous transition metal with a silver color, low density, and high strength, resistant to corrosion in ...
(Ti,  = 22) with electron configuration . In this case the open shell is and the allowed terms include three singlets (1S, 1D, and 1G) and two triplets (3P and 3F). (Here the symbols S, P, D, F, and G indicate that the total orbital angular momentum quantum number has values 0, 1, 2, 3 and 4, respectively, analogous to the nomenclature for naming atomic orbitals.) We deduce from Hund's first rule that the ground state term is one of the two triplets, and from Hund's second rule that this term is 3F (with L = 3) rather than 3P (with L = 1). There is no 3G term since its (M_L = 4, M_S = 1) state would require two electrons each with (M_L = 2, M_S = +1/2), in violation of the Pauli principle. (Here M_L and M_S are the components of the total orbital angular momentum L and total spin S along the z-axis chosen as the direction of an external magnetic field.)


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 \Delta E & = \xi (L,S) \ \\ & = \ (1/2) \xi (L,S) \ \end The value of \xi (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 \, .


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 Sulfur ( American spelling and the preferred IUPAC name) or sulphur ( Commonwealth spelling) is a chemical element; it has symbol S and atomic number 16. It is abundant, multivalent and nonmetallic. Under normal conditions, sulfur atoms ...
 (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 is the lowest energy state. For example, in
phosphorus Phosphorus is a chemical element; it has Chemical symbol, symbol P and atomic number 15. All elemental forms of phosphorus are highly Reactivity (chemistry), reactive and are therefore never found in nature. They can nevertheless be prepared ar ...
the lowest energy state has S = 3/2,\ L = 0 for three unpaired electrons in three 3p orbitals. Therefore, J = S = 3/2 and the ground state is ^4\!S_\, .


Excited states

Hund's rules work best for the determination of the
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than 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 A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with two neutrons, depending on the isotope, held together by the strong ...
, Hund's first rule correctly predicts that the 1s2s
triplet state In quantum mechanics, a triplet state, or spin triplet, is the quantum state of an object such as an electron, atom, or molecule, having a quantum spin ''S'' = 1. It has three allowed values of the spin's projection along a given axis ''m''S = � ...
(3S) is lower than the 1s2s singlet (1S). Similarly for organic molecules, the same rule predicts that the first triplet state (denoted by T1 in
photochemistry Photochemistry is the branch of chemistry concerned with the chemical effects of light. Generally, this term is used to describe a chemical reaction caused by absorption of ultraviolet (wavelength from 100 to 400 Nanometre, nm), visible ligh ...
) is lower than the first excited
singlet state In quantum mechanics, a singlet state usually refers to a system in which all electrons are paired. The term 'singlet' originally meant a linked set of particles whose net angular momentum is zero, that is, whose overall spin quantum number s=0. A ...
(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 Titanium is a chemical element; it has symbol Ti and atomic number 22. Found in nature only as an oxide, it can be reduced to produce a lustrous transition metal with a silver color, low density, and high strength, resistant to corrosion in ...
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.


References


External links

*
Multiplets in Transition Metal ions
in E. Pavarini, E. Koch, F. Anders, and M. Jarrell: Correlated Electrons: From Models to Materials, Jülich 2012, * E. Scerri, The Periodic Table, Its Story and Its Significance, 2nd ed. (Oxford University Press, 2020) {{ISBN, 978-0-19-091436-3 Atomic physics Eponymous chemical rules Quantum chemistry Spectroscopy Rules