Hund's rules

Hund's rules , named after Friedrich Hund , make a statement about the angular momentum configuration in which the electrons in the orbitals of an atom are in the ground state. These rules apply in the context of the LS coupling , which is particularly true for light elements . But the rules can also achieve good results for heavier atoms.

background

If one wants to theoretically determine the structure of the electron shell of an atom with electrons, then in principle the Schrödinger equation must be solved for this problem. However, analytically, this is only possible if one neglects the electrostatic interaction of the electrons with one another (see three-body problem ). Instead, an approximately shielded Coulomb potential is used for the interaction between electrons and nucleus in order to do justice to the weakened attraction of external electrons . ${\displaystyle Z}$

Such a calculation describes the height of the energy levels sufficiently well, but only provides a dependence on the main quantum number and the secondary quantum number (with ). Each such subshell can be filled with electrons which, according to the Pauli principle , have to differ in terms of the magnetic quantum number or the spin quantum number . In order to determine the basic state of the atom, one imagines according to the structural principle , the energy levels one after the other, starting from the lowest, with electrons. Are not sufficient electrons left to the highest energy level to fill completely, there is a mapping problem: since it is energetically indifferent, the electrons which magnetic or spin quantum number on a lower shell assume theoretically any could (allowed) values are assigned. Such a ground state is called degenerate . ${\displaystyle n}$ ${\displaystyle l}$ ${\displaystyle l<n}$ ${\displaystyle 2(2l+1)}$ ${\displaystyle m_{l}}$ ${\displaystyle m_{s}}$

In practice one places z. B. by measuring the magnetic susceptibility that the assignment is energetically not indifferent. The degeneracy of the ground state is therefore an artifact from the above-mentioned neglect of the electron-electron interaction. In order to be able to make a theoretical prediction about the electron distribution on the magnetic and spin quantum number, it turns out that a set of simple rules, Hund's rules, is sufficient.

Russell-Saunders coupling ( LS coupling)

→ Main article : LS coupling

The magnetic moments of orbital angular momentum  L and spin  S of the electrons of an atom do not interact separately, but by adding them to a total angular momentum as a whole with an external magnetic field . If the Coulomb interaction between the electrons is large compared to their own spin-orbit interaction, the total angular momentum can be determined within the framework of the - or Russell-Saunders coupling (after Henry Norris Russell and Frederick Albert Saunders ). The following rules apply here: ${\displaystyle LS}$

• The orbital angular momentum of the electrons add up to the total orbital angular momentum ${\displaystyle {\boldsymbol {l}}_{i}}$

${\displaystyle {\boldsymbol {L}}=\sum _{i}{\boldsymbol {l}}_{i}}$

with . ${\displaystyle {\boldsymbol {L}}^{2}=L(L+1)\hbar ^{2}}$

• The spins of the electrons add up to the total spin ${\displaystyle {\boldsymbol {s}}_{i}}$

${\displaystyle {\boldsymbol {S}}=\sum _{i}{\boldsymbol {s}}_{i}}$

with . ${\displaystyle {\boldsymbol {S}}^{2}=S(S+1)\hbar ^{2}}$

• Total orbital angular momentum and total spin add up to the total angular momentum

${\displaystyle {\boldsymbol {J}}={\boldsymbol {L}}+{\boldsymbol {S}}}$

with and . ${\displaystyle {\boldsymbol {J}}^{2}=J(J+1)\hbar ^{2}}$ ${\displaystyle J_{z}=M_{J}\hbar ,\;M_{J}\in \{-J,\,-J+1,\,\ldots ,\,+J\}}$

Instead of the previous description of its state by quantum numbers , which was separate for each electron , one now has the new quantum numbers for the overall electronic system. They belong to as described above. The overall electronic state is usually noted as a term symbol in the form (with an explicit indication of J, but not of , since the energy also depends on in the magnetic field , see Zeeman effect or Landé factor and Paschen-Back effect ). ${\displaystyle l,s\,\,(=1/2),m_{l},m_{s}}$ ${\displaystyle L,S,J,M_{J}}$ ${\displaystyle {\boldsymbol {L}}^{2},{\boldsymbol {S}}^{2},{\boldsymbol {J}}^{2},J_{z}}$ ${\displaystyle ^{2S+1}L_{J}}$ ${\displaystyle M_{J}}$ ${\displaystyle M_{j}}$

The rules

The division into four rules made here is consistent with the widely used textbooks on atomic physics . However, especially in older books, there are fewer, usually two, Hund's rules that correspond to the 2nd and 3rd rule listed here. The exchange interaction provides a reason for the 2nd and 3rd rule .

Hund's first rule

"Full shells and lower shells have zero total angular momentum."

This rule results directly from the Pauli principle . For a filled shell all possible quantum numbers have to be occupied, so there are as many positive as negative orientations of the orbital angular momentum and spins of the electrons. The resulting total angular momentum and the associated quantum numbers can only have the value zero. Consequently, only the non-closed shells have to be taken into account when calculating . Strictly speaking, this fact also results from the other rules, but since it represents an important result, it is often cited as a separate rule. ${\displaystyle {\boldsymbol {J}}}$ ${\displaystyle J,m_{J}}$ ${\displaystyle {\boldsymbol {J}}}$

Hund's second rule

"The total spin assumes the maximum possible value, so the spins of the individual electrons are as parallel as possible." ${\displaystyle {\boldsymbol {S}}}$ ${\displaystyle {\boldsymbol {s}}_{i}}$

Explanation

In order to comply with this rule, the electrons must first be assigned different values ​​for the magnetic quantum number , so that the same spin quantum numbers are possible in accordance with the Pauli principle. There are different values ​​for , therefore the total spin quantum number can at most assume the value . This value is reached when the bowl is exactly half full. In the case of a larger filling, the spins of the electrons must be antiparallel to those of the electrons already installed due to the Pauli principle. ${\displaystyle m_{l}}$ ${\displaystyle 2l+1}$ ${\displaystyle m_{l}}$ ${\displaystyle S}$ ${\displaystyle {\tfrac {2l+1}{2}}=l+{\tfrac {1}{2}}}$

background

Originally the explanation for this rule was the following assumption: According to the Pauli principle, the wave function of the electrons must be totally antisymmetric. Spins standing in parallel mean a symmetrical spin component of the wave function. The antisymmetry must then come from the orbit portion. However, an antisymmetrical path component describes a state in which the electrons are as far apart as possible. This property ensures that the Coulomb interaction energy is as small as possible . The Coulomb repulsion of the electrons when the shells are not fully occupied is also behind the constant J _Hund , which is assigned to the so-called “Hund's rule exchange”. This constant represents an effective intra-atomic “ exchange interaction ”, which is responsible for the parallel positioning of neighboring spins of the same shell when the shells are not fully occupied and has the form ( Heisenberg model , scalar product of vectorial spin operators), with a positive J _dog , which is represented by a double integral over a certain combination of the quantum mechanical wave functions involved and can be expressed by the associated Coulomb denominator: ${\displaystyle -J_{\text{Hund}}\,{\hat {\boldsymbol {S_{1}}}}\cdot {\hat {\boldsymbol {S_{2}}}}}$

${\displaystyle J_{\text{Hund}}\propto e^{2}\int _{x}\ \int _{y}\mathrm {d} ^{3}x\ \mathrm {d} ^{3}y\ u^{*}({\boldsymbol {x}})v^{*}({\boldsymbol {y}})v({\boldsymbol {x}})u({\boldsymbol {y}})/|{\boldsymbol {x}}-{\boldsymbol {y}}|}$

Here and are the two quantum mechanical functions considered. The starred functions are their conjugate complexes. Finally is the elemental charge. ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle e}$

The name "exchange interaction" comes from the reversal of the role of and in the second factor: Behind it is the quantum mechanical Pauli principle . So one has to do with an interplay of the Coulomb interaction and the Pauli principle. ${\displaystyle u}$ ${\displaystyle v}$

Since the Coulomb interaction is larger than the spin-orbit coupling according to the assumption, the state with as many parallel spins as possible is also the one with the lowest energy.

In fact, however, quantum mechanical calculations have shown that electrons in singularly occupied orbitals are less shielded from the charge of the nucleus, as a result of which the orbitals contract. This then leads to an energetically more favorable configuration of the entire atom.

Hund's third rule

"If the Pauli principle allows several constellations with maximum total spin , then the sub-states are occupied with the magnetic quantum number in such a way that the total orbital angular momentum is maximal." ${\displaystyle {\boldsymbol {S}}}$ ${\displaystyle m_{l}}$ ${\displaystyle {\boldsymbol {L}}}$

Explanation

According to the rule, the first electron of a new shell will have the maximum value of . Because of the Pauli principle and the second rule, the second electron must not have the same value for , i.e. it has the second largest value . The total angular momentum quantum number for this case is thus . If the bowl is half full, then, according to the second rule, all must be taken once, so it is zero here. When filling the second half, they are then assigned in the same order as when filling the first half. ${\displaystyle |m_{l}|=l}$ ${\displaystyle m_{l}}$ ${\displaystyle l-1}$ ${\displaystyle L}$ ${\displaystyle L=l+(l-1)=2l-1}$ ${\displaystyle m_{l}}$ ${\displaystyle L}$ ${\displaystyle m_{l}}$

background

In the single-particle case, the mean distance between an electron and the nucleus increases with the quantum number , the component of the angular momentum. Since electrons that are far from the nucleus also tend to be far apart, (as in the case of Hund's second rule) the Coulomb interaction becomes small. The effect is smaller than that caused by parallel spins. Therefore Hund's second rule also takes precedence over the third. ${\displaystyle m_{l}}$ ${\displaystyle z}$

Hund's fourth rule

“If a subshell is at most half full, then the state with the minimum total angular momentum quantum number is most strongly bound. If the lower shells are more than half full, the opposite is true. " ${\displaystyle J}$

Explanation

According to the second and third rule, the quantum numbers for total spin and total orbital angular momentum are determined. Therefore all integer values ​​between and remain for the total angular momentum . According to this rule it is determined that it is always calculated as follows: ${\displaystyle |L-S|}$ ${\displaystyle L+S}$ ${\displaystyle J}$

• If the bowl is less than half full, it is . ${\displaystyle J=|L-S|}$
• If the bowl is more than half full, it is . ${\displaystyle J=L+S}$

If the bowl is exactly half full, then the third rule applies , so both calculations give the same value. ${\displaystyle L=0}$

This rule need not be considered if one is only interested in the distribution of for the electrons of a shell. For the magnetic behavior of the atoms, however, the overall configuration of the electrons and thus the overall angular momentum is decisive. ${\displaystyle m_{l},m_{s}}$

background

The LS coupling shows that for a maximum half-full shell, an anti-parallel setting of spin and orbital angular momentum is energetically more favorable. If the shell is more than half full, the electrons required to fill the shell can be interpreted as “holes”, the rotation of which creates a magnetic field with a polarity reversed compared to electrons. This means that parallel settings for and are preferred. ${\displaystyle L}$ ${\displaystyle S}$

The Hund's rule of chemistry

In chemistry , only one single Hund's rule is often used, which was found purely empirically by Friedrich Hund himself in 1927 and which corresponds in content to the second of the rules listed above. It says: If several orbitals / secondary quanta with the same energy level are available for the electrons of an atom, these are first occupied by one electron each with a parallel spin (formal term: “maximum multiplicity”). Only when all orbitals of the same energy level are filled with one electron each will they be completed by the second electron.

The distinction between Hund's rule in chemistry and Hund's rules in physics only refers to the nomenclature  - of course, the same rules and principles apply in chemistry and physics.

Since Hund's rule describes the position of the terms belonging to a certain configuration of the electrons , it has an influence on the chemical behavior of atoms.

An electron configuration caused by a strong ligand that does not comply with Hund's rule is referred to as magnetically abnormal ( low spin configuration ).

application

In the following table, the quantum numbers for total spin, total orbital angular momentum and total angular momentum only refer to the relevant subshell. ${\displaystyle S,L,J}$

Legend
${\displaystyle l}$	${\displaystyle s,p,d,f,\dotsc }$ ${\displaystyle (=0,1,2,3,\dotsc )}$	Minor quantum number
${\displaystyle m_{l}}$	${\displaystyle -l,\dotsc ,l}$	Magnetic quantum number
${\displaystyle m_{s}}$	↑, ↓ ½, −½ ${\displaystyle (=}$ ${\displaystyle )}$	Spin quantum number
${\displaystyle S}$		Total spin quantum number
${\displaystyle L}$	${\displaystyle S,P,D,F,\dotsc }$ ${\displaystyle (=0,1,2,3,\dotsc )}$	Total orbital angular momentum quantum number
${\displaystyle J}$		Total angular momentum quantum number

${\displaystyle e^{-}}$	${\displaystyle m_{l}}$					${\displaystyle S}$	${\displaystyle L}$	${\displaystyle J}$
	2	1	0	−1	−2	${\displaystyle |\Sigma m_{s}|}$	${\displaystyle |\Sigma m_{l}|}$
s -shell ( ) ${\displaystyle l=0}$
s 1			↑			½	0	½
s 2			↑ ↓			0	0	0
p -shell ( ) ${\displaystyle l=1}$
p 1				↑		½	1	½
p 2			↑	↑		1	1	0
p 3		↑	↑	↑		1½	0	1½
p 4		↑ ↓	↑	↑		1	1	2
p 5		↑ ↓	↑ ↓	↑		½	1	1½
p 6		↑ ↓	↑ ↓	↑ ↓		0	0	0
d shell ( ) ${\displaystyle l=2}$
d 1					↑	½	2	1½
d 2				↑	↑	1	3	2
d 3			↑	↑	↑	1½	3	1½
d 4		↑	↑	↑	↑	2	2	0
d 5	↑	↑	↑	↑	↑	2½	0	2½
d 6	↑ ↓	↑	↑	↑	↑	2	2	4th
d 7	↑ ↓	↑ ↓	↑	↑	↑	1½	3	4½
d 8	↑ ↓	↑ ↓	↑ ↓	↑	↑	1	3	4th
d 9	↑ ↓	↑ ↓	↑ ↓	↑ ↓	↑	½	2	2½
d 10	↑ ↓	↑ ↓	↑ ↓	↑ ↓	↑ ↓	0	0	0

Example: We are looking for the ground state of an atom with 8 electrons on the 3 -shell in the 4th period (the element nickel ). The shell has space for 10 electrons, 5 of them with spin ↑, the others with spin ↓. According to Hund's first rule, the fully occupied inner shells make no contribution and do not need to be considered. ${\displaystyle d}$

Spin ↑ and spin ↓ have the same rights, but in the literature they usually begin with spin ↑.

Since, according to Hund's second rule, the spins should be arranged as parallel as possible, all 5 places are initially occupied with spin ↑. The remaining 3 electrons must then have spin ↓. The following applies to the total spin quantum number . The total spin results in (corresponding to 2 unpaired spins). This results in a spin multiplicity of (upper left index on the term symbol). ${\displaystyle S=\left|\Sigma m_{s}\right|}$ ${\displaystyle S=1}$ ${\displaystyle 2S+1=3}$

According to Hund's third rule, it must be maximum. The remaining spin ↓ electrons occupy the states . Overall will . ${\displaystyle L=|\Sigma m_{l}|}$ ${\displaystyle m_{l}=2,1,0}$ ${\displaystyle L=3}$

Hund's fourth rule finally provides the total angular momentum of the electron shell. Since the bowl is more than half full, the following applies here . ${\displaystyle J=L+S}$ ${\displaystyle J=4}$

The overall electronic state is thus characterized by the term symbol ³ F ₄ .

On the origin of Hund's rules

After a colloquium lecture on the history of quantum mechanics, Friedrich Hund was publicly asked in 1985 at the University of Regensburg how he came up with Hund's rules. The answer, in a nutshell: "Well, simply by" staring at "the spectra." (To put it more elegantly, at first simply by attempting to interpret the findings of the experimental physicists; the mathematical-physical justification for the rules followed only later.) In fact, Hund's rules emerged only a little earlier (1925 to 1927) than z. B. the work mentioned in the text on the Heisenberg model (1928).

See also

• Hund's coupling cases

literature

• Bergmann-Schaefer textbook on experimental physics . Volume IV, Part 1: Structure of Matter. 2nd edition, Walter de Gruyter, Berlin, 1987, ISBN 3-11-008074-5 .
• Dieter Meschede (Ed.): Gerthsen Physik . 23rd edition. Springer, Heidelberg 2006, ISBN 3-540-25421-8 .

Individual evidence

1. ^ HN Russell, FA Saunders: New Regularities in the Spectra of the Alkaline Earths . In: Astrophysical Journal . tape 61 , no. 38 , 1925.
2. ↑ F. Hund: Interpretation of Molecular Spectra, I and II. In: Journal of Physics. 40, pp. 742-764 (1927) and 42, pp. 93-120 (1927), based on Hund's unpublished habilitation thesis from 1925.
3. ↑ W. Heisenberg: On the theory of ferromagnetism . In: Journal of Physics . tape 49 , 1928, pp. 619-636 , doi : 10.1007 / BF01328601 .