Hund's rules

Named after Friedrich Hund Hund's rules make a statement, in which angular momentum configuration, the electrons are in the orbitals of an atom in the ground state. These rules apply here as part of the LS- coupling, which is fulfilled in particular for light elements. But for heavier atoms, the rules can achieve good results.

Background

If one wants to determine the structure of the electron shell of an atom with electrons in theory, the Schrödinger equation for this problem must be solved in principle. This is analytically only possible when the electrostatic interaction between the electrons neglected (see Three - Body Problem ). Instead, they used approximately a screened Coulomb potential for the interaction between electrons and core to the weakened attraction of outer electrons to meet.

Russell - Saunders coupling (LS coupling)

The magnetic moments of the orbital angular momentum L and spin S of the electrons in an atom do not interact separately but. By adding to a total angular momentum as a whole with an external magnetic field Here, when the Coulomb interaction of the electrons with one another is large compared to their own spin -orbit interaction, can the total angular momentum in the context of - determine or Russell - Saunders coupling (after Henry Norris Russell and Frederick Albert Saunders ). The following rules apply:

• The spins of the electrons add up to the total spin

Instead of the previous, separate for each electron, description of his condition by the quantum numbers, you now have the new quantum numbers for the total electronic system. They belong to as described above. The overall electronic state is usually listed as term symbol in the form ( with explicit specification of J, but not, as only in the magnetic field, the energy also depends on, see Zeeman effect or Landé factor and Paschen-Back effect).

The rules

Note that this division into four rules is consistent with the widely used textbooks on atomic physics. However, one finds especially in older books even less, usually two, Hund's rules, which then correspond to the 2nd and 3rd listed here generally. A justification for the 2nd and 3rd rule provides the exchange interaction.

First Hund's rule

This rule follows directly from the Pauli principle. For a filled shell, all the possible quantum numbers must be established, therefore, there are an equal number of positive and negative orientations of the orbital angular momenta and spins of the electrons. The resulting total angular momentum quantum numbers and the associated can only have the value zero. Consequently, so only the non- closed shells must be taken into account in the calculation of. Strictly speaking there is this fact as a consequence of the other rules, but as it is an important result, it is often cited as a separate rule.

" The total spin assumes the maximum possible value, the spins of the individual electrons are therefore as parallel as possible. "

Explanation

To be with this rule, the electrons initially different values ​​for the magnetic quantum number must be assigned so that the same spin quantum numbers are possible in accordance with the Pauli principle. There are different values ​​for, so the total spin quantum number can be up to accept the value. This value is reached when the tray is filled exactly in half. With greater need filling because of the Pauli principle, the spins of the electrons are antiparallel to those of the already installed electrons.

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. Standing parallel spins mean a symmetric spin part of the wave function. The antisymmetry must then originate from the rail share. However, an anti- symmetrical track portion describes a state in which the electrons are as far as possible away from each other. This property ensures the smallest possible Coulomb interaction energy. The Coulomb repulsion of the electrons are not fully occupied shells is also behind the constant JHund, associated with the so-called " Hund's -rule exchange". This constant represents an intra- atomic effective " exchange interaction ", which is responsible for the parallel orientation of neighboring spins of the same dish at not fully occupied shells and has the form ( Heisenberg model, the scalar product of vector spin operators), with positive JHund, which by a double integral given by a combination of the involved quantum mechanical wave functions and can be expressed by the corresponding Coulomb denominator:

Here, u and v are both considered quantum mechanical functions. Marked with stars whose functions are conjugate complex. Finally, e is the elementary charge.

The name " exchange interaction " comes from interchanging the role of u and v in the second factor: Behind the quantum mechanical Pauli principle hides. So you have to do it with an interplay of Coulomb interaction and Pauli principle.

In fact, quantum mechanical calculations have shown that electrons are shielded in singularly occupied orbitals less opposite the charge of the nucleus, so that the orbitals contract. This leads to a more energetically favorable configuration of the overall atom.

Third Hund's rule

"Allows the Pauli principle multiple configurations with maximum total spin, then the sub-states are occupied with the magnetic quantum number, the total orbital angular momentum is maximized. "

According to the rule the first electron shell is a new take on the maximum value of. The second electron may be due to the Pauli principle and the second usually do not get the same value for, so get the second largest value. The total angular momentum quantum number for this case is it. Is the cup half full, so all must be assigned once after the second rule here is therefore equal to zero. When filling the second half are then awarded according to the same order as when filling the first half.

In Einteilchenfall the average distance of an electron from the nucleus with the quantum number of the component of the angular momentum is growing. Since electrons that are far removed from the core, tend to are also far apart (as in the case of the Second Hund's rule) the Coulomb interaction small. But the effect is smaller than the parallel spin. Therefore, the second Hund's rule has precedence over the others.

" Is a lower shell filled more than half, then the state is the most bound with minimal total angular momentum quantum number. With more than half-full subshells, it is vice versa. "

In chemistry, a single Hund's rule is often used, which was found in 1927 by Friedrich dog itself purely empirical and content corresponds to the second of the above rules. It says: ( " Maximum multiplicity of" formal term ) if there are several orbitals / secondary quanta with the same energy level for the electrons of an atom are available, this first with one electron with parallel spin are occupied. Only when all the orbitals of the same energy levels are filled with one electron, they are completed by the second electron.

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

As Hund's rule describes the location belonging to a specific configuration of the electron terms, it has influence on the chemical behavior of atoms.

Application

The total angular momentum finally delivers the fourth Hund's rule. Since the shell is more than half full, true, in this case.

The overall electronic state is thus characterized by the term symbol 3F4.

After a colloquium lecture on the history of quantum mechanics Friedrich Hund in 1985 was asked publicly at the University of Regensburg, how he had come because of the Hund's rules. The answer in a nutshell: ". Well, simply by" stare at " spectra " ( said Posh, first simply by attempting to interpret the findings of the experimental physicist, and the mathematical and physical justification for the rules followed only later. ) In indeed emerged Hund's rules only little earlier ( 1925-1927 ) than, for example mentioned in the text work for Heisenberg model (1928 ).