Racah parameter

When an atom has more than one electron there will be some electrostatic repulsion between those electrons. The amount of repulsion varies from atom to atom, depending upon the number and spin of the electrons and the orbitals they occupy. The total repulsion can be expressed in terms of three parameters A, B and C which are known as the Racah parameters after Giulio Racah, who first described them. They are generally obtained empirically from gas-phase spectroscopic studies of atoms.[1]

They are often used in transition-metal chemistry to describe the repulsion energy associated with an electronic term. For example, the interelectronic repulsion of a ³P term is A + 7B, and of a ³F term is A - 8B, and the difference between them is therefore 15B.

Definition

The Racah parameters are defined as

${\begin{pmatrix}A\\B\\C\\\end{pmatrix}}={\begin{pmatrix}1&0&-49\\0&1&-5\\0&0&35\\\end{pmatrix}}{\begin{pmatrix}F_{0}\\F_{2}\\F_{4}\\\end{pmatrix}}$

where $F_{k}$ are Slater integrals

${\begin{pmatrix}F_{0}\\F_{2}\\F_{4}\\\end{pmatrix}}={\begin{pmatrix}F^{0}\\{\frac {1}{49}}F^{2}\\{\frac {1}{441}}F^{4}\\\end{pmatrix}}$

and $F^{k}$ are the Slater-Condon parameters

$F^{k}:=\int _{0}^{\infty }r_{1}^{2}dr_{1}\int _{0}^{\infty }r_{2}^{2}dr_{2}R^{2}(r_{1})R^{2}(r_{2}){\frac {r_{<}^{k}}{r_{>}^{k+1}}}$

where $R(r)$ is the normalized radial part of an electron orbital, $r_{>}=\max\{r_{1},r_{2}\}$ and $r_{<}=\min\{r_{1},r_{2}\}$ [2].

References

Ballhausen, C.J. (1980). Molecular Electronic Structure of Transition Metal Complexes. McGraw-Hill. ISBN 0070034958.
Sugano, Satoru; Tanabe, Yukito; Kamimura, Hiroshi (1970). Multiplets of Transition Metal Ions in Crystals. Academic Press.

External links

Multiplets in Transition Metal Ions in E. Pavarini, E. Koch, F. Anders, and M. Jarrell (eds.): Correlated Electrons: From Models to Materials, Jülich 2012, ISBN 978-3-89336-796-2

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[1] Ballhausen, C.J. (1980). Molecular Electronic Structure of Transition Metal Complexes. McGraw-Hill. ISBN 0070034958.

[2] Sugano, Satoru; Tanabe, Yukito; Kamimura, Hiroshi (1970). Multiplets of Transition Metal Ions in Crystals. Academic Press.

Racah parameter

Definition

See also

References

External links