Slater Type Orbitals

This article deals with special single-particle orbitals . For the construction method of many-particle wave functions , see Slater determinant .

In quantum chemistry are Slater-type orbitals (STOs) or Slater functions (named after John C. Slater , who in 1930 introduced) wave functions , which in quantum chemistry for the construction of atomic orbitals and in the LCAO - approximation for molecular orbitals can be used. They are given in spherical coordinates , with the radial part corresponding to an exponential function and the angular function to a spherical surface function .

In the hydrogen atom, Slater functions are the exact solutions of the Schrödinger equation for the wave functions of the electron . For atoms with several electrons the functions are well suited to describe the behavior of the wave functions at a large distance from the atomic nucleus. In the formalism of quantum mechanics, integrals have to be solved, which with Slater functions are associated with considerable computational effort. They are therefore often replaced by a linear combination of several Gaussian functions , the so-called Gaussian Type Orbitals (GTOs), e.g. B. the minimum basic set  STO-3G consists of three GTOs to approximate an STO.

Radial part

${\displaystyle R(r)=r^{n^{*}-1}\cdot e^{-\zeta r}\cdot N\,}$

With

• the distance of the electron from the nucleus ${\displaystyle r}$
• a constant derived from the principal quantum number ${\displaystyle n=1,2,\dotsc }$ ${\displaystyle n^{*}}$
• a shielding constant  for the effective nuclear charge ${\displaystyle \zeta }$
• a normalization constant . ${\displaystyle N}$

The constant is up to identical with the principal quantum number. For larger values ​​of n it is increasingly smaller. ${\displaystyle n^{*}}$ ${\displaystyle n=3}$

The normalization constant  is calculated from the normalization of the above. Equation: ${\displaystyle N\,}$

${\displaystyle N^{2}\cdot \int _{0}^{\infty }\left(r^{n-1}\cdot e^{-\zeta r}\right)^{2}r^{2}\,\mathrm {d} r=1}$

with the help of the general integral

${\displaystyle \int _{0}^{\infty }(x^{n}\cdot e^{-\alpha x})\,\mathrm {d} x={\frac {n!}{\alpha ^{n+1}}}}$

to

${\displaystyle \Rightarrow N=(2\zeta )^{n}{\sqrt {\frac {2\zeta }{(2n)!}}}.}$

Angle-dependent part

For the angle-dependent part of the STOs, i.e. H. those who by and dependent, are usually spherical harmonics used, because of their zero points for the necessary nodal surfaces provide. ${\displaystyle \vartheta }$ ${\displaystyle \varphi }$

Individual evidence

1. ^ JC Slater: Atomic Shielding Constants . In: Physical Review . tape 36 , no. 1 , 1930, p. 57-64 , doi : 10.1103 / PhysRev.36.57 .
2. ^ Attila Szabo, Neil S. Ostlund: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory . Courier Corporation, 1996, ISBN 978-0-486-69186-2 , pp. 154 .
3. ^ ^{A b} Peter W. Atkins: Quantum: Terms and Concepts for Chemists . VCH, ISBN 978-0-486-69186-2 .