Slater-type orbitals (STOs) or Slater-type functions (STFs) are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method. They are named after the physicist John C. Slater, who introduced them in 1930. They possess exponential decay at long range and Kato's cusp condition at short range (when combined as hydrogen-like atom functions, i.e. the analytical solutions of the stationary Schrödinger equation for one electron atoms). Unlike the hydrogen-like ("hydrogenic") Schrödinger orbitals, STOs have no radial nodes (neither do Gaussian-type orbitals).
Definition STOs have the following radial part:
R ( r ) = N r n − 1 e − ζ r {\displaystyle R(r)=Nr^{n-1}e^{-\zeta r}\,}
where
n is a natural number that plays the role of principal quantum number, n = 1,2,..., N is a normalizing constant, r is the distance of the electron from the atomic nucleus, and
ζ {\displaystyle \zeta } is a constant related to the effective charge of the nucleus, the nuclear charge being partly shielded by electrons. Historically, the effective nuclear charge was estimated by Slater's rules. The normalization constant is computed from the integral
∫ 0 ∞ x n e − α x d x = n ! α n + 1 . {\displaystyle \int _{0}^{\infty }x^{n}e^{-\alpha x}\,\mathrm {d} x={\frac {n!}{~\alpha ^{n+1}\,}}~.}
Hence
N 2 ∫ 0 ∞ ( r n − 1 e − ζ r ) 2 r 2 d r = 1 ⟹ N = ( 2 ζ ) n 2 ζ ( 2 n ) ! . {\displaystyle N^{2}\int _{0}^{\infty }\left(r^{n-1}e^{-\zeta r}\right)^{2}r^{2}\,\mathrm {d} r=1\Longrightarrow N=(2\zeta )^{n}{\sqrt {\frac {2\zeta }{(2n)!}}}~.}
It is common to use the spherical harmonics Y l m ( r ) {\displaystyle Y_{l}^{m}(\mathbf {r} )} depending on the polar coordinates of the position vector r {\displaystyle \mathbf {r} } as the angular part of the Slater orbital.
STO-nG Basis Sets All the spherical coordinates of the Slater-type orbital can be written as the function:
S n l m ( r , θ , ϕ ) = N r n − 1 e − ζ r Y l m ( θ , ϕ ) {\displaystyle S_{nlm}(r,\theta ,\phi )=N\,r^{n-1}e^{-\zeta r}Y_{l}^{m}(\theta ,\phi )}
Where the constant N = ( 2 ζ ) n 2 ζ ( 2 n ) ! {\displaystyle N=(2\zeta )^{n}{\sqrt {\frac {2\zeta }{(2n)!}}}}
Below is the Gaussian-type function:
G n l m ( r , θ , ϕ ) = N r n − 1 e − α r 2 Y l m ( θ , ϕ ) {\displaystyle G_{nlm}(r,\theta ,\phi )=N\,r^{n-1}e^{-\alpha r^{2}}Y_{l}^{m}(\theta ,\phi )}
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