Slater orbital: Difference between revisions

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It is common to use the real form of  [[spherical harmonics]] depending on the polar coordinates &theta; and &phi;  as the angular part of the Slater orbital.
It is common to use the real form of  [[spherical harmonics]] as the angular part of the Slater orbital. A list of cartesian real spherical harmonics is given in this [[solid harmonics|article]].
 
The first few Slater type orbitals are (where we use ''s'' for ''l'' = 0, ''p'' for ''l'' = 1 and ''d'' for ''l'' = 2):
:<math>
\begin{align}
1s  &= 2\zeta^{3/2} e^{-\zeta r} \\
2s  &= \frac{2}{3}\sqrt{3} \zeta^{5/2}\,  r \,e^{-\zeta r} \\
2p_x &= \frac{2}{3}\sqrt{3} \zeta^{5/2}\,  r \,e^{-\zeta r} \Big[\sqrt{\frac{3}{4\pi}} \frac{x}{r}                \Big] = \sqrt{\frac{\zeta^{5}}{\pi}}\, x \,e^{-\zeta r} \\
2p_y &= \frac{2}{3}\sqrt{3} \zeta^{5/2}\,  r \,e^{-\zeta r} \Big[\sqrt{\frac{3}{4\pi}} \frac{y}{r}                \Big] =  \sqrt{\frac{\zeta^{5}}{\pi}}\, y \,e^{-\zeta r} \\
2p_z &= \frac{2}{3}\sqrt{3} \zeta^{5/2}\,  r \,e^{-\zeta r} \Big[\sqrt{\frac{3}{4\pi}} \frac{z}{r}                \Big] =  \sqrt{\frac{\zeta^{5}}{\pi}}\, z \,e^{-\zeta r} \\
3s  &= \frac{2}{15}\sqrt{10} \zeta^{7/2}\, r^2\,e^{-\zeta r}\\
3p_x &= \frac{2}{15}\sqrt{10} \zeta^{7/2}\, r^2\,e^{-\zeta r}\Big[\sqrt{\frac{3}{4\pi}} \frac{x}{r}                \Big] =  \sqrt{\frac{2\zeta^7}{15 \pi}} \,r\, x\, e^{-\zeta r} \\
3p_y &= \frac{2}{15}\sqrt{10} \zeta^{7/2}\, r^2\,e^{-\zeta r}\Big[\sqrt{\frac{3}{4\pi}} \frac{y}{r}                \Big] =  \sqrt{\frac{2\zeta^7}{15 \pi}} \,r\, y\, e^{-\zeta r} \\
3p_z &= \frac{2}{15}\sqrt{10} \zeta^{7/2}\, r^2\,e^{-\zeta r}\Big[\sqrt{\frac{3}{4\pi}} \frac{z}{r}                \Big] =  \sqrt{\frac{2\zeta^7}{15 \pi}} \,r\, z\, e^{-\zeta r} \\
3d_{z^2} &= \frac{2}{15}\sqrt{10} \zeta^{7/2}\, r^2\,e^{-\zeta r} \Big[\sqrt{\frac{5}{4\pi}} \frac{3z^2-r^2}{2r^2}\Big] = \frac{1}{3}\sqrt{\frac{\zeta^7}{2\pi}}e^{-\zeta r} (3z^2-r^2) \\
3d_{xz} &= \frac{2}{15}\sqrt{10} \zeta^{7/2}\, r^2\,e^{-\zeta r} \Big[\sqrt{\frac{5}{4\pi}}\sqrt{3}\frac{xz}{r^2}\Big] = \sqrt{\frac{2\zeta^7}{3\pi}} \,xz\,e^{-\zeta r} \\
3d_{yz} &= \frac{2}{15}\sqrt{10} \zeta^{7/2}\, r^2\,e^{-\zeta r} \Big[\sqrt{\frac{5}{4\pi}}\sqrt{3}\frac{yz}{r^2}\Big] = \sqrt{\frac{2\zeta^7}{3\pi}} \,yz\,e^{-\zeta r} \\
3d_{xy} &= \frac{2}{15}\sqrt{10} \zeta^{7/2}\, r^2\,e^{-\zeta r} \Big[\sqrt{\frac{5}{4\pi}}\frac{2}{3}\sqrt{3}\frac{xy}{r^2}\Big] = \frac{4}{3}\sqrt{\frac{2\zeta^7}{15\pi}} \,xy\,e^{-\zeta r} \\
\end{align}
</math>
==Reference==
==Reference==
<references />
<references />

Revision as of 04:58, 9 October 2007

Slater-type orbitals (STOs) 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[1].

STOs have the following radial part:

where

n is a natural number that plays the role of principal quantum number, n = 1,2,...,
N is a normalization constant,
r is the distance of the electron from the atomic nucleus, and
is a constant related to the effective charge of the nucleus, the nuclear charge being partly shielded by electrons.

The normalization constant is computed from the integral

Hence

It is common to use the real form of spherical harmonics as the angular part of the Slater orbital. A list of cartesian real spherical harmonics is given in this article.

The first few Slater type orbitals are (where we use s for l = 0, p for l = 1 and d for l = 2):

Reference

  1. J.C. Slater, Atomic Shielding Constants, Phys. Rev. vol. 36, p. 57 (1930)