Slater-type orbitals

Slater-type orbitals (STO) are mathematical functions that resemble hydrogenic wavefunctions. They were introduced by John Slater in 1930 and are used as trial wavefunctions in computational chemistry to approximate energies of systems. The general form of STOs is:

$\psi(r,\theta,\phi)=Nr^{n-1}e^{-\zeta r}Y_{l,ml}(\theta,\phi)\;\;\;\;\;\;\;\;(26)$

where

N is a normalisation constant

n, $l$ and $m_l$ are quantum numbers

$\zeta$ is the effective charge of the nucleus

$Y_{l,ml}(\theta,\phi)$ represents the spherical harmonics

STOs with different n but the same $l$ and $m_l$ are not orthogonal to one another. When working with such STOs, the Gram-Schmidt process is used to convert the non-orthogonal orbitals to orthogonal ones.

Question

Show that the normalisation constant of a 1s orbital is $\sqrt\frac{\zeta^3}{\pi}$ .

Answer

$N^2\int_0^{2\pi}d\phi\int_0^{\pi}sin\theta d\theta\int_0^{\infty}e^{-2\zeta r}r^2dr=1$
Using the identity $\int_0^{\infty}x^ne^{-\alpha x}dx=\frac{n!}{\alpha^{n+1}}$ , we have $\sqrt\frac{\zeta^3}{\pi}$ .

Question

Using the identity $\int_0^{\infty}x^ne^{-\alpha x}dx=\frac{n!}{\alpha^{n+1}}$ , show that the 1s orbitals are orthogonal to each other but not orthogonal to the 2s orbital.

Answer

$\frac{\zeta_1^3}{\pi}\int e^{-\zeta_1r}e^{-\zeta_1r}d\tau=1$
$\sqrt\frac{\zeta_1^3\zeta_3^5}{3\pi^2}\int e^{-\zeta_1r}re^{-\zeta_3r}d\tau=\frac{\sqrt{192\zeta_1^3\zeta_3^5}}{(\zeta_1+\zeta_3)^4}$

For the orbitals to be orthogonal to each other, either $\zeta_1$ or $\zeta_3$ needs to be zero, which is forbidden because the effective nuclear charge is not zero. Hence, the orbitals are not orthogonal to each other.

Slater-type orbitals

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