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:

where

*N* is a normalisation constant

*n*, and are quantum numbers

is the distance between an electron and the nucleus of an atom.

is the effective charge of the nucleus

represents the spherical harmonics

STOs with different *n* but the same and 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 1*s* orbital is .

###### Answer

Using the identity , we have .

###### Question

Using the identity (see this article for proof), show that the 1*s* orbitals are orthogonal to each other but not orthogonal to the 2*s* orbital.

###### Answer

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