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