An antisymmetriser is an operator that makes a wavefunction antisymmetric.
Consider a system of two-electrons, each of which is described by a spin-orbital.
Question
What is the difference between an orbital and a spin-orbital?
Answer
An orbital is a one-electron spatial wavefunction , i.e. a function that gives the best estimated energy eigenvalue of an electron when operated on by the Hamiltonian. For a given spatial orbital
, we can form two spin-orbitals
and
, where
and
are spin functions. We say that the spatial orbital is doubly occupied by two electrons with the same energy. The abbreviated symbol of a spin-orbital is
, where
represents the set of all 4 coordinates (3 spatial coordinates and 1 spin coordinate) associated with the electron.
The symmetric and antisymmetric forms of the total wavefunction of the system are and
respectively. We have
where
-
-
-
.
-
is the identity operator.
, like the exchange operator, swaps the labels of any two identical particles, i.e.
-
-
Eq59 implies that the antisymmetriser transforms the symmetric form of the wavefunction into a linear combination of states, each with a distinct permutation of spin-orbital labels.
We can also expressed the antisymmetriser as , where an even (odd) numbered r represents an even (odd) number of times the labels are permutated. When the antisymmetriser acts on the wavefunction of an n-electron system,
, we would expect the linear combination to have
terms since there are
ways to permutate the labels. The antisymmetriser is therefore:
where N is the normalisation constant.
To evaluate N, we have
Due to the orthonormal property of , we find, after expanding the product of the linear combinations of
and its complex conjugate, that the integrals result in
terms of unity. Hence,
Substitute eq61 in eq60,
