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.



What is the difference between an orbital and a spin-orbital?


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


      1.  .
      2.   is the identity operator.
      3. , 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,


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