Slater Determinants and Fermi Correlation

The Hartree product wavefunction described in equation 5.1 can easily be antisymmetrised  

The form of the wavefunction given in equation 5.3 is known as a Slater determinant [12] and can be generalised for an N-electron wavefunction, giving 

This form of the many-electron wavefunction, satisfying the antisymmetry principle, exhibits correlation between electrons of like spin. To see this, we take the two-electron wavefunction given in equation 5.3. For two electrons of unlike spin, i.e., 

For two electrons of like spin, the integration over spin degrees of freedom produces a different probability distribution. Setting 

Using equation 5.10, P(n, ri) = 0, demonstrating that the motion of the like-spin electrons is correlated and, for a given electron, the probability of finding another like-spin electron in its immediate vicinity is lower the that of an unlike-spin electron. This reduction in probability is known as a Fermi hole, and the correlation is known as Fermi or exchange correlation. 

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