If we integrate over the spin variables (, ), we get the spatial probability density function of the two electrons (i.e. the probability of finding one electron at and the other at whatever may the spin angular momenta of the electrons are)
(3)
To arrive at , we have used the usual orthonormality conditions for the spin functions, and .
Equation immediately tells us that is a product of two spatial probability density functions and which are mutually independent. That is, the probability of finding electron 1 at is uncorrelated with the probability of finding electron 2 at . But that is clearly unphysical. Electrons are negatively charged and repel each other by Coulomb forces (repulsion energy=). They would naturally try to avoid each other and not be at the same point in space (). They would exclude a certain volume of space around each of them (due to repulsion) where the probability of finding the other would be small. In other words the probability density would be correlated. The two-electron HF spatial probability density is, on the other hand, totally uncorrelated a property that arises clearly from the single determinant approximation used in the HF description. We suspect therefore the neglect of electron correlation (more precisely Coulomb correlation) in the HF description is at the root of the inadequacy of HF wavefunction in describing the real atoms and molecules. We note here that vanishes when which means two electrons having the same spin cannot be at same point (r) in space. The HF description has Fermi or spin correlation built into the wave function,, but Coulomb correlation is missing.
Post-Hartree Fock methods attempt to take care of the deficiency by switching over to many determinant description of the state being probed so that the (?hitherto) neglected (Coulomb) correlation appears in the spatial probability density function. We would now examine the issue thoroughly and find out what can be done to improve the quality of the wave function that the HF method provides.
, 
), we get the spatial probability density function 
of the two electrons (i.e. the probability of finding one electron at 
and the other at 
whatever may the spin angular momenta of the electrons are) 
, we have used the usual orthonormality conditions for the spin functions, 
and 
.
immediately tells us that 
is a product of two spatial probability density functions 
and 
which are mutually independent. That is, the probability of finding electron 1 at 
is uncorrelated with the probability of finding electron 2 at 
. But that is clearly unphysical. Electrons are negatively charged and repel each other by Coulomb forces (repulsion energy=
). They would naturally try to avoid each other and not be at the same point in space (
). They would exclude a certain volume of space around each of them (due to repulsion) where the probability of finding the other would be small. In other words the probability density would be correlated. The two-electron HF spatial probability density is, on the other hand, totally uncorrelated 
a property that arises clearly from the single determinant approximation used in the HF description. We suspect therefore the neglect of electron correlation (more precisely Coulomb correlation) in the HF description is at the root of the inadequacy of HF wavefunction in describing the real atoms and molecules. We note here that 
vanishes when 
which means two electrons having the same spin cannot be at same point (r) in space. The HF description has Fermi or spin correlation built into the wave function,, but Coulomb correlation is missing.