Full configuration interaction exact solution of approximate problem

Full configuration interaction exact solution of approximate problem 

At this point we are sufficiently equipped to consider briefly the methods used to approximate the wave functions constructed in the restricted subspace of orbitals. So far the only approximation was to restrict the orbital basis set. It is convenient to establish something that might be considered to be the exact solution of the electronic structure problem in this setting. This is the full configuration interaction (FCI) solution. In order to find one it is necessary to construct all possible Slater determinants for N electrons allowed in the basis of 2M spin-orbitals. In this context each Slater determinant bears the name of a basis configuration and constructing them all means that we have their full set. Then the matrix representation of the Hamiltonian in the basis of the configurations ( K is constructed  

The FCI approach, if one had decided to actually use it to study any realistic problem, would require enormous computing power since the dimensionality of the FCI problem increases factorially with the increase of the size of the system for N electrons in 2M spin-orbitals the number of basis configurations amounts to  

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