If we want to solve
a matrix problem, we need to find a suitable linear vector space. Now
is an N-electron function that must be
antisymmetric with respect to interchange of electronic coordinates.
As we just saw in the previous section, any such N-electron function
can be expressed exactly as a linear combination of Slater
determinants, within the space spanned by the set of orbitals
If we denote our Slater determinant basis
then we can express the eigenvectors as
If we solve this matrix equation, , in the space of all possible Slater determinants as just described, then the procedure is called full configuration-interaction, or full CI. A full CI constitues the exact solution to the time-independent Schrödinger equation within the given space of the spin orbitals . If we restrict the N-electron basis set in some way, then we will solve Schrödinger's equation approximately. The method is then called ``configuration interaction,'' where we have dropped the prefix ``full.'' For more information on configuration interaction, see the lecture notes by the present author  or the review article by Shavitt .