Symmetry Restrictions on the CI Space

and

then

First we show that is an eigenfunction of with eigenvalue

Now apply to

Where we have used the given that . We may now write

Now consider again equation (4.1). If we take the adjoint of this equation we obtain

Now use the fact that (we assumed was Hermitian) and that the eigenvalues of a Hermitian operator are real. This yields

Multiply on the right by

Now multiply equation (4.6) on the left by to obtain

If we subtract equation (4.10) from equation (4.9) we arrive at

Since we assumed , then . Recalling the definition of , we have

which was to be proven. Thus if our desired wavefunction is an eigenfunction of some Hermitian operator that commutes with the Hamiltonian, our CI space need not include those

(tex2html_deferredtex2html_deferred4.tex2html_deferred13S^2 | = S(S+1) | ) |

and any basis function of a different spin can be excluded from the CI. Slater determinants are generally not eigenfunctions of . However, we can take linear combinations of Slater determinants so that we do have eigenfunctions of ; such functions are generally called configuration state functions, or CSF's. The advantage of using CSF's is that we can throw out all functions with the wrong eigenvalue