We define a creation operator by its action on a Slater determinant as
Thus, 
creates an electron in spin-orbital 
. The order
of creation operators is important since
Thus, the anticommutation relation for and 
is
This means that if we swap the order of two creation operators, we must change the sign of the resulting expression. Similarly, an annihilation operator is defined as
The anticommutation relation for annihilation operators is
In addition, it may be shown that
For completeness, we note that
and
That is, we cannot create an electron in a spin-orbital already containing an electron, and we cannot annihilate an electron in a spin-orbital not containing an electron.
