The concept of p-orthogonality between electronic states, which generalizes common orthogonality and strong orthogonality, provides a natural hierarchy for group function methods such as the electronic mean field configuration interaction method. In this letter, this theoretical concept is applied in numerical calculations for the first time. The accuracy of the geminal mean field configuration interaction wave functions of simple molecular systems is studied as the orthogonality constraint between the geminals relaxes from 1-orthogonality (that is strong orthogonality) to 2-orthogonality, to no orthogonality constraint at all.