Geminal Product Wavefunctions: A General Formalism

We consider the construction of a wavefunction composed of $N_{\beta }$ singlet coupled geminals and $N_{\alpha }{\mathrm{\hspace{0.17em}-\hspace{0.17em}N}}_{\beta }\alpha$ spin orbitals where $N_{\alpha }{\mathrm{\hspace{0.17em}+\hspace{0.17em}N}}_{\beta }$ is the total number of electrons. An orthonormal basis $\mathrm{\{\chi \}}$ is introduced and each geminal and orbital is expanded in this basis. The wavefunction is expressed as a linear combination of determinants over the $\mathrm{\{\chi \}}$ basis and the energy expression is obtained. It is suggested that the geminal and orbital expansion coefficients be obtained using a numerical minimization technique. As an example we construct an APG function for LiH and BH (using only $\sigma$ orbitals) and compare the energies obtained with this model to the corresponding MO–SCF, APSG, and CI results over the same $\mathrm{(\sigma )}$ basis. We find that the APG function recovers 99.9% and 98.0% of the energy difference between the CI and MO–SCF, while our APSG function recovers 96.6% and 80% for these two molecules. Since most pair function models of electronic structure may be viewed as special cases of the parent APG model, this approach includes these models and in addition allows one to construct wavefunctions within a unified formalism in which different pairs in a given molecule may be treated at different levels of accuracy.