Graphical Approach to Interpreting and Efficiently Evaluating Geminal Wavefunctions

We consider wavefunctions built from antisymmetrized products of two-electron wavefunctions (geminals), which is arguably the simplest extension of the antisymmetrized product of one-electron wavefunctions (orbitals) (i.e., a Slater determinant). Extensive use of geminals in wavefunctions has been limited by their high cost stemming from the many combinations of the two-electron basis functions (orbital pairs) used to build the geminals. When evaluating the overlap of the APG wavefunction with an orthogonal Slater determinant, this cost can be interpreted as the cost of evaluating the permanent, resulting from the symmetry with respect to the interchange of orbital pairs, and the cost of assigning the occupied orbitals to the orbital pairs of the wavefunction. Focusing on the latter, we present a graphical interpretation of the Slater determinant and utilize the maximum weighted matching algorithm to estimate the combination of orbital pairs with the largest contribution to the overlap. Then, the cost due to partitioning the occupied orbitals in the overlap is reduced from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝒪</mml:mi> <mml:mo>(</mml:mo> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>!</mml:mo> <mml:mo>!</mml:mo> <mml:mo>)</mml:mo></mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝒪</mml:mi> <mml:mo>(</mml:mo> <mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow> <mml:mrow><mml:mn>3</mml:mn></mml:mrow> </mml:msup> <mml:mi>log</mml:mi> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> . Computational results show that many of these combinations are not necessary to obtain an accurate solution to the wavefunction. Because the APG wavefunction is the most general of the geminal wavefunctions, this approach can be applied to any of the simpler geminal wavefunction ansätze. In fact, this approach may even be extended to generalized quasiparticle wavefunctions, opening the door to tractable wavefunctions built using components of arbitrary numbers of electrons, not just two electrons.