I have been reading through [1] to get a better understanding of geminal-based methods. Some short passages are included below:
The occupation of each orbital in the expansion above (Eq. 5) is indicated by the summation index $m_i$, while the total number of occupied orbitals in each Slater determinant has to equal the number of electron pairs. The APIG wave function thus includes $K\choose P$ determinants ($K$ is number of orbitals, $P$ the number of electron pairs).
The AGP ansatz in eqn (8), a tensor product of the same geminal, represents a computationally attractive correlated wave function with very few parameters; in the most general case, the AGP scales as $\frac{K(K-1)}{2}$ where $K$ is the number of spin-orbitals.
Why does this 2nd paragraph (shortly after Expression (10) in the paper) say that AGP has a complexity of $O(K^2)$, which contradicts its own sentences for the early paragraph after expression (5) about having binomial complexity?
References
- P. Tecmer & K. Boguslawski Phys. Chem. Chem. Phys., 2022,24, 23026-23048 DOI