Analysis of two-orbital correlations in wave functions restricted to electron-pair states

Wave functions constructed from electron-pair states can accurately model strong electron correlation effects and are promising approaches especially for larger many-body systems. In this article, we analyze the nature and the type of electron correlation effects that can be captured by wave functions restricted to electron-pair states. We focus on the pair-coupled-cluster doubles (pCCD) ansatz also called the antisymmetric product of the 1-reference orbital geminal (AP1roG) method, combined with an orbital optimization protocol presented in Boguslawski et al. [Phys. Rev. B 89, 201106(R) (2014)], whose performance is assessed against electronic structures obtained form density-matrix renormalization-group reference data. Our numerical analysis covers model systems for strong correlation: the one-dimensional Hubbard model with a periodic boundary condition as well as metallic and molecular hydrogen rings. Specifically, the accuracy of pCCD/AP1roG is benchmarked using the single-orbital entropy, the orbital-pair mutual information, as well as the eigenvalue spectrum of the one-orbital and two-orbital reduced density matrices. Our study indicates that contributions from singly occupied states become important in the strong correlation regime which highlights the limitations of the pCCD/AP1roG method. Furthermore, we examine the effect of orbital rotations within the pCCD/AP1roG model on correlations between orbital pairs.

Figure 1

Orbital-pair mutual information for the half-filled 1D Hubbard model with periodic boundary conditions, 14 sites, and different on-site interaction strengths for the optimized AP1roG basis. The single-orbital entropy is site independent and given below each figure. The strength of the orbital-pair correlations for both the (a) DMRG (left panel) and the (b) AP1roG (right panel) correlation diagrams are color coded: the black lines indicate strong correlations, whereas the green lines indicate weak correlations.

Figure 2

Decaying values of the mutual information for the half-filled Hubbard model with 14 sites using the optimized AP1roG orbital basis (a). $I_{i|j}$ is sorted with respect to the DMRG reference values so that each value of $I_{i|j}$ is shown for the same orbital pair $i$ and $j$ in both DMRG and AP1roG calculations. Eigenvalues of the (b) one-orbital reduced density matrix and (c) two-orbital reduced density matrix for the half-filled Hubbard model with 14 sites obtained in DMRG and AP1roG calculations using the optimized AP1roG orbital basis. The eigenvalues of ${\rho }_{i,j}$ for each pair $i,j$ are ordered as in (a). The red lines and symbols indicate AP1roG data, whereas the blue lines and symbols mark the corresponding DMRG results.