Seniority-zero Quadratic Canonical Transformation Theory

We propose a method to solve the Schr\"odinger equation for systems with static/strong electron correlation using Hamiltonian transformations. Building on our previous work on seniority-zero canonical transformation theory, which seeks a unitary transformation that maps the Hamiltonian into the seniority-zero space, this method presents an alternative way of evaluating the Baker--Campbell--Hausdorff (BCH) expansion based on quadratic canonical transformation theory. The extension aims to relax the small-generator constraint by allowing approximate four-body contributions in the expansion, thus expanding the class of excitations previously allowed in SZ-LCT, where only approximate three-body operators were retained. Numerical tests reveal that the seniority-zero quadratic canonical transformation method (SZ-QCT) delivers good accuracy, with most errors within chemical accuracy. In particular, SZ-QCT shows sub-millihartree errors in cases where larger generators are needed to recover the residual dynamic correlation. The computational scaling of SZ-QCT is the same as that of SZ-LCT, $\mathcal{O}(N^8/n_c)$, where $n_c$ is the number of cores available for the computation