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This paper introduces a second-derivative-corrected extension of the Flexible Ansatz for N-body Perturbation Theory (FANPT) to enhance the robustness of solving nonlinear wavefunction equations. By retaining the overlap Hessian and neglecting only higher-order derivatives, the authors achieve a more accurate initial guess for the projected FANCI equations, particularly beneficial for larger $\lambda$-steps. Testing on Lithium Hydride and Beryllium in H$_2$ demonstrates that this correction significantly reduces parameter deviations and improves the reliability of FANPT as a continuation strategy.
The second-derivative correction to FANPT leads to a more reliable continuation strategy for nonlinear wavefunction equations, reducing parameter deviations by leveraging the overlap Hessian.
We present a second-derivative-corrected extension of the Flexible Ansatz for N-body Perturbation Theory (FANPT) for solving nonlinear Flexible Ansatz for N-body Configuration Interaction (FANCI) wavefunction equations. The original quasilinear FANPT approximation neglects second- and higher-order derivatives of the determinant overlap with respect to wavefunction parameters. In this work, we retain the overlap Hessian and neglect only third- and higher-order parameter derivatives, thereby including the leading nonlinear response of the wavefunction ansatz while preserving the same response-matrix structure used in the original FANPT formulation. The resulting additional terms enter only through the constant vector of the response equations and are implemented for coupled-cluster wavefunctions in the FanPy/FANCI framework. The new approximation is tested on the Lithium Hydride molecule and the insertion of Beryllium into H$_2$ using seniority-restricted coupled-cluster wavefunctions in the STO-6G basis. The main advantage of the second-derivative correction is that it provides a better initial guess for solving the projected FANCI equations at the next point along the adiabatic connection. This improvement is diagnosed by comparing the FANPT-propagated parameters with the independently optimized FANCI parameters at the same value of $\lambda$. For nonlinear coupled-cluster ansatzes, especially when larger $\lambda$-steps are used, the corrected approximation reduces the same-$\lambda$ parameter deviations and suppresses large parameter-space excursions. These results show that the leading nonlinear overlap correction improves the reliability of FANPT as a continuation strategy for nonlinear wavefunction equations.
