Assess the neglected higher-order pTC Hamiltonian terms

Determine the significance of the quadratic three-body term \(\hat{\mathcal H}_{\rm 3q}\) and the four-body term \(\hat{\mathcal H}_{\rm 4}\) in the projective transcorrelated Hamiltonian, particularly in small orbital spaces where these unexamined contributions may not be negligible.

Background

The projective transcorrelated Hamiltonian contains, in addition to the terms retained in the study, a three-body contribution quadratic in the transcorrelation generator, H^3q\hat{\mathcal H}_{\rm 3q}, and a four-body contribution, H^4\hat{\mathcal H}_{\rm 4}. The paper neglects these terms because they are third-order contributions and focuses on a Hamiltonian truncated after the terms linear in the generator.

The authors explicitly identify the treatment of these terms as unresolved and suggest that their importance may be especially pronounced in small orbital spaces. Establishing their numerical impact is necessary to determine the accuracy and robustness of the truncated projective transcorrelation model beyond the approximation examined in the paper.

References

Several questions still remain. The terms \hat{\mathcal H}{\rm 3q} and \hat{\mathcal H}{\rm 4} of Eq.~eq:hptc, which have not been examined, are unlikely to be negligible in a small orbital space.

— Construction of downfolded Hamiltonians from projective transcorrelation  (2609.08087 - Ten-no, 8 Sep 2026) in Section Conclusions

We will leave it to future investigations to determine whether this error has significant effect on properties and should be corrected.

— Examination of Cluster Based Methods to Fermionic Systems  (2609.08773 - Keyes et al., 8 Sep 2026) in Section 5, Results and Discussion, paragraph following Table 1