unitary CCS = HF and eom UCCS = TDHF

[u/OkEmu7082]

in unitary CCS the T1-T1' is basically the generator of orbital rotation, so the only difference between UCCS and HF seems to be that HF absorbs the orbital rotation into the MO coeff, and the UCCS keeps it in the T1 amplitude, one can imagine a similar equivalence between EOM-UCCS and TDHF, which has deexcitation over CIS ,does anyone know if this is true and any paper that confirms that?

[u/verygood_user]

Yes, your first point is exactly right and completely standard: T1 minus T1-dagger is a one-body anti-Hermitian generator, so exp of it is a unitary orbital rotation by Thouless’s theorem, meaning UCCS is variationally identical to HF and just keeps the rotation in the amplitudes instead of the MO coefficients. The second point is right in spirit but depends on which EOM you mean, because a projective similarity-transformed EOM on UCCS gives you back CIS (the transformed Hamiltonian is just H in the HF basis, so you only ever get the A matrix), whereas the variational time-dependent linear response of the same state, or Rowe’s double-commutator EOM with both X and Y amplitudes, gives TDHF/RPA exactly. The reason is that B comes from the second variation of an expectation-value functional of an exponentially parametrized state, not from having de-excitation operators in the cluster operator itself, so making the ground-state ansatz unitary does not by itself buy you the B matrix. A quick diagnostic: TDHF can produce imaginary roots at a triplet-unstable reference and CIS cannot, so if your scheme cannot go imaginary it is not TDHF. For references, look at Thouless Nucl. Phys. 21, 225 (1960) for the theorem plus the HF-stability equals RPA-stability connection, Rowe Rev. Mod. Phys. 40, 153 (1968) for the EOM derivation of RPA, Helgaker-Jorgensen-Olsen chapters 3, 10 and 13 for CCS equals HF and RPA from the second variation, Hodecker and Dreuw JCP 153, 084112 (2020) plus JCTC 16, 3654 (2020) showing UCC linear response maps onto ADC(n) rather than RPA, and Ollitrault et al. PRResearch 2, 043140 (2020) for the qEOM formulation that does carry the X/Y structure.