Comment on “Theoretical examination of QED Hamiltonian in relativistic molecular orbital theory” [J. Chem. Phys. 159, 054105 (2023)]

In a recent paper (called Paper I hereafter), 1 Inoue and coworkers sifted in total nine quantum electrodynamics (QED) Hamiltonians resulting from combinations of three types of contractions of fermion operators [constantly null contraction (CNC), 1 chargeconjugated contraction (CCC), 2 and conventional contraction (cC); vide post] and three representations of the vacuum [free-particle orbitals (FPO), Furry orbitals (FO), and molecular orbitals (MO)], based on four criteria (orbital rotation invariance, charge conjugation invariance, time reversal invariance, and nonrelativistic limit).The term "nonrelativistic limit" (nrl) means here that, in the limit of the infinite speed of light, a correct QED Hamiltonian should agree with the nonrelativistic one for a system composed of both electrons and (real) positrons.Their conclusion was that only the MO-CNC variant (H QED(MO-CNC) n) of the nine QED Hamiltonians, along with the MOs that give a stationary point of total energy and a counter term that suppresses divergence, is free of internal inconsistence and is hence the recommended QED Hamiltonian.However, the H QED(MO-CNC) nHamiltonian [also called Fock-space (FS) Hamiltonian 3 ] misses by construction the leading QED effect [vacuum polarization (VP) and electron self-energy (ESE)], at variance with the complete H QED(MO-CCC) n