Probing Collins Conjecture with correlation energies and entanglement entropies for the ground and excited states in the helium iso‐electronic sequence

In the present work, we present an investigation of Collins Conjecture, a hypothesis made by D. M. Collins in 1993 relating correlation energy and entanglement entropy, by calculating the ground state and singly‐excited triplet‐spin 1s2s 3S and 1s3s 3S state energies of the helium iso‐electronic sequence, with Z = 2–15. By using extensive orthonormal configuration interaction (CI) type wave functions with B‐Spline basis up to about 6000 terms, linear entropy and von Neumann entropy for the abovementioned atomic systems are determined. Together with the Hartree‐Fock energies obtained following a self‐consistent field theory, we have found that there exist linearly proportionalities between the renormalized correlation energies and entanglement entropies of both linear and von Neumann, showing a support for Collins Conjecture applicable to the ground and singly‐excited triplet‐spin states in the helium sequence, for a range of finite Z‐values. Correlation energy versus entanglement entropies for the excited 1s2s 3S and 1s3s 3S states in helium iso‐electronic sequency, with Z = 2–15, showing the existence of a proportionality between correlation energy and entanglement entropy for such states of the helium sequence. The squares, triangles are for SL and SvN of the 1s2s 3S states, respectively, and inverted triangles and tiled squares are for SL and SvN of the 1s3s 3S states, respectively.