Probing Collins conjecture with correlation energies and entanglement entropies for the ground state of the helium isoelectronic sequence

Correlation energy of a quantum system is defined as the difference between its exact energy $E_{\mathrm{ex}}$, and its Hartree-Fock energy $E_{\mathrm{HF}}$. In a recent related development, entanglement measures can be quantified with von Neumann entropy $S_{vN} (\rho )=-Tr(\rho \log_{2} \rho )$ or linear entropy $S_{L} (\rho )=1-Tr(\rho^{2})$, where $\rho $ is the one-particle reduced density matrix, and $Tr(\rho^{2})$ is defined as the purity of state. In the present work we calculate $S_{L}$ and $S_{vN}$ for the ground 1$s^{\mathrm{2\thinspace 1}}S$ states in helium-like ions for $Z=$ 2 to 15, using configuration interaction (CI) with $B$-Spline basis up to about 6000 terms to construct the wave functions, and with which density matrix, linear and von Neumann entropies are calculated [1]. We have found close relationship between the reduced correlation energy, defined as $E_{\mathrm{corr}}=$ ($E_{\mathrm{CI\thinspace }}$-- $E_{\mathrm{HF}})$/$E_{\mathrm{CI}}$ (with $E_{\mathrm{CI}}$ being our calculated energy), and $S_{L}$ or $S_{vN}$. Our results support Collins conjecture [2] that there is a linear relationship between correlation energy and entanglement entropy, i.e., $E_{\mathrm{corr}}=$ \textit{CS}, where $C$ is called Collins constant. Using the calculated ground state energies for $Z=$ 2 to $Z=$ 15, and the entanglement measured with linear entropy $S_{L}$ for such states, $C$ is determined as 0.90716. At the meeting, we will present result for Collins constant determined from von Neumann entropy, and details of our calculations. [1] Y.-C. Lin, C.-Y. Lin, and Y. K. Ho, \textit{Phys. Rev. A} \textbf{87}, 022316 (2013); \textit{Can. J. Phys}. \textbf{93}, 646 (2015). [2] D. M. Collins, \textit{Z. Naturforsch}, \textbf{48, }68 (1993).