Hartree-Fock implementation using a Laguerre-based wavefunction for the ground-state and correlation energies of two-electron atoms

An implementation of the Hartree-Fock (HF) method using a Laguerre-based wavefunction is described and used to accurately study the ground state of two-electron atoms in the fixed nucleus approximation, and by comparison with fully-correlated (FC) energies, used to determine accurate electron correlation energies. A variational parameter $A$ is included in the wavefunction and is shown to rapidly increase the convergence of the energy. The one-electron integrals are solved by series solution and an analytical form is found for the two-electron integrals. This methodology is used to produce accurate wavefunctions, energies and expectation values for the helium isoelectronic sequence, including at low nuclear charge just prior to electron detachment. Additionally, the critical nuclear charge for binding two electrons within the HF approach is calculated and determined to be $Z_C^{HF}$ = 1.031 177 528