  In this article, we explore the construction of Hamiltonians with long-range
interactions and their corrections using the short-range behavior of the wave
function. A key aspect of our investigation is the examination of the
one-particle potential, kept constant in our previous work, and the effects of
its optimization on the adiabatic connection.
  Our methodology involves the use of a parameter-dependent potential dependent
on a single parameter to facilitate practical computations. We analyze the
energy errors and densities in a two-electron system (harmonium) under various
conditions, employing different confinement potentials and interaction
parameters. The study reveals that while the mean-field potential improves the
expectation value of the physical Hamiltonian, it does not necessarily improve
the energy of the system within the bounds of chemical accuracy.
  We also delve into the impact of density variations in adiabatic connections,
challenging the common assumption that a mean field improves results. Our
findings indicate that as long as energy errors remain within chemical
accuracy, the mean field does not significantly outperform a bare potential.
This observation is attributed to the effectiveness of corrections based on the
short-range behavior of the wave function, a universal characteristic that
diminishes the distinction between using a mean field or not.
