  The present work introduces a new form of explicitly correlated factor in the
context of the transcorrelated methods. The new correlation factor is obtained
from the r 12 $\approx$ 0 mathematical analysis of the transcorrelated
Hamiltonian, and its analytical form is obtained such that the leading order in
1/r 12 of the scalar part of the effective two-electron potential reproduces
the long-range interaction of the range-separated density functional theory.
The resulting correlation factor exactly imposes the cusp and is tuned by a
unique parameter $\mu$ which controls both the depth of the coulomb hole and
its typical range in r 12. The transcorrelated Hamiltonian obtained with such a
new correlation factor has a straightforward analytical expression depending on
the same parameter $\mu$, and its physical contents continuously change by
varying $\mu$ : one can change from a non divergent repulsive Hamiltonian at
large $\mu$ to a purely attractive one at small $\mu$. We investigate the
convergence of the ground state eigenvalues and right-eigenvectors of such new
transcorrelated Hamiltonian as a function of the basis set and as a function of
$\mu$ on a series of two-electron systems. We found that the convergence
towards the complete basis set is much faster for a quite wide range values of
$\mu$. We also propose a specific value of $\mu$ which essentially reproduce
the results obtained with the frozen Gaussian geminal introduced by Ten-No
[CPL-330,169 (2000)].
