We investigate the possibility of using a transcorrelated (TC) Hamiltonian to describe electron correlation. A method to obtain TC wavefunctions was developed based on the mathematical framework of the bi-variational principle. This involves the construction of an effective TC Hamiltonian matrix, which can be solved in a self-consistent manner. This was optimized using a method we call second-order-moment minimization and demonstrate that it is possible to obtain highly accurate energies for some closed-shell atoms and helium-like ions. The effects of certain correlator terms on the description of electron-electron and electron-nuclear cusps were also examined graphically, and some TC wavefunctions were compared against near-exact Hylleraas wavefunctions.