Transcorrelated method for electronic systems coupled with variational Monte Carlo calculation

A Jastrow–Slater-type wave function is often used as a trial function for precise calculations of the total energy of electronic systems, where the correlation effect is taken into account by the Jastrow factor that directly depends on the distance between electrons. Since many-body integrals are inevitable there, the calculation totally depends on Monte Carlo sampling, and so, except for very simple cases, it is very difficult to optimize one-body wave functions in the Slater determinant which determine the nodal surfaces of the total wave function. Here we propose and demonstrate that the total wave function is efficiently optimized by coupling an ordinary variational Monte Carlo (VMC) technique with the transcorrelated method, in which the one-body wave functions are definitely obtained by solving Hartree–Fock-type self-consistent-field (SCF) equations derived from the similarity-transformed Hamiltonian. It is shown that the present method reproduces about 90% of the correlation energy for helium-like two-electron systems $({\mathrm{H}}^{-},$ He, ${\mathrm{Li}}^{+},$ and ${\mathrm{Be}}^{\text{2+}})$ and gives much better results than the conventional VMC method using the Hartree–Fock orbitals for a Li atom, a Be atom, and a ${\mathrm{H}}_2$ molecule. It is also shown that the orbital energy appearing in the SCF equations gives a good approximation to the ionization potential.