In this work, we report potential energy surfaces (PESs) of the sodium dimer calculated by variational (VMC) and lattice regularized diffusion Monte Carlo (LRDMC). The VMC calculation is accurate for determining the equilibrium distance and the qualitative shape of the experimental PES. Remarkably, after the application of the LRDMC projection to this single determinant ansatz, namely the Jastrow Antisymmetrized Geminal Product (JAGP), chemical accuracy (~ 1kcal/mol) is reached, and the obtained dissociation energy, equilibrium internuclear distance, and harmonic vibrational frequency are in very good agreement with the experimental ones. This outcome crucially depends on the quality of the optimization used to determine the best possible trial function within the chosen ansatz. The strategy adopted in this work is to minimize the variational energy by initializing the trial function with the DFT single determinant ansatz expanded exactly in the same atomic basis used for the corresponding VMC and LRDMC calculations. This atomic basis is ad-hoc reshaped for QMC calculations. Indeed, we multiply the standard Gaussian type atomic orbitals by a one-body Jastrow factor, satisfying in this way the electron-ion cusp conditions. This allows us to use a very small basis almost converged in the complete basis set limit, by reducing the computational effort as well as the statistical fluctuations on the total energy. In order to achieve these important advantages, we have defined a very efficient DFT algorithm in the mentioned basis, by estimating the corresponding matrix elements on a mesh, and by using a much finer mesh grid in the vicinity of nuclei.
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