Variational Monte Carlo Method with Dirichlet Boundary Conditions: Application to the Study of Confined Systems by...

Variational Monte Carlo method is a powerful tool to determine approximate wave functions of atoms, molecules, and solids up to relatively large systems. In the present work, we extend the variational Monte Carlo approach to study confined systems. Important properties of the atoms, such as the spatial distribution of the electronic charge, the energy levels, or the filling of electronic shells, are modified under confinement. An expression of the energy very similar to the estimator used for free systems is derived. This opens the possibility to study confined systems with little changes in the solution of the corresponding free systems. This is illustrated by the study of helium atom in its ground state (1)S and the first (3)S excited state confined by spherical, cylindrical, and plane impenetrable surfaces. The average interelectronic distances are also calculated. They decrease in general when the confinement is stronger; however, it is seen that they present a minimum for excited states under confinement by open surfaces (cylindrical, planes) around the radii values corresponding to ionization. The ground (2)S and the first (2)P and (2)D excited states of the lithium atom are calculated under spherical constraints for different confinement radii. A crossing between the (2)S and (2)P states is observed around rc = 3 atomic units, illustrating the modification of the atomic energy level under confinement. Finally the carbon atom is studied in the spherical symmetry by using both variational and diffusion Monte Carlo methods. It is shown that the hybridized state sp(3) becomes lower in energy than the ground state (3)P due to a modification and a mixing of the atomic orbitals s, p under strong confinement. This result suggests a model, at least of pedagogical interest, to interpret the basic properties of carbon atom in chemistry.