A numerical method was developed to solve the Schrödinger equation in the integral form referred to as Alternative Variational Monte Carlo. This method combines differentiation techniques, numerical integration and Schmidt orthonormalization to calculate fundamental and excited states of quantum systems. Previously, the method was applied successfully in simple systems as the particle in the box, confined harmonic oscillator, among others. This project aims at the application of the methodology in multielectronic atomic systems. Applying the method to the hydrogen atom in three-dimensional space represented by cartesian coordinates were obtained relative errors of 0.2% for the ground state and 1.1% for the first excited state regarding the exact results. In order to reduce the associated errors of the energy and minimize computational effort, we focus on the solution of the radial wave functions for the hydrogen atom and use a discretization based on a q-exponential distribution. This combination generated energy errors of 0.0006%. The method was adapted for many-body problems considering an open and closed-shell restricted Hartree-Fock approximation for atoms. Applying this approximation to He, Li and Be the ground state energies were obtained with relative errors of the order of 0.000003%, 0.000002% and 0.00000001%, respectively. The same methodology can be applied successfully in other atoms and for excited states showing excellent numerical performance.