### MATHEMATICAL MODELLING OF TECHNOLOGICAL PROCESSES AND SYSTEMS

## A VARIATIONAL SOLUTION OF THE SCHRÖDINGER EQUATIONS IN AN INHOMOGENEOUS СOULOMB FIELD

The present work is devoted to computer modeling of the emission processes from the surface of graphene. The pivotal obstacle for emission is a model of the unperturbed emission surface. The hydrogen-like atom model is one of the useful approaches describing the states of the emission surface. In [1] this model was used considering ion screening in the Brandt model [2]. To calculate the ground state of the electron, we used the variational solution of the Schrodinger equation, based on the minimization of the potential energy of an electron in the field of a homogeneous ion. However, the field of the screened singly ionized carbon atom in the Brandt model is not homogeneous. Therefore, it was shown in [3] that it is possible to obtain a binding energy error of up to 40% when using only the external screening parameter without taking into account the inhomogeneity. In this paper, we consider the effect of the ion screening parameter in the Brandt model λ and the algorithm for determining it by minimizing the total energy of the electron interaction in s state in two parameters: the effective ion charge and the ion screening parameter. The obtained solution of the Schrödinger equation is used to calculate the ground state of a hydrogen-like carbon atom in a graphene lattice at zero temperature and is compared with the results of [2, 4].