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We show that the central finite difference formula for the first and the second derivative of a function can be derived, in the context of quantum mechanics, as matrix elements of the momentum and kinetic energy operators on discrete coordinate eigenkets |xn〉 defined on a uniform grid. Starting from the discretization of integrals involving canonical commutations, simple closed-form expressions of the matrix elements are obtained. A detailed analysis of the convergence toward the continuum limit with respect to both the grid spacing and the derivative approximation order is presented. It is shown that the convergence from below of the eigenvalues in electronic structure calculations is an intrinsic feature of the finite difference method. © 2018 Wiley Periodicals, Inc.

In this paper we study the electron-electron correlation in silicon anisotropic quantum dots. A variational approach to the problem of the quantum confinement of two electrons in an ellipsoidal quantum dot is presented. All the calculations are done as a function of the dot aspect ratio, showing how the Coulomb repulsion and dielectric effects (related to the dielectric mismatch between the dot and the surrounding medium) give rise to shape-dependent correlation effects. The energy needed to add a second electron to the ellipsoidal quantum dot is calculated and discussed as well.

The single particle spectrum of a charged particle in a nanometric ellipsoidal quantum dot is calculated taking also into account the surface polarization effects. Furthermore the optical properties of the absorption coefficient also depending on the light polarisation are discussed and compared with the experimental data.