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Confining electrons with Hooke and Yukawa like potentials

The potential energy operator opens a gap and discretizes the energy levels of the otherwise continuous energy spectra of the electronic kinetic energy operator, giving rise to the wonderful and puzzling, in a way, quantum world. Quantum chemistry has dealt, up to now, with the Coulomb nucleus-electron attractive potential energy operator for the confining of the electrons of the systems of interest, namely, atoms and molecules. However, since the recent advent of the so-called quantum dot systems, confining potentials other than the Coulombic one have been seen to represent more precisely the physics of these man-made "artificial" atoms and molecules. We shall discuss some of the salient physical properties of electrons under these "new" confining potentials with an emphasis on their electron and electron-pair densities ​
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