Hookean systems and Quantum DotsExact solutions of the Schrodinger equation for multiparticle systems with interparticle Coulombic interactions are unknown. One can try to find such exact solutions by modeling the interparticle interaction potential. Within this context, a much studied model is the Hookean two-electron atom, a system possessing a nucleus with charge +2 interacting through a harmonic potential with the electrons which, in turn, repel each other through the usual Coulomb interaction.

The replacement of the Coulombic central confining electron-nucleus potential occurring in real systems by a harmonic potential makes the problem separable in terms of center-of-mass and relative coordinates. For the relative motion, the ensuing equation has analytic solutions only for discrete values of the harmonic confinement strength parameter. We have extended these models to  solve exactly a general three-body problem depending on the relation of masses and force constant of the harmonic potential. In addition, we have also developt quasi-exact solutions for the non-Born-Oppenheimer Hookean H2 molecule, in which we model the confining electron–nucleus Coulombic potential with a harmonic potential, while keeping the remaining interparticle interactions. The corresponding electron and nuclei pair densities are calculated from these solutions, and they are used to get insoght various phenomena, such as the relation between the different correlation regimes between electrons and nuclei and the emergence of molecular structure in 3-body systems as a function of the relative masses. In addition, we have observed that the inclusion of the proper Coulomb potential for the electron-electron interaction is the main relevant aspect affecting the dynamics of electronic motion, which pinpoints to the suitability of this toy-system as fas as modelization is concerned.

A particularly interesting  extension and application of these models is the description of the electronic structure in Quantum Dots, in which  a small number of electrons confined artifically using nanodevices can be simulated with harmonic potentials. There are various areas in which Quantum Dots show potential applications, from quantum computing to nano-medicine. However, the characteristics  of the electronic structure of  Quantum Dots is yet not fully understood. In particular, the effect of external magnetic fields on the spin transitions in QD’s, and the treatment of the screening effects caused by the external medium are two aspects of these articial atoms that need a careful investigation. On the other hand, the magnetic coupling of two or more localized QD’s has not been studied yet. For the 2 electron spherical Quantum Dots, we can use our exact solutions developed for the three body problem to understand how single-tripet transition in QD’s are dependent on the magnetic fields. However, the extension of this work to systems containing more than 2 electrons, incorporation of screening effects, and the consideration of multiple QD’s, requires the use of standard approximate quantum chemical methods. For this purpose,  the corresponding one electron confinement integrals over gaussian-type functions, and the one- and two-electron Yukawa integrals have been developed and interfaced with the GAMESS-US program, which opens the possibility to study various aspects of the electronic structure of Quantum Dots.

Please refer to the following publications for more information:

  • E. V. Ludeña, X. Lopez, J. M. Ugalde, Non-Born-Oppenheimer Treatment of the H2 Hookean Molecule, J. Chem. Phys. 123, 024102 (2005).
  • X. Lopez, J. M. Ugalde, E. V. Ludeña, Extracular Densities of the Non–Born–Oppenheimer Hookean H2 Molecule, Chem. Phys. Lett. 412, 381 (2005).
  • E. V. Ludeña, X. Lopez, J. M. Ugalde, Pair Densities for the Hooke and Hooke-Calogero models of the non-Born-Oppenheimer hydrogen molecule, Lecture Series on Computer and Computational Sciences 4, 1217 (2005).
  • X. Lopez, J. M. Ugalde, E. V. Ludeña, Exact non-Born-Oppenheimer wave function for the Hooke-Calogero model of the H2 molecule. An useful exact molecular model for elecrton correlation studies. European Journal of Physics D 37, 351 (2006).
  • X. Lopez, J. M. Ugalde, L. Echevarria, E. V. Ludeña, Exact non-Born-Oppenheimer wave functions for three-particle Hookean systems with arbitrary masses, Phys. Rev. A 74, 042504 (2006).
  • V. V. Karasiev, X. Lopez, J. M. Ugalde, E. V. Ludeña, Kinetic Energy Functionals: Exact Ones From Analytic Model Wave Functions and Appoximate Ones in Orbital-Free Molecular Dynamics, Int. J. Mod. Phys. B 24, 5139 (2010).
  • E. V. Ludeña, L. Echevarria, J. M. Ugalde, X. Lopez, A. Corella-Madueño, Model for a biexciton in a lateral quantum dot based on exact solutions for the Hookean H2 molecule, Int. J. Quantum Chem.  111 1808 (2011).
  • E. V. Ludeña, L. Echevarria, X. Lopez, J. M. Ugalde, Non-Born-Oppenheimer electronic and nuclear densities for a Hooke-Calogero three-particle model: Non-uniqueness of density-derived molecular structure?, J. Chem. Phys. , (2012, in press).