## Calculation Of The Electronic Properties Of Quantum Drops

The current and emerging technologies for the production of these polymer nanoparticles generally result in the quantum drop having a residual static electric charge. The polarity of the residual charge may be either positive or negative, however, for the purposes of this contribution the residual charge will be taken to consist of an excess of electrons confined within the polymer nanoparticle. The computation of the electronic properties of quantum drops involves modeling a large number of atoms and electrons with no periodic behavior in any direction. To solve this problem we choose to implement a semi-classical calculation of the electronic states of polymer quantum drops, as has recently been applied to semiconductor quantum dots4-9 We examine a model Hamiltonian that includes the effect of electric and magnetic field, but which neglects the role of the electron spin.

It is clear from classical electrodynamics that the excess electrons would find their equilibrium configuration at the surface of the polymer quantum drop. For any deviation in the radial direction away from the surface of the drop, the electron would experience a restoring force that could be represented in the form of an image charge located a similar distance from the surface as the electron in the opposite radial direction. This classical electrodynamic model would provide the forces under which a single electron on the surface of a quantum drop would move. For the situation in which there is more than one electron, we need to take into account the electron-electron repulsion as well as the restoring force. We have developed a Hamiltonian that allows us to treat both the mutual repulsion of the electrons and, in an approximate way, the confinement of the electrons near the surface of the polymer quantum drop. The semi-classical energy spectrum of the system can be generated by a Poincare surface-of-section technique!10-13 Inclusion of electric and magnetic fields in the model Hamiltonian gives a first view into the unique electronic properties of quantum drops.

Owing to the interaction between the electron and the substrate, the effective mass of the electron in a semiconductor quantum dot is less than 10% of the actual mass of the electron.4 We do not know the extent to which this substrate-electron interaction changes the electron's effective mass in the polymer quantum drop. In any event, the effective mass of the electron will be a strong function of the material that composes the quantum drop. Hence, in developing the model Hamiltonian, we have chosen to take the effective mass of the electron on our polymer quantum drops to be the actual mass of the electron. A second concern that must be addressed in semiconductor q uantum dots is the exchange interaction among electrons confined in the same potential 14-16 . In a polymer quantum drop, each electron is confined by its own image, so that the overlap between electron orbitals is seen to be quite small and the interaction of the electrons is appropriately modeled as the Coulombic interaction between electrons.

In this investigation, the model Hamiltonian is designed to describe the interaction of n electrons when the electrons are confined to the neighborhood of the surface of a sphere. We begin with the minimum energy configuration of the electrons on the sphere surface and then fix all of the electrons to the minimum energy configuration at the surface of the sphere, except for one. The lone remaining electron, the active electron, is allowed to move under the influence of the fixed electron(s). The Hamiltonian that determines the motion of the active electron is then:

! ^Jggj where n is the canonical momentum of the active electron in the presence of a magnetic field:

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