Hamiltonians and solution techniques

Chapter 1 introduces several of the common principles, techniques and approximations that will be employed throughout the text. Classically, the Hamiltonian is the sum of kinetic energy and potential energy. In quantum mechanics, it is an operator that, by operating on the statefunction, leads to the energy observable. The chapter begins with a preliminary description of the crystal’s Hamiltonian and then introduces approximation techniques that permit useful solutions. Beginning with the simple jellium model, Hartree and Hartree-Fock approaches are developed, exchange interactions and correlation effects are explored, and both time-independent perturbation and time-dependent perturbation techniques discussed. Examples illustrate scattering by perturbation as well as adiabatic evolution. The centrality of fast-and-slow interactions is stressed, the Born-Oppenheimer approximation is illustrated through the configuration coordinate diagram, and interacting electron systems are analyzed. The multi-electron aspects are stressed by discussing static screening, dynamic screening and the meaning of permittivity therein.

How metallic are noble-metal clusters? Static screening and polarizability in quantum-sized silver and gold nanoparticles

Small noble-metal clusters of about 2 nm are strongly metallic, even one atomic shell screens 96% of external fields, whereas electronic shell-closings and Friedel oscillations modify the classical picture.

Properties of many-polaron in fractional dimensional space

Polaron properties such as the binding energy and the effective mass are calculated in the fractional dimensional space approach using the second-order perturbation theory. The effect of carrier density on the static screening correction of the electron–phonon interaction is included in the Hubbard’s local field factor. It is found that the polaron properties decrease with increase in dimension due to quantum confinement. Since the screening of the electron–phonon interaction is enhanced with increase in doping, the polaron properties decrease with increasing density.