On the Thermodynamic Limit for Responses to Static Electromagnetic Fields

For molecular systems, the quantum-mechanical treatment of their responses to static electromagnetic fields usually employs a scalar-potential treatment of the electric field and a vector-potential treatment of the magnetic field. Although the potential for each field separately is associated with the choice of an (unphysical) origin, the precise choice of the origin for the electrostatic field has little consequences for the results. This is different for the magnetic field where the introduction of so called Gauge-Invariant Atomic Orbitals (GIAOs) poses one approach for removing the origin-dependence of the responses. For large, extended, regular systems (consisting of a large number of periodically repeated, identical units with deviations from this regularity only at the boundaries) additional problems occur. Thus, for an efficient treatment of such systems, it is practice to treat them as being infinite and periodic. However, the presence of the additional potentials from static electromagnetic fields breaks the translational symmetry. Moreover, these potentials are unbounded. In the present contribution we shall start with the large, finite, regular system and for this require that a thermodynamic limit exists also when the system is exposed to a static electromagnetic field. This gives information on the properties of the electronic wavefunctions of the system and on its charge distribution. Subsequently, we study the infinite, periodic system with the requirement that the responses of this are identical to those of the large, finite system in its thermodynamic limit. From this requirement, the Hamilton operator for the periodic system is derived. Properties of the resulting single-particle equations are discussed and results of model calculations presented.