Density Functional Theory for the Response of Periodic Systems to Electric Fields Based on the Vector Potential Approach

The electronic and nuclear (structural/vibrational) response of 1D-3D nanoscale systems to electric fields gives rise to a host of optical, mechanical, spectral, etc. properties that are of high theoretical and applied interest. Due to the computational difficulty of treating such large systems it is convenient to model them as infinite and periodic (at least, in first approximation). The fundamental theoretical/computational problem in doing so is that the position operator, normally employed to describe the interaction with the field, breaks translational symmetry and is also unbounded. Several solutions have been suggested. Our own approach is to replace the usual scalar interaction potential with the time-dependent vector potential, which results in the so-called vector potential approach (VPA). Besides restoring translational symmetry, this approach has the advantage that static and dynamic fields fall within the same framework. This talk will focus on the recent development and implementation of the VPA for computations carried out by means of Kohn-Sham density functional theory (KS-DFT). A self-consistent field equation for the KS-DFT crystal orbitals will be presented and it will be shown how the phases of the crystal orbitals, as a function of wave-vector, are related to surface charges. With appropriate modifications the calculation of electronic linear and nonlinear optical properties, as well as excitation energies, will be seen to follow along the same general lines as for ordinary molecules. Efficient treatments of structural and vibrational response properties such as infrared intensities, Raman intensities, vibrational contributions to static (hyper)polarizabilities and to dynamic nonlinear optical processes, have also been formulated. All of these developments are now implemented in the CRYSTAL code, which utilizes Gaussian-type orbitals and accounts for exact exchange in a straightforward manner. A number of calculations done with this code will be shown for illustrative purposes.