Linear response properties of nonconducting periodic systems from first principles
Izmaylov, Artur F.
Scuseria, Gustavo E.
Doctor of Philosophy
This work is devoted to extending Kohn-Sham density functional theory (KS-DFT) with Gaussian-type orbitals (GTOs) and periodic boundary conditions (PBC) to linear response properties of nonconducting periodic systems, namely harmonic vibrational frequencies, infrared intensities, static and dynamic polarizabilities, and optical absorption spectra. In order to accomplish this, four things must be considered. First, theoretical modeling of response properties implies some knowledge of excited states. Thus, to stay within DFT framework one needs to use the time-dependent (TD) DFT formalism. TD-DFT imposes additional requirements on the accuracy of density functional approximations, because not only the exchange-correlation energy and potential are involved but also the exchange-correlation kernel. We consider one of the most promising directions towards improved accuracy: combining available non-empirical semi-local functionals with the non-local Hartree-Fock exchange (HFx) functional. Several different ways of introducing HFx are discussed in this work: global hybrids, range-separated hybrids, optimized effective potentials, and effective local potentials. Second, solving problems within the TD-DFT framework for periodic systems also requires implementing efficient computational tools. The TD-DFT equations for all properties considered have many similarities. Moreover, from the algebraic point of view the most computationally demanding step is the same. It is a contraction of two-electron integrals over atomic GTOs with various density-like quantities. The use of a GTO basis in this step is crucial for adapting different linear scaling techniques developed for large molecules. Third, introducing PBC requires additional care as some phenomena have no analogue in the molecular case. For vibrational frequencies, this can be seen from the existence of two types of vibrations: “in-phase” and “out-of-phase”, depending on the phase difference between oscillations in different unit cells. Since the position operator is unbound in extended systems, the dipole moment per unit cell becomes a nontrivial quantity. To evaluate the periodic dipole and its derivatives we use the projection technique developed by Blount and a discretized form of the Berry phase. Another interesting problem occurring in periodic systems is a collapse of the lowest semi-local TD-DFT excitation to the minimal KS direct band gap. We propose a simple analytical model that explains why this failure takes place.