Stochastic Density Functional Theory: Real- and Energy-Space Fragmentation for Noise Reduction*

Real-space/plane-wave based density function theory (DFT) is an important approach to understand electronic, optical, and magnetic properties of semiconductor and metallic materials. However, the computational scaling of conventional DFT methods is relatively high. Such a high scaling limits DFT applications in modeling complex materials such as semiconductor/metallic devices and nanomaterials. Therefore, developing linear scaling DFT methods is necessary for studying complex materials. While most linear scaling DFT assumes a localized density matrix, this assumption is not required for stochastic DFT method which utilizes stochastic orbitals instead of deterministic Kohn-Sham orbitals. However, noise in stochastic DFT limits the efficiency of this approach. Various noise reduction techniques that use fragmentations in  real space and/or energy space have been developed. These techniques can significantly reduce the noise level in stochastic DFT to enhance the computational efficiency. These noise-reduction stochastic DFT methods have been applied to geometry optimization of semiconductor materials which is a challenging problem for stochastic DFT with noisy atomic forces.