Ab initio methods play a crucial role in unraveling the energetic, optical, magnetic, and electrical properties of molecules and materials. They strive to tackle the ambitious objective of solving both time-dependent and time-independent Schrödinger equations. However, due to the complexity of the many-body Schrödinger equation, the computational resources required to solve it directly are often impractical, limiting its application to systems comprising only a few electrons. Among the multitude of approximations devised to circumvent this issue, Density Functional Theory (DFT), including both ground-state and time-dependent approaches (TD-DFT), has emerged as a highly effective strategy. It offers a promising balance between computational efficiency and predictive accuracy. Yet, even with such an approach, tackling experimental-relevant systems hosting between 103 and 104 atoms remains a computational challenge.

 

To address these limitations, we're in the process of developing innovative stochastic electronic structure methods. By introducing random vectors, we're able to construct a low-rank approximation of the identity operator. This innovative approach can significantly decrease the computational scaling of DFT and TD-DFT, leading to linear scaling methods. Here, the computational effort scales linearly with the system size (N), where N represents the number of atoms. As a result, stochastic electronic structure methods offer exciting potential for modeling intricate materials such as nanostructures, metamaterials, and interfaces. This could be a major step forward in handling large, experimentally relevant systems, thus pushing the boundaries of computational materials science.