First Principles Derivation of the Effect of Geometric Noise on Distributions of Electronic Properties using the Effective Stochastic Kohn-Sham Potential Method

Obtaining ensemble averages by sampling many conformations is vital for an accurate description of temperature-dependent properties of chemical systems. However, constructing distributions of 10⁵ - 10⁶ samples is computationally challenging due to the high computational cost of performing calculations. In this work, we present a new approach called the effective stochastic Kohn-Sham potential (ESKS) method to address this challenge. Using the classical nuclear-nuclear repulsion energy as a metric, we derive statistical relationships between the distribution of nuclear-nuclear repulsion energies and the distributions of ground state quantum mechanical properties for a general chemical system. The results from this analysis show that the geometric noise experienced by molecules due to solvent interaction and thermal motion can be effectively captured by an effective stochastic operator. This allows for introducing a stochastic variable in the Kohn-Sham potential. Comparison of the analytical results with numerical DFT calculations on small molecules, semiconductor clusters, and large organic molecules with be presented. Both analytical and numerical results demonstrate the advantage of using an effective stochastic operator for performing large scale sampling of conformations.