Variation-Free Approach for Density Functional Theory: Data-Driven Stochastic Optimization

Abstract

Density functional theory (DFT) is an efficient instrument for describing a wide range of nanoscale phenomena: wetting transition, capillary condensation, adsorption, etc. In this paper, we suggest a method for obtaining the equilibrium molecular fluid density in a nanopore using DFT without calculating the free-energy variation─Variation-Free Density Functional Theory (VF-DFT). This technique can be used to explore confined fluids with a complex type of interactions, additional constraints, and, to speed up calculations, which might be crucial in an inverse problem. The fluid density in VF-DFT approach is represented as a decomposition over a limited set of basis functions. We applied principal component analysis (PCA) to extract the basic patterns from the density function and take them into account in the construction of a set of basis functions. The decomposition coefficients of the fluid density by the basis were sought by stochastic optimization algorithms: genetic algorithm (GA) and particle swarm optimization (PSO), to minimize the free energy of the system. In this work, two different fluids were studied: nitrogen at a temperature of 77.4 K and argon 87.3 K, at a pore size of 3.6 nm, and the performance of optimization algorithms was compared. We also introduce the Hybrid Density Functional Theory (H-DFT) approach based on stochastic optimization methods and the classical Picard iteration method to find the equilibrium fluid density starting from the physically appropriate solution. The combination of Picard iteration and stochastic algorithms helps to significantly speed up the calculations of equilibrium density in the system without losing the quality of the solution, especially in cases with the high relative pressure and expressed layering structure.