Accelerate stochastic calculation of random-phase approximation correlation energy difference with an atom-based correlated sampling

A kernel polynomial method is developed to calculate the random phase approximation (RPA) correlation energy. In the method, the RPA correlation energy is formulated in terms of the matrix that is the product of the Coulomb potential and the density linear response functions. The integration over the matrix's eigenvalues is calculated by expanding the density of states of the matrix in terms of the Chebyshev polynomials. The coefficients in the expansion are obtained through stochastic sampling. Since it is often the energy difference between two systems that is of much interest in practice, another focus of this work is to develop a correlated sampling scheme to accelerate the convergence of the stochastic calculations of the RPA correlation energy difference between two similar systems. The scheme is termed the atom-based correlated sampling (ACS). The performance of ACS is examined by calculating the isomerization energy of acetone to 2-propenol and the energy of the water-gas shift reaction. Using ACS, the convergences of these two examples are accelerated by 3.6 and 4.5 times, respectively. The methods developed in this work are expected to be useful for calculating RPA-level reaction energies for the reactions that take place in local regions, such as calculating the adsorption energies of molecules on transition metal surfaces for modeling surface catalysis.