Adaptive Sampling of Singularities and Long Tails for Monte Carlo Integration of Electron Correlation

The accuracy of stochastic algorithms for electron correlation energy calculations critically depends on the proper treatment of singularities and long-range tails arising from the two-electron Coulomb operator. In this work, an enhanced Monte Carlo approach is developed that constructs a tailored sampling distribution via a progressive residual fitting (PRF) strategy within the importance-sampling framework, termed MC@PRF. Two key techniques, selective scaling for tail amplification and the Top 5 Rule for singularity regularization, enable robust and accurate construction of the sampling probability density function without additional computational cost. Comprehensive numerical tests demonstrate that MC@PRF substantially reduces statistical errors and accelerates convergence while maintaining high accuracy, from low-dimensional benchmark functions to twelve-dimensional second-order Møller-Plesset correlation energy calculations. Moreover, MC@PRF naturally supports automatic stratified sampling, adaptively allocating computational effort between singular and regular regions without prior knowledge of the integrand's structure, thereby establishing a general and efficient Monte Carlo framework for complex integrals in quantum chemistry and related fields.