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.
Yan et al. (Wed,) studied this question.