In this paper, an importance sampling method based on the Generalized Feynman–Kac (GFK) method has been used to calculate the mean values of quantum observables from quantum correlation functions for many-body systems with the Born–Oppenheimer approximation in the nonrelativistic limit both at zero and finite temperature. Specifically, the expectation values $〈r_i^n〉$, $〈r_{ij}^n〉$, $〈r_i^{-n}〉$ and $〈r_{ij}^{-n}〉$ for the ground state of the lithium and beryllium and the density matrix, the partition function, the internal energy and the specific heat of a system of quantum harmonic oscillators are computed, in good agreement with the best nonrelativistic values for these quantities. Although the initial results are encouraging, more experimentation will be needed to improve the other existing numerical results beyond chemical accuracies specially for the last two properties for lithium and beryllium. Also more work needs to be done to improve the trial functions for finite temperature calculations. Although these results look promising, more work needs to be done to achieve the spectroscopic accuracy at zero temperature and to estimate the finite temperature effects from the non-Born–Oppenheimer calculations. Also more experimentation will be needed to study the convergence criteria for the inverse properties for atoms at zero temperature.

在本文中，基于广义 Feynman-Kac (GFK) 方法的重要性采样方法已被用于计算具有非相对论极限下的 Born-Oppenheimer 近似的多体系统的量子相关函数的量子可观测量的平均值。在零和有限的温度。具体来说，期望值$〈r_{\mathrm{一世}}^n〉$,$〈r_{\mathrm{一世}j}^n〉$,$〈r_{\mathrm{一世}}^{-n}〉$和$〈r_{\mathrm{一世}j}^{-n}〉$对于锂和铍的基态和密度矩阵，计算了量子谐振子系统的配分函数、内能和比热，与这些量的最佳非相对论值非常吻合。尽管最初的结果令人鼓舞，但需要更多的实验来改进其他现有的数值结果，超出化学精度，特别是锂和铍的最后两个特性。还需要做更多的工作来改进有限温度计算的试验功能。尽管这些结果看起来很有希望，但需要做更多的工作来实现零温度下的光谱精度，并从非 Born-Oppenheimer 计算中估计有限温度效应。