He@C$_{60}$的平移本征态：基于高斯过程回归插值的四维从头算势能面
Translational eigenstates of He@C$_{60}$ from four-dimensional \textit{ab initio} Potential Energy Surfaces interpolated using Gaussian Process Regression
AI总结：
本研究构建四维势能面并利用高斯过程回归插值，计算He@C$_{60}$的平移本征态，比较多种电子结构方法，推荐RPA作为高效替代方案。
摘要：
我们利用包含氦原子三个平动自由度和C$_{60}$笼半径的四维势能面（PES）研究内嵌富勒烯体系$^3$He@C$_{60}$。我们比较了MP2、SCS-MP2、SOS-MP2、RPA@PBE和C(HF)-RPA方法，以校准并增强对电子结构方法选择的信心。由于这些计算成本高昂，我们利用高斯过程回归（GPR）对势能面进行插值，因其在处理稀疏训练数据时具有有效性。势能面被分解为一个二维径向表面，并施加修正以获得整体四维表面。核哈密顿量被对角化以生成笼内平动/振动本征态。由于径向势的非谐性，主量子数为$n$的三维谐振子能量的简并性被解除。角动量态的$(2l+1)$重简并也因势的角依赖性而被轻微解除。我们计算出基频范围在96cm$^{-1}$至110cm$^{-1}$之间，具体取决于所用的电子结构方法。本征态能量的误差棒由GPR计算得出，约为±1.5cm$^{-1}$。我们还通过考虑波函数与一维经验势的重叠和Hellinger距离来比较波函数。与能量结果一致，两种从头算方法MP2和RPA@PBE显示出最佳一致性。虽然MP2的一致性优于RPA@PBE，但鉴于其更高的计算效率和相当的性能，我们推荐RPA作为这些体系中MP2的替代电子结构方法选择。
英文摘要：
We investigate the endofullerene system $^3$He@C$_{60}$ with a four-dimensional Potential Energy Surface (PES) to include the three He translational degrees of freedom and C$_{60}$ cage radius. We compare MP2, SCS-MP2, SOS-MP2, RPA@PBE and C(HF)-RPA to calibrate and gain confidence in the choice of electronic structure method. Due to the high cost of these calculations, the PES is interpolated using Gaussian Process Regression (GPR), owing to its effectiveness with sparse training data. The PES is split into a two-dimensional radial surface, to which corrections are applied to achieve an overall four-dimensional surface. The nuclear Hamiltonian is diagonalised to generate the in-cage translational/vibrational eigenstates. The degeneracy of the three-dimensional harmonic oscillator energies with principal quantum number $n$ is lifted due to the anharmonicity in the radial potential. The $(2l+1)$-fold degeneracy of the angular momentum states is also weakly lifted, due to the angular dependence in the potential. We calculate the fundamental frequency to range between 96cm$^{-1}$ and 110cm$^{-1}$ depending on the electronic structure method used. Error bars of the eigenstate energies were calculated from the GPR and are on the order of approximately $\pm$ 1.5cm$^{-1}$. Wavefunctions are also compared by considering their overlap and Hellinger distance to the one-dimensional empirical potential. As with the energies, the two \textit{ab initio} methods MP2 and RPA@PBE show the best agreement. While MP2 has better agreement than RPA@PBE, due to its higher computational efficiency and comparable performance, we recommend RPA as an alternative electronic structure method of choice to MP2 for these systems.
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