均匀电子气体：III。三维球体上的低密度气体
Uniform electron gases: III. Low-density gases on three-dimensional spheres
摘要：
结合变分蒙特卡罗（VMC）和完全基组极限Hartree-Fock（HF）计算，我们获得了三维球（3-sphere）上低密度同自旋电子的近精确关联能，即~四维球体的表面。在VMC计算中，我们比较了两种单电子基函数对这些强关联体系的计算效率，并分析了能量收敛与Jastrow因子质量的关系。HF计算采用球面高斯函数（SGF），这是弯曲的空间类似物的Carnival高斯函数。在低密度下，电子相对地局限于维格纳晶体中，通过求解3-球上的汤姆逊问题（即$n$点电荷的最小能量排列），对于不同的$n$值，可以找到自然的SGF中心。我们发现了11个$n$的特殊值，它们的Thomson位置是等价的。其中三个是四维柏拉图立体的顶点--超四面体（$n=5$）、超八面体（$n=8$）和24胞体（$n=24$）--第四个是以前没有报道过的高度对称结构（$n=13$）。通过计算电子在其平衡位置附近的谐振频率，我们还发现了汤姆逊能量的一阶振动修正。
英文摘要：
By combining variational Monte Carlo (VMC) and complete-basis-set limit Hartree-Fock (HF) calculations, we have obtained near-exact correlation energies for low-density same-spin electrons on a three-dimensional sphere (3-sphere), i.e.~the surface of a four-dimensional ball. In the VMC calculations, we compare the efficacies of two types of one-electron basis functions for these strongly correlated systems, and analyze the energy convergence with respect to the quality of the Jastrow factor. The HF calculations employ spherical Gaussian functions (SGFs) which are the curved-space analogs of cartesian Gaussian functions. At low densities, the electrons become relatively localized into Wigner crystals, and the natural SGF centers are found by solving the Thomson problem (i.e. the minimum-energy arrangement of $n$ point charges) on the 3-sphere for various values of $n$. We have found 11 special values of $n$ whose Thomson sites are equivalent. Three of these are the vertices of four-dimensional Platonic solids --- the hyper-tetrahedron ($n=5$), the hyper-octahedron ($n=8$) and the 24-cell ($n=24$) --- and a fourth is a highly symmetric structure ($n=13$) which has not previously been reported. By calculating the harmonic frequencies of the electrons around their equilibrium positions, we also find the first-order vibrational corrections to the Thomson energy.
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