The Hartree–Fock (HF) equation for atoms with closed (sub)shells is transformed with the pseudospectral (PS) method into a discrete eigenvalue equation for scaled orbitals on a finite radial grid. The Fock exchange operator and the Hartree potential are obtained from the respective Poisson equations also discretized using the PS representation. The numerical solution of the discrete HF equation for closed-(sub)shell atoms from He to No is robust, fast and gives extremely accurate results, with the accuracy superior to that of the previous HF calculations. A very moderate number of 33 to 71 radial grid points is sufficient to obtain total energies with 14 significant digits and occupied orbital energies with 12 to 14 digits in numerical calculations using the double precision (64-bit) of the floating-point format.The electron density at the nucleus is then determined with 13 significant digits and the Kato condition for the density and s orbitals is satisfied with the accuracy of 11 to 13 digits. The node structure of the exact HF orbitals is obtained and their asymptotic dependence, including the common exponential decay, is reproduced very accurately. The accuracy of the investigated quantities is further improved by performing the PS calculations in the quadruple precision (128-bit) floating-point arithmetic which provides the total energies with 25 significant digits while using only 80 to 130 grid points.

中文翻译：

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Hartree-Fock 方程的高精度数值解与闭壳原子的拟谱法

具有封闭（子）壳原子的 Hartree-Fock (HF) 方程用伪谱 (PS) 方法转换为有限径向网格上缩放轨道的离散特征值方程。Fock 交换算子和 Hartree 势是从各自的 Poisson 方程中获得的，这些方程也使用 PS 表示进行了离散化。从 He 到 No 的封闭（子）壳原子的离散 HF 方程的数值解是稳健、快速的，并且给出了极其准确的结果，其精度优于之前的 HF 计算。在使用浮点格式的双精度（64 位）的数值计算中，33 到 71 个非常适中的径向网格点的数量足以获得 14 位有效数字的总能量和 12 至 14 位数字的占用轨道能量。然后用 13 位有效数字确定原子核处的电子密度，密度和 s 轨道的加藤条件满足 11 到 13 位的精度。获得了精确 HF 轨道的节点结构，并且非常准确地再现了它们的渐近相关性，包括常见的指数衰减。通过在四倍精度（128 位）浮点运算中执行 PS 计算，可以进一步提高研究量的准确性，该运算提供 25 位有效数字的总能量，同时仅使用 80 到 130 个网格点。包括常见的指数衰减，都非常准确地再现。通过在四倍精度（128 位）浮点运算中执行 PS 计算，可以进一步提高研究量的准确性，该运算提供 25 位有效数字的总能量，同时仅使用 80 到 130 个网格点。包括常见的指数衰减，都非常准确地再现。通过在四倍精度（128 位）浮点运算中执行 PS 计算，可以进一步提高研究量的准确性，该运算提供 25 位有效数字的总能量，同时仅使用 80 到 130 个网格点。