The mapping of an electronic state on a real-space support lattice may offer advantages over a basis set ansatz in cases where there are linear dependences due to basis set overcompleteness or when strong internal or external fields are present. Such discretization methods are also of interest because they allow for the convenient numerical integration of matrix elements of local operators. We have developed a pseudo-spectral approach to the numerical solution of the time-dependent and time-independent Schrödinger equations describing electronic motion in atoms and atomic ions in terms of a spherical coordinate system. A key feature of this scheme is the construction of a Variational Basis Representation (VBR) for the non-local component and of a Generalized Finite Basis Representation (GFBR) for the local component of the electronic Hamiltonian operator. Radial Hamiltonian eigenfunctions ${\chi }_{\mathit{nl}\text{;}\beta }\left(r\right)$ of the H atom-like system and spherical harmonics form the basis set. Two special cases of this approach are explored: In one case, the functions of the field-free H atom are used as the elements of the basis set, and in the second case, each radial basis function has been obtained by variationally optimizing a shielding parameter $\beta$ to yield a minimum energy for a particular eigenstate of the H atom in a uniform magnetic field.

We derive a new quadrature rule of nearly Gaussian accuracy for the computation of matrix elements of local operators between radial basis functions and perform numerical evaluation of local operator matrix elements involving spherical harmonics. With this combination of radial and angular quadrature prescriptions we ensure to a good approximation the discrete orthogonality of Hamiltonian eigenfunctions of H atom-like systems for summation over the grid points. We further show that sets of ${\chi }_{\mathit{nl}\text{;}\beta }\left(r\right)$ functions are linearly independent, whereas sets of the polar-angle-dependent components of the spherical harmonics, i.e., the associated Legendre functions, are not and provide a physical interpretation of this mathematical observation.

The pseudo-spectral approach presented here is applied to two model systems: the field-free H atom and the H atom in a uniform magnetic field. The results demonstrate the potential of this method for the description of challenging systems such as highly charged atomic ions.

在由于基组超完备性而存在线性相关性或存在强大的内部或外部场的情况下，电子状态在真实空间支撑晶格上的映射可能比基组ansatz更具优势。这种离散化方法也很有趣，因为它们可以方便地对局部算子的矩阵元素进行数值积分。我们已经开发了一种伪光谱方法，用于求解与时间相关和与时间无关的Schrödinger方程的数值解，这些方程根据球坐标系描述了原子和原子离子中的电子运动。该方案的关键特征是针对非局部分量的变分基础表示（VBR）和广义有限基础表示的构造（GFBR），用于电子哈密顿算子的本地组件。径向哈密顿特征函数${\chi }_{\mathit{nl}\text{;}\beta }（\mathrm{[R}）$H原子状系统的原子团和球谐函数构成了基础集。探索了这种方法的两种特殊情况：一种情况下，将无场H原子的功能用作基础集的元素，第二种情况下，通过对屏蔽进行了可变优化来获得每个径向基础函数范围$\beta$ 在均匀磁场中产生H原子特定本征态的最小能量。

我们推导了一种新的近似高斯精度的正交规则，用于计算径向基函数之间的局部算子矩阵元素，并对涉及球谐函数的局部算子矩阵元素进行了数值评估。通过径向和角度正交公式的这种组合，我们可以确保很好地近似H原子状系统的哈密顿本征函数的离散正交性，以便在网格点上求和。我们进一步证明${\chi }_{\mathit{nl}\text{;}\beta }（\mathrm{[R}）$ 函数是线性独立的，而球谐函数的极角相关分量的集合（即相关的勒让德函数）则不是，并且不能为这种数学观察提供物理解释。

这里介绍的伪谱方法适用于两个模型系统：无场H原子和均匀磁场中的H原子。结果证明了该方法在描述具有挑战性的系统（例如高电荷原子离子）中的潜力。