作为三维泰勒展开的加法定理。II。$B$函数和其他指数衰减函数
Addition Theorems as Three-Dimensional Taylor Expansions. II. $B$ Functions and Other Exponentially Decaying Functions
摘要：
加法定理可以通过根据$f (\mathbf{r} + \mathbf{r}') = \exp (\mathbf{r}' \cdot \mathbf{\nabla}) f (\mathbf{r})$进行三维泰勒展开来构造。然而，由于人们通常对不可约球面张量的加法定理感兴趣，因此应用笛卡尔形式的平移算子$\exp (x' \partial /\partial x) \exp (y' \partial /\partial y) \exp (z' \partial /\partial z)$将导致巨大的技术问题。一个更好的替代方案包括使用平移算子$\exp (\mathbf{r}' \cdot \mathbf{\nabla})$的级数展开，其涉及拉普拉斯算子$\mathbf{\nabla}^2$和球面张量梯度算子$\mathcal{Y}_{\ell}^{m} (\nabla)$的幂，它们分别是秩为零和$\ell$的不可约球面张量[F.D.\ Santos，Nucl.Phys.A { \bf 212}，341（1973）]。以这种方式，确实可以通过进行三维泰勒展开来导出加法定理[E. J. Weniger，Int. J. Quantum Chem. { \bf 76}，280（2000）]。在$B$函数的情况下，球面形式的平移算子的应用特别简单，并导致一个结构比较紧凑的加法定理。由于其他指数衰减函数，如Slater型函数，束缚态氢本征函数和其他基于广义Laguerre多项式的函数可以用$B$函数的简单有限和表示，这些函数的加法定理可以立即写下来。
英文摘要：
Addition theorems can be constructed by doing three-dimensional Taylor expansions according to $f (\mathbf{r} + \mathbf{r}') = \exp (\mathbf{r}' \cdot \mathbf{\nabla}) f (\mathbf{r})$. Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form $\exp (x' \partial /\partial x) \exp (y' \partial /\partial y) \exp (z' \partial /\partial z)$ would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator $\exp (\mathbf{r}' \cdot \mathbf{\nabla})$ involving powers of the Laplacian $\mathbf{\nabla}^2$ and spherical tensor gradient operators $\mathcal{Y}_{\ell}^{m} (\nabla)$, which are irreducible spherical tensors of ranks zero and $\ell$, respectively [F.D.\ Santos, Nucl. Phys. A {\bf 212}, 341 (1973)]. In this way, it is indeed possible to derive addition theorems by doing three-dimensional Taylor expansions [E.J. Weniger, Int. J. Quantum Chem. {\bf 76}, 280 (2000)]. The application of the translation operator in its spherical form is particularly simple in the case of $B$ functions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slater-type functions, bound-state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums of $B$ functions, the addition theorems for these functions can be written down immediately.
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