球面张量梯度算子
The Spherical Tensor Gradient Operator
摘要：
球张量梯度算子${\mathcal{Y}}_{\ell}^{m} (\nabla)$是秩为$\ell$的不可约球张量，它是用正则体调和${\mathcal{Y}}_{\ell}^{m} (\bm{r})$中$\nabla$的笛卡尔分量代替$\bm{r}$的笛卡尔分量而得到的.因此，将其应用于标量函数会产生秩为$\ell$的不可约球面张量。因此，原则上只考虑标量函数的多中心积分就足够了：更高的角动量态可以通过相对于核坐标的微分产生。${\mathcal{Y}}_{\ell}^{m} (\nabla)$的许多性质可以借助于霍布森关于微分的一个老定理容易地理解[Proc. London Math. Soc. { \bf 24}，54 - 67（1892）]。从霍布森定理可以看出，如果将${\mathcal{Y}}_{\ell}^{m} (\nabla)$应用于某些具有相当相关性的标量函数，例如库仑势、高斯函数或约化贝塞尔函数，则会产生特别紧凑的结果。傅立叶变换产生了${\mathcal{Y}}_{\ell}^{m} (-\mathrm{i} \bm{p})$，因此对理解${\mathcal{Y}}_{\ell}^{m} (\nabla)$的性质很有帮助。也可以将${\mathcal{Y}}_{\ell}^{m} (\nabla)$应用于广义函数，例如产生球面δ函数$δ_{\ell}^{m} (\bm{r})$。微分算子${\mathcal{Y}}_{\ell}^{m} (\nabla)$也可用于推导逐点收敛加法定理。通过推导$r^ν {\mathcal{Y}_{\ell}^{m}} (\bm{r})$与$ν\in \mathbb{R}$的加法定理证明了这种方法的可行性。
英文摘要：
The spherical tensor gradient operator ${\mathcal{Y}}_{\ell}^{m} (\nabla)$, which is obtained by replacing the Cartesian components of $\bm{r}$ by the Cartesian components of $\nabla$ in the regular solid harmonic ${\mathcal{Y}}_{\ell}^{m} (\bm{r})$, is an irreducible spherical tensor of rank $\ell$. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank $\ell$. Thus, it is in principle sufficient to consider only multicenter integrals of scalar functions: Higher angular momentum states can be generated by differentiation with respect to the nuclear coordinates. Many of the properties of ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. London Math. Soc. {\bf 24}, 54 - 67 (1892)]. It follows from Hobson's theorem that some scalar functions of considerable relevance as for example the Coulomb potential, Gaussian functions, or reduced Bessel functions produce particularly compact results if ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ is applied to them. Fourier transformation is very helpful to understand the properties of ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ since it produces ${\mathcal{Y}}_{\ell}^{m} (-\mathrm{i} \bm{p})$. It is also possible to apply ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ to generalized functions, yielding for instance the spherical delta function $δ_{\ell}^{m} (\bm{r})$. The differential operator ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ can also be used for the derivation of pointwise convergent addition theorems. The feasibility of this approach is demonstrated by deriving the addition theorem of $r^ν {\mathcal{Y}_{\ell}^{m}} (\bm{r})$ with $ν\in \mathbb{R}$.
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