关于Hylleraas坐标
On the Hylleraas Coordinates
摘要：
Hylleraas坐标$s=r_{1}+r_{2}$、$t=r_{1}-r_{2}$、$u=|{\bf r}_{1}-{\bf r}_{2}|$是确定氦原子、其等电子序列的正离子和负氢离子的性质的自然坐标。在本文中，我们推导出一个新的积分表示的属性，如能量，规范化和期望的任意运营商，写在$(s,t,u)$坐标。导出的表达式对\emph{finite}和\emph{infinite}空间都是有效的。各种性质的积分在每种情况下包括两个分量$A$和$B$。这些组件的贡献的体积的整合和规范化的波函数为有限的空间，并在变分计算的基态能量的氦原子限制在一个有限的体积中的例子证明。我们证明，当集成空间是\emph{infinite}，Hylleraas所采用的能量和其他属性的表达式对应\emph{only}的积分$A$。我们进一步证明了对于Hylleraas和其他作者所采用的近似变分波函数，项$B$的贡献为零。对于精确的波函数，这种贡献也为零。值得注意的是，文献中没有提到积分的分量$B$。因此，本文的一个主要目的是指出这一术语的存在。
英文摘要：
The Hylleraas coordinates $s=r_{1}+r_{2}$, $t=r_{1}-r_{2}$, $u=|{\bf r}_{1}-{\bf r}_{2}|$ are the natural coordinates for the determination of properties of the Helium atom, the positive ions of its isoelectronic sequence, and the negative Hydrogen ion. In this paper, we derive a new expression for integrals representing properties such as the energy, normalization and expectation of arbitrary operators, as written in the $(s,t,u)$ coordinates. The expression derived is valid for both \emph{finite} and \emph{infinite} space. The integrals for the various properties are comprised in each case of two components $A$ and $B$. The contribution of these components to the volume of integration and the normalization of a wave function for finite space, and in variational calculations of the ground state energy of the Helium atom confined in a finite volume is demonstrated by example. We prove that when the integration space is \emph{infinite}, the expression for the energy and other properties employed by Hylleraas corresponds \emph{only} to that of integral $A$. We further prove that for the approximate variational wave functions employed by Hylleraas and other authors, the contribution of the term $B$ vanishes. This contribution also vanishes for the exact wave function. It is interesting to note that the component $B$ to the integral is not mentioned in the literature. A principle purpose of the paper, therefore, is to point out the existence of this term.
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