dorsal/arxiv
View SchemaOn the Hylleraas Coordinates
| Authors | Xiao-Yin Pan, Viraht Sahni, Lou Massa |
|---|---|
| Categories | |
| ArXiv ID | physics/0310128 |
| URL | https://arxiv.org/abs/physics/0310128 |
Abstract
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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"abstract": "The Hylleraas coordinates $s=r_{1}+r_{2}$, $t=r_{1}-r_{2}$, $u=|{\\bf\nr}_{1}-{\\bf r}_{2}|$ are the natural coordinates for the determination of\nproperties of the Helium atom, the positive ions of its isoelectronic sequence,\nand the negative Hydrogen ion. In this paper, we derive a new expression for\nintegrals representing properties such as the energy, normalization and\nexpectation of arbitrary operators, as written in the $(s,t,u)$ coordinates.\nThe expression derived is valid for both \\emph{finite} and \\emph{infinite}\nspace. The integrals for the various properties are comprised in each case of\ntwo components $A$ and $B$. The contribution of these components to the volume\nof integration and the normalization of a wave function for finite space, and\nin variational calculations of the ground state energy of the Helium atom\nconfined in a finite volume is demonstrated by example. We prove that when the\nintegration space is \\emph{infinite}, the expression for the energy and other\nproperties employed by Hylleraas corresponds \\emph{only} to that of integral\n$A$. We further prove that for the approximate variational wave functions\nemployed by Hylleraas and other authors, the contribution of the term $B$\nvanishes. This contribution also vanishes for the exact wave function. It is\ninteresting to note that the component $B$ to the integral is not mentioned in\nthe literature. A principle purpose of the paper, therefore, is to point out\nthe existence of this term.",
"arxiv_id": "physics/0310128",
"authors": [
"Xiao-Yin Pan",
"Viraht Sahni",
"Lou Massa"
],
"categories": [
"physics.atom-ph",
"physics.chem-ph"
],
"title": "On the Hylleraas Coordinates",
"url": "https://arxiv.org/abs/physics/0310128"
},
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