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Dimensionality dependence of total energy of closed shells in a bare Coulomb field for large atomic number
Author(s) -
March N. H.,
Cizek J.
Publication year - 1988
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560330405
Subject(s) - coulomb , curse of dimensionality , limit (mathematics) , generalization , physics , energy (signal processing) , total energy , field theory (psychology) , field (mathematics) , atomic physics , quantum mechanics , quantum electrodynamics , mathematical physics , mathematics , electron , mathematical analysis , statistics , pure mathematics , displacement (psychology) , psychology , psychotherapist
For Z particles moving independently in an unscreened Coulomb potential energy −Ze 2 /r, it is known that the total energy E = −(3/2) 1/3 Z 7/3 in nonrelativistic theory as Z→∞. The generalization of this result to d dimensions in the same asymptotice limit is obtained as\documentclass{article}\pagestyle{empty}\begin{document}$$E(Z,d) = - \frac{{2\left({\frac{{d!}}{4}} \right)^{1 - (2/d)} }}{{(d - 1)!(d - 2)}}Z^{3 - (2/d)},$$\end{document}which includes the known result above for the three‐dimensional case.