12 lines
1.1 KiB
TeX
12 lines
1.1 KiB
TeX
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\section{Summary}
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In the first section of this chapter, we have reported exact solutions of a Coulomb correlation problem, consisting of two electrons on a $D$-dimensional sphere.
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The Coulomb problem can be solved exactly for an infinite set of values of the radius $R$ for both the ground and excited states, on both the singlet and triplet manifolds.
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The corresponding exact solutions are polynomials in the interelectronic distance $\ree$.
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The cusp conditions, which are related to the behaviour of the wave function at the electron-electron coalescence point, have been analysed and classified according to the natural or unnatural parity of the state considered.
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In the second section, we proved that the leading term in the large-$D$ expansion of the high-density correlation energy of an electron pair is invariant to the nature of the confining potential.
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For any such system, the correlation energy is given by $\Ec \sim -\gamma^2/8$, where $\gamma = 1/(D-1)$ is the Kato cusp factor in a $D$-dimensional space.
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