We discuss alternative homogeneous electron gas systems in which a finite number $n$ of electrons are confined to a $D$-dimensional sphere. We derive the first few terms of the high-density ($r_s\to0$, where $r_s$ is the Seitz radius) energy expansions for these systems and show that, in the thermodynamic limit ($n\to\infty$), these terms become identical to those of $D$-dimensional jellium.
Deep Dive into Thinking outside the box: the uniform electron gas on a hypersphere.
We discuss alternative homogeneous electron gas systems in which a finite number $n$ of electrons are confined to a $D$-dimensional sphere. We derive the first few terms of the high-density ($r_s\to0$, where $r_s$ is the Seitz radius) energy expansions for these systems and show that, in the thermodynamic limit ($n\to\infty$), these terms become identical to those of $D$-dimensional jellium.
The D-dimensional uniform electron gas (UEG), or D-jellium, is the foundation of most density functionals. It consists of interacting electrons in an infinite volume and in the presence of a uniformly distributed background positive charge. Traditionally, in its paramagnetic version, the system is constructed by allowing the number n of paired electrons in a D-dimensional box of volume V to approach infinity with ρ = n/V held constant. 1,2 Using atomic units, the high-density (r s → 0, where r s is the Seitz radius) expansion of the reduced energy (i.e. energy per electron) of D-jellium is
where ε T and ε X are kinetic 3,4 and exchange 5,6 energies
and ε C is the correlation energy. After many important contributions, [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] it is known that, for 2-and 3-jellium, the correlation energy takes the form
[a j (D) + b j (D) ln r s ] r j s .
The constant term in (3) is usually decomposed as a 0 (D) = a 0,J (D) + a 0,K (D),
where a 0,J is the direct (“ring-diagram”) contribution, and a 0,K is the second-order exchange part. The first few a j and b j and are known analytically or numerically for the important D = 2 and D = 3 cases (see Table I).
In this Article, we introduce an alternative paradigm, in which the electrons are confined to a D-sphere, that is, the surface of a (D + 1)-dimensional ball. These systems a) Electronic mail: loos@rsc.anu.edu.au b) Corresponding author; Electronic mail: peter.gill@anu.edu.au possess uniform densities, even for finite n, and because all points on a D-sphere are equivalent, their mathematical analysis is relatively straightforward. [27][28][29][30][31] Electronic properties of the UEG on a 2-sphere have been previously studied in modeling multielectron bubbles in liquid helium (see Ref. 32), and similarities between this system and 2-jellium have been noticed by Longe and Bose. 33 However, the UEG on a 3-sphere has not been considered before, and this Article presents the first study of correlation effects in a spherically-confined threedimensional UEG.
The orbitals for an electron on a D-sphere of radius R are the normalized hyperspherical harmonics Y µ , where is the principal quantum number and µ is a composite index of the remaining quantum numbers. 34,35 We confine our attention to systems in which every orbital with = 0, 1, . . . , L is occupied by two electrons, thus yielding an electron density that is uniform on the sphere (see Eq. ( 9) below). The resulting model is defined completely by the three parameters D, L and R.
The volume of a D-sphere is
where Γ is the gamma function, 36 the number of orbitals with quantum number is
and each of these has energy
Because the total number of electrons is
arXiv:1101.3131v5 [physics.chem-ph] 14 Nov 2011
it follows that the uniform electron density is
and the Seitz radius is
with
Using the hyperspherical harmonic addition theorem, 35 one finds that the one-particle density matrix is
where
is a Lth degree Jacobi polynomial. 36 The angle θ is that subtended by the electrons at the origin and is related to the interelectronic distance by the relation 37
The density matrix decays rapidly with interelectronic separation when L is large (Fig. 1), illustrating the “shortsightedness” of matter. 38,39 Many properties of the UEG on a D-sphere can be found from Eqs. ( 6) - (12). Its kinetic energy, for example, is
and it can be shown that its exchange energy is
where 4 F 3 is a generalized hypergeometric function. 36
The above expressions are exact for all L but, in the thermodynamic limit (n, L → ∞), each simplifies signifi-
ρ → 2 Γ(D/2 + 1)
where J n is the nth-order Bessel function. 36 We note that (19) reduces to the usual density matrices in 2-jellium 40 and 3-jellium. 5 The kinetic and exchange energies become
Equations ( 20) and ( 21) yield the two terms in (2), and are identical to the D-jellium expressions. Particular cases are given in Table I. These results were originally discovered by Glasser and Boersma, 41 and Iwamoto 42 for D-jellium, but our derivation for the UEG on a D-sphere is more compact than theirs.
We now turn our attention to the study of the correlation energy of the spherically-confined UEGs. By applying perturbation theory to UEG on a 2-sphere, we find that the reduced energy coefficient corresponding to the lowest-order ring-diagram contribution is
(2i + 1)(2j + 1)(2a + 1)(2b + 1)
where ij|ab are two-electron integrals and the brackets are 3j symbols. 36 For the UEG on a 3-sphere, the coupling coefficient in SO( 4) is much simpler than in SO(3) 43 and the energy coefficient from the lowest-order ring-diagram is
(i + 1)(j + 1)(a + 1)(b + 1)
where the sum over respects the same restrictions as in the 3j symbols in (22).
The second-order exchange part for the UEG on a 2-sphere is
where the curly brackets denote 6j symbols, 36 and for the UEG on a 3-sphere, we found
where we have used the SO(4) version of the 6j
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