We study the $D$-dimensional high-density correlation energy $\Ec$ of the singlet ground state of two electrons confined by a harmonic potential with Coulombic repulsion. We allow the harmonic potential to be anisotropic, and examine the behavior of $\Ec$ as a function of the anisotropy $\alpha^{-1}$. In particular, we are interested in the limit where the anisotropy goes to infinity ($\alpha\to0$) and the electrons are restricted to a lower-dimensional space. We show that tuning the value of $\alpha$ from 0 to 1 allows a smooth dimensional interpolation and we demonstrate that the usual model, in which a quantum dot is treated as a two-dimensional system, is inappropriate. Finally, we provide a simple function which reproduces the behavior of $\Ec$ over the entire range of $\alpha$.
The two-electron problem is one of the fundamental problems of quantum physics [1][2][3][4] and, although it looks simple, it has only been solved in certain very special cases [5][6][7][8][9][10][11][12][13]. Many of the methods that have been developed to provide approximate solutions to the twoelectron problem have been central in the development of molecular physics and quantum chemistry [14,15].
The familiar Hartree-Fock (HF) model [16] treats a system as a separable collection of electrons, each moving in the mean field of the others. The HF solution provides us with a good approximation to the energy and is widely applied to model complex molecular systems [17]. However, it is essential to understand its error [18]
which Wigner called the correlation energy [19]. Studies of correlation effects in two-electron systems are interesting in their own right, but also provide simple examples to test computational models [20] and shed light on more complicated systems [21][22][23]. They have been extensively studied, for various confining external potentials, interacting potentials and degrees of freedom [24][25][26][27]. However, most previous studies have focussed on spherically symmetric external potentials, for anisotropy significantly complicates the mathematical analysis. This is unfortunate, for most real systems are not isotropic, and it is therefore important to understand how anisotropy affects the correlation energy.
Quantum dots are often modeled by electrons in a harmonic potential with Coulombic repulsion [28][29][30][31]. Be-cause experimental conditions strongly confine the electrons in one dimension, the model potentials are usually spherical and two-dimensional. Calculations on such quantum dots have been used extensively in the development of exchange-correlation density functionals for low-dimensional systems in the framework of densityfunctional theory (DFT) [32][33][34][35][36][37].
In addition to experimental progress [38][39][40], many theoretical investigations have studied the effects of the confinement strength in the third dimension on the energy of the quantum dot. These studies have used DFT [41,42], HF [43], exact diagonalization [44] and exact solutions [45][46][47]. However, despite the importance of the correlation energy, only a few studies [48][49][50] have explored the confinement effect on E c .
In this paper, we examine the effects of anisotropy on the energy of the nodeless ground state of two electrons in a D-dimensional harmonic potential, using atomic units throughout. We define the external potential by
where λ governs the overall strength of the potential and α j ∈ (0, 1] is the force constant for the Cartesian coordinate x j . The isotropic case is obtained when all α j ’s are equal. We are particularly interested in the behavior where one or more of the α j approach 0 for, in such limits, the system is constrained towards a lower dimensionality. We restrict our attention to the high-density (λ → ∞) limit [51][52][53][54] for it has been found that the high-density behavior of electrons is surprisingly similar to that at typical atomic and molecular electron densities [24][25][26][27].
The Hamiltonian describing this system is
where r 12 = |r 1 -r 2 | is the interelectronic distance and ∇ 2 is the D-dimensional Laplace operator [55,56]. Scal-ing all lengths by λ, we obtain
which is suitable for large-λ perturbation theory with the zeroth-order Hamiltonian
and the perturbation
The Hamiltonian Ĥ(0) is separable and its eigenfunctions and energies are (7) and
where n i = (n i,1 , . . . , n i,D ) holds the quantum numbers of the ith electron, x i,j is the ith coordinate of the jth electron, and H n (x) is the nth Hermite polynomial [57].
Expanding the exact and HF energies as power series
HF + E
HF + O(λ -1 ), (10) one finds [52,53,58,59] that
and, therefore, that the limiting correlation energy is
HF .
In this paper, we show that E c is strongly affected by the anisotropy and dimensionality of the potential. In Sec. III, we use perturbation theory to obtain integral expressions for E (2) and
HF in an anisotropic quantum dot. In Sec. IV, we use the integral to express E c as a infinite sum in a special case. Finally, in Sec. V, we present numerical results and discuss some of the implications with regard to quantum dots and dimensional interpolation.
The exact and HF second-order energies are [24,59]
Whereas the summation for the exact energy includes all states, the summation for the HF energy includes only singly-excited states [59].
Employing the Fourier representation
one finds
where Γ is the gamma function [57].
We now try to solve the integral
for special values of α j . The isotropic case, i.e. where all α j are equal, has been considered in detail in Ref. [24].
In the present paper, we generalize this to case where the α j take two distinct values. Without loss of generality, we let δ be an integer such that 0 < δ < D, and set
We a
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