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We demonstrate that the Schr\"odinger equation for two electrons on a ring, which is the usual paradigm to model quantum rings, is solvable in closed form for particular values of the radius. We show that both polynomial and irrational solutions can be found for any value of the angular momentum and that the singlet and triplet manifolds, which are degenerate, have distinct geometric phases. We also study the nodal structure associated with these two-electron states.
We discuss the model of a $D$-dimensional confined electron gas in which the particles are trapped by a harmonic potential. In particular, we study the non-interacting kinetic and exchange energies of finite-size inhomogeneous systems, and compare the resulting Thomas-Fermi and Dirac coefficients with various uniform electron gas paradigms. We show that, in the thermodynamic limit, the properties of this model are identical to those of the $D$-dimensional Fermi gas.
Using the spherical geometry, we introduce a novel model to study excitons confined in a three-dimensional space, which offers unparalleled mathematical simplicity while retaining much of the key physics. This new model consists of an exciton trapped on the 3-sphere (i.e. the surface of a four-dimensional ball), and provides a unified treatment of Frenkel and Wannier-Mott excitons. Moreover, we show that one can determine, for particular values of the dielectric constant $\epsilon$, the close...
We introduce a new paradigm for finite and infinite strict-one-dimensional uniform electron gases. In this model, $n$ electrons are confined to a ring and interact via a bare Coulomb operator. In the high-density limit (small-$r_s$, where $r_s$ is the Seitz radius), we find that the reduced correlation energy is $\Ec(r_s,n) = \eps^{(2)}(n) + O(r_s)$, and we report explicit expressions for $\eps^{(2)}(n)$. In the thermodynamic (large-$n$) limit of this, we show that $\Ec(r_s) = - \pi^2/360 + O...
We introduce a generalization (gLDA) of the traditional Local Density Approximation (LDA) within density functional theory. The gLDA uses both the one-electron Seitz radius $\rs$ and a two-electron hole curvature parameter $\eta$ at each point in space. The gLDA reduces to the LDA when applied to the infinite homogeneous electron gas but, unlike the LDA, is is also exact for finite uniform electron gases on spheres. We present an explicit gLDA functional for the correlation energy of electron...
We report benchmark results for one-dimensional (1D) atomic and molecular systems interacting via the Coulomb operator $|x|^{-1}$. Using various wavefunction-type approaches, such as Hartree-Fock theory, second- and third-order M{\o}ller-Plesset perturbation theory and explicitly correlated calculations, we study the ground state of atoms with up to ten electrons as well as small diatomic and triatomic molecules containing up to two electrons. A detailed analysis of the 1D helium-like ions is...
We introduce a new basis function (the spherical gaussian) for electronic structure calculations on spheres of any dimension $D$. We find \alert{general} expressions for the one- and two-electron integrals and propose an efficient computational algorithm incorporating the Cauchy-Schwarz bound. Using numerical calculations for the $D = 2$ case, we show that spherical gaussians are more efficient than spherical harmonics when the electrons are strongly localized.
We introduce the two-particle probability density $X(x)$ of $x=\bm{r}_{12}\cdot\bm{p}_{12}=\left(\bm{r}_1-\bm{r}_2\right) \cdot \left(\bm{p}_1-\bm{p}_2\right)$. We show how to derive $X(x)$, which we call the Posmom intracule, from the many-particle wavefunction. We contrast it with the Dot intracule [Y. A. Bernard, D. L. Crittenden, P. M. W. Gill, Phys. Chem. Chem. Phys., 10, 3447 (2008)] which can be derived from the Wigner distribution and show the relationships between the Posmom intracul...
We introduce a new paradigm for one-dimensional uniform electron gases (UEGs). In this model, $n$ electrons are confined to a ring and interact via a bare Coulomb operator. We use Rayleigh-Schr\"odinger perturbation theory to show that, in the high-density regime, the ground-state reduced (i.e. per electron) energy can be expanded as $\eps(r_s,n) = \eps_0(n) r_s^{-2} + \eps_1(n) r_s^{-1} + \eps_2(n) +\eps_3(n) r_s + \ldots$, where $r_s$ is the Seitz radius. We use strong-coupling perturbation...
We show that the exact solution of the Schr\"odinger equation for two electrons confined to two distinct concentric rings or spheres can be found in closed form for particular sets of the ring or sphere radii. In the case of two concentric rings, we report exact polynomial and irrational solutions. The same methodology is applied to the case of two concentric spheres for which we report exact polynomial solutions for the ground state and the excited states of $S$ symmetry. For these concentri...
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