In a two-dimensional parabolic quantum dot charged with N electrons, Thomas-Fermi theory states that the ground-state energy satisfies the following non-trivial relation: Egs/( ω) ≈ N 3/2 fgs(N 1/4 β), where the coupling constant, β, is the ratio between Coulomb and oscillator ( ω) characteristic energies, and fgs is a universal function. We perform extensive Configuration Interaction calculations in order to verify that the exact energies of relatively large quantum dots approximately satisfy the above relation. In addition, we show that the number of energy levels for intraband and interband (excitonic and biexcitonic) excitations of the dot follows a simple exponential dependence on the excitation energy, whose exponent, 1/Θ, satisfies also an approximate scaling relation a la Thomas-Fermi, Θ/( ω) ≈ N −γ g(N 1/4 β). We provide an analytic expression for fgs, based on two-point Padé approximants, and two-parameter fits for the g functions.
In a quantum dot with dozens of electrons, an approximation beyond Tamm-Dankoff is used to construct the quantum states with an additional electron-hole pair, i.e., the "excitonic" states. The lowest states mimic the noninteracting spectrum, but with excitation gaps renormalized by Coulomb interactions. At higher excitation energies, the computed density of energy levels shows an exponential increase with energy. In the interband absorption, we found a background level in the quasicontinuum of states rising linearly with the excitation energy. Above this background, there are distinct peaks related to single resonances or to groups of many states with small interband dipole moments.
We show through numerical investigations that the ground-state correlation energies of atomic ions follow an unexpectedly simple scaling relation, E c ≈ Z 4/3 f c (Z/N), where N is the number of electrons, Z is the atomic number, and f c is a universal function, for which an analytic expression with a one-parameter fit can be provided. The relation agrees well with several sets of correlation energies obtained from different methods for atomic ions with N = 2, . . . ,18 and Z = 2, . . . ,28. Moreover, our relation gives a good agreement with neutral atoms up to N ≈ 90. Our main result is readily applicable to estimating correlation energies of heavy elements, for which there are no available data in the literature. The simplicity of the relation may also have implications in the development of correlation functionals within density-functional theory.
We derive scaling relations for the electrochemical potential and addition energy of 2D quantum dots charged with N electrons. In the derivation we apply the Thomas-Fermi theory for the harmonic model of a quantum dot in the effective mass approximation. We demonstrate that the resulting scaling relations yield excellent agreement with measured chemical potentials and addition energies for both lateral and vertical quantum dots. Moreover, we show that the scaling relation has predictive power in estimating the confinement strength and the number of electrons trapped in the quantum dot.
We find an unexpected scaling in the correlation energy of artificial atoms, i.e., harmonically confined two-dimensional quantum dots. The scaling relation is found through extensive numerical examinations including Hartree-Fock, variational quantum Monte Carlo, density functional, and full configuration interaction calculations. We show that the correlation energy, i.e., the true ground-state total energy minus the Hartree-Fock total energy, follows a simple function of the Coulomb energy, confinement strength and number of electrons. We find an analytic expression for this function, as well as for the correlation energy per particle and for the ratio between the correlation and total energies. Our tests for independent diffusion Monte Carlo and coupled-cluster results for quantum dots-including open-shell data-confirm the generality of the scaling obtained. As the scaling also applies well to ≳100 electrons, our results give interesting prospects for the development of correlation functionals within density functional theory.
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