Excited states of incipient Wigner molecules

An accurate configuation-interaction method employing a mean-field basis set is used to study the excitation spectrum of localized Wigner states in the strongly interacting regime of a quasi-two-dimensional parabolic quantum dot with $N=6$ electrons. The approach achieves errors of order 1 part in 10${}^4$ (or better) in the energies of low-lying states for a Wigner-Seitz radius $r_s=12\hspace{0.5em}{a}_0^{*}$–$16\hspace{0.5em}{a}_0^{*}$, and is used to study low-lying spin, rotational, vibrational, and isomeric excitations. The vibrational excitations at $r_s=50\hspace{0.5em}{a}_0^{*}$ are shown to correspond qualitatively with the classical normal modes of the $N=6$ electron dot, although the excitation energies agree only semiquantitatively with the classical normal-mode frequencies as a result of residual quantum fluctuations.

Figure 1

Excitation energy relative to the ${}^{1}S$ ground state of the lowest-lying ${}^{3}S$, ${}^{5}S$, ${}^{7}S$, and ${}^{3}P$ states of the $N=6$ electron dot versus Wigner-Seitz radius $r_s$ [Eq. (3)].

Figure 2

Stable configurations of the classical six-electron parabolic dot (from Ref. 20): (a) pentagonal ground-state configuration and (b) excited isomer (staggered hexagon).

Figure 3

Excitation energies of the six-electron dot versus Wigner-Seitz radius $r_s$ (partly taken from Ref. 17). The quantities $\Delta E_{\text{rot}}$, $\Delta E_{\text{vib}}$, and $\Delta E_{\text{iso}}$ are approximate rotational, vibrational, and isomeric excitation energies, respectively, inferred from a classical model (see text); $\Delta E({}^{3}S)$ and $\Delta E({}^{3}P)$ are excitation energies to the lowest ${}^{3}S$ and ${}^{3}P$ states calculated by CI. All excitation energies are scaled by $r_s$.

Figure 4

Charge-charge pair-correlation functions $g(\mathbf{r},{\mathbf{r}}_0)$ of low-lying $S$-wave states for $N=6$ electrons for $r_s=50\hspace{0.5em}{a}_0^{*}$ (partly taken from Ref. 17). We show representative PCFs for each spin multiplet. (a) Ground-state spin multiplet; (b)–(e) first to fourth excited $S$-wave spin multiplets, respectively. The position ${\mathbf{r}}_0$ of the reference electron is indicated by a dot, and the set of total spins $S$ present in each multiplet is shown.

Figure 5

Normal modes of the ground-state isomer of the classical six-electron dot. The normal-mode frequency $\Omega$ is indicated as a multiple of the frequency $\omega$ of the parabolic confinement potential [Eq. (2)].

Figure 6

Charge-charge pair-correlation functions $g(\mathbf{r},{\mathbf{r}}_0)$ of (a) the ${}^{1}S$ ground state, and (b) the first excited ${}^{1}S$ state of the $N=6$ electron dot at high-to-intermediate electron densities. The position ${\mathbf{r}}_0$ of the reference electron is indicated by a dot.

Figure 7

Spin configurations for $S_z=3$ (top row) and $S_z=2$ (bottom row) for $C_{5v}$ (left group) and $C_{6v}$ (right group) molecular symmetries. A full circle indicates spin up, and an empty circle spin down.