Eur. Phys. J. B 27, 385–392 (2002) DOI: 10.1140/epjb/e2002-00169-x T HE EUROPEAN P HYSICAL JOURNAL B Investigation of excitation energies and Hund’s rule in open shell quantum dots by diffusion Monte Carlo L. Colletti 1 , F. Pederiva 2, a , E. Lipparini 2 , and C.J. Umrigar 3 1 Lawrence Livermore National Laboratory, Livermore, CA 94551-0808, USA 2 Dipartimento di Fisica and INFM, Universit`a di Trento, via Sommarive 14, 38050 Povo, Trento, Italy 3 Cornell Theory Center, Cornell University, Ithaca, NY 14853, USA Received 20 November 2001 and Received in final form 20 February 2002 Published online 6 June 2002 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2002 Abstract. We use diffusion Monte Carlo to study the ground state, the low-lying excitation spectrum and the spin densities of circular quantum dots with parabolic radial potentials containing N = 16 and N = 24 electrons, each having four open-shell electrons and compare the results to those obtained from Hartree-Fock (HF) and density functional local spin density approximation (LSDA) calculations. We find that Hund’s first rule is obeyed in both cases and that neither HF nor LSDA correctly predict the ordering of the energy levels. PACS. 73.21.La Electron states in quantum dots 1 Introduction It is possible to make solid-state structures, called quan- tum dots, at semiconductor interfaces that confine a small number of mobile electrons. Quantum dots [1–4], contain- ing one to several tens of electrons are analogous to atoms with tunable properties, exhibiting shell structure and obeying Hund’s first rule. They are both of considerable technological interest and of theoretical interest because it is possible to go from a weak correlation to a strong correlation regime by tuning the relative strength of the external potential to the electron-electron potential. A variety of theoretical methods have been used to study the electronic structure of quantum dots. Some of the simpler methods that have been used are the (spin- unrestricted) Hartree-Fock method (HF) and the space- and spin-unrestricted Hartree-Fock method (UHF) [5,6], that treat exchange exactly but totally ignore correlation, and the local spin density approximation (LSDA) and its space-unrestricted version, ULSDA [7,8], to density func- tional theory that treat both exchange and correlation ap- proximately. In contrast to the situation in atoms, where the exchange energy is much larger than the correlation energy, in dots it is possible to be in a low density regime where exchange and correlation are equally important. Consequently it has been found [9,10] that in contrast to atoms, the Hartree-Fock approximation is a poor approxi- mation for dots and that the local spin density approxima- tion is significantly better but nevertheless inadequate for predicting the correct ordering and values of the ground a e-mail: pederiva@science.unitn.it and excited state energies. Another disadvantage of the lo- cal spin density approximation is that the wave functions are not eigenstates of total spin ˆ S 2 . The configuration in- teraction or exact diagonalization method [11–14] has the advantage of being systematically improbable but suffers from an exponential increase in the computer time as a function of the number of electrons for fixed accuracy in the energy. Hence this method in practice yields accurate energies for only N ≤ 6 electrons. Some of the most accu- rate results to date have been obtained by the stochastic variational method with correlated Gaussian basis func- tions [15], but this method too scales badly with N and is limited to a similar or slightly larger number of elec- trons. Another method that has been used [16,17] is the path-integral Monte Carlo method but it has the disad- vantage that one cannot differentiate between states that differ only in the orbital angular momentum, L, quantum number. Instead, the diffusion Monte Carlo method is our method of choice because it offers the best compromise between computational time and accuracy. For bosonic ground states it yields essentially exact results, aside from statistical errors. For fermionic states it suffers from the “fixed-node error” but this error can in many cases be made very small by choosing well optimized trial wave functions. In our earlier work [9], we calculated ground and low-lying excited states of dots with N ≤ 13 electrons. The energies of ground and excited states with N ≤ 6 electrons were checked by the stochastic variational method [15] and excellent agreement was found for all states except one [18].