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Cell["Electronic Structure of Multi-Electron Quantum Dots", "Title"],
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Ranga D. Muhandiramge
Jingbo Wang\
\>", "Author"],
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Quantum dots are artificially fabricated atoms, in which charge carriers are \
confined in all three dimensions just like electrons in real atoms. \
Consequently, they exhibit properties normally associated with real atoms \
such as quantised energy levels and shell structures. These properties are \
described by the electron wavefunctions whose evolution is governed by the \
Schr\[ODoubleDot]dinger equation and the Pauli exclusion principle. \
\>", "Text"],
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"There are many methods available to solve the Schr\[ODoubleDot]dinger \
equation for multiple electrons. They roughly fall into the categories of the \
diagonalisation method, mean-field density-functional theory, and the \
self-consistent field approach. One of the first theoretical studies of \
quantum dots was by Pfannkuche et al. [",
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calculations and exact diagonalisation of the Hamiltonian for two electrons \
in a circularly symmetric parabolic potential. They found good agreement \
between the two methods for the triplet state but marked differences for the \
singlet state, indicating important spin correlations were not included \
properly in their Hartree\[Dash]Fock model. This suggests that the proper \
treatment of electron spins is crucial for correctly obtaining the electronic \
structures in quantum dots."
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potential was approximated by an empirical polynomial expression. Lee et al. \
[",
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"-electron quantum dot using density-functional theory, where the \
generalised-gradient approximation was used for exchange-correlation \
potentials. Exchange interaction comes directly from the antisymmetrisation \
of wavefunctions as required by Pauli\[CloseCurlyQuote]s exclusion principle. \
In density-functional theory this is a major problem since the mathematical \
object is the electron density rather than the electron wavefunction, making \
evaluation of the exchange interaction intrinsically difficult."
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",
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eigenstates as the basis functions. This approach, namely the configuration \
interaction (CI) method, takes into account the full interaction and \
correlation of the electrons in the system as long as the numerical results \
converge with an increasing number of basis functions. However, such an \
approach involves the calculation of a very large number of interaction \
integrals and the inversion of large matrices, which can be prohibitively \
expensive in terms of computer resources. Reimann et al. [",
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elements for a six-electron quantum dot. Calculations for any higher number \
of electrons were not considered numerically viable using the conventional CI \
formalism, even with state-of-the-art computing facilities."
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"We have recently developed a spin-adapted configuration interaction (SACI) \
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antisymmetrised products of spatial wavefunctions and spin eigenfunctions. \
The SACI method has an advantage over using Slater determinants in that a \
smaller basis is used. This reduces the computational resources required, \
allowing calculations for dots with more than six electrons on a desktop \
computer. "
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"After some theoretical background, we present results from a ",
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wavefunctions of multiple electrons confined by a 2D harmonic well and under \
the influence of a perpendicular magnetic field. ",
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improves the speed and accuracy of calculations."
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can select only a finite number of basis elements out of an infinite set, we \
need to have some method of selecting appropriate basis elements. As we are \
usually interested in the ground and first few excited states of a quantum \
dot, we have chosen to use basis elements in increasing order of the sum of \
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are in maximum correspondence), then the matrix element is zero; when the \
spatial orbitals differ by two or less orbitals, a different rule applies for \
each of the three cases (i.e., zero, one, and two orbitals differ). These \
formulas are quite general and can be used for any interaction Hamiltonian \
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In the Hamiltonian element formulas we must calculate the two-electron \
interaction integral. For the case of electrons in a 2D harmonic well \
interaction through a Coulomb potential under the influence of a \
perpendicular magnetic field, the integral is in the following form: \
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Here are some examples of interaction Hamiltonian matrix elements indicating \
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It can also be proved that the interaction element is zero if the sum of the \
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The preceding calculations show that the spin eigenfunction with which a \
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In this section we will demonstrate how the package can be used to do various \
calculations such as energy levels, wavefunctions, convergence plots, and \
energy level evolutions with magnetic fields.\
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To calculate energy levels, we need to choose a basis. For larger basis sizes \
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As mentioned, the accuracy of our result as well as the computational time \
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As the interaction Hamiltonian for lower cutoff energies is a submatrix of \
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The list basis size can be specified manually, but a convenient method is to \
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We can see that we end up with a numerical approximation to the wavefunction. \
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