## Final Report Summary - 3DQDOTS (Many-Electron Wigner Molecules: 3-D quantum dots)

Exactly (quasi-)solvable models of quantum mechanics that pertain to real physical systems are few and far between. Among nontrivial instances of such models, the two-electron harmonium atom has attracted much attention from both chemists and physicists. Harmonium atoms differ from their ordinary counterparts only in the external potential and thus they are ideally suited for testing, calibration and benchmarking of approximate electronic structure methods of quantum chemistry. Despite of being of even greater interest, harmonium atoms with more than two electrons have been studied far less than their two-electron counterparts. The main objective of this research project is the study of few-electron harmonium atoms and its application as benchmark for electronic structure methods such as density functional theory (DFT).

The study of few-electron harmonium atoms required the development of the appropriate software to perform its calculations. In the first stage of the project we develop new computer software that permitted the full-configuration interaction (FCI) calculations of few-electron harmonium atom energies. In order to obtain very accurate estimates of the exact energies for these systems, we constructed basis sets that consisted of spherical Gaussian primitives with even-tempered exponents. The latter exponents were optimized using our own algorithm. Finally, these energies were extrapolated for both the number of primitives and the angular momentum providing a final energy value. We published the description the whole procedure and test its performance on two-electron harmonium atom. The ground-state energy was obtained with a few micro-Hartree accuracy.

The following project was devoted to the study of the two lowest lying states of three-electron harmonium, which needed of revision of the aforementioned algorithm in order to reduce the computational cost of our method. Meanwhile these calculations were performed, we obtained the analytical formulae for the second-order energy corrections of the weak-correlation limit of these two electronic states. The combination of the numerical results obtained for three-electron harmonium atoms and the analytical formulae were used to extrapolate the electronic energies. The latter possess accuracy of tens of micro-Hartree.

We are currently involved in the calculation of four-electron harmonium. Due to the computational cost of these calculations we needed to modify both the algorithm and the computer software. With the experience gained in the previous projects we have tuned our algorithm. We also designed computer software that could run in parallel (MPI protocol) on up to 64 cores.

Finally, we have used the results obtained to calibrate many DFT functionals available in the literature. Interestingly, none of the functionals tested can reproduce the energy of three-electron harmonium for the whole range of confinement parameters we used. Most importantly, the cost of these test calculations is computationally inexpensive, providing a formidable alternative to current benchmark sets, which involve large computational cost.

Overall, the results obtain in this project will be relevant for the calibration of DFT functionals. The next natural step is to study the wavefunction of three- and four-electron harmonium, which will be relatively simple to obtain from the data generated in this project; this information will be used to guide the construction of new DFT functionals. The forthcoming research will be devoted to this goal. Since the vast majority of computational calculations nowadays (we speculate 90%) are performed within the framework DFT, the results of this Marie Curie project are expected to have great impact on the field.

The study of few-electron harmonium atoms required the development of the appropriate software to perform its calculations. In the first stage of the project we develop new computer software that permitted the full-configuration interaction (FCI) calculations of few-electron harmonium atom energies. In order to obtain very accurate estimates of the exact energies for these systems, we constructed basis sets that consisted of spherical Gaussian primitives with even-tempered exponents. The latter exponents were optimized using our own algorithm. Finally, these energies were extrapolated for both the number of primitives and the angular momentum providing a final energy value. We published the description the whole procedure and test its performance on two-electron harmonium atom. The ground-state energy was obtained with a few micro-Hartree accuracy.

The following project was devoted to the study of the two lowest lying states of three-electron harmonium, which needed of revision of the aforementioned algorithm in order to reduce the computational cost of our method. Meanwhile these calculations were performed, we obtained the analytical formulae for the second-order energy corrections of the weak-correlation limit of these two electronic states. The combination of the numerical results obtained for three-electron harmonium atoms and the analytical formulae were used to extrapolate the electronic energies. The latter possess accuracy of tens of micro-Hartree.

We are currently involved in the calculation of four-electron harmonium. Due to the computational cost of these calculations we needed to modify both the algorithm and the computer software. With the experience gained in the previous projects we have tuned our algorithm. We also designed computer software that could run in parallel (MPI protocol) on up to 64 cores.

Finally, we have used the results obtained to calibrate many DFT functionals available in the literature. Interestingly, none of the functionals tested can reproduce the energy of three-electron harmonium for the whole range of confinement parameters we used. Most importantly, the cost of these test calculations is computationally inexpensive, providing a formidable alternative to current benchmark sets, which involve large computational cost.

Overall, the results obtain in this project will be relevant for the calibration of DFT functionals. The next natural step is to study the wavefunction of three- and four-electron harmonium, which will be relatively simple to obtain from the data generated in this project; this information will be used to guide the construction of new DFT functionals. The forthcoming research will be devoted to this goal. Since the vast majority of computational calculations nowadays (we speculate 90%) are performed within the framework DFT, the results of this Marie Curie project are expected to have great impact on the field.