The potential energy curve and the electron density distribution for low-lying electronic states of the helium and lithium atoms confined by an anisotropic harmonic oscillator potential have been studied for different confinement parameters (ω x , ω y , ω z ) by using the quantum chemical configuration interaction (CI) method employing a Cartesian anisotropic Gaussian basis set supplemented by a quantum chemical standard Cartesian Gaussian basis set, respectively. In the case of the helium atom full CI wavefunctions have been used, and in the case of the lithium atom multi-reference CI wavefunctions have been employed. The results for the two confined quantum systems are compared to each other: parabolic-and non-parabolic-type potential energy curves are observed for states of both atoms in different type harmonic oscillator confinement. The non-parabolic potentials always appear in pairs, one with a double minimum and the other with a sharp edge at the minimum. They result from the interaction of energetically nearby states of the same symmetry. The electron density distributions of the states of the two atoms show deformation from spherical symmetry according to the symmetry and strength of the confining potential at the position of the atoms. The electron distribution of the states having non-parabolic potential energy curves is strongly deformed and occasionally shows curvilinear nodal planes of parabolic shape.
The potential energy curve and the electron density distribution for low-lying electronic states of the helium and lithium atoms confined by an anisotropic harmonic oscillator potential have been studied for different confinement parameters (ω x , ω y , ω z ) by using the quantum chemical configuration interaction (CI) method employing a Cartesian anisotropic Gaussian basis set supplemented by a quantum chemical standard Cartesian Gaussian basis set, respectively. In the case of the helium atom full CI wavefunctions have been used, and in the case of the lithium atom multi-reference CI wavefunctions have been employed. The results for the two confined quantum systems are compared to each other: parabolic-and non-parabolic-type potential energy curves are observed for states of both atoms in different type harmonic oscillator confinement. The non-parabolic potentials always appear in pairs, one with a double minimum and the other with a sharp edge at the minimum. They result from the interaction of energetically nearby states of the same symmetry. The electron density distributions of the states of the two atoms show deformation from spherical symmetry according to the symmetry and strength of the confining potential at the position of the atoms. The electron distribution of the states having non-parabolic potential energy curves is strongly deformed and occasionally shows curvilinear nodal planes of parabolic shape.
The three-dimensional Schrodinger equation for three electrons in a parabolic confinement potential (with strength measured by the frequency ω) can be decoupled into three pair problems, provided the expectation value of the center of mass vector R is small compared with the average distance between the electrons. This should be fulfilled at the strong correlation limit (small ω), where the electron system tends to crystallize. The remaining part of the Hamiltonian, which is not included in the independent pair model, is taken into account in perturbation theory. The complementary treatment of the weak correlation limit (large ω) considers noninteracting electrons as a zeroth-order approximation and includes the electron–electron interaction in perturbation theory. It turns out that both approaches match satisfactorily for intermediate ω. Our results are compared with those obtained with the Hartree–Fock, configuration interaction, multiconfigurational complete active space, and stochastic variational method approaches and the data from a Wigner crystal treatment in a harmonic approximation.
The present approach is an advancement of the author’s former attempts to develop an atom model of Helium with well-defined electron trajectories. Thus it calls in question the traditional quantum mechanics which assume—in contrast and as a consequence of Heisenberg’s uncertainty principle—electronic probabilities of presence. Its basic idea consists of the assumption that the motions of the two electrons are influenced by their spins exhibiting the value h/2π, but in two different ways: on the one hand, one spin induces a rotation; and on the other hand, the other spin induces a harmonic oscillation. A second important relation is given by the fact that the retroactive force of the oscillation process is due to the centrifugal force when the process runs along the surface of a sphere, whereas in usual oscillation processes—such as the one of a spring pendulum—it is due to a permanent shift between potential and kinetic energy. Therefore, in the present case, the potential energy remains constant since the distance between the nucleus and the—diametrically positioned—electrons remains constant. Considering these two conditions and the usual physical relations such as Coulomb attraction, centrifugal force, and the conservation laws of the angular momentum and of the energy, it was possible to compute the respective key values. Thereby, the deflection of the oscillation angle ψ = 45˚ is remarkable. Finally, the process is described using a Cartesian coordinate system with z as the rotation axis, a variable oscillation distance d and variable rotation velocities rrot. Thereby, the projections onto the x-axis and on the y-axis are not identically equal, leading to an elliptic projection shape. Thus this system is anisotropic, in contrast to the isotropic array of the conventional quantum mechanics according to Schrödinger, where the 1s-orbital is spherically symmetrical. This anisotropy explains the existence of interatomic Van der Waals forces, which enable the condensation of Helium, even though the condensation temperature is very low. But in particular, it exhibits well-defined electron waves, thus finally delivering the explanation of the hypothesis of Louis de Broglie, which has been established 100 years ago.
Three low lying energy levels of a 3D two-electron quantum dot (QD) with parabolic confinement are obtained by the variational method. The proposed interpolation formulas for the variation parameters allow one to recover the energy levels in a wide range of the Coulomb interaction constant. The quantum states of the QD are divided into the para- and ortho-states like in the theory of helium atom. The quantum transitions from the ortho-state to the para-state are possible only with account of the spin-orbit interaction. At low temperatures, an ensemble of two-electron QDs contains dots in the ground para-state and in the first excitedortho-state, which is metastable. These QDs have the entangled spin wave functions that are related to the Einstein–Podolsky–Rosen (EPR) states desirable for the quantum information protocol.
The problem of two Coulombically interacting electrons confined in a cylindrical potential box with impenetrable walls is solved by the method of full configuration interaction (CI) for states with zero axial component of the total orbital angular momentum, including the ground state ${}^{1}{\ensuremath{\Sigma}}_{g}^{+}$. The number of variables in the Schr\"odinger equation is reduced from six to five by separating the ${\mathrm{L\ifmmode \hat{}\else \^{}\fi{}}}_{z}$ operator from the complete Hamiltonian. Two-electron basis sets of symmetry-adapted configuration state functions are constructed from real eigenfunctions of the one-electron Hamiltonian for a cylindrical potential box. Kinetic energy integrals are evaluated analytically. Electron repulsion integrals are computed by developing a separable expression for the 1/r12 operator in the form of a double Fourier-Bessel-Fourier-cosine series and using enough terms to achieve a relative accuracy of ${10}^{\ensuremath{-}10}$ for the eigenvalues. Energy-level ordering and electron densities of selected electronic states are reported for cylindrical cavities of varying size with a fixed 1:1 length-to-diameter ratio. A remarkable feature of this system is that convergence of the full CI energy with respect to the one-electron basis set varies dramatically with the extent of confinement: For small cavities, the rate of approaching the basis-set limit is similar to that for two-electron atoms but is significantly faster for large cavities. Formation of singlet and triplet Wigner molecules is observed in cylindrical cavities with radii greater than about 5 bohrs.
Atomic systems subjected to external confinement exhibit a range of intriguing physical properties. In this work, we employ a well-established work-function-based Kohn-Sham density functional theory (DFT) within a generalized pseudospectral (GPS) method to determine the energy eigenvalues and eigenfunctions of Hookium (a two-electron system bound by a harmonic potential). We consider the two cases, viz. (i) Hookium and (ii) Hookium under the influence of a spherical cavity confinement. Two correlation energy functionals like Wigner and Lee-Yang-Parr (LYP) are considered to include explicit correlation energy in the calculation. Furthermore, we provide a comprehensive analysis of the quantum information-theoretic aspects of confined systems by examining the position-space Shannon and Fisher entropies.
At the limit of an infinite confinement strength ω, the ground state of a system that comprises two fermions or bosons in harmonic confinement interacting through the Fermi-Huang pseudopotential remains strongly correlated. A detailed analysis of the one-particle description of this "contactium" reveals several peculiarities that are not encountered in conventional model systems (such as the two-electron harmonium atom, ballium, and spherium) involving Coulombic interparticle interactions. First of all, none of the natural orbitals (NOs) {ψn(ω;r)} of the contactium is unoccupied, which implies nonzero collective occupancies for all the angular momenta. Second, the NOs and their non-ascendingly ordered occupation numbers {νn} turn out to be related to the eigenfunctions and eigenvalues of a zero-energy Schrödinger equation with an attractive Gaussian potential. This observation enables the derivation of their properties, such as the n-4/3 asymptotic decay of νn at the n→∞ limit (which differs from that of n-8/3 in the Coulombic systems), the independence of the confinement energy vn=⟨ψn(ω;r)|12ω2r2|ψn(ω;r)⟩ of n, and the n-2/3 asymptotic decay of the respective contribution νntn to the kinetic energy. Upon suitable scaling, the weakly occupied NOs of the contactium turn out to be virtually identical to those of the two-electron harmonium atom at the ω → ∞ limit, despite the entirely different interparticle interactions in these systems.