H.E. Montgomery

Exact Relations for Confined One-Electron Systems

Publisher Summary This chapter describes integrals of motion and boundary value problems. The chapter describes the mutual connections among the commutation relations and exponential transformations, for example, in the context of Hausdorfs relations. The coordinate transformation is a standard method for the analysis of boundary value problems. An important type of commutation relation is naturally connected with the scaling procedure. The possibility of using the virial/hypervirial relations for approximate wavefunctions depends on the form and nature of the classes of approximate functions. The chapter considers the one-parameter family of regions Ω (λ) and the corresponding boundary value problems, and also presents the main results concerning the energy changes with region modifications. Perturbation theory can be used to describe the splitting of electron energy levels for a hydrogen atom under small shifts from the center of an impenetrable spherical cavity, for example, it is demonstrated that first-order perturbation theory shows that for any s -state, the position of the atom in the center corresponds to a minimum.

Electron correlation energy in confined two-electron systems

Abstract Radial, angular and total correlation energies are calculated for four two-electron systems with atomic numbers Z = 0 – 3 confined within an impenetrable sphere of radius R. We report accurate results for the non-relativistic, restricted Hartree–Fock and radial limit energies over a range of confinement radii from 0.05 – 10 a 0 . At small R, the correlation energies approach limiting values that are independent of Z while at intermediate R, systems with Z ⩾ 1 exhibit a characteristic maximum in the correlation energy resulting from an increase in the angular correlation energy which is offset by a decrease in the radial correlation energy.

H2+ embedded in a Debye plasma: Electronic and vibrational properties

Abstract The effect of plasma screening on the electronic and vibrational properties of the H 2 + molecular ion was analyzed within the Born–Oppenheimer approximation. When a molecule is embedded in a plasma, the plasma screens the electrostatic interactions. This screening is accounted in the Schrodinger equation by replacing the Coulomb potentials with Yukawa potentials that incorporate the Debye length as a screening parameter. Variational expansions in confocal elliptical coordinates were used to calculate energies of the 1 s σ g and 2 p σ u states over a range of Debye lengths and bond distances. When the Debye length is comparable to the equilibrium bond distance, the dissociation energy is reduced while the equilibrium internuclear separation is increased. Expectation values, static dipole polarizabilities and spectroscopic constants were calculated for the 1 s σ g state.

Confined two-electron systems: excited singlet and triplet S states

Energies are reported for the first four singlet and triplet spherically symmetric states of 2e systems confined in an impenetrable sphere. All calculations used explicitly correlated Hylleraas basis sets. These calculations identify a series of level crossings and avoided crossings for Coulombic systems with a positive charge at the center of the sphere. Similar crossings do not occur for the 2e quantum dot. Configuration interaction calculations conducted in parallel provide a description of the system within the more traditional atomic orbital picture. The general shape of the energy versus confinement graphs and the locations of level crossings are shown to be consequences of scaling properties of the interparticle potentials.

Variational perturbation treatment of the confined hydrogen atom

The Schrodinger equation for the ground state of a hydrogen atom confined at the centre of an impenetrable cavity is treated using variational perturbation theory. Energies calculated from variational perturbation theory are comparable in accuracy to the results from a direct numerical solution. The goal of this exercise is to introduce the student to the effects of confinement on atomic systems using a tractable problem from which insight into variational perturbation theory may be gained.

Spherically symmetric states of Hookium in a cavity

When a two-electron atom or ion is enclosed in an impenetrable spherical cavity, level crossings and avoided crossings are observed as the cavity radius changes. The locations of those crossings depend on the cavity radius and the nuclear charge of the system. The question arises as to whether this crossing behavior is unique to the one-electron Coulomb potential or whether it can be observed in other confined single-particle electron potentials. In this work we examined some low-lying singlet and triplet states of two-electron systems with isotropic harmonic single-particle 3D potentials. The spherically symmetric S states are analyzed using variational energies calculated with Hylleraas-type function. The energy dependence of low-lying states is considered as a function of the cavity radius and the harmonic force constant. For positive force constants, there exist cavity radii where the 21S and 13S states are degenerate. Analogous points do not exist for the two-electron quantum dot where the one-electron potential corresponds to an infinite rectangular box. The structure of the energy spectrum for negative force constants is also studied. The similarities and differences of the two-electron S states for the Coulomb and harmonic potentials are considered.

How to spoil a good basis set for Rayleigh-Ritz calculations

For model quantum mechanical systems such as the harmonic oscillator and a particle in an impenetrable box, we consider the set of exact discrete spectrum functions and define the modified basis set by subtraction of the ground state wavefunction from all the other wavefunctions with some real weights. It is demonstrated that the modified set of functions is complete in the space of square integrable functions if and only if the series of the squared weights diverges. A similar, but nonequivalent criterion is derived for convergence of Rayleigh-Ritz ground state energy calculations to the exact ground state energy value with the basis set extension. Some numerical illustrations are provided which demonstrate a wide variety of possible situations for model systems.

Monotonicity in confined system problems

When an atom is confined at the center of a spherically symmetric impenetrable sphere, the electron density is altered. Of particular interest are the changes in the electron density at the center of the cavity as they affect important physical properties such as the isotropic constants of the hyperfine structure. This work presents four simple theorems that facilitate understanding the behavior of the electron density of the ground state with variations in the potential or in the radius of the impenetrable spherical cavity. The general statements are illustrated by discussion of the known results for confined and free systems and the results of simple evaluations for model hydrogen-like atoms in a cavity.

Frequency-dependent polarizabilities and shielding factors for confined one-electron systems

Frequency-dependent dipole polarizabilities and shielding factors are calculated for the ground state of spherically symmetric screened one-electron systems embedded in an impenetrable spherical cavity. Coulomb, Yukawa, Hulthen and exponential cosine-screened Coulomb potentials are considered. In contrast to free systems, Dirichlet boundary conditions introduce a contribution to the shielding factor that results from an integral over the surface of the confining boundary. This is a fundamental difference between free and confined systems and results in unexpected modifications to some of the classic relations for free systems. The methods derived also give a simple expression for the polarizability of the confined harmonic oscillator as an example of extending the methods of this work to potentials beyond the four studied.

One- and multiconfigurational study of excited states of He atom in a small impenetrable cavity

AbstractThe system of ten low-lying electronic states of a helium atom placed in the center of an impenetrable spherical cavity of radius Rc is studied for the case of small cavities. The methods for numerical and qualitative study of the atomic systems in small cavities are analyzed. The problem is studied by the finite-difference variants of the restricted SCF, the simple restricted CI and explicitly correlated Hylleraas-type approximations (for S-states). The limited applicability of the SCF approximation for excited states is demonstrated. The importance of the relatively low-lying configurations 2p2 and corresponding states 1S, 3P and 1D is confirmed by numerical calculations and correlation diagrams.