Endohedrally confined hydrogen atom with a moving nucleus

Abstract

We studied the hydrogen atom as a system of two quantum particles in different confinement conditions; a spherical-impenetrable-wall cavity and a fullerene molecule cage. The motion is referred to the center of spherical cavities, and the Schrödinger equation solved by means of a Generalized Sturmian Function expansion in spherical coordinates. The solutions present different properties from the ones described by the many models in the literature, where the proton is fixed in space and only the electron is considered as a quantum particle. Our results show that the position of the proton (i.e. the center of mas of the H atom) is very sensitive to the confinement condition, and could vary substantially from one state to another, from being sharply centered to being localized outside the fullerene molecule. Interchange of the localization characteristics between the states when varying the strength of the fullerene cage and mass occurred through crossing phenomena. I INTRODUCTIONBeing the simplest atomic system, hydrogen is one of the most extensively studied elements of the periodic table. The atom has a relatively simple mathematical description through the analytical solution of the twobody Schrödinger equation, which allows a comprehension of its electronic structure, its quantum states, its discrete nature of energy levels, and other related properties. From the experimental side, it has been bombarded with electrons [1], protons [2, 3] and photons [4,5], combined with other chemical elements, and more recently [6,7] confined in fullerene structures.The wave functions which theoretically describe these processes are usually obtained from the hydrogen-like Schrödinger equation, by separating the center of mass and relative coordinates [8]. The problem maps to a central field problem for a particle of reduced mass µ, and since µ is very similar to the electron mass, it is usually interpreted as if the nucleus were fixed at the center of the coordinate system. Actually, the center of mass has also a quantum behavior, but it is not generally considered, since the interesting properties such as ionization energies or the excited states structure, only depend on the relative dynamics.The same approach is sometimes applied when the atomic confinement is modeled [9][10][11][12], where the proton is generally considered as an infinitely massive particle fixed in space, acting as a Coulomb center for the electrons clamped somewhere in the box [13,14]. In the case where the proton is not centered, the separation of coordinates is not possible as discussed by Tanner [15] and Amore (and Fernández) [16] for the harmonic oscillator. The way to deal with the system while keeping the two-body simplicity is considering the electron-wall interactions of a spherical box or fullerene molecule through boundary conditions [17] or central potentials [18] respectively, as a function of the radial coordinates. The movement of the nucleus can then be considered perturbatively [13], or in the Born-Oppenheimer approximation, where the ene...

On the shell‐confined atom problem

“…We use here the description of the problem [45] (see eqs. [15][16][17][18] for the case of the ks-state of the spherically symmetric one-electron system with the exact stationary wave-function Ψ = Ψ ks . In the presence of the external electric field with constant strength F and the potential −Fz one can use for the first-order correction to the wave function as an approximation by linear combination of M functions of the form f j Ψ ( j = 1/M)…”

“…Ludlow et al have investigated the polarizability of noble gas atoms inside fullerene shells and pointed out the possibility of using confinement to tune the polarizability of an atomic system. Very recently Randazzo and Rios have shown that the position of the proton in an endohedrally confined hydrogen atom is very sensitive to the election density inside the cage, while Aquino et al have calculated the energies of a hydrogen atom with a nucleus of finite size confined within a penetrable sphere. The papers mentioned allow the conclusion that a detailed study of the dependence of the energy and polarizability of a shell‐confined system on the confinement geometry will be a useful addition to the growing literature on shell‐confined systems.…”

Quantum mechanical virial-like theorem for confined quantum systems

“…Finally, in the penetrable, smooth case it becomes, V = V (r) + V c (r) [14]. In recent years, various models were proposed and investigated by many authors [3,[15][16][17][18], especially in the context of H atom, maintaining these confinement conditions, revealing numerous striking features [1,3,[19][20][21].…”

Three‐body molecular states of the system in the Born–Oppenheimer approximation

“…Basically, it allows us to deal with the bound states of two‐electron atoms, as well as the two electron continuum associated to single and double ionization by electron, ion or photon impact . Recently, we have applied to determine the collective dynamics of a confined proton‐electron pair without resorting to the BO approximation . The method is based on the Sturmian expansion in two of the three interparticle distances or Jacobi pairs (tested here for the first time), which is able to adequately impose a variety of asymptotic conditions such as the stationary ones associated to bound states or the outgoing (incoming) wave behavior corresponding to the scattering states.…”

“…[13,14] Recently, we have applied to determine the collective dynamics of a confined proton-electron pair without resorting to the BO approximation. [15] The method is based on the Sturmian expansion in two of the three interparticle distances or Jacobi pairs (tested here for the first time), which is able to adequately impose a variety of asymptotic conditions such as the stationary ones associated to bound states or the outgoing (incoming) wave behavior corresponding to the scattering states. Due to the masses of the particles involved in these systems, for low energies the partial wave expansion converges with a relatively low number of terms.…”