Helium, confined energies

Recently, Banerjee, Kamal and Chowdhury [118] computed confined helium atom energies for the ground and three low-lying excited states using a two parameter wave function that represents a generalization of an ansatz proposed by Le Sech [195] for free atomic systems. 

In this work we have also presented some of the most relevant methods utilized in various studies of the confined hydrogen with the nucleus located off-centre and when the atom is confined by imposing Neumann boundary conditions. Likewise, the most important methods applied in the analysis of a helium atom confined in spherical boxes of penetrable and impenetrable boxes, and by placing the nucleus off-centre in such cavities. Unlike the CHA problem, the accuracy attained in a confined helium atom energy calculation is around 1 x 10-5 hartrees. At present, only a few of the low-lying state energies for the helium atom have been obtained with the latter precision. 

Considering the overall impurity content we have also to take into account seeded impurities (see Sect. 1.3) and the fusion product helium (the fusion ash ). A stationary burning fusion plasma can only be obtained when the helium is exhausted sufficiently fast. The corresponding figure of merit is given by the ratio of the effective He exhaust time over the energy confinement time Pile = THe/Ve [9],... 

Here we consider [25] the properties of H at the centre of a spherical box of radius R, using a numerical approach to obtain the energies and polarizabilities. We also develop some model wave functions, simple expressions for the energies and polarizability, deduce the critical radius R for which E = 0, and extend the analysis to the confined helium atom with effective screening. 

Then the ground state energies of the confined helium atom are given by... 

The reported calculations on confined helium focus on studying the associated electronic properties under conditions of extreme pressure, in which the electron clouds, unlike those in free atoms, are forced to remain spatially restricted. A particularly important aspect refers to the systematic analysis of how energy and electronic correlation vary as a function of the confining cavity dimension. 

Table 9 Ground state energy of the helium atom confined by an impenetrable spherical box of radius R obtained by ten Seldam and de Groot [99]. Distances and energies are given in bohrs and hartrees, respectively...

Table 11 Energy for the confined helium atom lowest triplet state 1 3S as a function of the box radius R obtained by Aquino et al. [117], Patil and Varshni [49] and Banerjee et al. [118], Energies and distances are given in hartrees and bohrs, respectively. The exact energy for the free (unconfined) system is —2.17523 hartrees...

In the late 60 s Gimarc [101] analyzed the confined helium atom problem by systematically studying the correlation energy in a two electron atom. The correlation energy is defined as the difference between the Hartree-Fock energy and the exact value,... 

The correlation energies for free (unconfined) H, He, Li+ and Be++ were well known at that time, and so Gimarc wanted to analyze, in particular, how the correlation energy changes as a function of the box radius for the confined helium atom isoelectronic series. Gimarc performed a number of variational calculations based on the following wave functions ... 

In 1978, Ludena [102] carried out a Hartree-Fock calculation by using a wave function consisting of a single Slater determinant for the closed-shell atoms, whereas he used a linear combination of the Slater determinants for the open-shell atoms. Each Slater-type orbital times a cut-off function of the form (1 — r/R) to satisfy the boundary conditions. Ludena studied pressure effects on the electronic structure of the He, Li, Be, B, C and Ne neutral atoms. The energies he obtained for the confined helium atom are slightly lower than those Gimarc obtained, especially for box radii in the range R > 1.6 au. 

In a first report, Marin and Cruz [16] studied the helium atom confined in an impenetrable spherical box where they used the direct variational method to optimize the energy value. The Hamiltonian for a spherically confined helium atom within a hard box is given by... 

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