Confined many-electron atoms

Garza, J., Vargas, R. and Vela, A. 1988. Numerical self-consistent-field method to solve the Kohn-Sham equations in confined many-electron atoms. Phys. Rev. E. 58 3949-54. 

For this kind of confinement, the solution of the non-relativistic time independent Schrodinger equation has been tackled by different techniques. For confined many-electron atoms the density functional theory [6], using the Kohn-Sham model [7], has given some estimations of the non-classical effects [8-11], through the exchange-correlation functional. An elementary review of this subject can be found in Ref [12], where numerical techniques are discussed to solve the Kohn-Sham equations. Furthermore, in this reference, some chemical predictors are analyzed as a function of the confinement radii. 

For two-electron atoms, many approaches have been applied a review made by Aquino reported the techniques used up to 2009 [20], To date, the expansion of the wave function in terms of Hylleraas-type functions is the technique that gives the lowest energies for several confinement radii [21-23], which can be used as reference when other techniques are proposed for the study of these systems. However, such a technique has not been used for atoms with several electrons, for example, beryllium. In this sense, in this chapter we test the many-body perturbation theory to second order, as a technique to estimate the CE for confined many-electron atoms. In the next section, we discuss the theory behind of the HF method, and the basis set proposed for its implementation for confined atoms. In the same section, the many-body perturbation theory to second order proposed by Moller and Plesset (MP2) [24] also is discussed, and we give some details about the implementation of our code implemented in GPUs [25]. Finally, we contrast our results for helium-like atoms with more sophisticated techniques in order to know the percent of correlation energy recovered by the MP2 method. 

In Chapter 5, a successful application of second order many-body perturbation theory to estimate the CE for confined many-electron atoms has been depicted. The second order many-body perturbation theory proposed by Moller and Plesset and detailed implementation of code in GPU... 

The specific structure of the states for Hp was described in detail in [79], where it is mentioned as a well-known physical effect. For example, it was noted in the theory of disordered semiconductors that a similar "ladder" structure of states is realized for the system where the Coulomb potential is modified within a sphere as a constant potential (see [86,87] for a qualitative discussion and analytical solution of the problem). For quantum chemistry, the situation is interesting, as was shown in a series of publications of Connerade, Dolmatov and others (see e.g. [19,88-91] note that the series of publications on confined many-electron systems by these authors is much wider). The picture described is realized to some extent for the effective potential of inner electrons in multi-electron atoms, as it is defined by orbital densities with a number of maximal points. The existence of a number of extrema generates a system of the type described above [89]. This situation was modeled and described for the one-electron atom in [88] it is similar to that one described in Sections 5.2 and 5.3. 

C. Diaz-Garda, S. A. Cruz, Many-electron atom confinement by a penetrable spherical box, Inti. J. Quantum Chem. 108 (2008) 1572-1588. 

The confinement model has been extensively used to analyze the hydrogen atom enclosed by hard and soft spherical boxes [1-98], with confining boxes of diverse geometrical shapes [31,88-98], and it has also been applied to studies of the helium atom [99-122], many-electron atoms [50, 52,55,123-131], molecules [132-142] and the harmonic oscillator [143-171], among others. 

An atom confined within a sphere, of radius Rc, with rigid walls has been used as a model to give an insight into the behavior of electrons confined under high pressures [1-3]. For many-electron atoms, this model has been applied by using the Hartree-Fock (HF) [4,5] and the Kohn-Sham (KS) model [6], some applications of these methods can be found in Refs. [7-12]. 

Thomas-Fermi-Dirac-Weizsacker Density Functional Formalism Applied to the Study of Many-electron Atom Confinement by Open and Closed Boundaries... 

Many-electron Atom Confinement by Closed Boundaries 257... 

In this contribution, the adequacy of the TFDA.W method for the study of many-electron atom confinement within different confinement conditions -by closed and open boundaries - is explored. We begin by making a brief review of the main strategy followed in the TFDA.W method to account for the study of many-electron atoms confined by hard and soft spherical boxes. Here, important quantitative corrections to our previous studies are introduced, leading to better agreement with other reference calculations,... 

TFDAW Density Functional Formalism Applied to Many-electron Atom Confinement... 

MANY-ELECTRON ATOM CONFINEMENT BY CLOSED BOUNDARIES... 

We now consider a many-electron atomic system confined by a spherical cage of radius R with a confining barrier potential Vc, such that... 

Let us consider a many-electron atom of nuclear charge Z confined by a hard prolate spheroidal cavity. In this study the nuclear position will correspond to one of the foci as shown in Figure 4. In terms of prolate spheroidal coordinates, the nuclear position then corresponds to one of the foci for a family of confocal orthogonal prolate spheroids and hyperboloids defined, respectively, by the variables f and rj as [73] ... 

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