The Helium-Like Atom

The helium nucleus is even more massive than the hydrogen nucleus, and we will assume that the nucleus is stationary while studying the motion of the electrons. Of course, the nucleus can move, but as it moves, the electrons follow along, adapting to a new location of the nucleus almost as though it had always been there. This is similar to ignoring the motion of the earth around the sun when we describe the motion of the moon relative to the earth. 

Assuming a stationary nucleus, the classical Hamiltonian function of the electrons in a helium-like atom with Z protons in the nucleus is 

The Hamiltonian operator of Eq. (18.1-2) gives a Schrodinger equation that cannot be solved exactly. No three-body system can be solved exactly, either classically or 

We substitute the trial solution ofEq. (18.1-6) intoEq. (18.1-5) and use the fact that treated as a constant when //hl(2) operates and 2(2) is treated as a constant when Hhl(1) operates. The result is  

The variables are now separated. That is, each of the terms on the left-hand side of the equation depends only on a set of variables not occurring in the other term. Because each set of variables can be allowed to range while the other is fixed, the first term must equal a constant, which we call Ei, and the second term must equal a constant, Ez. These constants must add to the approximate energy eigenvalue 

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Using the finite-size scaling method, study of the analytical behavior of the energy near the critical point shows that the open-shell system, such as the lithium-like atoms, is completely different from that of a closed-shell system, such as the helium-like atoms. The transition in the closed-shell systems from a bound state to a continuum resemble a first-order phase transition, while for the open-shell system the transition of the valence electron to the continuum is a continuous phase transition [9]. 

The antiprotonic helium system was used as a model when developing our nonzero angular momentum 3D finite element method. This is an example of a system for which the wave function cannot exactly be decomposed into an angular and a radial part. Besides the helium like atoms it is the experimentally most accurately known three-body system. 

An effective Hamiltonian, that contains all QED corrections of the order ma aZY including radiative and nonradiative ones, was derived by Araki [48] and Sucher [49 for the helium-like atom. Later it was generalized to the N-electron atom [50]. 

For a prototypical three body problem, the Helium-like atom, the procedure is well known since the early days of quantum mechanics. More recently, Fano, Macek and Klar [18-23] identified a near separable variable p = (rf -I- rD, where rj and T2 are the two Jacobi vectors of the system, named hyperradius , corresponding to the radius of a six-dimensional hypersphere pareuneterized by five hyperangles . However, note that the hyperradius is independent of the numbering of particles and is therefore very useful for rearrangement problems. 

We first consider the helium-like atom, with a nucleus of charge Z fixed at the origin. The electronic Hamiltonian defined in Section 1.1 takes the form... 

In Cartesian coordinates with the origin at the nucleus, the nonrelativistic electronic Hamiltonian of the helium-like atoms may be written in the usual manner as... 

Wilets, L., and Cherry, I. J., Phys. Rev. 103, 112, Lower bound to the ground state energy and mass polarization in helium-like atoms. ... 

The molal diamagnetic susceptibilities of rare gas atoms and a number of monatomic ions obtained by the use of equation (34) are given in Table IV. The values for the hydrogen-like atoms and ions are accurate, since here the screening constant is zero. It was found necessary to take into consideration in all cases except the neon (and helium) structure not only the outermost electron shell but also the next inner shell, whose contribution is for argon 5 per cent., for krypton 12 per cent., and for xenon 20 per cent, of the total. 

For all intents and purposes then, we are concerned here with the CCSD (coupled cluster with all single and double substitutions [30]) correlation energy. Its convergence is excruciatingly slow Schwartz [31] showed as early as 1963 that the increments of successive angular momenta l to the second-order correlation energy of helium-like atoms converge as... 

We have already dealt with the calculation of the wave functions of the hydrogen atom. We now proceed to consider many-electron atoms, first dealing with the simplest such example, the helium atom which possesses two electrons. The Hamiltonian for a helium-like atom with an infinitely heavy nucleus can be obtained by selecting the appropriate terms from the master equation in chapter 3. The Hamiltonian we use is... 

An important advantage of hrst method is the possibility of using different expressions for the atomic potential, and the calculations can be done not only for a purely Coulomb interaction, but in the multiconhguration interaction approximation, the Hartree-Fock-Dirac approximation, and the relativistic random phase approximation with exchange effects. The most exact relativistic calculations were done in [12] for the polarizability of the ground state of a helium-like atom. 

From the present calculations, the expectation value of the operator r 2 may provide a direct physical picture about the thermodynamic stability and dissociation of Hj-like molecules. As shown in Fig. 16, there is a vertical jump of the mean value ru at Xc. We note that there are similarities and differences between helium-like atoms and Hj-like molecules. In Section V.A of heliumlike systems, based on an infinite mass assumption, we show that the electron at the critical point leaves the atom with zero kinetic energy in a first-order phase transition. This limit corresponds to the ionization of an electron as the nuclear charge varies. For the Hj-like molecules, the two protons move in an electronic potential with a mass-polarization term. They move apart as X approaches its critical point and the system approaches its dissociation limit through a first-order phase transition. 

Still the agreement was not satisfactory. We also noted that some QED corrections could be of the same order of magnitude as the relativistic corrections. Lamb shift for one electron in a helium-like atom with a nuclear charge Z can be expressed as ... 

The terms give the fe-th order contributions to the energy of the helium-like core, and are independent of the valence state. These are precisely the terms evaluated in the previous section. The terms are the fe-th order contributions to the energy of the atom relative to the ionic core in other words is the fc-th order contribution to the... 

The hyperspherical coordinates, which will be discussed in detail in the present article, represent a generalization to any mass of the near separability for the hyperradius explored for Helium-like atom. Jacobi vectors, besides the reduction of the dimensionality of the problem eliminating the center of mass coordinates, allow us, after a proper mass-seeding, to express the kinetic energy of the system in a diagonal form which depends only on the reduced mass of the system. 

The Hylleraas expansion is well known to be an efficient tool for obtaining accurate results for two-electron systems. The complex-rotation transformation does not effect the angular properties of this expansion Its capability of representing the angular electron correlation effects to the infinite order is preserved. Computations for helium-like atoms employing the Hylleraas-type expansion have been performed for two decades and have given numerous accurate results (6,98-111). 

This classic text is the standard reference for the Dirac theory of hydrogen- and helium-like atoms. Bethe had already published a review article on one- and two-electron atoms in Handbuch der Physik in 1933 [72], which is very readable — especially because it demonstrates the early difficulties with a quantum theory of boimd-state electrons and their interactions. 

This classic monograph by Bethe and Salpeter should also be recommended as further reading. Since it considers two-electron, i.e., helium-like atoms, it contains a discussion of the Breit equation. However, this discussion is mostly from the point of view of quantum electrodynamics. It does not rest on Darwin s classical potential energy expression subjected to the correspondence principle. While the former is the more fundamental p>oint of view, in quantum chemistry the latter is sufficient as we neglect or model many physical effects of little importance in chemistry (such as radiative corrections) an5rway. However, this book also contains much material on how radiative corrections can be considered, if this is desired. 

The helium atom contains two electrons and a nucleus containing two protons. We define a helium-like atom to have two electrons and a nucleus with Z protons, so that Z = 2 represents the He atom, Z = 3 represents the Li+ ion, and so on. The heliumlike atom is shown in Figure 18.1. Our treatment will apply to any value of Z. With the hydrogen-like atom, the motion of the electron relative to the nucleus was equivalent to the motion of a fictitious particle with mass equal to the reduced mass of the electron and the nucleus. Replacing the reduced noass by the mass of the electron in our formulas was equivalent to assuming a stationary nucleus. This was a good approximation, because the mass of the nucleus is large compared to mass of the electron, which means that the nucleus remains close to the center of mass. 

A more serious omission is that we have not discussed the application of any of these techniques to the study of the Lamb shift in hydrogenic systems and the fine structure of the light elements H, He, Li, etc. The development of our understanding of the fine structure of hydrogen-like and helium-like atoms is reported in the proceedings of the International Conferences on Atomic Physics which are published under the title Atomic Physics, Vols. 1-4. 

Two-electron systems are the most studied systems in quantum mechanics due to the fact that they are the simplest systems that contain the electron-electron interaction, which is a challenge for the solution of the Schrodinger equation [1], In particular, helium-like atoms are used many times as a reference to apply new theoretical and computational techniques. Additionally, in recent years the study of many-electron atoms confined spatially have a particular interest since the confinement induces important changes on the electronic structure of these systems [2, 3], The confinement imposed by rigid walls has been quite popular from the Michels proposal made 76 years ago [4], followed by Sommerfeld and Welker one year later [5]. Such a model assumes that the external potential has the expression... 

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. 

For example, nitrogen ( N ) has five valence electrons and needs three more electrons to complete its octet. Chlorine (-CL) has seven valence electrons and needs one more electron to complete its octet. Argon OArO already has a complete octet and has no tendency to share any more electrons. Hydrogen (H-) needs one more electron to reach its helium-like duplet. Because hydrogen completes its duplet by sharing one pair of electrons, we say that it has a valence of 1 in all its compounds. In general, the valence of an element is the number of bonds that its atoms can form. 

As an example we may calculate the energy of the helium atom in its normal state (24). Neglecting the interaction of the two electrons, each electron is in a hydrogen-like orbit, represented by equation 6 the eigenfunction of the whole atom is then lt, (1) (2), where (1) and (2) signify the first and the second electron. 

A treatment of the hydrogen molecule by the Ritz method, applied to helium by Kellner (25), has been reported by S. C. Wang (Phys. Rev., 31, 579 (1928)). With this method the individual eigenfunctions p and

hydrogen-like eigenfunctions of an atom with atomic number Z differing from unity. The value found for Z is 1.166, and the... 

Evidence has been advanced8 that the neutral helium molecule which gives rise to the helium bands is formed from one normal and one excited helium atom. Excitation of one atom leaves an unpaired Is electron which can then interact with the pair of Is electrons of the other atom to form a three-electron bond. The outer electron will not contribute very much to the bond forces, and will occupy any one of a large number of approximately hydrogen-like states, giving rise to a roughly hydrogenlike spectrum. The small influence of the outer electron is shown by the variation of the equilibrium intemuclear distance within only the narrow limits 1.05-1.13 A. for all of the more than 25 known states of the helium molecule. 

The elements helium, neon, argon, krypton, xenon, and radon—known as the noble gases—almost always have monatomic molecules. Their atoms are not combined with atoms of other elements or with other atoms like themselves. Prior to 1962, no compounds of these elements were known. (Since 1962, some compounds of krypton, xenon, and radon have been prepared.) Why are these elements so stable, while the elements with atomic numbers 1 less or 1 more are so reactive The answer lies in the electronic structures of their atoms. The electrons in atoms are arranged in shells, as described in Sec. 3.6. (A more detailed account of electronic structure will be presented in Chap. 17.)... 

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