Helium quantum mechanical model

The electron correlation problem occurs with all polyelectronic atoms. To treat these systems using the quantum mechanical model, we must make approximations. The simplest approximation involves treating each electron as if it were moving in a field of charge that is the net result of the nuclear attraction and the average repulsions of all the other electrons. To see how this is done, let s compare the neutral helium atom and the He+ ion ... 

Although the helium atom can be readily described in terms of the quantum mechanical model, the Schrodinger equation that results cannot be solved exactly. The difficulty arises in dealing with the repulsions between the electrons. Since the electron pathways are unknown, the electron repulsions cannot be calculated exactly. This is called the electron correlation problem. 

At the begirming of this chapter, we learned that the quantum-mechanical model explained the chemical properties of the elements such as the inertness of helium, the reactivity of hydrogen, and the periodic law. We can now see how The chemical properties of elements are largely determined by the number cf valence electrons they contain. Their properties vary in a periodic fashion because the number of valence electrons is periodic. 

Since elements within a column in the periodic table have the same niunber of valence electrons, they also have similar chemical properties. The noble gases, for example, all have 8 valence electrons, except for heliiun, which has 2 ( Figure 9.29). Although we don t get into the quantitative (or numerical) aspects of the quantum-mechanical model in this book, calculations show that atoms with 8 valence electrons (or 2 for helium) are particularly low in energy, and therefore stable. The noble gases are indeed chemically stable, and thus relatively inert or nonreactive as accounted for by the quantum model. 

Fig. 32 shows on the left a conventional, localized domain model of the electronic environment of an atom that satisfies the Octet Rule. Each domain is occupied by two electrons. It is well known, however, that the assumption of two electrons per orbital is unnecessarily restrictive 27>122>. Better energies are obtained in quantum mechanical calculations if different orbitals are used for electrons of different spins, a fact first demonstrated in quantitative calculations on helium by Hylleraas 123> and Eckart 124>. Later, this "split-orbital method was applied to 71-electron systems 27,125) Its general application to chemical systems has been developed by Linnett 126>. 

One of the best reasons for studying the one-dimensional model of helium is the search for the quantum mechanical manifestations of classical chaos in the helium atom. Since (10.2.4) contains much of the essential physics of the three-dimensional helium atom, it is a natural starting point for quantum chaos investigations. 

Fig. 10.10 proves that a close connection exists between the classical mechanics and the quantum mechanics of the simple one-dimensional two-electron model. On the basis of the evidence provided by Fig. 10.10, there is no doubt that classical periodic orbits determine the structure of the level density in an essential way. The key element for establishing the one-to-one correspondence between the peaks in R and the actions of periodic orbits is the scaling relations (10.3.10). Similar relations hold for the real helium atom. Therefore, it should be possible to establish the same correspondence for the three-dimensional helium atom. First steps in this direction were taken by Ezra et al. (1991) and Richter (1991). 

The quantum mechanics of the s-wave model of the helium atom was investigated by Draeger et al. (1994). It turns out that for certain classes of states the energy levels of s-wave helium are very close to the energy levels of the real helium atom. 

The collinear model (Eq. (15)) has been successfully used in the semiclassical description of many bound and resonant states in the quantum mechanical spectrum of real helium [49-52] and plays an important role for the study of states of real helium in which both electrons are close to the continuum threshold [53, 54]. The quantum mechanical version of the spherical or s-wave model (Eq. (16)) describes the Isns bound states of real helium quite well [55]. The energy dependence of experimental total cross sections for electron impact ionization is reproduced qualitatively in the classical version of the s-wave model [56] and surprisingly well quantitatively in a quantum mechanical calculation [57]. The s-wave model is less realistic close to the break-up threshold = 0, where motion along the Wannier ridge, = T2, is important. 

Claiming the periodic table is explained by traditional quantum mechanics is flat wrong (Scerri 1998a, b, c).20 The new model, however, has much to say. One problem is the issue of the Group II elements. By all rights they should be chemically inert. The valence electrons are paired. A rather subtle result from the new model is the issue of the spin-zero boson. If paired electrons approach closely enough, as in the core of heavier atoms, or, even as simple a system as helium, the constructive interference... 

The Bohr model gave the correct energies for the hydrogen atom but failed when applied to helium. Hence, in the early days of quantum mechanics, it was important to show that the new theory could give an accurate treatment of helium. The pioneering work on the helium ground state was done by Hylleraas in the years 192 1930. To allow for the effect of one electron on the motion of the other, Hylleraas used variational functions that contained the interelectronic distance ri2- One function he used is... 

Molecular quantum chemistry and quantum mechanical simulation of solids have followed substantially independent paths and strategies for many years, with almost no reciprocal influence. In the implementation of computational schemes and formalisms, they started from different elementary models either the hydrogen or helium atom like, for example, the parameterization of a correlation functional based on accurate He atom calculations by Colle and Salvetti, or the electron gas, which is the reference system of the local density approximation "" (LDA) to density functional theory (DFT). Moreover, if we compare the simplest real crystals, like lithium metal or sodium chloride, with the smallest molecule, H2, the much greater complexity of the solid system is... 

More than 80 years ago, helium was found to exhibit a liquid-liquid phase transition at very low temperature T < 3K) [19]. However, this phase transition is due to quantum mechanical effects (and it is not a first-order phase transition either). That first-order phase transitions could exist in classical liquids, at much higher temperatures than that characterizing helium s LLPT, was not realized until much more recently. In 1967, Rapoport published an article on the anomalous melting curve maxima observed in systems such as cesium and rubidium [20]. His work was based on statistical mechanics calculations using a two-species model for liquids. In this work, he noticed that for particular parameterizations, the model predicted the existence of an LLPT. However, due to lack of experimental evidence at that time, he did not explore the predictions of the model for polymorphic liquids [20,21]. Similar models to that used in Ref. [20] were studied by Aptekar and Ponyatovsky (see Ref. [22] and references therein). 

K. Raghavachari and L. A. Curtiss, in Quantum-Mechanical Prediction of Thermochemical Data, J. Cioslowski (Ed.) Vol. 22 of, Understanding Chemical Reactivity, Kluwer Academic Publishers, 2001 chapter 3, pp. 67-98, Complete Basis Set Models for Chemical Reactivity From the Helium Atom to Enzyme Kinetics. 

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... 

Before discussing in detail the numerical results of our computational work, we describe the theoretical and computational context of the present calculations apart from deficiencies of models employed in the analysis of experimental data, we must be aware of the limitations of both theoretical models and the computational aspects. Regarding theory, even a single helium atom is unpredictable [14] purely mathematically from an initial point of two electrons, two neutrons and two protons. Accepting a narrower point of view neglecting internal nuclear structure, we have applied for our purpose well established software, specifically Dalton in a recent release 2.0 [9], that implements numerical calculations to solve approximately Schrodinger s temporally independent equation, thus involving wave mechanics rather than quantum... 

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