Schrodinger equation helium atom

The coordinates used to describe the helium atom are shown in Fig. 8.1. The Schrodinger equation, using atomic units and assuming infinite nuclear mass, can be written... 

The time-independent SchrOdinger equation for atomic, ionic or molecular systems containing two or more electrons does not yield exact solutions and approximation methods must be employed. To consider the simplest such atomic case, the helium atom, the time-independent SchrSdinger equation for the system is,... 

Unfortunately, the Schrodinger equation for multi-electron atoms and, for that matter, all molecules cannot be solved exactly and does not lead to an analogous expression to Equation 4.5 for the quantised energy levels. Even for simple atoms such as sodium the number of interactions between the particles increases rapidly. Sodium contains 11 electrons and so the correct quantum mechanical description of the atom has to include 11 nucleus-electron interactions, 55 electron-electron repulsion interactions and the correct description of the kinetic energy of the nucleus and the electrons - a further 12 terms in the Hamiltonian. The analysis of many-electron atomic spectra is complicated and beyond the scope of this book, but it was one such analysis performed by Sir Norman Lockyer that led to the discovery of helium on the Sun before it was discovered on the Earth. 

The many-electron Schrodinger equation cannot be solved exactly (or at least has not been solved) even for a simple two-electron system such as helium atom or hydrogen molecule. Approximations need to be introduced to provide practical methods. 

During this period, accurate solutions for the electronic structure of helium (1) and the hydrogen molecule (2) were obtained in order to verify that the Schrodinger equation was useful. Most of the effort, however, was devoted to developing a simple quantum model of electronic structure. Hartree (3) and others developed the self-consistent-field model for the structure of light atoms. For heavier atoms, the Thomas-Fermi model (4) based on total charge density rather than individual orbitals was used. 

Following the development of quantum theory by Heisenberg [1] and Schrodinger [2] and a few further discoveries, the basic principles of the structure of atoms and molecules were described around 1930. Unfortunately, the complexity of the Schrodinger equation increases dramatically with the number of electrons involved in a system, and thus for a long time the hydrogen and helium atoms and simple molecules as H2 were the only species whose properties could really be calculated from these first principles. In 1929, Dirac [3] wrote ... 

The simplest kind of ab initio calculation is a Hartree-Fock (HF) calculation. Modem molecular HF calculations grew out of calculations first performed on atoms by Hartree1 in 1928 [3]. The problem that Hartree addressed arises from the fact that for any atom (or molecule) with more than one electron an exact analytic solution of the Schrodinger equation (Section 4.3.2) is not possible, because of the electron-electron repulsion term(s). Thus for the helium atom the Schrodinger equation (cf. Section 4.3.4, Eqs. 4.36 and 4.37) is, in SI units... 

Schrodinger s equation is not analytically solvable for anything more complicated than a hydrogen atom. Even a helium atom, or the simplest possible molecule (H+) requires a numerical calculation by computer. However, these calculations give results which agree extremely well with experiment, so their validity is not doubted. 

In contrast to the Schrodinger equation for the hydrogen atom, the Schrodinger equation for a polyelectronic atom cannot be solved exactly. For example, although the hydrogen and helium atoms are similar in many respects, the mathematical descriptions of these atoms are fundamentally... 

Figure 8.1 Coordinates for helium atom Schrodinger equation.

As we move from one-electron to many-electron atoms, both the Schrodinger equation and its solutions become increasingly complicated. The simplest many-electron atom, helium (He), has two electrons and a nuclear charge of +2e. The positions of the two electrons in a helium atom can be described using two sets of Cartesian coordinates, (xi, yi, Zi) and (xi, yz, Zz), relative to the same origin. The wave function tf depends on all six of these variables if = (x, y, Zu Xz, yz Zz)-... 

Joslin and Goldman [105] in 1992 studied this problem by using the Diffusive Quantum Monte Carlo Methods. By resorting to the hard spherical box model, they performed calculations, not only on the ground state of helium atom, but also for H- and Li+. In this method the Schrodinger equation is... 

The one-particle equations for the wave functions p z, p) are solved by means of the fully numerical mesh method described in refs. [3,18,20]. In our first works on the helium atom in magnetic fields [18,20] we calculated the Coulomb and exchange integrals by means of a direct summation over the mesh nodes. But this direct method is very expensive with respect to the computer time and due to this reason we obtained in the following works [28-31] these potentials as solutions of the corresponding Poisson equation. The problem of the boundary conditions for the Poisson equation as well as the problem of simultaneously solving Poisson equation on the same meshes with Schrodinger-like equations for the wave functions p z, p) have been discussed in ref. [20],... 

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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