Many-electron atoms Schrodinger equation

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

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. 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

In many-electron atoms, the Schrodinger equation cannot be solved exactly, so approximations must be made. The simplest and crudest approximation is to neglect entirely electron-electron interactions (repulsions) and electron spin. In this way, hydrogenic orbitals are found as solutions. Into these orbitals we then place the electrons, according to the aufbau principle, and thus derive electron... 

Having decided to use AOs (or combinations of them) for yrA and pB> we will now look at the form these take. They are approximate solutions to the Schrodinger equation for the atom in question. The Schrodinger equation for many-electron atoms is usually solved approximately by writing the total electronic wavefunction as the product of one-electron functions (these are the AOs). Each AO 4>i is a function of the polar coordinates r, 0, and single electron and can be written as... 

Because of interelectronic repulsions, the Schrodinger equation for many-electron atoms and molecules cannot be solved exactly. The two main approximation methods used are the variation method and perturbation theory. The variation method is based on the following theorem. Given a system with time-independent Hamiltonian //, then if

well-behaved function that satisfies the boundary conditions of the problem, one can show (by expanding

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Better approximations can be made, and numerical calculations leave no doubt that Schrodinger s equation works very accurately for many-electron atoms, as it does for hydrogen. However, the orbital approximation is good enough for most purposes, and it leads to the very appealing picture of a many-electron atom in which each electron occupies an orbital which is similar to, although not identical with, the orbitals which form the exact solutions of the hydrogen atom. 

In 1972 T. L. Allen used Monte Carlo for FSGO method by least squares solution of the Schrodinger equation for many electron atoms and molecules. The least squares solution of the Schrodinger equation was introduced by D. H. Weinstein in 1934, and developed by others. Let us define the local energy, for our system of interest as... 

While a great deal of progress has proved possible for the case of the hydrogen atom by direct solution of the Schrodinger wave equation, some of which will be summarized below, at the time of writing the treatment of many-electron atoms necessitates a simpler approach. This is afforded by the semi-classical Thomas-Fermi theory [4-6], the first explicit form of what today is termed density functional theory [7,8]. We shall summarize below the work of Hill et al. [9], who solved the Thomas-Fermi (TF) equation for heavy positive ions in the limit of extremely strong magnetic fields. This will lead naturally into the formulation of relativistic Thomas-Fermi (TF) theory [10] and to a discussion of the role of the virial in this approximate theory [11]. 

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)-... 

There is a fourth quantum number that is necessary but does not result from the solution to the Schrodinger equation as we have written it. Rather, it results from a relativistic form of the equation. This is the spin quantum number, ms, which is needed for many-electron atoms and has values... 

Solutions of Schrodinger s wave equation give the allowed energy levels and the corresponding wavefunctions. By analogy with the orbits of electrons in the classical planetary model (see Topic AT), wavefunctions for atoms are known as atomic orbitals. Exact solutions of Schrodinger s equation can be obtained only for one-electron atoms and ions, but the atomic orbitals that result from these solutions provide pictures of the behavior of electrons that can be extended to many-electron atoms and molecules (see Topics A3 and C4-C7). 

The first ionisation limit of a many-electron atom corresponds to the ground state of the corresponding or parent ion. Higher thresholds correspond to excited states of the parent ion. Apart from the special case of He, which has a hydrogenic parent ion, they are not simply related to fundamental constants. The many-electron Schrodinger equation must also be solved for the parent ion in order to determine the energies of the thresholds. 

Centrifugal barrier effects have their origin in the balance between the repulsive term in the radial Schrodinger equation, which varies as 1/r2, and the attractive electrostatic potential experienced by an electron in a many-electron atom, whose variation with radius differs from atom to atom because of screening effects. In order to understand them properly, it is necessary to appreciate the different properties of short and of long range potential wells in quantum mechanics. 

Like the Bohr model, the Schrodinger equation does not give exact solutions for many-electron atoms. However, unlike the Bohr model, the Schrodinger equation gives very good approximate solutions. These solutions show that the atomic orbitals of many-electron atoms resemble those of the H atom, which means we can use the same quantum numbers that we used for the H atom to describe the orbitals of other atoms. 

For many-electron atoms, no exact solutions to the corresponding Schrodinger equation exist because of the electron-electron repulsions. However, various approximations can be used to locate the electrons in these atoms. The common procedure for predicting where electrons are located in larger atoms is the Aufbau (building up) principle. 

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