Structure of Many-Electron Atoms

When is turned on, the total Hamiltonian i becomes Hq + gaining a term proportional to L S. In this case, it can be shown (Problem 2.4) that 

We now briefly consider the magnetic dipole (Ml) selection rules for transitions in hydrogenlike and alkali atoms. The relevant matrix element is (not as has sometimes been implied, because the 

We now extend our discussion of hydrogenlike atoms to complex atoms with a total of p electrons. The nonrelativistic Hamiltonian operator for such atoms in the absence of external fields is 

Since the Schrodinger equation using this two-electron Hamiltonian cannot be exactly solved, we use as a trial wave function for ground-state He the product of one-electron orbitals i(l) and 02(2)  

An improved wave function can be obtained by replacing the fixed atomic number Z = 2 in iAioo(0 with a single variational parameter C- The trial energy Wq is then calculated in a manner analogous to Eq. 2.56, and is minimized with respect to C by setting = 0. This procedure yields C = 27/16 = 1.688 in 

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As discussed in Section 5.1, the structure of many-electron atoms can be understood only by assuming that no more than two electrons can occupy each separate orbital. Taking account of the electron spin allows a deeper interpretation of this fact. One way of expressing the Pauli exclusion principle is no two electrons can have the same values of all four quantum numbers, n, l, m, and ms. As only two values of ms are permitted, it follows that each orbital, specified by a given set of values of n, l, and m, can hold... 

Another measure of the size of an orbital is the most probable distance of the electron from the nucleus in that orbital. Figure 5.4c shows that the most probable location of the electron is progressively farther from the nucleus in ns orbitals for larger n. Nonetheless, there is a finite probability for finding the electron at the nucleus in both 2s and 3s orbitals. This happens because electrons in s orbitals have no angular momentum ( = 0), and thus can approach the nucleus along the radial direction. The ability of electrons in s orbitals to penetrate close to the nucleus has important consequences in the structure of many-electron atoms and molecules (see later). 

The next section shows that these general statements are important for determining the electronic structure of many-electron atoms even though they are deduced from the one-electron case. 

Although the proper point of departure for relativistic atomic structure calculations is quantum electrodynamics (QED), very few atomic structure calculations have been carried out entirely within the QED framework. Indeed, almost all relativistic calculations of the structure of many-electron atoms are based on some variant of the Hamiltonian introduced a half century ago by Brown and Ravenhall [1] to understand the helium fine structure. By decoupling the electron and radiation fields in QED to order a (the fine-structure constant) using a contact transformation. Brown and Ravenhall obtained a relativistic momentum-space Hamiltonian in which the electron-electron Coulomb interaction was surrounded by positive-energy projection operators. Owing to the fact that contributions from virtual electron-positron pairs are automatically projected out of... 

As we shall see, the exclusion principle is an essential part of our understanding of the structure of many-electron atoms. 

The Hartree-Fock approximation [13, 14] plays a central role in the molecular electronic structure theory. In most cases, it provides a qualitatively correct description of the electronic structure of many electron atoms and molecules in their ground electronic state. In addition, it constitutes a basis upon which more accurate methods can be developed. A detailed derivation and discussion of the method can be found in textbooks such as [10, 11]. The Hartree-Fock approximation assumes the simplest possible form for the electronic wavefunction, i.e a single Slater determinant given by Eq. (2.41). Starting from the electronic TISE Eq. (2.5), the Hartree-Fock energy is simply... 

These are the simplest processes in spectroscopy. The principles of spectroscopy will be a recurring theme as we probe the microscopic structure of many-electron atoms and molecules, because spectroscopy remains the most precise and adaptable tool for controlling and measuring the quantum mechanical characteristics of a chemical system. From spectroscopy comes our most precise molecular geometries and successful theories of chemical bonding, as well as many of our most powerful analytical techniques. 

In principle, a description of the electronic structure of many-electron atoms and of polyatomic molecules requires a solution of a Schrodinger equation for stationary states quite similar to equation 3.36 [2]. Even for a simple molecule like, say, methane, however, such an equation would be enormously more complicated, because the hamiltonian operator would include kinetic energy terms for all electrons, plus coulombic terms for the electrostatic interaction of all electrons with all nuclei and of all electrons with all other electrons. The QM hamiltonian operator for the electrons in a molecule reads ... 

The observed structure of the spectra of many-electron atoms is entirely accounted for by the following postulate Only eigenfunctions which are antisymmetric in the electrons , that is, change sign when any two electrons are interchanged, correspond to existant states of the system. This is the quantum mechanics statement (26) of the Pauli exclusion principle (43). 

In recent years the old quantum theory, associated principally with the names of Bohr and Sommerfeld, encountered a large number of difficulties, all of which vanished before the new quantum mechanics of Heisenberg. Because of its abstruse and difficultly interpretable mathematical foundation, Heisenberg s quantum mechanics cannot be easily applied to the relatively complicated problems of the structures and properties of many-electron atoms and of molecules in particular is this true for chemical problems, which usually do not permit simple dynamical formulation in terms of nuclei and electrons, but instead require to be treated with the aid of atomic and molecular models. Accordingly, it is especially gratifying that Schrodinger s interpretation of his wave mechanics3 provides a simple and satisfactory atomic model, more closely related to the chemist s atom than to that of the old quantum theory. 

Non-relativistic (1.14) and relativistic (2.15) wave functions are widely used for theoretical studies of the structure and spectra of many-electron atoms and ions. However, it has turned out that such forms of wave functions in the case of the jj coupling scheme are not optimal. Their utilization, particularly in the relativistic approximation, is rather inconvenient and tedious. 

Operators H4 and H f corresponding to the spin-orbit and spin-spin interactions, are in charge of the fine structure of the terms. As a rule, operator H4 plays the main role. The one-electron part of (19.13) is often called the spin-own-orbit interaction operator. In the case of many-electron atoms it is also called the simplified operator of the spin-orbit interaction. 

This monograph presents a complete, up-to-date guide to the theory of modern spectroscopy of atoms. It describes the contemporary state of the theory of many-electron atoms and ions, the peculiarities of their structure and spectra, the processes of their interaction with radiation, and some of the applications of atomic spectroscopy. 

The data of atomic spectroscopy are of extreme importance in revealing the nature of quantum-electrodynamical effects. For the investigation of many-electron atoms and ions, it is of great importance to combine theoretical and experimental methods. Therefore, the methods used must be universal and accurate. A number of physical characteristics of the many-electron atom (e.g., a complete set of quantum numbers) may be found only on the basis of theoretical considerations. In many cases the mathematical modelling of physical objects and processes using modern computers may successfully replace the corresponding experiments. In this book we shall describe the contemporary state of the theory of many-electron atoms and ions, the peculiarities of their structure and spectra as well as the processes of their interaction with radiation, and some applications. 

The starting point of the creation of the theory of the many-electron atom was the idea of Niels Bohr [1] to consider each electron of an atom as orbiting in a stationary state in the field, created by the charge of the nucleus and the rest of the electrons of an atom. This idea is several years older than quantum mechanics itself. It allows one to construct an approximate wave function of the whole atom with the help of one-electron wave functions. They may be found by accounting for the approximate states of the passive electrons, in other words, the states of all electrons must be consistent. This is the essence of the self-consistent field approximation (Hartree-Fock method), widely used in the theory of many-body systems, particularly of many-electron atoms and ions. There are many methods of accounting more or less accurately for this consistency, usually named by correlation effects, and of obtaining more accurate theoretical data on atomic structure. 

UV Photoelectron Spectra. These in general have provided impressive support for both the qualitative ideas and the explicit quantitative results concerning the multiple M-M bonds (23, 28) and closely related ones such as the formal single bond (see Table 1) in dirhodium species (29). Such spectra have also strongly supported the accepted views on the electronic structures of many metal atom cluster compounds (30, 31). 

Because of their greater complexity, the electronic structures and spectra of many-electron atoms cannot be rigorously determined by quantum mechanical methods. Rather, they are determined approximately by a building up process using the properties of the hydrogenic orbitals just discussed. 

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