The structures of many-electron atoms

The Schrodinger equation for a many-electron atom is highly complicated because all the electrons interact with one another. Even for a He atom, with its two electrons, no mathematical expression for the orbitals and energies can be given and we are forced to make approximations. Modern computational techniques, however, are able to refine the approximations we are about to make and permit highly accurate numerical calculations of energies and wavefunctions. 

The periodic recurrence of analogous groimd state electron configurations as the atomic number increases accounts for the periodic variation in the properties of atoms. Here we concentrate on two aspects of atomic periodicity—atomic radius and ionization energy—and see how they can help to explain the different biological roles played by different elements. 

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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). 

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 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). 

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 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. 

This section barely scratches the surface of many-electron atomic structure calculations. They have mushroomed in complexity to multiconfigurational SCF calculations in which linear combinations of atomic orbitals (LCAOs) are used for each of the spatial orbitals With atomic orbital basis sets of sufficient... 

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 ... 

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... 

The octet rule accounts for the valences of many of the elements and the structures of many compounds. Carbon, nitrogen, oxygen, and fluorine obey the octet rule rigorously, provided there are enough electrons to go around. However, some compounds have an odd number of electrons. In addition, an atom of phosphorus, sulfur, chlorine, or another nonmetal in Period 3 and subsequent periods can accommodate more than eight electrons in its valence shell. The following two sections show how to recognize exceptions to the octet rule. 

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. 

Note that in this case the unshared pairs of electrons are in equatorial positions, which results in a linear structure for IF2 even though the hybrid orbital type is sp3d. It is the arrangement of atoms, not electrons, that determines the structure for a molecule or ion. It is apparent that the simple procedures described in this section are adequate for determining the structures of many molecules and ions in which there are only single bonds and unshared pairs of electrons. 

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. 

The non-relativistic wave function (1.14) or its relativistic analogue (2.15), corresponds to a one-electron system. Having in mind the elements of the angular momentum theory and of irreducible tensors, described in Part 2, we are ready to start constructing the wave functions of many-electron configurations. Let us consider a shell of equivalent electrons. As we shall see later on, the pecularities of the spectra of atoms and ions are conditioned by the structure of their electronic shells, and by the relative role of existing intra-atomic interactions. 

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. 

In this book we are particularly interested in simple descriptions of structures that are easily visualized and providing information of the chemical environment of the ions and atoms involved. For metals, there is an obvious pattern of structures in the periodic table. The number of valence electrons and orbitals are important. These factors determine electron densities and compressibilities, and are essential for theoretical band calculations, etc. The first part of this book covers classical descriptions and notation for crystals, close packing, the PTOT system, and the structures of the elements. The latter and larger part of the book treats the structures of many crystals organized by the patterns of occupancies of close-packed layers in the PTOT system. 

The scientific interests of Huzinaga are numerous. He initially worked in the area of solid-state theory. Soon, however, he became interested in the electronic structure of molecules. He studied the one-center expansion of the molecular wavefunction, developed a formalism for the evaluation of atomic and molecular electron repulsion integrals, expanded Roothaan s self-consistent field theory for open-shell systems, and, building on his own work on the separability of many-electron systems, designed a valence electron method for computational studies on large molecules. 

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). 

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