The many-electron atom

How the correspondence principle should be applied to an atomic system thus depends critically on whether or not there exists a multiperiodic representation of the classical trajectories - the question first raised by Einstein. If the system possesses multiperiodic orbits, then its motion becomes separable, i.e. it becomes equivalent to as many independent modes as there are degrees of freedom. Dynamical separability is assumed in all independent particle and perturbative models of the many-electron atom. It is, however, not strictly applicable and the successes of simple quantum theory for many-electron systems are, to say the least, surprising. It was pointed out by Einstein, who based his arguments on the work of Poincare [519], that there exists no true separation of the three-body problem. 

In general, a Hamiltonian is neither completely integrable, nore purely chaotic, which means that, in practice, phase space fragments itself into islands of stability, inside which the motion is quasiperiodic, and regions, 

the unperturbed H atom becomes the unique example of perfectly regular quantisation for atoms. For many-electron atoms, complete separability cannot be assumed, and Bohr-Sommerfeld quantisation cannot apply exactly. In the semiclassical limit, one may expect to find at least some situations where the corresponding classical system will exhibit chaos, at least in some domain of parameter space, and where, for the quantum system, some related complications due to this breakdown will persist. 

As a practical guide, we may follow the line that any gross breakdown in n characterisation signals difficulties in applying the Bohr-Sommerfeld quantisation condition and should be investigated. It is important (as indicated above) to select groups of levels which interact with each other, since otherwise, their behaviour may be ruled by complexity rather than 

We therefore seek systems in which the independent particle model breaks down and many different excitation channels are strongly coupled together. 

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In addition to the three quantum numbers discussed above, experimental evidence reqnires an additional quantum number ms, which by analogy to classical mechanics is attributed to an intrinsic (i.e.. position-independent) property of the electron called spin. Unlike the other quantum numbers, however, it can assume only two values ( ). As we shall see, this fact determines the orbital population of the many-electron atom. 

Exact solutions such as those given above have not yet been obtained for the usual many-electron molecules encountered by chemists. The approximate method which retains tile idea of orbitals for individual electrons is called molecular-orbital theory (M. O. theory). Its approach to the problem is similar to that used to describe atomic orbitals in the many-electron atom. Electrons are assumed to occupy the lowest energy orbitals with a maximum population of two electrons per orbital (to satisfy the Pauli exclusion principle). Furthermore, just as in the case of atoms, electron-electron repulsion is considered to cause degenerate (of equal energy) orbitals to be singly occupied before pairing occurs. 

As we have already seen in Chapters 11 and 12, the realization of one or another coupling scheme in the many-electron atom is determined by the relation between the spin-orbit and non-spherical parts of electrostatic interactions. As the ionization degree of an atom increases, the coupling scheme changes gradually from LS to jj coupling. The latter, for highly ionized atoms, occurs even within the shell of equivalent electrons (see Chapter 31). 

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. 

The Pauli exclusion principle is a simplified exposition, intended for chemists with no understanding of quantum mechanics, and applied to a particular system, the many-electron atom. Like all simplified explanations it should not be taken too literally. 

We now move to the many-electron atom or molecule. Within the Bom-Oppenheimer38 approximation (i.e., neglect of nuclear motion) the Hamiltonian H becomes... 

Despite its importance of principle, one should not overstate the role of chaos in the spectroscopy of highly excited atoms although favourable circumstances can arise, they are rare. There are two fundamental reasons for this. The first is the Pauli principle as noted in chapter 1, the shell structure of atoms restores spherical symmetry to the many-electron atom at each new row of the periodic table, and spherical symmetry, which helps the independent particle model, inhibits chaos. Secondly, as the excitation energy is increased, autoionisation and the Auger effect also become obstacles to the emergence of chaos, because the lifetimes are so short that instabilities in the underlying classical dynamics do not have time to develop. 

The total relativistic and QED energy shift for many-electron atoms consists of two parts. The first part contains the Bethe logarithm and the other is the average value of some effective potential. Throughout the exact nonrelativistic (Schrddinger) wave functions for the many-electron atom are used. The energy shift is [50] ... 

Relativistic Effects in the Many-Electron Atom, Lloyd Annstrong, Jr. and Serge Feneuille The First Bom Approximation, K. L. Bell and A. E. Kingston... 

In the many-electron atom, the 3/> orbital electrons are more effectively shielded by the iimer electrons of the atom (that is, the l5, 2s, and 2p electrons) than the 35 electrons. The 35 orbital is said to be more penetrating than the 3p and 3d orbitals. In the hydrogen atom there is only one electron, so the 35, 3p, and 3d orbitals have the same energy. 

When the discussion is limited to a single open subshell and to perturbations within such a shell, an interesting formulation of the many-electron atomic problem can be achieved that exhibits useful particle-hole symmetry. The summations in the perturbation term of the hamiltonian then run only over electron states nlmu) of the open subshell (nQ. This means that the electron repulsion integrals can be expressed as... 

In this chapter we apply dimensional scaling techniques to the problem of electronic structure in many-electron atoms. As usual in the dimensional scaling approach, the motivating idea is to generalize the problem to spaces of arbitrary dimensionality Z , treat it at one or more values of D where it s particxilarly easy to do so, and finally relate the results obtained back to jD = 3. The D - oo limit turns out to be the easiest place to treat the many-electron atom. In fact, one can obtain [1] analytic solutions for this limit, as well as for the first-order corrections at finite D. With some work these results can be used to calculate approximate solutions at D = Z. However, the raw jD— oo solutions do not correspond very well with our common notions of what an atom looks like. 

We now review the D— oo solution for the many-electron atom problem. Since D, the limit is obtained by simply dropping the second derivative terms from Eq. (4),... 

From now we have at our side a simple analytical radial density frame in which the valence properties of the many-electron atomic systems can be fairly treated. In such, the Table 4.4 presents the particular constants that are relevant for the atomic systems under actual study. 

After having transformed all operators of the many-electron atomic Hamiltonian in Eq. (9.1) to polar coordinates, we may write the Dirac-Coulomb Hamiltonian for an atom explicitly as... 

To keep the math manageable, we limit ourselves to atoms with only one electron for now. This is a stepping stone between the semiclassical Bohr model and the many-electron atoms that follow. For the one-electron atom, we can solve our Schrodinger equation exactly. That capability will last us only until we get to the next chapter, so let s enjoy it while we can ... 

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