It took a very long time to write this book, especially to bring it to a relatively consistent and complete form. The journey of the reader to these final pages was also not easy and straightforward. What are your feelings after getting through the jungle of more than 1300 many-storeyed formulas? Perhaps, twofold. At first – relief and satisfaction: it is all over now, I made it! But secondly – are all these formulas correct? The answer is not so simple. I tried to do my utmost to be able to answer ‘yes’: compared with the original papers, deduced some of them again, checked numerically, looked for special cases, symmetry properties, etc. But I cannot assert that absolutely all signs, phases, indices, etc. are correct. Therefore, if you intend to do some serious research starting with one or other formula from the book, it is worthwhile carrying out additional checks, making use of one of the above mentioned methods.

Not all aspects of the theory are dealt with in equal depth. Some are just mentioned, some even omitted. For example, the method of effective (equivalent) operators deserves mentioning. It allows one to take into account the main part of relativistic effects but at the same time to preserve the LS coupling used for classification of the energy spectra of the atoms or ions considered.

While studying the energy spectra or other spectral quantities of atoms and ions having complex electronic configurations, one ought to consider the expressions for the matrix elements of the operators both within each shell of equivalent electrons and between each pair of these shells. For example, in order to find the energy spectrum of the ground configuration 1s22s2 of the beryllium atom, we have to calculate the interaction energy in each shell ls2 and 2s2 as well as between them. The last case will be discussed in this chapter. If there are more shells, then according to the two-particle character of interelectronic interactions, we have to account for this interaction between all possible pairs of shells.

Let us notice that momenta of each shell may be coupled into total momenta by various coupling schemes. Therefore, here, as in the case of two non-equivalent electrons, coupling schemes (11.2)–(11.5) are possible, only instead of one-electronic momenta there will be the total momenta of separate shells. To indicate this we shall use the notation LS, LK, JK and JJ. Some peculiarities of their usage were discussed in Chapters 11 and 12 and will be additionally considered in Chapter 30. Therefore, here we shall restrict ourselves to the case of LS coupling for non-relativistic and JJ (or jj) coupling for relativistic wave functions. We shall not indicate explicitly the parity of the configuration, consisting of several shells, because it is simply equal to the sum of parities of all shells.

It has taken a very long time for this book to appear. For many years I had in my mind the idea of publishing it in English, but practical implementation became possible only recently, after drastic changes in the international political situation. The book was started in the framework (a realistic one) of the former USSR, and finished soon after its collapse, after my motherland, the Republic of Lithuania, regained its independence.

Academician of the Lithuanian Academy of Sciences, Adolfas Jucys, initiated the creation of a group of scientists devoted to the theory of complex atoms and their spectra. Later it was named the Vilnius (or Lithuanian) school of atomic physicists, often called by his name. However, for many years the results of these studies were published largely only in Russian and, therefore as a rule they were not known among Western colleagues, particularly those in English-speaking countries. A large number of the papers were published in Russian in the main Lithuanian physical journal Lietuvos fizikos rinkinys – Lithuanian Journal of Physics, translated into English by Allerton Press, Inc. (New York) as Soviet Physics – Collection (since 1993 – Lithuanian Physics Journal).

Recently the situation has become incomparably better. There is no problem publishing the main ideas and results in prestigious international journals in English. However, it would be very useful to collect, to analyse and to summarize the main internationally recognized results on the theory of many-electron atoms and their spectra in one monograph, written in English. This book is the result of the long process of practical realization of that idea.

In previous chapters we considered the wave functions and matrix elements of some operators without specifying their explicit expressions. Now it is time to discuss this question in more detail. Having in mind that our goal is to consider as generally as possible the methods of theoretical studies of many-electron systems, covering, at least in principle, any atom or ion of the Periodical Table, we have to be able to describe the main features of the structure of electronic shells of atoms. In this chapter we restrict ourselves to a shell of equivalent electrons in non-relativistic and relativistic cases.

A shell of equivalent electrons

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.

N electrons with the same values of quantum numbers nili (LS coupling) or niliji (jj coupling) are called equivalent. The corresponding configurations will be denoted as nlN (a shell) or nl jN (a subshell). A number of permitted states of a shell of equivalent electrons are restricted by the Pauli exclusion principle, which requires antisymmetry of the wave function with respect to permutation of the coordinates of the electrons.

The wave function for the particular case of two equivalent electrons may be constructed, using vectorial coupling of the angular momenta and antisymmetrization procedure.

Methods of accounting for correlation effects

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree–Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions.

In fact, electrons do not move independently, they ‘feel’ each other, there is a certain correlation between the electrons in their mutual Coulomb field (many-body effects).

Many-body calculations which go beyond the Hartree–Fock model can be performed in two ways, i.e. using either a variational or a perturbational procedure. There are a number of variational methods which account for correlation effects: superposition-of-configurations (or configuration interaction (CI)), random phase approximation with exchange, method of incomplete separation of variables, multi-configuration Hartree–Fock (MCHF) approach, etc. However, to date only CI and MCHF methods and some simple versions of perturbation theory are practically exploited for theoretical studies of many-electron atoms and ions.

The CI method is often based on the use of analytic functions, which form the basis set.

Least squares fitting

The methods of theoretical study of the energy spectra of atoms and ions, described in the previous chapters, do not always ensure the required accuracy. Therefore, in a number of cases, when we are not interested in the accuracy of the theory and when part of the data is available (e.g. experimental measurements of the energy levels, which are usually known to high precision), then we can utilize the latter for the so-called semi-empirical evaluation of wave functions and other spectroscopic characteristics. The most widespread is the semi-empirical method of least squares fitting. As a rule, it is used together with diagonalization of the total energy matrix, built in a certain coupling scheme, to find the whole energy spectrum and corresponding eigenfunctions of an atom or ion. The eigenfunctions are used further on to calculate the oscillator strengths, transition probabilities and other spectral characteristics. Let us sketch its main ideas.

As we have seen while considering the matrix elements of various operators, any expression obtained consists of radial integrals and coefficients. These coefficients can be calculated using the techniques of irreducible tensorial sets and CFP, whereas the values of radial integrals are found starting with analytical or numerical radial wave functions. Uncertainties of these quantities calculated in a given approximation (it is well known, for example, that the Hartree-Fock values of radial integrals of electrostatic interactions in some cases exceed the exact ones by 1.5 times) are the main reason for discrepancies between calculations and experimental measurements of spectral characteristics. In a semi-empirical approach we consider these integrals as unknown parameters which can be determined from experimental data by extrapolation or interpolation.

Electronic transitions in intermediate coupling

As we have seen in Chapter 11, the energy levels of atoms and ions, depending on the relative role of various intra-atomic interactions, are classified with the quantum numbers of different coupling schemes (11.2)–(11.5) or their combinations. Therefore, when calculating electron transition quantities, the accuracy of the coupling scheme must be accounted for. The latter in some cases may be different for initial and final configurations. Then the selection rules for electronic transitions are also different. That is why in Part 6 we presented expressions for matrix elements of electric multipole (Ek) transitions for various coupling schemes.

In various pure coupling schemes the intensities of spectral lines may differ significantly. Some lines, permitted in one coupling scheme, are forbidden in others. Comparison of such theoretical results with the relevant experimental data may serve as an additional criterion of the validity of the coupling scheme used.

Studies of spectra of many-electron atoms and ions show that the presence of a pure coupling scheme is the exception rather than the rule. Therefore, their energy spectra must be calculated, as a rule, in intermediate coupling via diagonalization of the total energy matrix, starting with the coupling scheme assumed to be closest to reality. In such a case the electronic transitions must also be calculated in intermediate coupling.

The wave function in intermediate coupling is a linear combination of the relevant quantities of pure coupling (see (11.10)).

The word ‘atom’ introduced by Democritus more than 2000 years ago in Greek means ‘unseparable’. Only in the 20th century was it shown by Rutherford that the atom possesses a complex structure. The discovery of the complex inner structure of an atom, in fact, has led to the emergence of the main branch of physics describing the structure of a microworld, i.e. quantum mechanics, which, in its turn, has stimulated the development of many other domains of physics, neighbouring sciences and technology. Quantum mechanics continues to be of great importance for their future progress.

However, it is very far from enough to know that any atom consists of a nucleus and of electrons orbiting around it like planets around the Sun. The inner structure of an atom and its main fundamental characteristic – spectra – hide many fundamental laws of nature, the discovery of which has been a challenge for many generations of scientists. Among these laws it is worth mentioning parity violation and the manifestation of a number of fine quantum electrodynamical effects. This is particularly the case for complex atoms with many electronic shells and for highly ionized atoms, whose shell structure and spectra differ considerably from those of neutral or just a few times ionized atoms. These differences and changes are caused by the interplay of the relative role of existing intra-atomic (electron–nucleus or electron–electron) interactions. Therefore, the ability to describe them precisely in order to take them into account, is very important.

For many reasons atomic spectroscopy continues to be one of the most rapidly developing branches of physics.