Expression of the energy operator in terms of irreducible tensors

While calculating the energy spectra of atoms or ions one ought to be able to find numerical expressions for all terms of the energy operator (e.g. (1.16) or (2.1)) with regard to the wave functions of the system under consideration. The total matrix element of each term of the energy operator in the case of complex electronic configurations will consist of matrix elements, describing the interaction inside each shell (in relativistic case – subshell) of equivalent electrons as well as between these shells. However, as a rule, it is impossible to find directly formulas for the matrix elements of the corresponding operators. For this purpose each operator must be expressed in terms of the so-called irreducible tensors. This will enable us, further on, to find their matrix elements, to exploit very efficiently the mathematical apparatus of the angular momentum theory, second-quantization as well as the methods of the coefficients of fractional parentage (CFP) and the irreducible tensorial operators, composed of the unit tensors.

In practice, the transformation of any operator to irreducible form means in atomic spectroscopy that we employ the spherical coordinate system (Fig. 5.1), present all quantities in the form of tensors of corresponding ranks (scalar is a zero rank tensor, vector is a tensor of the first rank, etc.) and further on express them, depending on the particular form of the operator, in terms of various functions of radial variable, the angular momentum operator L(1), spherical functions C(k) (2.13), as well as the Clebsch-Gordan and 3nj-coefficients.

In the previous chapter we discussed the problem of classification of the energy levels of many-electron atoms using various coupling schemes, including intermediate ones. For this purpose we start with the coupling scheme closest to reality and optimize it. In practice this may be achieved in two different ways, namely, utilizing the expressions for matrix elements of the energy operator in a different pure coupling scheme with subsequent diagonalization of the energy matrices, or starting with the energy matrices in one definite coupling scheme, not necessarily supposed to be the closest to reality, with further use of the special transformation matrices, converting the wave functions of a given coupling scheme (including an intermediate one) to the other. These transformation matrices have to cover the cases of complex electronic configurations, consisting of several open shells.

Usually, the first way is utilized in practice. This is due to the well developed mathematical technique necessary, by the presence of the expressions for both the matrix elements of the energy operator and of the electronic transitions in various coupling schemes. However, the second method is much more universal and easier to apply, provided that there are known corresponding transformation matrices. Now we shall briefly describe this method.

Let |ψi) and |φj) denote two full orthonormal sets of wave functions, corresponding to a given energy spectrum of a definite many-electron system, classified with the aid of quantum numbers of two different coupling schemes. Let us suppose that these functions are non-relativistic, then they will have one and the same electronic configuration. In a relativistic case this transformation will couple different electronic configurations, and this coupling will be approximate.

Methods of accounting for correlation effects

Correlation effects have already been mentioned a number of times. Let us discuss them here in more detail. In the Hartree-Fock approach (to be more exact, in its single-configuration version, which we have been considering so far) it is assumed that each electron in an atom is orbiting independently in the central field of the nucleus and of the remaining electrons (one-particle approach). However, this is not exactly so. There exist effects, usually fairly small, of their correlated, consistent movement, causing the so-called correlation energy.

In the language of quantum numbers it may be explained in the following way. Only the parity of the configuration and total angular momentum are exact quantum numbers. From the parity conservation law it follows that the energy operator matrix elements connecting configurations of opposite parity vanish. Matrix elements of the one- and two-particle operators (the operators we are considering in this book) also vanish if they connect wave functions of configurations differing by the quantum numbers of more than one and two electrons, respectively. However, the non-diagonal with respect to configurations matrix elements of, for example, an electrostatic interaction, are in general non-zero. Hence, the configuration, considered as a quantum number, is not always an exact quantum number, and then a single-configuration approach is not sufficiently accurate. For this reason, while forming the energy matrix, non-diagonal, with respect to configurations, matrix elements must also be accounted for.

Role of relativistic effects

Calculations using the methods of non-relativistic quantum mechanics have now advanced to the point at which they can provide quantitative predictions of the structure and properties of atoms, their ions, molecules, and solids containing atoms from the first two rows of the Periodical Table. However, there is much evidence that relativistic effects grow in importance with the increase of atomic number, and the competition between relativistic and correlation effects dominates over the properties of materials from the first transition row onwards. This makes it obligatory to use methods based on relativistic quantum mechanics if one wishes to obtain even qualitatively realistic descriptions of the properties of systems containing heavy elements. Many of these dominate in materials being considered as new high-temperature superconductors.

It is apparent that progress in our understanding of the properties of neutral heavy elements and their ions, including very highly ionized ones, as well as their role as constituents of molecules and solids, will depend on the development of theoretical methods and computational techniques, which are based on relativistic quantum mechanics. Fairly efficient methods of this kind have already been elaborated and many versions of relativistic codes for work with isolated atoms and ions are already available and in daily use by internationally known theoretical and experimental physicists and chemists.

General remarks

Ten years have elapsed since first publication of this monograph. This may, at first glance, seem a very short time in what has been a very stable area of fundamental physics. But the last decade has seen unprecedented technological progress and socio-economic changes. Have there been radical changes in “Theoretical Atomic Spectroscopy” in that time?

The answer is very much “yes”. Just a few examples driving current interest in this subject are:

• In the explosively growing nanoscience area we are approaching the molecular and atomic scales where quantum mechanical effects become significant. This is opening new horizons in the nanotechnologies, including quantum computing and quantum cryptography.

• In astrophysics, there has been a requirement for much improved spectroscopy of highly charged ions.

• The practical considerations necessary for building thermonuclear power plants (driven through ITER, DEMO and other projects) require deeper insight into the structure and properties of ions, atoms and their separate isotopes.

• Progress in fission research, particularly in the development of higher efficiency reactors, in radioactive waste management, in transmutation studies and transuranic elements have contributed substantially to a renewed interest in the structure and properties of complex many-electron atoms and ions of arbitrary ionization.

Even before publication of the first edition of this book it became obvious to me that there was more that needed to be said. A very important development had been the formulation and practical implementation of a new version of Racah algebra based on the second-quantization method in the coupled tensorial form, angular momentum theory in three spaces (orbital, spin and quasispin) and a generalized graphical technique.

Interaction of atomic electrons with electromagnetic radiation

Until now we have studied the methods of theoretical description of the energy of separate, non-interacting atoms or ions. Interaction was present indirectly while speaking about many states or excited configurations, saying nothing about the mechanisms of their creation or their lifetimes.

The wavelengths and intensities of the electronic transitions from the higher to lower (or, finally, ground) states are measured experimentally. From these data the energy levels (energy spectra) are deduced. Therefore, it is very important to have efficient and fairly general methods of theoretical description of the main quantities of electronic transitions in many-electron atoms and ions, including very highly ionized ones.

An atom in a ground state can exist for an infinitely long time, whereas any excited state exists only for a finite time interval. This time interval (lifetime of the excited state), together with the energy spectrum of the excited configuration, is one of the basic spectral characters of the quantum mechanical system considered. There exist various channels of decay of the excited states, and the realization of each of them depends on concrete physical conditions.

The finite lifetime of each excited state is the reflection of a fundamental law of nature – tendency towards minimum total energy of a system. The quantum mechanical system tends to occupy the state in which its total energy would be minimal. However, the transition of an atom to the lowest (ground) state depends on many circumstances (first of all, on the sort of excited configuration, on the presence of external fields, on the character of the matter itself – density of gas, vapours or plasma, etc.).

Autoionization

Methods of theoretical studies of energy spectra and electron transitions, described in previous chapters, are also applicable to configurations with vacancies in inner shells. However, such types of configurations possess a number of peculiarities, mainly connected with problems of ensuring the orthogonality of the appropriate wave functions, with the changes of relative role played by relativistic and correlation effects. These peculiarities are manifest in the energy and radiation spectra of such systems. As a rule we shall only touch the main aspects of this problem. More detailed information may be found in the monograph by Karazija. From a physical point of view, processes involving inner electronic shells of free atoms and ions lead to X-ray and electronic (photoelectron and Auger) spectra. Appropriate theory may be used to interpret the spectra of gases and vapours as well as such spectra of solid state, containing atoms with open d- and f-shells and corresponding to transitions in inner shells. Comparison of such spectra of free atoms with the spectra of the same atoms in molecules or solid states allows one to elucidate the effects of free atoms and chemical surroundings, e.g. of the crystalline field.

The single-configuration approach remains the main method used, allowing one to describe not only qualitatively, but in many cases quantitatively, many features of various such spectra and the regularities in them. However, having in mind that relativistic effects grow very rapidly with increase of nuclear charge and with shrinking electronic shells, accounting for relativistic effects for this kind of phenomena is of paramount importance. Again, they may be taken into consideration as corrections in the Hartree–Fock–Pauli approximation or, starting with a relativistic approach, in the Dirac–Hartree–Fock approximation.

Relativistic approach and quasispin for one subshell

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).

With jj coupling, the spin-angular part of the one-electron wave function (2.15) is obtained by vectorial coupling of the orbital and spin-angular momenta of the electron. Then the total angular momenta of individual electrons are added up. In this approach a shell of equivalent electrons is split into two subshells with j = 1 ± 1/2. The shell structure of electronic configurations in jj coupling becomes more complex, but is compensated for by a reduction in the number of electrons in individual subshells.

The energy spectrum of atoms and ions with jj coupling can be found using the relativistic Hamiltonian of N-electron atoms (2.1)–(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)).

The graphs of Jucys, Levinson and Vanagas (JLV graphs)

The role of graphical methods to represent angular momentum, 3nj- and jm-coefficients was emphasized in the Introduction as one of the most important milestones in the theory of many-electron atoms and ions.

As was already mentioned, in theoretical atomic spectroscopy, while considering complex electronic configurations, one has to cope with many sums over quantum numbers of the angular momentum type and their projections (3nj- and jm-coefficients). There are collections of algebraic formulas for particular cases of such sums. However, the most general way to solve problems of this kind is the exploitation of one or another versions of graphical methods. They are widely utilized not only in atomic spectroscopy, but also in many other domains of physics (nuclei, elementary particles, etc.).

The main advantage of graphic methods, compared to algebraic ones, is their general character: in fact, you can easily exploit them in any case. They are extremely efficient if you account for the peculiarities and symmetry properties of the quantities considered. The first version of graphical technique (graphs of Jucys, Levinson and Vanagas) was based on the Wigner coefficients (symmetric part of the Clebsch–Gordan coefficient (6.7)) and their symmetry properties. The main element of this technique was the symbol, presented in Fig. 8.1, describing the Wigner coefficient (Eq. (6.8)).

The sign by the vertex (Fig. 8.1) indicates whether the arguments of the Wigner coefficient are ordered in a counter-clockwise direction (positive sign) or a clockwise direction (negative sign). A change of the cyclic order of lines around a vertex corresponds to interchanging two of the columns of the Wigner coefficient.

Angular momentum

The main ideas of angular momentum theory (e.g. central field approximation, self-consistent field method, etc.) are fairly general and are equally applicable for both the non-relativistic and relativistic theories of many-electron atoms and ions in spite of the fact that the corresponding energy operators differ significantly in nature. This distinction manifests itself in different kinds of wave functions (non-relativistic (1.14) and relativistic (2.15) or (2.18)). Strong interaction of orbital L and spin s momenta of an electron in relativistic approximation leads, as we shall see in Chapter 9, to the splitting of a shell of equivalent electrons into subshells according to j = L + s values and to the occurrence of jj coupling inside these subshells and between them. In order to use the mathematical apparatus of the angular momentum theory, we have to write all the operators in j-representation, i.e. express them in terms of the quantities which transform like the eigenfunctions of the operators j2 and jz. From a mathematical point of view the main difference between non-relativistic and relativistic momenta is that in a non-relativistic case we have one-electron angular and spin momenta, which can acquire integer and half-integer numerical values, correspondingly, whereas in a relativistic case we have only one-electron total momentum, acquiring half-integer numerical values. However, they all satisfy the same commutation conditions (5.2), and all formulas of Chapter 5 on tensorial algebra are equally applicable in both cases.

Hyperfine interactions in non-relativistic approximation

The hyperfine structure (splitting) of energy levels is mainly caused by electric and magnetic multipole interactions between the atomic nucleus and electronic shells. From the known data on hyperfine structure we can determine the electric and magnetic multipole momenta of the nuclei, their spins and other parameters.

In a non-relativistic approximation the usual fine structure (splitting) of the energy terms is considered as a perturbation whereas the hyperfine splitting – as an even smaller perturbation, and they both are calculated as matrix elements of the corresponding operators with respect to the zero-order wave functions.

The representation of hyperfine interactions in the form of multipoles follows from expansion of the potentials of the electric and magnetic fields of a nucleus, conditioned by the distribution of nuclear charges and currents, in a series of the corresponding multipole momenta. It follows from the properties of the operators obtained with respect to the inversion operation that the nucleus can possess non-zero electric multipole momenta of the order k = 0,2,4,…, and magnetic ones with k = 1,3,5,…. The energy of the hyperfine interactions decreases rapidly with the increase of multipolarity, therefore, it is usually enough to take into consideration only the first non-zero term in the expansion over multipoles. However, in a number of cases, due to the existence of selection rules and other effects, the higher multipoles must be accounted for. Indeed, in the hexadecapole nuclear momentum for 165Ho was found.

As usual, the operators describing hyperfine interactions are to be expressed in terms of irreducible tensors. Then we are in a position to find formulas for their matrix elements.

Peculiarities of multiply charged ions

Extensive studies of energy spectra and other characteristics of atoms and ions allow one to reveal general regularities in their structure and properties. For example, by considering the lowest electronic configurations of neutral atoms, we can explain not only the structure of the Periodical Table of elements, but also the anomalies. The behaviour of the ionization energy of the outer electrons of an atom illustrates a shell structure of electronic configurations.

Multiply charged ions represent a peculiar world, essentially differing from the world of neutral atoms or their first ions. In this chapter we shall try to describe these peculiarities. Main attention will be paid to the need for a combination of theoretical and experimental studies of the spectra of highly ionized atoms, and to the role of relativistic and correlation effects for multiply charged ions, and to the regularities and irregularities of the behaviour of their spectral characteristics along isoelectronic sequences.

Recently intense beams of ultracold highly ionized atoms up to naked uranium (U92+) have been obtained. Single ions were confined in ion traps, ion crystals were observed, and the phase transition from ion cloud to crystal was noticed. With increase of ionization degree the radiation of the ions moves to a shorter and shorter wavelength region, thus requiring special measurement techniques. For very highly ionized atoms (e.g. hydrogen-like bismuth), transitions between the hyperfine structure components occur in the optical region. It is very useful, to know the wavelength domain of the radiation of such ions. The relevant calculations here are very helpful.

General remarks

As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics – the Schrödinger equation – lies beyond the potentialities of modern mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties.

Symmetry concepts come about in physics in two ways. First, since any physical process occurs in a real space, we have to make use of one or another coordinate system. The isotropy and homogeneity of space make physically meaningful only those mathematical relationships that remain unchanged under rotations of the axes of the coordinate system, and that impose fairly rigorous constraints on the possible physical laws. Second, every physical object and process features a symmetry which should be taken into account by the physical theory.

A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum.

The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modern theory of complex atoms and ions is based.