Quantum number Relativistic effects

Having stated the limitations (non-relativistic Hamilton operator and the Bom-Oppenheimer approximation), we are ready to consider the electronic Schrodinger equation. It can only be solved exactly for the Hj molecule, and similar one-electron systems. In the general case we have to rely on approximate (numerical) methods. By neglecting relativistic effects, we also have to introduce electron spin as an ad hoc quantum effect. Each electron has a spin quantum number of 1 /2. In the presence of an... 

Schrodinger s equation has solutions characterized by three quantum numbers only, whereas electron spin appears naturally as a solution of Dirac s relativistic equation. As a consequence it is often stated that spin is a relativistic effect. However, the fact that half-integral angular momentum states, predicted by the ladder-operator method, are compatible with non-relativistic systems, refutes this conclusion. The non-appearance of electron... 

The methods used to describe the electronic structure of actinide compounds must, therefore, be relativistic and must also have the capability to describe complex electronic structures. Such methods will be described in the next section. The main characteristic of successful quantum calculations for such systems is the use of multiconfigurational wave functions that include relativistic effects. These methods have been applied for a large number of molecular systems containing transition metals or actinides, and we shall give several examples from recent studies of such systems. 

The inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

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. 

There exist a number of methods to account for correlation [17, 45, 48] and relativistic effects as corrections or in relativistic approximation [18]. There have been numerous attempts to account for leading radiative (quantum-electrodynamical) corrections, as well [49, 50]. However, as a rule, the methods developed are applicable only for light atoms with closed electronic shells plus or minus one electron, therefore, they are not sufficiently general. 

Table 2. Electronic origins (all in cm-1) at very low temperatures in crystalline caesium and rubidium uranyl chloride, caesium uranyl nitrate and sodium uranyl acetate. The quantum number Q characterizing many-electron states in linear chromophores (subject to perceptible relativistic effects) may correspond to two energy levels because of the 4 or 6 ligating atoms in the equatorial plane...

Table 3. M.O. energies (all in eV) calculated for the uranium hexafluoride molecule. When effects of spin-orbit coupling are neglected, the symmetry types tlu,. .. of the point-group Oh are indicated. The quantum numbers y6, y7 and y8 refers to the corresponding double-group (used for describing relativistic effects) in which case the parity can be seen from the main component of the one-electron function given at first in parenthesis...

To each degree of freedom we now have an independent quantum number, of which cme, the space quantum number, has no effect on the energies of the system in free space. The relativistic energy is now... 

Radii. The filling of the 4f orbitals (as well as relativistic effects) through the lanthanide elements cause a steady contraction, called the lanthanide contraction (Section 19-1), in atomic and ionic sizes. Thus the expected size increases of elements of the third transition series relative to those of the second transition series, due to an increased number of electrons and the higher principal quantum numbers of the outer ones, are almost exactly offset, and there is in general little difference in atomic and ionic sizes between the two heavy atoms of a group, whereas the corresponding atoms... 

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