Quantum numbers, relativistic atomic orbital

Relativistic All-Electron Approaches 65 Table 3.1 Quantum numbers for relativistic atomic orbitals... 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

The wavefunction of an electron associated with an atomic nucleus. The orbital is typically depicted as a three-dimensional electron density cloud. If an electron s azimuthal quantum number (/) is zero, then the atomic orbital is called an s orbital and the electron density graph is spherically symmetric. If I is one, there are three spatially distinct orbitals, all referred to as p orbitals, having a dumb-bell shape with a node in the center where the probability of finding the electron is extremely small. (Note For relativistic considerations, the probability of an electron residing at the node cannot be zero.) Electrons having a quantum number I equal to two are associated with d orbitals. 

Thus, formula (2.18) represents a new form of the non-relativistic wave function of an atomic electron (to be more precise, its new angular part in jj coupling). It is an eigenfunction of the operators I2, s2, j2 and jz, and it satisfies the one-electron Schrodinger equation, written in j-representation. Only its phase multiplier depends on the orbital quantum number to ensure selection rules with respect to parity. 

Spin-orbit coupling arises naturally in Dirac theory, which is a fully relativistic one-particle theory for spin j systems.11 In one-electron atoms, spin s and orbital angular momentum l of the electron are not separately conserved they are coupled and only the resulting total electronic angular momentum j is a good quantum number. 

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... 

Thus, the most common assumption was that a material s properties are governed by quantum theory and that relativistic effects are mostly minor and of only secondary importance. Quantum electrodynamics and string theory offer some possible ways of combining quantum theory and the theory of relativity, but these theories have only very marginally found their way into applied quantum theory, where one seeks, from first principles, to calculate directly the properties of specific systems, i.e. atoms, molecules, solids, etc. The only place where Dirac s relativistic quantum theory is used in such calculations is the description of the existence of the spin quantum number. This quantum number is often assumed to be without a classical analogue (see, however, Dahl 1977), and its only practical consequence is that it allows us to have two electrons in each orbital. 

The definition of basis sets involves the selection of a set of functions for each angular momentum of the atom. In non-relativistic calculations, is a good quantum number, but in relativistic calculations it is j or k which is the good quantum number. However for the lighter elements where the effects of relativity are small, is an approximately good quantum number, and basis sets for the spin-orbit components of a non-relativistic subshell can share exponents. Basis sets that are optimized with the same exponents for the two spin-orbit components are called -optimized. Similarly, basis... 

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