S, P, D, and F states

TABLE 10 The magnetic properties of neodymium on a variety of crystal lattices (fee, bee, simple cubic, and hexagonal lattices) calculated as described in the text. The magnetic moments are all written in units of the Bohr magneton. The second column is the spin magnetic moment of the valence s, p, d, and f-states. The third column is the orbital moment associated with the valence electrons. The final two columns are the spin and orbital magnetic moments of the Sl-coirected f-states... 

The above definitions must be qualified by stating that for principal quantum number I there are only s orbitals for principal quantum number 2 there are only s and p orbitals for principal quantum number 3 there are only s, p and d orbitals for higher principal quantum numbers there are s, p, d and f orbitals. 

In a manner similar to that by which the atomic states were designated as s, p, d, or /, the letters S, P, D, and F correspond to the values of 0, 1, 2, and 3, respectively, for the angular momentum vector, L. After the values of the vectors L, S, and / have been determined, the overall angular momentum is described by a symbol known as a term symbol or spectroscopic state. This symbol is constructed as Ps+1)Lj where the appropriate letter is used for the L value as listed earlier, and the quantity (2S + 1) is known as the multiplicity. For one unpaired electron, (2S + 1) = 2, and a multiplicity of 2 gives rise to a doublet. For two unpaired electrons, the multiplicity is 3, and the state is called a triplet state. 

The orbitals of the unexcited state atom include s, p, d, and f orbitals. The electrons fill these orbitals in a pre-determined pattern filling the lowest energy orbitals first. S orbitals are the lowest energy so they fill with a maximum of two electrons first, followed by the p orbitals. 

The second chapter deals with quantum chemical considerations, s, p, d and f orbitals, electronic configurations, Pauli s principle, spin-orbit coupling and levels, energy level diagrams, Hund s mles, Racah parameters, oxidation states, HSAB principle, coordination number, lanthanide contraction, interconfiguration fluctuations. This is followed by a chapter dealing with methods of determination of stability constants, stability constants of complexes, thermodynamic consideration, double-double effect, inclined w plot, applications of stability constant data. 

Electrons in an atom are contained within atomic orbitals, each of which may only hold a maximum of two electrons. The principal quantum shell is indicated by a number, e g. 1, 2, 3, etc. Within each shell, there may exist a number of subshells s, p, d and f, depending upon which shell is being considered. There is only one s subshell, while there are three p subshells, which are degenerate pr, pv and pz Hund s rule states that if electrons are placed in degenerate orbitals, they... 

Essentially, this Table is based upon the distribution of electrons amongst four sets of orbitals labelled s, p, d, and f and is comprised of the main group elements, with the completion of s and p orbitals, the transition elements, with the completion of electron shells for the d orbitals, and then the inner-transition elements, known as the lantha-nons and actinons, with the completion of f orbitals. All of these transition and inner-transition elements are metals in their native state, whereas the elements to the top right-hand side of the main group of the Periodic Table tend to be non-metals. 

The complete list of computations and results can be found in the original work [122]. Here we recall that the SSEA results revealed in a transparent and systematic way the relative role of all types of states in the spectrum. Furthermore, and most important as regards the original motivation of this work, it was demonstrated that, for this nonperturbative problem, in order to describe the phenomena of ATI, of PADs, and of HHG, conclusions based on the consideration of just the two symmetries S and P° are unreliable. Instead, for Helium interacting with the A, = 248 nm (5 eV) pulse, when the intensity reaches about 10 W / cm, in order for reasonably converged results to be obtained, well-represented states with symmetries up to at least S, P°, D, and F must be included. We note that in the SSEA calculations of Ref. [122] we studied convergence quantitatively with symmetries up to L = 15. 

The values of L correspond to atomic states described as S, P, D, F, and higher states in a manner similar to the designation of atomic orbitals as s, p, d, and f. The values of S (arising from Ms) are used to calculate the spin multiplicity, defined as 25 -I- 1. For example, states having spin multiplicities of 1, 2, 3, and 4 are desaibed as singlet, doublet, triplet, and quartet states. The spin multiplicity is designated as a left superscript. Examples of atomic states are given in Table 11.3 and in the examples that follow. ... 

For the central ion of a complex the character of the electronic shell is connected with the nature of the atomic orbital. The s, p, d, and f electrons are characterized by an increasing distance to the kernel and therefore can be seen as increasingly polarizable electrons. In the language of Pearson the character of these electrons varies from extremely hard (s-electrons) to extremely soft (f-electrons). The hardness increases with ionization and for different valance states with an increased oxidation state. 

Any two electrons occupying the same orbital would have a symmetric spatial wave function. Therefore, their spin wave functions must be antisymmetric, as is the case for Equation (4.17). In other words, the Pauli exclusion principle states that no two electrons in the same atom may have all four quantum numbers the same. Each electron in an atom must possess a unique set of quantum numbers. As a result, every hydrogenic orbital in a polyelectronic atom can hold at most two electrons, and then if and only if their electron spins are opposite. Hence, the sets of s, p, d, and f orbitals for a given value of n can hold a maximum of 2, 6, 10, and 14 electrons, as suggested by the blocks of elements shown in Figure 4.12. 

The observation of spectral lines under high resolution revealed that many possessed a fine structure, and this led to the concept of electronic sublevels. These were named s, p, d and f levels, the letters having their origin in the atomic spectra of the alkali metals in which four series of lines were observed, which were known as sharpy principal, diffuse and fundamental. In 1896 some lines had been found to be split in a magnetic field by Pieter Zeeman (1865-1943), and this phenomenon was now explained in terms of electron spin. Each electron was now described in terms of four quantum numbers principal (n), orbital (/), magnetic m) and spin (5). In 1925 Wolfgang Pauli (1900-1958) put forward his exclusion principle, which stated that no two electrons in a given atom could have all four quantum numbers the same. 

Differences in lanthanide and actinide hydration thermodynamics have been discussed by Bratsch and Lagowski (1986) who attributed the difierences to relativistic effects in the actinides which cause changes in the energies of the s, p, d, and f orbitals. For example, the first and second ionization potentials of the electrons of the 7s state of the actinides are higher than those of the 6s state of the lanthanides whereas the third ionization potentials are similar for both families and the fourth ionization potential is lower for the actinides than the lanthanides. The small decrease in IP3 and IP4 for the f configuration in the actinides results in smooth variations in the relative stabilities of the adjacent oxidation states across the actinide series while the greater spatial extension of the 5f orbitals increases the actinide susceptibility to environmental efiects (Johnson 1982). 

Thus, we see the first shell contains only the It orbital the second shell contains the 2s and three 2p orbitals the third shell contains the 3s, three 3p, and five 3d orbitals and the fourth shell consists of a 4t, three 4p, five 4d, and seven orbitals. All subsequent shells contain s, p, d, and f subshells as well as others that are not occupied in any presently known elements in their lowest energy states. 

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