Electronic multiplets

Under the second topic of Ligand-field Theory and its Extensions we describe the basic concepts behind the various versions of LFT - the angular-overlap model (AOM) and its extensions. In the section named The physical background conditions for the applicability of the ligand-field approach we sketch briefly the theoretical foundation and limits of applicability of the effective-hamiltonian approach with special attention to electronic multiplets. In the theory section, we describe various approaches in current calculations of electronic structure, such as LFDFT, SORCI and TDDFT, with the various applications detailed in the following section, before an outlook for further developments. 

LFT originated as a purely electrostatic model - crystal-field theory (CFT) [18], in which d-electronic multiplets of transition metals are perturbed by ligands as point charges or point dipoles. The CF operator (Equation 1) acts within the space of Slater determinants (SD) composed of purely d-spin-orbitals in which two-electron energies are taken into account with the Coulomb operator and one-electron energies with a crystal-field potential (vcp), the first and second terms in Equation 1, respectively. 

Transitions between crystal-field-split f-electron multiplet levels of various trivalent lanthanide ions (R) have also been observed by Raman scattering in the insulating phase of high-Tc cuprates (Heyen et al. 1991, Ruf et al. 1992, Dufour et al. 1995, Sanjuijo et al. 1995, Jandl et al. 1995, Cardona 1999), particularly those containing Nd " ... 

These two functions combine into two electronic multiplet states and P. P is ninefold degenerate and P threefold degenerate. The spin-triplet has the lowest energy. This rule is called Hund s rule (after Friedrich Hund who started his career as an assistant of Werner Heisenberg in Gottingen). The rule is not absolute, but if some approximations are made, it may be proven mathematically. 

FIGURE 2.3 Electronic multiplet states belonging to the carbon atom in the ground configuration [He]2s 2p. ... 

Carrier Number of outer electrons Multiplet structure... 

Wliile the earliest TR-CIDNP work focused on radical pairs, biradicals soon became a focus of study. Biradicals are of interest because the exchange interaction between the unpaired electrons is present tliroiighoiit the biradical lifetime and, consequently, the spin physics and chemical reactivity of biradicals are markedly different from radical pairs. Work by Morozova et al [28] on polymethylene biradicals is a fiirther example of how this method can be used to separate net and multiplet effects based on time scale [28]. Figure Bl.16.11 shows how the cyclic precursor, 2,12-dihydroxy-2,12-dimethylcyclododecanone, cleaves upon 308 mn irradiation to fonn an acyl-ketyl biradical, which will be referred to as the primary biradical since it is fonned directly from the cyclic precursor. The acyl-ketyl primary biradical decarbonylates rapidly k Q > 5 x... 

There are two further rules for ground terms which tell us whether a multiplet arising from equivalent electrons is normal or inverted. 

Normal multiplets arise from equivalent electrons when a partially filled orbital is less than half full. 

Also shown in Figure 8.15 is the line due to removal of a 2p electron from the sulphur atom in thiophene. Spin-orbit coupling is sufficient to split the resulting core term, as it is called, into 3/2 and 1/2 states, the multiplet being inverted. 

The ground configuration of Ar is KL3s 3p, giving an inverted P /2 multiplet. The excited states involved in laser action involve promotion of an electron from the 3p orbital into excited As,5s,Ap,5p,3d,Ad,... orbitals. Similarly, excited states of Kr involved arise from promotion of an electron from the Ap orbital. In Ar the KL3s 3p configuration gives rise to 5, V, terms (see Section 7.1.2.3). Most laser transitions involve the core in one of the states and the promoted electron in the Ap orbital. 

Apart from the A-methyl group, three double-bond equivalents and three multiplets remain in the chemical shift range appropriate for electron rich heteroaromatics, Sh = 6.2 to 6.9. A-Methyl-pyrrole is such a compound. Since in the multiplets at Sh = 6.25 and 6.80 the Jhh coupling of 4.0 Hz is appropriate for pyrrole protons in the 3- and 4-positions, the pyrrole ring is deduced to be substituted in the 2-position. 

The only electronic states that can be treated are those that correspond to the highest spin multiplet of a given orbital configuration. 

The same equatorial preference is also manifested in the 3,3-disubstituted thietane oxides66,194. Thus, the NMR spectra of 5e,f contain two Me singlets at 1.23 and 1.30 ppm and two methylene multiplets at 3.03 and 3.53 ppm (in CDC13). The large difference in the chemical shifts of the axial and equatorial a-methylene hydrogens is characteristic of an axial nonbonded electron pair on sulfur (conformation 5e equation 73). This conformational preference is corroborated by the small differences in the chemical shifts of the two methyl groups, and fits the contention that 1,3-diaxial interactions are responsible for this ultimate result. Certainly, these interactions are greater in the conformer 5f. 

Here L, S, and J are the quantum numbers corresponding to the total orbital angular momentum of the electrons, the total spin angular momentum, and the resultant of these two. Hund predicted values of L, S, and J for the normal states of the rare-earth ions from spectroscopic rules, and calculated -values for them which are in generally excellent agreement with the experimental data for both aqueous solutions and solid salts.39 In case that the interaction between L and S is small, so that the multiplet separation corresponding to various values of J is small compared with kT, Van Vleck s formula38... 

The first example of chemically induced multiplet polarization was observed on treatment of a solution of n-butyl bromide and n-butyl lithium in hexane with a little ether to initiate reaction by depolymerizing the organometallic compound (Ward and Lawler, 1967). Polarization (E/A) of the protons on carbon atoms 1 and 2 in the 1-butene produced was observed and taken as evidence of the correctness of an earlier suggestion (Bryce-Smith, 1956) that radical intermediates are involved in this elimination. Similar observations were made in the reaction of t-butyl lithium with n-butyl bromide when both 1-butene and isobutene were found to be polarized. The observations were particularly significant because multiplet polarization could not be explained by the electron-nuclear cross-relaxation theory of CIDNP then being advanced to explain net polarization (Lawler, 1967 Bargon and Fischer, 1967). 

The existing SCF procedures are of two types in restricted methods, the MO s, except for the hipest (singly) occupied MO, are filled by two electrons with antiparallel spin, while in unrestricted methods, the variation procedure is performed with individual spin orbitals. In the latter, a total wave function is not an eigenvalue of the spin operator S, which is disadvantageous in many applications because of a necessary annihilation of higher multiplets by the projection operator. Since in practical applications the unrestricted methods have not proved to be remarkably superior, we shall call our attention in this review mainly to the restricted methods. 

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