Factor orbital reduction

The experimental data, in accordance with theoretical predictions, show that the orbital angular momentum in real metal complexes is somewhat reduced so that it is legitimate to operate with L - tcL, 0 tc 1 being an orbital reduction factor. There are two sources of orbital reduction  

Terms of the magnetic susceptibility formula for multiplets of cT-metal ions 

In metal complexes a set of molecular orbitals can be written in the form 

Since the action of the angular momentum operator to a ket dj ) yields a (complex) number Sj and another ket, say djy) 

The last term can be neglected owing to the combined effect of the small coefficient (cL)2 and the small integral (xf/i xf)- Using the normalisation condition for molecular orbitals 

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A detailed treatment of the temperature dependence and anisotropy of the magnetic moments of all the dx configurations in pseudo-axial (CooV) symmetry has though now been given by Warren (101), in which variation of the orbital reduction factor, k, and distortions from effective Cv symmetry were also considered. This has lately been followed by a similar treatment due to Cerny (102) of the d d2, d8, and d9 configurations but, although some sophistications were included the results are essentially equivalent to those of the author, and furthermore only the undistorted situation, with k = 1, was considered. Consequently the author s own treatment (101) is here briefly summarised, the theoretical approach being that most appropriate for the sandwich complexes of the 3 d series, to which the bulk of the available experimental material relates. 

In interpreting the experimentally determined moments however (which with the sole exception of Fe(Cp)2+ relate only to average moments), the part played by two parameters is of great importance, namely the distortion parameter, A, and the orbital reduction factor,... 

Orbitally Degenerate Ground States. For all orbitally degenerate ground states the g values are calculated to be markedly anisotropic and to deviate substantially from 2. Thus, writing the effective orbital reduction factor as k (equivalent to k of Section 4), the g values are found (101) to be as follows for the systems shown -... 

The ligand field was parameterised in terms of ea only, and values of this parameter, together with the spin-orbit coupling constant X, the orbital reduction factor k and the Racah parameter B were obtained by fitting the d-d spectra, zero-field splittings, principal magnetic susceptibilities and e.s.r. g-values. 

The eventual covalence effect can be accounted for via an orbital reduction factor k < 1 that reduces the orbital angular momentum... 

The Hamiltonian for interacting T-terms can be modified to the form accounting for the orbital reduction factor... 

The magnetic susceptibility components are functions of the Racah parameters (B, C), CF strengths (F2(L) and F4(L)), the spin-orbit coupling constant d> and the orbital reduction factors kx, ny, kz. 

The 6x6 secular equation within the ground2 T2g term on the symmetry descent (Table 13) is fully factored and yields analytical formulae for the energy levels (Table 30) also in the case of the anisotropic orbital reduction factors. Then the energy gap S67 = / (/ 7)a - (/, ) is < 67 = A.[- 2v + 3kz - (Av2 + 4kzv + 8/c2 + k2 )1/2 ] /4 and disappears upon recovery of the octahedral geometry. 

Fig. 23 Effective magnetic moment for octahedral d2 complexes like V(III). Left-. Figgis theory dotted - strong field reference (A = 1), solid - weak field reference (A = 3/2). Complete d2 calculation dash-dot-dot. Effect of the orbital reduction factor dashed - k = 0.4, dash-dot - k = 0 (spin-only magnetism). Inset - enlarged low-temperature window. Right the calculated energy levels (not to scale, values in cm-1)...

Orbital reduction factor k 1 tends to switch off orbital angular momentum toward spin-only case (S = 1). 

These gaps collapse properly to zero when both orbital reduction factors are isotropic. The derivation is consistent with the result of Solomon et al. [86]. 

Within the SH formalism the MPs (gx, gy, gz, D, E, /tip) are thought of as physical constants associated with each particular system. The electronic-magnetic theory beyond the SH formalism reveals that there are only the electronic-structure parameters (like B, C, ) associated with the electron configuration and the CF parameters [like F2(L) and F4(I)] for each ligand. A more realistic approach brings the orbital reduction factors k (which must be anisotropic) and in a particular case of the degenerate electronic states also the force-field and vibronic coupling parameters (like Kee, Xe, Xee, and eventually Ktt, Xt, Xtt, or even more parameters). 

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