Valence contributions

Another typical example for anisotropic covalency is found in five-coordinate ferric compounds with intermediate spin S = 3/2 (also discussed in Sect. 8.2). Crystal field theory predicts a vanishing valence contribution to the EFG, whereas large quadrupole splittings up to more than 4 mm s are experimentally found. 

Fig. 5.5 Lack of correlation between the As orbital population and the valence contribution to the isomer shift (taken from [19])...

Fig. 5.6 Changes in the shape of the valence contribution due to geometric and electronic relaxation in [FeF4]" . Full line [FeF4] at its equilibrium geometry, dashed line [FeF4] at its equilibrium geometry. The square of the valence orbital that mainly contributes to p(0) along the Fe-F bond (distances are in units of the Bohr radius) is also drawn (from [19])...

For the one-center valence contribution, there are essentially three factors that control its value (a) the radial wavefunction of the 3d orbitals, (b) the covalent dilution of the 3d orbitals with ligand orbitals, and (c) the occupation pattern of the 3d shell. An additional factor may be low-symmetry induced 3d/4p mixing. We will focus on the first three factors here. 

From the ligand field considerations outlined previously, and the values in Table 5.6, the EFG along the Fe-O bond should be dominated by the one-center valence contributions around the iron and given by ... 

In a crystal-field picture, the electronic structure of iron in the five-coordinate compounds is usually best represented by a (d yf idyz, 4cz) ( zO configuration [66, 70], as convincingly borne out by spin-unrestricted DFT calculations on the Jager compound 20 [68]. The intermediate spin configuration with an empty d 2 yi orbital in the CF model, however, has a vanishing valence contribution to the... 

Nevertheless, core-correlation contributions to AEs are often sizeable, with contributions of about 10 kJ/mol for some of the molecules considered here (CH4, C2H2, and C2H4). For an accuracy of 10 kJ/mol or better, it is therefore necessary to make an estimate of core correlation [9, 56]. It is, however, not necessary to calculate the core correlation at the same level of theory as the valence correlation energy. We may, for example, estimate the core-correlation energy by extrapolating the difference between all-electron and valence-electron CCSD(T) calculations in the cc-pCVDZ and cc-pCVTZ basis sets. The core-correlation energies obtained in this way reproduce the CCSD(T)/cc-pCV(Q5)Z core-correlation contributions to the AEs well, with mean absolute and maximum deviations of only 0.4 kJ/mol and 1.4 kJ/mol, respectively. By contrast, the calculation of the valence contribution to the AEs by cc-pCV(DT)Z extrapolation leads to errors as large as 30 kJ/mol. 

The latter equation achieves a separation of the core and valence contributions. The terms in brackets are individually zero at the hmits = 0 and N = 0. Here we postulate that physically meaningful core populations exist that allow such a core-valence separation and proceed with... 

The addition of 0.18 interstitial ions to the formula unit of La2Ni04 requires that the oxidation state of Ni be increased to +2.36. Given that the equatorial Ni-O bonds have a length of 194 pm and therefore a bond valence of 0.46 vu, this increase in the oxidation state of Ni allows the axial bond valences to be increased from 0.08 to 0.26 vu reducing the length of the Ni-Oa iai bonds from 259 pm to the more acceptable value of 215 pm. This in turn reduces the valence required for the axial La-O bond by 0.18 vu which, together with the extra valence contributed by the interstitial 0 , reduces the distortion around La " " to an acceptable level. It is difficult to calculate the BSI and GII for this compound since one needs to know how the interstitial 0 ions are ordered within the LaO double layer, but clearly the BSI will be considerably reduced from the value 0.29 vu that it had before the introduction of the defect and subsequent electronic relaxation. This form of the structure is stable and is the form normally found when the material is prepared in air. 

In this chapter, we use exclusively relativistically optimized or experimental geometries. Hence, we concentrate on direct relativistic effects only. They can be separated into scalar and spin-orbit/Fermi contact effects. In addition, there are, in both cases, core and valence contributions. 

The quadrupolar splittings depend mainly on the electric field gradient at the nucleus, which is caused by the spatial distribution of the electrons around it, via electric Coulomb forces. Both electrons (valence contribution f/yai) and neighboring anions and cations (lattice contribution U x) in the vicinity of the nucleus contribute to this electric field gradient U, which can be expressed as follows ... 

Molecules with Several Atomic Cores.—From the above discussion it is seen that, in principle, the effective hamiltonian for atomic valence electrons is dependent on the valence state of the atom, this dependence arising from the valence contribution to the all-electron Fock operator F. In practice this dependence is very weak unless the atom is multiply ionized, and can usually be safely neglected, so that a single effective hamiltonian can suffice for many valence states. However, for a molecular system in which there is more than one core region additional approximations must be introduced to maintain a simple form of the effective hamiltonian. For two atomic cores defined in terms of orbital sets and and a valence set < F) the equation equivalent to (21) is... 

The quadrupole splitting is sensitive to temperature primarily through the valence contribution, which reflects the temperature dependence of electrons between different... 

Doublet 1 may be assigned unambiguous to P4 symmetry, since the hyperfme parameters follow the expected behaviour for Fe in a relatively undistorted tetrahedral site. The high value of quadrupole splitting indicates a high valence contribution, and displays the expected temperature variation for tetrahedral environments bb 1968). The characteristic Mossbauer temperature 0m is in the range expected for Fe (De Grave and Van Alboom 1991). 

It Is, In particular, not common practice to describe the continuum electronic states of molecules In terms of Rydberg and valence contributions, nor to clarify their spatial characteristics In terms of atomic compositions, largely as a consequence of the absence of theoretical procedures for constructing continuum states In such fashion. 

Owing to the similarity of the mass-polarization operator and the Breit operator, the mass-polarization corrections in MBPT can be classified using the scheme described for the Breit interaction in the previous subsection. Correspondingly, we write the valence contribution to the expectation value of P as... 

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