In this chapter, we will restrict our attention to the ferrites and a few other closely related materials. The great interest in ferrites stems from their unique combination of a spontaneous magnetization and a high electrical resistivity. The observed magnetization results from the difference in the magnetizations of two non-equivalent sub-lattices of the magnetic ions in the crystal structure. Materials of this type should strictly be designated as “ferrimagnetic” and in some respects are more closely related to anti-ferromagnetic substances than they are to ferromagnetics in which the magnetization results from the parallel alignment of all the magnetic moments present. We shall not adhere to this special nomenclature except to emphasize effects, which are due to the existence of the sub-lattices.

Ferromagnetic materials

The surface and materials science of tin oxide

Two three-dimensional numerical schemes are presented for molecular integrands such as matrix alements of one-electron operators occuring in the Fock operator and expectation values of one-electron operators describing molecular properties. The schemes are based on a judicious partitioning of space so that product-Gauss integration rules can be used in each region. Convergence with the number of integration points is such that very high accuracy (8–10 digits) may be obtained with obtained with a modest number of points. The use of point group symmetry to reduce the required number of points is discussed. Examples are given for overlap, nuclear potential, and electric field gradient integrals.

Three-dimensional numerical integration for electronic structure calculations

Prediction and interpretation of the 57Fe isomer shift in Mössbauer spectra by density functional theory

Figure 18 presents the C-O stretching vibrational frequencies of the first-row transition-metal monocarbonyl cations, neutrals, and anions in solid neon; similar diagrams have been reported for neutral MCO species in solid argon, but three of the early assignments have been changed by recent work and one new assignment added. The laser-ablation method produces mostly neutral atoms with a few percent cations and electrons for capture to make anions; in contrast, thermal evaporation gives only neutral species. Hence, the very recent neon matrix investigations in our laboratory provide carbonyl cations and anions for comparison to neutrals on a level playing field. Several trends are very interesting. First, for all metals, the C-O stretching frequencies follow the order cations > neutrals > anions with large diagnostic 100-200 cm-1 separations, which is consistent with the magnitude of the metal d to CO pi * donation. Second, for a given charge, there is a general increase in C-O stretching vibrational frequencies with increasing metal atomic number, which demonstrates the expected decrease in the metal to CO pi * donation with increasing metal ionization potential. Some of the structure in this plot arises from the extra stability of the filled and half-filled d shell and from the electron pairing that occurs at the middle of the TM row; the plot resembles the "double-humped" graph found for the variation in properties across a row of transition metals. For the anions, the variation with metal atom is the smallest since all of the metals can easily donate charge to the CO ligand. Third, for the early transition-metal Ti, V, and Cr families, the C-O stretching frequencies decrease when going down the family, but the reverse relationship is observed for the late transition-metal Fe, Co, and Ni families. In most of the present discussion, we have referred to neon matrix frequencies; however, the argon matrix frequencies are complementary, and useful information can be obtained from comparison of the two matrix hosts. In most cases, the neon-to-argon red shift for neutral carbonyls is from 11 to 26 cm-1, but a few (CrCO) lie outside of this range. In the case of FeCO and Fe(CO)2, it appears that neon and argon trap different low-lying electronic states. In general, the carbonyl neutrals and anions have similar shifts but carbonyl cations have larger matrix shifts. For example, the FeCO+ fundamental is at 2123.0 cm-1 in neon and 2081.5 cm-1 in argon, a 42.5 cm-1 shift, which is larger than those found for FeCO- (11.7 cm-1) and FeCO (11.7 cm-1). It is unusual for different low-lying electronic states to be trapped in different matrices, but CUO provides another example. The linear singlet state (1047.3, 872.2 cm-1) is trapped in solid neon, and a calculated 1.2 kcal/mol higher triplet state is trapped in solid argon (852.5, 804.3 cm-1) and stabilized by a specific interaction with argon. The bonding trends are well described by theoretical calculations of vibrational frequencies. Table 5 compares the scale factors (observed neon matrix/calculated) for the C-O stretching modes of the monocarbonyl cations, neutrals, and anions of the first-row transition metals observed in a neon matrix using the B3LYP and BP86 density functionals. Most of the calculated carbonyl harmonic stretching frequencies are within 1% of the experimental fundamentals at the BP86 level of theory, while calculations using the B3LYP functional give frequencies that are 3-4% higher as expected for these density functionals and calculations on saturated TM-carbonyls. For second- and third-row carbonyls using the BP86 density functional and the LANL effective core potential in conjunction with the DZ basis set, the agreement between theory and experiment is just as good. For example, the 16 M(CO)1-4 neutral and anion and 2 MCO+ cation (M = Ru, Os) carbonyl frequencies are fit within 1.5%. The 16 species (M = Rh, Ir) are fit within 1%, but the Rh(CO)1-4+ calculations are 2-3% too low and Ir(CO)1-4+ computations are 1-2% too low. In addition to predicting the vibrational frequencies, DFT can be used to calculate different isotopic frequencies, and isotopic frequency ratios can be computed as a measure of the normal vibrational mode in the molecule for an additional diagnostic. For diatomic CO, the 12CO/13CO ratio 1.0225 and C16O/C18O ratio 1.0244 characterize a pure C-O stretching mode. In a series of molecules such as RhCO+, RhCO, and RhCO-, where the metal-CO bonding varies, the Rh-C, C-O vibrational interaction is different and the unique isotopic ratios for the carbonyl vibration are characteristic of that particular molecule. Table 6 summarizes the isotopic ratios observed and calculated for the RhCO+,0,- species. Note that RhCO+ exhibits slightly more carbon-13 and less oxygen-18 involvement in the C-O vibration than CO itself and that this trend increases to RhCO and to RhCO- as the Rh-C bond becomes shorter and stronger. Note also how closely the calculated and observed ratios both follow this trend. In a molecule with two C-O stretching modes, for example, bent Ni(CO)2 exhibits a strong b2 mode at 1978.9 cm-1 and a weak a1 mode at 2089.7 cm-1 in solid neon, and these two modes involve different C and O participations. The symmetric mode shows substantially more C (1.0242) and less O (1.0217) participation than does the antisymmetric mode with C (1.0228) and O (1.0238) involvement, based on the given isotopic frequency ratios, which are nicely matched by DFT calculations (a1 1.0244, 1.0224 and b2 1.0232, 1.0241, respectively). These investigations of vibrational frequencies in unsaturated transition-metal carbonyl cations, neutrals, and anions clearly demonstrate the value of a close working relationship between experiment and theory to identify and characterize new molecular species.

Spectroscopic and theoretical investigations of vibrational frequencies in binary unsaturated transition-metal carbonyl cations, neutrals, and anions.

First-principles electronic-structure calculations were performed for embedded clusters representing an Fe impurity in the metal hosts Be, Mg, Ca, Sr, and Al. The discrete-variational method was employed in the framework of local-spin-density theory. BeFe was found to be nonmagnetic, as well as AlFe, this last if local relaxation is taken into account. For Fe in the other alkaline-earth metals, local magnetic moments g2${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ were encountered. Mechanisms leading to moment stability or moment quenching are discussed. The contact hyperfine field at the Fe nucleus in CaFe was calculated, and a small positive value was found. Magnetic susceptibilities were estimated employing the virtual-bound-state model, with parameters derived from the calculations.

Magnetism of Fe impurities in alkaline-earth metals and Al.

The Discrete Variational method for molecules and clusters (DVM), in the framework of Density Functional theory, is described in detail. The numerical grids utilized, basis functions and potential are discussed, as well as spin-polarization for magnetic systems, total energy and dynamics. The relativistic version of the DV method is also described. Applications to large molecules range from porphyrins, a transition metal complex of thiophene, the circular molecule “ferric wheel” containing ten Fe atoms and other transition metal complexes investigated by fragments. Examples of relativistic calculations are given for 5d-metal complexes. Calculations for solids, represented by embedded clusters as large as 65–75 atoms, include transition metals, perovskites, silicates and rare-earth borocarbides. Properties investigated and analysed are structural, optical, hyperfine, magnetic and superconducting. The electronic structure and chemical bonds are also studied by Mulliken populations and charges, bond order, density of states and spin density maps; results are related to experimentally observed characteristics.

The Discrete Variational Method in Density Functional Theory and its Applications to Large Molecules and Solid-State Systems

Electronic structure and electric-field gradients of crystalline Sn(II) and Sn(IV) compounds

Electronic structure, magnetic properties, Mössbauer isomer shifts, and hyperfine fields of disordered Fe-rich Fe-Al alloys.

First-principles self-consistent spin-polarized electronic structure calculations were performed for the nanoscale magnetic molecules Mn12O12(CH3COO)16(H2O)4 and Fe11O6(OH)6(O2CPh)15 . The numerical Discrete Variational method was employed, within Density Functional theory. Charges and magnetic moments were obtained for the atoms, as well as Density of States diagrams, and charge and spin density maps. For Mn12O12(CH3COO)16(H2O)4 , values of the Heisenberg exchange parameters J were derived from the calculations; Mossbauer hyperfine parameters were calculated for Fe11O6(OH)6(O2CPh)15 and compared to reported experimental values. Key-words: Nanoscale; Molecular magnets.

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Diana Guenzburger

Papers

Magnetism of Fe impurities in alkaline-earth metals and Al.

The Discrete Variational Method in Density Functional Theory and its Applications to Large Molecules and Solid-State Systems

Electronic structure and electric-field gradients of crystalline Sn(II) and Sn(IV) compounds

Electronic structure, magnetic properties, Mössbauer isomer shifts, and hyperfine fields of disordered Fe-rich Fe-Al alloys.

Electronic structure, spin couplings, and hyperfine properties of nanoscale molecular magnets