Polarization spin, calculations

Table 5.8 Contributions to the Fe hyperfine interaction in [Fe(0)(NH3)5] in the lowest 5=1 and 5 = 2 states from spin-polarized B3LYP calculations (in MHz) (from [60])...

It is relatively common for DFT calculations to not explicitly include electron spin, for the simple reason that this approximation makes calculations faster. In materials where spin effects may be important, however, it is crucial that spin is included. Fe, for example, is a metal that is well known for its magnetic properties. Figure 8.10 shows the energy of bulk Fe in the bcc crystal structure from calculations with no spin polarization and calculations with ferromagnetic spin ordering. The difference is striking electron spins lower the energy substantially and increase the predicted equilibrium lattice constant by 0.1 A. 

In Fig. 6, it is seen that the americium metallic volume at T = 0, p = 0, is well reproduced by spin-polarized band calculations. The same band-calculations predict the pressure p in which the transition from localized to itinerant behaviour occurs for americium. Here, we want to present the simplified Friedel-type model (Skriver, Anderson and Johansson ) by which the spin-polarized americium system is described and both transitions, i and ii, predicted. One word of caution the model, as well as the band calculations, do not account for the change in structure accompanying the transition, and which is an important fact in actinide metals, which will be discussed in the following... 

In order to calculate the magnetic moment it is necessary to perform spin polarized band calculations. The extension of the local density approximation to cover the spin polarized case has been made by von Barth and Hedin and Guimarsson and Lund-qvist ... 

Fig. 21. Experimental and theoretical form factors for NiO [after Ref. 26)]. After inclusion of the effects of covalency, the orbital moment, and inner shell spin polarization, the calculated curve is in good agreement with the experimental figure [Ref. (27)], especially for x/4ji<0.3...

A 62-atoms embedded cluster in cubic geometry was considered to represent 7-Fe (see Fig. 17), and spin-polarized SCF calculations were performed for both the ferromagnetic (FM) and antiferromagnetic (AFM) states, at several lattice constants. For the AFM, a layered arrangement of up and down spins (illustrated in Fig. 17) was considered. 

The LSD calculations of HJ predict a l(Tg2ffg l7r 15g l(5 l7ig lUy ground state with a second state derived by transferring an electron from 5 to Ttg about 0.2 eV higher. The calculated bond length, 2.07 A, is a reasonable interpolant of the experimental values for Ni2 and Fe2 while the calculated frequency, 360 cm S is only in moderate accord with the experimental value. No fully self-consistent spin-polarized LSD calculations have yet been performed for C02. 

In this section we will illustrate the gap concept by some explicit applications. We exclude applications to small molecules as well as spin-polarized band calculations, which fall outside the scope of this article. Several of the examples to be discussed are related to the concepts of Mott and Peierls gaps,21 28 but we will not discuss these very interesting aspects here. A number of—direct or indirect—applications on polyenes have been carried out these have been treated in another paper.29... 

Fig. 3.8. Direct magnetoabsorption in germanium at RT. The polarization condition is usually referred as n polarization. The calculated peaks 1 and 2 correspond to transitions from spin-split levels of the Landau ladder of the heavy hole valence band and the 1+ and 2+ ones to corresponding transitions for light hole VIS. With the ordinate scale used, the indirect absorption is barely visible (after [13])...

Electronic structure and intramolecular exchange constants have been calculated (spin-polarized DFT calculations) for three cyanide-bridged molecular magnets, Fem(Tp )(CN)3Mn(DMF)412-(OTf)2-2DMF (Mn=Mn, Co, Ni).264... 

Prior to the work of Ref [84], a series of LMTO calculations had studied UO2. A full-potential, non-spin-polarized LMTO treatment with a gradient-corrected XC model gave a lattice constant (fluorite structure) of 9.81 au, well below the experimental 10.34 au (see [84] for references to the other literature). An approximate LMTO (LMTO-ASA) non-spin-polarized treatment with LDA XC gave 9.92 au and a bulk modulus of 5 = 291 GPa versus the experimental value 207 GPa. Spin-polarized LDA calculations did give anti-ferromagnetic ordering (foimd experimentally) at the experimental lattice constant but the moment vanished at the LDA equilibrium value. All the results were metallic. 

Figure 16 Calculated potential energies for an isolated H2 molecule as a function of interatomic distance. The closed and open circles are results from spin-unpolarized and spin-polarized GGA calculations, respectively, the solid curve is a fit to a Morse potential, and the dashed curve shows the highly accurate results of Kolos and Wolniewicz from ref. 43...

The main motivation of the spin polarized band calculation was to study the spatial distribution or form factor of the conduction electron spin. This will be discussed more thoroughly in ch. 7, section 4. 

The zero-field moment of 7.63/iB permits a direct estimate of the conduction electron polarization as 0.63/xb, since the anisotropy-induced zero point motion is negligible for Gd. This increased estimate of the conduction electron polarization is of particular significance for energy band calculations. Harmon and Freeman s spin-polarized APW calculation (1974) requires little adjustment to concur with the experimental result. 

As the first step, B3LYP spin-polarized LCAO calculations for the cubic LaMnOs and SrMnOs (with one formula unit per primitive cell) are performed, using the maximal spin projection Sz = 2 for four d-electrons of the Mn + ion. As we have seen in Chap. 9, such a spin projection ensures the lowest total energy compared with Sz = 0,1. The optimized cubic lattice constants are a=3.967 A and a=3.840 A for LaMnOa and SrMnOs, respectively. These values are in a reasonable agreement with the experimental lattice constants a=3.947 A and a=3.846 A, respectively. The two optimized cubic lattice constants for LaMnOs and SrMnOs were used for calculating the lattice constants of their solid solutions according to Vegaid s law (linear dependence of the lattice parameters on the composition). As follows from the WIEN-2k calculations, this is fulfilled quite well in this system. 

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