Spin-polarized calculations

The conclusion is that the effects of spin polarization on the total energy are very small. Spin-polarized calculations are still useful and necessary, however, because they produce spin densities, which contain valuable information about the electronic structure of the impurity at different sites. They also allow the calculation of hyperfine parameters, which can be compared directly with experiment (see Section IV.2). 

The impurity interacts with the band structure of the host crystal, modifying it, and often introducing new levels. An analysis of the band structure provides information about the electronic states of the system. Charge densities, and spin densities in the case of spin-polarized calculations, provide additional insight into the electronic structure of the defect, bonding mechansims, the degree of localization, etc. Spin densities also provide a direct link with quantities measured in EPR or pSR, which probe the interaction between electronic wavefunctions and nuclear spins. First-principles spin-density-functional calculations have recently been shown to yield reliable values for isotropic and anisotropic hyperfine parameters for hydrogen or muonium in Si (Van de Walle, 1990) results will be discussed in Section IV.2. 

The charge density in a (110) plane for neutral H at the bond-center site in Si, as obtained from pseudopotential-density-functional calculations by Van de Walle et al. (1989), is shown in Fig. 7a. In the bond region most of the H-related charge is derived from levels buried in the valence band. It is also interesting to examine the spin density that results from a spin-polarized calculation, as described in Section II.2.d. The difference between spin-up and spin-down densities is displayed in Fig. 7b. It is clear... 

The spin, in this case, is a good quantum number and spin polarized calculations are easy to perform,... 

Fig. 13. Lattice parameters of the actinide nitrides from LMTO (labelled Pauli pramagnetic), RLMTO (labelled Dirac) and LMTO spin polarized (labelled Pauli spin polarized) calculations. The black filled circles are the experimental lattice parameters...

It becomes particularly, marked for AmN where the f-band is almost half-filled and polarizes completely. The result is that the lattice parameter of AmN from the spin polarized calculation is within 0.7% of the experimental value ... 

The description of the chemisorption in terms of cycloaddition reactions is useful if it leads to reliable predictions of the reaction products for most reactions, a variety of products are possible, yet only one will result from a particular cycloaddition mechanism. Central to the applicability of such schemes is the notion that the silicon dimers contain a weak 7r bond responsible for the enforced concerted motion of the two electrons involved. However, in reality there is little evidence to support the presence of even a weak 7r bond within the dimers. While DFT calculations that enforce spin pairing depict the bond as a singlet biradical [32], spin-polarized calculations predict a triplet ground state for the unbuckled dimer [33] with no 7T character whatsoever. The decoupling of the two silicon electrons means that their motion is not likely to be concerted so that a [2+2] cycloaddition reaction becomes better represented as an independent [1 + 2+1] process, a notation that recognizes the independence of the silicon free radicals. This mechanism is also illustrated in Fig. 3. In practice such a reaction is unlikely to proceed in a concerted fashion, and a key signature for it would be the... 

One can dissociate the NO dimer simply by increasing the N-N bond distance to infinity. One can also require that during that process the molecule remain on the singlet surface, which by definition has a wavefunction and thus density that has equal spin-up and spin-down components everywhere in space. We are not interested in spin-restricted dynamics. We are interested in the much more balanced chemical dynamics that treats each half of the dissociated dimer correctly in DFT via a spin-polarized calculation. This decision must be made independent of whether or not one wants to use spatial symmetry to reduce the cost of the calculation. Spin-unrestricted DFT chemical dynamics will be called balanced in the following. 

The interconfigurational energy AE=E(dn 1s1) — E(dn 2s2) calculated by Harris and Jones96 is shown in Figures 15a and b for the Ad and 5d series, respectively. The squares show the non spin-polarized results, the full circles the spin-polarized calculations and the triangles show the experimental values. The trends in the density calculations are seen to agree well with experimental values. Earlier work of Slater et al. considered mixed configurations Zdn xAsx of Co and Ni. 

Figure 15 Interconfigurational energy AE= E(dn 1s1)—E(dn 2s2). (a) For the 4d series, (b) For the 5d series, taken from the work of Harris and Jones.96 , Nonspin-polarized results O, spin-polarized calculations, A, experimental values...

From the calculated adsorbate-substrate bond length a substantial covalent character of alkali bonding to transition metal surfaces is deduced in the low coverage limit. A spin-polarized calculation shows that the unpaired spin of the alkali atom is almost completely quenched upon chemisorption. 

Figure 2. Cluster density of states (in arbitrary units) generated by Gaussian broadening of the one-electron energies and cluster Fermi energy Cf a) icosahedral Niia (spin-polarized calculation, majority spin above the axis a = 0.2 eV). b) NiirNa (solid line a = 0.3 eV). Also shown are the sum of the contnbutions from the s and p populations of the nickel atoms (dashed line) and the population of the sodium atom (dotted line).

Table V. Mulliken populations n and spin polarization s of the valence orbitals of NiirNa obtained from a spin polarized calculation...

In section 2 the theory of ensembles is reviewed. Section 3 summarizes the parameter-free theory of G par[ll]. The self-consistently determined ensemble a parameters of the ensemble Xa potential are presented. In section 4 spin-polarized calculations using several ground-state exchange-correlation potentials are discussed. In section 5 the w dependence of the ensemble a parameters is studied. It is emphasized that the excitation energy can not generally be calculated as a difference of the one-electron energies. The additional term should also be determined. Section 6 presents accurate... 

Table II presents the first excitation energies obtained from spin- polarized calculations [24]. As ground-state exchange-correlation potentials were used the extra term in Eq.(20) does not appear. This is, certainly, one of the reasons for the difference between the calculated and the experimental excitation energies. There is a definite improvement comparing with the nonspin-polarized results [13]. Still, in most cases the calculated excitation energies are highly overestimated. The results provided by the different local density approximations are quite close to each other. The best one seems to be the Gunnarson-Lundqvist-Wilkins approximation. (In non-spin-polarized case the Perdew-Zunger parametrization gives results closest to the experimental data[30].)...

ACKN0WLECX3EMENTS. We thank P. Kasai. R. MacFarlane, H. E. Hunziker, M. Crowder and R. E. Smalley for discussions, P. S. Bagus for the spin-polarized calculation on La. 0. Hird for XPS data and R. 0. Kendrick for help with the g-vaiue measurement. 

In the case of a spin-polarized calculation, a similar fit is performed for the spin density... 

In spin-polarized calculations for magnetic systems, V c will be different for each spin a. One of the earliest approximations, known as Xa, included only the first term in Eq. (37) with an empirical parameter Xa [39] ... 

Electronic structure SCF spin-polarized calculations were performed with the DV method for the cluster [Fe(OC)2(02CC)]io formed by stripping the ferric wheel molecule of its peripheral H and Cl atoms (see Fig. 8) [85]. Mossbauer spectroscopy measurements have been reported [84] calculations of the hyperfine parameters were performed and compared to experiment. The magnetic moment obtained on the Fe was 4.3/ib and the charge +2.3 [85]. 

A final comment is required about the problem of the description of isolated TM atoms on oxide surfaces in general. TM atoms have complex spin states which are not properly described within the DFT approach. Spin-polarized calculations provide a way to take into account the spin properties of the system but some details of the interaction may be described incorrectly. In this respect, the use of wave function-based methods is particularly important for benchmarks and comparisons. This problem is less severe when one considers small clusters where several electronic states exist separated by small energies. But one should be well aware of the fact that the treatment of isolated atoms, dimers, and very small aggregates with DFT methods requires spin-polarized approaches and special care in evaluating the results [199,200]. 

Within the framework of the local density method, which is strictly an orbital theory, the antiferromagnetic state can be attained by reducing the symmetry constraints imposed on spin-polarized calculations, hence allowing the spin orbitals to localize and local magnetic moments to persist, if it is variationally favorable to do so. While I do not know of any formal justification for this type of symmetry breaking (one cannot just mix determinants within DFT to obtain proper spin and space eigenfunctions), the results discussed below for Cr2, M02, and Mn2 certainly indicate that it is a reasonable approach. A rough rationalization can be obtained if one reasons... 

Fig. 8. Calculated surface energy for fcc(lll) surfaces of 3d and 4d metals (solid squares), compared with experiment (open circles) (Skriver and Rosengaard, 1992 tight-binding L.MTO-ASA, with Green function method). For the 3d metals, the dashed line connecting solid circles gives results from spin-polarized calculations. For the 4d metals, the dashed line connecting open triangles gives results from Methfessel et al. (1992 full potential LMTO, slab geometry).

The structural similarity of MgAgAs-type compounds with Heusler alloys and with the transition-metal based half-metallic ferromagnets (de Groot et al. 1983) has provoked band-structure calculations of UNiSn (Mueller et al. 1987, Albers et al. 1987). Self-consistent-field scalar relativistic spin-polarized calculations neglecting the spin-orbit coupling revealed the following features of the valence band ... 

In the latter case, one has to be aware of solutions with broken spatial symmetry. This problem arises also in NCSDFT the (initial) symmetry of a system, as described by a scalar Hamiltonian, is destroyed by the vector field term proportional to as, which, similarly to an external magnetic field, reduces the spatial symmetry of the one-electron Hamiltonian. In spin-polarized calculations including SO interaction, the conventional collinear approach, where only one component of the spin-density s = Tr a p) is used in the definition of the xc energy functional, has the major drawback of breaking the spatial symmetry of the energy functional [18,64]. 

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