Polarization calculation

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... 

Stemming from a comparison between calculated and experimental methods, the proposed calculations appear quite precise and thus open up new perspectives in polarization calculations for natural crystalline phases. As we will see in chapter 4, polarization energy is of fundamental importance in the evaluation of defect equilibria and consequent properties. 

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 ... 

Figure 11.15 shows Asano s calculations of extinction by nonabsorbing spheroids for an incident beam parallel to the symmetry axis, which is the major axis for prolate and the minor axis for oblate spheroids. Because of axial symmetry extinction in this instance is independent of polarization. Calculations of the scattering efficiency Qsca, defined as the scattering cross section divided by the particle s cross-sectional area projected onto a plane normal to the incident beam, are shown for various degrees of elongation specified by the ratio of the major to minor axes (a/b) the size parameter x = 2ira/ is determined by the semimajor axis a. 

To account for this phenomenon the hybrids used in the bond polarization calculations were described in agreement with an idea taken from the maximum overlap method (17). Atomic hybrids with a rigid relative orientation were optimized so that the overlap of the hybrids that form a bond is at maximum. The estimated... 

The semi-empirical bond polarization model is a powerful tool for the calculation of, 3C chemical shift tensors. For most molecules the errors of this model are in the same order of magnitude as the errors of ab initio methods, under the condition that the surrounding of the carbon is not too much deformed by small bond angles. A great advantage of the model is that bond polarization calculations are very fast. The chemical shift tensors of small molecules can be estimated in fractions of a second. There is also virtually no limit for the size of the molecule. Systems with a few thousand atoms can be calculated with a standard PC within a few minutes. Possible applications are repetitive calculations during molecular dynamics simulations for the interpretation of dynamic effects on 13C chemical shift distribution. 

If the electron solvent polarization is neglected, the study of electron transitions and the determination of the solvent shift do not require appreciable modifications in the basic scheme of ASEP/MD. During a Franck-Condon transition the solute and solvent nuclei remain fixed and hence the ASEP obtained for the initial state can be used for the rest of the states of interest. However, it is known that the electron degrees of freedom of the solvent can respond to the sudden change of the solute electron charge distribution. In fact, the polarization component can contribute appreciably to the final value of the solvent shift. The determination of this component requires additional calculations where the solute and solvent charge distributions are equilibrated. Each electronic state requires a separate calculation of the solvent polarization component. It is hence necessary to perform as many polarization calculations as electronic states being considered. 

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. 

In other words, to obtain a closed system to solve numerically, we must require that the nonlinear polarization is well approximated by the nonlinear polarization calculated only from the forward propagating field. This means that the equation is only applicable when the back-reflected portion of the field is so small that its contribution to the nonlinearity can be neglected. 

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].)...

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