Projected density of states

From the Kronig-Kramer relation it immediately follows that A(e) = 0, as the function to be integrated is odd, and hence the resulting projected density of states becomes... 

Figure 6.21. Projected density of states Ha( ) when an adsorbate level located at Eg = 12.0 eV approaches a surface with an sp band. The function A(e) follows the shape of an sp band at low energies, but decreases at higher energies due to a vanishing overlap. See text for further explanation.

Figure 4.9 Total and projected density of states forthe hydroxylated (top) and reduced (bottom) TiOjfl 1 0) surface, calculated using the B3LYP hybrid functional. The Ti3+ states are localized on (a) the Ti ion between the two bridging OH groups, Tilton (d) the Ti ion nearest to the oxygen vacancy, Tij (b), (c) on a five-...

In principle, valence band XPS spectra reveal all the electronic states involved in bonding, and are one of the few ways of extracting an experimental band structure. In practice, however, their analysis has been limited to a qualitative comparison with the calculated density of states. When appropriate correction factors are applied, it is possible to fit these valence band spectra to component peaks that represent the atomic orbital contributions, in analogy to the projected density of states. This type of fitting procedure requires an appreciation of the restraints that must be applied to limit the number of component peaks, their breadth and splitting, and their line-shapes. 

Fig. 10. Hybridized model densities of states for UN. The full rectangles are the original unhybridized densities of states (see Fig. 48). The broken rectangles are the additional projected densities of states due to hybridization. In a vertical line are the contributions to the local atomic and angular momentum projected densities of states. The electron transfer, in terms of the fractional occupancy (F) of the unhybridized f-band, is shown...

Figure 2.7. Theoretical adatom-projected density of states of different atoms on jellium with an electron density that corresponds to A1 metal. Reproduced from [30].

Figure 4.2. The density of states projected onto the d states of the surface atoms for the surfaces considered in Figure 4.1 (grey). Also shown (black) is the oxygen 2px projected density of states for adsorbed on the same surfaces. The formation of bonding and anti-bonding states below and above the metal d states is clearly seen. Adapted from Ref. [4].

This immediately gives the projected density of states ... 

Figure 4.5 shows solutions to the Newns-Anderson model using a semi-elliptical model for the chemisorption function. The solution is shown for different surface projected density of states, nd(e), with increasing d band centers sd. For a given metal the band width and center are coupled because the number of d electrons must be conserved. 

Figure 4.5. Calculated change in the sum of the one-electron energies using the Newns-Anderson model. The parameters are chosen to illustrate an oxygen 2p level interacting with the d states of palladium with a varying d band center, ed. In all cases, the number of d electrons is kept fixed. The corresponding variations in the metal and adsorbate projected densities of states are shown above. Notice that the adsorbate-projected density of states has only a small weight on the antibonding states since it has mostly metal character. Adapted from Ref. [4].

Figure 4.9. Projected densities of states onto the d states of the surface atoms for different Pt surfaces with decreasing atom density The hexagonally reconstructed (100) surface, the close-packed (111) surface, the step atoms on a (211) surface and the kink atoms on a (11 8 5) surface. Adapted from Ref. [19].

Figure 4.11. Calculated changes in the adsorption energy of atomic H and on a series of Pt(lll) surfaces, where the second layer has been replaced by a layer of 3d transition metals. To the right the variations in the d-projected density of states for the Pt surface atoms are shown. Adapted from Ref. [33].

Fig, 13 Projected Density of States (PDOS) for the optimised (111 surface, for the a) for the Pd/Zr02, and in b) for the Pt/ZrOi interfaces when the metal is adsorbed on top of O.,. The Fermi level is marked as a dashed line, and energies are given in eV. Note the scaling of the PDOS in comparison to the total DOS. Calculations performed with the Hartree-Fock Hamiltonian. 

Fig.3f. Density of states plot and projected densities of states for the Si,2 ring states the major contribution at Ep arises from the e2 state which is a n-antibonding state within one ring but bonding between adjacent rings...

Figure 25. Projected density of states on the d band of Au(lll) (dotted line), and on the Is orbital of hydrogen when the atom approaches to the surface (Data obtained from Ref. 55.)...

The projected density of states on the J-band of Pts for bare [Xbuik-Xs]-Pt3 (red) and oxygen-chemisorbed [Xbuik-Xs]-Pt3-02 (blue) surfaces (Fig. 3) reveals in each case a J-DOS spectram of the adsorbed Pt3 island narrower and shifted to higher energies... 

A physical quantity of remarkable interest is the projected density of states, defined for Hermitian operators as... 

For a discussion of the form of the distribution N(E), see, for instance, P. G. Nevai, Orthogonal Polynomials, Memoires American Mathematical Society, Providence, RI, 1979. In essence, in handling dN(E) = n(E) dE, one can consider n( ) as the (projected) density of states of a Hermitian operator and N(E) as the corresponding integrated density of states. 

The parameters a and can be obtained from the moments by evaluating Hankel determinants or working out product-difference algorithms. The projected density of states n( ) is then given by... 

For extended impurities it is convenient to use the recursion method and to calculate directly the needed matrix elements of G, from which the projected density of states can be obtained and the effect of the impurity inferred. 

Authors do not really agree on the charge redistribution which takes place at the MgO(lOO) surface, because the ionicity of the oxide appears to be nearly total in HF approaches [61,63], while in DFT and semi-empirical methods, the Mg-0 bonds present a small but non-negligible covalent character [60,66]. The surface projected Density of States (DOS) displays surface states just at the top of the VB and bottom of the conduction band (CB), originating from a reduction of the Madelung potential on the surface atoms. These states also exist on NiO(lOO) [74] and CoO(100)[75] and were shown to play a role in the formation of STM images (Fig. 2). A small reduction of the HOMO-LUMO gap results in MgO(lOO) as well as CaO(lOO) [60,64,67,68], and NiO(lOO) [74], which is qualitatively consistent with experimental observation [76-79]. 

The calculations involved determination of the projected densities of states (ProDOS)(5). The ProDOS were calculated near the Fermi energy, with approximate intensity for dipole allowed transitions from the core levels of selected site and parity. With the one-electron molecular orbitals of the clusters, as in Eq. (5), the ProDOS are defined in terms of the eigenvector coefficients of the orbitals and a Lorentzian (a localized line shape) of width a, as... 

Fig. 5.3. Computed (DFT-GGA) projected density of states (PDOS), cf. (5.2), of the clean basal surfaces of the late 4d TMs (gray shading), and of the 0(1 x 1) adlayer (black shading). Shown are the Na s) contributions of the Ru 4d and O 2p states. The decrease of the oxygen PDOS bandwidth (from left to right) and the increased filling of antibonding states (the higher energy oxygen PDOS region) are clearly visible...

The spectra of metallic crystals can be solved in k-space. Theoretical calculations of the partial and projected density of states of the crystal band structure were reported by several groups to interpret the XANES of metals " >. We discuss here the band structure approach developed by Muller et al. by which it is possible... 

XANES spectra of different systems have been interpreted with the band structure aproximation As an example for a transition metal we discuss here palladium absorption edges. The comparison between the K and Lj edge of Pd metal with the theoretical band approach is shown in Fig. 21. We can observe that the K and Li edges present the same spectral features and therefore contain identical information. In fact, the selection rule for electronic transitions selects the same I = 1 projected density of states. Because the L, edge occurs at lower energy a better instrumental energy resolution is obtained and the structures are better resolved. 

Because of the selection rules A1 = +1, AS = 0 and AJ = +1 the edge spectra select the angular momentum T and the total angular J of the local density of states. In Fig. 22 the edge spectrum of Pd metal is compared with the one-electron band calculation of Muller The good agreement between theory and experiment is evident. The total density of states is plotted in Fig. 22 to show that the total DOS does not explain the experimental spectra because only the local 1 = 2 and T = 0 projected density of states contribute to the spectrum. 

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