Renormalized atom

An approach that is very closely related to the Atomic Sphere Approximation is the Renormalized Atom Theory, introduced first by Watson, Ehrenreich, and Hodges (1970) (sec also Watson and Ehrenreich, 1970, Hodges et al., 1972, and particularly Gelatt, Ehrenreich, and Watson, 1977). The name derives from the way the potential is constructed a charge density for each atom is constructed on the basis of atomic wave functions that are truncated at the Wigner-Seitz, or atomic, sphere. The charge density from each state is then scaled up (renormalized) to make up for that density beyond the sphere which has been dropped. 

Fig. 10. Comparison of photoionization energies of 4/electrons relative to Ep calculated by Herbst et al. (23) using the renormalized atom scheme with experimental values of Baer and Busch (16) for the metals and in part from Ref. 17 for RE-Sb and RE-Te.

A renormalized-atom method has been worked out by authors of [52] in order to investigate in detail and calculate the cohesive energy of the 3d and 4d transition metals. This method is structured in such a way that it is possible to separate the energy of formation of the metal starting from free atoms into a number of terms and examine the importance of each. 

The difference in one-electron energy between this band and the renormalized atom s level is —0.033 Ryatom (—0.45eVatom ). The largest contribution to the cohesion of a transition metal with a particularly filled d band such as titanium is due to the broadening of the renormalized-atom d level into the d band. For titanium, the calculated contribution of d band broadening is —0.375 Ryatom ... 

The calculation of 4f promotion energy requires the combination of an atomic calculation for the 4f shell and a band calculation for the 5d band. The two calculations are based on such drastically different approximations that the combination of the two under one algorithm is a seemingly impossible task. Herbst et al. (1972) overcame this difficulty by using the renormalized atom method first proposed for the d band metals by Watson et al. (1970) and reported in detail by Hodges et al. (1972). We will review here the philosophy of the method, with particular emphasis on the meaning of the various approximations. The computational details are found in the original article. The relativistic version of the calculation has been published recently by Herbst et al. (1976). 

The Hartree-Fock method leaves out the correlation energy. This is compensated by assuming that the correlation energy in the renormalized atom equals that in the free atom, and the latter is obtained by comparing the Hartree-Fock... 

The approximation here is that the change in correlation due to the redistribution of the outer shell charges is negligible. Then we obtain the total energy of the renormalized atom... 

Now we are ready to pack the renormalized atoms together into a crystal. The outer shell states must broaden into bands, so the energy in the metal must differ from that of the renormalized atom because of this redistribution of electrons in the band states. Written explicitly... 

Fig. 3.61. Comparison between the calculated and the measured 4f promotion energies of the lanthanide metals (Baer and Busch, 1974). The calculation was made by Herbst et al. (1972) using the renormalized atom method.

Using unscreened Coulomb interaction the parameter U was found by the authors to be 2 Ry. In contrast, the renormalized atom calculation, which took into consideration the screening and relaxation effects, yielded a value U s 0.5 Ry for all rare earth metals. 

Fig. 9. Comparison of experimental and calculated minimum energies A and A required to modify the 4f population in the pure lanthanide metals. Circles values from combined XPS-BIS spectra (Lang et al. 1981). Squares renormalized atom calculations (Herbst et al. 1976, 1978). Triangles Bom-Haber cycles (Johansson 1979b). Since the 4f state of Ce is not atomic-like, instead of zl the fitted value of Ef (star) has been plotted (see section 6). The dashed horizontal lines on the right-hand side indicate approximately where the ground state line is shifted for atoms in the indicated situations (see section 5).

We calculate the total RHF energy of the ionized metallic final state, the second term on the right side of eq. (25), by the methods described in section 2.1. RHF computations are performed for the 4f" Sd"" 6s free ions, renormalized atom crystal potentials are constructed, and self-consistent band calculations are carried out. Normalization of the wave functions to the WS sphere ensures that the final state cell has charge -l-lle. The q = 0 component of the full crystal potential, which arises from the charge of the other WS cells, is not included in the total energy since our intent is to compare to the completely screened limit where no such term appears (each cell in that case being neutral). Multiplet theory is again employed to place the 4f electrons into their Hund-rule states.. 

The anion-cation bond-breaking produced by the formation of a surface is responsible for several phenomena. Some have an electrostatic origin. This is the case for polarization which is induced by surface electric fields which are much larger than those in the bulk. This is also the case for shifts of the renormalized atomic energies of surface atoms and shifts of surface bands, which are induced by reduction of the Madelung potential at the surface. On the other hand, as far as covalent effects are concerned, a modification of the anion-cation hybridization takes place, due to the lower coordination of the surface atoms. This has important consequences on the gap width and on the electron distribution. 

The values of A are fixed by the positions of the renormalized atomic levels of the surface atoms, which partly determine the transport properties of these films, but also their chemical reactivity. Fig. 3.5 shows the large variations of the lowest cation ec and highest anion 6a renormalized atomic energies, as a function of the film thickness n, for different orientations. 

Under the most simplifying Hartree approximation, when the eigenstates of the Hamiltonian are developed on an atomic orbital basis set, the diagonal terms of the Hamiltonian matrix, which represent effective renormalized atomic orbital energies, read (in atomic units) ... 

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