Renormalized atom method

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 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). 

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.

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... 

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.. 

Although P3 procedures perform well for a variety of atomic and molecular species, caution is necessary when applying this method to open-shell reference states. Systems with broken symmetry in unrestricted Hartree-Fock orbitals should be avoided. Systems with high multireference character are unlikely to be described well by the P3 or any other diagonal approximation. In such cases, a renormalized elec-... 

MCT can be best viewed as a synthesis of two formidable theoretical approaches, namely the renormalized kinetic theory [5-9] and the extended hydrodynamic theory [10]. While the former provides the method to treat both the very short and the very long time responses, it often becomes intractable in the intermediate times. This is best seen in the calculation of the velocity time correlation function of a tagged atom or a molecule. The extended hydrodynamic theory provides the simplicity in terms of the wavenumber-dependent hydrodynamic modes. The decay of these modes are expressed in terms of the wavenumber- and frequency-dependent transport coefficients. This hydrodynamic description is often valid from intermediate to long times, although it breaks down both at very short and at very long times, for different reasons. None of these two approaches provides a self-consistent description. The self-consistency enters in the determination of the time correlation functions of the hydrodynamic modes in terms of the... 

Another simple method that allows for the SE calculations in heavy atoms is the partial wave remormalization (PWR) approach first proposed in [69], [70]. Within this approach the renormalized expression for the lowest-order SE is presented in the form of the partial wave expansion ... 

The same approach was proposed in Ref. [669] in the sense of an atomic approximation to the projection on electronic states. If the electronic spinors are then transformed to a two-component picture by a renormalization matrix, a local X2C method emerges. 

The results of Refs. [26,27] related to heats of formation and equilibrium geometries obtained by the APSLG-MINDO/3 method are somewhat more accurate than those of the standard (SCF-)MINDO/3 method. The APSLG form of the trial wave function also ensures its correct asymptotic behavior under cleavage of chemical bonds which indirectly justifies some level of bonae fidelitatis of the wave function employed. The APSLG form of the wave function also allows to represent a renormalization of the bare R-subsystem Hamiltonian in terms of well-defined characteristics like atomic charges, bond polarizabilities and ionization potentials for the chemical bonds [74,91]. 

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