Ab-initio Pseudo-potentials

Besides (6.55), there are still two other conditions imposed on the pseudo-potential the pseudo wave-functions should not have nodal smfaces and the pseudo energy-eigenvalues should match the true valence eigenvalues, i.e.. 

The potentials thus constructed are called norm-conserving pseudo-potentials, and are semi-local potentials that depend on the energies of the reference electronic levels, 

In summary, to obtain the pseudo-potential the procedme is i) The free atom Kohn-Sham radial equations are solved taking into accoimt all the electrons, in some given reference configuration 

The resulting pseudo-potential, mgscn still includes screening effects due to the valence electrons that have to be subtracted to yield 

The cutoff radii, r , are not adjustable pseudo-potential parameters. The choice of a given set of cutoff radii establishes only the region where the pseudo and true wave-functions coincide. Therefore, the cutoff radii can be considered as a measure of the quality of the pseudo-potential. Their smallest possible value is determined by the location of the outermost nodal surface of the true wave-functions. For cutoff radii close to this minimum, the pseudopotential is very realistic, but also very strong. If very large cutoff radii are chosen, the pseudo-potentials will be smooth and almost angular momentum independent, but also very unrealistic. A smooth potential leads to a fast convergence of plane-wave basis calculations [58]. The choice of the ideal cutoff radii is then the result of a balance between basis-set size and pseudopotential accuracy. 

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Dolg M, Stoll H, Preuss H (1989) Energy-adjusted ab initio pseudo potentials for the rare earth elements. J Chem Phys 90 1730... 

In the calculations based on effective potentials the core electrons are replaced by an effective potential that is fitted to the solution of atomic relativistic calculations and only valence electrons are explicitly handled in the quantum chemical calculation. This approach is in line with the chemist s view that mainly valence electrons of an element determine its chemical behaviour. Several libraries of relativistic Effective Core Potentials (ECP) using the frozen-core approximation with associated optimised valence basis sets are available nowadays to perform efficient electronic structure calculations on large molecular systems. Among them the pseudo-potential methods [13-20] handling valence node less pseudo-orbitals and the model potentials such as AIMP (ab initio Model Potential) [21-24] dealing with node-showing valence orbitals are very popular for transition metal calculations. This economical method is very efficient for the study of electronic spectroscopy in transition metal complexes [25, 26], especially in third-row transition metal complexes. 

Melius, C. F. and Goddard, W. A. Ill, Ab initio effective potentials for use in molecular quantum mechanics, Phys. Rev. AlO 1528 (1974) Kahn, L. R., Baybutt, P., and Truhlar, D. G., Ab initio effective core potentials Reduction of all-electron molecular structure calculations to calculations involving only valence electrons, J. Chem. Phys. 65 3826 (1976). For calculations involving atoms beyond the second long row of the periodic table, it is common to exclude inner-shell electrons from the calculation and to introduce an effective one-electron potential (a pseudo-potential) which accounts for these electrons. These two papers describe an ab initio procedure for deriving such a pseudo-potential. 

The 3d ab initio simulations [4] for Na3 are based, in a similar way, on three ab initio potential-energy surfaces for Na3(X), Na3(B), and Na3(X), with 3d ab initio dipole coupling between Na3(X) and Na3(B) evaluated by V. Bonacic-Koutecky et al. [5] plus Condon-type coupling between Na3(B) and Na3(X). Additional potential-energy surfaces interfere at the conical intersections of the pseudo-Jahn-Teller distorted Na3(B) state (see Ref. 6), but we have tested carefully [4] that these interferences are negligible in the frequency domains of the experimental femtosecond/picosecond laser pulse experiments [7] as well as in the continuous-wave experiments [8]. 

The assignment of the intense band in the beam to a LMCT transfer is supported by molecular orbital calculations [31]. Pseudo-potential ab initio... 

Figure 4. Plot of the pseudo-potential and the probability density for the pure bending states (vi = V3 = 0) of the HCN CNH system versus bending angle 0. These results were obtained by applying sixth-order CPT to the ab initio surface of Tennyson and co-workers [7,8]. The vertical scale is the same for all probability plots, and the baseline for each plot coincides with the energy of the corresponding state.

Solution of the Kohn-Sham equations as outlined above are done within the static limit, i.e. use of the Born-Oppenheimer approximation, which implies that the motions of the nuclei and electrons are solved separately. It should however in many cases be of interest to include the dynamics of, for example, the reaction of molecules with clusters or surfaces. A combined ab initio method for solving both the geometric and electronic problem simultaneously is the Car-Parrinello method, which is a DFT dynamics method [52]. This method uses a plane wave expansion for the density, and the inner ions are replaced by pseudo-potentials [53]. Today this method has been extensively used for studies of dynamic problems in solids, clusters, fullerenes etc [54-61]. We have recently in a co-operation project with Andreoni at IBM used this technique for studying the existence of different isomers of transition metal clusters [62,63]. 

Because of the difficulty of performing reliable ab initio calculations, it is often necessary to resort to more approximate methods especially for larger systems. These methods could be just the standard ab initio methods using basis sets of limited size or they could be one of the semi-theoretical methods (atoms-in-molecules or pseudo-potential) outlined in Section III. 

Another option that reduces the number of functions, particularly when heavy atoms are involved, is the replacement of inner shell electrons by effective (or pseudo) potentials. Such procedures have been incorporated into many ab initio program systems including ACES II. Since the core electrons are not explicitly considered, effective potentials can drastically reduce the computational effort demanded by the integral evaluation. However, because the step is an inexpensive part of a correlated calculation, the role of effective potentials in correlated calculations is less important, due to the fact that dropping orbitals is tantamount to excluding them via effective potentials. An exception occurs when relativistic effects are important, as they would be in a description of heavy atom systems. Most such chemically relevant effects are due to inner shell elearons their important physical effects, like expanding the Pt valence shell, can be introduced via effective potentials that are extracted from Dirac-Fock or other relativistic calculations on atoms. ° Similarly, some effeaive potentials introduce some spin-orbital effects as well. Thus, besides simplifying the computation, effective potential calculations could include important physical effects absent from the ordinary nonrelativistic methods routinely applied. 

Ph. Durand, J.-P. Malrieu, Effective Hamiltionians and pseudo potentials as tools for rigorous modeling, in K.P. Lawley (Ed.), Ab Initio Methods in Quantum Chemistry I, John Wiley Sons, New York, 1987, p. 352. 

TABLE 4-6. Pseudo potential used for ab-initio band structure calculation. 

The lattice defect of Ce02 is always an issue in the field of cosmetic application for UV shield materials. In order to reduce the photo catalytic activity, a small amount of metal ions are doped to keep the stoichiometry. Similarly this doping method is taken in the sintering process of CeOa to control the grain growth [21, 22]. In such cases, qualitatively, computational analysis gives an interpretation for the mechanism of functional properties. Table 4-8 shows pseudo potential used for ab-initio band stmcture calculation and Figure 4-10 shows the results of band stmcture... 

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