Pseudopotential Adjustment

The traditional and most widely applied approach uses pseudopotentials adjusted in atomic calculations for specific chemical elements, which are usually superimposed in the subsequent molecular calculations to give an approximate molecular pseudopotential Fcv, i.e. 

The leading point-charge terms usually have to be augmented by correction terms AV, which may be formulated in a local, semilocal or nonlocal form. The advantage of these element-specific pseudopotentials is their economy regarding the number of necessary parameter sets to cover the whole Periodic Table. Once they have been generated for every element separately, they may be applied in calculations on compounds with arbitrary composition. Moreover, the pseudopotential adjustment itself is further simplified when the central-field approximation is used in the necessary atomic calculations. 

To summarize, the RPPA is a method that can accurately describe relativistic effects, even though the relativistic perturbation operator used in the pseudopotential procedure is acting on the valence space and not the region dose to the nudeus, as this is the case for the correct all-electron relativistic perturbation operator. That is, relativistic effects are completely transferred into the valence space. These effects are also completely transferable from the atomic to the molecular case as the results for Au2 show. If relativistic pseudopotentials are carefully adjusted, they can produce results with errors much smaller than the errors originating from basis set incompleteness, basis set superposition or from the electron correlation procedure applied. 

Figgen, D., Rauhut, G., Dolg, M. and StoD, H. (2005) Energy-consistent pseudopotentials for group 11 and 12 atoms adjustment to multi-configuration Dirac-Hartree-Fock data. Chemical Physics, 311, 227-244. 

The most important approach to reducing the computational burden due to core electrons is to use pseudopotentials. Conceptually, a pseudopotential replaces the electron density from a chosen set of core electrons with a smoothed density chosen to match various important physical and mathematical properties of the true ion core. The properties of the core electrons are then fixed in this approximate fashion in all subsequent calculations this is the frozen core approximation. Calculations that do not include a frozen core are called all-electron calculations, and they are used much less widely than frozen core methods. Ideally, a pseudopotential is developed by considering an isolated atom of one element, but the resulting pseudopotential can then be used reliably for calculations that place this atom in any chemical environment without further adjustment of the pseudopotential. This desirable property is referred to as the transferability of the pseudopotential. Current DFT codes typically provide a library of pseudopotentials that includes an entry for each (or at least most) elements in the periodic table. 

The shape-consistent (or norm-conserving ) RECP approaches are most widely employed in calculations of heavy-atom molecules though ener-gy-adjusted/consistent pseudopotentials [58] by Stuttgart team are also actively used as well as the Huzinaga-type ab initio model potentials [66]. In plane wave calculations of many-atom systems and in molecular dynamics, the separable pseudopotentials [61, 62, 63] are more popular now because they provide linear scaling of computational effort with the basis set size in contrast to the radially-local RECPs. The nonrelativistic shape-consistent effective core potential was first proposed by Durand Barthelat [71] and then a modified scheme of the pseudoorbital construction was suggested by Christiansen et al. [72] and by Hamann et al. [73]. 

The method employed was similar to that of Ref. 35, but with several improvements. ab initio, norm-conserving, nonlocal pseudopotential were used to represent the metal ions. This capability enables reliably realistic representation of the metal s electronic structure. Thus the cadmium pseudopotential was able, for example, to reproduce the experimental cadmium-vacuum work function using no adjustable parameters (unlike the procedure followed in Ref. 35). Pseudopotentials of the Troullier and Martins form [53] were used with the Kleinman-Bylander [54] separable form, and a real space... 

Energy-Adjusted Ab Initio Pseudopotentials for the Second and Third Row Transition Elements. 

The theoretical calculations of the band structure of InN can be grouped into semi-empirical (pseudopotential [10-12] or tight binding [13,14]) ones and first principles ones [15-22], In the former, form factors or matrix elements are adjusted to reproduce the energy of some critical points of the band structure. In the work of Jenkins et al [14], the matrix elements for InN are not adjusted, but deduced from those of InP, InAs and InSb. The bandgap obtained for InN is 2.2 eV, not far from the experimentally measured value. Interestingly, these authors have calculated the band structure of zincblende InN, and have found the same bandgap value [14]. 

Other complications are associated with the partitioning of the core and valence space, which is a fundamental assumption of effective potential approximations. For instance, for the transition elements, in addition to the outermost s and d subshells, the next inner s and p subshells must also be included in the valence space in order to accurately compute certain properties (54). A related problem occurs in the alkali and alkaline earth elements, involving the outer s and next inner s and p subshells. In this case, however, the difficulties are related to core-valence correlation. Muller et al. (55) have developed semiempirical core polarization treatments for dealing with intershell correlation. Similar techniques have been used in pseudopotential calculations (56). These approaches assume that intershell correlation can be represented by a simple polarization of one shell (core) relative to the electrons in another (valence) and, therefore, the correlation energy adjustment will be... 

G. Igel, and H. PreulS. Int. J. Quantum Chem., 26, 725 (1984). Pseudopotential Calculations Including Core-Valence Correlation Alkali and Noble-Metal Compounds. M. Dolg, U. Wedig, H. Stoll, and H. PreulS, /. Chem. Phys., 86, 866 (1987). Energy-Adjusted Ab Initio Pseudopotentials for the First Row Transition Elements. M. Dolg, H. Stoll, A. Savin, and H. PreulS, Theor. Chim. Acta, 75,173 (1989). Energy-Adjusted Pseudopotentials for the Rare Earth Elements. 

A. Nicklass, et al. Ab-initio energy-adjusted pseudopotentials for the noble-gases Ne through Xe—Galculation of atomic dipole and quadrupole polarizabilities, /, Chem. Phys. 102 (22) (1995) 8942-8952. 

Eq. (20-4.3) and is the desired result. It has been used with a calculation of the body-centered cubic bands just as it was in Eq. (20-8), but including Km as well as and to obtain the relation, in Eq. (20-9), between W, and r,. In treating the bands, and in making the Solid State Table,, which is found here to be one sixth of, was taken equal to zero, and and were adjusted slightly, but such as to retain the same relation between W, and. Exactly the same ratios between the different matrix elements were obtained by Andersen, Klose, and Nohl (1978), by using the Muffin-Tin Orbital theory, and they obtained the same -dependence. Here we have added the relation to the r, from pseudopotential theory. 

A remark should be made here with respect to the generation and adjustment of the widely used effective core potentials (ECP, or pseudopotentials) [85] in standard non-relativistic quantum chemical calculations for atoms and molecules. The ECP, which is an effective one-electron operator, allows one to avoid the explicit treatment of the atomic cores (valence-only calculations) and, more important in the present context, to include easily the major scalar relativistic effects in a formally non-relativistic approach. In general, the parameters entering the expression for the ECP are adjusted to data obtained from numerical atomic reference calculations. For heavy and superheavy elements, these reference calculations should be performed not with the PNC, but with a finite nucleus model instead [86]. The reader is referred to e.g. [87-89], where the two-parameter Fermi-type model was used in the adjustment of energy-conserving pseudopotentials. 

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