Empirical Pseudo-potentials

Until the late 70 s the method employed to construct a pseudo-potential was based on the Phillips and Kleinman cancellation idea. A model analytic potential was constructed and its parameters were fitted to experimental data. However, these models did not obey condition (6.43). 

One of the most popular model potentials was introduced by Heine and Abarenkov in 1964 [36,37,38]. The Heine-Abarenkov potential is 

A simplification of the Heine-Abarenkov potential was proposed in 1966 by Ashcroft [39,40] 

In this model potential it is assumed that the cancellation inside the core is perfect, i.e., that the kinetic term cancels exactly the Coulomb potential for r R. To adjust R, Ashcroft used data on the Fermi surface and on liquid phase transport properties. 

The above mentioned and many other model potentials are discontinuous at the core radius. This discontinuity leads to long-range oscillations of their Fourier transforms, hindering their use in plane-wave calculations. A recently proposed model pseudo-potential overcomes this difficulty the evanescent core potential of Fiolhais et al. [41] 

---

EPM Empirical pseudo-potential method Heine and Weaire (1971)... 

FIGURE 6.15 Fermi surfaces of LiC6 calculated using empirical pseudo-potentials and a self-consistent determination of the charge transfer the Fermi surfaces for the lower (a) and upper (b) bands. (From Ohno, T., J. Phys. Soc. Jpn. 49(Suppl. A), 899, 1980. With permission.)... 

The calculated and measured electron effective mass m c and its k-dependency for WZ and ZB GaN and AIN are summarised in TABLES 1 and 2, respectively. Suzuki et al derived them with a full-potential linearised augmented plane wave (FLAPW) band calculation [4,5], Miwa et al used a pseudopotential mixed basis approach to calculate them [6]. Kim et al [7] determined values for WZ nitrides by the full-potential linear muffin-tin orbital (FP-LMTO) method. Majewski et al [8] and Chow et al [9,10] used the norm-conserving pseudo-potential plane-wave (PPPW) method. Chen et al [11] also used the FLAPW method to determine values for WZ GaN, and Fan et al obtained values for ZB nitrides by their empirical pseudo-potential (EPP) calculation [12],... 

Recently, a new kind of analytic pseudo-potentials, directly in a fully nonlocal form, has been proposed [129,160]. Coefficients are fitted to minimize penalty functions, like atomic properties, to ensure that these properties are well reproduced for the pseudoatom. It has been further generalized for use in the context of QM/MM situations [161] or to include semi-empirically the long-range van der Waals attraction [161,162]. 

An approach to elucidate new empirical restraints from PDB coordinates and chemical shift in the Biological Magnetic Resonance Data Bank was applied by Kuszewski et al.89,90 for a structure determination automation program. The relation between the chemical shift and coordinates can be used for an NMR assignment validation.91 The torsion angle ranges of (j) and ijj that are favorable in Ramachandran plot and their mean values in PDB are also useful as pseudo-potentials.92,93... 

Diatomics-in-Molecules (DIM) Method Semi-Empirical Valence-Bond Methods Approximate Pseudo-Potential Theories... 

Pseudo-potential calculations The Schrodinger equation is applied, the core electrons are described by empirical potentials, and only the valence electrons are taken into account in the calculations. 

KLEIN - We do indeed use a semi-empirical model for the various interaction potentials. First, we model the ammonia inter molecular potential with an effective pair potential which ignores many body polarization. Models of this type are remarkably successful in explaining the physical properties of polar fluids. Of course, we really should include many body forces, but at this stage we ignore them. The ammonia potential is fitted to the heat of evaporation and the zero-pressure density. The electron-alkali metal (Lithium) potential is represented by the Shaw pseudo potential fitted to the ionization energy. This is the simplest and crudest model possible. We have explored the effect of using (a) Heine-Abarenkov, (b) Ashcroft, and (c) Phillips-Kleinman forms. Our results are not very sensitive to the choice of pseudo potential. (In the case of Cs metal, which I did not discuss, the sensitivity to the potential is crucial). 

To parameterize the new quantities occurring in these equations a few semi-empirical relations from the literature were adopted. The asymptotic value of bubble induced turbulent kinetic energy, fesia, is estimated based on the work of [3]. By use of the so-called cell model assumed valid for dilute dispersions, an average relation for the pseudo-turbulent stresses around a group of spheres in potential flow has been formulated. Prom this relation an expression for the turbulent normal stresses determining the asymptotic value for bubble Induced turbulent energy was derived ... 

The column headed HSE uses an approximation made originally by Mansoori and Leland (3) that the diameter used in the hard sphere equations of state is c0o-, the LJ a parameter for each molecule multiplied by a universal constant for conformal fluids. This approximation then requires that be replaced by equations defining the HSE pseudo parameters, Equations 10 and 11. The results in the HSE column use c0 = 0.98, the value for LJ fluids obtained empirically by Mansoori and Leland. This procedure is correct only for a Kihara-type potential and it is not consistent with the LJ fluids in Table I. Furthermore, this causes only the high temperature limit of the repulsion effects to be included in the hard-sphere calculation. Soft repulsions are predicted by the reference fluid. 

Note that our purpose of rigorous modelling cannot be completely separated from earlier research on semi-empirical or model Hamiltonians. On one side these Hamiltonians could be parametrized by theoretical simulation techniques and on the other some experimental data could also be introduced in the simulation techniques, for example in the characterization of truncated Hamiltonians. Finally it should be emphasized that research on pseudo-Hamiltonians and model Hamiltonians is always guided by some intuitive knowledge of the passive and active constituents of the system (atomic cores, atoms in molecules, functional group,...) and by the assumption of transferability of their potentials and interactions. 

This example demonstrates practically independent development of ECPs for molecules and crystals when for the same ECP property different terms are used. The work by PhiUips and Kleinman (PK) [473] is an important step in the ECP apphcations for sohds. PK developed the pseudopotential formalism as a rigorous formulation of the earher empirical potential approach. They showed that ECP that has the plane-wave pseudo wavefunctions as its eigenstates could be derived from the all-electron potential and the core-state wavefunctions and energies. Thus a nonempirical approach to finding ECP was introduced. 

---