Pseudopotential approaches

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

All calculations are scalar relativistic calculations using the Douglas-Kroll Hamiltonian except for the CC calculations for the neutral atoms Ag and Au, where QCISD(T) within the pseudopotential approach was used [99], CCSD(T) results for Ag and Au are from Sadlej and co-workers, and Cu and Cu from our own work, using an uncontracted (21sl9plld6f4g) basis set for Cu [6,102] and a full active orbital space. 

Jaros(1975a) o a k.2 Yes Approximated via Penn model Expanded in band functions, with subsequent use of approximate pseudopotential approach Parabolic... 

Jaros(1977) 0 ap2 Yes Matrix elements approximated via Penn model pseudopotential approach and Calculated... 

It is possible, by using the LCAO approach or a pseudopotential approach, to make a calculation of energy bands for the distorted lattice. There are still two atoms per primitive cell, so no serious difficulty is encountered. The sum of the... 

The expansion in F/( , - tj.) would suggest the possibility of incorporating the effect of the resonance in perturbation theory and thus extending the pseudopotential perturbation theory of simple metals to transition metals. This has in fact been done (Harrison, 1969) and the pseudopotential approach has been extensively developed by Moriarty (1970, I972a,b,c), but the application.s have been largely restricted to the ends of the transition scries where the expansion is clearest. 

The choice of one or the other of these approaches depends not only on the computing resources but more specially on the nature of the problem. It is obvious that properties relying on the density near the nucleus, hyperfine fields for instance, require the use of an all-electron method. In contrast, efficient structural optimizations when the unit cell changes are much more easily performed in pseudopotential approaches. 

Note that, in the above discussion, we have neglected methods that generate band structures from empirical data. Most band calculations before the 1970s were of this type. The considerable contributions to knowledge made through use of the empirical pseudopotential approach, for example, have been discussed by Cohen (1979). Such approaches have... 

Zunger, A., and M. L. Cohen (1978). Density-functional pseudopotential approach to crystal phase stability and electronic structure. Phys. Rev. Lett. 41, 53-56. 

Plenkiewicz B, Frongillo Y, Lopez-Castillo J-M, Jay-Gerin J-R (1996) A simple but accurate core-tail pseudopotential approach to the calculation of the conduction-band energy V of quasifree excess electrons and positrons in nonpolar fluids. J Chem Phys 104 9053-9057. 

Within the pseudopotential approach, one replaces N all-electron equations 1... 

In order to further check the dependence of the difference in the eigenvalues calculated in the all-electron and pseudopotential approaches, we compare the... 

Briefly summarizing the results presented in Tables 1-5, one could conclude that the pseudopotential approach is promising for simulations of large-scale systems. A special advantage appears to be presented by a combined approach when some region of a large system is considered all-electronically, whereas the rest is replaced with pseudoatoms. The latter will lead to a drastic reduction in computational efforts without the loss of accuracy. 

Table 4. DV-X orbital energies (in eV) of the PdCl dianion calculated within the all-electron and pseudopotential approaches. The latter includes the atomic pseudopotentials and pseudoorbitals on the chlorine atoms only. R Pd — O) = 2.30 A.

This method is probably as accurate as some other simple pseudopotential approaches. However, there appear to be some difficulties in improving it to the standard of some of the recent pseudopotential calculations. Attempts to use larger than minimal basis sets required the inclusion of a Phillips-Kleinman term in addition to the orthogonality procedure in order to prevent collapse of the valence orbitals into the core space. Thus in calculations on AlaQ, Vincait had to include not only the A1 3s and Cl 3j and 3p shells but also the A12p and Cl 2s and 2p shells explicitly in the valence-electron basis in order to obtain good results. Consequently this calculation was not substantially less expensive in computing time than an equivalent all-electron calculation. 

A lot of theoretical work has been done in order to explain the size dependent properties of semiconducting nanocrystals. These methods are primarily based on the effective mass approximation, pseudopotential approaches or the tight binding scheme. Each of these methods has certain advantages and disadvantages. We shall explore these methods in some detail in Section 11.5. 

The way i>f p is generated from the atomic calculation is not unique. Common pseudopotentials are generated following the prescription of, e.g., Bachelet, Hamann and Schlriter [82], Kleinman and Bylander [83], Vanderbilt [84] or Troullier and Martins [85]. Useful reviews are Refs. [86, 87, 88]. The pseudopotential approach is very convenient because it reduces the number of electrons treated explicitly, making it possible to perform density-functional calculations on systems with tens of thousands of electrons. Moreover, the pseudopotentials upp are much smoother than the bare nuclear potentials vext. The remaining valence electrons are thus well described by plane-wave basis sets. 

In the most recent version of the energy-consistent pseudopotential approach the reference data is derived from finite-dilference all-electron multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian. As an example the first parametrization of such a potential,... 

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