Pseudopotential calculation methods

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].)... 

The traditional valence-only MO schemes are Extended Hiickel and CNDO with its subsequent modifications. Present-day computing facilities make it possible to move one step further, to the ab initio treatment of valence electrons through the use of pseudopotential (PP) methods. The essentials of such methods will be illustrated in the following, through a description of the NOCOR formulation3, which will then be used for extensive calculations on sulphoxide and sulphone systems. The general concepts exposed in the foregoing sections will be illustrated by many examples. 

Physical and Structural Aspects.—Perhaps the most significant theoretical paper comes from an application of pseudopotential SCF methods to PX3 molecules.1 This method neglects core orbitals, and hence shortens computer-time requirements, in comparison with more conventional SCF calculations. The results1 are really most encouraging, and compare favourably with standard SCF results, in relation to experimental values for bond angles and lengths, and for dipole moments. 

The first explanation and use of such a pseudopotential is due to Heilman5 (1935) who used it in atomic calculations. More recently the pseudopotential concept was reformulated by Phillips and Kleinman7 who were interested in its application to the solid state.8-10 Research in both solid- and liquid-state physics with pseudopotentials was reviewed by Ziman,11 and work in the fields of atomic spectroscopy and scattering has been discussed by Bardsley.12 For an earlier review on applications to the molecular environment the reader is referred to Weeks et a/.13 In this article we shall concentrate on molecular calculations, specifically those of an ab initio nature. Our objective in Section 2 has been to outline the theoretical origins of the pseudopotential approximation, and in Section 3 we have described some of the techniques which have been used in actual calculations. Section 4 attempts to present results from a representative sample of pseudopotential calculations, and our emphasis has been to concentrate on particular molecules which have been the subjects of investigation by the various approaches, rather than to catalogue every available calculation. Finally, in Section 5, we have drawn some conclusions on the relative merits of the different methods and implementations of pseudopotentials. Some of the possible future developments are outlined in the context of the likely progress in quantum chemistry. 

The pseudopotential method has various advantages. Eliminating the core electrons from the problem reduces the number of particles that must be considered in the Kohn-Sham (KS) equations for the effective one-electron potential. For example, a pseudopotential calculation for bulk silicon (with 10 core and 4 valence electrons) requires the calculation of 4 occupied bands at each k-point, while an all-electron approach would require the calculation of 14 occupied bands. More importantly, the smooth spatial variation of the pseudopotential and pseudowavefunction allows the use of computationally convenient and unbiased basis, such as plane wave basis sets or grids in space. 

The delocalized (right-hand) side of Fig. 1.1 involves some form of calculation on the full lattice such as a band-theory calculation. Again, the Hartree-Fock wave function may be employed in an ab initio method or some approximate method such as Huckel band theory, or the local-exchange approximations employed leading to augmented-plane-wave or ab initio pseudopotential (PP) methods. As an alternative to band theory, the development of the ionic approach using pair potentials or modified electron-gas (MEG) theory has proved effective for certain crystalline species. 

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. 

They applied the same pseudopotential as before in the paper with title ab initio molecular calculations with pseudopotentials calculations of double-zeta quality on ethylene, acetylene, and water , but a linear combination of FSGO has been used in the following articles. A description of C-C bond for double and triple bond has been given in the pseudopotential-FSGO method. 

A comparison of different methods was undertaken for the hydride of element 111 (Seth et al. 1996). The conclusion of this study was that Dirac-Fock calculations, all-electron DKH calculations and relativistic pseudopotential calculations give very similar results, showing that relativistic effects are also well described in the more approximate methods. A large relativistic bond length contraction of about 50 pm was found, which makes the bond length of (111)H even slightly shorter than that of AuH, which is 152.4 pm, with a relativistic effect of the order of 20 pm (see Kaldor and Hess 1994). 

From the experimental side, the band-structure parameters are mainly determined from the cyclotron resonance (CR) spectra of electron and holes (see for instance [4]). Some of these parameters can also be obtained from the Zeeman splitting of electronic transitions of shallow impurities involving levels for which the electronic masses can be taken as those of free electrons or holes, or from the magnetoreflectivity of free carriers. Average effective masses can also be deduced from the Hall-effect measurements or from other transport measurements. Calculation methods that have been used to obtain band-structure parameters free from experimental input are the ab-initio pseudopotential method, the k-p method and a combination of both. These theoretical methods are presented in Chap. 2 of [107]. VB parameters at k = 0 including k and q have been calculated for several semiconductors with diamond and zinc-blended structures by Lawaetz [55]. 

New Species. - Pseudopotential calculations were used to predict the existence of the new species AuF,195 NUO+ 196 and the first noble-metal-noble-gas chemical bonds in the species AuXe+ and XeAuXe+.197 All these species were later prepared by mass-spectroscopic methods.198,199 The first thermodynamically stable diatomic trication UF3+ 200 or the new hydride CdH2201 are further examples. 

Once again the quality of the valence correlation treatment is crucial, making the large-core pseudopotential calculation (even without a CPP correction in this case) more accurate than a fiilly relativistic all-electron calculation. Another remark is relative to the comparison of the two pseudopotential SOCI calculations. A DGCI treatment, which in principle is better than a CI/SO one when large spin-orbit interaction is involved, does not guaranty by itself an accurate SOCI calculation, since in this example a CI /SO method gave much more re-... 

Before reviewing applications to large or extended systems (Section 4), we will illustrate the performance of various one- and two-component DKH-DF approaches using atoms and small molecules of heavy elements as benchmark systems. We will compare results of DKH calculations to those of four-component relativistic methods and other approaches based on transformed Hamiltonians. In many cases, results obtained with pseudopotentials are also available, but will be not considered here because of their computational efficiency, pseudopotential calculations are by far the most widely used strategy for heavy-element compounds. 

In this work we have performed ab initio pseudopotential calculations of the electronic and magnetic properties of pure and mixed Fe systems with the Siesta method. We have compared our results with available data obtained through different well known ab initio methods in order to demonstrate the capabilities of this method for describing magnetic systems. 

The second DFT LCAO linear-scaling method by Scuseria and Kudin (SK method) [379] uses Gaussian atomic orbitals and a fast multipole method, which achieves not only linear-scaling with system size, but also very high accuracy in aU infinite summations [397]. This approach allows both all-electron and pseudopotential calculations and can be applied also with hybrid HF-DFT exchange-correlation functionals. 

---