Pseudopotential techniques, quantum

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. 

AIMD, CPMD still needs to treat a large number of integrals related to electronic coordinates. For these integrals, special tricks should be used. That is, the time-consuming integral treatments in conventional quantum chemistry should be avoided. For this reason, the current implementations of CPMD use two techniques plane waves as a basis set and the pseudopotential approximation. Of course, it should be borne in mind that CPMD was originally developed for solid state applications, an area where the use of plane waves is a traditional technique. 

A further reduction of the computational effort in investigations of electronic structure can be achieved by the restriction of the actual quantum chemical calculations to the valence electron system and the implicit inclusion of the influence of the chemically inert atomic cores by means of suitable parametrized effective (core) potentials (ECPs) and, if necessary, effective core polarization potentials (CPPs). Initiated by the pioneering work of Hellmann and Gombas around 1935, the ECP approach developed into two successful branches, i.e. the model potential (MP) and the pseudopotential (PP) techniques. Whereas the former method attempts to maintain the correct radial nodal structure of the atomic valence orbitals, the latter is formally based on the so-called pseudo-orbital transformation and uses valence orbitals with a simplified radial nodal structure, i.e. pseudovalence orbitals. Besides the computational savings due to the elimination of the core electrons, the main interest in standard ECP techniques results from the fact that they offer an efficient and accurate, albeit approximate, way of including implicitly, i.e. via parametrization of the ECPs, the major relativistic effects in formally nonrelativistic valence-only calculations. A number of reviews on ECPs has been published and the reader is referred to them for details (Bala-subramanian 1998 Bardsley 1974 Chelikowsky and Cohen 1992 Christiansen et... 

The final consequence of such a strategy would be to try to eliminate the degrees of freedom of the core electrons as well and to introduce a possibly nonlocal effective potential (pseudopotential), the parameters of which are adjusted either to experiments, which are relativistic from the very beginning, or suitable atomic properties derived from relativistic calculations. This method has developed to the real working horse of relativistic quantum chemistry, and several variants are known as relativistic pseudopotentials, effective core potentials (ECPs) or ab initio model potentials. See Relativistic Effective Core Potential Techniques for Molecules Containing Very Heavy Atoms. 

Unfortunately there are no simple theories to predict the cohesive energies of the metals like the coulomb attraction in ionic crystals. More sophisticated quantum mechanical theories using pseudopotential or other modeling techniques are generally required. There are some interesting correlations, however. 

Classical simulations often lack the crucial insight into the problem, because one cannot simply use the force to characterize all the possible interactions. Fortunately, with decades of development, theoretical calculations have become quite sophisticated for crystals and molecules, although not yet for realistic nanometer-sized materials. For solids, the pseudopotential as well as the full-potential linearized augmented plane-wave (FLAPW) method within the density functional theory are well developed. Modern quantum chemical techniques (Gaussian98 [5] and MOLPRO [6]) are quite efficient to compute the potential surfaces for a given molecule. In order to illustrate those possibilities, we show some of our own results in simulating the reaction path for a segment of the retinal molecule in rhodopsin [7]. 

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