Pseudopotential approximation

The first reliable energy band theories were based on a powerfiil approximation, call the pseudopotential approximation. Within this approximation, the all-electron potential corresponding to interaction of a valence electron with the iimer, core electrons and the nucleus is replaced by a pseudopotential. The pseudopotential reproduces only the properties of the outer electrons. There are rigorous theorems such as the Phillips-Kleinman cancellation theorem that can be used to justify the pseudopotential model [2, 3, 26]. The Phillips-Kleimnan cancellation theorem states that the orthogonality requirement of the valence states to the core states can be described by an effective repulsive... 

The basic idea of the pseudopotential theory is to replace the strong electron-ion potential by a much weaker potential - a pseudopotential that can describe the salient features of the valence electrons which determine most physical properties of molecules to a much greater extent than the core electrons do. Within the pseudopotential approximation, the core electrons are totally ignored and only the behaviour of the valence electrons outside the core region is considered as important and is described as accurately as possible [54]. Thus the core electrons and the strong ionic potential are replaced by a much weaker pseudopotential which acts on the associated valence pseudo wave functions rather than the real valence wave functions (p ). As... 

We briefly discuss the performance of the relativisitic pseudopotential approximation with respect to all-electron methods, as this is the most widely used relativistic... 

Schwerdtfeger, P., Fischer, T Dolg, M Igel-Mann, G., Nicklass, A., Stoll, H. and Haaland, A. (1995) The Accuracy of the Pseudopotential Approximation. 1. An Analysis of the Spectroscopic Constants for the Electronic Ground States of InCl and InCh. Journal of Chemical Physics, 102, 2050-2062. 

Hamiltonian which, in a nonlocal pseudopotential approximation, can be written as... 

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 use of a pseudopotential is one of the ways to remedy for the more notorious deficiency of plane waves basis sets the impossibihty of describing correctly the rapid oscillation of the valence wave functions in the region around the ion cores. However plane waves offer several advantages (unbiased representation, arbitrarily good convergence accmacy, easiness of mathematical and computational implementation) which lie at the heart of their widespread use. In the pseudopotential approximation, the nncleus and core electrons are... 

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. 

An important step towards accurate descriptions of solid ionic energetic materials has been done by Sorescu and Thompson.[104-106] They used DFT and the pseudopotential method to investigate the structural and electronic properties of AND [104, 105] and AN [106] in solid phases. The advantage of using the pseudopotential approximation is that only the valence... 

Heavy neutral species In order to model the electron density of heavy elements for subsequent use in the construction of pseudopotential representations of the inner-shell regions near heavy nuclei, atomic DHF calculations are often employed as a starting point [158]. These pseudopotential approximations are used in order to reduce the computational cost of molecular or solid-state calculations of extended systems containing heavy elements. Similarly, atomic DHF calculations are used in the design of atom-centred basis sets, either by the direct... 

P. Schwerdtfeger, T. Fischer, M. Dolg, G. Igel-Mann, A. Niddass, H. Stoll, A. Haa-land, The accuracy of the pseudopotential approximation. I. An analysis of the spectroscopic constants for the electronic ground states of InCl and InCls using various three valence electron pseudopotentials for indium, J. Chem. Phys. 102 (1995) 2050-2062. 

Another realistic approach is to construct pseudopotentials using density fimctional theory. The implementation of the Kohn-Sham equations to condensed matter phases without the pseudopotential approximation is not easy owing to the dramatic span in length scales of the wavefunction and the energy range of the eigenvalues. The pseudopotential eliminates this problem by removing the core electrons from the problem and results in a much simpler problem [27]. 

Figure 17. Plot of the centroid velocity correlation function for liquid neon. The solid line is the CMD result calculated with the centroid pseudopotential approximation, while the dashed line is the classical MD result. The self-diffusion constant is proportional to the time integral of the centroid velocity correlation functions.

Within the local density-pseudopotential approximation, the total electronic potential for the valence electrons is written as... 

The MD simulations are a flexible and efficient tool the effect of various forces can be studied individually or in conjunction with other forces. In particular, the simulations can be performed with the time-varying radio-frequency potential, or in the pseudopotential approximation and the results compared. For many purposes the simulations need to compute the ion dynamics only for a few tens of milliseconds. Therefore, for not too large ion numbers ( 1000) the time required to obtain useful information using a personal computer is quite reasonable (hours), making the simulations a practical tool. 

Full ab initio treatments for complex transition metal systems are difficult owing to the expense of accurately simulating all of the electronic states of the metal. Much of the chemistry that we are interested in, however, is localized around the valence band. The basis functions used to describe the core electronic states can thus be reduced in order to save on CPU time. The two approximations that are typically used to simplify the basis functions are the frozen core and the pseudopotential approximations. In the frozen core approximations, the electrons which reside in the core states are combined with the nuclei and frozen in the SCF. Only the valence states are optimized. The assumption here is that the chemistry predominantly takes place through interactions with the valence states. The pseudopotential approach is similar. 

The valence electrons oscillate in the core region as is shown in Fig. A5, which is difficult to treat using plane wave basis functions. Since the core electrons are typically insensitive to the environment, they are replaced by a simpler smooth analytical function inside the core region. This core can also now include possible scalar relativistic effects. Both the frozen core and pseudopotential approximations can lead to significant reductions in the CPU requirements but one should always test the accuracy of such approximations. 

G. Igel-Mann, A. Nicklass, H. StoU. The accuracy of the pseudopotential approximation. I. An analysis of the spectroscopic constants for the electronic groimd... 

T. Leininger, A. Niddas, H. Stoll, M. Dolg, P. Schwerdtfeger. The accuracy of the pseudopotential approximation. 11. A comparison of various core sizes for indium pseudopotentials in calculations for spectroscopic constants of InH, InF, and InCl. /. Chem. Phys., 105 (1996) 1052-1059. 

The normconserving pseudopotentials have proven to work well for many systems, notably semiconductors (see, e.g., Kune, this volume) and their surfaces (see, e.g., Morthrup and Cohen, 1982), ionic compounds (Froyen and Cohen, 1984), and simple metals (see, e.g., Lam and Cohen, 1981). Applications to transition metals also exist (see, e.g., Greenside and Schluter, 1983). The pseudopotential approximation becomes less satisfactory when valence and core electrons begin to have large overlap, both because of the pseudo-wavefunctions lacking nodes, and because the x-c potential ih the core region should also account for the presence of the core electrons. The latter problem can in many cases be treated well by "nonlinear" pseudopotentials (Louie et al., 1982). 

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