Accuracy pseudopotential approximation

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. 

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. 

The use of a pseudopotential is one of the ways to remedy for the more notorious deficiency of plane waves basis sets the impossibility 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 accuracy, easiness of mathematical and computational implementation) which lie at the heart of their widespread use. In the pseudopotential approximation, the nucleus and core electrons are... 

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. 

One more important difference between the GTO and PW approaches is that whereas in the former case the representation of both core and valence orbitals is the same, in the PW formalism the number of PW components needed to correctly describe the behavior of the wave function near the nucleus is prohibitively large. This problem is solved by modeling the core electrons using the pseudopotential approximation, in which it is assumed that the core electrons do not significantly influence the electronic structure and chemistry of atoms. The valence electrons of a particular atom are then considered to move in an effective ionic potential due to the core electrons and the nucleus. Same approach can also be applied to reduce the size of the GTO basis set in calculations without dramatic loss of accuracy. Furthermore, the use of pseudopotentials allows the inclusion of nonrelativistic effects in the calculations, which are particularly important for the chemistry of heavy elements. 

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].)... 

Two methods are mainly responsible for the breakthrough in the application of quantum chemical methods to heavy atom molecules. One method consists of pseudopotentials, which are also called effective core potentials (ECPs). Although ECPs have been known for a long time, their application was not widespread in the theoretical community which focused more on all-electron methods. Two reviews which appeared in 1996 showed that well-defined ECPs with standard valence basis sets give results whose accuracy is hardly hampered by the replacement of the core electrons with parameterized mathematical functions" . ECPs not only significantly reduce the computer time of the calculations compared with all-electron methods, they also make it possible to treat relativistic effects in an approximate way which turned out to be sufficiently accurate for most chemical studies. Thus, ECPs are a very powerful and effective method to handle both theoretical problems which are posed by heavy atoms, i.e. the large number of electrons and relativistic effects. 

Another pertinent question is related to the accuracy of the common approximation to describe relativistic effects at the pseudopotential level. Our AE scalar relativistic DKH scheme allows to evaluate the precision of the latter scheme. A relativistic pseudopotential [196] was utilized to treat the heavy element Pd in the Pd-0 complexes employing extended EPE-embedded cluster models of the quality comparable to that for the AE cluster model. This resulted in the adsorption energy value 123 kJ/mol and the Pd-0 bond length 213 pm. For the Pd-0 complexes under scrutiny the deviations from the corresponding scalar relativistic values, by 3 kJ/mol and 2 pm respectively, are rather small. Clearly, relativistic pseudopotentials for heavier atoms have to be constructed with due care [8]. The AE scalar relativistic DKH approach certainly provides an attractive alternative. 

Early LSDA static pseudopotential approaches to sodium microclusters date back approximately 20 years [122], see Appendix C. It would be misleading to consider LDA calculations as the natural extension of jellium models. However, the global validity of the latter cannot but anticipate the success of the former. Clearly, these should also clarify the role of the atomic structure in determining the electronic behavior of the clusters and the extent to which the inhomogeneity of the electron distribution is reflected in the measurable properties. Many structural determinations are by now available for the smaller aggregates, made at different levels of approximation and of accuracy (e.g. [110, 111], see Appendix C). The most extensive investigation of sodium clusters so far is the LDA-CP study of Ref. [123] (see Appendix C), which makes use of all the features of the CP method. Namely, it uses dynamical SA to explore the potential-energy surface, MD to simulate clusters at different temperatures, and detailed analysis of the one-electron properties, which can be compared to the predictions of jellium-based models. 

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