Pseudopotential Small-Core

In the present work, correlation consistent basis sets have been developed for the transition metal atoms Y and Hg using small-core quasirelativistic PPs, i.e., the ns and (nA)d valence electrons as well as the outer-core (nA)sp electrons are explicitly included in the calculations. This can greatly reduce the errors due to the PP approximation, and in particular the pseudo-orbitals in the valence region retain some nodal structure. Series of basis sets from double-through quintuple-zeta have been developed and are denoted as cc-pVwZ-PP (correlation consistent polarized valence with pseudopotentials). The methodology used in this work is described in Sec. II, while molecular benchmark calculations on YC, HgH, and Hg2 are given in Sec. III. Lastly, the results are summarized in Sec. IV. 

New correlation consistent basis sets have been developed for Y and Hg in conjunction with accurate small-core relativistic pseudopotentials. A few allelectron basis sets have also been optimized both with and without the inclusion... 

Usually the atoms on the local region include all electrons or a small core pseudopotential whereas the atoms in the outer region include a larger core and most often only the outermost ns electrons necessary to represent the metal conduction band are explicitly included in the quantum mechanical calculation. This is nowadays a standard procedure to model Ni, Cu, Ag and Pt surfaces. " " In addition, a mixed basis scheme is usually employed to further reduce the computational cost while attempting to preserve the quality of the cluster model ab initio calculations. In this scheme, the atoms in the outer region are treated with a rather limited, even minimal, basis set whereas atoms in the local region are treated with a more extended basis set. 

Besides the reduction of frozen-core errors when going from large-core to medium-core or small-core potentials also the valence correlation energies obtained in pseudopotential calculations become more accurate since the radial nodal structure is partially restored [97,98]. Clearly the accuracy of small-core potentials is traded against the low computational cost of the large-core po-... 

The results of this procedure for alkaline and alkaline-earth systems were quite good [186,187], at least for atoms, and pseudopotentials of this type were generated [188] and applied [189] for most of the main group elements. However, due to the limited validity of the frozen-core approximation when going from a medium or highly charged one-valence electron ion to a neutral atom or nearly neutral ion, the approach is bound to fail for most other elements. This is especially the case for transition metals, lanthanides and actinides, where small cores are indispensable for accurate pseudopotentials. More recent calibration studies of alkaline and alkaline earth elements exhibited however, that for accu-... 

Parameters of energy-consistent ab initio pseudopotentials and corresponding valence basis sets are available for almost all elements of the periodic table [93,94,117,190-192,194-201]. A compilation of parameters for the use within the MOLPRO program system also exists on the internet under the address http //www.theochem.uni-stuttgart.de. Special care has to be taken when spin-orbit coupling is included in calculations with small-core PPs some SO operators are constructed (similar to the large-core case) for a fully variational two-component treatment, whereas in some cases effective valence SO operators are defined. The latter have to be applied in SO-CI calculations for the valence electrons, in which the semi-core shells (outside the PP core) are frozen in their scalar-relativistic form. 

Modern relativistic effective core potentials provide a useful tool for accurate quantum chemical investigations of heavy atom systems. If sufficiently small cores are used to minimize frozen-core and other errors, they are able to compete in accuracy with the more rigorous all-electron approaches and still are, at the same time, economically more attractive. Successful developments in the field of valence-only Hamiltonians turned relativistic effects into a smaller problem than electron correlation in practical calculations. Both the model potential and the pseudopotential variant have advantages and disadvantages, and the answer to the question which approach to follow may be a matter of personal taste. Highly accurate correlated all-electron calculations are becoming... 

The free-electron approximation described in Chapter 15 is so successful that it is natural to expect that any effects of the pseudopotential can be treated as small perturbations, and this turns out to be true for the simple metals. This is only possible, however, if it is the pseudopotential, not the true potential, which is treated as the perturbation. If we were to start with a free-electron gas and slowly introduce the true potential, states of negative energy would occur, becoming finally the tightly bound core states these are drastic modifications of the electron gas. If, however, we start with the valence-electron gas and introduce the pseudopotential, the core states are already there, and full, and the effects of the pseudopotential are small, as would be suggested by the small magnitude of the empty-core pseudopotential shown in Fig. 15-3. 

Recently, Stoll et al [94] used a very similar approach to EPCISO. One minor difference is the use of the DGCI Pitzer s code which works with CSFs basis functions instead of determinants. Apparently another difference here is the absence of a selection process of the spin-orbit matrix elements. In this study small-core and large-core energy-consistent pseudopotentials were combined for the calculation of spectroscopic constants of lead and bismuth compounds (BiH, BiO, PbX, BiX, (X=F, Cl, Br, I)). 

Large-core pseudopotentials are well known to lead to sizable errors [122-125] we thus recommend the use of small-core ECP s that include semicore orbitals in the valence space [120],... 

Schwerdtfeger et al [124] have systematically tested several pseudopotentials for the heaviest element of group I-B, gold atom and its hycWde. The variation between the results obtained from all valence electron small-core Stuttgart energy-adjusted pseudopotentials [122,127] and all electron Douglas-Kroll calculations for AuH is found to be small (Are = 0.001 A, ADe = 0.03 eV, AtOe = 9 cm ). This demonstrates that the pseudopotential approach is a reliable and... 

A priori it is not clear if effective core potentieds, which have for example been adjusted to reproduce atomic energy differences in wave function based calculations or to reproduce the shape of the valence orbitals outside the core, can successfully be used in density functional calculations. For so-called small core potentieds, where the atomic core has been chosen such that core and valence densities have little overlap, test calculations have shown that results from allelectron and pseudopotential calculations were virtually the same [74]. A related investigation on gold compounds comes to the same conclusion [75]. It is however not recommended to perform density functional investigations with large-core pseudopotentials that have been adjusted in wave function calculations. One example for a leirge-core situation is a transition metal where the vedence d orbiteds are (of course) treated explicitly, while the s emd p orbitals of the same principal quantum number are considered core orbitals. From an energetic view, such a separation seems well justified. However, problems arise since the densites of the s,p, and d orbitals of the same principal quantum number have considerable overlap. 

Other, scalar relativistic effects are usually minor. Among them, the most important is the contraction of s-orbitals caused by the increase in electron mass due to high velocity near the nucleus. Except in the most careful work, such effects are modeled using relativistic effective core potentials (ECPs), also called core pseudopotentials [76]. When an ECP is used, the corresponding valence basis set should be used for the remaining electrons. A small-core ECP, in which fewer electrons are replaced by the effective potential, is a weaker approximation and therefore more reliable than the corresponding large-core ECP. The selection of basis sets to accompany ECPs is more restricted than the selection of all-electron basis sets, but appropriate correlation-consistent basis sets are available for heavy p-block elements [77-80]. 

It is obvious from Table 1 that the quality of the valence basis sets of the available pseudopotentials varies considerably. The large-core ECPs by Hay and Wadt have a low number of electrons in the valence space. Also, the valence orbitals are described by a rather small basis set. We recommend use of the small-core ECPs, which give clearly better results.The same comment applies to the two sets of ECPs that have been published by Christiansen et al 85-88 xhe (n - l)s and (n - l)p electrons should be treated as part of the valence electrons (see also below). 

Table 9.26. The theoretical and experimental structural parameters for rutile (in A), numbers in parentheses indicate the percent deviation from low-temperature neutron-diffraction experiments. The references to the theoretical and experimental data, given in this table, can be found in [597]. SC,LC mean small-core and large-core pseudopotentials, AE -all electron calculations...

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