Pseudopotentials basis

For systems with heavy atoms we often employ pseudopotential basis sets (frequently relativistic), which reduce the computational burden of large numbers of electrons. Transition metals present problems beyond those of main-group heavy atoms not only can relativistic effects be significant, but electron d- or f-levels, variably perturbed by ligands, make possible several electronic states. DFT calculations, with pseudopotentials, are the standard approach for computations on such compounds. 

All calculations were performed with the Gaussian 98 suite of programs [4] using the hybrid DFT/HF B3LYP method. The all-electron 6-311+G(2df) basis set was used for H, O, Si, and Ge, whereas the valence triple zeta pseudopotential basis set of Stoll et al. [5] was used for Sn. The Sn basis set was augmented with diffuse function exponents one-third in size of the outermost valence exponents, the two-membered d-polarization set of Huzinaga, [6] and f(QSn = 0.286. This basis set combination will be called (v)TZ throughout this paper. 

In this section we include a description of the computational scheme Siesta (Spanish Initiative for Electronic Structure of Thousands of Atoms) taken from the recent review by Artacho et al Details about the choice of the pseudopotential, basis set, and computational parameters are also given. 

Hartree-Fock LCAO method. A model of a finite crystal with periodical boundary conditions (cyclic model) with the main region composed of 4 x 4 x 4 = 64 primitive cells was adopted. The first basis consists of 13 s- and p- atomic-like functions per atom, the pseudopotential basis consists of 2 s-, 6 p- and 5 d-functions per atom. The weight function p(r) = S(r — q) has been taken in (3.115). 

X. Cao and M. Dolg, Segmented contraction scheme for smaU-core lanthanide pseudopotential basis sets, /. Mol. Struct. (Theochem), 581, 139-147 (2002). 

Cao XY, Dolg M. Segmented contraction scheme for small-core actinide pseudopotential basis sets. Journal of Molecular Structure THEOCHEM. 2004 673(l-3) 203-9. 

There are a variety of other approaches to understanding the electronic structure of crystals. Most of them rely on a density functional approach, with or without the pseudopotential, and use different bases. For example, instead of a plane wave basis, one might write a basis composed of atomic-like orbitals ... 

An approach closely related to the pseudopotential is the orthogonalizedplane wave method [29]. In this method, the basis is taken to be as follows ... 

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].)... 

Split Valence Basis Sets Polarized Basis Sets Diffuse Functions Pseudopotentials... 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

G.P. Francis and M.C. Payne, Finite basis set corrections to total energy pseudopotential calculations, J. 

Ultrasoft pseudopotentials have now been constructed for all elements for all elements from H to Bi [10]. It has been shown that without any loss of accuracy even for first-row and transition-elements small PW basis-sets comparable in size to those necessai y for soft normconserving pseudopotentials for A1 or Si can be used. For any further details, see [9,. 37]. 

All calculations are scalar relativistic calculations using the Douglas-Kroll Hamiltonian except for the CC calculations for the neutral atoms Ag and Au, where QCISD(T) within the pseudopotential approach was used [99], CCSD(T) results for Ag and Au are from Sadlej and co-workers, and Cu and Cu from our own work, using an uncontracted (21sl9plld6f4g) basis set for Cu [6,102] and a full active orbital space. 

To summarize, the RPPA is a method that can accurately describe relativistic effects, even though the relativistic perturbation operator used in the pseudopotential procedure is acting on the valence space and not the region dose to the nudeus, as this is the case for the correct all-electron relativistic perturbation operator. That is, relativistic effects are completely transferred into the valence space. These effects are also completely transferable from the atomic to the molecular case as the results for Au2 show. If relativistic pseudopotentials are carefully adjusted, they can produce results with errors much smaller than the errors originating from basis set incompleteness, basis set superposition or from the electron correlation procedure applied. 

Stuttgart pseudopotential for Au with a uncontracted (lls/10p/7d/5f) valence basis set and a Dunning augmented correlation consistent valence triple-zeta sets (aug-cc-pVTZ) for both C and N, but with the most diffuse f function removed, was used. 

Peterson, K.A. and Puzzarini, C. (2005) Systematically convergent basis sets for transition metals. II. Pseudopotential-based correlation consistent basis sets for the group 11 (Cu, Ag, Au) and 12 (Zn, Cd, Hg) elements. Theoretical Chemistry Accounts, 114, 283-296. 

In molecular DFT calculations, it is natural to include all electrons in the calculations and hence no further subtleties than the ones described arise in the calculation of the isomer shift. However, there are situations where other approaches are advantageous. The most prominent situation is met in the case of solids. Here, it is difficult to capture the effects of an infinite system with a finite size cluster model and one should resort to dedicated solid state techniques. It appears that very efficient solid state DFT implementations are possible on the basis of plane wave basis sets. However, it is difficult to describe the core region with plane wave basis sets. Hence, the core electrons need to be replaced by pseudopotentials, which precludes a direct calculation of the electron density at the Mossbauer absorber atom. However, there are workarounds and the subtleties involved in this subject are discussed in a complementary chapter by Blaha (see CD-ROM, Part HI). 

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