Shape-consistent pseudopotentials

The shape-consistent (or norm-conserving ) RECP approaches are most widely employed in calculations of heavy-atom molecules though ener-gy-adjusted/consistent pseudopotentials [58] by Stuttgart team are also actively used as well as the Huzinaga-type ab initio model potentials [66]. In plane wave calculations of many-atom systems and in molecular dynamics, the separable pseudopotentials [61, 62, 63] are more popular now because they provide linear scaling of computational effort with the basis set size in contrast to the radially-local RECPs. The nonrelativistic shape-consistent effective core potential was first proposed by Durand Barthelat [71] and then a modified scheme of the pseudoorbital construction was suggested by Christiansen et al. [72] and by Hamann et al. [73]. 

Thble3.6 Bond length Rc (A), vibrational constant coe (cm-1) and binding energy De (eV) of Eka-Au hydride (111)H without (with) counterpoise correction of the basis-set superposition error. All-electron (AE) values based on die Dirac-Coulomb-Hamiltonian (Seth and Schw-erdtfeger 2000) are compared with valence-only results obtained with energy-consistent (EC) (Dolg etal. 2001) and shape-consistent (SC) (Han and Hirao 2000) pseudopotentials (PP). The numbers 19 and 34 in parentheses denote the number of valence electrons for the Eka-Au PP. 

The origin of shape-consistent pseudopotentials [131,160] lies in the insight that the admixture of only core orbitals to valence orbitals in order to remove the radial nodes leads to too contracted pseudo valence orbitals and finally as a consequence to poor molecular results, e.g., to too short bond distances. It has been recognized about 20 years ago that it is indispensable to have the same shape of the pseudo valence orbital and the original valence orbital in the spatial valence region, where chemical bonding occurs. Formally this requires also an admixture of virtual orbitals in Eq. 37. Since these are usually not obtained in finite difference atomic calculations, another approach was developed. Starting point... 

Shape-consistent pseudopotentials including spin-orbit operators based on Dirac-Hartree-Fock AE calculations using the Dirac-Coulomb Hamiltonian have been generated by Christiansen, Ermler and coworkers [161-170]. The potentials and corresponding valence basis sets are also available on the internet under http //www.clarkson.edu/ pac/reps.html. A similar, quite popular set for main group and transition elements based on scalar-relativistic Cowan-Griffin AE calculations was published by Hay and Wadt [171-175]. 

The functional form of energy-consistent pseudopotentials is identical to the one of shape-consistent pseudopotentials, both types of pseudopotentials can be used in standard quantum chemical program packages (e.g., COLUMBUS, GAUSSIAN, GAMESS, MOLPRO, TURBOMOLE) as well as polymer or solid state codes using Gaussian basis sets (e.g., CRYSTAL). 

Shape-Consistent Pseudopotentials. - While with model potentials the wavefunction is (ideally) not changed with respect to the valence part of an AE frozen-core wavefunction, such a change is desirable for computational reasons. The nodal structure of the valence orbitals in the core region requires highly localized basis functions these are not really needed for the description of bonding properties in molecules but rather for the purpose of core-valence orthogonalization. The idea to incorporate this Pauli repulsion of the core into the pseudopotential is as old as pseudopotential theory itself.62,63 Modem ab initio pseudopotentials of this type have been developed since the end of the seventies, cf. e.g. refs. 64-68. 

DFT-Based Pseudopotentials. - The model potentials and shape-consistent pseudopotentials as introduced in the previous two sections can be characterized by a Hartree-Fock/Dirac-Hartree-Fock modelling of core-valence interactions and relativistic effects. Now, Hartree-Fock has never been popular in solid-state theory - the method of choice always was density-functional theory (DFT). With the advent of gradient-corrected exchange-correlation functionals, DFT has found a wide application also in molecular physics and quantum chemistry. The question seems natural, therefore Why not base pseudopotentials on DFT rather than HF theory ... 

This is perfectly possible, indeed, and has been worked out for model potentials ( /, e.g. ref. 107) as well as for shape-consistent pseudopotentials (usually called norm-conserving PP in this context).108-110 In the first case, the main change is to replace the HF core-valence exchange operator in equation (5)... 

Within the molecular quantum-chemical regime, various strategies on how to generate sets of pseudopotentials have been followed, such as the shape-consistent pseudopotentials for which the pseudo-valence orbital is identical... 

The Toulouse quantum-chemistry group created quasirelativistic pseudopotentials based on the atomic structure method developed by Barthelat et al. (1980) and the general procedure of adjustment devised by Durand and Barthelat (1974, 1975). The pseudopotential is constructed in such a way that the pseudo-orbitals coincide best with the all-electron valence orbitals and are smooth in the core region, i.e. an approach which has later been termed norm-conserving (Hamaim et al. 1979) or shape-consistent (Christiansen et al. 1979, Rappe et al. 1981). No parameter sets for the lanthanides and actinides have been published up to now. 

Stevens and coworkers have published shape-consistent one-component quasirelativistic pseudopotentials, i.e. sLi-igAr (Stevens et al. 1984), igK-syLa and 72Hf-86Rn (Stevens et al. 1992). Cundari and Stevens (1993) presented the corresponding parameters for the lanthanides sgCe-jiLu. Spin-orbit potentials have not been published but may be derived (Stevens and Krauss 1982) since the reference data are taken from all-electron DHF calculations. 

This effect presents some serious problems for the development of pseudopotentials. A pseudopotential that depends critically on the shape of the pseudospinor and for which the results are sensitive to the valence occupation is of no value. The problem was overcome (Christiansen et al. 1979) by the definition of the so-called shape-consistent pseudospinors and the corresponding pseudopotentials. 

The introduction of shape-consistent pseudospinors solved the problems in the pseudopotential that were caused by the use of the Philips-Kleinman pseudospinors. However, the other characteristics of pseudospinors that were discussed in the previous section still apply to shape-consistent pseudospinors. The inclusion of virtual spinors in the expansion of the core tail does not alter the conclusions drawn all but the lowest pseudospinor mix, and the eigenvalue spectrum is compressed. 

In some of the early studies, the pseudospinors were obtained by minimizing a function involving the kinetic energy (Kahn et al. 1976). The shape-consistent pseudopotentials of Hay and Wadt (Hay and Wadt 1985, Wadt and Hay 1985) and Christiansen, Ermler, and coworkers (Pacios and Christiansen 1985, Hurley et al. 1986, La John et al. 1987, Ross et al. 1990, 1994, Ermler et al. 1991) are obtained by fitting a polynomial function to the core tail with the requirements that it have no nodes and the minimum number of inflection points and must match the derivatives to the order of the polynomial at the join point. While this procedure guarantees the smoothness of the function, especially after the pseudospinor is expanded in a Gaussian basis set, the choice of the join point must be made with care. If it is too far out, the results can be unsatisfactory, as was found for the 6p elements (Wildman et al. 1997). 

There have been a number of basis sets for lanthanide and actinide elements previously reported in the literature that are based on relativistic effective core (ECP) potentials, or pseudopotentials (PP). These can be most easily categorized by the type of underlying ECP used (a) shape consistent pseudopotentials, (b) energy consistent pseudopotentials, and (c) model potentials. 

It may be asked how accurate energy-consistent pseudopotentials will reproduce the shape of the valence orbitals/spinors and their energies. Often radial expectation values < r > are used as a convenient measure for the radial shape of orbitals/spinors. Due to the pseudo-valence orbital transformation and the simplified nodal structure it is clear that values n < 0 are not suitable, since the resulting operator samples the orbitals mainly in the core region. Table 2 lists orbital energies, < r > and < > expectation values for the Db [Rn] 5f 6d ... 

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