Energy-consistent pseudopotentials

Figgen, D., Rauhut, G., Dolg, M. and StoD, H. (2005) Energy-consistent pseudopotentials for group 11 and 12 atoms adjustment to multi-configuration Dirac-Hartree-Fock data. Chemical Physics, 311, 227-244. 

M. Burkatzki, C. Filippi, M. Dolg, Energy-consistent pseudopotentials for quantum Monte Carlo calculations. J. Chem. Phys. 126, 234105 (2007) (and references contained therein)... 

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. 

Stoll H, Metz B, Dolg M. Relativistic energy-consistent pseudopotentials—recent developments. J Comput Chem 2002 23 767-778. 

In the most recent version of the energy-consistent pseudopotential approach the reference data is derived from finite-dilference all-electron multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian. As an example the first parametrization of such a potential,... 

Figure 13. Valence spinors of the Db atom in the 6d 7s ground state configuration from average-level all-electron (AE, dashed lines) multiconfiguration Dirac-Hartree-Fock calculations and corresponding valence-only calculations using a relativistic energy-consistent 13-valence-electron pseudopotential (PP, solid lines). A logarithmic scale for the distance r from the (point) nucleus is us in order to resolve the nodal structure of the all-electron spinors. The innermost parts have been truncated.

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 ... 

The formalism described here to derive energy-consistent pseudopotentials can be used for one-, two- and also four-component pseudopotentials at any desired level of relativity (nonrelativistic Schrbdinger, or relativistic Wood-Boring, Douglas-Kroll-Hess, Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian implicit or explicit treatment of relativity in the valence shell) and electron correlation (single- or multi-configurational wavefunctions. The freedom... 

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. 

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). 

As an example for the typical quality of molecular properties obtained with the newly developed basis sets of Martin and Sundermann [241] the results for the energy-consistent large-core Ge pseudopotential for the diatomics GeH, GeO and GeF is displayed in Table 8. In addition to their work on main group PP basis sets Martin and Sundermann [241] also proposed (2flg) correlation sets to be used with the (8s7p6d)/[5s4p3d] valence basis sets of the transition... 

Molecular constants of selected Ge diatomics obtained with energy-consistent four-valence electron pseudopotential (PP) [197] and a core-polarization potential (CPP) [188] in connection with the optimized cc-pVnZ (n=T,Q) valence basis sets of Martin and Sundermann [241]. The label > denotes the result of an extrapolation to the basis set limit. 

Atomization energies (in a.u.), bond lengths (in A), and force constants k of the breathing mode (in a.u.), from one-component SCF calculations using energy-consistent scalar-relativistic pseudopotentials (EC-PP), ab initio model potentials (AIMP) and valence basis sets of double zeta (DZ) and polarized double-zeta (DZP) quality, in comparison to all-electron (AE) relativistic SCF calculations. Numbers in parentheses are differences to corresponding non-relativistic results. 

Bond lengths R (A), binding energies D. (eV) and vibrational constants a>e (cm ) of the homonuclear halogen dimers from dl-electron (AE) Douglas-Kroll-HeB (DKH) and valence-only energy-consistent pseudopotential (EC-PP) Hartree-Fock self-consistent field (SCF) calculations. The effects of static and dynamic core-polarization at the valence-only level are modelled by a core-polarization potential (CPP). 

Bond lengths (A), vibrational constants (Og (cm ) and binding energies D (eV) of ThO in the ground state from energy-consistent pseudopotential (EC-PP) [198,243], model potential (MP) [269] and ab initio model potentiM (AIMP) [115] calculations in comparison to experimentd data. The values are without/with counter poise correction of the basis set superposition error. 

The author is grateful to H. Stoll (Stuttgart) and P. Schwerdtfeger (Auckland) many years of cooperation on the field of energy-consistent pseudopotentials. Financial support of the Deutsche Forschungsgemeinschaft (DFG) and the Fonds der Chemischen Industrie (FCI) is gratefully acknowledged. 

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)). 

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