Reliable Effective Core Potentials

Once a nodeless orbital has been generated the one-electron atomic Fock equation is easily inverted to produce a (radially) local operator, the EP, which represents the core-valence interactions (22,23). 

A separate EP must be generated for each Z value. When generalized to take into account the various orbital symmetries the total (nonrelativistic) EP has the form 

Although the goal here is the extension of molecular electronic structure techniques into the realm of heavy elements, it is important that as much as possible of the reliability of light-element work be retained. The accuracy of the effective potential approximation can most readily be determined by careful comparisons of molecular EP results with those obtained from allelectron calculations. At the present time this can be done easily only for the nonrelativistic case. Although comparisons can be made with experimentally determined properties, it should be kept in mind that in general highly accurate valence wave functions are required. 

Bond Lengths (a.u.) and Dissociation Energies (kcal/mol) of F2 and Cl2 from Two-Configuration MCSCF Wave Functions 1 

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The methods listed thus far can be used for the reliable prediction of NMR chemical shifts for small organic compounds in the gas phase, which are often reasonably close to the liquid-phase results. Heavy elements, such as transition metals and lanthanides, present a much more dilficult problem. Mass defect and spin-coupling terms have been found to be significant for the description of the NMR shielding tensors for these elements. Since NMR is a nuclear effect, core potentials should not be used. 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

In earlier papers [6-8] we have proposed a procedure for evaluating core ioniza-tion/excitation chemical shifts in molecules from computed core ionization/excitation energies for the relevant isolated atom in neutral and valence-ionized states and from computed charge transfer relative to this atom within the molecule. The atomic calculations involved relaxation, possibly correlation and, when appropriate, relativity and other effects, while in the molecule one could use any approximate method (possibly involving effective core potentials) yielding reliable charges. 

Dynamic correlation, at the MP2 level of theory, is very important for the prediction of accurate energetics for these species. While MP2 was not very important for the prediction of geometries for the fluorine dimer, it was essential for the geometry predictions for the heavier congeners. The effective core potential basis set overestimates the dimerization energy of TiF4 by about 4 kcal/mol, but it was quite reliable for the heavier species. 

The relativistic effective core potential method is reviewed. The basic assumptions of the model potential and pseudopotential variants are discussed and the reliability of both approaches in electronic structure calculations for heavy element systems is demonstrated for selected examples. 

We review the Douglas-Kroll-Hess (DKH) approach to relativistic density functional calculations for molecular systems, also in comparison with other two-component approaches and four-component relativistic quantum chemistry methods. The scalar relativistic variant of the DKH method of solving the Dirac-Kohn-Sham problem is an efficient procedure for treating compounds of heavy elements including such complex systems as transition metal clusters, adsorption complexes, and solvated actinide compounds. This method allows routine ad-electron density functional calculations on heavy-element compounds and provides a reliable alternative to the popular approximate strategy based on relativistic effective core potentials. We discuss recent method development aimed at an efficient treatment of spin-orbit interaction in the DKH approach as well as calculations of g tensors. Comparison with results of four-component methods for small molecules reveals that, for many application problems, a two-component treatment of spin-orbit interaction can be competitive with these more precise procedures. 

Despite this ubiquitous presence of relativity, the vast majority of quantum chemical calculations involving heavy elements account for these effects only indirectly via effective core potentials (ECP) [8]. Replacing the cores of heavy atoms by a suitable potential, optionally augmented by a core polarization potential [8], allows straight-forward application of standard nonrelativistic quantum chemical methods to heavy element compounds. Restriction of a calculation to electrons of valence and sub-valence shells leads to an efficient procedure which also permits the application of more demanding electron correlation methods. On the other hand, rigorous relativistic methods based on the four-component Dirac equation require a substantial computational effort, limiting their application in conjunction with a reliable treatment of electron correlation to small molecules [9]. 

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]. 

Effective core potential (ECP) or pseudo-potential approximation, has been proved to be very useful for modeling of heavy atoms in the ab initio methods (Hay and Wadt 1985). In this approximation, core electrons are replaced by an effective potential, thereby reducing the number of electrons to be considered and hence requiring fewer basis functions. The ECP method takes into account the relativistic effect on valence electrons, thus making it applicable to heavy atoms (e.g., second- and third-row transition metals, lanthanides and actinides). It is relatively cheap, works very well, and has very little loss in reliability. 

In recent years, there has been an increasing interest in the inclusion of relativistic effects for molecules containing heavy atoms. One of the most practical yet reliable methods is to use relativistically derived effective core potentials. Major relativistic effects such as the Darwin and mass-velocity effects are easily taken into account in the form of a spin-free (SF) one-electron operator. The spin-orbit (SO) interaction is in general too strong to be considered as a small perturbation, and therefore should be treated explicitly as a part of the total Hamiltonian. 

The drive toward reliable quantum mechanical predictions for large molecular systems is well represented in ECC articles by George Bacskay Solvation Modeling), Krishnan Balasub-ramanian Relativistic Effective Core Potential Techniques for Molecules Containing Very Heavy Atoms), Margareta Blomberg Configuration Interaction PCI-X and Applications), and Keiji Morokuma Hybrid Methods). 

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