Core-polarization potentials

It was recently suggested by Nicklass and Peterson [60] that the use of core polarization potentials (CPPs) [61] could be an inexpensive and effective way to account, for the effects of inner shell correlation. The great potential advantage of this indeed rather inexpensive method over the MSFT bond-equivalent model is that it does not depend on... 

As we have seen, the basis set requirements for CC and CV correlation axe very stringent. There is therefore considerable attraction in methods that treat these effects semi empirically. One approach is to treat CV correlation effects by an effective operator. The core polarization potential used by Muller et al. for CV correlation in the alkali atoms and alkali dimers is one such approach [101]. This method has been used successfully for other atoms, such as copper [98]. 

The assumption that an electron can be described as a quasi-free particle implies that the interaction between the electron and any atom in the liquid is weak. It is then necessary that the attractive potential of the nucleus experienced by the electron penetrating the atomic core and the long range core polarization potential will be balanced by the electron s increased kinetic energy in the nuclear region. This restriction implies that the pseudopotential of the atom should be small. 

A further reduction of the computational effort in investigations of electronic structure can be achieved by the restriction of the actual quantum chemical calculations to the valence electron system and the implicit inclusion of the influence of the chemically inert atomic cores by means of suitable parametrized effective (core) potentials (ECPs) and, if necessary, effective core polarization potentials (CPPs). Initiated by the pioneering work of Hellmann and Gombas around 1935, the ECP approach developed into two successful branches, i.e. the model potential (MP) and the pseudopotential (PP) techniques. Whereas the former method attempts to maintain the correct radial nodal structure of the atomic valence orbitals, the latter is formally based on the so-called pseudo-orbital transformation and uses valence orbitals with a simplified radial nodal structure, i.e. pseudovalence orbitals. Besides the computational savings due to the elimination of the core electrons, the main interest in standard ECP techniques results from the fact that they offer an efficient and accurate, albeit approximate, way of including implicitly, i.e. via parametrization of the ECPs, the major relativistic effects in formally nonrelativistic valence-only calculations. A number of reviews on ECPs has been published and the reader is referred to them for details (Bala-subramanian 1998 Bardsley 1974 Chelikowsky and Cohen 1992 Christiansen et... 

Although the frozen-core approximation underlies all ECP schemes discussed so far, both static (polarization of the core at the Hartree-Fock level) and dynamic (core-valence correlation) polarization of the core may accurately and efficiently be accounted for by a core polarization potential (CPP). The CPP approach was originally used by Meyer and co-workers (MUller etal. 1984) for all-electron calculations and adapted by the Stuttgart group (Fuentealba et al. 1982) for PP calculations. The... 

In cases where ns and np valence orbitals are present together with (n—1 )d and (n—2)f valence orbitals, for example, for Cs, it proved to be more accurate to augment the core-polarization potential by a short-range local potential (Dolg 1996a)... 

Figure 15. First (IPj) and second (IP2) ionization potentials of the lanthanide elements j La -2jLu. Experimental values are compared to results from 4f-in-core pseudopotential (PP) calculations with and without account of core-valence correlation effects by means of a core polarization potential (CPP) [95].

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. 

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). 

The outer electron (rg) in Li is less tightly bound and therefore it moves more slowly than an inner electron (fg). In such a case, the "adiabatic approximation can be used. The outer electron is considered to be instantaneously at rest and the energy of the core is determined for various fixed values of Tg. This energy depends on Tg parametrically and in turn acts as a potential energy ["core polarization potential E,(rg)] for the motion of the outer electron Tg. The resulting wave function is... 

For 77-electron spectra, one may start at least formally with H.F. based on the average energy of a configuration , with symmetry and equivalence restrictions (see Section VI). As results on atoms also indicate, the e,/s of S cores as well as core-polarization potentials will not be much affected by the valence electrons, so that one deals with just the E part of Eq. (128). 

What remedies do we have The brute-force device tried in pioneer days, of incorporating core- and core-valence correlation effects into pseudopotentials just by fitting to experimental reference data containing these effects, does not work since the one-electron/one-center PP ansatz is insufficient for this purpose, cf. below. Certainly more reliable is a DFT description of core contributions to correlation effects which is possible with (and actually implied in) the non-linear core corrections discussed in Section 1.4. Another device, which has shown excellent performance in the context of quantum-chemical ab initio calculations180 and has later been adapted to PP work cf. e.g. refs. 139, 181-184), is that of core-polarization potentials (CPP)... 

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]. 

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