Frozen core potentials

There are several variations of this method. The PRDDO/M method is parameterized to reproduce electrostatic potentials. The PRDDO/M/FCP method uses frozen core potentials. PRDDO/M/NQ uses an approximation called not quite orthogonal orbitals in order to give efficient calculations on very large molecules. The results of these methods are fairly good overall, although bond lengths involving alkali metals tend to be somewhat in error. 

In a recent series of articles [179-181], Seijo and Barandiaran have investigated the spectroscopy of several actinide impurities (Pa" - -, and in crystal environments. In particular, they discuss the relative position of the 5 and 5/ " 6i/ manifolds (see also chapter 7 of this book). All calculations use relativistic large-core AIMPs on the actinide centres and on the chlorine ligands. The transferability of these frozen core potentials from the neutral / elements to their cation has been discussed in Ref [182]. The crystal environment is described by the AIMP embedding cluster method. Electron and spin-orbit interactions are treated simultaneously by the three-step spin-fi e-state-shifted method detailed in section 2.2.5, using either MRCI or CASSCF/MS-CASPT2 methods in the spin-fi ee step. The active space includes the 5/ and 6d orbitals of the actinide centre, as well as the Is orbitals in order to avoid the prob-... 

Effective potentials also depend on the type of basis set used, hi atomic orbital calculations, they are sometimes referred to as frozen-core potentials. In most cases, only the highest-energy s, p and d electrons are included in the calculation. In plane-wave calculations, effective potentials are known as pseudopotentials They come in different varieties soft or ultrasoft pseudopotentials need only a relatively low energy cut-off as they involve a larger atomic core. ... 

FCP = frozen core potential MBS = minimum basis set NQOAO = not quite orthogonal atomic orbital OAO = orthogonal atomic orbital PESP = parametrized electrostatic potential PRDDO = partial retention of diatomic differential... 

This section presents an outline of the methodological aspects of the method. First, the PRDDO/M basis set is described. Then, the fundamental approximations of the method are presented and the parametrization scheme is outlined. This is followed by a discussion of recent modifications of the method to make it suitable for very large molecules. This includes the introduction of not quite orthogonal atomic orbitals (NQOAOs) and frozen core potentials (FCPs). Finally, the accuracy of the method relative to the underlying ab initio calculations is demonstrated and some timing comparisons are presented. 

Table 4 Fixed Atomic Parameters in SINDOl. Ionization Potentials / (eV), Frozen Core Potentials e (Hartree), and Orbital Exponents of Inner Shells x...

It is not possible to use normal AO basis sets in relativistic calculations The relativistic contraction of the inner shells makes it necessary to design new basis sets to account for this effect. Specially designed basis sets have therefore been constructed using the DKH Flamiltonian. These basis sets are of the atomic natural orbital (ANO) type and are constructed such that semi-core electrons can also be correlated. They have been given the name ANO-RCC (relativistic with core correlation) and cover all atoms of the Periodic Table.36-38 They have been used in most applications presented in this review. ANO-RCC are all-electron basis sets. Deep core orbitals are described by a minimal basis set and are kept frozen in the wave function calculations. The extra cost compared with using effective core potentials (ECPs) is therefore limited. ECPs, however, have been used in some studies, and more details will be given in connection with the specific application. The ANO-RCC basis sets can be downloaded from the home page of the MOLCAS quantum chemistry software (http //www.teokem.lu.se/molcas). 

Pettersson,L.G.M., Wahlgren,U. and Gropen,0. (1983), Effective core potential calculations using frozen orbitals. Applications to transition metals Chem.Phys. 80, 7... 

The Pauli operator of equations 2 to 5 has serious stability problems so that it should not, at least in principle, be used beyond first order perturbation theory (20). These problems are circumvented in the QR approach where the frozen core approximation (21) is used to exclude the highly relativistic core electrons from the variational treatment in molecular calculations. Thus, the core electronic density along with the respective potential are extracted from fully relativistic atomic Dirac-Slater calculations, and the core orbitals are kept frozen in subsequent molecular calculations. 

An efficient way to solve a many-electron problem is to apply relativistic effective core potentials (RECP). According to this approximation, frozen inner shells are omitted and replaced in the Hamiltonian hnt by an additional term, a pseudopotential (UREP)... 

An overview of quantum Monte Carlo electronic structure studies in the context of recent effective potential implementations is given. New results for three electron systems are presented. As long as care is taken in the selection of trial wavefunctions, and appropriate frozen core corrections are included, agreement with experiment is excellent (errors less than 0.1 eV). This approach offers promise as a means of avoiding the excessive configuration expansions that have plagued more conventional transition metal studies. 

Equation 3 obviously adds ajpraximations. These include the usual effective potential assumptions (frozen core, etc.) in addition to the localization shown in the equation. However in one sense it is a trade-off in that the local potential effectively eliminates the fixed node approximation in the core region. 

A frozen-core calculation involves choosing a particular state (for example the one lowest in energy), performing a Hartree—Fock calculation to find the best orbitals, then using the orbitals of the core to generate a nonlocal potential (5.27), which is taken to represent the core in calculations of further states. 

Table 5.1 illustrates the frozen-core approximation for the case of sodium using a simple Slater (4.38) basis in the analytic-orbital representation. The core (Is 2s 2p ) is first calculated by Hartree—Fock for the state characterised by the 3s one-electron orbital, which we call the 3s state. The frozen-core calculation for the 3p state uses the same core orbitals and solves the 3p one-electron problem in the nonlocal potential (5.27) of the core. Comparison with the core and 3p orbitals from a 3p Hartree—Fock calculation illustrates the approximation. The overwhelming component of the 3p orbital agrees to almost five significant figures. 

Table 5.3. One-electron separation energies for the lower-energy states of sodium (units eV). Experimental data (EXP) are from Moore (1949). The calculations are FCHE, frozen-core Hartree—Fock and POL, frozen-core Hartree—Fock with the phenomenological core-polarisation potential (5.82)...

An appropriate potential for many cases is the frozen-core Hartree—Fock potential with the addition of a core-polarisation term. 

Scattering from alkali-metal atoms is understood as the three-body problem of two electrons interacting with an inert core. The electron—core potentials are frozen-core Hartree—Fock potentials with core polarisation being represented by a further potential (5.82). 

Essentially-complete agreement with experiment is achieved by the coupled-channels-optical calculation. We can therefore ask if scattering is so sensitive to the structure details in the calculation that it constitutes a sensitive probe for structure. The coupled-channels calculations in fig. 9.3 included the polarisation potential (5.82) in addition to the frozen-core Hartree—Fock potential. Fig. 9.4 shows that addition of the polarisation potential has a large effect on the elastic asymmetry at 1.6 eV, bringing it into agreement with experiment. However, in general the probe is not very sensitive to this level of detail. 

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