Frozen density approximation

Table 1 Various properties for a water molecule in the gas phase and in an water solvent. The FD results were obtained with a frozen-density approximation, and the QM with a polarizable-molecules approximation. Listed are the size of the dipole moment, the three independent components of the quadrupole moment, excitation energies, static polarizability, and static hyperpolarizability. All results are from ref. 28... 

In the frozen MO approximation the last terms are zero and the Fukui functions are given directly by the contributions from the HOMO and LUMO. The preferred site of attack is therefore at the atom(s) with the largest MO coefficients in the HOMO/LUMO, in exact agreement with FMO theory. The Fukui function(s) may be considered as the equivalent (or generalization) of FMO methods within Density Functional Theory (Chapter 6). 

The usual way chemistry handles electrons is through a quantum-mechanical treatment in the frozen-nuclei approximation, often incorrectly referred to as the Born-Oppenheimer approximation. A description of the electrons involves either a wavefunction ( traditional quantum chemistry) or an electron density representation (density functional theory, DFT). Relativistic quantum chemistry has remained a specialist field and in most calculations of practical... 

Fukui functions and other response properties can also be derived from the one-electron Kohn-Sham orbitals of the unperturbed system [14]. Following Equation 12.9, Fukui functions can be connected and estimated within the molecular orbital picture as well. Under frozen orbital approximation (FOA of Fukui) and neglecting the second-order variations in the electron density, the Fukui function can be approximated as follows [15] ... 

As Fig. 12 shows, the inner shell electrons of the alkaline ions behave classically like a polarizable spherical charge-density distribution. Therefore it seemed promising to apply a "frozen-core approximation in this case 194>. In this formalism all those orbitals which are not assumed to undergo larger changes in shape are not involved in the variational procedure. The orthogonality requirement is... 

The most important approach to reducing the computational burden due to core electrons is to use pseudopotentials. Conceptually, a pseudopotential replaces the electron density from a chosen set of core electrons with a smoothed density chosen to match various important physical and mathematical properties of the true ion core. The properties of the core electrons are then fixed in this approximate fashion in all subsequent calculations this is the frozen core approximation. Calculations that do not include a frozen core are called all-electron calculations, and they are used much less widely than frozen core methods. Ideally, a pseudopotential is developed by considering an isolated atom of one element, but the resulting pseudopotential can then be used reliably for calculations that place this atom in any chemical environment without further adjustment of the pseudopotential. This desirable property is referred to as the transferability of the pseudopotential. Current DFT codes typically provide a library of pseudopotentials that includes an entry for each (or at least most) elements in the periodic table. 

If the electron density were known at high resolution, the antishielding effects would be represented in the experimental distribution, and the correction in Eq. (10.31a) would be superfluous. However, the experimental resolution is limited, and the frozen-core approximation is used in the X-ray analysis. Thus, for consistency, the Rcore shielding factor should be applied in the conversion of the... 

The suitability of light-atom crystals for charge density analysis can be understood in terms of the relative importance of core electron scattering. As the perturbation of the core electrons by the chemical environment is beyond the reach of practically all experimental studies, the frozen-core approximation is routinely used. It assumes the intensity of the core electron scattering to be invariable, while the valence scattering is affected by the chemical environment, as discussed in chapter... 

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. 

We see from Eq. (3) that in addition to its use of approximate orbitals, Fukui s frontier density is a frozen-orbital approximation to the Fukui function, as indicated by the second term on the right-hand side of Eq. (3) [32]. 

All calculations in Ref. [22] were performed utilizing the Gaussian-98 code [30]. The potential energy scan was performed by means of the Mqller-Plesset perturbation theory up to the fourth order (MP4) in the frozen core approximation. The electronic density distribution was studied within the population analysis scheme based on the natural bond orbitals [31,32], A population analysis was performed for the SCF density and MP4(SDQ) generalized density determined applying the Z-vector concept [33]. 

With the exception of Ligand Field Theory, where the inclusion of atomic spin-orbit coupling is easy, a complete molecular treatment of relativity is difficult although not impossible. The work of Ellis within the Density Functional Theory DVXa framework is notable in this regard [132]. At a less rigorous level, it is relatively straightforward to develop a partial relativistic treatment. The most popular approach is to modify the potential of the core electrons to mimic the potential appropriate to the relativistically treated atom. This represents a specific use of so-called Effective Core Potentials (ECPs). Using ECPs reduces the numbers of electrons to be included explicitly in the calculation and hence reduces the execution time. Relativistic ECPs within the Hartree-Fock approximation [133] are available for all three transition series. A comparable frozen core approximation [134] scheme has been adopted for... 

Systematic analysis of the data with respect to different schemes of density functional calculations given in Table 1 shows clearly that the model core potential (MCP) describing inner electrons in molybdenum gives rise to the results of the same quality as the all-electron calculations. Not only is the use of MCP for heavy atom calculations justified by numerical efficiency but also it may incorporate part of relativistic effects, which may be crucial in describing properties of such atoms. This tendency has been checked and found to be valid also for other states of MoO thus from now on only the frozen core approximation for molybdenum will be presented. 

As was discussed, infrared and raman spectra for organometallic systems can typically be computed to within 5% of the experiment. Unlike adsorption energy predictions, structure and vibrational frequencies are fairly insensitive to differences in the DFT methods (local vs. nonlocal spin density). Even some of the earliest reported local-spin-density approximation (LDA) DFT calculations which ignored adsorbate and surface relaxation predicted frequencies to within 10 percent of the measured values. For example, Ushio et al. have shown that LDA calculations for formate on small Nia clusters (frozen at its bulk atomic positions) provide very good agreement with experimental HREELS studies on Ni(lll) [72]. Unlike adsorption energy predictions, structure and vibrational frequencies are fairly insensitive to gradient-corrections. 

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