Mean-field approximation reliability

In order to perform the calculation., of the conductivity shown here we first performed a calculation of the electronic structure of the material using first-principles techniques. The problem of many electrons interacting with each other was treated in a mean field approximation using the Local Spin Density Approximation (LSDA) which has been shown to be quite accurate for determining electronic densities and interatomic distances and forces. It is also known to reliably describe the magnetic structure of transition metal systems. 

The mean field approximation captures some essence of particle interactions (this is what sets it apart from the ideal gas model) but it neglects correlations by its assumption that each particle interacts with an unperturbed equilibrium distribution of particles. The accuracy of the mean field depends on the strength of correlations [9], Among theoretical models for which the mean field is considered exact are hard-spheres in the infinite dimension [10], Another example are particles with the pair interaction U(r) = X u Xr) taken to the limit A 0, and where the function u(r) is bounded and of finite range [11], For experimentally relevant systems A remains finite, and so the corrections to the mean field are always present and real. An example of a system regarded as weakly correlated is the Gaussian core model at room temperature. For this model and these conditions the mean field approximation is a reliable theoretical description [12]. 

As has been shown in Section 3.4, COSMO-RS provides access to the thermodynamic parameters of solvents as well as of solute molecules. It provides the knowledge of the chemical potential of any compound in any fluid. Thus COSMO-RS enables the handling of almost any partition problem of compounds between liquid and gaseous phases. In this context it is important that COSMO-RS handles multicomponent mixtures without additional complications, because it does not make any mean-field approximation. Therefore multicomponent mixtures are described almost as reliably as binary mixtures or pure liquids. 

The asymmetric spin boson model presents a significantly more challenging non-adiabatic condensed phase test problem due to the asymmetry in forces from the different surfaces. Approximate mean field methods, for example, will fail to reliably capture the effects of these different forces on the dynamics. 

To study the range of possibilities the first molecular dynamics simulations of a DNA duplex tethered to a surface was performed [35,36] The technical aspects of simulations near surfaces are nontrivial, especially as concerns reliable boundary conditions [37], Molecular dynamics provides a more quantitative picture of the salt gradients and DNA structures near the surface responsible for changes in hybridization affinities and specificities than approximate (PB level mean field) theory and so may be used as a check on the simple analytical picture derived above. In addition simulation provides mechanistic clues which can form additional hypotheses for testing. 

Most band-stmcture calculations in soHd-state physics do not employ any of these techniques (which are computationally very expensive for solids), but are actually calculations of the Kohn Sham eigenvalues e,-. This simplification has proved enormously successful, but one must be aware of the fact that it takes the auxiliary single-body equation (62) literally as an approximation to the many-body Schrodinger equation. DFT, practised in this mode, is not a rigorous many-body theory anymore, but a mean-field theory (albeit one with a very sophisticated mean field Ws(r)). A more reliable alternative is provided by the GW method, which in principle is independent of DFT, but in practice normally takes DFT results as a starting point. ... 

So if this all sounds a bit bleak, what s the good news Well, strangely, there is quite a lot. For a start, let s not forget that had the 13C nucleus been the predominant carbon isotope, the development of the whole NMR technique itself would have been held back massively and possibly even totally overlooked as proton spectra would have been too complex to interpret. Whimsical speculation aside, chemical shift prediction is far more reliable for 13C than it is for proton NMR and there are chemical shift databases available to help you that are actually very useful (see Chapter 14). This is because 13 C shifts are less prone to the effects of molecular anisotropy than proton shifts as carbon atoms are more internal to a molecule than the protons and also because as the carbon chemical shifts are spread across approximately 200 ppm of the field (as opposed to the approx. 13 ppm of the proton spectrum), the effects are proportionately less dramatic. This large range of chemical shifts also means that it is relatively unlikely that two 13C nuclei are exactly coincident, though it does happen. 

In the above paragraphs, we have already introduced several approximations in the description of the shift and relaxation rates in transition metals, the most severe being the introduction of the three densities of states Dsp E ),Dt2g(E ), and Deg E ). The advantage is that these values can be supplied by band structure calculations and that the J-like hyperfine field can sometimes be found from experiment. We have no reliable means to calculate the effective Stoner factors ai that appear in Eq. (2), and the disenhancement factors ki in the expression for the relaxation rate, Eq. (4), are also unknown. It is often assumed that k/ can be calculated from some /-independent function of the Stoner parameter k (x), thus k/ = k((X/). A few models exist to derive the relation k((x), all of them for simple metals [62-65]. For want of something better they have sometimes been applied to transition metals as well [66-69]. We have used the Shaw-Warren result [64], which can be fitted to a simple polynomial in rx. There is little fundamental justification for doing so, but it leads to a satisfactory description of, e.g., the data for bulk Pt and Pd. 

Table 1 contains some further information useful to characterize the different contributions to the molecule/surface interaction orientation dependence and the typical strength of the different contributions, and whether or not they can be understood on a purely classical basis. If one wants to calculate molecule/surface interactions by means of quantum-mechanical or quantum-chemical methods, the most important question is whether standard density functional (DPT) or Hartree-Fock theory (self consistent field, SCF) is sufficient for a correct and reliable description. Table 1 shows that all contributions except the Van der Waals interaction can be obtained both by DPT and SCF methods. However, the results might be connected with rather large errors. One famous example is that the dipole moment of the CO molecule has the wrong sign in the SCF approximation, with the consequence that SCF might yield a wrong orientation of CO on an oxide surface (see also below). In such cases, the use of post Hartree-Fock methods or improved functionals is compulsory. 

Finally, some types of experiments require the detection of the FISH signals to be quantitative. Several laboratories have shown that wide-field deconvolution microscopy can detect signals over approximately 2.5-3 orders of magnitude in intensity with excellent linearity. This means that both weak and intense hybridization signals can be reliably detected and quantified in the same sample volume. 

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