Pseudo-energy

Other forms for the pseudo-energy penalty term have also been investigated (61,62). In any case, pseudo-energy penalty term acts as a constraint on the overall energy of the system, which is a balance between favorable conformational energies and overall molecular alignment as measured by field-based similarity (dissimilarity). 

As described in the Introduction, it is usually possible to consider the modeling of experimental data separately from the scheme actually used to move atoms about. Ideally, the different models should be able to be used in the different minimization or dynamics schemes. Thus, the subsequent sections describe the kind of data offered by NMR and the kinds of penalty functions or pseudo-energy terms that can be used to represent them. For convenience, we use nomenclature common to force field-based approaches where one refers to a distance constraint potential Vdc r) as a function of intemudear distance. 

Rather than simply adjusting a power term, Scarsdale et al.47 noted that the pseudo-energy term was being based on a distance r, but what was measured was more directly related to r 6. With this reasoning, they used a term of the form... 

A more difficult question is the relative balance between the conventional force field and the pseudo-energy terms representing experimental data.81 If... 

When making such a decision, some considerations should be borne in mind. First, if one is using a refinement scheme that produces a known distribution of structures, then one can calculate the likely deviation that the pseudo-energy terms will permit. For example, if an MD refinement is used, structures will be able to cross barriers of about BT. Then, if one is using a quadratic form to enforce distance restraints, one could recast Eq. [9] as follows to get an idea of the violations that would be permitted for a given force constant and temperature ... 

Another important effect of pseudo-energy force constants is controlling the distribution of errors. For example, a misassigned NOE may show up as a residual violation if a small value for Kdc is used, but it may cause a distortion of the structure and high potential energy if Kdc is large. 

The calculation of the energy or the fitting of the test sequence in the fold of the template is no easy matter. The utilization of a full force field with complete atom representation does not properly discriminate between the different folds [31]. This seems to be related to an energy surface that is too fine and the presence of numerous local minima. In its place a potential function based on a statistical analysis of known protein structures has been developed [34], The pair-wise penalty function provides a pseudo-energy based on the number of times the specific interaction has been observed in known protein structures. This function provides amino acid-amino acid interactions as well as a measure for the solvent exposure of each amino acid [34],... 

Results of pyrolysis of propane, n-butane, and n-hexane at a wide range of temperatures and conversions, including the range of commercial operation, are presented. Extensive product inhibition is evident in all cases. The rates of decomposition can be characterized by two pseudo energies of activation E, calculated by comparing data at constant decomposition, variable time and temperature, and E (always less than E) at constant time, variable decomposition and temperature. Both E and E are relatively constant over the conversion range studied. Near atmospheric pressure data fit the equation X = exp —sl0 [1 + 1-3 (Nc — 2)] t1/re — E /RT where X = fraction feed paraffin undecomposed, Nc = carbon number of feed paraffin, r = E/E = 1.68, t = reaction time (seconds), T = reaction temperature (K), a0 = 2.85 109 (from propane data) or 2.66 109 (better to n-butane and n-hexane data), and E = 46.0 kcal/ mol. 

In geometry optimizations or molecular dynamics calculations the NMR data are used as target functions, by defining proper pseudo-energies as a function of measurements and calculated data. From these energies forces have to be derived that are used to drive the system into the desired direction. In the case of chemical shifts these pseudo-energies depend on the calculated values, hence we need to calculate the derivatives of the chemical shifts with respect to the molecular coordinates. 

To account for experimental constraints, in most calculations the following pseudo-energy term is added to the total energy ... 

In MD simulations or geometry optimizations pseudo-forces are needed in addition to the pseudo-energies ... 

These forces which drive the system under investigation into the direction of the minimum pseudo-energy contain derivatives of the theoretical chemical shifts with respect to the coordinates. As pointed out in Section 5, the calculation of chemical shift derivatives is even more time consuming than the calculation of the chemical shifts itself. The calculations should be performed at least on the same theoretical level as the chemical shifts. If theoretical or empirical chemical shift contour maps have been worked out in advance, their derivatives can be calculated numerically. If the contour maps are constructed as a function of the dihedral angles (see Sections 6.3-6.4), only the forces with respect to these inner coordinates are readily obtained. 

To predict proton chemical shifts, methods had been developed using theoretically derived formulae with an empirical parametrization (see Section 4.6). Despite the relatively large scatter of the predictions when plotted versus the experimental values, it was found that the deviations were reduced when calculating the shifts from better refined structures. Proton chemical shifts were thus included as target functions into force field calculations. Kuszewski et included a harmonic pseudo-energy term into XI LOR. " The ll... 

A further consequence of the fluctuations is the opening of a so-called pseudo energy gap. This refers to a strong decrease of the density of states compared to the metallic density of states in the temperature range between Tp and T. This pseudo energy gap is also observed in one-dimensional organic metals above the real phase-transition temperature Tp (see Sect. 9.6). 

Fa)2PF6 crystal. Agff is the effective or pseudo-energy gap for T> Tp. A(T) was determined both from the magnetic susceptibility of the charge carriers (solid curve) and from the conductivity cr(T) (dashed curve). Tp is the transition temperature of the Peierls phase transition. From [24]. 

Table 3. The pseudo-energy value of relocated sites and natural sites.

Besides (6.55), there are still two other conditions imposed on the pseudo-potential the pseudo wave-functions should not have nodal smfaces and the pseudo energy-eigenvalues should match the true valence eigenvalues, i.e.. 

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