Functional models, local

At the beginning of a project, the model system must be determined. Oligomers can be used to model properties that are a function of local regions of the chain only. Simulations of a single polymer strand can be used to determine the tendency to fold in various manners and to hnd mean end-to-end distances and other properties generally considered the properties of a single mol-... 

Takesue [takes87] defines the energy of an ERCA as a conserved quantity that is both additive and propagative. As we have seen above, the additivity requirement merely stipulates that the energy must be written as a sum (over all sites) of identical functions of local variables. The requirement that the energy must also be propagative is introduced to prevent the presence of local conservation laws. If rules with local conservation laws spawn information barriers, a statistical mechanical description of the system clearly cannot be realized in this case. ERCA that are candidate thermodynamic models therefore require the existence of additive conserved quantities with no local conservations laws. A total of seven such ERCA rules qualify. ... 

Halls, M. D., Schlegel, H. B., 1998, Comparison of the Performance of Local, Gradient-Corrected, and Hybrid Density Functional Models in Predicting Infrared Intensities , J. Chem. Phys., 109, 10587. 

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

Recent progress in protein dynamics studies by NMR was greatly facilitated by the invention of the model-free formalism [28, 32]. In this approach, the local dynamics of a protein are characterized by an order parameter, S, measuring the amplitude of local motion on a scale from 0 to 1, and the correlation time of the motion, T oc. The model-free expression for the correlation function of local motion reads... 

Calculated equilibrium geometries for hydrogen and main-group hydrides containing one and two heavy (non-hydrogen) atoms are provided in Appendix A5 (Tables A5-1 and A5-10 for molecular mechanics models, A5-2 and A5-11 for Hartree-Fock models, A5-3 and A5-12 for local density models, A5-4 to A5-7 and A5-13 to A5-16 for BP, BLYP, EDFl and B3LYP density functional models, A5-8 and A5-17 for MP2 models and A5-9 and A5-18 for MNDO, AMI and PM3 semi-empirical models). Mean absolute errors in bond lengths are provided in Tables 5-1 and 5-2 for one and two-heavy-atom systems, respectively. 

Calculated heavy-atom bond distances in molecules with three or more first and/or second-row atoms are tabulated in Appendix A5 molecular mechanics models (Table A5-21), Hartree-Fock models (Table A5-22), local density models (Table A5-23), BP, BLYP, EDFl and B3LYP density functional models (Tables A5-24 to A5-27), MP2 models (Table A5-28), and MNDO, AMI and PM3 semi-empirical models (Table A5-29). Results for STO-3G, 3-21G, 6-31G and 6-311+G basis sets are provided for Hartree-Fock models, but as in previous comparisons, only 6-3IG and 6-311+G basis sets are employed for local density, density functional and MP2 models. 

None of the semi-empirical models perform as well as Hartree-Fock models (except STO-3G), local density models, density functional models or MP2 models. PM3 provides the best overall description, although on the basis of mean absolute errors alone, all three models perform to an acceptable standard. Given the large difference in cost of application, semi-empirical models clearly have a role to play in structure determination. 

Results from local density models and BP, BLYP and EDF 1 density functional models are, broadly speaking, comparable to those from 6-3IG models, consistent with similarity in mean absolute errors. As with bond length comparisons, BLYP models stand out as inferior to the other non-local models. Both B3LYP/6-31G and MP2/6-31G models provide superior results, and either would appear to be a suitable choice where improved quality is required. 

Consistent with earlier remarks made for bond length comparisons, little if any improvement results in moving from the 6-3IG to the 6-311+G basis set for Hartree-Fock, local density and density functional models, but significant improvement results for MP2 models. 

Comparative data for heavy-atom bond lengths and skeletal bond angles for molecules incorporating one or more third or fourth-row, main-group elements are provided in Appendix A5 Table A5-39 for Hartree-Fock models with STO-3G, 3-2IG and 6-3IG basis sets. Table A5-40 for the local density model, BP, BLYP, EDFl andB3LYP density functional models and the MP2 model, all with the 6-3IG basis set, and in Table A5-41 for MNDO, AMI and PM3 semi-empirical models. 6-31G, local density, density functional and MP2 calculations have been restricted to molecules with third-row elements only. Also, molecular mechanics models have been excluded from the comparison. A summary of errors in bond distances is provided in Table 5-8. 

All density functional models (including the local density model) and the MP2/6-31G model perform admirably in describing the structures of these compounds. In terms of mean absolute errors, the local density model fares best and the BLYP model fares worse. The former observation is consistent with the favorable performance of Hartree-Fock models for these systems and of the previously noted parallels in structural results for Hartree-Fock and local density models. Figures 5-32 to 5-37 provide an overview. 

Density functional models provide a much better account. The local density model does the poorest and BP and B3LYP models do the best, but the differences are not great. As with metal-carbon (carbon monoxide) lengths, bond distances from all-electron 6-3IG calculations are usually (but not always) shorter than those obtained... 

As with metal-carbon monoxide bonds, the MP2/6-3IG model does not lead to results of the same calibre as those from density functional models (except local density models). The model actually shows the opposite behavior as 6-3IG, in that bond lengths are consistently shorter than experimental values, sometimes significantly so. In view of its poor performance and the considerable cost of MP2 models (relative to density functional models), there seems little reason to employ them for structural investigations on organometallics. 

Comparative data for a few particularly interesting systems is provided in Table 5-15. STO-3G, 3-21G and 6-3IG Hartree-Fock models, local density models, BP, BLYP, EDFl and B3LYP density functional models all with the 6-3IG basis set, the MP2/6-31G model and MNDO,AMl andPM3 semi-empirical models have been examined. 

Triplet methylene is known to be bent with a bond angle of approximately 136°. This is closely reproduced by all Hartree-Fock models (except for STO-3G which yields a bond angle approximately 10° too small), as well as local density models, BP, BLYP, EDFl and B3LYP density functional models and MP2 models. Semi-empirical models also suggest a bent structure, but with an HCH angle which is much too large. 

There is a very wide variation in the quality of results from the different models. MNDO and AMI semi-empirical models, the ST0-3G model and both local density models are completely unsatisfactory. The 3-2IG model, all density functional models with the 6-3IG basis set and the PM3 model fare better, while 6-3IG ... 

Density functional models and MP2 models show more consistent behavior. With the 6-311+G basis set, calculated basicities are generally very close to experimental values. The corresponding results with the 6-3IG basis set are generally not as good, although the differences are not that great. In terms of mean absolute errors, local density models perform the worst, and B3LYP/6-311+G and MP2/ 6-311+G models perform the best. 

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