Root-mean-squared deviation

The ternary diagrams shown in Figure 22 and the selectivi-ties and distribution coefficients shown in Figure 23 indicate very good correlation of the ternary data with the UNIQUAC equation. More important, however, Table 5 shows calculated and experimental quarternary tie-line compositions for five of Henty s twenty measurements. The root-mean-squared deviations for all twenty measurements show excellent agreement between calculated and predicted quarternary equilibria. 

This subroutine also prints all the experimentally measured points, the estimated true values corresponding to each measured point, and the deviations between experimental and calculated points. Finally, root-mean-squared deviations are printed for the P-T-x-y measurements. 

A typical molecular dynamics simulation comprises an equflibration and a production phase. The former is necessary, as the name imphes, to ensure that the system is in equilibrium before data acquisition starts. It is useful to check the time evolution of several simulation parameters such as temperature (which is directly connected to the kinetic energy), potential energy, total energy, density (when periodic boundary conditions with constant pressure are apphed), and their root-mean-square deviations. Having these and other variables constant at the end of the equilibration phase is the prerequisite for the statistically meaningful sampling of data in the following production phase. 

Example Crippen and Snow reported their success in developing a simplified potential for protein folding. In their model, single points represent amino acids. For the avian pancreatic polypeptide, the native structure is not at a potential minimum. However, a global search found that the most stable potential minimum had only a 1.8 Angstrom root-mean-square deviation from the native structure. 

In order to examine whether this sequence gave a fold similar to the template, the corresponding peptide was synthesized and its structure experimentally determined by NMR methods. The result is shown in Figure 17.15 and compared to the design target whose main chain conformation is identical to that of the Zif 268 template. The folds are remarkably similar even though there are some differences in the loop region between the two p strands. The core of the molecule, which comprises seven hydrophobic side chains, is well-ordered whereas the termini are disordered. The root mean square deviation of the main chain atoms are 2.0 A for residues 3 to 26 and 1.0 A for residues 8 to 26. 

In analytical chemistry one of the most common statistical terms employed is the standard deviation of a population of observations. This is also called the root mean square deviation as it is the square root of the mean of the sum of the squares of the differences between the values and the mean of those values (this is expressed mathematically below) and is of particular value in connection with the normal distribution. 

Figure 3. A comparison of the aC backbone of PelC with PelE. The aCs which superimpose within a root-mean-square deviation of 1.5 A are shown in black and those, within 3.0 A are shown in dark gray. The remaining backbone regions are shown in light The largest structural differences occur in the loops capping one end of the parallel p as well as in those comprising the putative substrate binding groove.

Figure 2-4. Comparison of optimized and X-ray structures for the active site of RNR. The X-ray structure of R2met is superimposed on the optimized structures from active-site QM-only (left) and ONIOM2 0middle) models. The plot shows the quality of the optimizations evaluated as the root-mean-square deviations (in A) compared to the X-ray structures of RNR and MMO (right). (Adapted from Torrent et al. [24]. Reprinted with permission. Copyright 2002 Wiley Periodicals, Inc.)...

In all the studied systems addition of the surrounding protein in an ONIOM model clearly improves the calculated active-site geometries. This is clearly illustrated in Figure 2-13, which shows the root-mean-square deviation between calculated and experimental structures for four of the studied enzymes. 

Figure 13-6. The root mean square deviation (A) of the backbone of Orf2 over time (ps)...

H20, CH3OH-. . H20, CN-. . HzO, HCC-. . H20, HCOCT. . H20. The DFT(B3LYP) and the DFT(BLYP) results were in a fair agreement with the MP2 results. The root mean square deviation of the DFT and the MP2 complexation enthalpies amounted to 0.7 and 1.1 kcal/mol, for B3LYP and BLYP, respectively. From the basis set dependence of the DFT results, it was concluded that the nonlocal DFT calculations require diffuse atomic functions. 

To compute the results shown in Tables 34-3 and 34-4, the precision of each set of replicates for each sample, method, and location are individually calculated using the root mean square deviation equation as shown (Equations 34-1 and 34-2) in standard symbolic and MathCad notation, respectively. Thus the standard deviation of each set of sample replicates yields an estimate of the precision for each sample, for each method, and for each location. The precision is calculated where each ytj is an individual replicate (/ ) measurement for the ith sample yt is the average of the replicate measurements for the ith sample, for each method, at each location and N is the number of replicates for each sample, method, and location. The results of these computations for these data... 

The analytical results for each sample can again be pooled into a table of precision and accuracy estimates for all values reported for any individual sample. The pooled results for Tables 34-7 and 34-8 are calculated using equations 34-1 and 34-2 where precision is the root mean square deviation of all replicate analyses for any particular sample, and where accuracy is determined as the root mean square deviation between individual results and the Grand Mean of all the individual sample results (Table 34-7) or as the root mean square deviation between individual results and the True (Spiked) value for all the individual sample results (Table 34-8). The use of spiked samples allows a better comparison of precision to accuracy, as the spiked samples include the effects of systematic errors, whereas use of the Grand Mean averages the systematic errors across methods and shifts the apparent true value to include the systematic error. Table 34-8 yields a better estimate of the true precision and accuracy for the methods tested. 

Tables 1, 2, and 3 present a set of five alcohols. In Table 1, it should be noted that while MM3(96) calculates the magnitude of the dipole moment to be essentially the same for the entire set of molecules, MM3(2000) is superior in reproducing the experimental dipole moments. This is demonstrated by comparing the root mean squared deviation of 0.0878 Debye in MM3 to the 0.0524 Debye deviation in MM3(2000). (All of the experimental values except where notes are stark effect measurements determined from microwave spectra.)...

It is known that ab initio methods are not accurate in reproducing or predicting molecular dipole moments. For example, a typical basis set minimization with no additional keywords was carried out, and the results show that the computed magnitude of the dipole moment is not particularly accurate when compared with experimental values. For alcohols, MP2 has a root mean squared deviation of 0.146 Debye, while HF had a deviation of 0.0734 Debye when measured against the experimental values. 

The differences between ab initio and molecular mechanics generated dipole moments were discussed. The MM3(2000) force field is better able to reproduce experimental dipole moments for a set of forty-four molecules with a root mean squared deviation (rmsd) of 0.145 Debye compared with Hartree-Fock (rmsd 0.236 Debye), M0ller-Plesset 2 (rmsd 0.263 Debye) or MM3(96) force field (rmsd 0.164 Debye). The orientation of the dipole moment shows that all methods give comparable angle measurements with only small differences for the most part. This consistency within methods is important information and encouraging since the direction of the dipole moment cannot be measured experimentally. 

Each abundance was divided by the abundance of that element (except for Rh) in Type / carbonaceous chondrites. Rh abundances were divided by Rh abundances in other types of chondrites as Cl values were not available. Errors in the LBL measurements reflect 1 a values of the counting errors, except for the Au error. The latter is the root-mean-square deviation of six measurements, because the six values were not consistent within counting errors. The Os measurement was on a HNO,-insoluble residue that had been fired to 800°C. Key , this work and O, previous work of Ganapathy. 

The Tb abundance in meteorites is assumed to be 0.5 ppm. Errors for the Danish Cretaceous (3 samples) and Tertiary (3 samples) HNO,-insoluble residues are root-mean-square deviations. Errors for the HNO,-insoluble residues from the Gubbio and Danish boundary layers are 1 a values of the counting errors. Key , Gubbio boundary layer residue O, Danish Cretaceous residues < >, Danish Tertiary residues and , Danish... 

There is considerable similarity between domains 1, 3, and 5 with excellent matching of the (3-strands 168 a-carbon atoms of domain 3 and 149 atoms of domain 5 can be superimposed on domain 1 with a root mean square fit of only 1.0 A. Major deviations occur in the loops between the first and second, and the fourth and sixth strands. The similarity of the smaller mononuclear copper binding domains, 2,4, and 6, is even more pronounced with 147 atoms from domain 4 and 143 from domain 6 fitting domain 2 with a root mean square deviation of only 0.9 A in each case. However, although all 6 domains are based on a cupre-doxin-type fold, the various loop regions deteriorate the match between an even and an odd domain with a typical fit of 1.8 A for only 91 atoms when domain 2 is superposed onto domain 1. The superposition results are summarized as part of Table 2. 

From a structural point of view the OPLS results for liquids have also shown to be in accord with available experimental data, including vibrational spectroscopy and diffraction data on, for Instance, formamide, dimethylformamide, methanol, ethanol, 1-propanol, 2-methyl-2-propanol, methane, ethane and neopentane. The hydrogen bonding in alcohols, thiols and amides is well represented by the OPLS potential functions. The average root-mean-square deviation from the X-ray structures of the crystals for four cyclic hexapeptides and a cyclic pentapeptide optimized with the OPLS/AMBER model, was only 0.17 A for the atomic positions and 3% for the unit cell volumes. 

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