Least squares method, molecular

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

Reflection intensity in the SAED negatives was measured with a microdensitometer. The refinement of the structure analysis was performed by the least square method over the intensity data (25 reflections) thus obtained. A PPX single-crystal is a mosaic crystal which gives an "N-pattem". Therefore we used the 1/d hko as the Lorentz correction factor [28], where d hko is the (hkO) spacing of the crystal. In this case, the reliability factor R was 31%, and the isotropic temperature factor B was 0.076nm. The molecular conformation of the P-form took after that of the P-form since R was minimized with this conformation benzene rings are perpendicular to the trans-zigzag plane of -CH2-CH2-. 

XPLOR is a modern package of refinement programs that includes powerful procedures for energy refinement by simulated annealing, in addition to more traditional tools like least-squares methods and molecular-replacement searches. The package is available for use on many different computer systems. Simulated annealing for large molecules usually requires supercomputers. 

In 1961 a molecular model of a (7/2) helix [Figure 6(a)] was proposed by the author and his coworkers (13.) based on the information from x-ray, infrared, and Raman spectroscopy, but the crystal structure could not be determined at that time. After ten years, owing to the development of methods and apparatus, especially the constrained least-squares method and a vacuum cylindrical camera with a radius of 10 cm, the crystal structure has been determined as shown in Figure 6(c) (22.). The internal rotation angles are considerably distorted from the uniform helix, although the molecular conformation is essentially the (7/2) helix and close to the TTG sequences. 

There has been no controversy about the structure of fluorene (31) but its true conformation was in doubt for a number of years. From an early X-ray analysis, Iball (1936a) concluded that the fluorene molecule had a folded conformation and, in a review, Cook and Iball (1936) discussed further evidence for a non-planar conformation, provided by optical activity studies of unsymmetrically substituted fluorene derivatives. Later stereochemical studies (Weisburger et al., 1950) suggested that fluorene had, in fact, a planar conformation. A reinvestigation of the crystal structure by Burns and Iball (1954, 1955) and, independently, by Brown and Bortner (1954) showed that the early X-ray work was in error and confirmed the planar conformation. The refinement of the crystal structure (Burns and Iball, 1954, 1955), by two-dimensional Fourier and least-squares methods, reveals that the maximum deviation of the carbon atoms from the mean molecular plane is 0-030 A, the r.m.s. deviation being 0-017 A. This deviation, 0-017 A, is taken by Burns and Iball to be a measure of the accuracy of their analysis, assuming now that the molecule is strictly planar. 

The structure of the closely related molecule, 1,2-cyclopentenophen-anthrene, has been determined and refined with partial three-dimensional data by least-squares methods by Entwhistle and Iball (1961). Independent confirmation of the correctness of this structure has been provided by Basak and Basak (1959) who did not, however, carry out any refinement of the structure. Entwhistle and Iball s results show that the molecule is not planar the deviations of the carbon atoms from the mean molecular plane are shown in Fig. 9 (the standard deviations of the atomic coordinates lie between 0-009 and 0-015 A). The three aromatic rings appear to be linked in a slightly twisted arrangement. Atoms H and K, which are bonded to the overcrowded hydrogen atoms, are displaced almost the same distance on opposite sides of the mean plane. In the five-membered ring, atoms C and E are below the molecular plane by about 0-10 A while atom D lies 0-18 A... 

Very recently, new descriptors have successfully been derived from 3-D molecular fields. These descriptors were correlated with the experimental permeation data by discriminant partial least-squares methods. The training set consisted of 44 compounds. The authors were able to deduce a simple mathematical model that allows external prediction. More than 90% of blood-brain permeation data were correctly predicted [77]. 

According to a complete X-ray diffraction analysis, Se6 consists of ring molecules with the molecular symmetry of Dzd the crystal and molecular parameters are listed in Table II (17) and the crystal structure is shown in Fig. 2. Refinement by the least squares method resulted in the following atomic parameters of the single atom in the asymmetric unit x = 0.1602 0.00048, y = 0.20227 0.00047, z = 0.12045 0.00120 calculated density, 4.71 g/cm3. An earlier investigation of selenium vapor by electron diffraction led to an internuclear distance of 234 1 pm and an average bond angle of 102 0.5° for the chairlike cyclic Se6 molecule (23). 

Fig. 23. Molecular weight dependence of the mean-square weight average radius of gyration in the unperturbed state relative to Mw (

In MoQSAR, the internal nodes include the sum, quadratic and cubic power operators and the terminal nodes consist of the molecular descriptors available for the dataset. A chromosome is translated into a QSAR in two steps (1) the expression encoded in a chromosome is extracted to determine the descriptors that will be used in the QSAR model (2) optimum values for the coefficients and the intercept are calculated using the least-squares method. 

It is very important to obtain accurately the first 2 constants (Ci and C2 or Ki and K ) from the experimental data in order to compare them with those calculated from the molecular-statistical theory. Here, Cl and C2 constants were determined from the experimental isotherm by the least squares method with the aid of a computer. However, for practical purposes, it is necessary to describe the isotherms to a higher adsorption level. From Figures 3 and 6, it can be seen that in order to reproduce the isotherm up to — 75% saturation of the adsorbent and to calculate a at different values of p and T, it is sufficient to determine only 2 more constants, C3 and C4. 

Kim, K.H. and Martin, Y.C. (1991c). Evaluation of Electrostatic and Steric Descriptors for 3D-QSAR the H and CH3 Probes Using Comparative Molecular Field Analysis (CoMFA) and the Modified Partial Least Squares Method. In QSAR Rational Approaches to the Design of Bioactive Compounds (Silipo, C. and Vittoria, A., eds.), Elsevier, Amsterdam (The Netherlands), pp. 151-154. 

Frequently, with smaller and well-diffracting structures (W , < 700 and dp > 10), all atoms of the structure can be written out as the initial model by the program and they just have to be named correctly (as in Fig. 9.11) and refined. The refinement process (see Refs. 110 —1121) uses incremental movement of the atom coordinates and atomic displacement parameters (commonly called as thermal parameters ) of the structure solution model using the so-called least-squares method. The model (the calculated structure factors) is fitted against the measured data (the observed structure factors) and the R-factor (see above. Section 9.2.1) is calculated. With larger structures or if the unit cell contains light atom solvent molecules (C, H, O, N atoms only), some atoms, sometimes even 50% of all atoms, cannot be located from the first very crude electron density map (calculated from the ab initio phase set). However, those atoms which are chemically feasible (based on the proposed molecular structure) can be fed into the calculation of the calculated structure factors Peak ( cafc will approach Fobs when a more accurate model is... 

To calculate the contributions of all molecular groups to retention (adsorption equilibrium constant) ai, ARmi under some standard conditions, it is necessary to determine the retention volumes Vm (Va) or capacity factor k or Rm values from Rp data for a number of compounds possessing different amounts of such groups and to solve the systems of linear equations (2 or 3 or 5) by the least -squares method. 

The molecular term jIm, or sometimes the ratio j(Im/Ib) = sM(s), is analyzed by a least-squares method and the bond distances rg, bond angles and some of the amplitudes of vibration are determined (see more about this in the next section). The rg distance, sometimes called an operational parameter, is an ill-defined parameter since it refers to the maximum position of any peak of the P(r)/r function. The better parameter, the Vg distance. 

Molecular structures determined directly from the observed groimd state moments of inertia, /, are called tq structures. They depend strongly on the specific set of isotopomer data used because the isotopic dqiendence of the vibration-rotation interaction terms is different from the dependence of the moments of inertia. This quickly leads to contradictions in the results from different isotopic species. Alternately, data from many isotopic molecules can be averaged by the least-squares method. The results still depend somewhat on the isotopic data set and they have low relative precision typically in the range of 1%. 

Least-squares methods have been used to determine molecular structures successfiilly first by Nosberger et al. [31], Schwendeman [28], and Typke [32]. Nosberger et al. fitted structural parameters (internal coordinates) to isotopic differences of moments of inertia. To solve the normal equations, they used the singular value decomposition of real matrices to calculate the pseudo-inverse of such matrices with the option to omit nearzero singular values in illdetermined systems. Schwendeman [28] fitted internal coordinates to moments of inertia or isotopic differences of these... 

A systematic presentation of least-squares methods applied to the determination of molecular structures has been published by Rudolph [33]. 

A much more detailed account of the particular problems of least-squares methods in the determination of molecular structures has been given by Rudolph [35]. In this review, Rudolph calls all methods mentioned in the preceding paragraph tq or /"o-derived methods. He reserves the term method strictly to least-squares methods based on the equations of Kraitchman [15], Chutjian [16] and Nygaard [17] and their extensions to multiply substituted species [32,35,21] despite the fact that the p-Kr methods A/> He result in structures much closer to the traditional... 

In the columns identifying the experimental method, MW stands for any method studying the pure rotational spectrum of a molecule except for rotational Raman spectroscopy marked by the rot. Raman entry. FUR stands for Fourier transform infhired spectroscopy, IR laser for any infiured laser system (diode laser, difference frequency laser or other). LIF indicates laser induced fluorescence usually in the visible or ultraviolet region of the spectrum, joint marks a few selected cases where spectroscopic and diffraction data were used to determine the molecular structure. A method enclosed in parentheses means that the structure has been derived from data that were collected by this method in earlier publications. The type of structure determined is shown by the symbols identifying the various methods discussed in section II. V/ refers to determinations using the Kraitchman/Chutjian expressions or least squares methods fitting only isotopic differences of principal or planar moments (with or without first... 

Fig. 3.7. Frequency dependency of NTj of the CH2 (rrr) carbon for PMMA/CDCI3 solution at 50°C. The curves are the results obtained by the least-squares method using different models for the molecular motion. (Reproduced from Ref. 10 with permission.)...

In general, there is an art and a science to molecular mechanics parameterization. On one extreme, least-squares methods can be used to optimize the parameters to best fit the available data set, and reviews on this topic are avail-able. Alternatively, parameters can be determined on a trial-and-error basis. The situation in either case is far from straightforward because the data usually available come from a variety of sources, are measured by different kinds of experiments in different units, and have relative importances that need subjective assessment. Therefore, straightforward applications of least-squares methods are not expected to give optimum results. 

Carlson, C.M., Gilkeson, J., Linderman, K., LeBlanc, S., and Hefferlin, R. 1997. Global Forecasting of Data Using Least-Squares Methods and Molecular Databases A Feasibility Study Using Triatomic Molecules. Croatica Chemica Acta 70 479-508. 

The ease with which a molecular model can be fitted to the observed density depends on the resolution at which the map is calculated (compare figures 2.1(c), 2.1 (d), 10.4 and 10.5) and its quality. The resolution limit of the calculation is set by the phase determination method. For the method of isomorphous replacement, phasing is successful usually to =2.5-3.0 A. In the case where a related structure is already known the method of molecular replacement (Rossmann (1972)) can be used whereby rotation and translation matrices are determined and then calculated phases used. Clearly, these two procedures are both approximate methods. The model is usually improved by using least-squares methods of refinement (for a collection of papers see Machin, Campbell and Elder (1980)) and higher resolution data (better than 2 A or so). Refinement methods involve the determination of shifts to the atomic parameters (coordinates and thermal parameters) so as to agree better with the observed diffraction data whilst preserving the known stereochemical features of proteins and nucleic acids. This is achieved by minimising a composite observational function ... 

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