Least squares, method inertia

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... 

Let us assume that the rotational spectra of the torsional states v = 0 and v = 1 have been measured and assigned. The information available is sets of line frequencies p, splittings Ap, and an intensity ratio I0/Ix. From the frequencies v of the component lines of the multiplets, frequencies of fictitious unsplit lines are calculated by a weighted mean using formulas resulting from the Hamiltonian of Eq. (4). These lines are used to fit by a least squares method the rotational constants A> B, C or, implicitly, the effective principal moments of inertia / = fl2f2A, etc., of /fa, which are different from Ig [compare Eq. (8)]. Experience shows that for the two rotational spectra, v = 0 and v = 1, two different sets of rotational constants must be used. Here a limitation of the model... 

Partial least square regression a regression method that maximizes the co-inertia of a table of independent and a table of dependent variables. Volume 2(2). 

For further confirmation of the mode-softening and a possible identification of the molecular nature of the over-damped mode, we used the rigid-body motion analysis of the thermal- parameters of the room temperature x-ray diffraction study. A thermal-motion analysis (TMA) program was used to calculate the components of the librational (L) and the translational (T) tensors with a least-square fit of the published thermal parameters ( ) of all nonhydrogen atoms of the molecule. The librational frequencies were calculated by the method of Cruickshank (7), using the appropriate eigenvalues of the L-tensor and the corresponding moments of inertia. 

Several of the procedures for deriving structural parameters from moments of inertia make use of the method of least squares. Since the relation between moments of inertia and Cartesian coordinates or internal coordinates is nonlinear, an iterative least squares procedure must be used.18 In this procedure an initial estimate of the structural parameters is made and derivatives of the n moments of inertia with respect to each of the k coordinates are calculated based on this estimate. These derivatives make up a matrix D with n rows and k columns. We then define a vector X to be the changes in the k coordinates and a vector B to be the differences between the experimental moments and the calculated moments. We also define a weight matrix W to be the inverse of the ma-... 

For monatomic, diatomic, and polyatomic gases, the TDF code may be used to calculate the free energy, enthalpy, and entropy as functions of temperature. The moments of inertia can be calculated for the polyatomic case. For one-Debye Theta and two-Debye Theta solids, the heat capacity is computed as a function of temperature in addition to the free energy, enthalpy, and entropy. These functions are fit to a fourth degree polynomial by the method of least squares. The integration constant, IC, in the equation H° — H° = J T dS/dT) + IC is computed from the fit of entropy as a function of temperature. 

Once the set of L scaled moments of inertia F have been evaluated, the molecular structural parameters are derived by means of a standard least-squares fitting of the F s. This is found to provide the best averaging of small residual vibrational effects. Eor a linear friatomic molecule XYZ, the four moments of inertia would be analyzed for the parameters dYx and d z (see Table II). Importantly, the method employs a minimal set of isotopic substitution data compared to the mass-dependence method. It is, however, necessary to select the parent such that all isotopic substitutions satisfy either Am, > 0 or Am, < 0 for all atoms i. This minimizes residual vibrational effects. For the general case, there are moments 7 and I (a = a,b,c) associated with each axis, and these are used to calculate the corresponding Pa, Pt, Pc and the la, Ic- The moments of inertia 7 are then analyzed by least squares for the structural parameters. Table XXIII compares several structures for OCS. Results for SO2 are summarized in Table XVIII. It is apparent that the structures compare most favorably with the sfructures. Similar results are found for other molecules. 

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