Force constants, in normal coordinate analysis

Evidently, the values of force constants depend on the force field initially assumed. Thus, a comparison of force constants between molecules should not be made unless they are obtained by using the same force field. The normal coordinate analysis developed in Secs. 1.12-1.14 has already been applied to a number of molecules of various structures. Appendix Vll lists the G and F matrix elements for typical molecules. 

For polyatomic molecules, the stretching force constant for a particular bond cannot in general be obtained in an unambiguous manner because any given vibrational mode generally involves movements of more than two of the atoms, which prevent the expression of the observed frequency in terms of the force constant for just one bond. The vibrational modes of a polyatomic molecule can be analyzed by a method known a normal coordinate analysis to... 

With the aid of a normal coordinate analysis involving different isotopomers a linear structure of the Pd-Si-0 molecule is deduced. The results of ab initio MP2 calculations (Tab. 4) confirm the experimentally obtained IR spectra and their interpretation. The Pd-C bond in PdCO is similar to the Pd-Si bond in PdSiO which means, that the donor bond is strengthened by x acceptor components. This conclusion is in line with the high value of the Pd-Si force constant (exp. f(PdSi) = 2.69, f(SiO) = 8.92 mdyn/A) as well as with the energy of PdSiO (Pd + SiO —> PdSiO + 182 kJ/mol for comparison Pd + CO —> PdCO + 162 kJ/mol, MP2 level of theory). 

DFT calculations were performed on Mo dinitrogen, hydra-zido(2-) and hydrazidium complexes. The calculations are based on available X-ray crystal structures, simplifying the phosphine ligands by PH3 groups. Vibrational spectroscopic data were then evaluated with a quantum chemistry-assisted normal coordinate analysis (QCA-NCA) which involves calculation of the / matrix by DFT and subsequent fitting of important force constants to match selected experimentally observed frequencies, in particular v(NN), v(MN), and 8(MNN) (M = Mo, W). Furthermore time-dependent (TD-) DFT was employed to calculate electronic transitions, which were then compared to experimental UVATs absorption spectra (16). As a result, a close check of the quality of the quantum chemical calculations was obtained. This allowed us to employ these calculations as well as to understand the chemical reactivity of the intermediates of N2 fixation (cf. Section III). 

The N-N and metal-N force constants resulting from normal coordinate analysis of the N2-, NNH-, NNH2-, and NNH3-complexes l ppe, 2 ppeF, 34ppeF, and 4 epeF as well as the nitrido and imido complexes 5, Ns and 6 pe C1 are graphically represented in Fig. 4 (15-17). Upon protonation of l pe to the NNH complex 2 peF, the N-N force constant decreases from... 

The MP2/TZDP optimized structures were then used to calculate the stationary state geometry force constants and harmonic vibrational frequencies, also at the MP2 level. These results serve several purposes. Firstly, they test that the calculated geometry is really an energy minimum by showing all real frequencies in the normal coordinate analysis. Secondly, they provide values of the zero-point energy (ZPE) that can be used... 

The normal coordinate analysis also allowed the authors to quantify the relative weakness of the metal-carbon force constant in comparison with other metal-ethylene complexes (0.33 vs 0.58 mdyneA ) . 

In mass-weighted coordinates, the hessian matrix becomes the harmonic force constant matrix, from which a normal coordinate analysis may be carried out to yield harmonic frequencies and normal modes, essentially a prediction of the fundamental IR transition... 

A Young s modulus of 380 GPa at 0 K is found for the chain under these assumptions. Inasmuch as the force constants used in McCullough et al were derived by Lifson and Warshel from normal coordinate analysis of infra-red spectra and in this derivation the C-C bond was not assumed to be rigid the use of Lifson and Warshel s force constants in the way in which McCullough et al. use them would seem to be questionable. 

Molecular Cl(H)Si=S (126) was also formed in an argon matrix in a photochemically induced reaction of SiS with HC1. From the isotopic splittings (H/D and 35C1/37C1) of the IR absorptions the Cs structure of the species with silicon as the central atom is deduced. By a normal coordinate analysis a value of 4.83 mdynA-1 is obtained for the SiS force constant, a value which was confirmed by ab initio SCF calculations of the IR spectrum51. 

Raman and UV-visible spectroscopy, but no precise characterization was made. A report was made in 1981 where the IR spectrum of Cu atoms deposited with C02 at 80 K was interpreted in terms of the formation of a -coordinated complex between C02 and zerovalent copper [32]. Almond et al. [33] prepared a (C02) M(CO)5 molecule (M = Cr, W), that led to the formation of CO and oxometal carbonyl under UV irradiation. The first complete study of the reactivity of C02 with the first row of transition metals was made by Mascetti et al. [34, 35]. Here, it was shown that the late transition metal atoms (Fe, Co, Ni, and Cu) formed one-to-one M(C02) complexes, where C02 was bonded in a side-on (Ni), end-on (Cu), or C-coordinated (Fe, Co) manner, while the earlier metal atoms (Ti, V, and Cr) spontaneously inserted into a CO bond to yield oxocarbonyl species OM(CO) or 0M(C0)(C02). Normal coordinate analysis showed that the force constants of CO bonds were significantly decreased by 50%, compared to free C02, and that the OCO angle was bent between 120 and 150°. 

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