Intramolecular force constants

It would be most desirable to have a direct link between the geometrical relaxations and the force constant changes. From molecular vibrational spectroscopy it is well known that the intramolecular force constants show a scaling behavior, Badger s rule ... 

Intramolecular force constants Intermolecular force constants... 

Both intramolecular force constants are lowered somewhat through complex formation (Table 6). As expected this effect is larger in the proton-donor than in the proton-acceptor molecule. In Table 7 we present calculated and experimental data on the vibrational spectrum of (HF)2. General agreement is obtained. The most remarkable feature is the strict separation of intra- and intermolecular modes on the frequency axis. Hydrogen bond formation is a weak interaction compared to the formation of a chemical bond hence, the normal frequencies are well separated. However, Hartree-Fock calculations of bond stretching force constants... 

Table 3.81 Computed intramolecular force constants (mdyn/A) of the cyclic formamide dimer ...

The sound velocity in a fiber, and the sonic modulus calculated therefrom, are related to molecular orientation (De Vries ). As shown by Moseley ), the sonic modulus is independent of the crystallinity at temperatures well below the T (which means that the inter- and intramolecular force constants controlling fiber stiffness are not measurably different for crystalline and amorphous regions at these temperatures). An orientation parameter a, calculated from the sonic modulus, is therefore taken as a measure for the average orientation of all molecules in the sample, regardless of the degree of crystallinity. The parameter is called the total orientation , as contrasted to crystalline and amorphous orientation, from X-ray data. 

Neutron time-of-flight spectra (2<5) for Nylon-6 are given in Fig. 10. A comparison of NIS and infrared data for Nylon-6 (5, 16, 18, 37), and NIS and infrared spectra 30) for n-heptane is given in Table 3. The vibrations of the -( 112)5- segments in Nylon-6 are expected to occur below 600 cm-, where the intramolecular C-C-C deformation and C-C torsional modes in n-paraffins have been shown to occur (7). In view of the transferability established for the intramolecular force constants among the n-paraffins 33), it was of interest to test the degree to which they could be transferred to the —( 112)5 groups of Nylon-6. 

Table 5. Intramolecular force constants of water molecules in the vapour and in some condensed systems (calculated from the harmonic frequencies We) (see also Eriksson and Lindgren " )...

In conclusion, we present optical evidence for a transition of molecular nitrogen to a nonmolecular state. The transition occurs on compression when the ratio of inter-to-intramolecular force constants reaches 0.1 [37]. This is small compared to the highest ratio reached for hydrogen (in its molecular phase). It suggests that the destabilization of the triple-bonded nitrogen molecule is the driving force of the nonmolecular transition. Vibrational and... 

The antiparallel-chain structure (Figure 5-8) has only a screw axis of symmetry, and its modes are distributed as follows A-62 modes, Raman, IR B-61 modes, Raman, IR. As for PGI, extensive IR spectra on isotopic derivatives (NH, CD2 ND, CH2 and ND, CD2) were available [97] as well as Raman spectra on the N-deuterated molecule [98]. The PGI force field [96] was used as a starting point, and refinement required small adjustments in 10 of 70 intramolecular force constants [123]. Amide mode frequencies are given in Table 5-12 the overall rms frequency error is 5.4 cm" b... 

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

The thirty-two silent modes of Coo have been studied by various techniques [7], the most fruitful being higher-order Raman and infra-red spectroscopy. Because of the molecular nature of solid Cqq, the higher-order spectra are relatively sharp. Thus overtone and combination modes can be resolved, and with the help of a force constant model for the vibrational modes, various observed molecular frequencies can be identified with specific vibrational modes. Using this strategy, the 32 silent intramolecular modes of Ceo have been determined [101, 102]. 

The Raman and infrared spectra for C70 are much more complicated than for Cfio because of the lower symmetry and the large number of Raman-active modes (53) and infrared active modes (31) out of a total of 122 possible vibrational mode frequencies. Nevertheless, well-resolved infrared spectra [88, 103] and Raman spectra have been observed [95, 103, 104]. Using polarization studies and a force constant model calculation [103, 105], an attempt has been made to assign mode symmetries to all the intramolecular modes. Making use of a force constant model based on Ceo and a small perturbation to account for the weakening of the force constants for the belt atoms around the equator, reasonable consistency between the model calculation and the experimentally determined lattice modes [103, 105] has been achieved. 

The pressure dependence of wavenumbers has been investigated theoretically by LD methods on the basis of a Buckingham 6-exp potential. In the studies of Pawley and Mika [140] and Dows [111] the molecules were treated as rigid bodies in order to obtain the external modes as a function of pressure. Kurittu also studied the external and internal modes [141] using his deformable molecule model [116]. The force constants of the intramolecular potential (modified UBFF) were obtained by fitting to the experimental wavenumbers. The results of these studies are in qualitative agreement with the experimental findings. 

A relationship between the intramolecular metal-metal separation and metal-metal force constants (estimated from the metal-metal stretching frequencies) was also determined for the halo complexes as ... 

The final step in the MM analysis is based on the assumption that, with all force constants and potential functions correctly specified in terms of the electronic configuration of the molecule, the nuclear arrangement that minimizes the steric strain corresponds to the observable gas-phase molecular structure. The objective therefore is to minimize the intramolecular potential energy, or steric energy, as a function of the nuclear coordinates. The most popular procedure is by computerized Newton-Raphson minimization. It works on the basis that the vector V/ with elements dVt/dxn the first partial derivatives with respect to cartesian coordinates, vanishes at a minimum point, i.e. = 0. This condition implies zero net force on each atom... 

This way of expressing the overall modes for the pair of molecular units is only approximate, and it assumes that intramolecular coupling exceeds in-termolecular coupling. The frequency difference between the two antisymmetric modes arising in the pair of molecules jointly will depend on both the intra- and intermolecular interaction force constants. Obviously the algebraic details are a bit complicated, but the idea of intermolecular coupling subject to the symmetry restrictions based on the symmetry of the entire unit cell is a simple and powerful one. It is this symmetry-restricted intermolecular correlation of the molecular vibrational modes which causes the correlation field splittings. 

For the normal modes that contribute to vibrational trapping, AQe 0. If there is a series of such modes, X, is given by equation (22) where the summation is over all of the trapping vibrations. For the general case of unequal force constants, the contribution of the intramolecular vibrations to AG is not given by but rather by a more complex expression arising from the energy minimization procedure described in the derivation of equation (16a). 

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