Force constants of benzene

Using the computed VB diagrams in Figure 15, it is apparent that the jr-force constant of benzene is negative in the ground state and positive in the twin excited state, eq 15. 

The valence and symmetry force constants of benzene calculated using density functional theory were first reported by us [10c,d]. These results are summarized in this section. We discuss the vibrational frequencies (Table 5), isotopic shifts, and absorption intensities (Table 6). Selected force constants in symmetry-coordinate representations are listed and compared to the fields due to the Pulay [10b] et al. as well as OG [10a] in Table 7. 

As the comparison of theoretical and experimental force constants of benzene shows, it is a very difficult task to determine complete force fields, even for highly symmetrical medium-size molecules. We are primarily interested in the determination of force fields, for substantially larger molecules, like the transition metal complexes of benzene for example. For these larger systems, empirical determination of the force field is impractical, and the theoretical determination of force constants based on density functional theory seems to be a viable method. This application will be discussed in the next sections. 

Table 7. Disputed symmetry-force constants of benzene. Comparison with the benchmark empirical- and scaled ab initio fields ...

BzCr(CO)3. The skeletal internal coordinates, e.g. ligand-metal stretching, ligand tilting, etc., are based on our previous recommendations [59]. The significant force constants of benzene, Bz2Cr, BzCr(CO)3 and Cr(CO)6 are compared in Tables 15-17. 

The empirical data concerning the force constants for BzjCr is even more limited. Therefore, we do not make a quantitative comparison with the empirical force constants of Bz2Cr. However, our data clearly support the qualitative observation by Snyder [52] who reported, that CC-stretching force constants of benzene change considerably upon coordination and the CH-out-of-plane constants are close to those of ferrocene. 

Another trend which is apparent from the data is that the final geometry is obtained as a fine balance between the a-resistance to distortion and the T-dis-tortivity. Thus, hexagonal benzene is a D h hexagon because its (/-resistance exceeds the. T-distortivity by a significant margin. In contrast, N6 is almost indifferent to the distortion because its T-component is more distortive while its c/-frame is less resistant to the distortion than those of benzene. No doubt, the o-lone pair repulsion affects the force constants of the o-frame in N6. Similarly, while cyclobutadiene and N4 are clearly distortive as a whole, the P4 and Si4H4 squares are pretty indifferent to distortion. This is because the T-distortivity gets very small and the... 

As the energy depends on 8, the implication is that it will also depend on bond length, and so the prediction is that, if tt electronic effects are given free rein, the equilibrium structure of the molecular framework will be one in which there has been distortion towards a short-long bond-alternated Kekule structure. That benzene does not in fact distort in this way is then explained by the relative strengths and force constants of a and tt bonds - the u framework of benzene is simply too stiff to be distorted by the relatively weak tt forces [9,12],... 

Berezin162 has argued that vibrational force constants are a quantitative characteristic of the properties of the electron shells of molecules and has proposed that the coefficients of influence (where the matrix of the coefficients of influence is the inverse of that of the force constants) may serve as a quantitative criterion for aromaticity. The coefficients of influence for benzene, pyridine, pyrazine, s-triazine, and s-tetrazine have been calculated from the force constants of the molecules and the sum of the diagonal coefficients of influence of all six internal ring angles and seem to increase as Balaban s K values (Section II,F, 1) for these compounds decrease. 

This means that the point symmetry of H changes from to D2h ( L and H stand for low- and high -symmetry site, respectively, as should now be evident). The first step in the construction of the benzene dimer is to modify force constants of the two sites according to Eqs. (5.1) and (5.2). A similar study can be easily and systematically implemented within the one-dimensional algebraic model by taking the Hamiltonian operator for CH stretching modes of the benzene molecule (Section III.C.2),... 

The combination of a Hartree - Fock calculation and experimental information laid the groundwork for the first theoretical force field, due to Pulay et al. [10b]. In this calculation, nine parameters, which incorporated the effect of neglected electron correlation, were fitted to the observed frequencies and Coriolis constants of benzene. The accuracy of the determined fitting parameters was demonstrated by simulating the effect of electron correlation on the calculated HF force field of pyridine [34], naphthaline [35] and other benzene analogs. More elaborate calculations [33c, 33d], including a very recent high level (CCSD(T)) ob initio calculations by Zhou et al. [33d] have substantiated the scaled HF force field of Pulay et al. 

The internal coordinates were selected so as to make it physically meaningful to compare force constants of different molecules. For the free and the coordinated benzene ring, the internal coordinates selected were based on suggestions by Pulay et al. [10b, 13c]. The C-Cr-C bending coordinates of Cr(CO)g were chosen in a way that allowed analogous definitions in the case of... 

To achieve such an agreement, we took an inter-ring force constant Fr2 = 3.40 mdyn/A in PPS (rc-s = 1-75 A) and 5.42 in PPO (rc-o = 1-36 A). Compared to PPP or PPV, we had to differentiate the force constants of the benzene ring. Those close to the inter-ring bonds Ft 2 = 6.07 nxlyn/A in PPS and 6.37 in PPO differ slightly fiom those which are not close to them Ft2 equal to 6.20 mdyn/A. 

The force constant of the Si—Si bond has been calculated as 1.3 x 10 dynes cm however, it is believed that this value is probably in error (f04). The dipole moments of several aromatic disilanes have been reported (2) which allowed one to calculate an aryl—Si—aryl valence angle of 115°. The dipole moment of 1,2-dichlorotetramethyldisilane was found (105) to be 1.75 debye in carbon tetrachloride and 1.35 debye in benzene. On the basis of the dipole moments, the infrared and the Raman spectra (in the gas, liquid, and solid state), information on the rotation about the Si—Si axis in 1,2-dichlorotetramethyldisilane was obtained. In the solid state, the chlorine atoms assume the tram position, whereas in the liquid and gas state the molecule exerts torsional oscillations about the Si—Si axis to a certain extent. The phase transformations of hexamethyldisilane were studied by NMR (80) and thermodynamically by means of differential thermal analysis (25). From such studies it appears that at higher temperatures rotations about both the Si—Si and Si—CH3 axes occur in combination with the overall molecular rotation about the molecular axis, whereas at lower temperatures all movements are hindered except for the Si—CH3 axial rotation. 

Dielectric Constant The dielectric constant of material represents its ability to reduce the electric force between two charges separated in space. This propei ty is useful in process control for polymers, ceramic materials, and semiconduc tors. Dielectric constants are measured with respect to vacuum (1.0) typical values range from 2 (benzene) to 33 (methanol) to 80 (water). TEe value for water is higher than for most plastics. A measuring cell is made of glass or some other insulating material and is usually doughnut-shaped, with the cylinders coated with metal, which constitute the plates of the capacitor. 

A nonlinear molecule consisting of N atoms can vibrate in 3N — 6 different ways, and a linear molecule can vibrate in 3N — 5 different ways. The number of ways in which a molecule can vibrate increases rapidly with the number of atoms a water molecule, with N = 3, can vibrate in 3 ways, but a benzene molecule, with N = 12, can vibrate in 30 different ways. Some of the vibrations of benzene correspond to expansion and contraction of the ring, others to its elongation, and still others to flexing and bending. Each way in which a molecule can vibrate is called a normal mode, and so we say that benzene has 30 normal modes of vibration. Each normal mode has a frequency that depends in a complicated way on the masses of the atoms that move during the vibration and the force constants associated with the motions involved (Fig. 2). 

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