Principal force constants

The principal force constants for TC2O7 and Re207 in the gas phase were calculated using a simplified molecular model (Tabic 10.3. A). 

There is insufficient data to determine the six principal force constants and ten interactions in the most general quadratic valence force field for HN3. Thompson and Fletcher, however, applied a procedure of varying interaction constants to determine a set of force constants. These force constants are also listed in Table I. Coriolis coupling (caused by an acceleration directed at right angles to the particle displacement) between the parallel and perpendicular modes was not taken into account in this analysis. When frequencies only were used as constraints, it was possible to vary one interaction constant plus the principal constants, and the most significant interaction constant turned out to be that between F3 and F4. The importance of this constant has been explained by Thompson and Fletcher in terms of the following resonance structures ... 

We use the convention, from spectroscopy, that "principal" force constants describe the potential within a single normal or internal-coordinate mode whereas "interaction" force constants describe mode-mode coupling. This convention avoids confusion with "diagonal" and "cff-diagonal" matrix elements in a perturbation theory treatment of arhar-monicity. 

A valence force field includes only valence coordinates and principal force constants a general force field also includes interactions. Thus the harmonic valence terms are the principal ones in the first sum in eq. (35), the harmonic general field consists of the whole first sum, and the anharmonic valence terms comorise the second sum. 

Spectroscopists call the force constants that have all indices the same the principal force constants, while the anharmonicity associated with the principal force constants is called principal anharmonicity. The Morse and quadratic-quartic approximations treat only principal anharmonicity. However, as mentioned in Eq. [95], neglecting the cross terms between modes is a much more serious approximation in rectilinear coordinates.Explicitly including cross terms in rectilinear coordinates is expensive and cumbersome... 

The r vectors are the principal axes of inertia determined by diagonalization of the matrix of inertia (eq. (12.14)). By forming the matrix product P FP, the translation and rotational directions are removed from the force constant matrix, and consequently the six (five) trivial vibrations become exactly zero (within the numerical accuracy of the machine). 

The amonotonic order for Co and Ni, compared to the monotonic order for Cu, may best be rationalized in terms of a finite, positive kc=c,c=c interaction force constant for Co and Ni, rather than as the outcome of amonotonicity in the principal c=c force-constants. 

In the construction of the matrix F of Eq. (63), the symmetrical equivalence of the two O-H bonds was taken into account. Nevertheless, it contains four independent force constants. As the water molecule has but three fundamental vibrational frequencies, at least one interaction constant must be neglected or some other constraint introduced. If all of the off-diagonal elements of F are neglected, the two principal constants, f, and / constitute the valence force field for this molecule. However, to reproduce the three observed vibrational frequencies this force field must be modified to include the interaction constant... 

The slope of the potential is zero for all directions and only one of the 3N-6 principal curvatures is negative. As it is an important fact that only one curvature is negative we must define what is meant by principal curvatures. At any point on the surface we can establish a matrix of second derivatives of the potential (force constants)... 

Equation 2.17 is of the form A = PDP-1. The 9x9 Hessian for a triatomic molecule (three Cartesian coordinates for each atom) is decomposed by diagonalization into a P matrix whose columns are direction vectors for the vibrations whose force constants are given by the k matrix. Actually, columns 1, 2 and 3 of P and the corresponding k, k2 and k3 of k refer to translational motion of the molecule (motion of the whole molecule from one place to another in space) these three force constants are nearly zero. Columns 4, 5 and 6 of P and the corresponding k4, k5 and k6 of k refer to rotational motion about the three principal... 

Apart from the ill-definitiveness of the inverse vibrational problem two principal objections are usually posed against the standard VFF model [5], First, the neglect of the long range interactions is not always physically justified and often contradicts with the real electronic structure of the molecule under study. Second, the transferability of force-constants is still a disputable topic, especially when the force-constants are transferred between neutral molecules and the corresponding ionized forms, or between conformational isomers [6, 7],... 

The principal idea, which underlies different schemes of empirical corrections to Fq"m is that even if the quantum mechanical force fields fail to reproduce the experimental vibrational frequencies in absolute scale, they still keep the right information about the relative magnitude and sign of the different elements in the force-constant matrix. 

The calculation of vibration spectra in terms of force constants is similar to the calculation of energy bands in terms of interatomic matrix elements. Force constants based upon elasticity lead to optical modes, as well as acoustical modes, in reasonable accord with experiment, the principal error being in transverse acoustical modes. The depression of these frequencies can be understood in terms of long-range electronic forces, which were omitted in calculations tising the valence force field. The calculation of specific heat in terms of the vibration spectrum can be greatly simplified by making a natural Einstein approximation. 

The large difference between the angular force constants Ci determined in the two ways appears to be the principal defect in application of the valence force field theory to the treatment of covalent solids. The defect is not readily repaired by the... 

Here, Z, are the atomic numbers and are the principal quantum numbers of its valence electrons. The force constant for a C-C single bond is calculated to be 4.05 N/cm, the experimental value is 4.45 N/cm. Siebert also describes the relation between the force constantfy of a multiple bond of the order N to that of a single bond/) and the lengths I N and -) of these bonds ... 

We observed in Section 2.4 that we expect there to be some relationship between force constants and bond distances, that is, bonds with greater force constants should be shorter. Furthermore, because the force constant (given our previous assumptions) is dominated by the repulsive part of the potential function, and the repulsions are primarily due to nonbonding core electrons which are more numerous row by row down the periodic table, we expect that the repulsive wall of the core is further from the nucleus in each successive row (i.e., increasing principal quantum number). So if a functional relationship between force constants and bond distances exists, we expect that the parameters of the relationship, whatever its functional form, will depend on which row(s) of the periodic table the bonded atoms occupy. 

The percentages of the three principal resonance hybrid structures of the cyanate ion are given in Table X. Nelson and Nelson [557) studied the effect on von and vco as the force constants are altered by including progressively increasing contributions from, first, the resonance form N=C—0 and, second, "N—C=0 their results in the former case are repeated in Table XIII The calculations were... 

As in standard molecular dynamic simulations, the box is gradually heated, i.e., velocities of the atoms are increased, and eventually the size of the box is scaled up or down to the conditions of interest (pressure and temperature). In contrast to standard simulations, the present one is stopped periodically to perform DFT calculations on a small sample of the box. This is basically done to obtain new charges but eventually it can be modified to obtain new geometric parameters and force constants, depending on the specific properties that are target of the calculations. This procedure continues during heating principally and at the initial part of equilibration until a self-consistent force field compatible with the real conditions, is obtained for the equilibrated box. 

Two of the vibrational frequencies were observed and assigned by Lindeman and Wilson ( ). The remaining frequencies were calculated by a normal coordinate treatment using force constants obtained by comparison with the other mixed trihalides. The bond distances were reported by Wentink and Tiensuu (2). The bond angles are estimated by analogy with the other trihalides. The principal moments of inertia are - 14.8465 x lO g cm, Ig = 69.5984 x lO g cm and 1 - 84.4449 x 10 g cm. ... 

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