Effective force constant

And, using Equation 9.1, an effective force constant, g, can be obtained ... 

A potentially much more adaptable technique is force-field vibrational modeling. In this method, the effective force constants related to distortions of a molecule (such as bond stretching) are used to estimate unknown vibrahonal frequencies. The great advantage of this approach is that it can be applied to any material, provided a suitable set of force constants is known. For small molecules and complexes, approximate force constants can often be determined using known (if incomplete) vibrational specha. These empirical force-field models, in effect, represent a more sophisticated way of exhapolating known frequencies than the rule-based method. A simple type of empirical molecular force field, the modified Urey-Bradley force field (MUBFF), is introduced below. 

The 5cr level is primarily located on the carbon atom and can be identified with the carbonyl lone pair orbital, but in addition it is somewhat antibonding between carbon and oxygen. The population of this orbital has a significant influence on the carbonyl stretching force constant and, therefore, changes in the carbonyl force constant cannot be assigned solely to back donation effects. Force constants k in the M(CO)5L molecules can be described by the following equation ... 

The advantage of the effective Hamiltonian (7), beside the reduced space of nuclear coordinates, is that it involves effective force constants (elements of the matrix K) which rapidly decrease with the distance between centres. This allows for efficient models involving only few force constants. Furthermore, the force constants can be calculated. For this purpose the local nuclear coordinates, at the centre n, are expanded in Cartesian displacements, uKa n ), of surrounding atoms ... 

It is possible to distinguish the contributions from phonons and uniform strains in the effective force constants. After some transformations (see Appendix A) we obtain... 

Due to the dependence of / on active nuclear distortions, the Heisenberg exchange interaction, equation (38), summed up over all nearest neighbour centres in a chain, should be added from the start to the energy functional in equation (32). Equation (41) shows, however, that this addition merely leads to the renormalization of the effective force constant Kt in equation (32) by... 

The proposed approach to static cooperative Jahn-Teller effect is based on the exact effective Hamiltonian (7), acting in the reduced space of active one-centre distortions only. It involves effective force constants, which are analytically related to the parameters of the full microscopic Hamiltonian. Direct electronic interactions between sites, such as orbital-dependent electrostatic and exchange interactions [28], can be added to the effective Hamiltonian without modifying it. This approach proves to be especially efficient in the case of strong Jahn-Teller distortions, when the effects of second-order Jahn-Teller coupling become important. 

The potential energy function for a chemical bond is far more complex than a harmonic potential at high energies, as discussed in Chapter 3. However, near the bottom of the well, the potential does not look much different from the potential for a harmonic oscillator we can then define an effective force constant for the chemical bond. This turns out to be another problem that can be solved exactly by Schrodinger s equation. Vibrational energy is also quantized the correct formula for the allowed energies of a harmonic oscillator turns out to be ... 

Reduction of Polyatomic to Diatomic Oscillators. When one employs a vector representation for a potential energy surface, the vector addition of the oscillator force constants projected on a mass weighted RC leads to a single effective force constant for reactants or products. 

From the free energy all thermodynamic properties of the system can be calculated. For example, the entropy is S = -3A/dTand the energy is E = A + TS. For more details we refer to the review by Werthamer (1976). One important point should be mentioned. Expanding the potential VppiQp,. Qp ) as a Taylor series in the displacement coordinates Qp, we observe, using the analog of Eq. (87) for the force constants (88), that the odd power terms do not contribute to the effective force constants the SCP method neglects these terms. Their relative importance can, of course, be estimated by perturbation techniques as described in Section III,B. 

It is interesting to note, however, that there is a relation between the chemical isomer shift of a tin impurity atom and the effective force constant... 

Fig. 14.13 Relation between the chemical isomer shift of Sn impurity atoms in various metals and the effective force constant of the host, 6a M. [Ref. 245, Fig. 1]...

In the harmonic approximation in which a parabolic well near the equilibrium bond length is assumed, the frequencies will not be altered, but inclusion of anharmonic (cubic and higher order) terms in the intraionic potential function will further modify the effective force constant of the impurity ion. F was introduced via an expression by Bauer and Magat [26] and modified by Maki and Decius [27] based on Onsager s model of a homogeneous solvent of dielectric constant e with the solute contained in a spherical cavity of radius a. The frequency shift equation is given by... 

It is interesting to notice, based on Eq. (38), that, in relation to lifting its central point, an infinite membrane behaves like a spring with the effective force constant ... 

Pack and Hirschfelder (133) discuss the energy corrections to the Bom-Oppenheimer approximation, while Bunker (134) discusses the partial breakdown of the Bom-Oppenheimer approximation. The Hell-mann-Feynman theorem is also discussed by Tuan (135) with reference to multiconfiguration SCF-theory, while Loeb and Rasiel (136) discuss constraints upon the LiH molecule with reference to the Hellmann-Fe5mman theorem. King (137) developed a theory of effective Cartesian force constants which relate to the sum of the squares of the normal frequencies these effective force constants are independent of the inter-nuclear repulsion of the nuclei in a molecule, and thus removes one of the indeterminacy of the Hellmann-Feynman method described above. 

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