Central forces force constants

Very slight changes in wave numbers observed when moving within the same group, from Mo to W, from Nb to Ta and from Zr to Hf (see Fig. 46), indicate that the contribution of the change in the central atom s mass is compensated for by respective changes in the metal-ligand force constant. 

The relationship (equation (5.81)) between M and L depends only on fundamental constants, the electronic mass and charge, and does not depend on any of the variables used in the derivation. Although this equation was obtained by applying classical theory to a circular orbit, it is more generally valid. It applies to elliptical orbits as well as to classical motion with attractive forces other than dependence. For any orbit in any central force field, the angular... 

The summations in Eq. (8) and (9) usually extend over all internal parameters, independent and dependent, i.e. the potential constants in these expressions are also not all independent. For example, the nonsymmetric tetrasubstituted methane CRXR2R3R4 possesses five independent force constants for angle deformations at the central carbon atom, whereas in our calculations we sum over the potential energy contributions of the six different angles (only five are independent ) at this atom using six different potential constants for angle deformations. The calculation of the independent force constants, which is necessary for the evaluation of the vibrational frequencies, will be dealt with in Section 2.3. 

The model presented here develops these ideas and introduces features which make the calculation of mixture properties simple. For a polar fluid with approximately central dispersion forces together with a strong angle dependent electrostatic force we may separate the intermolecular potential into two parts so that the virial coefficients, B, C, D, etc. of the fluid can be written as the sum of two terms. The first terms B°, C°, D°, etc, arise from dispersion forces and may include a contribution arising from the permanent dipole of the molecule. The second terms contain equilibrium constants K2, K, K, etc. which describe the formation... 

This expression is used in all cases where atom i is connected to exactly three other atoms, /. k and l. Here, Kiju is the force constant, ( ijki is the angle between the i-l axis and the i-j-k plane and the expansion coefficients depend on the central atom type ... 

Neither of these vibrations corresponds to stretching vibrations of AH or BH. The antisymmetric vibrational mode represents translational motion in the transition state and has an imaginary force constant. The symmetric transition-state vibration has a real force constant but the vibration may or may not involve motion of the central H(D) atom2,12 13. If the motion is truly symmetric, the central atom will be motionless in the vibration and the frequency of the vibration will not depend on the mass of this atom, i.e. the vibrational frequency will be the same for both isotopically substituted transition states. It is apparent that under such circumstances there will be no zero-point energy difference... 

This cancellation (always marked in exchanges of acid-base type) can be ascribed to the decrease in force constant C of the central atom with increasing coordination number n, and is sometimes not so significant in other types of isotopic exchange reactions. 

Raman and i.r. spectra of [MgOig] - (M = Nb or Ta) salts have been determined and normal-co-ordinate analyses suggest that the ratio of the force constants of the terminal, bridging, and central M—O bonds is close to 8 4 1. 

Fig. 8a and b. First neighbour bonds of a central atom which is a bulk atom of a fee crystal. It has 12 first-neighbour bonds, b A cobalt atom in a monolayer of cobalt on copper (111). In this case the three upper bond force constants are reduced to zero (surface effect) while the effect of the copper substrate is to reduce the values of the three lower bonf force constants by about 15%... 

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. 

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