Modified valence force field

From the above example it is apparent that there may be (and usually are) more unknown force constants than observed frequencies. If no additional sources of data are available, it is necessary to make some assumptions to simplify the force field. Often all or some of the off-diagonal elements in the F matrix [Eq. (63)] are neglected, leading to the so-called valence force field (VFF) or modified valence force field (MVFF), respectively. 

A normal coordinate analysis was carried out (55) for CIF2O+ assuming the following geometry Bcio = 1.41 A rciF = 1-62 A, /5(0C1F) = 108° and a(FClF) = 93°. A modified valence force field was computed, and the results are given in Table X. As can be seen from Table I, the CIO-... 

Draeger, J.A. (1985) The methylbenzenes. 1. Vapor-phase vibrational fundamentals, internal rotations and a modified valence force-field, Specwochim. Acta., 41A, 607-627. Bemshtein, V. and Oref, I. (2001) Dependence of collisions lifetimes on translational energy, J. Phys. Chem. A. 105, 3454-3457. 

A modified valence force field was computed for the ion SF O", which showed a value of the SO stretching force constant lower than those found for S=0 double bonds and suggested an SO bond order of about 1.5. Likewise, low values were found for S—F stretching force constants, thus indicating significant ionic contributions to the SF bonds in SFjO". These results were therefore interpreted in terms of the resonance structures (19) and (20). 

Draeger, J.A. (1985). The methylbenzenes—I. Vapor-phase vibrational fundamentals, internal rotations and a modified valence force field. Spectrvchim. Acta A 41, 607. 

Takayama, T., M. Yuri, K. Itoh, T. Baba and J. S. Harris, Jr. 2000, Theoretical analysis of unstable two-phase region and microscopic structure in wurtzite and zinc-blende InGaN using modified valence force field model. J. Appl. Phys, 88(2) pp. 1104-1110. 

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... 

General quartic force field with three quartic constraints. c Tentative values based on a modified valence-force model with nine parameters. 

In addition to simple, modified, or general valence force fields with no, some, or all interaction force constants retained, a force field referred to as the Urey-Bradley force field is sometimes used to reduce the number of force constants needed. It uses simple valence type force constants but also includes some force constants between nonbonded atoms representing van der Waals forces. For further discussion, see Herzberg," Chapter 2 Wilson, E>ecius, and Cross, Chapter 8 and Steele, Chapter 5. 

The final general form of the VFF- and UBFF-expressions (2) and (4) as modified for our purposes is thus as follows (Central force fields are less efficient than valence- and UB-force fields, and are not considered here.) ... 

Finally, for this section we note that the valence interactions in Eq. [1] are either linear with respect to the force constants or can be made linear. For example, the harmonic approximation for the bond stretch, 0.5 (b - boV, is linear with respect to the force constant If a Morse function is chosen, then it is possible to linearize it by a Taylor expansion, etc. Even the dependence on the reference value bg can be transformed such that the force field has a linear term k, b - bo), where bo is predetermined and fixed, and is the parameter to be determined. The dependence of the energy function on the latter is linear. [After ko has been determined the bilinear form in b - bo) can be rearranged such that bo is modified and the term linear in b - bo) disappears.] Consequently, the fit of the force constants to the ab initio data can be transformed into a linear least-squares problem with respect to these parameters, and such a problem can be solved with one matrix inversion. This is to be distinguished from parameter optimizations with respect to experimental data such as frequencies that are, of course, complicated functions of the whole set of force constants and the molecular geometry. The linearity of the least-squares problem with respect to the ab initio data is a reflection of the point discussed in the previous section, which noted that the ab initio data are related to the functional form of empirical force fields more directly than the experimental data. A related advantage in this respect is that, when fitting the ab initio Hessian matrix and determining in this way the molecular normal modes and frequencies, one does not compare anharmonic and harmonic frequencies, as is usually done with respect to experimental results. 

Force constants of TcO calculated in a modified valence field are compared with the force constants of MnOj and ReO in Table 11.6.A. The calculation is based on frequencies measured in aqueous solutions 64], except for the frequencies of V2( ) which were deduced from IR spectra of the crystalline salts. The Raman spectrum of the aqueous I cOj ion shows only two lines and represents the spectrum of tetrahedral TCO4, perturbed by the close association of water molecules [64]. 

MM 1 -CARB is the MM 1 force field modified to include the carbohydrate parameters reported by Jeffrey and Taylor in 1980. MMl itself contained all the valence parameters necessary for carbohydrates, and the MMl-CARB version represents a modest tuning of the MMl parameters. As in the cases of MM2 and MM3, the values of the glycosidic torsion coefficients (V 3) in MMl were chosen so as to reproduce the experimentally determined AH for the axial to equatorial conversion of 2a to 2e. In contrast to MM2, MMl lacks a torsion-stretch cross term, and the MMl-GARB parameterization introduced different valence stretching and... 

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