Harmonic force field

Force constants of S2O were calculated from infrared and microwave data for a general harmonic force field using the equihbrium geometry the following data were obtained [34] (1 aj = 1 mdyn = 1 N cm ) ... 

Berces, A., Ziegler, T., 1992, The Harmonic Force Field of Benzene. A Local Density Functional Study , J. Chem. Phys., 98, 4793. 

Florian, J., Johnson, B. G., 1994, Comparison and Scahng of Hartree-Fock and Density Functional Harmonic Force Fields. 1. Formamide Monomer , J. Phys. Chem., 98, 3681. 

The revalues are distances between atoms separated by a chain of three (four) or more bonds (Section 2.1.5.). Mainly because of the introduction of the nonbonded interactions, Eq. (8) and (9) no longer represent simple harmonic force fields. We therefore denote the constants of these expressions as potential constants and not as force constants. In principle, all the constants of the force fields (2), (3), (4), (8), and (9) are different, as indicated by different indices (V FFF , U f/BFF , v = vibrational (understood in the sense of standard vibrational-spectroscopic computational techniques)). In what follows we shall be primarily concerned with force fields of the type of Eq. (8) which we therefore formulate with the simplest symbols. 

How one obtains the three normal mode vibrational frequencies of the water molecule corresponding to the three vibrational degrees of freedom of the water molecule will be the subject of the following section. The H20 molecule has three normal vibrational frequencies which can be determined by vibrational spectroscopy. There are four force constants in the harmonic force field that are not known (see Equation 3.6). The values of four force constants cannot be determined from three observed frequencies. One needs additional information about the potential function in order to determine all four force constants. Here comes one of the first applications of isotope effects. If one has frequencies for both H20 and D20, one knows that these frequencies result from different atomic masses vibrating on the same potential function within the Born-Oppenheimer approximation. Thus, we... 

Fig. 5.8 VPIE s of Waters and Ices. The points are experimental from various sources. The lines are calculated using an isotope independent harmonic force field consistent with spectroscopic information (Van Hook, W. A. J. Phys. Chem 72,1234 (1968))... 

Black GM, Law MM (2001) The general harmonic force field of methyl chloride. J Mol Spectrosc 205 280-285... 

Murrell 2S6> shows theoretically that for MX6 compounds the three bond stretching force constants can be calculated sufficiently accurately from IR and Raman spectra from this the complete harmonic force field can be deduced. This is an important simplification, and bond strengths can be more conveniently calculated. Murrell shows that e.g. in WFSC1 that replacement of F by Cl strengthens the trans WF bond and weakens the cis WF bonds. 

Adiabatic potential in the presence of the harmonic force field... 

At a fixed geometry, however, the (harmonic) force field can be added by assuming the total Hamiltonian... 

The shrinkage effect1 is treated in more detail elsewhere in the present article. Due to molecular vibrations interatomic distances observed by electron diffraction do not correspond to a set of distances calculated from a rigid geometrical model. Usually the shrinkage effect is routinely included in electron-diffraction least-squares refinement. In order to do so, it has been found appropriate to introduce a third distance type r defined as the distance between mean positions of atoms at a particular temperature. If the harmonic force field is known, iQ may be calculated from ra according to Eq. (12) ... 

A theoretical justification of the scaling procedure was given by Pupyshev et al [14]. They compared the force field Fhf obtained in the Hartree-Fock (HF) limit with the force-field Fa obtained in the configuration interaction (Cl) technique, where the electron correlation effects are taken into account by mixing the HF ground state function with electronic excitations from the occupied one-electron HF states to the virtual (unoccupied) HF states. It was assumed that the force constants F01 obtained in the Cl approximation simulate the exact harmonic force field while those, extracted from the HF approximation FHF cast the quantum-mechanical force field F1-"1. It was demonstrated that under conditions listed below ... 

Within the HA, the prediction of a vibrational absorption spectrum amounts to the calculation of the harmonic normal mode frequencies, vi7 and dipole strengths, Dt. The frequencies are obtained from the harmonic force field (HFF). With respect to Cartesian displacement coordinates, this is the Hessian (d2WG/dXXadXx,a,)0. Diagonalization (after mass-weighting) yields the force constants ky the frequencies, vy and the normal coordinates, <2 , i.e. the transformation matrices, SXa4. The dipole strengths depend in addition on the APTs these require calculation of (dtpG/dXXa)0. 

In comparing the results of a computation with measured data, one must be aware that a normal mode analysis supposes a harmonic force field. In a normal mode the ratio of... 

To estimate the parameters of the harmonic force fields that may appear from the SLG-based semiempirical treatment, we consider the symmetric correlated single bond, where the energy can be obtained without any reference to its environment. In our case the derivative of the bond energy with respect to a geometry parameter q has the form ... 

The primary motive for attempting calculations of this kind is simply our desire to determine the potential function V(r) more accurately and over a wider range of co-ordinate space. Even if our immediate ambition is only to determine the equilibrium configuration and the harmonic force field, our ability to withdraw this information from spectroscopic data is limited by the need to make corrections arising from the cubic and quartic anharmonic force field. 

For harmonic force field calculations it is usual to express the force field in terms of internal co-ordinates Ri defined through a linear transformation from cartesian displacement co-ordinates 8xm (a = x, y, or z) in the molecule-fixed axis system.24 The molecule-fixed axes are located using the Eckart conditions.24 The equations are written in the form... 

The normal co-ordinates Qr are first determined from a harmonic force field calculation, in the usual way. They are chosen to diagonalize the kinetic... 

The transformation from cartesian co-ordinates to normal co-ordinates is determined in the preliminary harmonic force field calculation in the customary notation it is written... 

It is clear from equation (61) that if the harmonic force field of a molecule is known, each observed af constant gives a linear relation between the cubic anharmonic constants m, where s ranges over all normal modes in the molecule (subject to symmetry restrictions). 

The problem is similar to that involved in harmonic force field calculations, but more difficult in almost all respects. In simple cases one may attempt to solve directly, or graphically, for some of the anharmonic 0 values using the observed values of the spectroscopic constants in equations like (61) and (62). These may then be related to / values through the L tensor as described on pp. 124—132. However, such methods are of only limited value. The more general method of calculation is to attempt an anharmonic force field refinement, in which a trial force field is refined, usually in a large non-linear least-squares calculation, to give the best agreement between the observed and... 

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