Harmonic force constants, calculation

Kofraneck and coworkers24 have used the geometries and harmonic force constants calculated for tram- and gauche-butadiene and for traws-hexatriene, using the ACPF (Average Coupled Pair Functional) method to include electron correlation, to compute scaled force fields and vibrational frequencies for trans-polyenes up to 18 carbon atoms and for the infinite chain. 

Starting from a set of harmonic force constants calculated in terms of internal coordinates, one can determine in this way the normal coordinates of the different vibrations (phonons) of an infinite chain as a function of (the columns of matrices L , ) as well as the corresponding phonon dispersion curves One should note that, in order to... 

So in order to calculate the harmonic force constant from equation 14.6 above,... 

The Gl procedure consists of a number of standard calculation steps that aim to find an equilibrium geometry, an electronic energy and a set of harmonic force constants. The defects mentioned above are assumed additive, and they are investigated in turn. 

For comparison with experimental frequencies (which necessarily are anharmonic), there is normally little point in improving the theoretical level beyond MP2 with a TZ(2df,2pd) type basis set unless anharmonicity constants are calculated explicitly. Although anharmonicity can be approximately accounted for by scaling the harmonic frequencies by 0.97, the remaining errors in the harmonic force constants at this level are normally smaller than the corresponding errors due to variations in anharmonicity. 

Since vibrational spectra of S2O2 have not yet been observed, the force constants calculated by ab initio MO methods were used to predict the harmonic vibrational wavenumbers of ds-S202 (C2v) and trans-S202 (C2I1) see Table 3 [34, 57]. 

I. Florian, I. Leszczynski, B. G. lohnson, and L. Goodman, Coupled cluster and density functional calculations of the molecular structure, infrared spectra, Raman spectra, and harmonic force constants for methanol. Mol. Phys. 91, 439 447 (1977). 

For small molecules it is possible to calculate from the experimental IR spectrum the simple harmonic force constant k and the anharmonicity corrections. Using k, theoretical harmonic frequencies can be calculated [236]. These correspond to a... 

Free energy second derivatives are mainly used to analyse the nature of stationary points on the PES, and to compute harmonic force constants and vibrational frequencies to perform such calculations in solution, one needs analytical expressions for Qa second derivatives with respect to nuclear displacements (the alternative of using numerical differentiation of gradients is far too much expensive except for very small molecules). 

To compensate for steric effects slight modification of ionization radii allows the derivation of observed dissociation energies, using the same general diatomic function for all covalent bonds, irrespective of bond order. Comparison of the modified characteristic radii and the fundamental ionization radii enables the calculation of harmonic force constants and defines a functional relationship between bonds of different order. 

It is nice to have a distinctive notation for the curvilinear co-ordinates, which emphasizes their difference from and yet their one-to-one correlation with the Rt co-ordinates. Most authors reporting anharmonic calculations do not in fact make any distinction they denote the curvilinear co-ordinates by the same symbols customarily used to denote the corresponding rectilinear coordinates in harmonic calculations. For many purposes this is satisfactory, particularly since the harmonic force constants are not altered by the change from rectilinear to curvilinear co-ordinates. However, in a general discussion it is important to distinguish the two sets, and so for the remainder of this section we shall follow Hoy et al.12 and write the curvilinear co-ordinates with the symbol Hi. 

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... 

Rationalization of these discrepancies between calculated and observed molecular properties has been achieved by recognizing that the accuracy problems tend to be systematic for the computational methods [726], For the same levels of theory, the errors in molecular parameters tend to be similar [79] for molecules if a variety of types and size. Building on the ideas of similarities between similar chemical functional groups in different molecules, the SQM procedure develops empirical scaling factors to correct the overestimated harmonic force constants from the computations [77], These scale factors form a diagonal matrix C which modified the calculated force matrix Fcalc according to the relationship [727] ... 

Our first example is the widely discussed problem of CO chemisorption on transition metal surfaces. Ray and Anderson/106/ investigated CO chemisorption on a Pt (111) surface using the ASED-MO theory. In their work, Pt4 and Pt10 clusters were used to model the Pt (111) surface. The bulk Pt-Pt bond distance was used and other theoretical parameters are reproduced in Table II. The calculated Pt-Pt and C-O bond distances and harmonic force constants are within 8% of experimental values for isolated Pt2 and CO. The results for dissociation energies are less satisfactory 94.08 and 116 kcal/mole in comparison with experimental values of 84.5 and 256 kcal/mole, respectively. 

If this Morse function is used to represent any single bond, not necessarily of a diatomic molecule, the constant a calculated from the harmonic force constant may not be entirely appropriate, and especially not over the entire range of r. Before deriving multiple-bond properties from the single-bond curve it is therefore useful to optimize the Morse constant empirically to improve the match between calculated and observed single-bond values of De and re. 

Calculate the harmonic force constant 4 and the Morse parameter j8 for the two states. Using the known r J value (0.2666 nm), plot the Morse curve for the ground electronic state of I2. Compare this with the harmonic potential calcnlated from k. ... 

The term in Az is primarily determined by anharmonic force constants, whereas the terms in Ax2 and Ay2 are primarily functions of the harmonic force constants. For diatomic molecules where the vibrational force constants are usually well characterized, it is possible to calculate the vibrational averages and thus obtain rt from rg. For polyatomic... 

The leading terms involve the quadratic force constants ///, and for small displacements these are clearly dominant. In harmonic force field calculations all higher order terms are neglected, and this is the case in this paper. 

The force constants are calculated from the distortion constants by means of Eq. (2). This expression was derived assuming an equilibrium configuration for the molecules. The effects of molecular vibrations cause Eq. (2) not to be obeyed exactly so that force constants calculated using the spectroscopic constants of the ground state will be somewhat in error. One way to assess this error is to compare the harmonic vibrational fundamentals calculated from the centrifugal distortion constants with those calculated from the infrared spectrum (or, when the harmonic frequencies are unknown, with the observed infrared band origins). This comparison is presented in Table 1 for a variety of bent, symmetric triatomic molecules. 

Although evaluations of harmonic force constants [d E dq,dqj), elearic polarizabilities d EIdeide ), and dipole moment derivatives (d E/d ,dqj) are perhaps the most common applications of second-order properties (or, equivalently, second derivatives), other areas of interest to chemists can be treated with these techniques. One such field of application that holds great promise for the future is the calculation of nuclear magnetic resonance chemical shifts. 

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