Force constant cubic

Table I Harmonic frequencies and cubic force constants (in the reduced normal coordinate representation) for the normal isotopomer (1) of Sij (top). Measured rotational constants and effective equilibrium values (in MHz) for the five isotopomers described in the text (bottom).

The conclusion is that if the spectrum can be analysed in terms of equations (3)—(7), then the force constants can be determined. The bond length re can be determined from the equilibrium rotational constant Bc then the quadratic force constant /3 can be determined either from the harmonic wavenumber centrifugal distortion constant De then the cubic force constant /3 can be determined from aB and finally the quartic force constant /4 can be determined from x. It is necessary to determine the force constants in this order since in each case we depend upon already knowing the preceding constants of lower order. The values of re,f2,f3, and /4 calculated in this way for a number of diatomic molecules are shown in Table 2. 

Bent Triatomic Molecules.—Calculations have been reported for many bent triatomic molecules (see Table 4). The general force field contains 2re + 4/ + 6/3 + 9A parameters, the relation to the primary spectroscopic constants being shown in Table 9. The fact that these are asymmetric top molecules, for which otj, a , and a can all be determined (generally from the microwave spectrum for the heavier molecules), means that 9 a values are available from each isotopic species to determine the 6 cubic force constants, so that the cubic force field is generally well determined. For the quartic force field the situation is much less satisfactory the experimental data on the anharmonic constants xrs are generally incomplete, and are in any case insufficient to fix all the quartic constants without good isotopic data. 

The differential Raman scattering cross sections and depolarization ratios in the Fermi resonance region of carbon disulphide CS2 were measured and interpreted in terms of three bond polarizability parameters and the cubic force constant k 22 (Montero et al., 1984). 

Bouteiller, Y, Allavena, M., and Leclercq, J. M., Cubic force constants of the FH—OH2 and FH" 0(CH )2 hydrogen-bonded complexes. An analysis of computed ab initio SCF values, Chem. Phys. Lett. 69, 521-524 (1980). 

Solimannejad and Pakiari developed ab initio analytical force constant for FSGO, and applied their theory to H2, LiH and BH the results are 5.936 (3.21%), 1.208 (17.73%) and 3.168 (4.48%) mdyn respectively. Ab initio analytic cubic force constants have also been formulated by Solimannejad and Pakiari in FSGO and applied to H2O and CH2. 

The geometry of the H2O. .. HF complex was partially optimized using triple zeta (TZ) and TZ + P basis sets The respective optimum O. .. F distances arc 0.267 and 0.269 nm. Quadratic and cubic force constants were obtained for both basis sets. On passing from the first to the second basis set, the intermolecular force constant, Krr, decreased by 57 %. When passing from the 6-31G basis set to the TZ + P basis set the complex departs from the C2, symmetry. 

Vibrations typically correspond to small excursions from the equilibrium geometry, and the energy scale of the potential surface is defined with respect to a zero at that geometry, where by definition all first derivatives (9V/9Sj)e vanish. The higher derivatives d V/dSidSj)e, (d V/dSidSjdSk)e, are the quadratic, cubic,. .. force constants. As coefficients in the Taylor expansion... 

Quadratic terms in die property expansions are considered to be first-order in electrical anharmonicity, cubic terms are taken to be second-order, etc. Similarly, cubic terms in the vibrational potential are considered to be first-order in mechanical anharmonicity, quartic terms are second-order, and so forth. The notation (n, m) is used hereafter for the order of electrical (n) and mechanical (m) anharmonicity whereas the total order (n -I- m) is denoted by I, II,. Although our definition of orders is reasonable other choices are possible. Two key questions are (1) How important are anharmonicity contributions to vibrational NLO properties and (2) What is the convergence behavior of the double perturbation series in electrical and mechanical anharmonicity Both questions will be addressed later. Here we note that compact expressions, complete through order II in electrical plus mechanical anharmonicity, have been presented [19]. The formulas of order I contain either cubic force constants or second derivatives of the electrical properties with respect to the normal coordinates. Depending upon the level of calculation at least one order of numerical differentiation is ordinarily required to determine these anharmonicity parameters. For electrical properties, the additional normal coordinate derivative may be replaced by an electric field derivative using relations such as d p./dQidQj = —d E/dldAj.ACd, = —dk,/rjF where F is the field and k j is... 

E/eRiCRjdR Cubic force constant part of the anharmonic contribution to vibrational frequencies anharmonic contribution to vibrationally averaged structures... 

In another application, the quadratic and cubic force constants are calculated by ab initio methods. From the force field, the geometry and the atomic masses, it is possible to calculate the vibrational frequencies and the vibration rotation constants Q5 (Eq. (2-4)). If they are sufficiently accurate, they can be used in place of the e>q)erimental... 

Anharmonicity of molecular vibrations presents one of the most vexing problems in studies of molecular structure. Anharmonic corrections (of first order, involving the cubic force constants) are required in accurate determinations of the equilibrium structures of molecules from rotational spectra, as well as from electron diffraction measurements. To obtain accurate harmonic force fields, it is necessary first to correct the vibrational data for anharmonicity (using second-order corrections, involving cubic and quartic force constants). Information on anharmonic force fields obtained from experimental data is also important as a basis for comparison in quantum chemical investigations of molecular forces as well as in studies of high-temperature thermodynamic properties and of rate and dissociation processes. Yet detailed studies of anharmonic force fields have hitherto been limited to small molecules with N = 2-4 atoms (in isolated cases to N = 6). 

Microwave measurements of the rotational transitions in vibration-ally excited states from which the cubic force constants could be determined220 are available, e.g., for S02,115,116 Se02,221,222 03,223 OF2,224 SiF2,225 GeF2,226 HCP,141,142 NF3,159,160 HSiF3,227 as well as for many of the molecules mentioned above. 

The model potential is a function of the deviation (x-Aq) from the reference distance Xq, which is the equilibrium value when a is zero. The quadratic and cubic force constants /2 (positive) and (negative) are taken as the same for all molecules in a related series, whereas a is a perturbation expressing steric and electronic differences in ligand-metal interactions between related molecules (dotted line in Figure 5.16). The perturbation a affects AE 2md shifts the equilibrium distance to Xe = Xo + Aaq. In linear approximation the relationship is... 

We will now comment briefly on the force fields that have been computed for benzene. The first one to incorporate both harmonic and anharmonic force constants (some cubic force constants) was reported by Pulay et al. (146) in 1981. This force field was determined with the 4-21P basis set at the SCF level. The computed harmonic force constants, as expected, were larger than the empirical values, so Pulay et al. developed a scaled quadratic force field. The scale factors were adjusted so that the computed frequencies agreed well with the experimental fundamental frequencies. 

The Cl + HC1 quantized transition states have also been studied by Cohen et al. (159), using semiclassical transition state theory based on second-order perturbation theory for cubic force constants and first-order perturbation theory for quartic ones. Their treatment yielded 0), = 339 cm-1 and to2 = 508 cm"1. The former is considerably lower than the values extracted from finite-resolution quantal densities of reactive states and from vibrationally adiabatic analysis, 2010 and 1920 cm 1 respectively (11), but the bend frequency to2 is in good agreement with the previous (11) values, 497 and 691 cm-1 from quantum scattering and vibrationally adiabatic analyses respectively. The discrepancy in the stretching frequency is a consequence of Cohen et al. using second-order perturbation theory in the vicinity of the saddle point rather than the variational transition state. As discussed elsewhere (88), second-order perturbation theory is inadequate to capture large deviations in position of the variational transition state from the saddle point. 

Table 5.17). is computed in terms of the cubic force constant and dipole... 

The cubic force field of pyramidal XYg-type molecules contains 14 independent parameters. Usually, however, the number of spectroscopic constants dependent on the enharmonic force field, such as rotation-vibration constants, l-type doubling constants, or anharmonicity constants, is smaller than the number of parameters to be determined. Their number thus has to be reduced by introducing model potentials and imposing certain constraints. Some of the possible routes were presented by Morino et al. [50]. Cubic force constants for NF3 pertinent to model potentials (mostly Morse potentials) have been calculated by several groups of workers [11, 12, 19, 51, 52], Some of the principal quartic constants have been estimated as well [12, 51]. 

There have been many studies of the structure of thiophene over the years, using data from many experimental techniques. A new paper reports the equilibrium structure, determined from gas-phase electron diffraction, vibrational and microwave data. In addition, quadratic and cubic force constants were calculated theoretically (up to the B3LYP/6-311 -I- G level). Harmonic scale factors were included as refinable parameters in the analysis of the data. The outcome is a structure that is less precise than one determined earlier, which also made use of dipolar coupling constants, but the ra distances from the two studies agree well. The refined equilibrium parameters include distances C = C 137.2(3), C-C 142.1(4) and S-C 170.4(2) pm and angles CSC 92.4(2), SCC 111.6 and CCC 112.2°. 

C Domingo, S Montero. Experimental determination of CH-stretching-bending-bending cubic force constants of ethane from Raman intensity analysis. J Chem Phys 86 6046-6058, 1987. 

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