Quartic force fields

The harmonic frequencies and the anharmonic constants may be obtained from experimental vibrational spectra, although their determination becomes difficult as the size of the system increases. In Table 1.10, we have listed experimental harmonic and anharmonic contributions to the AEs. These contributions may also be obtained from electronic-structure calculations of quadratic force fields (for harmonic frequencies) and cubic and quartic force fields (for anharmonic constants). For some of the larger molecules in Table 1.11, we have used ZPVEs calculated at the CCSD(T)/cc-pVTZ level or higher, see Ref. 12. In some cases, both experimental and theoretical ZPVEs are available and agree to within 0.3 kJ/mol [12, 57],... 

In a recent benchmark study [57] on the CH2=NH molecule, we explicitly computed a CCSD(T)/VTZ quartic force field at great expense (the low symmetry necessitated the computation of 2241 energy points in Cs symmetry and 460 additional points in Ci symmetry). The resulting anharmonic Ezpv, 24.69 kcal/mol, is only 0.10 kcal/mol above the scaled B3LYP/VTZ estimate, 24.59 kcal/mol. At least for fairly rigid... 

Table 1. Computed and experimental equilibrium geometry and harmonic force constants for HjO (all results in atomic units). The results of reference 70 were obtained by fitting 36 points on the surface to a general quartic force field the first terms of which were...

Once the reliability of CCSD(T) had been established, we could proceed with confidence to use it to predict vibrational frequencies for Be3 and Be. In order to obtain the best possible prediction to aid experimentalists, a full quartic force field was generated for each molecule [76], using finite differences of computed energies, and fundamental frequencies were obtained via second-order perturbation theory. In Table 5.7 we list the CCSD(T) fundamental frequndes and, for comparison, the CCSD, CCSD(T) and MRCI harmonic frequencies. 

Here, Kfj is the Hessian calculated at displaced geometries where for the displacement we used AQ = 0.03 for the cubic and AQ = 0.04 for the quartic force field. Note that for the construction of the Hamiltonian we have neglected contributions which are off-resonant by more than 1000 cm-1. 

An important task for theory in the quest for experimental verification of N4 is to provide spectral characteristics that allow its detection. The early computational studies focused on the use of infrared (IR) spectroscopy for the detection process. Unfortunately, due to the high symmetry of N4(7)/) (1), the IR spectrum has only one line of weak intensity [37], Still, this single transition could be used for detection pending that isotopic labeling is employed. Lee and Martin has recently published a very accurate quartic force field of 1, which has allowed the prediction of both absolute frequencies and isotopic shifts that can directly be used for assignment of experimental spectra (see Table 1.) [16]. The force field was computed at the CCSD(T)/cc-pVQZ level with additional corrections for core-correlation effects. The IR-spectrum of N4(T>2 ) (3) consists of two lines, which both have very low intensities [37], To our knowledge, high level calculations of the vibrational frequencies have so far only been performed... 

The refinement calculation may be carried out in a variety of ways, and a few general remarks should be made before we consider particular examples. We wish to determine re, /2, /3, and /4, where these denote symbolically the equilibrium structure (which may be thought of as the linear force field), the quadratic, cubic, and quartic force field. (Terms higher than quartic are not considered here.) Each set of data depends on all constants up to a certain order, as shown in Table 3 for example, Ae, Be, and Ce depend only on re, the oo values depend on re and /2, the a values on re, /2, and /3, and the x values on re, /2, /3, and /4. Ideally one should refine all data simultaneously to all force constants (including the equilibrium structure), but in practice the calculation has to be broken down into steps. Thus usually the equilibrium structure re, or some approximation to re, is determined first from the rotational constants then the quadratic force field /2 is determined from the o>, , and r values holding re constrained then the cubic force field /, is determined from the values holding re and /2 constrained and finally the quartic force field /, is determined from the x values holding re, /2, and/3 constrained. (This should be compared with the discussion for diatomic molecules at the end of Section 3.)... 

Well determined general quadratic + cubic + quartic force field. 6 Moderately... 

Table 5 suggests that one might hope to determine all the constants in the most general anharmonic force field without too much difficulty. The comparison of Suzuki s with Chedin and Cihla s results in Table 6 gives some feel for the reliability of the results obtained. These two calculations were made in different ways (see the original references) although both refined the force field to fit all observed vibrational levels and rotational constants, Suzuki used an up-to-quartic force field, where Chedin and Cihla used an up-to-sextic force... 

Linear Unsymmetric Triatomic Molecules.—Reducing the symmetry from Daoh to Coot, as in NaO, OCS, and HCN, increases the number of parameters in the general quartic force field to 2re + 4/2 + 6f3 + 9/i Table 7 shows their relationship to the primary spectroscopic observables. It is clear that problems of insufficient data to determine the general force field are already on the horizon for example, data from at least two different isotopic species must be combined in order to determine frrr, frrit, fruit, and fium from the observed values of a and a . In practice, of course, substitutions like 14N for 15N tend to change the spectroscopic constants by only a small fraction, and conversely the observed data on the constants of such isotopic species tend to give nearly parallel information on the force field to that obtained from the parent species. For these reasons the anharmonic force field of molecules like N20 is much less well determined than that of C02. These effects are apparent in the uncertainties obtained on the force constants in the refinement calculations referred to in Table 4. 

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. 

Quartic force fields for thiophene have been generated using DFT to evaluate vibrational levels by second-order perturbation theory (PT) and also by the variational method. The results for the fundamental frequencies are in very good agreement with observation <2003SAA1881>. 

R.J. Bartlett, I. Shavitt and G.D. Purvis, The quartic force field of H2O determined by many-body methods that include quadruple excitation effects, J. Chem. Phys., 71 (1979) 281-291. 

General quartic force field with three quartic constraints. 

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

An example of the removal of the harmonic force constant approximation is a study of the quartic force field for H2O at various CC, MBPT, and Cl levels. The energy of 18 vibrational levels are correctly described to within 18 cm, and, with an empirical correction, to within 6 cm" at the CCSDT-1 level. ... 

Perturbation theory has been applied to anharmonic calculations of spectroscopy from ab initio potentials in a large number of studies [19-25,115-121]. In nearly all cases so far, second-order perturbation theory was employed. The representation of the anharmonic potential generally used in these studies is a polynomial in the normal modes, most often a quartic force field. A code implementing this vibrational method was recently incorporated by V. Barone in gaussian [24]. Calculations were carried out for relatively large molecules, such as pyrrole and furan [25], uracil and thiouracil [118], and azabenzenes [119]. We note that in addition to spectroscopy, the ab initio perturbation theoretic algorithms were also applied to the calculation of thermodynamic properties... 

Figure 9 Smoothed CH (v = 3) overtone spectrum (a) and survival probability (b) computed from the DZP/SCF quartic force field. In (a), the peak intensity has been scaled to the value 10. Also in (a), spectra are shown for two values of the anharmonic scaling parameters continuous curve, a = 0.95, b = 0.90 dashed curve, a = b = 1.0.

The procedure we followed is basically the same as that used to fit the HCN surface. Here we started with an ab initio quartic force field, calculated by Martin et al. (96). The fit from this potential proceeded in two steps. First we varied six of the quadratic force constants. Because these force constants determine Z/ 0, these constants could not be optimized using the approaches detailed above. Instead, we fit these constants to the six harmonic frequencies in H2CO. This modification leads to a mean absolute difference for the 138 observed states included in the fits of 4.9 cm. Following the procedures outlined above, we were able to further refine the potential to obtain a final mean absolute deviation of 1.5 cm 1 for these states. 

The vibrational frequencies of HCCF have been calculated with a quartic force field and the measured spectrum analysed using a vibration-rotation Hamiltonian The fitted... 

Internal coordinate quartic force fields have been computed for relatively large systems, e.g., for the 17-atom amino acid L-proline [70]. Nevertheless, despite the fact that electronic-structure programs to compute analytic geometric first and second derivatives of the energy have become available at almost any level [71-74], to the best of our knowledge [68], complete sextic force fields in internal coordinates are available only for a handful of triatomic systems, N2O [75-78], CO2 [79-82], and H2O [48,59]. This is due to several factors. First, it is exceedingly difficult to determine accurate higher-order force constants strictly from experimental information. Second, force fields computed from most electronic-structure... 

Martin, J.M.L., Taylor, PR. Accurate ab initio quartic force field for frans-HNNH and treatment of resonance polyads, Spectrochim. Acta 1997, 53A, 1039-50. 

Huang X, Lee TJ (2(X)8) A procedure for computing accurate ab initio quartic force fields application to HO2 and H2O. J Chem Phys 129 044312... 

R. J. Bartlett, 1. Shavitt, and G. D. Purvis, The Quartic Force Field of H2O Determined by Many-Body Methods that Include Quadruple Excitation Effects, J. Chem. Phys. 71, 281-291 (1979). 

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