Vibrational error statistics

The so called G2 set of molecules is nowadays considered a standard for the validation of new quantum chemical approaches [72]. Table V collects an error statistic for several quantum mechanical approaches concerning the geometric and thermodynamic parameters of 32 molecules belonging to the G2 set, together with dipole moments and harmonic vibrational frequencies. 

The error statistics in Table 9 shows that the harmonic vibrational frequencies predicted with CCSD(T) and QCISD... 

Vibrational broadening in [162] was taken into account under the conventional assumption that contributions of vibrational dephasing and rotational relaxation to contour width are additive as in Eq. (3.49). This approximation provides the largest error at low densities, when the contour is significantly asymmetric and the perturbation theory does not work. In the frame of impact theory these relaxation processes may be separated more correctly under assumption of their statistical independence. Inclusion of dephasing causes appearance of a factor... 

Notice, however, that the preceding analysis gives only an upper limit and an average, or rms value, of position errors, and further, that the errors result from the limits of accuracy in the data. There are also two important physical (as opposed to statistical) reasons for uncertainty in atom positions thermal motion and disorder. Thermal motion refers to vibration of an atom about its rest position. Disorder refers to atoms or groups of atoms that do not occupy the same position in every unit cell, in every asymmetric unit, or in every molecule within an asymmetric unit. The temperature factor Bj obtained during refinement reflects both the thermal motion and the disorder of atom j, making it difficult to sort out these two sources of uncertainty. 

Computational Techniques. For evaluation of kf by eqs. (7) and (8), the vibrational energy level sum at a given total energy must be found. It has been shown, both through experiment and computation,9 19 20 that for polyatomic molecules, even with energies above 100 kcal. mole-1, it is necessary to use a quantum statistical treatment to find this sum. Classical approximations are totally inadequate and drastically in error. High speed machine computational techniques and simplified approximation formulas have been developed, which allow this quantum-statistical summation to be done with relative ease these methods are described and summarized in Appendix I. 

According to Malyj and Griffiths (1983), determining the equilibrium rotational or vibrational temperature by the Stokes/anti-Stokes ratio is not as simple and straightforward as the equations imply. The authors discuss the problems which evolve as a result of using standard lamps and show how to meet these difficulties by using reference materials to measure the temperature as well as to determine the instrumental spectral response function. The list of suitable materials includes vitreous silica and liquid cyclohexane, which are easy to handle and available in most laboratories. The publication includes a detailed statistical analysis of systematic errors and also describes tests with a number of transparent materials. 

The failure of Eayleigh s law may be ascribed either to an error in the statistical calculation, or to the inapplicability of the ordinary laws of mechanics to very rapid vibrations. Adopt-... 

Minimization of the error is performed numerically, and the results are presented in Table 28-4. A lower characteristic vibrational temperature is required in the statistical expression for Cp as n increases. This is consistent with the following facts ... 

If one looks at a molecule such as heptane, for example, one can add all of the appropriate increments and calculate the heat of formation with acceptable accuracy by the method previously described. But there are a few things that are really not proper about that kind of calculation. Heptane in the gas phase at 25°C (where heats of formation are defined) is actually a complicated mixture (a Boltzmann distribution) of a great many conformations, most of which have different enthalpies and entropies. Additionally, each of these conformations is also a Boltzmann distribution over the possible translational, vibrational, and rotational states. The Benson method works adequately for many cases like this because these statistical mechanical terms can be lumped into the increments and averaged out, and they are not explicitly considered. By adjusting the values of the parameters in Eq. (11.1) or (11.2), much of the resulting error of neglecting the statistical mechanics can be canceled out, or at least minimized, in simple cases. But we would like for this scheme to work for more complex cases too. As the system becomes more complicated, errors tend to cancel out less well. So let us go back and approach this problem in a more proper way. 

Table 6.2 Statistical Analysis of Relative Errors (%) in Computed Rotational Constants with Respect to Values Derived from Experimental Bq Values by Subtracting Computed Vibrational and Electronic Contributions...

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