Zero-order frequencies

In the non-rigid bender approximation, we solved the inverse eigenvalue problem described by Eq. (5.4), i.e. we determined the potential function parameters given in Table 3 for NX3 (X = H, D, T). We have used the experimental infrared frequencies of transitions from the ground state to the i>2,2 2 > 2. and 41 2 inversion states and the zero-order frequencies of vibrations (Table 4). The zero-order frequencies have been obtained from the observed fundamental frequencies of NH3 [Ref. >], ND3 [Ref. °>], NTg [Refs." and [Ref.- 3)] corrected for... 

On the other hand, we were concerned with consistency within our calculations. Thus, we used the same values for atomic masses throughout the calculations. (For Cl and Br, the masses of the most abundant isotopes were used.) Therefore, some of our calculated frequencies do not agree exactly with previously published data. Further, again for consistency, the force fields used were based exclusively on fundamental frequencies even though, in a few cases, data for zero-order frequencies are available. In order to permit replication of our results, the force constants actually used in the calculations are reported to as many as nine significant figures, which is, of course, far beyond the precision justified from present spectroscopic methods. 

Two kinds of basic data can be used for calculating the anharmonic force constants. The zero-order frequencies G)s and anharmonic constants x , in the second-order formula for the vibrational energy v,... 

Figure 3.38. Zero-order (frequency independent) phase errors arise when the phase of the detected NMR signals does not match the phase of the receiver reference rf. All resonances in the spectrum are affected to the same extent.

For isotope exchange reactions written as above involving only isotopically pure molecules (e g. pure C 0 or C 0), the symmetry number of a molecule and its isotopic derivative are identical. Therefore, o/a = 1, and the term need not be included in the calculations. The vibrational frequencies used in our calculations are the same as those on which Urey s (1947) calculations are based. However, Urey corrected these frequencies for anharmonicity (zero-order frequencies), whereas we used observed (measured) fundamental frequencies with no anharmonicity correction (see discussions in Bottinga 1969a, p. 52 McMillan 1985, p. 15 and Polyakov and Kharlashina 1995, p. 2568). Vibrational frequencies are generally reported in wave numbers (co), which have units of cm . For partition function calculations, wave numbers must be converted to units of sec by multiplying by c, the velocity of light (v = cco). 

The calculations above are made on the basis of the harmonic oscillator approximation, and include no explicit corrections for anharmonicity. The effects of anharmonic-ity can be incorporated by using calculated zero-order frequencies rather than observed fundamental frequencies (see above), and by adding anharmonic corrections to the ZPE and energy level spacing terms of Equation (9) (Urey 1947 Bottinga 1968 Richet et al. 1977). Urey (1947) included anharmonic effects in his calculations of partition function... 

Observed fundamental vibrational frequencies for CO and CO2 corresponding to zero-order frequencies (corrected for anharmonicity) given in Urey (1947). The vibrational frequencies of the C 0 and C 02 molecules (co ) were calculated using Equation (4) (see text). 

Precise control of the relative phase between pulses is cmcial to the success of many multi-pulse NMR experiments, and some correction to the phase of a soft pulse may be required to maintain these relationships when both hard and soft pulses are to be applied to the same nucleus. When soft pulses are used on the observe channel, the phase difference (which may arise because of the potentially different rf paths used for high- and low-power pulses) may be determined by direct inspection of two separate ID pulse-acquire spectra recorded with high- tmd low-power pulses but under otherwise identical conditions. Using only zero-order (frequency-independent) phase correction of each spectrum, the difference in the resulting phase constants (soft minus hard) represents the phase difference between the high- and low-power rf routes. Adding this as a constant offset to the soft pulse phase should yield spectra of phase identical to that of the hard-pulse spectmm when processed identically, and this correction can be used in all subsequent experiments, provided the soft-pulse power remains unchanged. 

The zero-order frequencies must be close together (typically within 30 cm ). 

The zero-order frequencies are those of the unperturbed groups. These will be either the completely isolated groups or the groups in the static field of the rest of the polymer. 

If the solvent is changed, the interacting zero-order frequencies shift slightly bnt differently and the relative intensities of the two bands change a lot. In inert solvents, 1750 cm is the more intense. In polar and hydrogen-bonding solvents, 1730 cm is the more intense. ... 

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