*2.6. Local Mode Analysis*

The Local Mode Analysis allowed identification of force constants for several diatomic stretching motions for all species ONE through ELEVEN. These included the two C=O stretches for ONE (Table 4) and the single C=O stretches for TWO through ELEVEN (Table 5). CO and OH stretches were also defined for TWO through ELEVEN. The three methyl CH stretches and the two NH stretches were chosen for all species. For systems which can plausibly be assigned hydrogen bonding structures, the X ... HY stretches were included in the analysis.


**Table 4.** For species ONE, local force constants (millidynes/Å) and associated frequencies (cm−1).


**Table 5.** For species TWO–ELEVEN, local force constants (millidynes/Å) and associated frequencies (cm−1).

1 Atoms linked by the hydrogen bond. Local force constants are based on computed values for TWO and ELEVEN, which provided parameters for a power law fit to BCP densities. Local frequencies are inferred from scaling to the square root of local force constants.

Local CH stretches span a narrow range of frequencies, ca. 3100 to 3200 cm<sup>−</sup>1. There seems to be no grea<sup>t</sup> impact on these values even in systems that may have CH ... O or CH . . . N interactions.

C=O stretches extend from 1650 to near 1700. The exception is species TWO at 1550, which has strong =O ... HO hydrogen bonding. OH stretches which do not participate on hydrogen bonding have frequencies near 3800 to 3850 (THREE, SEVEN, EIGHT and NINE) while TWO and FIVE have seriously reduced OH stretching frequencies, in keeping with their participation in hydrogen bonding.

Species TWO, THREE, and SIX have both local NH force constants above 7.2 and both local frequencies above 3600. These systems do not involve the NH2 group in H bonding. ONE and ELEVEN engage one NH bond in hydrogen bonding which is reflected in one low NH stretching frequency (3564 and 3212). FOUR is a puzzling exception, with nearidentical NH force contants. TEN shows both NH force constants of 6.8 and frequencies near 3500, and FIVE has its two modes with force constants near 7.1 and frequencies near 3579. EIGHT and NINE have one local mode with low frequency (below 3600) with the other mode higher than 3600. These values are consistent with an interaction between substituents OH and NH2 on an unsaturated carbon.

Yannacone et al. [18] established an empirical power law relation between density at a bond critical point and local force constants, of the form

$$\ln\left(k\_{\rm HBOND}\right) = \text{A } \ln(\rho\_{\rm BCP}) + \text{B}\_{\prime} \tag{2}$$

We fit the computed local force constants for TWO and ELEVEN to this form, finding A = 0.740 and B = 0.934, and inferred the remaining values reported in Table 4. The sequence of H bond strength begins with the strongest interactions, found in TWO > ELEVEN > FIVE which have *kLOCAL* above 0.200. The central cluster includes ONE, FOUR, and SIX with NH ... O=, >NH ... OH, and CH ... O= interactions. The weakest interactions (in descending order) are CH ... N, CH ... N, and CH ... OH. It is interesting to see that the strongest CH . . . X interaction (with X a carbonyl oxygen) is comparable to NH . . . O interactions.
