Generalization of the Badger Rule Based on the Use of Adiabatic Vibrational Modes

Abstract

Empirical relationships relating bond lengths to the corresponding bond stretching frequencies or bond stretching force constants were first derived in 1920s (see Table 4.1 for a summary) and have ever since been a topic of research on the nature of the chemical bond . It is remarkable that in a time of easily accessible quantum chemical results, there remains a need for empirically based estimates of either bond lengths or stretching frequencies. There are three primary reasons why such empirical rules and relationships are still valuable tools for modern research:(1) Established relationships between bond properties add to our understanding of the chemical bond, especially if they can be rationalized on a quantum mechanical basis because bonding between atoms is a quantum mechanical phenomenon.(2) There are experimental situations in which it is relatively easy to measure one bond property but difficult to obtain other bond properties. For example, it is easier to measure the vibrational spectra of a compound than to carry out a structural analysis. This is especially true for solid materials that do not crystallize, molecules on a surface, or molecules in some form of aggregation. If quantum chemical calculations are feasible only for model systems rather than the actual targets of chemical research, then vibrational spectroscopy may be the only tool for obtaining information that provides an insight into bond properties. (3) In the realm of computational chemistry, there is also a need for empirical relationships. They may be used to determine suitable bonding force fields for molecular mechanics utilizing force constant -bond length relationships. For quantum chemical geometry optimization, there is the need to set up a guess matrix of energy second derivatives (the Hessian matrix corresponding to the force constant matrix of a molecule), which is best done with the help of available geometry information and a suitable force constant -bond length relationship. For example, the standard procedure to calculate the geometry of a molecule is based on an initial guess of the energy Hessian derived with the help of the Badger rule [42,55]. It is due to these three reasons that there is ongoing research exploring the relationships between bond length r, bond stretching Computational Spectroscopy: Methods, Experiments and Applications. Edited by J€ org Grunenberg

Strength of the Pnicogen Bond in Complexes Involving Group Va Elements N, P, and As

“…Therefore, the stretching force constants should provide a direct measure of the strength of the corresponding bond and the bending force constants about bond-bond interactions based on hybridization, electrostatic and polarization effects. [95][96][97][98]125,126 In this connection, one has to consider that normal vibrational modes are generally delocalized because of mode coupling. Specific bond information can only be extracted from the calculated or measured vibrational frequencies if the coupling between the vibrational modes is suppressed and local vibrational modes are obtained as has been shown by Cremer and co-workers.…”

“…As noted above, the local mode frequencies calculated can be in principal measured 98,126,127 although experimental means for realizing routine measurements of local frequencies are still limited. Any set of measured frequencies, e.g.…”

“…When comparing a large set of k a values the use of a relative bond strength order (BSO) n is convenient [132][133][134][135]. The relative BSO n(EE') is obtained by utilizing the extended Badger rule,126,135,136 according to which n is related to the local stretching force constant k a by a power relationship, which is fully determined by two reference values and the requirement that for a zero-force constant n becomes zero. In this work, we used…”

A stunning example for a spontaneous reaction with a complex mechanism: the vinylidene–acetylene cycloaddition reaction

“…g ðsÞ, generalized normal mode force constants k g ðsÞ, and generalized normal mode coordinates Q g ðsÞ [23]. It is of advantage to express normal vibrational modes in terms of adiabatic internal coordinate modes (AICoMs) [25,27]. An AICoM a n (s) is a localized, elementary vibrational mode associated with an internal coordinate q n as for example a bond length, a bond angle, or a dihedral angle [25,27].…”

“…Understanding of the reaction mechanism implies a powerful tool for its systematic investigation. The tool we apply in this connection is the Unified Reaction Valley approach (URVA) [14][15][16][17][18][19][20][21][22], which is based on four major assets: (1) the Reaction Path Hamiltonian of Miller et al [23], (2) the IRC of Fukui [24], (3) the generalized adiabatic mode concept of Cremer and co-workers [25][26][27], and (4) the electron density analysis of Kraka and Cremer [28,29], which is based on Bader's virial partitioning analysis [30]. URVA leads to a detailed analysis of the reaction mechanism providing information on both electronic and dynamic changes of the RC along the RP.…”