Empirical Determination of the Harmonic Force Constants in Benzene. 2. The C−H Stretch System

Abstract

The aim of this work is twofold. In the first part, a detailed description is given of a specific vibrational model, designed for calculations on the vibrational energy levels in benzene and benzene isotopic species of D 6h symmetry. For the description of the C-H stretch system in benzene, a local mode (LM) formalism was applied, while, for the remaining non-C-H stretch vibrations, a symmetrized mode (SM) treatment was applied: this was called the combined LM/SM model. The model is based on a set of complex symmetrized curvilinear vibrational coordinates, which can be expressed as simple linear combinations of Whiffen's coordinates. The description in terms of complex symmetrized coordinates and wave functions allows for the construction of a separable symmetrized infinite-dimensional vibrational basis set, which is of crucial importance for large-scale calculations. In the second part of this work, using the described complex symmetrized LM/ SM vibrational model, calculations have been carried out on a large number of vibrational energy levels of four benzene D 6h isotopomers. The aim of the calculations was to redetermine a reliable set of harmonic force constants for benzene. Some of the force constant values obtained in the present work are substantially different from previous determinations by other authors. Using the presently determined set of harmonic force constant values in the calculations, a very good fit has been obtained to a large number of experimentally measured vibrational (both fundamental and overtone) energy levels of various symmetries, belonging to all four D 6h benzene isotopomers: C

Determination of an improved set of harmonic force constants for benzene

“…This consists of the following. Complex symmetry species (CSS) have been defined for the benzene point group D 6h 24 (such CSS can be defined in analogous manner for all symmetric top point groups 23), which replace the conventional real symmetry species 1, 25. The main advantage of the former over the latter, is that they are multiplicative, behaving effectively as nondegenerate species, analogously to the case of the abelian groups, unlike the two‐dimensional (2‐D) representation of symmetric tops, for which character tables have to be used, instead of direct multiplication tables.…”

“…The main advantage of the former over the latter, is that they are multiplicative, behaving effectively as nondegenerate species, analogously to the case of the abelian groups, unlike the two‐dimensional (2‐D) representation of symmetric tops, for which character tables have to be used, instead of direct multiplication tables. The CSS types of D 6h as well as their multiplication rules have been explained in detail and tabulated in our recent work 24. Curvilinear complex symmetrized vibrational coordinates q k for benzene have been introduced, corresponding (and transforming according) to the CSS of D 6h , instead of the Whiffen's symmetrized coordinates S k 3 ( q k can be expressed as simple linear combinations of the S k coordinates).…”

“…Curvilinear complex symmetrized vibrational coordinates q k for benzene have been introduced, corresponding (and transforming according) to the CSS of D 6h , instead of the Whiffen's symmetrized coordinates S k 3 ( q k can be expressed as simple linear combinations of the S k coordinates). Expressions for q k in terms of the local bond stretches and angle distortion coordinates in benzene 1 have been discussed and tabulated in our preceding work 24.…”

“…In this LM description, the basis CH stretch vibrational wavefunctions φ CH , are obtained as appropriately symmetrized linear combinations of products of six (local bond) Morse oscillator eigenfunctions. The explicit form and algorithm for generation of these (complex) symmetrized wave functions φ CH , describing the CH stretch overtone system of states in benzene, have been presented and discussed in full detail in our recent work 24.…”

“…In a series of works, we have developed and consistently elaborated a vibrational method and computational code, especially designed for implementation of large‐scale calculations on the vibrational energy levels (of both fundamental as well as higher excited vibrational frequencies) of benzene and all its D 6h isotopomers 22–24. The method is characterized by the following key features, which will be detailed in the following section: (i) fully symmetrized, based on a symmetrized, separable (in product form), infinite dimensional vibrational basis set and vibrational coordinates; (ii) based on combined local mode (LM) treatment (for the CH stretch vibrations) and symmetrized mode (SM) description of the remaining (ring) modes; and (iii) nonperturbative and based on an artificial intelligence (AI) search and selection procedure, combined with large‐scale Hamiltonian matrix manipulations.…”