On the Empirical Determination of Some Harmonic and Anharmonic Force Constants in Benzene

Abstract

In this work, the problem for the quality of empirically determined harmonic force constants in ground electronic state benzene has been carefully reexamined, for the case when strongly anharmonic vibrations are involved and in particular for the A 1g (ν 1 and ν 2 ) vibrational system. A numerical procedure, based on a local bond Hamiltonian representation for the C-H stretch system and a symmetrized coordinate treatment for the ν 1 (C-C) mode, has been described and applied to the determination of the harmonic F 1,1 , F 2,2 , and F 1,2 , and some important (diagonal) anharmonic force constants, instead of the traditional FG analysis. As a reference data for the determination of the required force constants, the set of experimentally observed ν 1 and ν 2 fundamentals for four D 6h benzene species C 6 H 6 , C 6 D 6 , 13 C 6 H 6 , and 13 C 6 D 6 have been employed. The harmonic force constants as well as harmonic frequencies obtained in this work have substantial deviations from previous determinations. Using the presently determined force constant values, a very good fit of the calculated to the experimentally observed frequencies has been achieved.

Complex symmeterized analysis of benzene vibrations

“…For the description of the group of strongly anharmonic CH stretching modes, it is not convenient to use symmeterized or normal coordinates but far more practical to use local bond coordinates 5–8. Thus, the CH stretch modes in benzene are best described as a set of six identical Morse oscillators, one for each of the local bond CH stretches s i (characterized by harmonic f SS and cubic diagonal f SSS force constants), which are weakly coupled to each other (by the small harmonic nondiagonal force constants f 1,2 , f 1,3 , f 1,4 ) 1, 5–9. In this local mode (LM) picture, the anharmonicity is entirely concentrated in the local bond force constant f SSS of a single Morse oscillator, while in terms of symmetry‐adapted coordinates and moreover in normal coordinates a large number of strong (cubic) anharmonic force constants arise, which makes the description in such coordinates cumbersome and nontransparent 9–11.…”

“…Thus, the CH stretch modes in benzene are best described as a set of six identical Morse oscillators, one for each of the local bond CH stretches s i (characterized by harmonic f SS and cubic diagonal f SSS force constants), which are weakly coupled to each other (by the small harmonic nondiagonal force constants f 1,2 , f 1,3 , f 1,4 ) 1, 5–9. In this local mode (LM) picture, the anharmonicity is entirely concentrated in the local bond force constant f SSS of a single Morse oscillator, while in terms of symmetry‐adapted coordinates and moreover in normal coordinates a large number of strong (cubic) anharmonic force constants arise, which makes the description in such coordinates cumbersome and nontransparent 9–11. Following Zhang et al 8, we employed a combined LM (for the CH stretches) + SM (symmeterized modes, for the non‐CH stretch or “ring” modes) description of vibrational motion in benzene.…”

Hamiltonian description and 6D calculations on the ammonia vibrational levels

“…Basis sets of this type for symmetric top molecules have been been defined by Della Valle [27], for the investigation of the relationship between normal and local modes and by Hougen, Mills and others [33][34][35][36], for the study of vibrational-rotational interactions. A complex symmetrized basis approach was employed in our work on the benzene vibrational system, allowing for a crucial reduction in size of the Hamiltonian matrices involved in the vibrational calculations [37][38][39].…”

“…To obtain complex symmetry adapted orthogonal N-H stretch wavefunctions u, corresponding to a configuration p 1 k 2 l 3 , appropriate linear combinations must be taken (obtained by consecutive rotation of the original configuration by 2p/3 around the N-atom, each time multiplying by an appropriate factor C k , similarly to the symmetrization scheme for benzene [37][38][39]):…”

“…The vibrational computational method, previously developed and applied to benzene [37][38][39], has been adapted in the present work for the calculation of vibrational levels in ammonia and will be briefly outlined in this section. The method is nonperturbative and based on the well known AI search procedure, developed and mainly applied for large scale calculations on highly excited vibrational states in polyatomic molecules [42][43][44][45][46][47].…”