Complex symmeterized analysis of benzene vibrations

Abstract

ABSTRACT:The aim of this work is to introduce specific complex coordinates and wave functions (Hamiltonian eigenfunctions) for the description of vibrational motion in benzene. When suitably chosen, these complex functions are shown to possess interesting transformation behavior under the symmetry operations of the molecular symmetrical top point group D 6 . This behavior is analyzed and classified as "complex symmetry types" (CSTs). CSTs can be defined for all symmetrical top point groups. The description in terms of complex symmeterized coordinates and wave functions allows for the construction of a separable symmeterized vibrational basis set for benzene. New results from calculations on vibrational energies and redeterminations of harmonic force constants in benzene, obtained using the CST description, are presented.

Hamiltonian description and 6D calculations on the ammonia vibrational levels

“…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]):…”

“…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].…”

Complex symmetrized calculations on ammonia vibrational levels

“…Simultaneously, the (real symmetric) Hamiltonian matrix H i,k of the vibrational problem is being composed, whose diagonal elements are k| H 0 + H int |k and the off-diagonal elements of which are given by k| H int |k ′ . For this procedure, as described in detail in our previous work [30,31], the availability of a symmetrized separable vibrational basis set |L;S;p 1 k 2 l 3 ) × (n 4a ,n 4b ) |n 2 is of crucial importance. Next, H i,k is diagonalized making use of a Lanczos tridiagonalization routine [34], thus obtaining the molecular vibrational levels.…”

“…In a recent work series [28][29][30][31], we introduced the concept of complex symmetry species (CSS) to describe the transformation properties of symmetric top point groups (and particularly the benzene group, D 6h ), instead of the conventional real symmetry species [1,7,27]. This allowed to greatly reduce the computational effort for large scale vibrational calculations in symmetric top molecules, and particularly for benzene [31].…”

“…The basic asset of CSS is that they allow the straightforward construction of quantum mechanical basis sets in product form, which are particularly suitable to describe those molecules belonging to a symmetric top point group. Such a set is most helpful to explore highly excited molecular vibrational levels, in the range of excessively high vibrational level density, as demonstrated in our recent work on benzene [28][29][30][31]. The introduction of complex symmetry species for the symmetric top point group of ammonia D 3h and their transformation properties and multiplication rules, are displayed in Table 1.…”