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Boletín de la Sociedad Chilena de Química

versión impresa ISSN 0366-1644

Bol. Soc. Chil. Quím. v.47 n.4 Concepción dic. 2002 


Jorge Ricardo Letelier D.*a , Alejandro Toro-Labbéb
Ying-Nan Chiuc

aDepartamento de Química, Facultad de Ciencias Físicas y Matemáticas
Universidad de Chile, Ave. Tupper 2069, Casilla 2777, Santiago, Chile.
bFacultad de Química, Universidad Católica de Chile.
Vicuña Mackenna 4860, Santiago, Chile.
cDepartment of Chemistry, The Catholic University ofAmerica
Washington, DC 20064, USA

(Received: March 06, 2002 - Accepted: June 10, 2002)


The geometrical foundation of the Hückel-Möbius concept is revisited and used to represent approximate torsional wavefunctions of the hindered internal rotation of a coaxial (XYn)2 type molecule, constructing linear combinations of vibrational wavefunctions (LCVW) centered at the periodic potential minima.

Making use of the sign variation of the "resonance" interaction integral with different vibrational levels, we construct energy levels that form the cluster of properly alternating non-degenerate and degenerate levels for the high-barrier case.

Keywords: Torsional Vibration, Hindered rotation eigenvalues


Se revisan los fundamentos geométricos del concepto Hückel-Möbius utilizado para representar funciones de onda torsionales aproximadas para la rotación interna impedida de moléculas tipo (XYn)2 coaxiales, construyendo combinaciones lineales de funciones vibracionales (LCVW) centradas en lo mínimos del potencial periódico.

Haciendo uso de la variación de signo de la integral de interacción "resonante", construimos el patrón de niveles de energía que conforman el cluster de niveles de energía alternantes, degenerados y no degenerados, para el caso de una barrera alta.

PALABRAS CLAVES: Vibración torcional, autovalores, totación impedida.


Hindered internal rotation has been of interest for a long time since its discovery in ethane by Kemp and Pitzer1 and its early treatment by Pitzer and by Wilson2. It has been thoroughly reviewed from the point of view of microwave spectroscopy3. Comprehensive symmetry treatments generally make use of Longuet-Higgins' symmetry group of non-rigid molecules. These treatments were motivated by the work of Longuet-Higgins4, Bunker 5 and Hougen6 .These treatments use a molecular double point group and arrive at symmetry classifications of vibrational and rotational levels. Specifically, for hindered internal rotation of ethane, hydrazine, ethylene and (XY2)2 coaxial rotors7, these treatments lead to the cluster of energy levels with appropriate alternation of non-degenerate (A) and degenerate (E) levels required by symmetry as shown in the work of Harter and Patterson8.

The general pattern and energy clusters for two coaxial rotors of equal groups (XYn)2, that makes use of the Hückel-Möbius concept has been derived and thoroughly discussed by Chiu9. In this work, we construct torsional-vibrational wavefunctions by taking appropriate linear combination vibrational wavefunctions (LCVW) centered at potential minima of the hindered (torsional) rotator with the purpose of approximating these torsional wavefunctions at high barrier. The process is analogous to taking linear combinations of atomic orbitals (LCAO) to construct molecular orbitals. Chiu went further to show that these LCVW have the desired transformation properties under the appropriate operations of the double group of Bunker and Hougen.


We briefly review here the basis of the method; detailed analysis can be found elsewhere9. Following Hougen6 and Bunker5 we consider the combined rotational wavefunctions with respect to the common axis of the two groups ca and cb, with rotational angles c a and c b respectively.


Where K = ka + kb is the angular momentum for the overall rotation of the whole molecule around the common axis, and k = (ka - kb) is for the free relative internal rotation of the two groups during which the overall moment-of-inertia axes are not changed. When the barrier is high this free relative internal rotation becomes torsional vibration. Because of the 1/2 factor in the definition of the overall rotational angles and the n-fold rotation of either the a or b group of (XYn), Cn, will produce half of the expected (2p /n) change in c and g . This is the reason why the potential for internal rotation has 2n-fold rather than n-fold symmetry. Furthermore the n-times repeated n-fold rotation,, of a or b, while regenerating the same physical configuration of the molecule, may or may not lead to the same F rot or F tor depending on whether K or k is even or odd, viz.


The double-valued nature or , is the basis for the double-group treatment. However, because the rotational wavefunction as a whole must be invariant under either or we see immediately that the even must occur with even (and odd Ko with odd Ko). This is so that the change c and g to c + p and g + p will leave the total rotational wavefunction Y invariant which is a physical basis6 of the double group. In other words, two orientations are equivalent, and the number of symmetry operators is doubled,




where the combinations and will always be even, guaranteeing invariance of Y in (3b) under two orientations.

With the above as background, the LCVW torsional wavefunctions that we construct must have those with odd property combining with odd K (= 1, 3 ...) and those with even combining with even K (= 0, 2 ...). It is found that the Hückel combination will always have even property and Möbius combination will always have odd property and there is just the right number of each. To show this we recall that because of the mathematical definition of g = 1/2(c a - c b), there are 2n minima for the n-fold symmetry group XYn, in coaxial-(XYn)2. By simple cyclic group theory, there are 2n linear combinations, notoriously of the Hückel type, of the vibrational wavefunctions with appropriate energies similar to LCAO-MO for periodic cyclic systems, as follows

Where k = 0, 1, 2, ... (2n- 1). To reach equation (4b), it has been assumed only nearest neighbor interaction, where b is related to the transmission coefficient for tunneling from one potential minimum to the next. It is the analog of the resonance integral in LCAO theory. We then divide the linear combinations into two sets: one with even k e(= 2L ) and one with odd k o(= 2L ' + 1). We retrieve the n-fold symmetry in each set before we apply the or , operator to test their Hückel or Möbius property, viz.



k e= 2L



and where L' are the integers:

k o= 2L ' + 1

Where the superscripts o and e denote odd and even numbers respectively and where the n-fold symmetry is apparent, based on summation index from 0 to n-1 in the wavefunctions and based on the double-group characters .


The concepts and methods of calculation, developed in the previous section, are now applied specifically to the case of the (X-Yn)2 -type molecules, where n = 2, 3, 4, 5.

In the foregoing analysis, the overall rotational angles and , as well as the angles c a and c b are referred to laboratory coordinates. In this context, g truly gives rise to a double-valued problem. We simplify the analysis by choosing instead the internal (referred to molecular axes) relative torsional angle q and the problem becomes then a n-fold symmetry instead of a 2n-fold one. The torsional wavefunctions and energies are now:

The potential used has the general form


Where , is scaling constant used to express energy in units of the rotational energy of a rotor of whose moment of inertia is , a0 being the unit of atomic length (ie. first Bohr radius). In this manner, the handling of potential functions and energy values are dimensionless. As an example, a typical cyclic potential of 4-fold symmetry is shown in Figure 1.

Fig 1. The n-fold potential energy function, as given by equation (8), for the relative torsional oscillation. The basis functions for the LCVW are taken as the solutions of the portion A-B.

In this calculation, the wavefunctions F s of equation (7a) are taken as the solutions of a one-dimensional oscillator, whose potential corresponds to the portion around one of the minimum of the cyclic potential.

Fig. 2. Wave functions, as given by equation (7a) for k =1, 3, 5 (odd) for a torsional oscillator of 2-fold symmetry. Potential minima are found at q = p /2 and q = 3p /2.

With reference to Figure 1, we solved the eigenvalue problem within the portion A-B indicated there and because the potential function V(q ) in eq. (8) is far from harmonic, the solutions are found numerically using other numerical methods developed by us11. The eigenfunctions are then of numerical nature, therefore, the linear combinations of used in eq. (7a) are also formed numerically. Within this approximation, only the lowest clusters of eigenvalues are considered to be reasonably well described.

Fig. 3.-Wave functions, as given by equation (7a) for k =2, 4, 6 (even) for a torsional oscillator of 2-fold symmetry. Potential minima are found at q = p /2 and q = 3p /2.

Fig. 4.-Wave functions, as given by equation (7a) for k =1, 3, 5 (odd) for a torsional oscillator of 3-fold symmetry. Potential minima are found at q = p /4, p and q = 3p /4.

Fig. 5.-Wave functions, as given by equation (7a) for k =2, 4, 6 (even) for a torsional oscillator of 3-fold symmetry. Potential minima are found at q = p /4, p and q = 3p /4

Within a given cluster of eigenvalues, the parameter a is set equal to the corresponding eigenvalue (v = 1, 2, ...) of the local potential (as shown schematically in Figure 1). The cluster of eigenvalues then in eq. (7b) corresponds to the splitting due to interaction between the displaced torsional oscillators.

The "Resonance" Integral.

Due to the analogy of the LCVW method to LCAO-MO it seems natural and tempting to use a similar approximation for the resonance integral b by making it proportional to the overlap. There is a difference however, which is that these are Franck-Condon overlap integrals between displaced oscillators at neighboring minima (if nearest neighbor approximation is considered). This "resonance integral", which determines the splitting of torsional levels, is also related to the tunneling probability, for this reason Chiu9, has coined them "tunneling- interaction" integral.

Since the value of b depends on the width, as well as the height and shape of the barrier, there is no simple way of knowing nor estimating it a priori. The effect of overlap can be incorporated (and also the sign alternation that comes with it) by using tables of Franck-Condon overlap integrals Rvv’ for displaced harmonic oscillator12 (HO), if effectively harmonic oscillators wavefunctions are used. In our case, since we used a potential of the form give in equation (8), the wavefunctions in eq. (7a) are not HO. Again, the Franck-Condon overlaps were carried out numerically and b is made proportional to the corresponding Rvv’. In our calculations, we have considered every quantity equal to unity, that is, torsional moment of inertia =1.0, etc. to make it system independent.

Results for the lowest eigenvalue cluster for several n-fold symmetry potentials are given in Table I. In this table, the cluster pattern is readily observed. Their corresponding wavefunctions are depicted in figures 2 to 5 (for n = 2- and 3-fold symmetry only). In these figures, the alternation pattern of double-single level is apparent.

Table I

Clusters of torsional vibration eigenvalues for the n-fold multiple-well potential
for (X-Yn)2 type molecules ( n =2,3,4,5). Wn refers to the number of well minima.
(Energy is given in arbitrary (dimensionless) units.)









































These results, although contain many approximations, qualitatively describe the energy level pattern and are in reasonably good agreement with those calculated by direct numerical integration11.

Improvements of the method.

The energy values computed using this method can be improved by adding non- nearest neighbor interactions, much in the same way as it is found in the treatment of linear crystals with periodic boundary conditions10. This requires the inclusion of several types of "resonance integrals" b 1, b2, b3, appropriate to first-, second-, third-neighbor interaction, and so on. Energies and wave functions would be as follow:


As an extension, the method can also be adapted to include non-symmetric cyclic potentials. As an example, consider the case of periodic cyclic potentials where local potential of different depth alternate, we have then


Here F (1) and F (2) represent the vibrational wave functions of the two local potentials and only nearest-neighbor interaction has been considered.


Funding for this work by Fondecyt-Chile, Grant N° 1000971 and the Chiu-Feng Chia Research Fund are gratefully acknowledged.


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