Free internal rotors

If reduced moment of inertia in amu A - of a rotor bound to an infinite mass 

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Figure C 1.3.4. The real pattern of intennolecular bending energy levels for Ar-HCl (left) compared witli tire pattern expected for a free internal rotor (centre) and a near-rigid bender (right). The allowed transitions are shown in each case. (Taken from 1191.)...

Van der Waals complexes containing monomers with large rotational constants, such as HF, HCl, H2O and NH3, undergo very wide-amplitude bending motions even in low vibrational states. These states are best understood using quantum numbers derived from a free internal rotor picture, rather than those derived from the conventional near-rigid picture. The anisotropy of the potential splits and shifts the monomer free-rotor states, but the qualitative pattern of energy levels and allowed transitions remains. 

The next step in molecular complexity beyond "ball + stick quantum mechanical motion involves interaction between an inert gas with a monomer with internal rotational structure, e.g. Ar + H2O. For this system we have assistance from both the near and far IR. Cohen et al. in the Saykally laboratories have utilized direct absorption far IR spectroscopy to detect two bands (2-II and II-S) in the Ar-H20 complex. These were ascribed initially to rotation-tunneling transitions between K 0 and K=1 manifolds of a quasirigid complex, but more recently have been reinterpreted in terms of near free internal rotor motion of the H2O in the presence of the Ar. There have been recent efforts by Hutson to fit the two bands to an angular intermolecular potential, but there proved to be more important terms in the expansion than data and thus a family of possible curves could be inferred. 

The photodissociation spectrum of HeN recorded in the 391-nm region exhibits two major vibrational bands the stronger band is almost coincident with the B <-X origin transition of N2, while the weaker one, displaced 195 cm" to higher energy from the more intense band, is a hot band of HeN which is associated with the 1-1 transition of N2. The similarity of this spectrum with that of Nj was interpreted by the ion being essentially a free internal rotor in both X and B electronic states. These results [8] confirmed the predicted absence of an appreciable barrier to rotation [9]. 

For lower potentials, line bands appear instead of a clear multiplet fine structure. In this case, the rotational spectrum modified by torsion is analyzed by proceeding from the hmiting case of free internal rotation. In this low barrier method the free internal rotation problem is first solved and the potential is then treated as a perturbation using the free rotor basis set. In this case, the internal rotation states are labeled by m which is a good quantum mrmber when the potential is zero. 

The microwave spectram exhibits a tunnelling sphtting due to a large amphtude internal rotation of the H2O subunit that exchanges bonded and non-bonded atoms. The internal rotor states are labelled with the asymmetric rotor quantum number for the rotational levels of free water jkak. ... 

For example. East and Radom devised a procedure they call El, which calculates from the MP2/6-31G geometry (MP2 calculations are (Uscussed in Section 15.18) and Svib from HF/6-31G scaled vibrational frequencies and the harmonic-oscillator approximation, except that internal rotations with barriers less than 1.4R7 are treated as free rotations [A. L. L. East and L. Radom,/. Chem. Phys., 106,6655 (1997)]. For 19 small molecules with no internal rotors, their El procedure gave gas-phase 5S,298 values with a mean absolute deviation from experiment of only 0.2 J/mol-K and a maximum deviation of 0.6 J/mol-K. The El procedure was in error by up to Ij J/mol-K for molecules with one internal rotor and by up to 2 J/mol-K for molecules with two rotors. An improved procedure called E2 replaces the harmonic-osdllator potential for internal rotors by a cosine potential calculated using the MP2 method and a large basis set, and reduces the error to 1 J/mol-K for one-rotor molecules. 

In most quantum chemical program packages, these equations are used only to calculate the temperamre dependence of thermodynamic properties. Internal free and hindered rotation contributions to the partition functions are normally neglected or implicitly use the pseudo-vibration approach for the internal rotor. 

The effect of using different internal rotor treatments (harmonic oscillator or free rotator approximations) instead of hindered rotor treatment on the calculated reaction rate coefficient is also shown there [63]. 

To capture the flavor of the method, consider the limit of a M-HX complex in which the intermolecular potential is mostly isotropic, but with sufficient anisotropy so the nearly free internal HX rotor motion is oriented in the body fixed frame. The ground state of such a complex would look like a j=0 HX rotor (i.e. an "s" orbital) bound to M, whereas the three lowest excited HX bending states would approximate the three j l rotor wave functions (i.e. three "p" orbitals) oriented with respect to the end-over-end plane of rotation of the M-HX centers of mass, one in a S and two in a II configuration. The j=0 HX rotor state probes predominantly the isotropic part of the intermolecular radial potential, whereas j=l HX rotor states (the S and either one of the II configurations) begin to sample in addition the lowest order anisotropic parts of the potential. The radial dependence of the intermolecular potential for each of these three states can be determined from rotational RKR method. In principle, these curves contain sufficient information to determine the three lowest... 

In conjunction with data on Ar-H20 from the near IR, however, these ambiguities can be resolved. For H2O in a cold jet, complex formation only of the lowest para Ar-H20 (correlating with para Oqo water) and ortho Ar-H20 states (correlating to ortho Iq water) will be appreciable. The (2j+l) degenerate ortho Iq H2O states will be split by interaction with the Ar induced anisotropy to yield states which can be characterized approximately by the projection of the internal rotor angular momentum onto the body fixed axis of the complex (i.e. 2 and II), provided that the splittings are small with respect to the rotational spacings in free water. Note that this K is not the same as the projection of j onto the body fixed axis (k ) of the water monomer, which is a very poor quantum... 

The T-shaped groimd state was fitted as an asymmetric rotor with free internal rotation using the Hamiltonian... 

Internal rotors are accounted for by the factor is sufficient to treat these degrees of freedom as free rotors. If there are r internal free rotors, one has... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

The rotation angle between the two planar pyridyl-rings was found to vary between 18.2 (solid-state [293]) and 37.2 (gas-phase [294]). H-NMR experiments in several solvents of different dielectric constants revealed that 4,4 -BP appears either highly twisted, or as a free rotor. The barrier to internal rotation has been estimated to be 17.0 kJ mol-1. The two rings are rotating almost freely in most liquid environments [295-297]. 

Molecular rotors fluorescence quantum internal torsional very sensitive to free... 

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