Rigid-rotor model

For each value of , there is a degeneracy due to the 2 + 1 values of m. So the rotational partition function is 

For high temperatures, (T drotation)i Equation (11.29) can be approximated by an integral to give 

For nonlinear molecules with three principal moments of inertia L, h, and Ic, the rotational partition function is 

In general, molecules with large moments of inertia have large rotational partition functions. 

EXAMPLE 11.4 The rotational partition function of O2. The length of the bond in an oxygen molecule is R = 1.2074 A, and the mass of each atom is m = 16gmo so the moment of inertia per molecule is 

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Within this "rigid rotor" model, the absorption speetrum of a rigid diatomie moleeule should display a series of peaks, eaeh of whieh eorresponds to a speeifie J ==> J + 1 transition. The energies at whieh these peaks oeeur should grow linearally with J. An example of sueh a progression of rotational lines is shown in the figure below. 

Figure 3.57 Similar to Fig. 3.56(b), for the uncorrelated (RHF) rigid-rotor model. The Lewis (E(L>, squares, light solid line) and non-Lewis ( (NL), circles, light solid line) components of )totai) are shown for comparison. 

As demonstrated in Figure 10.9b for oxygen superrotors N = 69), even in light centrifuged molecules with strong chemical bond, the revival period may deviate by as much as 10% from that predicted by the rigid-rotor model. 

In addition to vibrational motions, the molecule can undergo rotational motion perpendicular to the bond axis. For linear molecules, the energy associated with rotational transitions is approximated by the rigid-rotor model. 

Fig. 3.1. Schematic illustration of the Jacobi coordinates R and 7 for the atom-rigid rotor model with the restriction J = 0.

We recall from our discussion (section 6.8.1) of the rigid rotor model of a diatomic molecule that the Schrodinger equation is... 

The energy of rotational motion can be obtained approximately on the basis of the quantum mechanical rigid-rotor model (9,244). The system under study is assumed to be rigid in its equilibrium configuration. The computation of the equilibrium geometry is also carried out by quantum chemistry. 

The rigid rotor model assumes that the intemuclear distance Risa constant. This is not a bad approximation, since the amplitude of vibration is generally of the order of 1% of i . The Schrbdinger equation for nuclear motion then involves the three-dimensional angular-momentum operator, written J rather than L when it refers to molecular rotation. The solutions to this equation are already known, and we can write... 

Taking into account corrections to the harmonic oscillator-rigid rotor model, vibrational-rotational energies for a diatomic molecule can be represented by... 

For ordinary molecules the rigid rotor model works quite succussfully in explaining pure rotation spectra as well as the rotational fine structure obtained in other fields of spectroscopy. The vibrational perturbations appear mostly in the change of effective rotational constants with the vibrational state and in the centrifugal distortion effects. Much useful information can be found from these perturbation effects, however5. ... 

Weakly bound complexes display unusual structural and dynamical properties resulting from the shape of their intermolecular potential energy surfaces. They show large amplitude internal motions, and do not conform to the dynamics and selection rules based on the harmonic oscillator/rigid rotor models (4). Consequently, conventional models used in the analysis of the spectroscopic data fail, and the knowledge of the full intermolecular potential and dipole/polarizability surfaces is essential to determine the assignments of the observed transitions. 

First, the error limits given in Table II. 1 follow from the experimental uncertainties by standard error propagation and do not account for possible deficiencies of the rigid rotor model (see below). 

Equations (3.28)—(3.31) are based on the harmonic-oscillator/rigid-rotor model. The nonseparable vibrational-rotational i>A, JA levels, with anhar-monicity and vibration-rotation coupling included, may be calculated from Eq. (3.19). The Boltzmann terms for the energy levels, with the (2JA + 1)... 

Microcanonical transition-state theory (TST) assumes that all vibrational-rotational levels for the degrees of freedom orthogonal to the reaction coordinate have equal probabilities of being populated [12]. The quasi-classical normal-mode/rigid-rotor model described above may be used to choose Cartesian coordinates and momenta for these energy levels. Assuming a symmetric top system, the TS energy E is written as... 

By starting from more complex operators, such as (4.104), (4.105), or (4.106), one obtains approximate solutions expressed as power series in the rovibrational quantum numbers. (For a complete discussion, see Ref. 11.) The rotational factor, corresponds to a rigid-rotor model. 

The vibrational and rotational components can be calculated from the harmonic oscillator and rigid rotor models, for example, whose expressions can be found in many textbooks of statistical thermodynamics [20]. If a more sophisticated correction is needed, vibrational anharmonic corrections and the hindered rotor are also valid models to be considered. The translational component can be calculated from the respective partition function or approximated, for example, by 3I2RT, the value found for an ideal monoatomic gas. 

A typical time evolution of fluorescence anisotropy is a monotonously decreasing function. However, the sum of several exponentials with both positive and negative prefactors derived on the basis of a rigid rotor model does not preclude increasing or even a non-monotonous time evolution. The non-monotonous time evolution has been observed for perylene excited to S2 quite far in the blue region with respect to the emission [11], It starts, as predicted for the perpendicular orientation of dipole moments, at ro = -0.2, but increases rapidly to a slightly positive transient value and then decreases more slowly to = 0. The non-monotonous r t) decay can be rationalized by the solvent effect on the rotation of the flat disc-like perylene around three different axes. 

The rigid rotor model has been currently used for small fluorophores, but it is not suitable for tagged or labeled polymers. In this case, r t) monitors a combination of a fast rotation of the fluorophore around one or several single bonds attaching it to the polymer backbone, and a slower complex motion of a part of the chain together with the fluorophore. The relatively slow rotation of the whole polymer coil proceeds on a longer timescale than the fluorescence decay and is invisible in the time-resolved fluorescence measurement. It can be detected as a... 

To predict the properties of materials from the forces on the atoms that comprise them, you need to know the energy ladders. Energy ladders can be derived from spectroscopy or quantum mechanics. Here we describe some of the quantum mechanics that can predict the properties of ideal gases and simple solids. This will be the foundation for chemical reaction equilibria and kinetics in Chapters 13 and 19. Our discussion of quantmn mechanics is limited. We just sketch the basic ideas with the particle-in-a-box model of translational freedom, the harmonic oscillator model for vibrations, and the rigid rotor model for rotations. 

Figure 11.10 In the rigid rotor model of a diatomic molecule, masses mi and m2 are separated by a rigid connector of length R. The origin is at the center of mass, and the angular degrees of freedom are 9 and .

A major success of statistical mechanics is the ability to predict the thermodynamic properties of gases and simple solids from quantum mechanical energy levels. Monatomic gases have translational freedom, which we have treated by using the particle-in-a-box model. Diatomic gases also have vibrational freedom, which we have treated by using the harmonic oscillator model, and rotational freedom, for which we used the rigid-rotor model. The atoms in simple solids can be treated by the Einstein model. More complex systems can require more sophisticated treatments of coupled vibrations or internal rotations or electronic excitations. But these simple models provide a microscopic interpretation of temperature and heat capacity in Chapter 12, and they predict chemical reaction equilibria in Chapter 13, and kinetics in Chapter 19. 

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