The rigid rotor

The reciprocal of t gives the number of cycles per unit time, which is the frequency v of the rotation. The velocity u, may then be expressed as 

Since the linear velocity vector v, is perpendicular to the radius vector r the magnitude Li of the angular momentum is 

We next apply these classical relationships to the rigid diatomic molecule. 

Since the molecule is rotating freely about its center of mass, the potential energy is zero and the classical-mechanical Hamiltonian function H is just the kinetic energy of the two particles, 

If we substitute equation (5.61) for each particle into (5.64) while noting that the angular velocity to must be the same for both particles, we obtain 

In general, moments of inertia are determined relative to an axis of rotation. In this case the axis is perpendicular to the interparticle distance R and passes through the center of mass. Thus, we have 

The calculation of the differential operators goes as above for example, 

Depending on the f orm of the potential, this equation is applicable to a number of problems. We consider them in turn. 

If the entire coordinate space is to be covered, the limits of integration are 0 0 tc  

Suppose the two masses are held rigidly apart at some fixed distance ro. Since there is no momentum in the r direction, the derivative with respect to r cannot appear in the equation. The potential energy is equal to zero since the system rotates freely. Then Eq. (21.60) becomes 

Suppose the center of mass is at the position R then the sum of the second moments of mass about the center of mass is the moment of inertia, /, about any axis perpendicular to the axis of the rotor (Fig. 21.8). From the figure, we have/ = m (0 — Ry + 2( 0 — i ).  

Just as with other angular momenta there is space quantization of rotational angular momentum so that the z component is given by 

Solution of the Schrddinger equation for a rigid rotor shows that the rotational energy is quantized with values 

Question. Using Equation (1.62) calculate, to four significant figures, the rotational energy levels, in joules, for J= 0, 1 and 2 for Then convert these to units of cm. [Use a bond 

Answer. In a question like this close attention to units is extremely important and helpful. This applies particularly to the calculation of the moment of inertia I. Since it has dimensions of mass X length we shall aim for SI base units of kg nr. 

Note that, because E,. is inversely proportional to I, the energy levels are more closely spaced for, for example, 2c 0. 

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The first three Stokes lines in the rotational Raman spectrum of 02 are separated by 14.4 cm, 25.8 cm and 37.4 cm from the exciting radiation. Using the rigid rotor approximation obtain a value for tq. 

Calculating the electronic barrier with an accuracy of 0.1 kcal/mol is only possible for very simple systems. An accuracy of 1 kcal/mol is usually considered a good, but hard to get, level of accuracy. The situation is slightly better for relative energies of stable species, but a 1 kcal/mol accuracy still requires a significant computational effort. Thermodynamic corrections beyond the rigid rotor/harmonic vibrations approximation are therefore rarely performed. 

Classical dynamics is studied as a special case by analyzing the Ehrenfest theorem, coherent states (16) and systems with quasi classical dynamics like the rigid rotor for molecules (17) and the oscillator (18) for various particle systems and for EM field in a laser. 

Thus, the operators H and have the same eigenfunctions, namely, the spherical harmonics Yj iO, q>) as given in equation (5.50). It is customary in discussions of the rigid rotor to replace the quantum number I by the index J m the eigenfunctions and eigenvalues. 

The R(0) transition in 12CO is at 115.271 GHz. Calculate the position of the R(0) transition in 13CO in the rigid rotor approximation. The separation between the energy levels in the R-branch is between levels E(J + 1) — E(J) and in the rigid-rotor approximation is given by ... 

In the rigid rotor approximation, the radial wave function is independent of /. The total wave function is... 

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

This simplification was already introduced in Chapter 2. In the rigid rotor approximation there is no rotational-vibrational interaction. The molecular Schrodinger... 

Table 4.1 Partition functions evaluated in the rigid rotor harmonic oscillator approximation... 

The statement applies not only to chemical equilibrium but also to phase equilibrium. It is obviously true that it also applies to multiple substitutions. Classically isotopes cannot be separated (enriched or depleted) in one molecular species (or phase) from another species (or phase) by chemical equilibrium processes. Statements of this truth appeared clearly in the early chemical literature. The previously derived Equation 4.80 leads to exactly the same conclusion but that equation is limited to the case of an ideal gas in the rigid rotor harmonic oscillator approximation. The present conclusion about isotope effects in classical mechanics is stronger. It only requires the Born-Oppenheimer approximation. 

A2 Corrections to the Rigid Rotor Harmonic Oscillator Approximation in the Calculation of Equilibrium Constants... 

Tables 1 and 2 show the lowest torsional energy levels of hydrogen peroxide and deuterium peroxide which have been determined variationally using as basis functions the rigid rotor solutions. Experimental data are from Camy-Peiret et al [15]. The first set of leval data are from Camy-Peiret et al [15]. The first set of levels (SET I) has been calculated without including the pseudopotential V = 0). The levels corresponding to the other sets (SET II, SET III and SET IV) were obtained including pseudopotentials calculated with different numerical and analytical algorithms. Finally, the zero point vibration energy correction was introduced in the SET V [14],...

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. 

The simplest approach to modeling rotational spectroscopy is the so-called rigid-rotor approximation. In this approximation, the geometry of the molecule is assumed to be constant at the equilibrium geometry qeq. In that case, V(qeq) in Eq. (9.37) becomes simply a multiplicative constant, so that we may write the rigid-rotor rotational Schrodinger equation as... 

Recall that semiempirical methods were parameterized in such a way that the computed electronic energies were equated with heats of formation, not computed enthalpies. Thus, when a semiempirical electronic structure program reports a 298 K heat of formation for AMI, for instance, the reported value derives from adding the atomization energy AE to the experimental 298 K heats of formation of the atoms. Inspection of Figure 10.1 indicates that this will differ from the rigid-rotor-hannonic-oscillator computed result by ZPVE and the differential thermal contributions to the enthalpy of the molecule compared to the atoms. 

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