Rigid rotor energy levels

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.)...

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

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. 

Molecules in the gas phase have rotational freedom, and the vibrational transitions are accompanied by rotational transitions. For a rigid rotor that vibrates as a harmonic oscillator the expression for the available energy levels is ... 

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 ... 

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],...

Treatment of the rotational motion at the zeroth-order level described above introduces the so-called rigid rotor energy levels and wavefunctions Ej = h2 J(J+l)/(2 4,Re2) and Yjm (0,( )) these same quantities arise when the diatomic molecule is treated as a rigid rod of length Re. The spacings between successive rotational levels within this approximation are... 

Fig. 4. Energy levels of 156Er, -i-, and 158Er, -9-, minus a rigid-rotor energy vs. spin, I.

For the rotational partition function, we first consider a linear rigid rotor. The energy levels are Ej = J J + l)ft2/(2/), with J = 0,1,..., and I is the moment of inertia. Each energy level has a degeneracy of mj = 2J + 1. The partition function takes the form... 

In practice values of B are also often quoted in cm-1. For the simple rigid rotor the rotational quantum number J takes integral values, J = 0, 1, 2, etc. The rotational energy levels therefore have energies 0, 2B, 6B, 12B, etc. Elsewhere in this book we will describe the theory of electric dipole transition probabilities and will show that for a diatomic molecule possessing a permanent electric dipole moment, transitions between the rotational levels obey the simple selection rule A J = 1. The rotational spectrum of the simple rigid rotor therefore consists of a series of equidistant absorption lines with frequencies 2B, 4B, 6B, etc. 

If there was no interaction between vibration and rotation, the energy levels would be given by the simple sum of the expression giving the vibrational levels for the anharmonic oscillator, equation (6.188), and that describing the rotational levels of the rigid rotor, equation (6.162). There is an interaction, however during a vibration the moment of inertia of the molecule changes, and therefore so also does the rotational constant. We may therefore use a mean value of Bv for the rotational constant of the vibrational level considered, i.e. 

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