1. ROTATIONAL PARTITION FUNCTIONS:  We will consider linear molecules only. Usually q Rotational » q Vibrational. This is because: 1. rotational energy level spacings are very small compared to vibrational spacings and 2. each rotational level has a 2J+1 fold degeneracy. Due to degeneracy the populations of higher J levels are much higher than would be otherwise expected.

2. ROTATIONAL PARTITION FUNCTIONS 

3. ROTATIONAL PARTITION FUNCTIONS: 

4. 

5.  The last formula is “valid” (i.e. a good approximation) for almost all unsymmetrical linear molecules. Aside: For symmetrical linear molecules rotational levels may not all be populated. Only half are populated for 16 O 2 (all are populated for 16 O 18 O!). We need a symmetry number, σ, equal to 1 normally, or 2 for symmetric linear molecules.

6. ROTATIONAL PARTITION FUNCTIONS: 

7. TYPICAL PARTITION FUNCTION VALUES: Molecule B(MHz) σ q Rot (300K) H2H2 1,824,300 2 1.71 H 35 Cl 312,991 1 20.0 D 35 Cl 161,656 1 38.7 16 O 2 43,101 2 72.5 CsI 708.3 1 8830 H-C≡C-F 9706 1 644

8. PARTITION FUNCTION COMMENTS: 

9. ROTATIONAL LEVEL POPULATIONS – CO: J2J+1PiPi 01110.00927 130.98162.9450.02729 250.94594.7300.04383 5110.75738.3300.07720 8170.51318.7230.08085 10210.36087.5780.07023 15310.10823.3530.03108 20410.002040.83660.00775 25510.002420.12350.00114

10. COMMENTS ON PREVIOUS SLIDE: 

11. COMMENTS – CONTINUED:  Less than 1% of CO molecules are in the J=0 level at 300K.(More than 99.99% of CO molecules are in the v=0 level at 300K)  P 0 = 1/q Rot The P 0 value is small for many linear molecules at room temperature. P 0 values can be increased by lowering the temperature of the molecules.

12. HCl AND DCl INFRARED SPECTRA:  The HCl and DCl spectra obtained in the lab show features consistent with the results presented here. These spectra are shown on the next slides for consideration/class discussion.

15. THE HYDROGEN ATOM: 

16.  For the 3-dimensional PIAB we have:  3 Cartesian coordinates  3 quantum numbers required to describe E.  With problems involving rotation (especially in 3 dimensions) and energies of electrons in atoms, spherical polar coordinates (r,θ,φ)are a more natural choice than Cartesian coordinates. Why?

17. ATOMS AND ELECTRONIC ENERGIES: 

18. COULOMBIC INTERACTIONS:  Class discussion of coulombic forces, energies and “work terms” (simple integration). Need for spherical polar coordinates in treating the H atom.