Question

Trifluoromethyliodide, CF3I, is a prolate symmetric top with rotational constants A=0.1910 cm-1, B=0.05081 cm-1. Neglecting centrifugal distortion, calculate all the rotational energy levels for J=2. Make a similar calculation for the oblate symmetric top NH3, for which B=9.443 cm-1, C=6.196 cm- 1, and make a qualitative comparison with those for CF3I.

Answer 1

As we know, in the case of linear molecules there is one degenerate rotational axis which in turn has a single rotational constant. With symmetrical tops, there is one unique axis and two degenerate axes. So an additional rotational constant is needed to describe the energy levels of a symmetrical top. In addition to the rotational constant, an additional quantum number K must be introduced to describe the rotational energy levels of the symmetric top. These two additions give us the following rotational energy levels of a prolate and oblate symmetric top

E(J,K)(cm⁻¹)=B∗J(J+1)+(A−B)∗K²

Where B is the rotational constant of the unique axis, A is the rotational constant of the degenerate axes, J is the total rotational angular momentum quantum number and K is the quantum number that represents the portion of the total angular momentum that lies along the unique rotational axis. This leads to the property that K is always equal to or less than J i.e. J=0,1,2....K=+J to - J

so, for J=2., K=+2....0....-2

K=2,

E(J,K)(cm⁻¹)=B∗J(J+1)+(A−B)∗K²

or, E(2,2)(cm⁻¹)=0.05081∗2(2+1)+(0.1910-0.0508)∗2²

E(2,2)(cm⁻¹)=0.86566

for, K=1

=E(2,1)(cm⁻¹)=0.05081∗2(2+1)+(0.1910-0.0508)∗1²

^=0.44506

^{for, K=0}

E(2,0)(cm⁻¹)=0.05081∗2(2+1)+(0.1910-0.0508)∗0²

^=0.30486

^{for, K=-1}

E(2,-1)(cm⁻¹)=0.05081∗2(2+1)+(0.1910-0.0508)∗(-1)²

=0..44506

for, K=-2

E(2,-2)(cm⁻¹)=0.05081∗2(2+1)+(0.1910-0.0508)∗(-2)²

=0.86566

For, oblate symmetric top NH₃,

E(J,K)(cm⁻¹)=B∗J(J+1)+(C−B)∗K²

K=2,

E(2,2)(cm⁻¹)=9.443∗2(2+1)+(6.196-9.443)∗2²

E(2,2)(cm⁻¹)=56.658-12.988

=43.67

for, K=1, E(2,1)(cm⁻¹)=9.443∗2(2+1)+(6.196-9.443)∗1²

^=53.411

^{for, K=0}

E(2,0)(cm⁻¹)=9.443∗2(2+1)+(6.196-9.443)∗0²

=56.658,

for K=-1, E(2,-1)(cm⁻¹)=53.411

for K=-2, E(2,-2)(cm^-1)=43.67

In case of prolate symmetric top molecule i.e CF3I, for a given J, energy increases with K whereas for oblate symmetric top molecules, i.e. NH3, for a given J, energy decreases with K