the transition between v0-V2 and v0-v3 gives rise to which shift
Answers
Answer:
The harmonic oscillator approximation is convenient to use for diatomic molecules with quantized vibrational energy levels given by the following equation:
Ev(cm−1)=(v+12)ωe(3)
A more accurate description of the vibrational energies is given by the anharmonic oscillator (also called Morse potential) with energy of
Ev(cm−1)=ωe(v+12)−ωexe(v+12)2+ωeye(v+12)3+...(4)
where ωe is the vibrational frequency for the re internuclear separation and ωe >> ωexe >> ωeye. This accounts for the fact that as the higher vibrational states deviate from the perfectly parabolic shape, the level converge with increasing quantum numbers. It is because of this anharmoniticity that overtones can occur.
While it may seem that the harmonic oscillator and the anharomic oscillator are closely related, this is in fact not the case. The differences in the wavefunctions lead to a breakdown of selection rules, specifically, Δv=±1 selection rule can not be applied, and higher order terms must be accounted in the energy calculations.
649px-Morse-potential.png
Figure 2: Pictured above is the HOA (green parabola) superimposed on the anharmonic oscillator (blue curve) on a potential energy diagram. V(R) is the potential energy of a diatomic molecule and R is the radius between the centers of the two atoms. Towards the left is compression of the bond, towards the right is extension. Image used with prmission (CC-By-SA-3.0; Created by Mark Somoza March 26 2006).
There is only a small correction from the ground state to the first excited state for the anharmonic correction, but it becomes much larger for more highly excited states which are populated as the temperature increases. The deviation from the harmonic oscillator to the anharmonic oscillator results in expanding the energy function with additional terms and treating these terms with perturbation theory. The results in the correct vibrational energies and also relaxes the selection rules. A Δv=±1 is still most predominant, however, weaker overtones with Δv=±2, ±3,… can occur. It should be noted that a Δv=2 transition does not occur at twice the frequency of the fundamental transition, but at a lower frequency. Overtone transitions are not always observed, especially in larger molecules, because the transitions become weaker with increasing Δv.
Overtones
Overtones occur when a vibrational mode is excited from v=0 to v=2 , which is called the first overtone, or v=0 to v=3, the second overtone. The fundamental transitions, \(v=±1\0, are the most commonly occurring, and the probability of overtones rapid decreases as the number of quanta (Δv=±n) increases. Based on the harmonic oscillator approximation, the energy of the overtone transition would be n times larger than the energy of the fundamental transition frequency, but the anharmonic oscillator calculations show that the overtones are less than a multiple of the fundamental frequency. This is demonstrated with the vibrations of the diatomic HCl in the gas phase:
Table 1: HCl vibrational spectrum.
0→1
fundamental
2,885.9
2,885.9
2,885.3
0→2
first overtone
5,668.0
5,771.8
5,665.0
0→3
second overtone
8,347.0
8,657.7
8,339.0
0→4
third overtone
10,923.1
11,543.6
10,907.4
0→5
fourth overtone
13,396.5
14,429.5
13,370
We can see from Table 1, that the anharmonic frequencies correspond much better with the observed frequencies, especially as the vibrational levels increase.
Special case
If one of the symmetries is doubly degenerate in the excited state a recursion formula is required to determine the symmetry of the vth wave function, given by,
χv(R)=12[χ(R)χv−1(R)+χ(Rv)](5)
Where χv(R) is the character under the operation R for the vth energy level; χ(R) is the character under R for the degenerate irreducible representation; χv-1(R) is the character of the (v-1)th energy level; and χ(Rv) is the character of the operation Rv. This is demonstrated for the D3h point group below.