Rotational Spectroscopy

Which of these molecules will show a pure rotational microwave absorption spectrum

• H2

• HCl

• CH2Cl2

• NH3

• CO2

• No, yes, yes, yes, no

• HCl has a changing dipole as it rotates therefore oscillation of charge

Rotational spectroscopy selection rules

• molecule must have a permanent dipole

• Deltas = ± 1 (+1 = absorption, -1 = emission)

Equation for B tilda (rotational constant)

B tilda = reduced plancks/4 x pi x c x I

• I = moment of inertia

Equation for moment of inertia, I

I = mu x r2

• r = equilibrium bond length

Equation for F tilde J

F tilde J = Btilde J (J+1)

Relationship between B tilde and nu tilde

Nu tilde(J+1←J) = 2Btilde(J+1)

Peak breadth

Rotational spectroscopy is only really useful for gas phase molecules so broadening is probably dominated by instrumental factors of Doppler broadening

What is the reduced planks constant

1.05457 × 10 -34 Js

• to get from h to reduced - divide h by 2 pi

Method for determining B tilde if given two peak locations

Subtract both values to get the value of 2B tilde - the difference between the J levels

• to know what transitions the peaks correspond to, substitute each separately into the equation relating nu tide and B tilde using the original peak location as nu and B tilde as calculated above - rearrange to get (J+1) then J

Equation for equilibrium bond length, r

r = sqrt(I/mu)

Boltzmann expression

Nj/N = gje(-Ej/kT)

Equation for the degeneracy or level J, gJ

gJ= 2J+1

Equation for EJ in joules

EJ = Btilde h c J(J+1)

Estimation of kT value at room temperature

200 cm-1

• much larger than a typical rotational constant

Equation for Jmax (might be on equation sheet)

Jmax = sqrt(kT/2hcBtilde) - ½

• gives you a J number for the starting state

• You then need to write expressions for absorption (J’=? ←J”=?) and emission

• J” = starting state

• J’ = final state

• So absorption J’ will be one greater, emission J’ will be one less

Equation for F tilde J including the correction factor for centrifugal distortion

• assume F, B and D all have a tilde above

FJ = BJ(J+1)-DJ x J2 x (J+1)2

Equation for D tilde J might be on equation sheet

D tilde J = (4 x Btilde cubed)/(nu tilde squared)

Equation for the effective rotational constant (used bc bonds aren’t rigid and can flex out)- might be on equation sheet

Btilde eff = Btilde - Dtilde J (J+1)

Equation for transition energy, deltaFtildeJ (J+1←J) including correction factor for centrifugal distortion

deltaFtildeJ (J+1←J) = 2Btilde(J+1) - 4DJtilde (J+1)cubed

• for a non rigid motor

• The spacings between rotational levels (transitions) are no longer constant at 2Btilde, they decrease with increasing J level

What type of molecule might have a small centrifugal distortion constant?

Light atoms with a really stiff bond not susceptible to stretching w spinning

What type of molecule might have a large centrifugal distortion constant?

• two heave masses with a single bond separating them

• Still need a dipole though

Explain why centrifugal distortion decreases the energy gap between rotational states

As a molecule rotates faster (at higher rotational quantum numbers) the centrifugal force stretched the bond effectively increasing the moment of inertia and decreasing the rotational constant B- reduction in B causes energy level to converge as the rotational quantum number increases

Classification of molecules as a linear rotor

• two equal moments of inertia

• Moment of inertia along inter nuclear axis is 0

• Ic=Ib, Ia=0

Classification of molecules as Spherical rotors

• all three moments of inertia are equal

• Highly symmetric molecules

• I a= Ib= Ic

Classification of molecules as Symmetric rotors

• two moments of inertia are equal

• Third is non zero

• Ic = Ib which is greater than or less that equal to Ia

Asymmetric rotor

• three different (and non zero) moments of inertia

• Not very symmetric molecules

• Ic is greater than or equal to Ib which is greater than or equal to Ia

Polyatomic rotational spectra

• if you only have one rotational spectra you can’t figure out two unknown bond lengths so microwave spectroscopists collect multiple spectra by changing he isotopes

• They assume the geometry of the molecule doesn’t change with isotopic substitution

• Take an isotopically pure rotational spectrum then change one of the atoms to a different isotope and retake spectrum to compare the differences

Using rotational spectroscopy for structure determination

• each vibrational level has a different B value but the electronic potential energy surface has a defined equilibrium geometry

• In practice spectroscopists are measuring a vibrationally averaged ro structure which is then extrapolated back to the re structure

• Seeing a vibrationally averaged ro structure and getting a Bo rotational constant