Linear molecule Rotational Transitions:

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Linear molecule Rotational Transitions

• J 4
  
• J 3
• J 2
• J 1
• J 0

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Linear molecule Rotational Spectrum

• Intensity J 4?3
• J 1?0
• 
  2B
• Absorption Frequencies ?

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Effects of mass and molecular siZe

• The slide that follows gives B values for a
  number of diatomic molecules with different
  reduced masses and bond distances. What is the
  physical significance of the very different B
  values seen for H35Cl and D35Cl? All data are
  taken from the NIST site.
• http//www.nist.gov/pml/data/molspec.cfm

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Reduced Masses and Bond Distances
Molecule Bond Distance (Å) B Value (MHz)
H35Cl 1.275 312989.3
H37Cl 1.275 312519.1
D35Cl 1.275 161656.2
H79Br 1.414 250360.8
H81Br 1.414 250282.9
D79Br 1.414 127358.1
24Mg16O 1.748 17149.4
107Ag35Cl 2.281 3678.04
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Real Life Working Backwards?

• In the real world spectroscopic experiments
  provide frequency (and intensity) data. It is
  necessary to assign quantum numbers for the
  transitions before molecular (chemically useful)
  information can be determined. Sometimes all of
  the data are not available!

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Spectrum to Molecular StructurE

• Class Example A scan of the microwave
  (millimeter wave!) spectrum of 6LiF over the
  range 350 ? 550 GHz shows lines at 358856.2 MHz,
  448491.1 MHz and 538072.7 MHz. Assign rotational
  quantum numbers for these transitions. Determine
  a B value and the bond distance for 6LiF. Are the
  lines identically spaced?

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Higher order energy terms

• The slightly unequal spacing of lines in the 6LiF
  spectrum occurs because very rapidly rotating
  diatomic molecules distort. A higher order
  energy expression accounts for this effect
• EJ hBJ(J1) hDJJ2(J1)2
• DJ is the (quartic) centrifugal distortion
  constant.

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Higher order frequency expression

• The energy expression on the previous slide can
  be used with the selection rule ?J 1 (for
  absorption) and ?E h? to give
• ? 2B(J1) - 4DJ(J1)3
• This expression will be used in the lab (HCl/DCl
  spectrum). A typical frequency calculation is
  shown on the next slide.

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Non-Rigid rotor calculation, 7LiF

• Here B 40,026.883 MHz DJ 0.3505 MHz

Transition 2B(J1) 4DJ(J1)3 Freq. Calc. Freq. Obs.
J1?0 80053.766 1.402 80052.36 No Data
J2?1 160107.53 11.216 160096.32 160096.33
J3?2 240161.30 37.854 240123.45 240123.47
J4?3 320215.07 89.728 320125.34 320125.36
J5?4 400268.83 175.25 40093.58 400093.62
J6?5 480322.60 302.83 480019.77 480019.73
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Non-Rigid Molecules

• Aside Every spectroscopic constant tells us
  something. A small DJ value suggests that a
  molecule does not distort easily. Comparisons can
  be made for inertially similar molecules.
  Explanation?

Molecule B (GHz) DJ (kHz)
6LiF 45.23 443
13C18O 52.36 151
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Spectra of Nonlinear molecules

• With the particle in the box energy expressions
  grew more complex as we moved from one to three
  dimensions.
• PIAB one dimension Energy (eigenvalues!)
  expression has one term and one quantum number.
• PIAB three dimensions Energy expression has (up
  to!) three terms and three quantum numbers.

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Rotations in three dimensions

• For nonlinear molecules the number of quantum
  numbers and rotational constant needed to
  describe rotational energies is greater than one.
  We also have more than one I value. In general,
  we have a (3x3) matrix (moment of inertia tensor)
  that cane be diagonalized to simplify the
  mathematics.

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Nonlinear rigid rotors

• After diagonalization the moment of inertia
  tensor has three elements with Ia Ib Ic.
• Types of Rotors
• 1. Spherical tops Ia Ib Ic. We need just one
  quantum number (J again) to describe rotational
  energies. Examples CH4, SF6 and C60.

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Nonlinear rigid rotors

• 2. Symmetric tops Ia Ib Ic (oblate top) and
  Ia Ib Ic (prolate top). Examples
• Oblate top CHF3, HSi79Br3.
• Prolate top CH3F, CH3-CN.
• For symmetric tops we need two quantum numbers, J
  and K, to describe rotational energies.

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Nonlinear rigid rotors

•  

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Degeneracy

• In organic chemistry courses you have discussed
  NMR spectra and removal of spin degeneracy
  using a magnetic field. For the spin case, I ½,
  there is a two-fold energy degeneracy in the
  absence of a magnetic field. In rotational
  spectroscopy there is, similarly, a (2J1) fold
  degeneracy.

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Dipoles and Electric Fields.

• From physics, the energy of a linear rod with an
  electric dipole moment (µ) placed in an electric
  field can be found as µEcos?. Similar to NMR, the
  degeneracy of rotational energy levels can be
  removed by applying an electric filed to a gas.
  This enables the size of a molecules dipole
  moment to be determined.

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Degeneracy and Dipole moments

• We will not use the degeneracy of rotational
  levels for several weeks. It will be useful in
  calculating the relative intensities of spectral
  lines (after Boltzmann!). By experiment, it is
  found that a molecule must have a permanent
  non-zero electric dipole moment to have a pure
  rotational spectrum.

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Molecular structure/dipole moments

• From first year chemistry courses you should be
  able to take a simple molecular formula and (a)
  draw a Lewis structure for the molecule, (b)
  determine a molecules shape and (c) predict
  whether a molecule has net polarity. Review
  examples on the next slide.

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Molecular shapes and Polarity
Molecular Formula Electrically Polar Pure Rotational Spectrum?
H2
CO
HCN
CO2
CH2Cl2
SF6
OCCCS
CHFCHF (cis)
CHFCHF (trans)
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