Rigid Diatomic molecule

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Rigid Diatomic molecule I = r2  = m1m2/(m1+m2) B (MHz) = 505391/I (u Å 2) B (cm-1) = 16.863/I (u A2) Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 0 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J = ±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B Harry Kroto 2004

A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J = ±1 E Transition Frequencies > F (J) = 2B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc Harry Kroto 2004

B(J+1)(J+2) J+1 BJ(J+1) J F(J) = 2B(J+1) Harry Kroto 2004

Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 14B 6 42B 12B 5 30B 10B 4 20B 8B 3 12B 6B 2 6B 4B 1 2B 2B Harry Kroto 2004

A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J = ±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc Harry Kroto 2004

Far infrared rotational spectrum of CO J= 12 15 20B 10 Far infrared rotational spectrum of CO J= 12 15 20B 23.0 cm-1 61.5 cm-1 Line separations 2B Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 50/3.85 = 12.99 = 13 so line at 50cm-1 is J=12 B = 16.863/ I I = 16.863/ B I = 8.76 uA2 I =  r2  = m1m2/(m1+m2)= 16x12/28 = 6.86 8.76/6.86 = 1.277 = r2 r = 1.277½ = 1.130 A (1.128 acc B value 1.921) Harry Kroto 2004

Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 10 J= 12 15 20B 23.0 cm-1 61.5 cm-1 Line separations 2B Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 50/3.85 = 12.99 = 13 so line at 50cm-1 is J=12 B = 16.863/ I I = 16.863/ B I = 8.76 uA2 I =  r2  = m1m2/(m1+m2)= 16x12/28 = 6.86 8.76/6.86 = 1.277 = r2 r = 1.277½ = 1.130 A (1.128 acc B value 1.921) Harry Kroto 2004

My ABC System of Spectroscopy Harry Kroto 2004

Nuclear Energies H + H E(r) Chemical Energies r  v=3 2 1 r  Harry Kroto 2004

Rotational Spectroscopy Harry Kroto 2004

Nuclear Energies H + H E(r) Chemical Energies Rotational levels r  r  Harry Kroto 2004

A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J = ±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc Harry Kroto 2004

m2 m1 Rotational Spectra Linear Molecules Rigid Diatomic molecule E = ½I2 Rigid Diatomic molecule Angular velocity  m2 m1 I = r2 = m1m2/(m1+m2) Harry Kroto 2004

Rotational Energy Linear Diatomic Molecules Rigid Diatomic molecule Angular velocity  m2 Rotational Energy Linear Diatomic Molecules E = ½I2 m1 I = r2 = m1m2/(m1+m2) Harry Kroto 2004

A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J = ±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc Harry Kroto 2004

Rotational Spectra Linear Molecules E = ½I2  J2/2I (J = I ) E = ½ mv2  p2/2m (p = mv) H = J2/2I (Note V= 0) Harry Kroto 2004

Rotational Spectra Linear Molecules E = ½I2  J2/2I (J = I ) E = ½ mv2  p2/2m (p = mv) H = J2/2I (Note V= 0) Harry Kroto 2004

H = J2/2I J J2 J  = ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1) F(J) = B J(J+1) B = ħ2/h2I MHz B = ħ2/hc2I cm-1 J J2 J   J* J2 Jd Harry Kroto 2004

A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J = ±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc Harry Kroto 2004

H = J2/2I J J2 J  = ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1) F(J) = B J(J+1) B = ħ2/h2I MHz B = ħ2/hc2I cm-1 J J2 J   J* J2 Jd Harry Kroto 2004

H = J2/2I J J2 J  = ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1) F(J) = B J(J+1) B = ħ2/h2I MHz B = ħ2/hc2I cm-1 J J2 J   J* J2 Jd Harry Kroto 2004

H = J2/2I J J2 J  = ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1) F(J) = B J(J+1) B = ħ2/h2I MHz B = ħ2/hc2I cm-1 J J2 J   J* J2 Jd Harry Kroto 2004

H = J2/2I J J2 J  = ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1) F(J) = B J(J+1) B = ħ2/h2I MHz B = ħ2/hc2I cm-1 J J2 J   J* J2 Jd Harry Kroto 2004

Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( Line separations 2B Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 50/3.85 = 12.99 = 13 so line at 50cm-1 is J=12 B = 16.863/ I I = 16.863/ B I = 8.76 uA2 I =  r2  = m1m2/(m1+m2)= 16x12/28 = 6.86 8.76/6.86 = 1.277 = r2 r = 1.277½ = 1.130 A (1.128 acc B value 1.921) Harry Kroto 2004

A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J = ±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc Harry Kroto 2004

Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( Line separations 2B Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 50/3.85 = 12.99 = 13 so line at 50cm-1 is J=12 B = 16.863/ I I = 16.863/ B I = 8.76 uA2 I =  r2  = m1m2/(m1+m2)= 16x12/28 = 6.86 8.76/6.86 = 1.277 = r2 r = 1.277½ = 1.130 A (1.128 acc B value 1.921) Harry Kroto 2004

Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( J= 12 23.0 cm-1 61.5 cm-1 Line separations 2B Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 50/3.85 = 12.99 = 13 so line at 50cm-1 is J=12 B = 16.863/ I I = 16.863/ B I = 8.76 uA2 I =  r2  = m1m2/(m1+m2)= 16x12/28 = 6.86 8.76/6.86 = 1.277 = r2 r = 1.277½ = 1.130 A (1.128 acc B value 1.921) Harry Kroto 2004

Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 10 15 Line separations 2B Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 50/3.85 = 12.99 = 13 so line at 50cm-1 is J=12 B = 16.863/ I I = 16.863/ B I = 8.76 uA2 I =  r2  = m1m2/(m1+m2)= 16x12/28 = 6.86 8.76/6.86 = 1.277 = r2 r = 1.277½ = 1.130 A (1.128 acc B value 1.921) Harry Kroto 2004

A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J = ±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc Harry Kroto 2004

Radiotelescope in Canada Harry Kroto 2004

A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J = ±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc Harry Kroto 2004

B(J+1)(J+2) – D(J+1)2(J+2)2 J+1 BJ(J+1) – DJ2(J+1)2 J F(J) = 2B(J+1) – 4D(J+1)3 Harry Kroto 2004

A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J = ±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc Harry Kroto 2004

Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 10 15 Line separations 2B Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 50/3.85 = 12.99 = 13 so line at 50cm-1 is J=12 B = 16.863/ I I = 16.863/ B I = 8.76 uA2 I =  r2  = m1m2/(m1+m2)= 16x12/28 = 6.86 8.76/6.86 = 1.277 = r2 r = 1.277½ = 1.130 A (1.128 acc B value 1.921) Harry Kroto 2004

Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 10 15 Line separations 2B Approximately 61.5 – 23 = 38.5 cm-1 = 20B 2B = 3.85 B = 1.925 cm-1 ( 50/3.85 = 12.99 = 13 so line at 50cm-1 is J=12 B = 16.863/ I I = 16.863/ B I = 8.76 uA2 I =  r2  = m1m2/(m1+m2)= 16x12/28 = 6.86 8.76/6.86 = 1.277 = r2 r = 1.277½ = 1.130 A (1.128 acc B value 1.921) Harry Kroto 2004

Harry Kroto 2004