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![PY3P05
Molecular vibrations
Molecular vibrations
o Consider simple case of a vibrating diatomic molecule,
where restoring force is proportional to displacement
(F = -kx). Potential energy is therefore
V = 1/2 kx2
o Can write the corresponding Schrodinger equation as
where
o The SE results in allowed energies
QuickTime™ and a
Graphics decompressor
are needed to see this picture.
€
h2
2μ
d2
ψ
dx2
+[E −V]ψ = 0
h2
2μ
d2
ψ
dx2
+[E −1/2kx2
]ψ = 0
€
μ =
m1m2
m1 + m2
€
Ev = (v +1/2)hω ω =
k
μ
⎛
⎝
⎜
⎞
⎠
⎟
1/ 2
v = 0, 1, 2, …](https://image.slidesharecdn.com/jsatomiclecture14-250226152842-b1ab6dbc/75/Atomic-structure-and-electronic-transitions-9-2048.jpg)
Atomic structure and electronic transitions
- 1. PY3P05 Lecture 14: Molecularstructure Lecture 14: Molecular structure o Rotational transitions o Vibrational transitions o Electronic transitions QuickTime™ and a TIFF (Uncompressed) decomp are needed to see this pict QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
- 2. PY3P05 Bohn-Oppenheimer Approximation Bohn-Oppenheimer Approximation oBorn-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated. o This leads to molecular wavefunctions that are given in terms of the electron positions (ri) and the nuclear positions (Rj): o Involves the following assumptions: o Electronic wavefunction depends on nuclear positions but not upon their velocities, i.e., the nuclear motion is so much slower than electron motion that they can be considered to be fixed. o The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the fast- moving electrons. € ψmolecule (ˆ ri, ˆ Rj ) =ψelectrons(ˆ ri, ˆ Rj )ψnuclei ( ˆ Rj )
- 3. PY3P05 Molecular spectroscopy Molecular spectroscopy oElectronic transitions: UV-visible o Vibrational transitions: IR o Rotational transitions: Radio Electronic Vibrational Rotational E
- 4. PY3P05 Rotational motion Rotational motion oMust first consider molecular moment of inertia: o At right, there are three identical atoms bonded to “B” atom and three different atoms attached to “C”. o Generally specified about three axes: Ia, Ib, Ic. o For linear molecules, the moment of inertia about the internuclear axis is zero. o See Physical Chemistry by Atkins. € I = miri 2 i ∑
- 5. PY3P05 Rotational motion Rotational motion oRotation of molecules are considered to be rigid rotors. o Rigid rotors can be classified into four types: o Spherical rotors: have equal moments of intertia (e.g., CH4, SF6). o Symmetric rotors: have two equal moments of inertial (e.g., NH3). o Linear rotors: have one moment of inertia equal to zero (e.g., CO2, HCl). o Asymmetric rotors: have three different moments of inertia (e.g., H2O).
- 6. PY3P05 Quantized rotational energylevels Quantized rotational energy levels o The classical expression for the energy of a rotating body is: where a is the angular velocity in radians/sec. o For rotation about three axes: o In terms of angular momentum (J = I): o We know from QM that AM is quantized: o Therefore, , J = 0, 1, 2, … € Ea =1/2Iaωa 2 € E =1/2Iaωa 2 +1/2Ibωb 2 +1/2Icωc 2 € E = Ja 2 2Ia + Jb 2 2Ib + Jc 2 2Ic € J = J(J +1)h2 € EJ = J(J +1)h 2I , J = 0, 1, 2, …
- 7. PY3P05 Quantized rotational energylevels Quantized rotational energy levels o Last equation gives a ladder of energy levels. o Normally expressed in terms of the rotational constant, which is defined by: o Therefore, in terms of a rotational term: cm-1 o The separation between adjacent levels is therefore F(J) - F(J-1) = 2BJ o As B decreases with increasing I =>large molecules have closely spaced energy levels. € hcB = h2 2I => B = h 4πcI € F(J) = BJ(J +1)
- 8. PY3P05 Rotational spectra selectionrules Rotational spectra selection rules o Transitions are only allowed according to selection rule for angular momentum: J = ±1 o Figure at right shows rotational energy levels transitions and the resulting spectrum for a linear rotor. o Note, the intensity of each line reflects the populations of the initial level in each case.
- 9. PY3P05 Molecular vibrations Molecular vibrations oConsider simple case of a vibrating diatomic molecule, where restoring force is proportional to displacement (F = -kx). Potential energy is therefore V = 1/2 kx2 o Can write the corresponding Schrodinger equation as where o The SE results in allowed energies QuickTime™ and a Graphics decompressor are needed to see this picture. € h2 2μ d2 ψ dx2 +[E −V]ψ = 0 h2 2μ d2 ψ dx2 +[E −1/2kx2 ]ψ = 0 € μ = m1m2 m1 + m2 € Ev = (v +1/2)hω ω = k μ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/ 2 v = 0, 1, 2, …
- 10. PY3P05 Molecular vibrations Molecular vibrations oThe vibrational terms of a molecule can therefore be given by o Note, the force constant is a measure of the curvature of the potential energy close to the equilibrium extension of the bond. o A strongly confining well (one with steep sides, a stiff bond) corresponds to high values of k. € G(v) = (v +1/2)˜ v € ˜ v = 1 2πc k μ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/2
- 11. PY3P05 Molecular vibrations Molecular vibrations oThe lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations. o Transition occur for v = ±1 o This potential does not apply to energies close to dissociation energy. o In fact, parabolic potential does not allow molecular dissociation. o Therefore more consider anharmonic oscillator.
- 12. PY3P05 Anharmonic oscillator Anharmonic oscillator oA molecular potential energy curve can be approximated by a parabola near the bottom of the well. The parabolic potential leads to harmonic oscillations. o At high excitation energies the parabolic approximation is poor (the true potential is less confining), and does not apply near the dissociation limit. o Must therefore use a asymmetric potential. E.g., The Morse potential: where De is the depth of the potential minimum and € V = hcDe 1−e−a(R−Re ) ( ) 2 € a = μω2 2hcDe ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/2
- 13. PY3P05 Anharmonic oscillator Anharmonic oscillator oThe Schrödinger equation can be solved for the Morse potential, giving permitted energy levels: where xe is the anharmonicity constant: o The second term in the expression for G increases with v => levels converge at high quantum numbers. o The number of vibrational levels for a Morse oscillator is finite: v = 0, 1, 2, …, vmax € G(v) = (v +1/2) ˜ v−( ˜ v +1/2)2 xe ˜ v € xe = a2 h 2μω
- 14. PY3P05 Vibrational-rotational spectroscopy Vibrational-rotational spectroscopy oMolecules vibrate and rotate at the same time => S(v,J) = G(v) + F(J) o Selection rules obtained by combining rotational selection rule ΔJ = ±1 with vibrational rule Δv = ±1. o When vibrational transitions of the form v + 1 v occurs, ΔJ = ±1. o Transitions with ΔJ = -1 are called the P branch: o Transitions with ΔJ = +1 are called the R branch: o Q branch are all transitions with ΔJ = 0 € S(v,J) = (v +1/2)˜ v + BJ(J +1) € ˜ vP (J) = S(v +1,J −1) − S(v,J) = ˜ v −2BJ € ˜ vR (J) = S(v +1,J +1) − S(v,J) = ˜ v + 2B(J +1) QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
- 15. PY3P05 Vibrational-rotational spectroscopy Vibrational-rotational spectroscopy oMolecular vibration spectra consist of bands of lines in IR region of EM spectrum (100 – 4000cm-1 0.01 to 0.5 eV). o Vibrational transitions accompanied by rotational transitions. Transition must produce a changing electric dipole moment (IR spectroscopy). P branch Q branch R branch
- 16. PY3P05 Electronic transitions Electronic transitions oElectronic transitions occur between molecular orbitals. o Must adhere to angular momentum selection rules. o Molecular orbitals are labeled, , , , … (analogous to S, P, D, … for atoms) o For atoms, L = 0 => S, L = 1 => P o For molecules, = 0 => , = 1 => o Selection rules are thus = 0, 1, S = 0, =0, = 0, 1 o Where = + is the total angular momentum (orbit and spin).
- 17. PY3P05 The End! The End! oAll notes and tutorial set available from http://www.physics.tcd.ie/people/peter.gallagher/lectures/py3004/ o Questions? Contact: o peter.gallagher@tcd.ie o Room 3.17A in SNIAM