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Lecture 14: Molecular structureLecture 14: Molecular structure
o Rotational transitions
o Vibrational transitions
o Electronic transitions
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Bohn-Oppenheimer ApproximationBohn-Oppenheimer Approximation
o Born-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.
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ψmolecule (ˆ r i, ˆ R j ) =ψ electrons( ˆ r i, ˆ R j )ψ nuclei( ˆ R j )
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Molecular spectroscopyMolecular spectroscopy
o Electronic transitions: UV-visible
o Vibrational transitions: IR
o Rotational transitions: Radio
Electronic Vibrational Rotational
E
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Rotational motionRotational motion
o Must 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.
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I = miri2
i
∑
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Rotational motionRotational motion
o Rotation 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).
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Quantized rotational energy levelsQuantized 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, …
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Ea =1/2Iaωa2
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E =1/2Iaωa2 +1/2Ibωb
2 +1/2Icωc2
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E =Ja
2
2Ia
+Jb
2
2Ib
+Jc
2
2Ic
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J = J(J +1)h2
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EJ =J(J +1)h
2I
, J = 0, 1, 2, …
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Quantized rotational energy levelsQuantized 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
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F(J) = BJ(J +1)
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Rotational spectra selection rulesRotational 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.
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Molecular vibrationsMolecular 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
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are needed to see this picture.
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h2
2μ
d2ψ
dx 2 + [E −V ]ψ = 0
h2
2μ
d2ψ
dx 2+ [E −1/2kx 2]ψ = 0
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μ =m1m2
m1 + m2
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Ev = (v +1/2)hω
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=k
μ
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
v = 0, 1, 2, …
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Molecular vibrationsMolecular vibrations
o The 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.
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G(v) = (v +1/2) ˜ v
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˜ v =1
2πc
k
μ
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
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Molecular vibrationsMolecular vibrations
o The 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.
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Anharmonic oscillatorAnharmonic oscillator
o A 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
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V = hcDe 1− e−a(R−Re )( )
2
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a =μω2
2hcDe
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
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Anharmonic oscillatorAnharmonic oscillator
o The 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
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G(v) = (v +1/2) ˜ v − ( ˜ v +1/2)2 xe ˜ v
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xe =a2h
2μω
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Vibrational-rotational spectroscopyVibrational-rotational spectroscopy
o Molecules 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
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S(v,J) = (v +1/2) ˜ v + BJ(J +1)
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˜ v P (J) = S(v +1,J −1) − S(v,J) = ˜ v − 2BJ
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˜ v R (J) = S(v +1,J +1) − S(v,J) = ˜ v + 2B(J +1)
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Vibrational-rotational spectroscopyVibrational-rotational spectroscopy
o Molecular 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
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Electronic transitionsElectronic transitions
o Electronic 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 => Po 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).
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The End!The End!
o All notes and tutorial set available fromhttp://www.physics.tcd.ie/people/peter.gallagher/lectures/py3004/
o Questions? Contact:o [email protected] Room 3.17A in SNIAM