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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.

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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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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 peter.gallagher@tcd.ieo Room 3.17A in SNIAM