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PY3P05 o Rotational transitions o Vibrational transitions o Electronic transitions PY3P05 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 (r i ) and the nuclear positions (R j ): 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. " molecule r i , ˆ R j ) = " electrons r i , ˆ R j ) " nuclei ( ˆ R j )
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Page 1: o Rotational transitions o Vibrational transitions o ... · o Rotational transitions o Vibrational transitions o Electronic transitions PY3P05 o Born-Oppenheimer Approximation is

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o  Rotational transitions

o  Vibrational transitions

o  Electronic transitions

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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.

!

"molecule (ˆ r i, ˆ R j ) ="electrons( ˆ r i, ˆ R j )"nuclei( ˆ R j )

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o  Electronic transitions: UV-visible

o  Vibrational transitions: IR

o  Rotational transitions: Radio

Electronic Vibrational Rotational

E

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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.

!

I = miri2

i"

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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 inertia (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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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 =1/2Ia"a2

!

E =1/2Ia"a2 +1/2Ib"b

2 +1/2Ic"c2

!

E =Ja2

2Ia+Jb2

2Ib+Jc2

2Ic

!

J = J(J +1)!2

!

EJ =J(J +1)!2I

, J = 0, 1, 2, …

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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 =!2

2I=> B =

!4"cI

!

F(J) = BJ(J +1)

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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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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

!

!2

2µd2"dx 2

+ [E #V ]" = 0

!2

2µd2"dx 2

+ [E #1/2kx 2]" = 0

!

µ =m1m2

m1 + m2

!

Ev = (v +1/2)!"

!

" =kµ

#

$ % &

' (

1/ 2

v = 0, 1, 2, …

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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.

!

G(v) = (v +1/2) ˜ v

!

˜ v = 12"c

#

$ % &

' (

1/ 2

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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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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

!

V = hcDe 1" e"a(R"Re )( )

2

!

a =µ" 2

2hcDe

#

$ %

&

' (

1/ 2

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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

!

G(v) = (v +1/2) ˜ v " ( ˜ v +1/2)2 xe ˜ v

!

xe =a2!2µ"

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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 !J = ±1 with vibrational rule !v = ±1.

o  When vibrational transitions of the form v + 1 ! v occurs, !J = ±1.

o  Transitions with !J = -1 are called the P branch:

o  Transitions with !J = +1 are called the R branch:

o  Q branch are all transitions with !J = 0

!

S(v,J) = (v +1/2) ˜ v + BJ(J +1)

!

˜ v P (J) = S(v +1,J "1) " S(v,J) = ˜ v " 2BJ

!

˜ v R (J) = S(v +1,J +1) " S(v,J) = ˜ v + 2B(J +1)

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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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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 => 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).


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