Atomic structure and electronic transitions

TO know about the electronic transitions

1. PY3P05
Lecture 14: Molecular structureLecture 14: Molecular structure
oRotational transitions
oVibrational transitions
oElectronic transitions
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.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.

2. PY3P05
Bohn-Oppenheimer ApproximationBohn-Oppenheimer Approximation
oBorn-Oppenheimer Approximation is the assumption that the electronic motion and the
nuclear motion in molecules can be separated.
oThis leads to molecular wavefunctions that are given in terms of the electron positions (r
i
) and
the nuclear positions (R
j
):
oInvolves the following assumptions:
oElectronic 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.
oThe nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the fast-
moving electrons.

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

3. PY3P05
Molecular spectroscopyMolecular spectroscopy
oElectronic transitions: UV-visible
oVibrational transitions: IR
oRotational transitions: Radio
Electronic Vibrational Rotational
E

4. PY3P05
Rotational motionRotational motion
oMust first consider molecular moment of inertia:
oAt right, there are three identical atoms bonded to
“B” atom and three different atoms attached to “C”.
oGenerally specified about three axes: I
a
, I
b
, I
c
.
oFor linear molecules, the moment of inertia about the
internuclear axis is zero.
oSee Physical Chemistry by Atkins.

€
I=m
ir
i
2
i
∑

5. PY3P05
Rotational motionRotational motion
oRotation of molecules are considered to be rigid rotors.
oRigid rotors can be classified into four types:
oSpherical rotors: have equal moments of intertia (e.g., CH
4
, SF
6
).
oSymmetric rotors: have two equal moments of inertial (e.g., NH
3
).
oLinear rotors: have one moment of inertia equal to zero (e.g., CO
2, HCl).
oAsymmetric rotors: have three different moments of inertia (e.g., H
2
O).

6. PY3P05
Quantized rotational energy levelsQuantized rotational energy levels
oThe classical expression for the energy of a rotating body is:
where 
a
is the angular velocity in radians/sec.
oFor rotation about three axes:
oIn terms of angular momentum (J = I):
oWe know from QM that AM is quantized:
oTherefore, , J = 0, 1, 2, …
€
E
a
=1/2I
a
ω
a
2

€
E=1/2I
aω
a
2
+1/2I
bω
b
2
+1/2I
cω
c
2

€
E=
J
a
2
2I
a
+
J
b
2
2I
b
+
J
c
2
2I
c


€
J=J(J+1)h
2

€
E
J
=
J(J+1)h
2I
, J = 0, 1, 2, …

7. PY3P05
Quantized rotational energy levelsQuantized rotational energy levels
oLast equation gives a ladder of energy levels.
oNormally expressed in terms of the rotational constant,
which is defined by:
oTherefore, in terms of a rotational term:
cm
-1
oThe separation between adjacent levels is therefore
F(J) - F(J-1) = 2BJ
oAs B decreases with increasing I =>large molecules
have closely spaced energy levels.

€
hcB=
h
2
2I
=>B=
h
4πcI
€
F(J)=BJ(J+1)

8. PY3P05
Rotational spectra selection rulesRotational spectra selection rules
oTransitions are only allowed according to selection rule for
angular momentum:
J = ±1
oFigure at right shows rotational energy levels transitions and
the resulting spectrum for a linear rotor.
oNote, the intensity of each line reflects the populations of the
initial level in each case.

9. PY3P05
Molecular vibrationsMolecular 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 kx
2
oCan write the corresponding Schrodinger equation as
where
oThe SE results in allowed energies
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Graphics decompressor
are needed to see this picture.


€
h
2
2μ
d
2
ψ
dx
2
+[E−V]ψ=0
h
2
2μ
d
2
ψ
dx
2
+[E−1/2kx
2
]ψ=0

€
μ=
m
1m
2
m
1+m
2

€
E
v
=(v+1/2)hω
€
ω=
k
μ
⎛
⎝
⎜
⎞
⎠
⎟
1/2
v = 0, 1, 2, …

10. PY3P05
Molecular vibrationsMolecular vibrations
oThe vibrational terms of a molecule can therefore
be given by
oNote, the force constant is a measure of the
curvature of the potential energy close to the
equilibrium extension of the bond.
oA 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 vibrationsMolecular 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.
oTransition occur for v = ±1
oThis potential does not apply to energies
close to dissociation energy.
oIn fact, parabolic potential does not allow
molecular dissociation.
oTherefore more consider anharmonic
oscillator.

12. PY3P05
Anharmonic oscillatorAnharmonic 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.
oAt high excitation energies the parabolic
approximation is poor (the true potential is less
confining), and does not apply near the dissociation
limit.
oMust therefore use a asymmetric potential. E.g.,
The Morse potential:
where D
e
is the depth of the potential minimum and

€
V=hcD
e
1−e
−a(R−R
e
)
( )
2

€
a=
μω
2
2hcD
e

⎛

⎝

⎜

⎞

⎠

⎟
1/2

13. PY3P05
Anharmonic oscillatorAnharmonic oscillator
oThe Schrödinger equation can be solved for the Morse potential, giving permitted energy
levels:
where x
e
is the anharmonicity constant:
oThe second term in the expression for G increases with v => levels converge at high quantum
numbers.
oThe number of vibrational levels for a Morse
oscillator is finite:
v = 0, 1, 2, …, v
max

€
G(v)=(v+1/2)˜ v −(˜ v +1/2)
2
x
e
˜ v

€
x
e=
a
2
h
2μω

14. PY3P05
Vibrational-rotational spectroscopyVibrational-rotational spectroscopy
oMolecules vibrate and rotate at the same time =>
S(v,J) = G(v) + F(J)
oSelection rules obtained by combining rotational
selection rule ΔJ = ±1 with vibrational rule Δv = ±1.
oWhen vibrational transitions of the form v + 1  v
occurs, ΔJ = ±1.
oTransitions with ΔJ = -1 are called the P branch:
oTransitions with ΔJ = +1 are called the R branch:
oQ branch are all transitions with ΔJ = 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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TIFF (Uncompressed) decompressor
are needed to see this picture.

15. PY3P05
Vibrational-rotational spectroscopyVibrational-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).
oVibrational transitions accompanied by rotational transitions. Transition must produce a
changing electric dipole moment (IR spectroscopy).
P branch
Q branch
R branch

16. PY3P05
Electronic transitionsElectronic transitions
oElectronic transitions occur between molecular
orbitals.
oMust adhere to angular momentum selection rules.
oMolecular orbitals are labeled, , , , …
(analogous to S, P, D, … for atoms)
oFor atoms, L = 0 => S, L = 1 => P
oFor molecules,  = 0 => ,  = 1 => 
oSelection rules are thus
 = 0, 1, S = 0, =0,  = 0, 1
oWhere  =  +  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/
oQuestions? Contact:
[email protected]
oRoom 3.17A in SNIAM