Chem 5314 homework #5 – out of 33 marks
Problem 1
– 10 marks
Relative to the
n
= 0 vibrational level of the ground (X) electron state of
79
Br
2
, the
vibrational levels (n) of the excited (B) state have energies (in cm

1
) as follows: (the table
is taken from J. Mol. Spectroscopy
51
, 428 (1974), which is in the library; the symbols X
and B are spectroscopic symbols we have not discussed in class)
n
E (cm

1
)
0
15823.47
1
15987.78
2
16148.73
3
16306.26
4
16460.33
5
16610.87
6
16757.83
7
16901.15
8
17040.78
9
17176.65
10
17308.65
11
17436.89
12
17561.14
13
17681.50
14
17797.88
15
17910.18
16
18018.37
17
18122.45
18
18222.46
19
18318.33
20
18410.07
21
18497.66
The dissociation limit (again relative to vibrational level n=0 of the ground (X) electronic
state) is 19579.76 cm

1
for the B state.
1
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The harmonic oscillator has energy levels
E
n
= (
n
+
1
2
)
~
ω
The Morse potential has energy levels
E
n
=
(
(
n
+
1
2
)

x
e
(
n
+
1
2
)
2
)
~
ω
where
x
e
is the
anharmonicity parameter, which is related to the depth of the well (for a Morse oscillator
the well is not infinitely deep because the molecule can dissociate into separate atoms).
How well does the data for the B state fit a harmonic approximation?
How well does
it fit the Morse approximation? You must do some kind of rigorous statistical analysis to
quantify your answer.
Solution:
If we plot Δ
E
vs (n+1), the harmonic oscillator model gives a horizontal line (zero
slope) with a yintercept of
~
ω
and the Morse oscillator model gives a straight line with
a yintercept of
~
ω
and a slope of

2
~
ωx
e
.
We can see this by doing the math: for the
harmonic oscillator,
E
n
= (
n
+ 1
/
2)
~
ω
so that Δ
E
=
~
ω
(independent of
n
). For the Morse
oscillator,
E
n
= [(
n
+1
/
2)

x
e
(
n
+1
/
2)
2
]
~
ω
so that Δ
E
=
~
ω

2
~
ωx
e
(
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 Fall '11
 Nielsen
 Physical chemistry, Atom, Electron, pH, rigid rotor, Molecular physics

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