Chem 3502/5502
Physical Chemistry II (Quantum Mechanics)
3 Credits
Fall Semester 2009
Laura Gagliardi
Answers to Homework Set 7
From lecture 25: Using the third root of the secular equation for the allyl system, verify
the orbital coefficients given in eq. 2516
The second root was
E
=
α
. Plugging that value into the linear equations
a
i
H
ki
"
ES
ki
(
)
i
=
1
N
#
=
0
$
k
recalling that for the allyl system
H
11
=
H
22
=
H
33
=
α
,
H
12
=
H
21
=
H
23
=
H
32
=
β
,
H
13
=
H
31
= 0,
S
11
=
S
22
=
S
33
= 1, and all other
S
values are 0, we have
a
1
"
–
" #
2
$
(
)
•1
[
]
+
a
2
$
–
" #
2
$
(
)
• 0
[
]
+
a
3
0 –
" #
2
$
(
)
• 0
[
]
=
0
a
1
$
–
" #
2
$
(
)
• 0
[
]
+
a
2
"
–
" #
2
$
(
)
•1
[
]
+
a
3
$
–
" #
2
$
(
)
• 0
[
]
=
0
a
1
0 –
" #
2
$
(
)
• 0
[
]
+
a
2
$
–
" #
2
$
(
)
• 0
[
]
+
a
3
"
–
" #
2
$
(
)
•1
[
]
=
0
These equations simplify to
a
1
2
"
+
a
2
"
=
0
a
1
"
+
a
2
2
"
+
a
3
"
=
0
a
2
"
+
a
3
2
"
=
0
If we subtract the third equation from the first, we obtain
a
1
2
" #
a
3
2
"
=
0
which gives
a
1
=
a
3
If we use this relationship in the second equation above we have
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HW72
a
1
"
+
a
2
2
"
+
a
1
"
=
0
which gives
a
2
=
"
2
a
1
Normalization requires that
a
i
2
i
=
1
3
!
=
1
which leads to the final result
a
13
=
1
2
,
a
23
=
–
2
2
,
a
33
=
1
2
QED.
From lecture 26:
What is the Hartreeproduct wave function for 2 noninteracting quantum
mechanical harmonic oscillators (QMHOs) of reduced mass 1 a.u. in a potential having a
force constant of 1 a.u., where the first QMHO is in the ground state and the second is in
the first excited state? Determine the energy of the two QMHO system as an expectation
value of the Hartreeproduct wave function. Is the correct Hamiltonian for this system
separable into oneQMHO terms? If the QMHOs
were
interacting, explain how you
could use perturbation theory to determine the energy of the system correct to first order
(you don’t have to actually do it, just explain
how
to do it).
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