Chemistry Reference
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of T , we have a pair of pure complex roots while all other eigenvalues have negative
real parts. Second, that the derivative of .T/ at the critical value has nonzero real
part.
At the critical values the inequalities (2.60) as well as (2.61) become equalities,
so the eigenvalue equation (2.59) reduces to
C
2T/
C
T
2
a.1
.N
1/r/
aT
2
0
D
3
T
2
C
2
.aT
2
C
a.1
.N
1/r/
C
2T
C
2T/
2
:
D
T
2
a.1
.N
1/r/
aT
2
C
.aT
2
C
C
2T
Therefore the eigenvalues are
s
a.1
.N
1/r/
aT
2
1;2
D˙
i
;
(2.63)
C
2T
and
aT
2
C
2T
T
2
3
D
<0:
So the first condition is satisfied at the critical values. In order to show that the
second condition is also satisfied we have to differentiate implicitly the eigenvalue
equation (2.59) with respect to T . A simple calculation shows that
with the notation
P
dT
d
D
P
T
2
C
2
3
T
C
2
P
.aT
2
C
2T/
C
2
.2aT
C
2/
C
P
.1
C
2aT/
C
2a
D
0;
3
2
implying that
2
3
T
2
.2aT
C
2/
2a
3
2
T
2
P
D
C
2T/
C
.1
C
2aT/
:
(2.64)
C
2.aT
2
For the sake of simplicity introduce the notation
:
a.1
.N
1/r/
aT
2
1
C
2aT
T
2
˛
2
D
D
C
2T
Then
1;2
D˙
˛i, and at these values
˙
2˛
3
iT
C
˛
2
.2aT
C
2/
2a˛i
3˛
2
T
2
P
D
˙
2˛i.aT
2
C
2T/
C
.1
C
2aT/
˛
2
.2aT
C
2/
C
.
˙
2˛
3
T
2a˛/i
2˛
2
T
2
D
:
˙
2˛i.aT
2
C
2T/
Multiplying both the numerator and denominator by the complex conjugate of the
denominator, after some simple calculations we find that
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