Chemistry Reference
In-Depth Information
Ta b l e 4 . 8
Vibrational spectrum of UF
6
ν(
cm
−
1
)
Symmetry
Type
Calc.
Technique
ν
1
(A
1
g
)
Breathing
667
669
Raman (very strong)
ν
2
(E
g
)
Stretching
533
534
Raman (weak)
ν
3
(T
1
u
)
Stretching
626
624
IR
ν
4
(T
1
u
)
Bending
186
181
IR
ν
5
(T
2
g
)
Bending
202
191
Raman (weak)
ν
6
(T
2
u
)
Buckling
142
140
Overtone
It will be assumed that the coordinates vary in a harmonic manner with an angular
frequency
ω
; hence,
Q
k
=
Q
max
k
cos
ωt
. The second derivative is then given by
Q
k
=−
ω
2
Q
k
=−
(
2
πν)
2
Q
k
(4.99)
where
ν
is the vibrational frequency in Hertz. The equation of motion then is turned
into a set of homogeneous linear equations:
V
ki
−
δ
ki
ω
2
Q
i
=
∀
k
:
0
(4.100)
i
This set of equations is solved in the standard way by diagonalizing the Hessian
matrix, as
V−
=
ω
2
I
0
(4.101)
The eigenvalues of the secular equation yield the frequencies of the normal modes,
which are usually expressed as wavenumbers,
ν
, preferentially in reciprocal cen-
¯
timetres, cm
−
1
by dividing the frequency by the speed of light,
c
.
ν
c
=
ω
2
πc
ν
¯
=
(4.102)
In Table
4.8
we present the experimental results [
4
]forU
238
F
6
, as compared with
the Hessian eigenvalues, based on extensive relativistic calculations [
5
]. The eigen-
functions of the Hessian matrix are the corresponding normal modes. The Hessian
matrix will be block-diagonal over the irreps of the group and, within each irrep,
over the individual components of the irrep. Moreover, the blocks are independent
of the components. All this illustrates the power of symmetry, and the reasons for it
will be explained in detail in the next chapter. As an immediate consequence, sym-
metry coordinates, which belong to irreps that occur only once, are exact normal
modes of the Hessian. Five irreps fulfil this criterion: the
T
1
g
mode, which corre-
sponds to the overall rotations, and the vibrational modes,
A
1
g
+
T
2
u
.
Only the
T
1
u
irrep gives rise to a triple multiplicity. In this case, the actual normal
modes will depend on the matrix elements in the Hessian. Let us study this in detail
E
g
+
T
2
g
+
