4.8 Application: The Vibrations of UF 6
79
Fig. 4.5 Ligand numbering
and displacement coordinates
for UF 6
Table 4.7 Symmetry coordinates for UF 6
UT 1u p x
X 0
UT 1u p y
Y 0
UT 1u p z
Z 0
FσA 1g s
1/
√
6((Z 1 + X 2 + Y 3 − X 4 − Y 5 − Z 6 )
FσE g d z 2
1/
√
12(2Z 1 − X 2 − Y 3 + X 4 + Y 5 − 2Z 6 )
FσE g d x 2 −y 2
1/2((X 2 − Y 3 − X 4 + Y 5 )
FσT 1u p x
1/
√
2((X 2 + X 4 )
FσT 1u p y
1/
√
2((Y 3 + Y 5 )
FσT 1u p z
1/
√
2((Z 1 + Z 6 )
FπT 1g x
1/2(−Y 1 + Z 3 − Z 5 + Y 6 )
FπT 1g y
1/2((X 1 − Z 2 − X 6 + Z 4 )
FπT 1g z
1/2((Y 2 − X 3 − Y 4 + X 5 )
FπT 2g d yz
1/2((Y 1 + Z 3 − Z 5 − Y 6 )
FπT 2g d xz
1/2((X 1 + Z 2 − X 6 − Z 4 )
FπT 2g d xy
1/2((Y 2 + X 3 − Y 4 − X 5 )
FπT 1u p x
1/2((X 1 + X 3 + X 5 + X 6 )
FπT 1u p y
1/2((Y 1 + Y 2 + Y 4 + Y 6 )
FπT 1u p z
1/2((Z 2 + Z 3 + Z 4 + Z 5 )
FπT 2u f x(y 2 −z 2 )
1/2(−X 1 + X 3 + X 5 − X 6 )
FπT 2u f y(z 2 −x 2 )
1/2((Y 1 − Y 2 − Y 4 + Y 6 )
FπT 2u f z(x 2 −y 2 )
1/2((Z 2 − Z 3 + Z 4 − Z 5 )
f z(x 2 −y 2 ) , and its cyclic permutations f x(y 2 −z 2 ) and f y(z 2 −x 2 ) . In Table 4.7 we list
all 21 symmetry coordinates by category, using labels that refer to the central harmonic functions. We may denote the 21 symmetry coordinates by a row vector S.