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example 2 - PF5


 

The vibrational modes associated with the P-F bonds in PF5

The PF5 molecule belongs to the D3h point group.

The molecule contains both axial and equatorial bonds, these are calculated separately as they are different. The number of bonds that remain unshifted by the operations are represented by the following:

 

 

E

2C3
3C2
sh

3S3

3sv

Gaxial

2

2
0
0

0

2

Gequatorial
3
0
1
3
0
1


The reduction is made somewhat easier by considering the two types of F hybrid separately.

The representation for the axial hybrids is generated by adding the A1' and A2" rows together. (This is readily seen by noting that the '0' for the sh column requires one row from the single prime and one row from the double prime irreps be used. The '2' for the 3sv column can then only come about from adding the A1' and A2" rows).

 

 

E

2C3
3C2
sh

3S3

3sv

 

 A1'

 1

1

 1

1 1

 +

A2''

 2

 -1

-1

-1

-1 1

 

 

 

 

 

 

   

 

 Gaxial

2

2

0

0 2

 

The representation for the equatorial hybrids is generated by adding together the A1' and E' rows together. (This is readily seen by noting that the '3' for the sh column means that only only single prime rows have to be considered. The '1' for the 3C2 and for the 3sv columns can then only come about from adding the A1' and E' rows).

 

 

E

2C3
3C2
sh

3S3

3sv

 

 A1'

 1

1

 1

1
1

 +

E'

 2

 -1

0

2

-1
0

 

 

 

 

 

 

   

 

 Gequatorial

3

0

1

3

0
1

 


Therefore

Gaxial = A1' + A2"        and       Gequatorial = A1' + E'

(or G5F = 2A1' + A2" + E')


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