Supplementary Problems

Vibrational (IR and Raman) Spectroscopy: Total Modes of Motion Analysis

Determine the number of (i) IR and Raman active signals and (ii) IR and Raman active vibrations for each molecule.
1.	GeCl₄ (T_d)	[Solution]
2.	trans-SF₂Cl₄ (D_4h)	[Solution]
3.	PF₅ (D_3h)	[Solution]
4.	CH₂Cl₂ (C_2v)	[Solution]
5.	IF₅ (C_4v)	[Solution]
6.	H₃PCPH₃ (D_3d)	[Solution]
7.	H₂CCCH₂ (D_2d)	[Solution]
8.	XeF₄ (D_4h)	[Solution]
9.	SCl₆ (O_h)	[Solution]
10.	Si(CH₃)₄ (T_d)	[Solution]
11.	(NC)₂CC(CN)₂ (D_2h)	[Solution]
12.	Ni(CO)₄ (T_d)	[Solution]
13.	C₆H₆ (D_6h)	[Solution]
14.	SBrF₅ (C_4v)	[Solution]
15.	Fe(CO)₅ (D_3h)	[Solution]

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Solutions

GeCl₄:

T_d	E	8C₃	3C₂	6S₄	6σ_d
Γ_total	15	0	−1	−1	3

Reduces to: 1A₁ + 1E + 1T₁ + 3T₂

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
1A₁			1A₁	No	Yes
1E			1E	No	Yes
1T₁		−(R_x,R_y,R_z)	0T₁	No	No
3T₂	−(T_x,T_y,T_z)		2T₂	Yes	Yes

	Signals	Vibrations
IR:	2	6
Raman:	4	9

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trans-SF₂Cl₄:

D_4h	E	2C₄	C₂	2C₂′	2C₂″	i	2S₄	σ_h	2σ_v	2σ_d
Γ_total	21	3	−3	−3	−1	−3	−1	5	5	3

Reduces to: 2A_1g + 1A_2g + 1B_1g + 1B_2g + 2E_g + 3A_2u + 1B_2u + 4E_u

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
2A_1g			2A_1g	No	Yes
1A_2g		−R_z	0A_2g	No	No
1B_1g			1B_1g	No	Yes
1B_2g			1B_2g	No	Yes
2E_g		−(R_x,R_y)	1E_g	No	Yes
3A_2u	−T_z		2A_2u	Yes	No
1B_2u			1B_2u	No	No
4E_u	−(T_x,T_y)		3E_u	Yes	No

	Signals	Vibrations
IR:	5	8
Raman:	5	6

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PF₅:

D_3h	E	2C₃	3C₂	σ_h	2S₃	3σ_v
Γ_total	18	0	−2	4	−2	4

Reduces to: 2A₁′ + 1A₂′ + 4E′ + 3A₂″ + 2E″

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
2A₁′			2A₁′	No	Yes
1A₂′		−R_z	0A₂′	No	No
4E′	−(T_x,T_y)		3E′	Yes	Yes
3A₂″	−T_z		2A₂″	Yes	No
2E″		−(R_x,R_y)	1E″	No	Yes

	Signals	Vibrations
IR:	5	8
Raman:	6	10

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CH₂Cl₂:

C_2v	E	C₂	σ_v	σ′_v
Γ_total	15	−1	3	3

Reduces to: 5A₁ + 2A₂ + 4B₁ + 4B₂

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
5A₁	−T_z		4A₁	Yes	Yes
2A₂		−R_z	1A₂	No	Yes
4B₁	−T_x	−R_y	2B₁	Yes	Yes
4B₂	−T_y	−R_x	2B₂	Yes	Yes

	Signals	Vibrations
IR:	8	8
Raman:	9	9

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IF₅:

C_4v	E	2C₄	C₂	2σ_v	2σ_d
Γ_total	18	2	−2	4	2

Reduces to: 4A₁ + 1A₂ + 2B₁ + 1B₂ + 5E

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
4A₁	−T_z		3A₁	Yes	Yes
1A₂		−R_z	0A₂	No	No
2B₁			2B₁	No	Yes
1B₂			1B₂	No	Yes
5E	−(T_x,T_y)	−(R_x,R_y)	3E	Yes	Yes

	Signals	Vibrations
IR:	6	9
Raman:	9	12

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H₃PCPH₃:

D_3d	E	2C₃	3C₂	i	2S₆	3σ_d
Γ_total	27	0	−1	−3	0	5

Reduces to: 3A_1g + 1A_2g + 4E_g + 1A_1u + 4A_2u + 5E_u

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
3A_1g			3A_1g	No	Yes
1A_2g		−R_z	0A_2g	No	No
4E_g		−(R_x,R_y)	3E_g	No	Yes
1A_1u			1A_1u	No	No
4A_2u	−T_z		3A_2u	Yes	No
5E_u	−(T_x,T_y)		4E_u	Yes	No

	Signals	Vibrations
IR:	7	11
Raman:	6	9

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H₂CCCH₂:

D_2d	E	2S₄	C₂	2C₂′	2σ_d
Γ_total	21	−1	−3	−1	5

Reduces to: 3A₁ + 1A₂ + 1B₁ + 4B₂ + 6E

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
3A₁			3A₁	No	Yes
1A₂		−R_z	0A₂	No	No
1B₁			1B₁	No	Yes
4B₂	−T_z		3B₂	Yes	Yes
6E	−(T_x,T_y)	−(R_x,R_y)	4E	Yes	Yes

	Signals	Vibrations
IR:	7	11
Raman:	11	15

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XeF₄:

D_4h	E	2C₄	C₂	2C₂′	2C₂″	i	2S₄	σ_h	2σ_v	2σ_d
Γ_total	15	1	−1	−3	−1	−3	−1	5	3	1

Reduces to: 1A_1g + 1A_2g + 1B_1g + 1B_2g + 1E_g + 2A_2u + 1B_2u + 3E_u

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
1A_1g			1A_1g	No	Yes
1A_2g		−R_z	0A_2g	No	No
1B_1g			1B_1g	No	Yes
1B_2g			1B_2g	No	Yes
1E_g		−(R_x,R_y)	0E_g	No	Yes
2A_2u	−T_z		1A_2u	Yes	No
1B_2u			1B_2u	No	No
3E_u	−(T_x,T_y)		2E_u	Yes	No

	Signals	Vibrations
IR:	3	5
Raman:	3	3

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SCl₆:

O_h	E	8C₃	6C₂	6C₄	3C₂	i	6S₄	8S₆	3σ_h	6σ_d
Γ_total	21	0	−1	3	−3	−3	−1	0	5	3

Reduces to: 1A_1g + 1E_g + 1T_1g + 1T_2g + 3T_1u + 1T_2u

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
1A_1g			1A_1g	No	Yes
1E_g			1E_g	No	Yes
1T_1g		−(R_x,R_y,R_z)	0T_1g	No	No
1T_2g			1T_2g	No	Yes
3T_1u	−(T_x,T_y,T_z)		2T_1u	Yes	No
1T_2u			1T_2u	No	No

	Signals	Vibrations
IR:	2	6
Raman:	3	6

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10.

Si(CH₃)₄:

T_d	E	8C₃	3C₂	6S₄	6σ_d
Γ_total	51	0	−1	−1	5

Reduces to: 3A₁ + 1A₂ + 4E + 5T₁ + 8T₂

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
3A₁			3A₁	No	Yes
1A₂			1A₂	No	No
4E			4E	No	Yes
5T₁		−(R_x,R_y,R_z)	4T₁	No	No
8T₂	−(T_x,T_y,T_z)		7T₂	Yes	Yes

	Signals	Vibrations
IR:	7	21
Raman:	14	32

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11.

(NC)₂CC(CN)₂:

D_2h	E	C₂(z)	C₂(y)	C₂(x)	i	σ(xy)	σ(xz)	σ(yz)
Γ_total	30	−2	0	0	0	0	10	2

Reduces to: 5A_g + 2B_1g + 5B_2g + 3B_3g + 2A_u + 5B_1u + 3B_2u + 5B_3u

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
5A_g			5A_g	No	Yes
2B_1g		−R_z	1B_1g	No	Yes
5B_2g		−R_y	4B_2g	No	Yes
3B_3g		−R_x	2B_3g	No	Yes
2A_u			2A_u	No	No
5B_1u	−T_z		4B_1u	Yes	No
3B_2u	−T_y		2B_2u	Yes	No
5B_3u	−T_x		4B_3u	Yes	No

	Signals	Vibrations
IR:	10	10
Raman:	12	12

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12.

Ni(CO)₄:

T_d	E	8C₃	3C₂	6S₄	6σ_d
Γ_total	27	0	−1	−1	5

Reduces to: 2A₁ + 2E + 2T₁ + 5T₂

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
2A₁			2A₁	No	Yes
2E			2E	No	Yes
2T₁		−(R_x,R_y,R_z)	1T₁	No	No
5T₂	−(T_x,T_y,T_z)		4T₂	Yes	Yes

	Signals	Vibrations
IR:	4	12
Raman:	8	18

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13.

C₆H₆:

D_6h	E	2C₆	2C₃	C₂	3C₂′	3C₂″	i	2S₃	2S₆	σ_h	3σ_d	3σ_v
Γ_total	36	0	0	0	−4	0	0	0	0	12	0	4

Reduces to: 2A_1g + 2A_2g + 2B_2g + 2E_1g + 4E_2g + 2A_2u + 2B_1u + 2B_2u + 4E_1u + 2E_2u

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
2A_1g			2A_1g	No	Yes
2A_2g		−R_z	1A_2g	No	No
2B_2g			2B_2g	No	No
2E_1g		−(R_x,R_y)	1E_1g	No	Yes
4E_2g			4E_2g	No	Yes
2A_2u	−T_z		1A_2u	Yes	No
2B_1u			2B_1u	No	No
2B_2u			2B_2u	No	No
4E_1u	−(T_x,T_y)		3E_1u	Yes	No
2E_2u			2E_2u	No	No

	Signals	Vibrations
IR:	4	7
Raman:	7	12

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14.

SBrF₅:

C_4v	E	2C₄	C₂	2σ_v	2σ_d
Γ_total	21	3	−3	5	3

Reduces to: 5A₁ + 1A₂ + 2B₁ + 1B₂ + 6E

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
5A₁	−T_z		4A₁	Yes	Yes
1A₂		−R_z	0A₂	No	No
2B₁			2B₁	No	Yes
1B₂			1B₂	No	Yes
6E	−(T_x,T_y)	−(R_x,R_y)	4E	Yes	Yes

	Signals	Vibrations
IR:	8	12
Raman:	11	15

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15.

Fe(CO)₅:

D_3h	E	2C₃	3C₂	σ_h	2S₃	3σ_v
Γ_total	33	0	−3	7	−2	7

Reduces to: 4A₁′ + 2A₂′ + 7E′ + 5A₂″ + 4E″

All Motions	− Translations	− Rotations	Vibrations	IR Active	Raman Active
4A₁′			4A₁′	No	Yes
2A₂′		−R_z	1A₂′	No	No
7E′	−(T_x,T_y)		6E′	Yes	Yes
5A₂″	−T_z		4A₂″	Yes	No
4E″		−(R_x,R_y)	3E″	No	Yes

	Signals	Vibrations
IR:	10	16
Raman:	13	22

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