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Chapter 6Answers to Problems

36.1 (a) NH

3v 3 vC E 2C 3F

iN 4 1 2

iP 3 0 1

3n' 12 0 2

3n 1 2' = 3A + A + 4E

trans 1 rot 2' = A + E ' = A + E

3n-6 1' = 2A + 2E = 4 frequencies

1Infrared 4 (2A + 2E)

1Raman 4 (2A + 2E)

1Polarized 2 (2A )

1Coincidences 4 (2A + 2E)

Silent modes 0

6(b) FeCl 3–

h 3 2 4 2 4 6 h dO E 8C 6C 6C 3C i 6S 8S 3F 6F

iN 7 1 1 3 3 1 1 1 5 3

iP 3 0 -1 1 -1 -3 -1 0 1 1

3n' 21 0 -1 3 -3 -3 -1 0 5 3

-87-

3n 1g g 1g 2g 1u 2u' = A + E + T + T + 3T + T

trans 1u rot 1g' = T ' = T

3n-6 1g g 2g 1u 2u' = A + E + T + 2T + T = 6 frequencies

1uInfrared 2 (2T )

1g g 2gRaman 3 (A + E + T )

1gPolarized 1 (A )

Coincidences 0

2uSilent modes 1 (T )

2(c) H CO

2v 2 v vC E C F (xz) F (yz)

iN 4 2 2 4

iP 3 -1 1 1

3n' 12 -2 2 4

3n 1 2 1 2' = 4A + A + 3B + 4B

trans 1 1 2 rot 2 1 2' = A + B + B ' = A + B + B

3n-6 1 1 2' = 3A + B + 2B = 6 frequencies

1 1 2Infrared 6 (3A + B + 2B )

1 1 2Raman 6 (3A + B + 2B )

1Polarized 3 (3A )

1 1 2Coincidences 6 (3A + B + 2B )

Silent modes 0

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5(d) PF

3h 3 2 h 3 vD E 2C 3C F 2S 3F

iN 6 3 2 4 1 4

iP 3 0 -1 1 -2 1

3n' 18 0 -2 4 -2 4

3n 1 2 2' = 2A ' + A ' + 4E' + 3A " + 2E"

trans 2 rot 2' = E' + A " ' = A ' + E"

3n-6 1 2' = 2A ' + 3E' + 2A " + E" = 8 frequencies

2Infrared 5 (3E' + 2A ")

1Raman 6 (2A ' + 3E' + E")

1Polarized 2 (2A ')

Coincidences 3 (3E')

Silent modes 0

2 6(e) C H (staggered configuration)

3d 3 2 6 dD E 2C 3C i 2S 3F

iN 8 2 0 0 0 4

iP 3 0 -1 -3 0 1

3n' 24 0 0 0 0 4

-89-

3n 1g 2g g 1u 2u u' = 3A + A + 4E + A + 3A + 4E

trans 2u u rot 2g g' = A + E ' = A + E

3n-6 1g g 1u 2u u' = 3A + 3E + A + 2A + 3E = 12 frequencies

2u uInfrared 5 (2A + 3E )

1g gRaman 6 (3A + 3E )

1gPolarized 3 (3A )

Coincidences 0

1uSilent modes 1 (A )

2 2(f) H O

2 2C E C

iN 4 0

iP 3 -1

3n' 12 0

3n' = 6A + 6B

trans rot' = A + 2B ' = A + 2B

3n-6' = 4A + 2B = 6 frequencies

Infrared 6 (4A + 2B)

Raman 6 (4A + 2B)

Polarized 4 (4A)

Coincidences 6 (4A + 2B)

Silent modes 0

-90-

56.2 (a) SeF –

4v 4 2 v dC E 2C C 2F 2F

iN 6 2 2 4 2

iP 3 1 -1 1 1

3n' 18 2 -2 4 2

3n 1 2 1 2' = 4A + A + 2B + B + 5E

trans 1 rot 2' = A + E ' = A + E

3n-6 1 1 2' = 3A + 2B + B + 3E = 9 frequencies


1 1 2Raman 9 (3A + 2B + B + 3E)

1Polarized 3 (3A )


Silent modes 0

4(b) AsF –


iN 5 1 3 3

iP 3 -1 1 1

3n' 15 -1 3 3

3n 1 2 1 2' = 5A + 2A + 4B + 4B

trans 1 1 2 rot 2 1 2' = A + B + B ' = A + B + B

3n-6 1 2 1 2' = 4A + A + 2B + 2B = 9 frequencies

-91-

1 1 2Infrared 8 (4A + 2B + 2B )

1 2 1 2Raman 9 (4A + A + 2B + 2B )

1Polarized 4 (4A )

1 1 2Coincidences 8 (4A + 2B + 2B )

Silent modes 0

3(c) BeF –

3h 3 2 h 3 vD E 2C 3C F 2S 3F

iN 4 1 2 4 1 2

iP 3 0 -1 1 -2 1

3n' 12 0 -2 4 -2 2

3n 1 2 2' = A ' + A ' + 3E' + 2A " + E"

trans 2 rot 2' = E' + A " ' = A ' + E"

3n-6 1 2' = A ' + 2E' + A " = 4 frequencies

2Infrared 3 (2E' + A ")

1Raman 3 (A ' + 2E')

1Polarized 1 (A ')

Coincidences 2 (2E')

Silent modes 0

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4(d) OSF

2v 2 v vC E C F F ’

iN 6 2 4 4

iP 3 -1 1 1

3n' 18 -2 4 4

3n 1 2 1 2' = 6A + 2A + 5B + 5B

trans 1 1 2 rot 2 1 2' = A + B + B ' = A + B + B

3n-6 1 2 1 2' = 5A + A + 3B + 3B = 12 frequencies

1 1 2Infrared 11 (5A + 3B + 3B )

1 2 1 2Raman 12 (5A + A + 3B + 3B )

1Polarized 5 (5A )

1 1 2Coincidences 11 (5A + 3B + 3B )

Silent modes 0

(e) trans-FNNF

2h 2 hC E C i F

iN 4 0 0 4

iP 3 -1 -3 1

3n' 12 0 0 4

-93-

3n g g u u' = 4A + 2B + 2A + 4B

trans u u rot g g' = A + 2B ' = A + 2B

3n-6 g u u' = 3A + A + 2B = 6 frequencies

u uInfrared 3 (A + 2B )

gRaman 3 (3A )

gPolarized 3 (3A )

Coincidences 0

Silent modes 0

(f) cis-FNNF


iN 4 0 0 4

iP 3 -1 1 1

3n' 12 0 0 4

3n 1 2 1 2' = 4A + 2A + 2B + 4B

trans 1 1 2 rot 2 1 2' = A + B + B ' = A + B + B

3n-6 1 2 2' = 3A + A + 2B = 6 frequencies

1 2Infrared 5 (3A + 2B )

1 2 2Raman 6 (3A + A + 2B )

1Polarized 3 (3A )

1 2Coincidences 5 (3A + 2B )

Silent modes 0

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2 3(g) S O 2–

3v 3 vC E 2C 3F

iN 5 2 3

iP 3 0 1

3n' 15 0 3

3n 1 2' = 4A + A + 5E

trans 1 rot 2' = A + E ' = A + E

3n-6 1' = 3A + 3E = 6 frequencies


1Raman 6 (3A + 3E)

1Polarized 3 (3A )


Silent modes 0

2 6(h) B H

2h 2 2 2D E C (z) C (y) C (x) i F(xy) F(xz) F(yz)

iN 8 2 2 0 0 2 6 4

P 3 -1 -1 -1 -3 1 1 1i

3n' 24 -2 -2 0 0 2 6 4

-95-

3n g 1g 2g 3g u 1u 2u 3u' = 4A + 2B + 3B + 3B + A + 4B + 3B + 4B

trans 1u 2u 3u rot 1g 2g 3g' = B + B + B ' = B + B + B

3n-6 g 1g 2g 3g u 1u 2u 3u' = 4A + B + 2B + 2B + A + 3B + 2B + 3B

1u 2u 3uInfrared 8 (3B + 2B + 3B )

g 1g 2g 3gRaman 9 (4A + B + 2B + 2B )

gPolarized 4 (4A )

Coincidences 0

uSilent modes 1 (A )

3 2 2h6.3 C O as C

2h 2 hC E C i F

iN 5 1 1 5

iP 3 -1 -3 1

3n' 15 -1 -3 5

3n g g u u' = 4A + 2B + 3A + 6B

trans u u rot g g' = A + 2B ' = A + 2B

3n-6 g u u' = 3A + 2A + 4B

Comparing these results with those developed in the text (pp. 190-191):

2h 4hC D

u u u uInfrared 6 (2A + 4B ) 4 (2G + 2A )+

g g gRaman 3 (3A ) 3 (2G + A )+

g gPolarized 3 (3A ) 2 (2G )+

Coincidences 0 0

Silent modes 0 0

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d 4h6.4 In the correlation diagrams shown below, only the species of T and D associated withnormal modes are shown on the left. The frequency numbering of the undistorted structurehas been retained for the distorted structure, with subscripts (a, b, c) for formerly degeneratemodes, to show the fate of individual modes on descent in symmetry.

4 2v(a) Distorted tetrahedron (XY E) - C

d 2vT C

1 1 1 1 2a 3a 4a< A A < , < , < , < i.r., Raman (pol)

2 2 2b< E A < Raman

3 4 2 1 3b 4b< , < T B < , < i.r., Raman

2 3c 4cB < , < i.r., Raman

2d(b) Slightly squashed tetrahedron - D

d 2dT D

1 1 1 1 2a< A A < , < Raman (pol)

2 1 2b< E B < Raman

3 4 2 2 3a 4a< , < T B < , < i.r., Raman

3bc 4bcE < , < i.r., Raman

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4v(c) Square pyramid with X at the apex - C

4h 4vD C

1 1g 1 1 3< A A < , < i.r., Raman (pol)

2 1g 1 2 5< B B < , < Raman

4 2g 2 4< B B < Raman

3 2u 6 7< A E < , < i.r., Raman

5 2u< B

6 7 u< , < E

5 2uNote that the formerly silent mode, < (B ), of the perfect square plane becomes Ramanactive as a result of this distortion.

-98-

4 2h 2 4h(d) Planar MX with two long trans positions - D (C ' of D retained)

4h 2hD D

1 1g< A

1g 1 2A < , < Raman (pol)

2 1g< B

4 2g 1g 4< B B < Raman

3 2u< A

1u 3 5B < , < i.r.

5 2u< B

2u 6a 7aB < , < i.r.

6 7 u< , < E

3u 6b 7bB < , < i.r.

-99-

6.5 Direct correlations from structure I to II to III can be made by using the correlation tables in

2vAppendix B. A correlation from III to IV cannot be made in this way. Although C is a

4vsubgroup of C , structure IV is not obtained by retaining pre-existing elements of structure

2III. Most significantly, the C axis of IV does not exist in III, but rather is newly created.

4h 2vHowever, it is possible to make a correlation from II (D ) to IV (C ), using the correlation

2 4h 2 2v 4vin Appendix B in which C " of D is retained as the C axis of C . If a correlation III (C )

2v6 IV (C ) is attempted, using the correlation tables without realizing the inconsistency of

1axis orientations in the two structures, the incorrect results for IV will be 6A (R - pol, ir) +

2 1 2A (R) + 4 B (R, ir) + 4B (R, ir). That the results from the correlation II 6 IV shown beloware correct can be verified by determining the selection rules for IV de novo. By contrast, astructure whose selection rules could be determined correctly by direct correlation from III

2 3would be trans-MA B C. (Structure IV in Fig. 3.1 may be changed to this in the future.)

In the correlation diagrams shown here (I 6 II 6 III, this page, and II 6 IV, next page) onlythose symmetry species associated with normal modes are shown. When two or morefrequencies occur with the same symmetry, the number of occurrences is indicated in frontof the Mulliken symbol.

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I II

1u 2u uInfrared 2 (2T ) 5 (2A + 3E )

1g g 2g 1g 1g 2g gRaman 3 (A + E + T ) 5 (2A + B + B + E )

1g 1gPolarized 1 (A ) 2 (A )

Coincidences 0 0

2u 2uSilent modes 1 (T ) 1 (B )

III IV

1 1 1 2Infrared 8 (4A + 4E) 13 (6A + 4B + 3B )

1 1 2 1 2 1 2Raman 11 (4A + 2B + B + 4E) 15 (6A + 2A + 4B + 3B )

1 1Polarized 4 (A ) 6 (6A )

1 1 1 2Coincidences 8 (4A + 4E) 13 (6A + 4B + 3B )

Silent modes 0 0

-101-

2 2h6.6 (a) CO [O=C=O] Use D as a working group.


iN 3 3 1 1 1 1 3 3

P 3 -1 -1 -1 -3 1 1 1i

3n' 9 -3 -1 -1 -3 1 3 3

3n g 2g 3g 1u 2u 3u' = A + B + B + 2B + 2B + 2B

trans 1u 2u 3u rot 2g 3g 1g z' = B + B + B ' = B + B (not B – R )

3n-5 g 1u 2u 3u' = A + B + B + B

4h 3n-5 g u uY In D , ' = G + G + A = 3 frequencies+ +

u uInfrared 2 (G + A )+

gRaman 1 ( G )+

gPolarized 1 ( G )+

Coincidences 0

Silent modes 0

2v(b) OCN [O–C/N] Use C as a working group.– –


iN 3 3 3 3

iP 3 -1 1 1

3n' 9 -3 3 3

3n 1 1 2' = 3A + 3B + 3B

trans 1 1 2 rot 1 2 2 z' = A + B + B ' = B + B (not A – R )

3n-5 1 1 2' = 2A + B + B

4v 3n-5In C , ' = 2E + A = 3 frequencies+

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Infrared 3 (2G + A)+

Raman 3 (2G + A)+

Polarized 2 (2G )+

Coincidences 3 (2G + A)+

Silent modes 0

2h(c) H–C/C–H Use D as a working group.


iN 4 4 0 0 0 0 4 4

P 3 -1 -1 -1 -3 1 1 1i

3n' 12 -4 0 0 0 0 4 4

3n g 2g 3g 1u 2u 3u' = 2A + 2B + 2B + 2B + 2B + 2B


3n-5 g 2g 3g 1u 2u 3u' = 2A + B + B + B + B + B

4h 3n-5 g g u uY In D , ' = 2G + A + G + A = 5 frequencies+ +

u uInfrared 2 (G + A )+

g gRaman 3 (2G + A )+

gPolarized 2 (2G )+

Coincidences 0

Silent modes 0

-103-

(d) Cl–C/C–H


iN 4 4 4 4

iP 3 -1 1 1

3n' 12 -4 4 4

3n 1 1 2' = 4A + 4B + 4B

trans 1 1 2 rot 1 2 2 z' = A + B + B ' = B + B (not A – R )

3n-5 1 1 2' = 3A + 2B + 2B

4v 3n-5In C , ' = 3E + 2A = 5 frequencies+

Infrared 5 (3G +2 A)+

Raman 5 (3G + 2A)+

Polarized 3 (3G )+

Coincidences 5 (3G + 2A)+

Silent modes 0

2h(e) H–C/C–C/C–H Use D as a working group.


iN 6 6 0 0 0 0 6 6

P 3 -1 -1 -1 -3 1 1 1i

3n' 18 -6 0 0 0 0 6 6

3n g 2g 3g 1u 2u 3u' = 3A + 3B + 3B + 3B + 3B + 3B


3n-5 g 2g 3g 1u 2u 3u' = 3A + 2B + 2B + 2B + 2B + 2B

4h 3n-5 g g u uY In D , ' = 3G + 2A + 2G + 2A = 9 frequencies+ +

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u uInfrared 4 (2G + 2A )+

g gRaman 5 (3G + 2A )+

gPolarized 3 (3G )+

Coincidences 0

Silent modes 0

3n-6 1 2 1 1 2 2 3 46.7 (a) ' = A ' + A " + 2E' = 4 frequencies Y < (A '), < (A "), < (E'), < (E')The systematic assignment of frequency numbers labels nondegenerate modes before

2 3hdegenerate modes, even though E' is listed before A " in the current D character table.

3 3 3v(b) Association along the C axis of the NO ion causes its symmetry to descend to C . The–

changes in spectra can be predicted from the following correlation diagram.

3h 3vD C

1 1 1 1 2R. (pol) < A ' A < , < i.r., R. (pol)

2 2i.r. < A "

3i.r., R. < ,

4<3 4E E < , < i.r., R.

The descent does not lift any degeneracies, but the spectroscopic activities change

1significantly. The totally symmetric stretching mode, < , becomes infrared active. The out-

2of-plane deformation mode, < , becomes a Raman-active polarized mode.

3 2vIn-plane association causes the NO ion's symmetry to descend to C . The changes in–

spectra can be predicted from the following correlation diagram.

3h 2vD C

1 1 1 1 3a 4aR. (pol) < A ' A < , < , < i.r., R. (pol)

2 2 1 2i.r. < A " B < i.r., R

3 4 2 3b 4bi.r., R. < , < E' B < , < i.r., R.

3 4 3 4In this case the E' degeneracies of both < and < are lifted. Thus, both < and < may split

2v 1 2into two frequencies each, with C symmetries A and B , giving a total of six frequenciesactive in both the infrared and Raman spectra.

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The predictions for both modes of association are summarized in the table below.

3h 3v 2vD C C

2 1 1 1 2Infrared 3 (A " + 2E') 4 (2A + 2E) 6 (3A + B + 2B )

1 1 1 1 2Raman 3 (A ' + 2E') 4 (2A + 2E) 6 (3A + B + 2B )

1 1 1Polarized 1 (A ') 2 (2A ) 3 (3A )

1 1 1 2Coincidences 2 (2E') 4 (2A + 2E) 6 (3A + B + 2B )

Silent modes 0 0 0

2v(c) The data are consistent with the predictions of the C model of association.

2 2(d) The 830 cm frequency is < (A "), which is infrared active but Raman inactive for the–1

2 1 3h"free" ion. The first overtone, 2< , has A ' symmetry in D , which is a Raman active

2v 3vspecies. This fundamental becomes Raman active in either the C or C models ofassociation. Regardless of the mode of association, the overtone would be Raman active,since all first overtones of nondegenerate modes are totally symmetric and therefore Raman

2 3 3active. Failure to observe < in the Raman spectrum of 0.3-2.3 M Al(NO ) solutions mightseem to suggest that there is no cation-anion or solvent-anion association. However,association cannot be ruled out, because the fundamental, although Raman allowed byassociation, might be very broad and have too weak an intensity to be observed above theinstrumental background.

6.8 (a) True. All even-number direct products of a nondegenerate species give the totallysymmetric representation. Thus, all even-number overtones will belong to the totallysymmetric representation, which is always Raman allowed.

1 s n nv(b) Not always true. Molecules belonging to point groups C , C , C , and C have at leastone unit vector transforming as the totally symmetric representation. In these groups, theeven-number overtones will be infrared allowed, as well as Raman allowed. Also, in groupswith degenerate representations, the direct products will be reducible representations, whichmay contain an irreducible representation allowing infrared activity.

(c) True. All odd-number direct products of any irreducible representation will be or containthe original representation. If a vibrational mode is active as a fundamental, it follows thatall its odd-number overtones will be infrared allowed, too.

(d) True. The direct product of any irreducible representation with the totally symmetricrepresentation is the non-totally symmetric representation. The symmetry-based selection

s irules for the non-totally symmetric mode will apply to any combination < ± < , because the

isymmetry of the combination is the same as < .

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(e) True. All first overtones belong to or contain the totally symmetric representation. Theproduct of the totally symmetric representation with itself is, of course, totally symmetric.

s iTherefore combinations of the type < ± 2< will be Raman allowed.

6 2u6.9 (a) The < frequency arises from three degenerate T modes, which are inactive in both theinfrared and Raman spectra. They are silent modes.

h(b) O is centrosymmetric, so the rule of mutual exclusion precludes infrared-active modesfrom being Raman-active, and vice versa.

4(c) 2< This should occur at approximately 2 x 262.7 cm = 525.4 cm . The observed-1 -1

frequency (531 ± 3 cm ) is a little high, but not unreasonably so. The symmetry-1

1u 1u 1g g 1g 2gof this overtone is T x T = A + E + T + T , making it Raman allowed on the

1g g 2gbasis of A , E , and T .

4 6< + < This should occur at approximately 262.7 cm + 117 cm = 379.7 cm , which is-1 -1 -1

good agreement with the observed 380 ± 3 cm . The symmetry of this-1

1u 2u 2g g 1g 2gcombination is T x T = A + E + T + T , making it Raman allowed on the

g 2gbasis of E and T .

6 6 62< If the band at 233 ± 2 cm is assigned as 2< , then < is approximately 117 cm . -1 -1

As shown above, this is consistent with the assignment of the 380 ± 3 cm band to-1

4 6 6 2u 2u 1g g 1g 2g< + < . The symmetry of the 2< overtone is T x T = A + E + T + T ,

1g g 2gmaking it Raman allowed on the basis of A , E , and T . [Note: The symmetry of

2u 2u 1u 1u 4T x T is identical to T x T , as shown for 2< above.] The observed

1gpolarization is consistent with the A component.

1 3< + < This should occur at approximately 741 cm + 739 cm = 1480 cm , which is-1 -1 -1

good agreement with the observed 1479.4 ± 0.5 cm . The symmetry of this-1

1g 1u 1u 1ucombination is A x T = T , making it infrared allowed on the basis of T .

2 3< + < This should occur at approximately 652 cm + 739 cm = 1391 cm , which is-1 -1 -1

good agreement with the observed 1386.9 ± 0.5 cm . The symmetry of this-1

g 1u 1u 2u 1ucombination is E x T = T + T , making it infrared allowed on the basis of T .

2 4< + < This should occur at approximately 652 cm + 263 cm = 915 cm , which is good-1 -1 -1

agreement with the observed 913.1 ± 0.5 cm . ± 0.5 cm . The symmetry of this-1 -1

g 1u 1u 2u 1ucombination is E x T = T + T , making it infrared allowed on the basis of T .

(d) The combination and overtones observed in the Raman spectrum have gerade symmetry,and the combinations observed in the infrared spectrum have ungerade symmetry. Consistent with the rule of mutual exclusion, the Raman-active frequencies cannot beinfrared-active, and vice versa.

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1u(e) In order to be infrared allowed, a combination would have to be or contain T . The

6possible combinations with < , their symmetries, and whether infrared allowed are shownbelow.

Combination Symmetries Allowed?

1 6 1g 2u 2u< + < A x T = T No

2 6 g 2u 1u 2u< + < E x T = T + T Yes

3 6 1u 2u 2g g 1g 2g< + < T x T = A + E + T + T No

4 6 1u 2u 2g g 1g 2g< + < T x T = A + E + T + T No

5 6 2g 2u 1u u 1u 2u< + < T x T = A + E + T + T Yes

The predicted frequencies for the allowed combinations are:

2 6< + < . 652 cm + 117 cm = 769 cm-1 -1 -1

5 6< + < . 317 cm + 117 cm = 434 cm-1 -1 -1

Although allowed, these combinations apparently have too weak intensities to be observed.

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