cambridge university chemistry databook

8/12/2019 Cambridge University Chemistry DataBook

1/18

Department of Chemistry

Data Book(revised October 2005)

Contents

Periodic table of the elements inside front cover

Physical constants and conversion factors 1

Greek alphabet 3

Series 3

Stirling's formula 3

Determinants 3

Integrals 4

Integration by parts 4Trigonometrical formulae 4

Cosine formula 4Spherical polar coordinates 5

Laplacian 5

Spherical harmonics 5

Ladder operators 5

Character tables 6 11

Selected tables for descent in symmetry 11

Reduction of a representation 12

Projection operators 12

Direct products 12

Antisymmetrized squares 12

Flow chart for determining molecular point groups 13

Space groups (GEPs and SEPs) 14

Parameters for selected magnetic nuclei 15

Amino acids 16

Nucleotide bases 17


2/18

H

1

1.008

He

2

4.003

Li

3

6.94

Be

4

9.01

symbolatomic number

mean atomic massB

5

10.81

C

6

12.01

N

7

14.01

O

8

16.00

F

9

19.00

Ne

10

20.18

Na

11

22.99

Mg

12

24.31

Al

13

26.98

Si

14

28.09

P

15

30.97

S

16

32.06

Cl

17

35.45

Ar

18

39.95

K

19

39.102

Ca

20

40.08

Sc

21

44.96

Ti

22

47.90

V

23

50.94

Cr

24

52.00

Mn

25

54.94

Fe

26

55.85

Co

27

58.93

Ni

28

58.71

Cu

29

63.55

Zn

30

65.37

Ga

31

69.72

Ge

32

72.59

As

33

74.92

Se

34

78.96

Br

35

79.904

Kr

36

83.80

Rb

37

85.47

Sr

38

87.62

Y

39

88.91

Zr

40

91.22

Nb

41

92.91

Mo

42

95.94

Tc

43

Ru

44

101.07

Rh

45

102.91

Pd

46

106.4

Ag

47

107.87

Cd

48

112.40

In

49

114.82

Sn

50

118.69

Sb

51

121.75

Te

52

127.60

I

53

126.90

Xe

54

131.30

Cs

55

132.91

Ba

56

137.34

La*

57

138.91

Hf

72

178.49

Ta

73

180.95

W

74

183.85

Re

75

186.2

Os

76

190.2

Ir

77

192.2

Pt

78

195.09

Au

79

196.97

Hg

80

200.59

Tl

81

204.37

Pb

82

207.2

Bi

83

208.98

Po

84

At

85

Rn

86

Fr

87

Ra

88

Ac+

89

*Lanthanides Ce58

140.12

Pr

59

140.91

Nd

60

144.24

Pm

61

Sm

62

150.4

Eu

63

151.96

Gd

64

157.25

Tb

65

158.93

Dy

66

162.50

Ho

67

164.93

Er

68

167.26

Tm

69

168.93

Yb

70

173.04

Lu

71

174.97

+Actinides Th90

232.01

Pa

91

U

92

238.03

Np

93

Pu

94

Am

95

Cm

96

Bk

97

Cf

98

Es

99

Fm

100

Md

101

No

102

Lr

103


3/18

Constants

Symbol and ValueName definition (uncertainty) Unit

! 3 141592653589

e 2 718281828459

ln 10 1 log10e 2 302585092994

Speed of light c 2 99792458 108m s 1

Plancks constant h 6 6260693 11 10 34 Js

h h 2! 1 05457168 18 10 34 Js

Avogadros constant NA 6 0221415 10 1023mol 1

Elementary charge e 1 60217653 14 10 19C

Electron rest mass me 0 91093826 16 10 30kg

Atomic mass unit mu 1gmol 1 NA 1 66053886 28 10 27kg

Proton rest mass mp 1 67262171 29 10 27kg

Neutron rest mass mn 1 67492728 29 10 27kg

Faraday constant F NAe 9 64853383 83 104Cmol 1

Boltzmann constant kB 1 3806505 24 10 23 JK 1

Molar Gas constant R NAkB 8 314472 15 J mol 1K 1

Permeability of vacuum 0 4! 10 7 Hm 1

Permittivity of vacuum "0 1 0c2 8 8541878 10 12Fm 1

4!"0 1 1126501 10 10Fm 1

Bohr magneton B eh 2me 9 27400949 80 10 24 JT 1

Nuclear magneton N eh 2mp 5 05078343 43 10 27 JT 1

Stefan-Boltzmann constant # 2!5

k4

B 15h3

c2

5 670400 40 10 8

Wm 2

K 4

Bohr radius a0 4!"0h2 mee2 0 5291772108 18 10 10m

Hartree energy Eh e2 4!"0a0 4 35974417 75 10 18 J

Fine structure constant $ e2 4!"0hc 7 297352568 24 10 3

$ 1 137 035986

CODATA recommended values, December 2002http://physics.nist.gov/constants

The estimated standard uncertainty, in parentheses after the value,applies to the least significant digits of the value.

Energy Conversion Factors

J kJ mol 1 cm 1 K

1 J 1 6 0221 1020 5 0341 1022 7 2429 1022

1hartree 4 35974 10 18 2625 5 219475 315773

1 eV 1 60218 10 19 96 485 8065 54 11604

1kJmol 1 1 66054 10 21 1 83 5935 120 27

1 cm 1 1 98645 10 23 11 963 10 3 1 1 4388

1 K 1 38065 10 23 8 3145 10 3 0 69504 1

1 Hz 6 62607 10 34 3 9903 10 13 3 3356 10 11 4 7992 10 11

Note: the energy of a photon with reciprocal wavelength (wavenumber) 1 %andfrequency &ishc % h&. The energy corresponding to a temperatureTis kBT.

Other conversion factors

Length 10 10m

Energy cal 4 184 J

Pressure atm = 760 Torr 101 325 Pa

Torr = mm Hg 133 3 Pa

bar 105Pa

Radioactivity becquerel, Bq 1 s 1

curie, Ci 3 7 1010Bq

Charge esu 3 33564 10 10C

Dipole moment debye 10 18esucm 3 33564 10 30C m

a u ea0 8 478358 10 30C m

Temperature C 0 C 273 15K

The entropy unit (e.u.) used for entropies of activation is usually the c.g.s. unitcal/mol/ C. Howeversome authors use the same symbol for the SI unit, Jmol 1K 1.

2


4/18

Greek Alphabet

A ! alpha H " eta N # nu T $ tau

B % beta & ' ( theta ) * xi + , upsilon

- . gamma I / iota O o omicron 0 1 2 phi

3 4 delta K 5 kappa 6 7 pi X 8 chi

E 9 epsilon : ; lambda P < rho = > psiZ ? zeta M mu @ A sigma B C omega

Series

Geometrical progression

Sn a az az2 azn 1 a

1 zn

1 z S!

a

1 z when z 1

Power series

expz 1 z

z2

2!

z3

3!

zn

n!

cosz eiz e iz

2 1

z2

2!

z4

4!

z6

6!

sinz eiz e iz

2i z

z3

3!

z5

5!

z7

7!

1 z p 1 pz p p 1

2! z2

p p 1 p 2

3! z3 z 1

ln 1 z z z2

2

z3

3

z n

n z 1

Stirlings formula

ln n! n ln n n for largen

Determinants

a b

c d ad bc

In general,

det A !j Ai jCji (ifixed at any value)

where the cofactor Cji is 1 i j times the determinant of the matrix obtained by deleting theith row

and the jth column ofA. For example,

a11 a12 a13a21 a22 a23a31 a32 a33

a11a22 a23a32 a33

a12a21 a23a31 a33

a13a21 a22a31 a32

a11a22a33 a11a23a32 a12a23a31 a12a21a33 a13a21a32 a13a22a31

3


5/18

Integrals

!

!exp ax2 dx 7 a a 0

!

!

x2n exp ax2 dx 1 3 5 2n 17 a

2a n n 1; a 0

!

0

rn exp ar dr n!

an 1 n 0; a 0

" 2

0

sinm ' cosn ' d' m 1

m n

" 2

0

sinm 2 ' cosn ' d'

n 1

m n

" 2

0

sinm ' cosn 2 ' d'

so that

" 2

0

sinm ' cosn ' d' m 1 m 3 n 1 n 3m n m n 2 m n 4

C

whereC 7 2 ifm andn are botheven, andC 1 otherwise. E.g.:

" 2

0

sin' cos3 ' d' 2

4 2

1

4;

" 2

0

sin2 ' cos2 ' d' 1 1

4 2

7

2

7

16

Integration by partsb

au

d

dxdx u ba

b

a

du

dx dx

Trigonometrical formulae

sin A B sinA cosB cosA sinB

cos A B cosA cosB sinA sinB

cosA cosB 12

cos A B cos A B

sinA sinB 12

cos A B cos A B

sinA cosB 12

sin A B sin A B

sinA sinB 2sinA B

2cos

A B

2

sinA sinB 2cosA B

2sin

A B

2

cosA cosB 2cosA B

2 cos

A B

2

cosA cosB 2sinA B

2 sin

A B

2

Cosine formula

a2 b2 c2 2bc cosA

4


6/18

Spherical Polar Coordinates

Relationship with Cartesian coordinates

x r sin' cos2 r x2 y2 z2

y r sin' sin 2 ' arccos z r

z r cos' 2 arctan y x

Integration

dV!

r 0

"

# 0

2"

$ 0r2 sin' dr d' d2

z

x

y

r'

2

Laplacian

D2> E2>

Ex2E2>

Ey2E2>

Ez2

1

r2E

Er r2

E>

Er

1

r2 sin '

E

E' sin '

E>

E'

1

r2 sin2 '

E2>

E22

Spherical Harmonics

Ylm ' 2 2l 1

47 Clm ' 2

where

Clm ' 2 l m !

l m !

1

2

Pm

l cos' eim$ 1

m form 0,

1 form 0.

HerePml is an associated Legendre polynomial. In particular,

C00 1

C10 cos' z r

C1 11

2sin 'e i$ 1

2 x iy r

C201

2 3cos2 ' 1 1

2 3z2 r2 r2

C2 13

2cos ' sin 'e i$ 3

2 zx izy r2

C2 23

8sin2 'e 2i$ 3

8 x2 y2 2ixy r2

Ladder Operators

J Jx iJy; J J M J J 1 M M 1 J M 1

5


7/18

Character tables for some important symmetry groups

Ci E i

Ag 1 1 Rx;Ry;Rz x2;y2;z2;xy;xz;yzAu 1 1 x;y;z

Cs E #h

A 1 1 x;y Rz x2;y2;z2;xyA 1 1 z Rx;Ry xz;yz

C2 E Cz2

A 1 1 z Rz x2;y2;z2;xyB 1 1 x;y Rx;Ry xz;yz

2

C2v E Cz2

#xz #yz

A1 1 1 1 1 z x2;y2;z2

A2 1 1 1 1 Rz xyB1 1 1 1 1 x Ry xzB2 1 1 1 1 y Rx yz

C2h E Cz2

i #xy

Ag 1 1 1 1 Rz x2;y2;z2;xyBg 1 1 1 1 Rx;Ry xz;yzAu 1 1 1 1 zBu 1 1 1 1 x;y

D2 E Cz2

Cy2

Cx2

A 1 1 1 1 x2;y2;z2

B1 1 1 1 1 z Rz xyB2 1 1 1 1 y Ry xzB3 1 1 1 1 x Rx yz

D2d E 2S4 Cz2 2C2 2#d

A1 1 1 1 1 1 x2 y2;z2

A2 1 1 1 1 1 RzB1 1 1 1 1 1 x2 y2

B2 1 1 1 1 1 z xyE 2 0 2 0 0 x y Rx Ry xz yz

6


8/18

D2h E Cz2

Cy2

Cx2

i #xy #xz #yz

Ag 1 1 1 1 1 1 1 1 x2;y2;z2

B1g 1 1 1 1 1 1 1 1 Rz xyB2g 1 1 1 1 1 1 1 1 Ry xzB3g 1 1 1 1 1 1 1 1 Rx yzAu 1 1 1 1 1 1 1 1B1u 1 1 1 1 1 1 1 1 zB2u 1 1 1 1 1 1 1 1 yB3u 1 1 1 1 1 1 1 1 x

C3 E C3 C23

' exp 2!i 3

A 1 1 1 z Rz x2 y2; z2

1 ' '2 x iy Rx iRy xz iyz; x2 2ixy y2E

1 '2 ' x iy Rx iRy xz iyz; x2 2ixy y2

3

C3v E 2Cz3 3#v

A1 1 1 1 z x2 y2;z2

A2 1 1 1 RzE 2 1 0 x y Rx Ry xz yz; x2 y2 2xy

D3 E 2Cz3

3C2

A1 1 1 1 x2 y2;z2

A2 1 1 1 z RzE 2 1 0 x y Rx Ry xz yz ; x2 y2 2xy

D3d E 2C3 3C2 i 2S6 3#d

A1g 1 1 1 1 1 1 x2 y2;z2

A2g 1 1 1 1 1 1 RzEg 2 1 0 2 1 0 Rx Ry xz yz ; x2 y2 2xyA1u 1 1 1 1 1 1A2u 1 1 1 1 1 1 zEu 2 1 0 2 1 0 x y

D3h E 2C3 3C2 #h 2S3 3#v

A1

1 1 1 1 1 1 x2 y2;z2

A2

1 1 1 1 1 1 RzE 2 1 0 2 1 0 x y x2 y2 2xyA

1 1 1 1 1 1 1

A2

1 1 1 1 1 1 zE 2 1 0 2 1 0 Rx Ry xz yz

7


9/18

C4v E 2C4 C24 2#v 2#d

A1 1 1 1 1 1 z x2 y2;z2

A2 1 1 1 1 1 RzB1 1 1 1 1 1 x2 y2

B2 1 1 1 1 1 xy

E 2 0 2 0 0 x y Rx Ry xz yz

4

Note: The#vplanes inC4vcoincide with thexzandyzplanes.

D4h E 2C4 C24 2C2 2C2 i 2S4 #h 2#v 2#d

A1g 1 1 1 1 1 1 1 1 1 1 x2 y2;z2

A2g 1 1 1 1 1 1 1 1 1 1 RzB1g 1 1 1 1 1 1 1 1 1 1 x2 y2

B2g 1 1 1 1 1 1 1 1 1 1 xyEg 2 0 2 0 0 2 0 2 0 0 Rx Ry xz yzA1u 1 1 1 1 1 1 1 1 1 1A2

u 1 1 1 1 1 1 1 1 1 1 z

B1u 1 1 1 1 1 1 1 1 1 1B2u 1 1 1 1 1 1 1 1 1 1Eu 2 0 2 0 0 2 0 2 0 0 x y

Note: The C2axes in D4hcoincide with thexandyaxes, and the#vplanes with thexzandyzplanes.

Note that the quantities ( 12

5 1 satisfy( 2 1 ( and( ( 1.

C5v E 2C5 2C25 5#v ( 1

2 5 1

A1 1 1 1 1 z x2 y2;z2

A2 1 1 1 1 RzE1 2 ( ( 0 x y Rx Ry xz yzE2 2 ( ( 0 x2 y2 2xy

5

D5 E 2C5 2C25 5C2 ( 1

2 5 1

A1 1 1 1 1 x2 y2;z2

A2 1 1 1 1 z RzE1 2 ( ( 0 x y Rx Ry xz yzE2 2 ( ( 0 x2 y2 2xy

D5d

E 2C5 2C25

5C2 i 2S310

2S105#d

( 12

5 1

A1g 1 1 1 1 1 1 1 1 x2 y2;z2

A2g 1 1 1 1 1 1 1 1 RzE1g 2 ( ( 0 2 ( ( 0 Rx Ry xz yzE2g 2 ( ( 0 2 ( ( 0 x2 y2 2xyA1u 1 1 1 1 1 1 1 1A2u 1 1 1 1 1 1 1 1 zE1u 2 ( ( 0 2 ( ( 0 x yE2u 2 ( ( 0 2 ( ( 0

8


10/18

D5h E 2C5 2C25 5C2 #h 2S5 2S3

5 5#v ( 12 5 1

A1

1 1 1 1 1 1 1 1 x2 y2;z2

A2

1 1 1 1 1 1 1 1 RzE

1 2 ( ( 0 2 ( ( 0 x y

E2

2 ( ( 0 2 ( ( 0 x2 y2 2xyA

1 1 1 1 1 1 1 1 1

A2 1 1 1 1 1 1 1 1 zE

1 2 ( ( 0 2 ( ( 0 Rx Ry xz yz

E2

2 ( ( 0 2 ( ( 0

C6v E 2C6 2C26 C3

6 3#v 3#d

A1 1 1 1 1 1 1 z x2 y2;z2

A2 1 1 1 1 1 1 RzB1 1 1 1 1 1 1B2 1 1 1 1 1 1E1 2 1 1 2 0 0 x y Rx Ry xz yzE2 2 1 1 2 0 0 x2 y2 2xy

6

D6 E 2C6 2C26 C3

6 3C2 3C2

A1 1 1 1 1 1 1 x2 y2;z2

A2 1 1 1 1 1 1 z RzB1 1 1 1 1 1 1B2 1 1 1 1 1 1E1 2 1 1 2 0 0 x y Rx Ry xz yzE2 2 1 1 2 0 0 x2 y2 2xy

D6h E 2C6 2C26 C3

6 3C2 3C2 i 2S3 2S6 #h 3#d 3#v

A1g 1 1 1 1 1 1 1 1 1 1 1 1 x2 y2;z2

A2g 1 1 1 1 1 1 1 1 1 1 1 1 RzB1g 1 1 1 1 1 1 1 1 1 1 1 1B2g 1 1 1 1 1 1 1 1 1 1 1 1E1g 2 1 1 2 0 0 2 1 1 2 0 0 Rx Ry xz yzE2g 2 1 1 2 0 0 2 1 1 2 0 0 x2 y2 2xyA1u 1 1 1 1 1 1 1 1 1 1 1 1A2u 1 1 1 1 1 1 1 1 1 1 1 1 zB1u 1 1 1 1 1 1 1 1 1 1 1 1B2u 1 1 1 1 1 1 1 1 1 1 1 1E1u 2 1 1 2 0 0 2 1 1 2 0 0 x y

E2u 2 1 1 2 0 0 2 1 1 2 0 0

9


11/18

T E 4C3 4C23 3C2 ' exp 2!i 3

A1 1 1 1 1 x2 y2 z2

E 1 ' '2 1 z2 '2x2 'y2

1 '2 ' 1 z2 'x2 '2y2

T2 3 0 0 1 x y z Rx Ry Rz yz xz xy

Cubic

Td E 8C3 3C2 6S4 6#d

A1 1 1 1 1 1 x2 y2 z2

A2 1 1 1 1 1E 2 1 2 0 0 2z2 x2 y2 3 x2 y2

T1 3 0 1 1 1 Rx Ry RzT2 3 0 1 1 1 x y z yz xz xy

O E 8C3 3C24 6C4 6C2

A1 1 1 1 1 1 x2

y2

z2

A2 1 1 1 1 1E 2 1 2 0 0 2z2 x2 y2 3 x2 y2

T1 3 0 1 1 1 x y z Rx Ry RzT2 3 0 1 1 1 xz xy yz

Oh E 8C3 3C24 6C4 6C2 i 8S6 3#h 6S4 6#d

A1g 1 1 1 1 1 1 1 1 1 1 x2 y2 z2

A2g 1 1 1 1 1 1 1 1 1 1Eg 2 1 2 0 0 2 1 2 0 0 2z2 x2 y2 3 x2 y2

T1g 3 0 1 1 1 3 0 1 1 1 Rx Ry RzT2g 3 0 1 1 1 3 0 1 1 1 xz xy yz

A1u 1 1 1 1 1 1 1 1 1 1A2u 1 1 1 1 1 1 1 1 1 1Eu 2 1 2 0 0 2 1 2 0 0T1u 3 0 1 1 1 3 0 1 1 1 x y zT2u 3 0 1 1 1 3 0 1 1 1

IcosahedralIh E 12C5 12C25 20C3 15C2 i 12S

3

10 12S10 20S6 15# ( 12 5 1

Ag 1 1 1 1 1 1 1 1 1 1 x2

y

2

z

2

T1g 3 ( ( 0 1 3 ( ( 0 1 Rx Ry RzT2g 3 ( ( 0 1 3 ( ( 0 1Gg 4 1 1 1 0 4 1 1 1 0

Hg 5 0 0 1 1 5 0 0 1 1 112 2z2 x2 y2

1

2 x2 y2 xz xy yz

Au 1 1 1 1 1 1 1 1 1 1T1u 3 ( ( 0 1 3 ( ( 0 1 x y zT2u 3 ( ( 0 1 3 ( ( 0 1Gu 4 1 1 1 0 4 1 1 1 0

Hu 5 0 0 1 1 5 0 0 1 1

10


12/18

C!v E 2Cz $ )#v

* A1 1 1 1 z x2 y2;z2

* A2 1 1 1 Rz+ E1 2 2 cos$ 0 x y Rx Ry xz yz, E2 2 2 cos 2$ 0 x2 y2 2xy- E3 2 2 cos 3$ 0

Linear

D!h E 2Cz $ )#v i 2Sz $ )C2

*g A1g 1 1 1 1 1 1 x2 y2;z2

*g A2g 1 1 1 1 1 1 Rz+g E1g 2 2 cos$ 0 2 2 cos$ 0 Rx Ry xz yz,g E2g 2 2 cos 2$ 0 2 2cos2$ 0 x2 y2 2xy-g E3g 2 2 cos 3$ 0 2 2cos3$ 0

*u A1u 1 1 1 1 1 1 z

*u A2u 1 1 1 1 1 1+u E1u 2 2 cos$ 0 2 2 cos$ 0 x y,u E2u 2 2 cos 2$ 0 2 2cos2$ 0-u E3u 2 2 cos 3$ 0 2 2cos3$ 0

Selected tables for descent in symmetry

C2v C2 Cs Cs

E #xz E #yz

A1 A A AA2 A A A

B1 B A A

B2 B A A

D3h C3v C2v Cs Cs

#h #yz E #h E #v

A1 A1 A1 A AA

2 A2 B2 A A

E E A1 B2 2A A A

A1

A2 A2 A A

A2

A1 B1 A A

E E A2 B1 2A A A

D!h C2v

x y z x z y

*g A1

*g B1

+g A2 B2

,g A1 B1

*u B2

*u A2

+u A1 B1

,u A2 B2

O 3 Oh Td

Sg A1g A1

Pg T1g T1

Dg Eg T2g E T2

Fg A2g T1g T2g A2 T1 T2

Gg A1g Eg T1g T2g A1 E T1 T2

Su A1u A2

Pu T1u T2

Du Eu T2u E T1

Fu A2u T1u T2u A1 T2 T1

Gu A1u Eu T1u T2u A2 E T2 T1

11


13/18

Reduction of a representation

If. a1. 1 a2. 2 an. n , then

ak1h!R

/ k R */ R

where/ R is the character of the operationR in the rep-

resentation .,/ k R is the character of the operationRinthe representation . k , andhis the number of elements inthe group.

Projection operators

The projection operator for representation . k is

P k nk

h!

R

/ k R *R

The projected function P k fobtained by applying P k to

any function fis either zero or a component of a basis forrepresentation . k .

Direct ProductsGenerally,

/" " R /" R /" R

and if the resulting representation is reducible it can be re-duced in the usual way. Alternatively the following rulescan be applied.

Treatg uand symmetry separately. For groups with

an inversion centre,

g g u u g and g u u

For groups with a horizontalplane #hbut no inversion cen-tre, single and double primes, and , are used to denotesymmetry and antisymmetry with respect to#h. Then

and

Direct products involving nondegenerate representa-tions are easily worked out from the character table. Theproduct of anyAor Bwith anyEis anE, and the productof anyAorBwith anyTis aT.

In the cubic groups Td,Oand Oh,

E E A1 A2 E

E T1 E T2 T1 T2

T1 T1 T2 T2 A1 E T1 T2

T1 T2 A2 E T1 T2

For products ofEiwithEjin the axial groups the rulesare complicated. If there is only one E representation,apart fromg uor labels, it should be considered asE1.We need the ordernof the principal axis, which is usuallyobvious e.g. n 5 for D5h butfor Dmdwithm even,n 2m(because Dmdhas anS2maxis whenmis even).

a ForEi Ei:

(i) IfE2iexists, then

Ei Ei A1 A2 E2i

(ii) Otherwise, if 4i nthen

Ei Ei A1 A2 B1 B2

(iii) Otherwise

Ei Ei A1 A2 E2i n

b ForEi Ejwithi j:

(i) IfEi jexists, then

Ei Ej Ei j Ei j

(ii) If 2 i j n, then

Ei Ej Ei j B1 B2

(iii) Otherwise

Ei Ej Ei j Ei j n

If there is only oneArepresentation, apart fromg uorlabels, readA1and A2above asA; similarly forB.

Examples

For E E in C4v: there is only one Erepresentation, sotreat it as E1. Rule a(i) doesnt apply, because E2doesntexist, buta(ii) applies, soE E A1 A2 B1 B2.

ForE1

g E2

uinD5

d, note first thatg u u

. Then weneedE1 E2, for which rule b(iii) applies, withn 5, sothe result isE1g E2u E1u E2u.

Antisymmetrized SquaresThe antisymmetric component ofE Eor Ei Ei is al-ways A2. In the cubic groups, the antisymmetric part ofT1 T1andT2 T2is T1.

12


14/18

nC2!Cn?

n"v?

Cnv

"h?

Cnh

S2n Cn?

S2n Cn

i?

C#v

D#h

i?Ci

"?

Cs

C1

i?

Ih I

i?

Oh O

i?

"?

Th Td

Linear?

Are there any Cnproperrotation axes, with n $2?

Find the order n of the highest-orderproper rotation axis Cn.

Is there more than one such axis?

Value of n?

5 (6C5) 4 (3C4) 3 (4C3) 2 (3C2)

T

"h?

n"d?

Dnh Dnd Dn

YesYes

Yes

Yes

Yes Yes

Yes

Yes

Yes

Yes Yes

YesYes

Yes

Yes

No

No

No

No

No

No

No

No

No

No

No NoNoNo

No

No

Yes

A.J.S. 1991

13


15/18

Space Groups

General Equivalent Positions (GEPs) and Special Equivalent Positions (SEPs)

Space group P21

GEPs:

2 @ xn yn zn xn1

2 yn zn

SEPs:

None

Space group P21/c

GEPs:

4 @ xn yn zn xn1

2 yn

1

2 zn xn

1

2 yn

1

2 zn xn yn zn

SEPs:

4 pairs:

2 @ 0 0 0 and 0 12

1

2

2 @ 0 0 12

and 0 12

0

2 @ 12

0 0 and 12

1

2

1

2

2 @ 12

1

2 0 and 1

2 0 1

2

Space group P212121

GEPs:

4 @ xn yn zn xn1

2 yn1

2 zn1

2 xn1

2 yn zn1

2 xn yn1

2 zn

SEPs:

None

14


16/18

15

Parameters for selected magnetic nuclei*

isotope natural abundance (%) spin, I

1H 100 12

2

H 1.5 !102

13H 0 12

6Li 7 1

7Li 93 32

10B 20 3

11B 80 32

13C 1 12

14

N 100 115

N 0.4 12

17O 3.7 !10

2 52

19F 100 12

23Na 100 32

27Al 100 52

29Si 5 12

31

P 1001

2 51

V 100 72

57Fe 2 12

77Se 8 12

103Rh 100 12

107Ag 52 12

109Ag 48 12

113

Cd 12

1

2 119

Sn 9 12

129Xe 26 12

195Pt 34 12

203Tl 30 12

205Tl 70 12

207Pb 23 12

*The list is not exhaustive.


17/18

16

Amino acids H3N OO

HR

Name Three-letter code Single-letter code Side chain, R =

Serine Ser S CH2OH

Threonine Thr T CH(CH3)OH

Cysteine Cys C CH2SH

Methionine Met M CH2CH2SMe

Aspartic acid Asp D CH2COO

Asparagine Asn N CH2CONH2

Glutamic acid Glu E CH2CH2COO

Glutamine Gln QCH

2CH

2CONH

2 Lysine Lys K CH2CH2CH2CH2NH3

+

Arginine Arg R CH2CH2CH2NHNH2

NH2+

Glycine Gly G H

Alanine Ala A Me

Leucine Leu L CH2CHMe2

Isoleucine Ile I CH(Me)CH2Me

Valine Val V CHMe2

Histidine His H

N

H

NCH2

Phenylalanine Phe F

CH2

Tyrosine Tyr Y

CH2

OH

Tryptophan Trp W

N

H

CH2

Proline* Pro P N

H

COO

HH

*For proline the complete structure of the amino acid is shown.


18/18

17

Nucleotide bases

Name Abbreviation Structure

Guanine G NH

N

N

N

O

NH2

Adenine A N

N

N

N

NH2

Cytosine C

N

N

NH2

O

Thymine T

N

NH

O

O

Me

Uracil U

N

NH

O

O

cambridge university chemistry databook

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