chemistry databook_2005

7/25/2019 Chemistry DataBook_2005

1/18

Department of Chemistry

Data Book(revised October 2005)

Contents

Periodic table of the elements inside front cover

Physical constants and conversion factors 1Greek 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

H1

1.008

He2

4.003

Li3

6.94

Be4

9.01

symbolatomic number

mean atomic mass

B5

10.81

C6

12.01

N7

14.01

O8

16.00

F9

19.00

Ne10

20.18

Na11

22.99

Mg12

24.31

Al13

26.98

Si14

28.09

P15

30.97

S16

32.06

Cl17

35.45

Ar18

39.95

K19

39.102

Ca20

40.08

Sc21

44.96

Ti22

47.90

V23

50.94

Cr24

52.00

Mn25

54.94

Fe26

55.85

Co27

58.93

Ni28

58.71

Cu29

63.55

Zn30

65.37

Ga31

69.72

Ge32

72.59

As33

74.92

Se34

78.96

Br35

79.904

Kr36

83.80

Rb37

85.47

Sr38

87.62

Y39

88.91

Zr40

91.22

Nb41

92.91

Mo42

95.94

Tc43

Ru44

101.07

Rh45

102.91

Pd46

106.4

Ag47

107.87

Cd48

112.40

In49

114.82

Sn50

118.69

Sb51

121.75

Te52

127.60

I53

126.90

Xe54

131.30

Cs55

132.91

Ba56

137.34

La*57

138.91

Hf72

178.49

Ta73

180.95

W74

183.85

Re75

186.2

Os76

190.2

Ir77

192.2

Pt78

195.09

Au79

196.97

Hg80

200.59

Tl81

204.37

Pb82

207.2

Bi83

208.98

Po84

At85

Rn86

Fr87

Ra88

Ac + 89

*Lanthanides Ce58

140.12

Pr59

140.91

Nd60

144.24

Pm61

Sm62

150.4

Eu63

151.96

Gd64

157.25

Tb65

158.93

Dy66

162.50

Ho67

164.93

Er68

167.26

Tm69

168.93

Yb70

173.04

Lu71

174.97

+Actinides Th90

232.01

Pa91

U92

238.03

Np93

Pu94

Am95

Cm96

Bk97

Cf98

Es99

Fm100

Md101

No102

Lr103

Text


3/18

Constants

Symbol and ValueName denition (uncertainty) Unit

! 3 141592653589e 2 718281828459ln 10 1 log10 e 2 302585092994

Speed of light c 2 99792458 108 ms 1

Plancks constant h 6 6260693 11 10 34 Js h h 2! 1 05457168 18 10 34 Js

Avogadros constant N A 6 0221415 10 1023 mol 1

Elementary charge e 1 60217653 14 10 19 CElectron rest mass me 0 91093826 16 10 30 kgAtomic mass unit mu 1gmol 1 N A 1 66053886 28 10 27 kgProton rest mass m p 1 67262171 29 10 27 kgNeutron rest mass mn 1 67492728 29 10 27 kgFaraday constant F N Ae 9 64853383 83 104 Cmol 1

Boltzmann constant k B 1 3806505 24 10 23 JK 1

Molar Gas constant R N Ak B 8 314472 15 J mol 1 K 1

Permeability of vacuum 0 4! 10 7 Hm 1

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

4!" 0 1 1126501 10 10 Fm 1

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

Nuclear magneton N e h 2m p 5 05078343 43 10 27 JT 1

Stefan-Boltzmann constant # 2!5

k 4 B 15h

3

c2

5 670400 40 10 8

Wm 2

K 4

Bohr radius a0 4!" 0 h2 mee2 0 5291772108 18 10 10 mHartree energy E h e2 4!" 0 a 0 4 35974417 75 10 18 JFine structure constant $ e2 4!" 0 hc 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 signicant digits of the value.

Energy Conversion Factors

J kJmol 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 43881 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& is hc % h&. The energy corresponding to a temperature T is k BT .

Other conversion factors

Length 10 10 mEnergy cal 4 184JPressure atm = 760 Torr 101 325 Pa

Torr = mm Hg 133 3 Pabar 105 Pa

Radioactivity becquerel, Bq 1s 1

curie, Ci 3 7 1010 BqCharge esu 3 33564 10 10 CDipole moment debye 10 18 esucm 3 33564 10 30 Cm

a u ea 0 8 478358 10 30 CmTemperature 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 1 K 1 .

2


4/18

Greek Alphabet

A ! alpha H " eta N # nu T $ tauB % beta & ' ( theta ) * xi + , upsilon- . gamma I / iota O o omicron 0 1 2 phi3 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

S n a az az2 az n 1 a

1 zn

1 z S !

a1 z

when z 1

Power series

exp z 1 z z2

2! z3

3! zn

n!

cos z eiz e iz

2 1

z2

2! z4

4! z6

6!

sin z 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 23!

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 large n

Determinants

a bc d

ad bc

In general,

det A ! j Ai j C ji (i xed at any value)

where the cofactor C ji is 1 i j times the determinant of the matrix obtained by deleting the ith rowand the j th column of A . For example,

a 11 a12 a13a 21 a22 a23a 31 a32 a33

a 11a 22 a23a 32 a33

a 12a 21 a23a 31 a33

a 13a 21 a22a 31 a32

a 11 a 22 a 33 a11 a 23 a 32 a12 a 23 a 31 a12 a 21 a 33 a13 a 21 a 32 a13 a 22 a 31

3


5/18

Integrals

!

!exp ax2 dx 7 a a 0

!

! x2 n exp ax2 dx 1 3 5 2n 1

7 a

2a n n 1; a 0

!

0r n exp ar dr

n!a n 1

n 0; a 0

" 2

0sinm ' cos n ' d '

m 1m n

" 2

0sinm 2 ' cosn ' d '

n 1m n

" 2

0sinm ' cos n 2 ' d '

so that

" 2

0sinm ' cos n ' d ' m 1 m 3 n 1 n 3

m n m n 2 m n 4 C

where C 7 2 if m and n are both even , and C 1 otherwise. E.g.:

" 2

0sin ' cos 3 ' d '

24 2

14

;" 2

0sin 2 ' cos 2 ' d '

1 14 2

72

716

Integration by partsb

au

dd x

d x u bab

a

dud x

d x

Trigonometrical formulae

sin A B sin Acos B cos Asin B

cos A B cos Acos B sin Asin B

cos Acos B 12 cos A B cos A B

sin Asin B 12 cos A B cos A B

sin Acos B 12 sin A B sin A B

sin A sin B 2sin A B

2cos

A B2

sin A sin B 2cos A B

2sin

A B2

cos A cos B 2cos A B

2 cos

A B2

cos A cos B 2sin A B

2 sin

A B2

Cosine formulaa 2 b2 c2 2bc cos A

4


6/18

Spherical Polar Coordinates

Relationship with Cartesian coordinates

x r sin ' cos 2 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 "

$ 0r 2 sin ' dr d ' d 2

z

x y

r'

2

Laplacian

D2 > E2 >

E x2E2 >E y2

E2 >E z2

1r 2

EEr

r2E>Er

1r 2 sin '

EE'

sin 'E>E'

1r 2 sin 2 '

E2 >E22

Spherical Harmonics

Y lm ' 2 2l 1

47 C lm ' 2

where

C lm ' 2 l m !

l m !

12

P ml cos ' eim$ 1

m for m 0,1 for m 0.

Here P ml is an associated Legendre polynomial. In particular,

C 00 1

C 10 cos ' z r

C 1 1 12 sin ' e i$ 1

2 x iy r

C 20 12 3cos2 ' 1 12 3 z

2 r 2 r 2

C 2 1 32 cos ' sin ' e i$ 3

2 zx izy r2

C 2 2 38 sin2 ' e 2 i$ 38 x

2 y2 2ixy r2

Ladder Operators J J x i J y; J J M J J 1 M M 1 J M 1

5


7/18

Character tables for some important symmetry groups

C i E i

Ag 1 1 R x; R y; R z x2 ; y2 ; z2 ; xy; xz; yz Au 1 1 x; y; z

C s E #h

A 1 1 x; y R z x2 ; y2 ; z2 ; xy A 1 1 z R x; R y xz; yz

C 2 E C z2

A 1 1 z R z x2 ; y2 ; z2 ; xy B 1 1 x; y R x; R y xz; yz

2

C 2 v E C z2 # xz # yz

A1 1 1 1 1 z x2 ; y2 ; z2 A2 1 1 1 1 R z xy B1 1 1 1 1 x R y xz B2 1 1 1 1 y R x yz

C 2 h E C z2 i # xy

Ag 1 1 1 1 R z x2 ; y2 ; z2 ; xy Bg 1 1 1 1 R x; R y xz; yz Au 1 1 1 1 z Bu 1 1 1 1 x; y

D 2 E C z2 C y2 C x2

A 1 1 1 1 x2 ; y2 ; z2 B1 1 1 1 1 z R z xy B2 1 1 1 1 y R y xz B3 1 1 1 1 x R x yz

D 2 d E 2S 4 C z2 2C 2 2# d

A1 1 1 1 1 1 x2 y2 ; z2 A2 1 1 1 1 1 R z B1 1 1 1 1 1 x2 y2 B2 1 1 1 1 1 z xy E 2 0 2 0 0 x y R x R y xz yz

6


8/18

D 2 h E C z2 C y2 C x2 i # xy # xz # yz

Ag 1 1 1 1 1 1 1 1 x2 ; y2 ; z2 B1 g 1 1 1 1 1 1 1 1 R z xy B2 g 1 1 1 1 1 1 1 1 R y xz B3 g 1 1 1 1 1 1 1 1 R x yz Au 1 1 1 1 1 1 1 1 B1 u 1 1 1 1 1 1 1 1 z B2 u 1 1 1 1 1 1 1 1 y B3 u 1 1 1 1 1 1 1 1 x

C 3 E C 3 C 23 ' exp 2! i 3

A 1 1 1 z R z x2 y2 ; z21 ' ' 2 x i y R x i R y xz i yz; x2 2i xy y2 E 1 ' 2 ' x i y R x i R y xz i yz; x2 2i xy y2

3

C 3 v E 2C z3 3# v

A1 1 1 1 z x2 y2 ; z2 A2 1 1 1 R z E 2 1 0 x y R x R y xz yz ; x2 y2 2 xy

D 3 E 2C z3 3C 2

A1 1 1 1 x2 y2 ; z2 A2 1 1 1 z R z E 2 1 0 x y R x R y xz yz ; x2 y2 2 xy

D 3 d E 2C 3 3C 2 i 2S 6 3# d

A1 g 1 1 1 1 1 1 x2 y2 ; z2 A2 g 1 1 1 1 1 1 R z E g 2 1 0 2 1 0 R x R y xz yz ; x2 y2 2 xy A1 u 1 1 1 1 1 1 A2 u 1 1 1 1 1 1 z E u 2 1 0 2 1 0 x y

D 3 h E 2C 3 3C 2 #h 2S 3 3# v

A1 1 1 1 1 1 1 x2 y2 ; z2 A2 1 1 1 1 1 1 R z E 2 1 0 2 1 0 x y x2 y2 2 xy A1 1 1 1 1 1 1 A2 1 1 1 1 1 1 z E 2 1 0 2 1 0 R x R y xz yz

7


9/18

C 4 v E 2C 4 C 24 2# v 2# d

A1 1 1 1 1 1 z x2 y2 ; z2 A2 1 1 1 1 1 R z B1 1 1 1 1 1 x2 y2 B2 1 1 1 1 1 xy E 2 0 2 0 0 x y R x R y xz yz

4

Note: The # v planes in C 4 v coincide with the xz and yz planes.

D 4 h E 2C 4 C 24 2C 2 2C 2 i 2S 4 #h 2# v 2# d

A1 g 1 1 1 1 1 1 1 1 1 1 x2 y2 ; z2 A2 g 1 1 1 1 1 1 1 1 1 1 R z B1 g 1 1 1 1 1 1 1 1 1 1 x2 y2 B2 g 1 1 1 1 1 1 1 1 1 1 xy E g 2 0 2 0 0 2 0 2 0 0 R x R y xz yz A1 u 1 1 1 1 1 1 1 1 1 1 A2

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

B1 u 1 1 1 1 1 1 1 1 1 1 B2 u 1 1 1 1 1 1 1 1 1 1 E u 2 0 2 0 0 2 0 2 0 0 x y

Note: TheC 2 axes in D 4 h coincide with the x and y axes, and the # v planes with the xz and yz planes.

Note that the quantities( 12 5 1 satisfy ( 2 1 ( and ( ( 1.

C 5 v E 2C 5 2C 25 5# v ( 12 5 1

A1 1 1 1 1 z x2 y2 ; z2 A2 1 1 1 1 R z E 1 2 ( ( 0 x y R x R y xz yz E 2 2 ( ( 0 x2 y2 2 xy

5

D 5 E 2C 5 2C 25 5C 2 ( 12 5 1

A1 1 1 1 1 x2 y2 ; z2 A2 1 1 1 1 z R z E 1 2 ( ( 0 x y R x R y xz yz E 2 2 ( ( 0 x2 y2 2 xy

D 5d

E 2C 5 2C 25

5C 2 i 2S 310

2S 10 5#d

( 12

5 1

A1 g 1 1 1 1 1 1 1 1 x2 y2 ; z2 A2 g 1 1 1 1 1 1 1 1 R z E 1 g 2 ( ( 0 2 ( ( 0 R x R y xz yz E 2 g 2 ( ( 0 2 ( ( 0 x2 y2 2 xy A1 u 1 1 1 1 1 1 1 1 A2 u 1 1 1 1 1 1 1 1 z E 1 u 2 ( ( 0 2 ( ( 0 x y E 2 u 2 ( ( 0 2 ( ( 0

8


10/18

D 5 h E 2C 5 2C 25 5C 2 #h 2S 5 2S 35 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 R z E 1 2 ( ( 0 2 ( ( 0 x y E 2 2 ( ( 0 2 ( ( 0 x2 y2 2 xy A1 1 1 1 1 1 1 1 1 A2 1 1 1 1 1 1 1 1 z E 1 2 ( ( 0 2 ( ( 0 R x R y xz yz E 2 2 ( ( 0 2 ( ( 0

C 6 v E 2C 6 2C 26 C 36 3# v 3# d

A1 1 1 1 1 1 1 z x2 y2 ; z2 A2 1 1 1 1 1 1 R z B1 1 1 1 1 1 1 B2 1 1 1 1 1 1 E 1 2 1 1 2 0 0 x y R x R y xz yz E 2 2 1 1 2 0 0 x2 y2 2 xy

6

D 6 E 2C 6 2C 26 C 36 3C 2 3C 2

A1 1 1 1 1 1 1 x2 y2 ; z2 A2 1 1 1 1 1 1 z R z B1 1 1 1 1 1 1 B2 1 1 1 1 1 1 E 1 2 1 1 2 0 0 x y R x R y xz yz E 2 2 1 1 2 0 0 x2 y2 2 xy

D 6 h E 2C 6 2C 26 C 36 3C 2 3C 2 i 2S 3 2S 6 #h 3# d 3# v

A1 g 1 1 1 1 1 1 1 1 1 1 1 1 x2 y2 ; z2 A2 g 1 1 1 1 1 1 1 1 1 1 1 1 R z B1 g 1 1 1 1 1 1 1 1 1 1 1 1 B2 g 1 1 1 1 1 1 1 1 1 1 1 1 E 1 g 2 1 1 2 0 0 2 1 1 2 0 0 R x R y xz yz E 2 g 2 1 1 2 0 0 2 1 1 2 0 0 x2 y2 2 xy A1 u 1 1 1 1 1 1 1 1 1 1 1 1 A2 u 1 1 1 1 1 1 1 1 1 1 1 1 z B1 u 1 1 1 1 1 1 1 1 1 1 1 1 B2 u 1 1 1 1 1 1 1 1 1 1 1 1 E 1 u 2 1 1 2 0 0 2 1 1 2 0 0 x y

E 2 u 2 1 1 2 0 0 2 1 1 2 0 0

9


11/18

T E 4C 3 4C 23 3C 2 ' exp 2! i 3

A1 1 1 1 1 x2 y2 z2

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

1 ' 2 ' 1 z2 ' x2 ' 2 y2T 2 3 0 0 1 x y z R x R y R z yz xz xy

Cubic

T d E 8C 3 3C 2 6S 4 6# d

A1 1 1 1 1 1 x2 y2 z2 A2 1 1 1 1 1 E 2 1 2 0 0 2 z2 x2 y2 3 x2 y2T 1 3 0 1 1 1 R x R y R zT 2 3 0 1 1 1 x y z yz xz xy

O E 8C 3 3C 24 6C 4 6C 2

A1 1 1 1 1 1 x2

y2

z2

A2 1 1 1 1 1 E 2 1 2 0 0 2 z2 x2 y2 3 x2 y2T 1 3 0 1 1 1 x y z R x R y R zT 2 3 0 1 1 1 xz xy yz

O h E 8C 3 3C 24 6C 4 6C 2 i 8S 6 3# h 6S 4 6# d

A1 g 1 1 1 1 1 1 1 1 1 1 x2 y2 z2 A2 g 1 1 1 1 1 1 1 1 1 1 E g 2 1 2 0 0 2 1 2 0 0 2 z2 x2 y2 3 x2 y2T 1 g 3 0 1 1 1 3 0 1 1 1 R x R y R zT 2 g 3 0 1 1 1 3 0 1 1 1 xz xy yz A1 u 1 1 1 1 1 1 1 1 1 1 A2 u 1 1 1 1 1 1 1 1 1 1 E u 2 1 2 0 0 2 1 2 0 0T 1 u 3 0 1 1 1 3 0 1 1 1 x y zT 2 u 3 0 1 1 1 3 0 1 1 1

IcosahedralI h E 12C 5 12C 25 20C 3 15C 2 i 12S 310 12S 10 20S 6 15# ( 12 5 1

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

y2

z2

T 1 g 3 ( ( 0 1 3 ( ( 0 1 R x R y R zT 2 g 3 ( ( 0 1 3 ( ( 0 1Gg 4 1 1 1 0 4 1 1 1 0 H g 5 0 0 1 1 5 0 0 1 1 112 2 z2 x2 y2

12 x

2 y2 xz xy yz Au 1 1 1 1 1 1 1 1 1 1T 1 u 3 ( ( 0 1 3 ( ( 0 1 x y zT 2 u 3 ( ( 0 1 3 ( ( 0 1Gu 4 1 1 1 0 4 1 1 1 0 H u 5 0 0 1 1 5 0 0 1 1

10


12/18

C ! v E 2C z $ )# v

* A1 1 1 1 z x2 y2 ; z2* A2 1 1 1 R z+ E 1 2 2cos$ 0 x y R x R y xz yz, E 2 2 2cos2$ 0 x2 y2 2 xy- E 3 2 2cos3$ 0

Linear

D ! h E 2C z $ )# v i 2S z $ ) C 2

* g A1 g 1 1 1 1 1 1 x2 y2 ; z2

* g A2 g 1 1 1 1 1 1 R z+ g E 1 g 2 2 cos$ 0 2 2 cos$ 0 R x R y xz yz, g E 2 g 2 2 cos 2$ 0 2 2cos2$ 0 x2 y2 2 xy- g E 3 g 2 2 cos 3$ 0 2 2cos3$ 0

* u A1 u 1 1 1 1 1 1 z

* u A2 u 1 1 1 1 1 1+ u E 1 u 2 2 cos$ 0 2 2 cos$ 0 x y, u E 2 u 2 2 cos 2$ 0 2 2cos2$ 0- u E 3 u 2 2 cos 3$ 0 2 2cos3$ 0

Selected tables for descent in symmetry

C 2 v C 2 C s C s

E # xz E # yz

A1 A A A A2 A A A

B1 B A A

B2 B A A

D 3 h C 3 v C 2 v C s C s

# h # yz E #h E #v

A1 A1 A1 A A A2 A2 B2 A A

E E A1 B2 2 A A A A1 A2 A2 A A

A2 A1 B1 A A

E E A2 B1 2 A A A

D ! h C 2 v

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 O h T d

S g A1 g A1

Pg T 1 g T 1

Dg E g T 2 g E T 2

F g A2 g T 1 g T 2 g A2 T 1 T 2

Gg A1 g E g T 1 g T 2 g A1 E T 1 T 2

S u A1 u A2

Pu T 1 u T 2

Du E u T 2 u E T 1

F u A2 u T 1 u T 2 u A1 T 2 T 1

Gu A1 u E u T 1 u T 2 u A2 E T 2 T 1

11


13/18

Reduction of a representationIf . a1 . 1 a 2 . 2 a n. n , then

a k 1h ! R

/ k R */ R

where / R is the character of the operation R in the rep-resentation. , / k R is the character of the operation R inthe representation. k , and h is the number of elements inthe group.

Projection operatorsThe projection operator for representation. k is

P k nk

h ! R/ k R * R

The projected function P k f obtained by applyingP k to

any function f is 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.

Treat g u and symmetry separately. For groups withan inversion centre,

g g u u g and g u u

For groups with a horizontalplane# h but 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 any A or B with any E is an E , and the productof any A or B with any T is a T .

In the cubic groupsT d , O and O h ,

E E A1 A2 E

E T 1 E T 2 T 1 T 2

T 1 T 1 T 2 T 2 A1 E T 1 T 2

T 1 T 2 A2 E T 1 T 2

For products of E i with E j in the axial groups the rulesare complicated. If there is only one E representation,apart from g u or labels, it should be considered as E 1 .We need the order n of the principal axis, which is usuallyobvious e.g. n 5 forD 5 h but forD md with m even ,n 2m (becauseD md has an S 2 m axis when m is even).a For E i E i:

(i) If E 2 i exists, then

E i E i A1 A2 E 2 i

(ii) Otherwise, if 4i n then

E i E i A1 A2 B1 B2

(iii) Otherwise

E i E i A1 A2 E 2 i n

b For E i E j with i j :(i) If E i j exists, then

E i E j E i j E i j

(ii) If 2 i j n, then

E i E j E i j B1 B2

(iii) Otherwise E i E j E i j E i j n

If there is only one A representation, apart from g u orlabels, read A1 and A2 above as A; similarly for B.

Examples

For E E in C 4 v: there is only one E representation, sotreat it as E 1 . Rule a (i) doesnt apply, because E 2 doesntexist, but a(ii) applies, so E E A1 A2 B1 B2 .

For E 1

g E 2

u in D 5

d , note rst that g u u

. Then weneed E 1 E 2 , for which rule b(iii) applies, with n 5, sothe result is E 1 g E 2 u E 1 u E 2 u.

Antisymmetrized SquaresThe antisymmetric component of E E or E i E i is al-ways A2 . In the cubic groups, the antisymmetric part of T 1 T 1 and T 2 T 2 is T 1 .

12


14/18

nC 2 ! C n ?

n " v?

C nv

" h?

C nh

S 2n C n ?

S 2n C n

i?

C # v

D # h

i?C i

" ?

C s

C 1

i?

I h I

i?

O h O

i?

" ?

T h T d

Linear?

Are there any C n properrotation axes, with n $ 2?

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

Is there more than one such axis?

Value of n ?

5 (6C 5) 4 (3C 4) 3 (4C 3) 2 (3C 2)

T

" h?

n " d?

D nh D nd D n

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

Space group P 21

GEPs:

2 @ xn yn zn xn12 yn zn

SEPs:None

Space group P 21 /c

GEPs:

4 @ xn yn zn xn12 yn

12 zn xn

12 yn

12 zn xn yn zn

SEPs:4 pairs:

2 @ 0 0 0 and 0 1212

2 @ 0 0 12 and 0 1

2 0

2 @ 12 0 0 and 1

212

12

2 @ 1212 0 and

12 0

12

Space group P 21 21 21

GEPs:

4 @ xn yn zn xn12 yn

12 zn

12 xn

12 yn zn

12 xn yn

12 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 17Li 93 32 10B 20 311B 80 32 13C 1 12 14

N 100 115N 0.4 12 17O 3.7 ! 10 2 52 19F 100 12

23Na 100 32 27Al 100 52 29Si 5 12 31

P 10012

51V 100 72 57Fe 2 12 77Se 8 12

103Rh 100 12 107Ag 52 12 109Ag 48 12 113

Cd 121

2 119Sn 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 H3 N O O

HR

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

Serine Ser S CH2 OH Threonine Thr T CH(CH 3 )OH Cysteine Cys C CH2SH

Methionine Met M CH2 CH2 SMe Aspartic acid Asp D CH2 COO Asparagine Asn N CH2 CONH 2

Glutamic acid Glu E CH2CH2 COO

Glutamine Gln QCH

2CH

2CONH

2 Lysine Lys K CH2 CH2CH2 CH2 NH3+

Arginine Arg R CH2 CH2 CH2 NHNH2

NH2+

Glycine Gly G H

Alanine Ala A Me

Leucine Leu L CH2 CHMe 2 Isoleucine Ile I CH(Me)CH 2 Me

Valine Val V CHMe 2

Histidine His H

N

HNCH2

Phenylalanine Phe F

CH2

Tyrosine Tyr Y

CH2

OH

Tryptophan Trp W

NH

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

chemistry databook_2005

Documents