DODATEK

TABLICY CHARAKTERÓW GRUP PUNKTOWYCH

C2	E	f^α
A	+1 +1	z, x^2, Rz
B	+1 -1	X, y, Rx, Ry

C3	E	f^α
A	+1 +1 +1	z, Rz
ε	+1 τ τ*	x+iy, Rx+iRy
ε*	+1 τ* τ	x-iy, Rx-iRy

C4	E	f^α
A	+1 +1 +1 +1	z, Rz
B	+1 -1 +1 -1
ε	+1 -i -1 +i	x+iy, Rx+iRy
ε*	+1 i -1 -i	x-iy, Rx-iRy

C5	E C5	f^α
A	+1 +1 +1 +1 +1	z, Rz
ε1	+1 τ τ^2 (τ^2)* τ*	x+iy, Rx+iRy
ε*1	+1 τ* (τ^*)^2 τ^2 τ	x-iy, Rx-iRy
ε2	+1 τ^2 τ* τ (τ^2)*
ε*2	+1 (τ^2)* τ τ* τ^2

C6	E C6	f^α
A	+1 +1 +1 +1 +1 +1	z, Rz
B	+1 -1 +1 -1 +1 -1
ε1	+1 τ -τ* -1 -τ τ*	x+iy, Rx+iRy
ε*1	+1 τ* -τ -1 -τ* τ	x-iy, Rx-iRy
ε2	+1 -τ* -τ +1 -τ* -τ
ε*2	+1 -τ -τ* +1 -τ -τ*

C2v	E	f^α
A1	+1 +1 +1 +1	z, x^2, y^2
A2	+1 +1 -1 -1	Rz, xy
B1	+1 -1 +1 -1	x, Ry, xz
B2	+1 -1 -1 +1	y, Rx, yz

C3v	E 2 3σv	f^α
A1	+1 +1 +1	z, x^2 + y^2
A2	+1 +1 -1	Rz
E	+2 -1 0	x,y; Rx,Ry; x^2 - y^2,xy; xz,yz.

C4v	E 2 2σv 2σd	f^α
A1	+1 +1 +1 +1 +1	z, x^2 + y^2
A2	+1 +1 +1 -1 -1	Rz
B1	+1 -1 +1 +1 -1	x^2 - y^2
B2	+1 -1 +1 -1 +1	xy
E	+2 0 -2 0 0	x,y; Rx,Ry; xz,yz

C5v	E 2 2σv	f^α
A1	+1 +1 +1 +1	z, x^2 + y^2
A2	+1 +1 +1 -1	Rz
E1	+2 0	x,y; Rx,Ry; xz,yz
E2	+2 0	x^2 - y^2,xy

C6v	E 2 2 3σv 3σv	f^α
A1	+1 +1 +1 +1 +1 +1	z, x^2 + y^2
A2	+1 +1 +1 +1 -1 -1	Rz
B1	+1 -1 +1 -1 +1 -1
B2	+1 -1 +1 -1 -1 +1
E1	+2 +1 -1 -2 0 0	x,y; Rx,Ry; xz,yz
E2	+2 -1 -1 +2 0 0	x^2 - y^2,xy

Ci	E i	f^α
Ag	+1 +1	xy,xz,yz, Rx,Ry,Rz
Au	+1 -1	x,y,z

S4	E S4	f^α
A	+1 +1 +1 +1	Rz, x^2 + y^2
B	+1 -1 +1 -1	z, x^2 - y^2,xy
ε	+1 τ -1 τ*	x+iy
ε*	+1 τ* -1 τ	X-iy

τ =

S6	E i S6	f^α
Ag	+1 +1 +1 +1 +1 +1	Rz
Au	+1 +1 +1 -1 -1 -1	z
εg	+1 τ τ* +1 τ τ*
ε*g	+1 τ* τ +1 τ* τ
εu	+1 τ τ* -1 -τ -τ*	x+iy
ε*g		x-iy

τ =

S8	E S8 C2	f^α
A	+1 +1 +1 +1 +1 +1 +1 +1	Rz
B	+1 -1 +1 -1 +1 -1 +1 -1	z
ε1	+1 τ i -τ* -1 -τ -i τ*	x+iy
ε1*	+1 τ* -i -τ -1 -τ* i τ	x-iy
ε2	+1 i -1 -i +1 i -1 -i
ε2*	+1 -i -1 i +1 -i -1 i
ε3	+1 -τ* -i τ -1 τ* i -τ
ε3*	+1 -τ i τ* -1 τ -i -τ*

τ =

Cs	E σ	f^α
A'	+1 +1	x,y,Rz, x^2 - y^2,xy
A''	+1 -1	z,xz,yz, Rx,Ry

C2h	E i σh	f^α
Ag	+1 +1 +1 +1	Rz, xy
Au	+1 +1 -1 -1	z
Bg	+1 -1 +1 -1	Rx,Ry, xz,yz
Bu	+1 -1 -1 +1	x,y

C3h	E σh S3	f^α
A'	+1 +1 +1 +1 +1 +1	Rz, x^2 + y^2
A''	+1 +1 +1 -1 -1 -1	z
ε1	+1 τ τ* +1 τ τ*	x+iy
ε1^*	+1 τ* τ +1 τ* τ	x-iy
ε2	+1 τ τ* -1 -τ -τ*	Rx + iRy
ε2^*	+1 τ* τ -1 -τ* -τ	Rx - iRy

C4h	E C2 i σh S4	f^α
Ag	+1 +1 +1 +1 +1 +1 +1 +1	Rz, x^2 + y^2
Au	+1 +1 +1 +1 -1 -1 -1 -1	z
Bg	+1 -1 +1 -1 +1 -1 +1 -1	x^2 - y^2,xy
Bu	+1 -1 +1 -1 -1 +1 -1 +1
εg	+1 +i -1 -i +1 +i -1 -i	Rx + iRy
εg^*	+1 -i -1 i +1 -i -1 +i	Rx - iRy
εu	+1 +i -1 -i -1 -i +1 +i	x+iy
εu^*	+1 -i -1 i -1 +i +1 -i	x-iy

D2	E	f^α
A	+1 +1 +1 +1	x^2, y^2, z^2
B1	+1 +1 -1 -1	z, xy, Rz
B2	+1 -1 +1 -1	y, xz, Ry
B3	+1 -1 -1 +1	x, yz, Rx

D3	E 2 3C2	f^α
A1	+1 +1 +1	x^2 + y^2, z^2
A2	+1 +1 -1	z, Rz
E	+2 -1 0	x^2 - y^2,xy; xz,yz; x,y; Rx, Ry

D4	E 2 2 2	f^α
A1	+1 +1 +1 +1 +1	x^2 + y^2, z^2
A2	+1 +1 +1 -1 -1	z, Rz
B1	+1 -1 +1 +1 -1	x^2 - y^2
B2	+1 -1 +1 -1 +1	xy
E	+2 0 -2 0 0	x,y; Rx,Ry; xz,yz

D5		E 2 2 5C2	f^α
A1		+1 +1 +1 +1	x^2 + y^2, z^2
A2		+1 +1 +1 -1	z, Rz
E1		+2 2 2 0	x,y; Rx,Ry; xz,yz
E2		+2 2 2 0	x^2 - y^2,xy
D6		E 2 2С3 3 3		f^α
A1		+1 +1 +1 +1 +1 +1		x^2 + y^2, z^2
A2		+1 +1 +1 +1 -1 -1		z, Rz
B1		+1 -1 +1 -1 +1 -1
B2		+1 -1 +1 -1 -1 +1
E1		+2 +1 -1 -2 0 0		x,y; Rx,Ry; xz,yz
E2		+2 -1 -1 +2 0 0		x^2 - y^2,xy

D2d	E 2S4 2 2σd	f^α
A1	+1 +1 +1 +1 +1	x^2 + y^2, z^2
A2	+1 +1 +1 -1 -1	Rz
B1	+1 -1 +1 +1 -1	x^2 - y^2
B2	+1 -1 +1 -1 +1	z, xy
E	+2 0 -2 0 0	x,y; Rx,Ry; xz,yz

D3d	E 2С3 3C2 i 2S6 3σd	f^α
A1g	+1 +1 +1 +1 +1 +1	x^2 + y^2, z^2
A2g	+1 +1 -1 +1 +1 -1	Rz
A1u	+1 +1 +1 -1 -1 -1
A2u	+1 +1 -1 -1 -1 +1	z
Eg	+2 -1 0 +2 -1 0	Rx,Ry; xz,yz; x^2 - y^2,xy
Eu	+2 -1 0 -2 +1 0	x,y

D4d	E 2S8 2C4 2 C2 4 4σd	f^α
A1	+1 +1 +1 +1 +1 +1 +1	x^2 + y^2, z^2
A2	+1 +1 +1 +1 +1 -1 -1	Rz
B1	+1 -1 +1 -1 +1 +1 -1
B2	+1 -1 +1 -1 +1 -1 +1	z
E1	+2 τ 0 -τ -2 0 0	x,y
E2	+2 0 -2 0 +2 0 0	x^2 - y^2,xy
E3	+2 -τ 0 τ -2 0 0	Rx,Ry; xz,yz

τ =

D5d	E 2C5 2 5C2 i 2 2S10 5σd	f^α
A1g	+1 +1 +1 +1 +1 +1 +1 +1	x^2 + y^2, z^2
A2g	+1 +1 +1 -1 +1 +1 +1 -1	Rz
A1u	+1 +1 +1 +1 -1 -1 -1 -1
A2u	+1 +1 +1 -1 -1 -1 -1 +1	z
E1g	+2 2τ 2θ 0 +2 2τ 2θ 0	Rx,Ry; xz,yz
E2g	+2 2θ 2τ 0 +2 2θ 2τ 0	x^2 - y^2,xy
E1u	+2 2τ 2θ 0 -2 -2τ -2θ 0	x,y
E2u	+2 2θ 2τ 0 -2 -2θ -2τ 0

τ =
; θ =
.

D6d	E 2S12 2C6 2S4 2С3 2 C2 6 6σd	f^α
A1	+1 +1 +1 +1 +1 +1 +1 +1 +1	x^2 + y^2, z^2
A2	+1 +1 +1 +1 +1 +1 +1 -1 -1	Rz
B1	+1 -1 +1 -1 +1 -1 +1 +1 -1
B2	+1 -1 +1 -1 +1 -1 +1 -1 +1	z
E1	+2 +τ +1 0 -1 -τ -2 0 0	x,y
E2	+2 +1 -1 -2 -1 +1 +2 0 0	x^2 - y^2,xy
E3	+2 0 -2 0 +2 0 -2 0 0
E4	+2 -1 -1 +2 -1 -1 +2 0 0
E5	+2 -τ +1 0 -1 τ -2 0 0	Rx,Ry; xz,yz

τ =

D2h	E i σ^(xy) σ^(xz) σ^(yz)	f^α
Ag	+1 +1 +1 +1 +1 +1 +1 +1	x^2; y^2; z^2
Au	+1 +1 +1 +1 -1 -1 -1 -1
B1g	+1 +1 -1 -1 +1 +1 -1 -1	xy; Rz
B1u	+1 +1 -1 -1 -1 -1 +1 +1	z
B2g	+1 -1 +1 -1 +1 -1 +1 -1	xz; Ry
B2u	+1 -1 +1 -1 -1 +1 -1 +1	y
B3g	+1 -1 -1 +1 +1 -1 -1 +1	yz; Rx
B3u	+1 -1 -1 +1 -1 +1 +1 -1	x

D3h	E 2 3C2 2S3 3σv	f^α
	+1 +1 +1 +1 +1 +1	x^2 + y^2, z^2
	+1 +1 -1 +1 +1 -1	Rz
	+1 +1 +1 -1 -1 -1
	+1 +1 -1 -1 -1 +1	z
E'	+2 -1 0 +2 -1 0	x,y; x^2 - y^2,xy
E''	+2 -1 0 -2 +1 0	Rx,Ry; xz,yz

D4h	E 2 C2 2 2 i 2S4 σh 2σv 2σd	f^α
A1g	+1 +1 +1 +1 +1 +1 +1 +1 +1 +1	x^2 + y^2, z^2
A1u	+1 +1 +1 +1 +1 -1 -1 -1 -1 -1
A2g	+1 +1 +1 -1 -1 +1 +1 +1 -1 -1	Rz
A2u	+1 +1 +1 -1 -1 -1 -1 -1 +1 +1	z
B1g	+1 -1 +1 +1 -1 +1 -1 +1 +1 -1	x^2 - y^2
B1u	+1 -1 +1 +1 -1 -1 +1 -1 -1 +1
B2g	+1 -1 +1 -1 +1 +1 -1 +1 -1 +1	xy
B2u	+1 -1 +1 -1 +1 -1 +1 -1 +1 -1
Eg	+2 0 -2 0 0 +2 0 -2 0 0	Rx,Ry; xz,yz
Eu	+2 0 -2 0 0 -2 0 +2 0 0	x,y

D5h	E 2 2 5C2 σh 2S5 2 5σv	f^α
	+1 +1 +1 +1 +1 +1 +1 +1	x^2 + y^2, z^2
	+1 +1 +1 +1 -1 -1 -1 -1
	+1 +1 +1 -1 +1 +1 +1 -1	Rz
	+1 +1 +1 -1 -1 -1 -1 +1	z
	+2 2τ 2θ 0 +2 2τ 2θ 0	x,y
	+2 2τ 2θ 0 -2 -2τ -2θ 0	Rx,Ry; xz,yz
	+2 2θ 2τ 0 +2 2θ 2τ 0	x^2 - y^2,xy
	+2 2θ 2τ 0 -2 -2θ -2τ 0

τ =
; θ =
.

D6h	E 2 2C3 C2 3 3 i 2S3 2S6 σh σd 3σv	f^α
A1g	+1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1	x^2 + y^2, z^2
A1u	+1 +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 -1
A2g	+1 +1 +1 +1 -1 -1 +1 +1 +1 +1 -1 -1	Rz
A2u	+1 +1 +1 +1 -1 -1 -1 -1 -1 -1 +1 +1	z
B1g	+1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1
B1u	+1 -1 +1 -1 +1 -1 -1 +1 -1 +1 -1 +1
B2g	+1 -1 +1 -1 -1 +1 +1 -1 +1 -1 -1 +1
B2u	+1 -1 +1 -1 -1 +1 -1 +1 -1 +1 +1 -1
E1g	+2 +1 -1 -2 0 0 +2 +1 -1 -2 0 0	Rx,Ry; xz,yz
E1u	+2 +1 -1 -2 0 0 -2 -1 +1 +2 0 0	x,y
E2g	+2 -1 -1 +2 0 0 +2 -1 -1 +2 0 0	x^2 - y^2,xy
E2u	+2 -1 -1 +2 0 0 -2 +1 +1 -2 0 0

C∞v	E 2R(ϕ,z) 2R(2ϕ,z)............ ∞σv	f^α
Σ^+	+1 +1 +1 ………… .+1	z; x^2 + y^2
Σ^-	+1 +1 +1 …………. -1	Rz
Π	+2 2cosϕ 2cos2ϕ…………..0	x,y; xz,yz; Rx,Ry
Δ	+2 2cos2ϕ 2cos3ϕ……….. ...0	x^2 - y^2,xy
Φ	+2 2cos3ϕ 2cos4ϕ…………..0
…..	…………………………………………

D∞v	E 2R(ϕ,z)....... ∞σv i 2S(ϕ,z)…. ∞ C2	f^α
	+1 +1 ……….+1 +1 +1 ………+1	x^2 + y^2 +z^2
	+1 +1 ……….-1 +1 +1 ………-1	Rz
	+1 +1 ……….+1 -1 -1 ………-1	z
	+1 +1 ……….-1 -1 -1 ………+1
Πg	+2 +2 cosϕ……...0 +2 +2 cosϕ…......0	xz,yz; Rx,Ry
Πu	+2 +2 cosϕ……...0 -2 -2 cosϕ……...0	x,y
Δg	+2 +2 cos2ϕ…….0 +2 +2 cos2ϕ……0	x^2 - y^2,xy
Δu	+2 +2 cos2ϕ…….0 -2 -2 cosϕ……..0
…….	…………………………………………..

Td	E 8C3 3 C2 6S4 6σd	f^α
A1	+1 +1 +1 +1 +1	x^2 + y^2 +z^2
A2	+1 +1 +1 +1 +1
E	+2 -1 +2 0 0	x^2 - y^2,3z^2-r^2
T1	+3 0 -1 -1 +1	Rx,Ry, Rz
T2	+3 0 -1 +1 -1	x,y,z; xy,xz,yz

Oh	E 8C3 6C2 6C4 3 i 6S4 8 S6 3σh 6σd	f^α
A1g	+1 +1 +1 +1 +1 +1 +1 +1 +1 +1	x^2 + y^2 +z^2
A2g	+1 +1 -1 -1 +1 +1 -1 +1 +1 -1
A1u	+1 +1 +1 +1 +1 -1 -1 -1 -1 -1
A2u	+1 +1 -1 -1 +1 -1 +1 -1 -1 +1
Eg	+2 -1 0 0 +2 +2 0 -1 +2 0	x^2 - y^2,3z^2-r^2
Eu	+2 -1 0 0 +2 -2 0 +1 -2 0
T1g	+3 0 -1 +1 -1 +3 +1 0 -1 -1	Rx,Ry, Rz
T2g	+3 0 +1 -1 -1 +3 +1 0 -1 +1	xy,xz,yz
T1u	+3 0 -1 +1 -1 -3 -1 0 +1 +1	x,y,z
T2u	+3 0 +1 -1 -1 -3 +1 0 +1 -1

Ih	E 12C5 12 20C3 15C2 i 12 12S10 20S3 15σv	f^α
Ag	+1 +1 +1 +1 +1 +1 +1 +1 +1 +1	x^2 + y^2 +z^2
Au	+1 +1 +1 +1 +1 -1 -1 -1 -1 -1
T1g	+3 +τ 1-τ 0 -1 +3 τ 1-τ 0 -1	Rx,Ry, Rz
T2g	+3 1-τ +τ 0 -1 +3 1-τ +τ 0 +1
T1u	+3 +τ 1-τ 0 -1 -3 -τ τ-1 0 +1	x,y,z
T2u	+3 1-τ +τ 0 -1 -3 τ-1 -τ 0 +1	x^3,y^3,z^3
Gg	+4 -1 -1 +1 0 +4 -1 -1 +1 0
Gu	+4 -1 -1 +1 0 -4 +1 +1 -1 0	y(x^2-z^2), x(z^2-y^2), z((y^2-x^2)
Hg	+5 0 0 -1 +1 +5 0 0 -1 +1	2z^2-x^2-y^2, x^2 - y^2,xy, xz,yz
Hu	+5 0 0 -1 +1 -5 0 0 +1 -1

τ =

BIBLOGRAFIA

1. Włodzimierz Kołos, Joanna Sadlej. Atom i cząsteczka. Wydawnictwo naukowo-techniczne, 2007, 440 stron.

2. F. Albert Cotton. Teoria grup - Zastosowania w chemii. PWN, Warszawa, 1973.

3. Ronald C. King, Mirosław Bylicki, Jacek Karwowski. Symmetry, Spectroscopy and SCHUR. Proceeding of the Professor Brian G. Wybourne. Toruń, 2005, 382 strony.

4. Alojzy Gołębiewski. Chemia kwantowa związków organicznych. PWN, Warszawa, 1973.

5. Marek T. Pawlikowski. Wstęp do teoretycznej spektroskopii molekularnej. Teoria grup. Wydawnictwo Uniwersytetu Jagiellońskiego, 2007, 360 stron.

6. Joanna Sadlej. Spektroskopia molekularna. Wydawnictwo naukowo-techniczne, 2002, 495 stron.

7. K. Nakamoto. Infrared spectra of Inorganic and Coordination Compounds. A. Willey, New York, 1997.

8. W. B. Smith. Introduction to Theoretical Organic Chemistry and molecular Modelling. Verlag Chemie. New York, 1996.

9. R. B. Woodward, R. Hoffmann. The Conservation of Orbital Symmetry. Verlag Chemie. Weinheim/Bergstr. 1970.

SPIS TREŚCI

Wstęp 3

Zasadnicze oznaczenia i skróty 4

Rozdział 1. Elementy formalnej teorii grup 5

§1. Grupa 5

§ 2. Właściwości najprostsze grup 5

§ 3. Przykłady grup 6

§ 4. Podgrupa 7

§ 5. Rząd elementu. Grupy cykliczne 8

§ 6. Elementy sprzężone i klasy 9

§ 7. Podgrupa niezmienna 10

§ 8. Czynnik-grupa 10

§ 9. Homomorfizm i izomorfizm grup 10

§ 10. Właściwości homomorfizmu

11

§ 11. Iloczyn kartezjański grup 12

Rozdział 2. Grupy punktowe

13

§ 1. Przykłady grup. C_n 13

§ 2. Grupy C_nv 14

§ 3. Grupy C_nh 18

§ 4. Grupy D_n 19

§ 5. Grupy D_nd 21

§ 6. Grupy D_nh 24

§ 7. Grupy S_2n 26

§ 8. Punktowe grupy sześcienne 26

§ 9. Punktowe grupy dwudziestościanowe 28

§ 10. Grupy nieprzerwane 29

Rozdział 3. Grupy macierzy

31

§ 1. Grupy kwadratowych macierzy nieosobliwych

31

§ 2. Macierzy punktowych zmian współrzędnych 32

§ 3. Niektóre specjalne rodzaje macierzy 35

§ 4. Iloczyn skalarny 36

Rozdział 4. Teoria reprezentacji liniowych 40

§ 1. Reprezentacja grupy 40

§ 2. Przykłady reprezentacji. Reprezentacja grupy symetrii równania Schroedingera 41

§ 3. Reprezentacje przywiedlne i nieprzywiedlne 43

§ 4. Pierwszy lemat Schura 46

§ 5. Drugi lemat Schura 47

§ 6. Relacje ortogonalności 48

§ 7. Charaktery reprezentacji 50

§ 8. Reprezentacja regularna 53

§ 9. Ilość nie ekwiwalentnych reprezentacji nieprzywiedlnych 53

§ 10. Obliczenie charakterów reprezentacji nieprzywiedlnych 55

Rozdział 5. Kompozycje reprezentacji 58

§ 1. Reprezentacje nieprzywiedlne iloczynu kartezjańskiego grup 58

§ 2. Kartezjański (kroneckerowski) iloczyn reprezentacji 60

§ 3. Symetryzowany i anty symetryzowany iloczyny

reprezentacji 61

§ 4. Funkcje podstawowe reprezentacji nieprzywiedlnych 66

§ 5. Twierdzenie Wignera 67

Rozdział 6. Wykorzystanie teorii reprezentacji grup skończonych 75

§ 1. Teoria zaburzeń 75

§ 2. Wykorzystanie teorii grup w problemach spektroskopii

drgającej 82

§ 3. Budowa współrzędnych symetryzowanych (współrzędnych

symetrii) 96

§ 4. Reguły wyboru dla widm IR i KR 101

§ 5. Przejawy spektralne oddziaływań międzymolekularnych

105

§ 6. Wyznaczenie kształtu molekuł 107

§ 7. Reguły wyboru dla częstotliwości składanych. Warunek

możliwości rezonansu Fermiego 108

§ 8. Elektronowe widma pochłaniania 111

§ 9. Klasyfikacja stanów 121

Rozdział 7. Symetria i zdolność reakcyjna związków

organicznych 126

§ 1. Reakcje elektrocykliczne 127

§ 2. Przegrupowania sigmatropne 132

§ 3. Reakcje cykłoprzyłączenia 135

§ 4. Uogólnienie. Reakcje perycykliczne 139

Rozdział 8. Elementy teorii reprezentacji grup

nieprzerwanych 143

§ 1. Określenie, przykłady 143

§ 2. Dwuwymiarowe grupy rotacji i odbić 147

§ 3. Reprezentacje nieprzywiedlne grupy rotacji

trуjwymiarowej 155

§ 4. Wyprowadzenie reprezentacji nieprzywiedlnych grupy O(3)

z wykorzystaniem operatorуw infinitezymalnych 158

§ 5. Wykorzystywanie teorii grupy SO(3) w teorii

atomуw 161

DODATEK. Tablicy charakterów grup punktowych 168

Bibliografia 178

180

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