Teoria Grupo1

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授課教師 : 林宗欣(ST219; 32116; 0952006340) 
遊戲規則: 
 上課教材 : 老師講義(請到數位教學平台下載，請每面印兩頁，雙面列印) 
 參考書目: 
--Beginning Group Theory for Chemistry, Paul H. Walton, Oxford. 1998, University press 
--無解方程式，馬里歐．李維歐著，蔡承志譯，城邦文化出版，2008 
--群論初步，倪澤恩著，五南圖書出版，2008 
 分數計算 
--期中考 55 分; 期末考 55 分; 課堂加分(上課指正老師的錯誤或其他有助於老師教學
及增進同學權益的行為) 
--期末絕對不調分數 
 補考 : 期中考後 2 週內；期末考後 1 週內(補考分數為參考分數，使用了補考分數，總
成績最高只有 60 分) 
 考試內容 : Exercise 的類題 
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Symmetry 
視覺上的效果(生物 圖畫 壁紙) 
物理上的效果 
實例:Benzene (苯) Is more symmetrical than 2-chlorophenol (2-氯酚) 
 1H-NMR spectra 
-- benzene → single peak 
-- 2-chlorophenol → several peaks split by a complicated pattern 
--Qualitatively (定性上): symmetry has something to do with the energy levels of a molecule 
− all benzene hydrogen atoms are in the same chemical environment 
 
 Quantitatively (定量上): How important is symmetry in determining the energy levels of 
molecules? 
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 We must provide some type of rules to define the symmetry of molecules 
Symmetry: 若你對某事物做了某種處置(操作)，完成後猶一如舊觀，則該事物便為對稱的 
 
Rotation axes 
 Axis: passes through the oxygen atom and 
bisects the H-O-H angle 
 The molecule is indistinguishable from the 
original one 
 The water molecule is symmetric to rotation 
by 1800 about this axis 
 
Furthermore: If the molecule is symmetric (the same) by rotation of x degree about an axis, 
then we have a rotation axis of order (360/x). 
ex. H2O → 3600/1800 = 2  second order axis  C2 
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TERMINOLOGY 
 The axis is know as the “symmetry element” 
 The actual transformation of the molecule to its symmetrical equivalent position is known as 
the “symmetry operation” 
 The rotation is clockwise 
 
For water molecule, the C2 axis is the symmetry element and clockwise rotation of 1800 about 
this axis is the symmetry operation. 
 
Exercise Sketch what happens to the water molecule (labeling the 
hydrogen atoms as above), when C2 is carried out on it twice in 
succession. 
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Doing nothing (E) is an important operation in group theory 
1. C1 = E; 2. C21．C21 = C22 = E; 3. Cnn = E 
Exercise Sketch ammonia molecule and work out if it has a rotation axis (axes) and the order of 
the axis (axes). Draw the axis (axes) on your sketch of molecule. (Only one rotation axis of 
symmetry) 
 
Exercise Figure out all of the rotation axes that can be 
found for benzene, and sketch the axes on your picture 
of benzene. 
(Contains a large number of rotation axes of 
symmetry) 
 
 In molecules which have more than one rotational axis of symmetry, then the axis of highest 
order is defined as “principle axis”. 
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--C6 is principle axis in benzene 
--C6 contains a C3 axis and C2 axis; 1326 CC  , 1236 CC  
 Even-order axis greater than order 2 contain lower-order even axes. 
 The other two type of C2 axes are called C2’, C2” 
 
Reflection Planes 
The reflection plane is another symmetry element of the water molecule. 
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In group theory a reflection plane is denote by . These reflection planes contain the principle 
axis within the plane, and can be said to be “vertical” along with the principle axis, are denoted 
by v 
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Exercise List all of the symmetry elements present in the water molecule 
 
Reflection planes of NH3 
 σv in Ha-N plane 
 σv in Hb-N plane 
 σv in Hc-N plane 
Exercise 
Identify the principle rotations in AuBr4- 
Reflection planes of -4AuBr 
× 2 
v contains the Br-Au bonds 
 
× 2 d, d stands for “dihedral”. 
d bisects the Br-Au-Br angles 
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× 1 h, h stands for “horizontal”. 
 
 
 
Center of inversion /inversion center (i) 
Good thing : a single molecule can only have one such centre, and we only need to decided 
whether a molecule has got one or not. 
Bad thing : The actual symmetry element it self is sometimes difficult to “see” 
 The first way -- by inspection 
 
Inversion center lies on Au atom 
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 The second way -- mathematical description if the inversion center is placed on the origin of 
three-dimensional Cartesian coordinate set of 
axes, and atoms are at (x,y,z), then an inversion 
centre is a symmetry element if one same atom 
at (-x,-y,-z). (x,y,z) (-x,-y,-z); (xa,ya,za) = 
(-xc,-yc,-zc); (xb,yb,zb) = (-xd,-yd,-zd) 
 The third way -- combination of specific rotation axis and a specific reflection plane. 
--rotation 1800 followed by reflection in a reflection plane perpendicular to the rotation axis. 
--If a molecule contains a C2 axis and a reflection plane perpendicular to this axis it has 
inversion center. 
--The inversion center element is placed at the meeting point of the axis and the reflection 
plane. 
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※ The presence of an inversion centre of symmetry does not necessary need that the molecule 
has a C2 axis and a perpendicular reflection plane as individual symmetry element. 
 
Exercise Using the third method, reason whether or not H2O has an inversion centre as a 
symmetry element? Do the same thing for AuBr4-. 
 
Improper rotation axes (Sn) 
Example : tetrahedral (四面體) molecule, CH4 
-- It is impossible to find a “single” symmetry element which 
will allow us to transform each of the hydrogen atom position 
into all of the other atom positions. 
-- We need another symmetry element which can describe a 
symmetry operation to relate all hydrogen atoms position to 
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each other 
→ improper rotation axis 
-- this symmetry element is the most difficult to identify. 
Check C2 axes of methane 
-- Carbon atom is in the plane of the 
paper 
-- Only Ha and Hc (Hd and Hb) can be 
interchange 
-- It is impossible to interchange the Ha 
and Hb 
For 900 rotation (not a symmetry 
operation) 
-- H atoms are on opposite sides of the 
plane of the paper. 
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-- Not right, but “almost” gives right result. 
→ Combine the rotation by 900 with 
reflection in a reflection plane 
perpendicular to the axis of rotation 
 
※ The molecule does not necessarily have a either C4 axis of rotation, or a reflection plane as 
individual symmetry elements. 
 
The combination of a rotation and reflection in a perpendicular reflection plane gives a new 
symmetry element, which is called an improper rotation axis. C4 + reflection → S4 
Exercise Check S42 and S43 of methane 
S42 is the same as the C21 operation 
S44 is the same as the E operation 
Only S41 and S43 are unique 
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An S4 axis must have a coaxial C2 axis 
 
For any even-order Sn axis, there must be a coaxial Cn/2 axis. 
 S62 is the same as C31; S63 is the same as i; S64 is the same as C32; S66 is the same as E 
Only S61 and S65 are unique. 
 S2 is the same as inversion operation 
 
How about odd-order improper rotation axes 
 S33 is not the same as E operation; S33 corresponds to  operation 
 S36 is the same as the E operation; S32is the same as C32; S34 is the same as C3 
Only S31 and S35 are unique operations 
Many of improper operations can be described by other symmetry operations. 
With odd-order improper rotation axes; Sn2n is the same as E. 
 
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Exercise PF5 has a trigonal bipyramidal structure. Sketch the molecule and indentify the 
improper rotation axis. 
 
Exercise Identify the S6 axis in CoCl63- (an octahedral complex). 
This can be difficult to “see”. 
 This is an even-order improper rotation 
axis and it must have a coaxial Cn/2 axis. 
 Let’s look for C3 axis first. 
Exercise Identify and list as many 
symmetry elements as you can in the following molecules: PCl5, (E)-1,2-dibromoethane, CH4. 
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[PCl5 : E, C3, 3C2, h, S3 and 3v; (E)-1,2-dibromoethane: E, C2, i, h; CH4: E, C3, C2, S4, d]The properties of symmetry operation 
1. Successive operation 
(1) Is the order in which we perform operation important? 
PH3 
v after C3 
(v × C3) 
PH3 
C3 after v 
(C3 × v) 
 
 (v × C3) ≠ (C3 × v) (non-commutative) 
(2) Any combination of symmetry operations can be described by a single symmetry 
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operation. 
v × C3 = v’, C3 × v = v” 
(3) What about performing three successive symmetry operations C3 × v’ × v? 
 
i. Check v’ × v 
 
ii. v’ × v = C32; C3 × v’ × v = C3 × C32 = E 
Exercise Check v’ × v × C3, (C3 × v) × v’, C3 × (v × v’) for PH3. [E, C32, C32] 
 As the last two examples show, A(BC) = (AB)C. (associative) 
 
2. Identity 
E : doing nothing 
E can be seen as the result of combinations of other symmetry operations. 
 
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3. Inverse 
A-1 : the inverse of the A operation. A-1A = E 
Any symmetry operation (A) performed on a molecule can be “reversed” or “undone”, by 
another operation to give back the original molecule. The A-1 operation is called the inverse of 
the A operation. 
Exercise Write the inverse of the following symmetry operations in PH3. E, C31, C32, v. [E, C32, 
C31, v] 
 Each symmetry operation has an inverse operation which is also a symmetry operation of 
the molecule. 
4. Class 
E, C32, C31, v, v’, and v’’ are symmetry operations in NH3. C32 and C31 are “similar”, v, 
v’,and v’ are “similar”. Similar perations can be related by mathematical procedure, called a 
“similarity transformation”. 
The similar symmetry operation can be group into “class”. NH3 : E, 2C3, 3v. 
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 Symmetry operations within a class have similar mathematical properties. 
Point groups 
The symmetry operations of PCl5 are 
E, C31, C32, C2, C2’, C2”, h, S31, S35, v, v’, 
v”  Too time-consuming 
 Just identify key symmetry elements within 
a molecule. The key symmetry elements 
then define a particular group, which has 
several different symmetry elements. 
 Each classification is given a symbol and 
stands for a collection of symmetry operation.  “point group” 
Point: all the symmetry elements associated with the symmetry operations pass through a single 
point in space. This point is not changed in position by any of the symmetry operations. 
The point in PCl5 molecule lies directly on the phosphorus atom. (Note: The point is not 
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necessary to lie on an atom.) 
Group: we have a group of symmetry operation 
※Classify a molecule into a point group by answering some simple question about the 
molecule. 
Q1: Is the molecule one of the following groups? 
Yes: Octahedral  Oh 
tetrahedral  Td 
linear, and does not have an symmetry element  C∞v 
linear, and has an symmetry element  D∞h 
No: Go to Q2 
 
Q2: Does the molecule possess a rotation axis of order ≧ 2 
No: If it has no other symmetry  C1 
If it has one reflection plane of symmetry  Cs 
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If it has a centre of inversion  Ci 
 Yes: Go to Q3 
 
Q3: Has the molecule more than one rotation axis? 
No: If it has no other symmetry element  Cn (n = the order of principle axis) 
If it also has one  h  Cnh 
 If it has n  v  Cnv 
If it also has an S2n coaxial with the principle axis  S2n 
Yes: Go Part 4. 
 
Part 4: The molecule can be assigned a point group. 
If it has no other symmetry elements  Dn 
If it has got n  d reflection planes bisecting C2 axes  Dnd 
If it also has one  h  Dnh 
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PCl5 
 
Q1  No, go to Q2 
Q2  Yes, there is C3 axis, go to Q3 
Q3  Yes, go to Part 4 
Part4  It has three v planes, it also has a h plane 
 D3h 
 
NH3 
Q1  No, go to Q2 
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Q2  Yes, there is C3 axis, go to Q3 
Q3  No, it has three v planes  C3v 
 
※ The point groups are shorthand for the symmetry elements within a molecule. Ex. C3v  E, 
2C3, 3v 
Exercise Assign point groups to the following molecules: H2O, PH3, SO2, HCl, AuBr4-, CoCl63-, 
(E)-1,2-dibromoethane, benzene, methylbenzene, trichloromethane, NO3-, SO4-, HCCH, B2H6, 
Co(en)33+ (en = 1,2-diaminoethane).[ H2O (C2v), PH3 (C3v), SO2 (C2v), HCl (C∞v), AuBr4- (D4h), 
CoCl63- (Oh), (E)-1,2-dibromoethane (C2h), benzene (D6h), methylbenzene (C2v), 
trichloromethane (C3v), NO3- (D3h), SO4- (Td), HCCH (D∞h), B2H6 (D2h), Co(en)33+ (D3).]