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## Physical Chemistry

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**Chapter IV Molecular Symmetry and**Point Group • Reference Books: • F. Albert, Cotton, Chemical Application of Group Theory, Wiley Press, New York, 1971. • (中译本:群论在化学中的应用,科学出版社,1984) • (2) David M. Bishop, Group Theory and Chemistry, Clarendon Press, Oxford, 1973. • (中译本:群论与化学,高等教育出版社,1984)**Symmetry is all around us and is a fundamental property of**nature.**Motivational Factors**• The nature and degeneracy of vibrations. • The legitimate AO combinations for MOs. • The appearances and absences of lines in a molecule’s spectrum. • The polarity and chirality of a molecule.**§4-1. Symmetry Elements and Operations**The term symmetry is derived from the Greek word “symmetria” which means “measured together”. We require a precise method to describe how an object or molecule is symmetric.**Symmetry Operation**A symmetry operation is a movement of a body such that, after the movement has been carried out, every point of the body is coincident with an equivalent point (or perhaps the same point) of the body in its original orientation. Symmetry Element A symmetry element is a geometrical entity such as a line, a plane, or a point, with respect to which one or more symmetry operations may be carried out.**§4-1-1. Symmetry Elements and Operations Required in**Specifying Molecular Symmetry**1. The Identity Operation**E Ê • No matter how asymmetrical a molecule is, it must have an identity operation, E • The symbol “E” comes from the German, “eigen,” meaning “the same”**2. Proper Axes and Proper Rotations**Cn • An n-fold rotation is symbolized by the element Cn, and represents n–1 rotational operations about the axis.**Molecules may have many rotation axes. But the axis with the**highest n is designated as theprincipal axis • The operation C1 is merely E.**3. Symmetry Planes and Reflections** • There are 3 types of planes: • Vertical, horizontal and dihedral**Vertical v**If the reflection plane contains the Principle Axis, it is called a “vertical plane.”**C**h Horizontal plane h If the reflection plane is perpendicular to the Principle Axis, it is called a “horizontal plane.” A molecule can have only one h**Dihedral d**Vertical planes which bisect the angles between adjacent pairs of C2 axes perpendicular to the principle axis**4. Improper Axes and Improper Rotations**Sn=Cn h • Rotations by 2/nfollowed byreflection in a plane to the Sn axis.**5. The Inversion Center**i i = S2 = C2h=hC2 (x, y, z) (-x, -y, -z)**§4-1-2. Multiplication Table**Multiplication Table of H2O C2v E C2sxzsyz E E C2sxzsyz C2 C2 E syzsxz sxzsxzsyz E C2 syzsyzsxz C2 E Operate first Operate second**§4-1-3. General Relations Among Symmetry Elements and**Operations (1) The product of two reflections, intersecting at an angle of = 2/2n, is a rotation by 2 about the axis defined by the line of intersection 1. Products**(2) When there is a rotation axis, Cn, and a plane**containing it, there must be n such planes separated by angles of 2/2n; (3)The product of two C2 rotations about axes which intersect at an angle is a rotation by 2 about an axis perpendicular to the plane of the C2 axes; (4) A proper rotation axis of even order and a perpendicular reflection plane generate an inversion center.**2. Commutation**• The identity operation and the inversion with any operations; • Two rotations about the same axis; • Reflections through planes perpendicular to each other; • Two C2 rotations about perpendicular axes; • Rotation and reflection in a plane perpendicular to the rotation axis.**§4-2. Molecular Point Group**§4-2-1. Definitions and Theorems of Group Theory 1. Definitions A group is a collection of elements which are interrelated according to certain rules.**The product of any two elements in the group must be an**element in the group; • AB=C (2)One element in the group must commute with all others and leave them unchanged; E----the identity elementEX=XE=X (3)The associative law of multiplication must hold; A(BC)=(AB)C (4)Every element must have a reciprocal, which is also an element of the groupAA-1= A-1A= E**2. Theorems of Group**(1) For a certain element, there is only one reciprocal in the group (2)There is only one identity element in one group. (3)The reciprocal of a product of two or more elements is equal to the product of the reciprocals, in reverse order. (ABC·····XY)-1 = Y-1X-1····C-1B-1A-1 (4) If A1, A2 and A3····· are group elements, their product, says B, must be a group element, A1A2A3····· =B**3. Some Important Conceptions**Order----the number of elements in a finite group Finite groups; Infinite groups Subgroup---the smaller groups,whose elements are taken from the larger group {1，-1，i，-i } {1，-1 } 矩阵乘法**Similarity Transform**A, B and X are elements of a group, if B=X-1AX We say B is conjugate with A. Class---A complete set of elements which are conjugate to one another**Point group-----All symmetry elements in a molecule**intersect at a common point, which is not shifted by any of the symmetry operations. Schoenflies Symbols §4-2-2. Molecular Point Groups**Cn groups---only one Cn Axis**n Cn symmetry operations, g=n C1 CFClBrI**C3**C3 C3, (E, C3, C32)**2. Cnh groups:**Cn add a horizontal plane h g=2n n=1, C1h= Cs Cnh =Sn**Cs**H2TiO HOCl CH3OH**C2h (E, C2, h, i)**Trans-C2H2Cl2 **C3h(E, C3, C32, h, S3, S32)**B(OH)3, planar**3. Cnv groups: g=2n**Cn add a vertical plane v C2v (E, C2, 1, 2) H2O**C3**staggered-C2H3F3 C3v (E, 2C3, 3v) NH3**C4v**C6v OXeF4**Cv: C+v**AB type of diatomic molecules C v**4. Sn groups---with only one Sn Axis**• When n is odd, Sn = Cnh • When n is even, the group is called Sn and consists of n elements • S2=Ci, S4, S6**** i trans-C2H2F2Cl2Br2 Ci**S4**S4**5. Dn groups**Cn axis add n C2 axes perpendicular to Cn (g=2n) D3**6. Dnh groups**Dn group + h nC2Cn, h Cnh =Sn C2h = n v g=4n**D2h E, 3C2, s2=i, h, 2v**ethylene B4(CO)2**D3h**Ph(Ph)3**D4h**PtCl4 2- CAl4- Mn2(CO)10**D6h**D5h**Dh: C v+h**A2 type of diatomic molecules C h**7. Dnd groups**Dn + d d Cn n d d C2 S2n g=4n