Chapter 1 Fundamental Concepts

Size: px

Start display at page:

Download "Chapter 1 Fundamental Concepts"

Meredith Stafford
5 years ago
Views:

1 Chapter 1 Fundamental Concepts 1-1 Symmetry Operations and Elements 1-2 Defining the Coordinate System 1-3 Combining Symmetry Operations 1-4 Symmetry Point Groups 1-5 Point Groups of Molecules 1-6 Systematic Point Group Classification 1-7 Optical Activity and Symmetry

2 Molecular Symmetry and Group Theory Focus in this text is the application of symmetry arguments to solve physical problems of chemical interest. As a first step, we must identify and catalogue the complete symmetry. Once this is done, we can employ the mathematics of groups to simplify the physical problem and subsequently to obtain chemically useful solutions. Gas, liquid, solution, and solid

3 1.1 Symmetry Operations and Elements Symmetry operations; it is a movement of an object about a symmetry element such that the object s orientation and position before and after the operation are indistinguishable. Symmetry elements; it is an imaginary geometrical entity such as a line, plane, or point about which a symmetry operation can be performed. The object need not be in the identical position it had before the operations. It is only necessary that the position be indistinguishable and therefore equivalent.

4 1.1 Symmetry Operations and Elements The symmetry of a molecule or ion can be described in terms of the complete collection of symmetry operations it possesses. Operation will be one of five types. -identity, rotation, reflection, inversion, rotation-reflection (or improper rotation) Elements are (1) the object itself, (2) a line (rotation axis or proper axis), (3) a plane (reflection plane or mirror plane), (4) a point (inversion center), and (5) a line (improper axis). All symmetry elements will pass through a common point at the center of the structure. For this reason, the symmetry of isolated molecules and ions is called point group symmetry.

5 1.1 Symmetry Operations and Elements Identity (E)- doing nothing, need it for mathematical requirements of group theory. The object is said to be asymmetric, if it possesses only E. Rotation (C n )- rotation about an axis by 2π/n radians, the value n is the order of rotation

6 1.1 Symmetry Operations and Elements Rotation (C n )- C 4 indicates a fourfold rotation. Two successive C 4 rotations (which we could designate C 4 ) about the same axis has the same effect as a single C 2 rotation. 4 C 4 = E n C n = E n-1-1 C n = C n This single element, a C 4 axis, is associated with three unique operations: C 4, C 4 = C 2, C 4 =C 4. 2

7 1.1 Symmetry Operations and Elements

8 1.1 Symmetry Operations and Elements Rotation (C n )- Rotations beyond full circle are expressed as the equivalent single rotation that is less than 2π. 5 C 4 = C 4 There also exists a C 2 axis collinear with the C 4 axis. There are four other C 2 axes. We can distinguish these by adding prime( ) and double prime ( ) Pass through more atoms than

9 1.1 Symmetry Operations and Elements Rotation (C n )- For square planar MX 4, the rotational operations grouped by 3 class are 2C 4 (C 4 and C 4 ), C 2 (collinear with C 4 ), 2C 2, and 2C 2. The highest-order rotational axis an object possesses is called the principal axis of rotation.

10 1.1 Symmetry Operations and Elements Reflection (σ)- For square planar MX 4, three kinds of mirror plane.

11 1.1 Symmetry Operations and Elements Reflection (σ)- Performing two successive reflections about the same plane brings the object back into original configuration; σσ= σ 2 = E The operation of σ h (horizontal mirror plane) appears to do nothing to the molecule. -however, if atoms had a directional property perpendicular to the plane, the property will tune to negative. -defined as perpendicular to the principal axis of rotation -if no principal axis exists, defined as the plane of the molecule

12 1.1 Symmetry Operations and Elements Reflection (σ)- A σ v (vertical mirror plane) and σ d plane (dihedral mirror plane) are defined as to contain a principal axis of rotation. -defining σ v to contain the greater number of atoms

13 1.1 Symmetry Operations and Elements Inversion (i)- Related to the central point within the molecule. -The central point (0, 0, 0) is called an inversion center or center of symmetry. -Molecules that have inversion symmetry are said to be centrosymmetric. -performing inversion twice in succession would bring every point back into itself; ii = i 2 = E

14 1.1 Symmetry Operations and Elements Inversion (i)- No inversion center for ethane in the eclipsed configuration

15 1.1 Symmetry Operations and Elements Rotation-reflection (S n )- -also called improper rotation -a proper rotation (C n ) followed by a reflection (σ h ) in a plane perpendicular to the axis of rotation. -if both C n and σ h exist, S n must also exist. -however, the presence of both C n and σ h is not requirement for the existence of S n

2 S 4 S 4 = S 4 = C 2 3-1 S 4 = S 4 4 S 4

16 1.1 Symmetry Operations and Elements Rotation-reflection (S n )- -like proper rotation, a series of improper rotations can be performed about the same axis. 2 S 4 S 4 = S 4 = C S 4 = S 4 4 S 4 = E 3 -there are two S 4 operation (S 4 and S 4 )

17 1.1 Symmetry Operations and Elements Rotation-reflection (S n )- -there equivalent S 4 axes -total six operations, which belong to a class designated 6S 4

18 1.2 Defining the Coordination System We will adopt a standard Cartesian coordination system with x, y, z defined by so-called right-hand rule.

19 1.3 Combining Symmetry Operations Written in a right-to-left order -BA means do A first, then B. -S 4 S 4 = C 2, σ h C 4 = S 4. -Commutation is nor generally observed does not mean that it is never observed. -Some cases AB = BA, however, some cases AB = BA

20 1.3 Combining Symmetry Operations Symmetry elements of CBr 2 Cl 2. -the complete set of symmetry operations for the molecule consists of identity (E), a twofold principal axis of rotation (C2), and two reflections about different mirror planes (σ v and σ v ). 1 X 4 matrix to describe the locations of atoms Represent operator matrices

21 1.3 Combining Symmetry Operations EE = E, C 2 E = EC 2 = C 2, σ v E = Eσ v = σ v Since E really does nothing to the molecule C 2 σ v C 2 σ v = σ v σ v

22 1.3 Combining Symmetry Operations EE = E, C 2 E = EC 2 = C 2, σ v E = Eσ v = σ v Since E really does nothing to the molecule σ v C 2 σ v C 2 = σ v σ v C 2 = C 2 σ v = σ v They are commutative σ v

23 1.3 Combining Symmetry Operations Multiplication table Order of combination is row element (top) first, followed by column element (side)

24 1.4 Symmetry Point Groups The complete set of symmetry operations exhibited by any molecule defines a symmetry point group. The set must satisfy the four requirements of a mathematical group: closure, identity, associativity, and reciprocality. Closure; If A and B are elements of the group G, and if AB =X, then X is also in the group. In the case of symmetry groups, the group elements are the symmetry operations, not the symmetry elements. The operations of CBr 2 Cl 2 shows that all binary combinations equal either E, C 2, σ v or σ v ; this complete set of elements of a point group called C 2v. The order of the group h = 4

25 1.4 Symmetry Point Groups Identity; Any group G must has an element E. Associativity; If A, B, C, and X are members of the group G and C(BA) =X, then (CB)A = X, too. C 2 (σ v σ v ) = (C 2 σ v )σ v Do not confuse to commutation. Not always allow this CBA = BAC. In the mathematics of groups, any group in which all combinations of elements commute is said to be Abelian.

26 1.4 Symmetry Point Groups C 1 C 2 C s C s Reciprocality; Every element A of the Group G has an inverse, A -1, such that AA -1 = A -1 A = E. Subgroup Among the operations that constitute a point group, there generally exist smaller sets that also obey the four requirements of a group. If g is the order of subgroup of a group whose order is h, then h/g = n, where n is an integer. For C 2v, where h = 4, only subgroups with orders g = 1 and g = 2 are possible. The group C1 has no symmetry and is therefore asymmetric.

27 1.5 Point Groups of Molecules Schönflies notation; We have seen so far C 2, C s. Four general categories n is the order of the principal axis

28 Nonrotaional Groups Single-Axis Groups Dihedral Groups

29 1.5 Point Groups of Molecules Nonrotational Group E E, σ h E, i

30 1.5 Point Groups of Molecules Single-Axis Rotational Groups C n groups E, C 2, Cyclic Groups They are Abelian Multiplication table for the point group C 4 The recognition of the pattern makes it easy to construct the multiplication table for any cyclic group

31 1.5 Point Groups of Molecules Single-Axis Rotational Groups C nv groups 2 E, C 3, C 3, 3σ v E, C 4, C 2, C 4, 4σ v C groups; all noncentrosymmetric linear molecules are belong to here (e.g., HCl, ClBeF)

32 1.5 Point Groups of Molecules Single-Axis Rotational Groups C nh groups 2 E, C 2, σ h E, C 3, C 3, σ h Since C n σ h = S n and C 2 σ h = S 2 = i, these groups also have n-fold improper axes when n>2, and they are centrosymmetric when n is even.

33 1.5 Point Groups of Molecules Single-Axis Rotational Groups S 2n groups 3 E, S 4, S 2 = I, S 4 As the 2n notation implies, only groups of this type with evenorder principal improper axes exist. An odd-order S n axis is the same as that generated by the combination C n and σ h, which defines groups of the type C nh.

34 1.5 Point Groups of Molecules Dihedral Groups The dihedral groups have n twofold axes perpendicular to the principal n-fold axis D n groups 2 E, C 3, C 3, 3C 2

35 1.5 Point Groups of Molecules Dihedral Groups D nd groups out E, C 3, C 3, 3C 2, S 6, S 6, S 6, S 6, S 6, 3σ d

36 1.5 Point Groups of Molecules Dihedral Groups D nh groups 2 E, C 3, C 3, 3C 2, σ h, 3σ v

37 1.5 Point Groups of Molecules Cubic Groups 24 operation, listed by classes: E, 8C 3 (4C 3, 4C 3 ) 3 3C 2, 6S 4 (3S 4, 3S 4 ), 6σ d ) 2 48 operation, listed by classes: E, 8C 3 (4C 3, 4C 3 ) 2 3 6C 4 (3C 4, 3C 4 ), 6C 2, 3C 2 (3C 4 ), i 6S 4 (3S 4, 3S 4 ), 5 8S 6 (4S 6, 4S 6 ), 3σ h (σ xy, σ yz, σ xz ) 6σ d ) operation, listed by classes: E, 12C C 5, 20C 3, 15C 2, i 12S 10, 12S 10, 12S 6, 15σ)

38 1.6 Systematic Point Group Classification Identifying the point group of a molecule is a necessary first step for almost all applications of group theory in chemistry. Check key symmetry elements in a prescribed sequence.

39 1.6 Systematic Point Group Classification

40 1.6 Systematic Point Group Classification

41 1.6 Systematic Point Group Classification

42 1.6 Systematic Point Group Classification

43 1.6 Systematic Point Group Classification

44 1.6 Systematic Point Group Classification

45 1.7 Optical Activity and Symmetry Enantiomers Compounds that can exist as enantiomeric pairs are called chiral; they are said to be dissymmetric. Not always dissymmetric = asymmetric.

46 1.7 Optical Activity and Symmetry A molecule is dissymmetric and may be chiral either if it is asymmetric or if it has no other symmetry than proper rotation. The possible chiral groups are C 1, C n, and D n. Belongs to C 2 point group, but not be chiral. Free rotation about O-O bond

Symmetry Operations and Elements

Symmetry Operations and Elements The goal for this section of the course is to understand how symmetry arguments can be applied to solve physical problems of chemical interest. To achieve this goal we