hapter Molecular symmetry and symmetry point group
Why do we study the symmetry concept? The molecular configuration can be expressed more simply and distinctly. The determination of molecular configuration is greatly simplified. It assists giving a better understanding of the properties of molecules. To direct chemical syntheses; the compatibility in symmetry is a factor to be considered in the formation and reconstruction of chemical bonds.
Symmetry elements and symmetry operations Symmetry exists all around us and many people see it as being a thing of beauty. A symmetrical object contains within itself some parts which are equivalent to one another. The systematic discussion of symmetry is called : Some objects are more symmetrical than others.
. Symmetry elements and symmetry operations symmetry operation A action that leaves an object the same after it has been carried out is called symmetry operation. Example: Any rotation of sphere around axis through center brings sphere over into itself
Example: (a) An NH molecule has a threefold ( ) axis (b) an H O molecule has a twofold ( ) axis.
symmetry elements Symmetry operations are carried out with respect to points, lines, or planes called symmetry elements. Example: (a) An NH molecule has a threefold ( ) axis (b) an H O molecule has a twofold ( ) axis. NH has higher rotation symmetry than H O
Symmetry elements Some of the symmetry elements of a cube, the twofold, threefold, and fourfold axes.
Symmetry operations are: The corresponding symmetry elements are:
) The identity (E) Operation by the identity operator leaves the molecule unchanged. All objects can be operated upon by the identity operation. F l I Br
) Inversion and the inversion center (i) An object has a center of inversion, i, if it can be reflected through a center to produce an indistinguishable configuration. A regular octahedron has a centre of inversion (i).
For example These have a center of inversion i. These do not have a center of inversion.
z y x z y x z y x i Its matrix representation Inverts all atoms through the centre of the object
) Rotation and the n-fold rotation axis ( n ) Rotation about an n-fold axis (rotation through 6 o /n) is denoted by the symbol n. Example: Rotation of trigonal planer BF. One three-fold ( ) rotation axes. (=/) The principle rotation axis is the axis of the highest fold.
The matrix representations: + (x,y) (x,y ) x y x =-rsin(+)= -rsincos -rcossin = (-/)x + (-/)y y =rcos(+)= rcoscos - rsinsin = (/)x + (-/)y z y x y x z y x z y x cos sin sin cos z y x z y x 4 4 4 4 z y x y x z y x z y x cos sin sin cos z y x z y x
The matrix representations: onditions: The centre of mass of the molecule is located at the origin of the artesian oordinate System Principle axis is aligned with the z-axis (-x,-y) x x cos sin x x x y y sin cos y y y z z z z z y (x,y) n x k n cos sin sin cos k n
For example The principle rotation axis is the axis of the highest fold. 6 5
If reflection of an object through a plane produces an indistinguishable configuration then that plane is a plane of symmetry (mirror plane) denoted. 4) Reflection and the Mirror plane () xy z y x z y x z y x xy (x, y,-z) (x,y,z) z y x
There are three types of mirror planes: If the plane is perpendicular to the vertical principle axis then it labeled h. If the plane contains the principle axis then it is labeled v. If a plane contains the principle axis and bisects the angle between two adjacent -fold axes then it is labeled d.
If the plane is perpendicular to the vertical principle axis then it labeled h. Example: BF also has a h plane of symmetry.
If the plane contains the principle axis then it is labeled v. Example: Water Has a principle axis. Has two planes that contain the principle axis, v and v. H O H v v
If a plane contains the principle axis and bisects the angle between two adjacent -fold axes then it is labeled d.(dihedral mirror planes ) Example: BF Has a principle axis Has three- axes. Has three d planes (?). v
Example: 6H6 6 Benzene has one mirror plane perpendicular to the principle 6 axis ( h) Dihedral mirror planes ( d) bisect the axis perpendicular to the principle axis. d v
Example: H ==H d d
5) The improper rotation axis a. n-fold rotation + reflection, Rotary-reflection axis (S n ) Rotate 6 /n followed by reflection in mirror plane perpendicular to axis of rotation S 4
Example: H -H S 6 h S 6 The staggered form of ethane has an S 6 axis composed of a 6 rotation followed by a reflection.
Special ases: S and S S h h S h i
Stereographic Projections o x x o o We will use stereographic projections to plot the perpendicular to a general face and its symmetry equivalents, to display crystal morphology o for upper hemisphere; x for lower
E S S S S S S 6 5 4 ; ; ; ; ; S h h S
E S S S S 4 4 4 4 4 4 4 ; ; ; E S S S S S S S S S S 5 4 5 9 5 5 8 5 5 7 5 5 6 5 5 5 4 5 4 5 5 5 5 5 5 5 ; ; ; ; ; ; ; ; ; x S 4 x S4 S 4 4 4 S h h h S 5 5 5 6 6 S h
I i I E 6 5 4 I i I I i I I i I i I i I i i I i I h n n, x x b. n-fold rotation + inversion, Rotary-inversion axis(i n ) Rotation of n followed by inversion through the center of the axis
Summary Element Name Operation n n-fold rotation Rotate by 6 /n Mirror plane Reflection through a plane i S n enter of inversion Improper rotation axis Inversion through the center E identity Do nothing Rotation as n followed by reflection in perpendicular mirror plane
. ombination rules of symmetry elements A. ombination of two axes of symmetry The combination of two axes intersecting at angle of /n, will create a n axis at the point of intersection which is perpendicular to both the axes and there are n axes in the plane perpendicular to the axis. n + ( ) n ( )
B. ombination of two planes of symmetry. If two mirrors planes intersect at an angle of /n, there will be a n axis of order n on the line of intersection. Similarly, the combination of an axis n with a mirror plane parallel to and passing through the axis will produce n mirror planes intersecting at angles of /n. n + v n v v v v v Ex. H O, NH v ' v v
. ombination of an even-order rotation axis with a mirror plane perpendicular to it. ombination of an even-order rotation axis with a mirror plane perpendicular to it will generate a centre of symmetry at the point intersection. Each of the three operations xy, n and i is the product of the other two operations xy xy (x,y,z) (-x,-y,-z) (-x,-y,z) y x i h m m h
Groups and group multiplications. Definition: A mathematical group, G = {G,}, consists of a set of elements G = {E, A,B,,D,...} (a) losure. The product of any two elements A and B in the group is another element in the group. (b) Identity operation. The set includes the identity operation E such that AE=EA=A for all the operations in the set. (c) Associative rule. If A, B, are any three elements in the group then (AB) = A(B). (d) Inversion. For every element A in G, there is a unique element X in G, such that XA = AX = E. The element X is referred as the inverse of A and is denoted A -.
'' ',,,, E, E ) ( ) ( E symmetry elements: losure. Identity operation. Associative rule. Inversion. E Example: NH Therefore, these symmetry elements constitute a group, V
Example: G = {E,,, v (), v (), v () } NH : V v () v () v () v () + v () v () + v () v () v () + v () v () v v v v
. Group Multiplication Example: H O z x y v xz yz Its total symmetry elements: E,, xz yz
. Group Multiplication Example: H O Multiplication table of v v E xz yz E E xz yz E yz xz xz xz yz E yz yz E xz yz xz
Multiplication table of v v E xz yz E E xz yz E yz xz xz xz yz E yz yz xz E (). In each row and each column, each operation appears once and only once. () We can identify smaller groups within the larger one. For example, {E, } is a group. () The group order is the total number of the group
Example: NH v Its total symmetry elements:e,,, v, v, v Multiplication table of v v E v v v E v v v
Group Multiplication v E v v v E E v v v E E v v v v v v - = E
Group Multiplication v E v v v E E v v v E v v v E v v v v v v v v v v v v v v v v = v
Group Multiplication v E v v v E E v v v E v v v E v v v v v v v E v v v v E v v v v E v B A v v =
Multiplication table of v v E v v v E E v v v E v v v E v v v v v v v E v v v v E v v v v E
Point groups, the symmetry classification of molecules Point group: All symmetry elements corresponding to operations have at least one common point unchanged.
. The groups, i, and s The group A molecule belongs to the group if it has no element of symmetry other than the identity. Example: BrlF F l I Br
The group i It belongs to i if it has the identity and inversion alone. Example: meso-tartaric acid, HlBr-HlBr
The group s It belongs to s if it has the identity and a mirror plane alone. N S l 4 O
A molecule belongs to if it has only the identity E. l I F Br OOH HO H A molecule belongs to i if it has only the identity E and i. A molecule belongs to s if it has only the identity E and a mirror plane. H HOO OH Meso-tartaric acid N Quinoline HlBr-HlBr H =lbr
. The groups n, nv, nh and S n The group n A molecule belongs to the group n if it possess an only n-fold axes. Example: H O
H O O H H l H O 6 H (H ) H l
The group nv If in addition to a n axis it also has n vertical mirror planes v, then it it belongs to the nv group. n, n v =O v v v
v v 6 H N H 4 O P 4 S v 4 H
The group nh Objects having a n axis and a horizontal mirror plane belong to nh. n, h trans-hl=hl
h H l I 7 - H 6 l
i The presence of a twofold axis and a horizontal mirror plane jointly imply the presence of a centre of inversion in the molecule.
h h 4h 5h 6h l H O H H l Trans Hl=Hl h h H O B O H B(OH)
The group S n Objects having a S n improper rotation axis belong to S n. Group S = i Group S = s S 4
S same as i OOH HO H S 4 S 6 S 8 S 4 : S 4,,S 4,,i,S6 5,S6 S ; 4 ;S 8, S 8 5, 4,S8 7 H HOO OH Meso-tartaric acid S 4 S 4 Single-axis group
. The group D n, D nh, D nd The group D n A molecule that has an n-fold principle axis and n twofold axes perpendicular to n belongs to D n. n, n
D D D 4 D5 D6 o(dien) H l (mediate state) H 6
The groups D nh A molecule with a Mirror plane perpendicular to a n axis, and with n two fold axes in the plane, belongs to the group D nh. D n, h H H H 4 H h D h l 4 l h Au D 4h l l '
D nh F F P F F h F D h
D h Dnh H 4 H SiF 4 ( 5 H 5 N) D h BF Pl 5 Tc 6 l 6
D 4h D nh [Ni(N) 4 ] - [M (OOR) 4 X ] Re l 8 6 H 6 D r( 6h 6 H 5 ) D h O==O
The group D nd A molecule that has an n-fold principle axis and n twofold axes perpendicular to n belongs to D nd if it posses n dihedral mirror planes. D n, n d The order of group=4n D d
D d D nd N 4 S 4 Pt 4 (OOR) 8
D d Til 6 - TaF 8 - D 4d S 8 D 5 d
4. High order point groups Molecules having three or more high symmetry elements may belong to one of the following: T: 4, (T h : + h ) (T d : +S 4 ) O: 4, 4 (O h : + h ) I: 6 5, (I h : +i)
T d Species with tetrahedral symmetry O h Species with octahedral symmetry (many metal complexes) tetrahedral symmetry group octahedral symmetry group Icosahedral symmetry group I h Icosahedral symmetry (Buckminsterful lerene, 6 )
ubic groups 4, T: 4, (T h : + h ) (T d : +S 4 ) Shapes corresponding to the point groups (a) T. The presence of the windmill-like structures reduces the symmetry of the object from Td.
ubic groups T h {E,4,4,,I,4S 6,4S 65,σ h }
ubic groups S 4 d T d {E,,8,6S 4,6σ d }
T T h T d (H ) 4 Ti 8 + H 4 T d o 4 (O) P 4 O 6
ubic groups O 4 O: 4, 4 (O h : + h ) Shapes corresponding to the point groups (b) O. The presence of the windmill-like structures reduces the symmetry of the object from O h.
ubic groups d S 6 O h S 4 4 F F F S F F F
ubic groups SF 6 8 H 8 OsF 8 O h Rh
I group 5 B H (with hydrogen omitted) I: 6 5, (I h : +i) H
I h {E, 5, 5,,5,i,S,S,S 6,5σ} 6, the bird-view from the 5-fold axis and 6-fold axis
Y molecule Linear? N Y D h i? N v I h? Two or more n? (n>) Y Y Y 5? Y 4? O h? Y N Y N h? N T Y d? D h? Y Y n? Is there n perpendicular to n? N Y Y d? N N Y,i? Y N? i? N h? N Y v? nh N s i I h O h T h T d D nh D nd D n nv n
4 Application of symmetry H H. hirality Br F l F l Br A chiral molecule is a molecule that can not be superimposed on its mirror image These molecules are: cannot be superimposed on its mirror image. a pair of enantiomers (left- and right-handed isomers) does not possess an axis of improper rotation, S n Ability to rotate the plane of polarized light (Optical activity ) S n (i=s ; )
Optical activity is the ability of a chiral molecule to rotate the plane of plane-polarized light. polarizer sample planepolarized dextrorotatory (d) or (+) levorotatory (l) or ( ) optically inactive optically active
Optical activity Optically inactive: achiral molecule or racemic mixture - 5/5 mixture of two enantiomers Optically pure: % of one enantiomer Optical purity (enantiomeric excess) = percent of one enantiomer percent of the other e.g., 8% one enantiomer and % of the other = 6% e.e. or optical purity
A chiral molecule does not possess S n (i, ) n and D n may be chiral (no S n improper axis)
. Polarity, Dipole Moments and molecular symmetry A polar molecules is one with a permanent electric dipole moment. Dipole Moments are due to differences in electronegativity depend on the amount of charge and distance of separation in debyes (D), = 4.8 (electron charge) d (angstroms) For one proton and one electron separated by pm, the dipole moment would be: d 9 D (.6 )( m) 4. 8D.4 m
Bond Dipole Moments
Molecular Dipole Moments Depend on bond polarity and bond angles Vector sum of the bond dipole moments Symmetric molecules may have zero net dipole --- O : O==O Lone pairs of electrons contribute to the dipole moment 8/*/8/*/
Molecular Dipole Moments 8/*/8/*/
Molecular Dipole Moments
Permanent Dipole Moments HO OOH H (a) A permanent dipole moment can not exist if inversion center is present. Only molecules belonging to the groups n, nv and s may have an electric dipole moment H HOO OH Meso-tartaric acid inversion (b) Dipole moment cannot be perpendicular to any mirror plane or n. ( h ) H x OH O H = x OH
Molecular Dipole Moments and molecular symmetry O H H l O I F O H Br N Quinoline in plane H O Trans Hl=Hl along along along inversion l H H O H B l H O B(OH) h symmetry