10/25/2018 1 1 Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy). Predict IR spectra or Interpret UV-Vis spectra Predict optical activity of a molecule 2 Molecular Symmetry • Symmetry impacts • Physical properties • Reactions • Molecular orbitals • Electronic structure • Molecular vibrations • Group theory • Behavior of molecule based on symmetry • Symmetry analysis • Application of symmetry • Orbital symmetry • Vibrational symmetry 3 Symmetry is all around us and is a fundamental property of nature. 4 5 6
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Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy).
Predict IR spectra or Interpret UV-Vis spectra Predict optical activity of a molecule
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Molecular Symmetry• Symmetry impacts
• Physical properties• Reactions• Molecular orbitals• Electronic structure• Molecular vibrations
• Group theory• Behavior of molecule based on symmetry
• Symmetry analysis• Application of symmetry• Orbital symmetry• Vibrational symmetry
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Symmetry is all around us and is a fundamental property of nature.
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Facial symmetry
No! Yes7
Invariance to transformation as an indicator of facial symmetry:
Mirror image8
What is symmetry?
invariance* to transformation in space
nature of the transformation determines the type of symmetry
determines crystal packing, orbital overlap, spectroscopic properties
*an invariance (meaning in mathematics and theoretical physics)is a property of a system which remains unchanged under some transformation.
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Symmetry operations and elements
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Symmetry operations and elements
Group Theory: mathematical treatment of symmetry.
Symmetry Operation = an operation performed on an object which leaves it in a configuration that is indistinguishable from the original appearance (or action which molecular symmetry unchanged). e.g. rotation through an angle, reflection.
Symmetry Elements = the points, lines, or planed to which a symmetry operation is carried out.
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Element Operation SymbolIdentity Identity E
Symmetry plane Reflection in the plane
Inversion center Inversion of a point x, y,
z to –x,-y,-z
i
Proper axis Rotation by (360/n) Cn
Improper axis 1. Rotation by (360/n)2. Reflection in plane perpendicular to rotation axis
Sn
Symmetry operations and elements
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1. Identity operation (E)
The identity operation, E, states that the object exists
Also denoted as C1 or Cnn
Its existence is demanded by the math of group theory, and common sense
CHFClBr D-Glucose and L-Phenylalanine
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2. Planes and Reflection (σ) or Mirror plane (m)
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symmetry element is a plane all of the points of a molecule are passed
through the plane
Eyes glasses
hands
chair
Mirror plane
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H2O
Mirror plane
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Reflection across a plane of symmetry, (mirror plane)
v
v
Handedness is changed by reflection!
2 = E
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Examples: Mirror plane
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3. Inversion, Center of Inversion (i) • The inversion operation takes a points through the center of symmetry of the molecule to an equal distance on the other side.• A point at the center of the molecule
•(x, y, z) ---> (-x, -y, -z)
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Center of Inversion (i)
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Examples: Center of Inversion (i)
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Examples: Center of Inversion (i)
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4. Proper axes of rotation (Cn)
rotation by an angle , such that n = 360°
n is the order of the rotation Cn
the symmetry element is a line, about which the rotation takes place
a Cn axis generates n operations, which form a cyclic group or subgroup i. e. C4 generates C4
1, C42=C2, C4
3, C44=E 23
Rotational symmetry
n = 2 n = 5 n = 6
360/2 360/5 360/6
180o 72o 60o
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Examples: Proper axes of rotation (Cn)
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120 120
180
180
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Cnm = E
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C42 = C2
1C41
C43
Note: C42 = C2
1, C62 = C3
1, C63 = C2
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Sn - where n indicates the order of the rotation-composed of two successive geometry transformations: first, a rotation through 360°/n about the axis of that rotation (Cn), and second, reflection through a plane perpendicular to Cn (h).
symmetry element is both a line and a plane
5. Rotation-reflection, Improper axis (Sn)
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Improper axis (Sn)
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Improper axis (Sn)
CH4
1) 90 rotation
2) reflection
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CH4
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n-fold improper rotation, Snm
S41
90° h
Note that: S1 = , S2 = i, and sometimes S2n = Cn (e.g. in box) this makes more sense if you examine the final result of each of the operations.
S41
S42
C21
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rotation axes and mirror plane molecules: (H2O)
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rotation axes and mirror plane molecules: (C6H6)
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rotation axes and mirror plane molecules: (BF3)
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rotation axes and mirror plane molecules
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Examples for the different basic symmetry operations and symmetry elements
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Point Groups
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Point Groups
Point Group = the set of symmetry operations for a molecule.
Group Theory = the mathematical treatment of the properties of groups, can be used to determine the molecular orbitals, vibrations, and other properties of the molecules.
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Non‐rotation group (Low Symmetry)
Group Symmetry Examples
C1 E CHFClBr
Cs E, h H2C=CClBr
Ci E, i HClBrC=CHClBr
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Single axis groupGroup Symmetry Examples
Cn E, Cn H2O2
Cnv E, Cn , nσv H2O
Cnh E, Cn , σh, Sn, i B(OH)3
Cn subgroup + h
Cn subgroup + nv
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Cn Point Groups
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Cnv Point Groups
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Cnh Point Groups
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Group Symmetry Examples
S2n E, S2n 1,3,5,7-tetrafluoracyclooctatetrane
C∞v E, C∞, ∞σv HCl
Single axis group
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Sn Point Groups
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Dihedral groupsGroup Symmetry Examples
Dn E, Cn, nC2 NiN6
Dnd E, Cn, nC2, S2n S8
Dnh E, Cn, nC2,σh, nσv
D∞h E, i, C∞v, ∞σv
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adding a C2 perpendicular to the Cn requires that there must be n C2 axes perpendicular to the Cn
Dn: Cn + nC2 axes (molecules in this group must have a zero dipole moment and be optically active
Dnh: Dn + h
Dnd : Dn + v. The v operations will bisect the adjacent C2 axes
Dihedral groups
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Dn and Dnh Point Groups
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h
v57
Dnd Point Groups
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Cubic groupsGroup Symmetry Examples
Td E, C3, C2, S4, σd CCl4
Oh E, C3, C2, C4, i, S4, S6, σd, σh
Ih E, C3, C2, C5, i, S10, S6,σ
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Platonic solids
High Symmetry molecules
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or E, C2, 2v
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Point Groups
p-dichlorobenzene: E, 3, 3C2, i
Ethane (staggered): E, 3, C3, 3C2, i, S6
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Perpendicular C2 axes
Horizontal Mirror Planes
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Vertical or Dihedral Mirror Planes and S2n Axes
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Application of Symmetry
• Construction and labeling of molecular orbitals• Molecular properties
• Polarity• Chirality
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Polar Molecule
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Chiral Molecules
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Properties and Representations of Groups
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Representation of Point Groups Matrices Why Matrices? The matrix representations of
the point group’s operations will generate a character table.
We can use this table to predict properties.
Cij = Aik x Bkj
Cij = product matrix; i rows and j columnsAik = initial matrix; i rows and k columnsBkj = initial matrix; k rows and j columns
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1) Choose set of x,y,z axes- z is usually the Cn axis- xz plane is usually the plane of the
molecule2) Examine what happens after the molecule
undergoes each symmetry operation in the point group (E, C2, 2)
1. Matrix Representations of C2v
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E Transformation Matrix
x,y,z x,y,z
What matrix times x,y,z doesn’t change anything?
E Transformation Matrix
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C2 Transformation Matrix
x,y,z -x, -y, z
Correct matrix is:
v(xz) Transformation Matrix
x,y,z x,-y,z
Correct matrix is:
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These 4 matrices are the “Matrix Representation” of the C2v point group
All point group properties transfer to the matrices as well
Example: Ev(xz) = v(xz)
v(yz) Transformation Matrix
x,y,z -x,y,z
Correct matrix is:
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E C2 v(xz) v(yz)
3 -1 1 1
2. Reducible and Irreducible Representations
1). Character = sum of diagonal from upper left to lower right (only defined for square matrices)
• The set of characters = a reducible representation ()or shorthand version of the matrix representation
•For C2v Point Group:
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E C2 v(xz) v(yz)
2). Reducible and Irreducible Representations
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Axis used
E C2 v(xz) v(yz)
x 1 ‐1 1 ‐1
y 1 ‐1 ‐1 1
z 1 1 1 1
3 ‐1 1 1
IrreducibleRepresentations
Reducible Repr.
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C4v E 2C4 (z) C2 2sv 2sd
A1 1 1 1 1 1 z x2+y2, z2 z3, z(x2+y2)
A2 1 1 1 -1 -1 Rz - -
B1 1 -1 1 1 -1 - x2-y2 z(x2-y2)
B2 1 -1 1 -1 1 - xy xyz
E 2 0 -2 0 0 (x, y) (Rx, Ry) (xz, yz) (xz2, yz2) (xy2, x2y) (x3, y3)
Point groupClasses of symmetry operations
Symmetry or Mulliken labels, each corresponding to a different irreducible representation
Characters (of the IRs of the group)“1” indicate that operation leave the function unchange“-1” indicate that operation reverses the function
Basis functions having the same symmetry as the IR
linear functionstranslations along specified axisR, rotation about specified axis
quadratic functionscubic functions
Totally symmetric representation of the group
Symmetries of the s, p, d, and f orbitals can be found here (by their labels). Ex: the dxy orbital shares the same symmetry as the B2 IR.The s orbital always belongs to the totally symmetric representation (the first listed IR of any point group).
Character Tables
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Irreducible Representation Labels
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Character Tables
R = any symmetry operation= character (#)i,j = different representations (A1, B2, etc…)h = order of the group (4 total operations in the C2v case
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87 88
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The C3v character table
Irreducible representations
Symmetry operations
The order h is 6There are 3 classes
NH3
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The complete C4v character table
A1 transforms like z.E does nothing, C4 rotates 90o about the z-axis, C2 rotates 180o
about the z-axis, v reflects in vertical plane and d in a diagonal plane.
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A2 transforms like a rotation around z.
E+Rz
C4+Rz
C2+Rz
v-Rz
d-Rz
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Molecular Vibrations
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Molecular Vibrations
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• Electron repulsion• Bond breaking• Vibrational modes
• Depends upon number of atoms and degrees of freedom(*Degrees of Freedom = possible atomic movements in the molecule)
- 3N degrees of freedom for a molecule of N atoms
Theory
• Constraints due to• Translational and rotational motion of molecule• Motion of atoms relative to each other
- Linear moleculesOnly 2 rotations change the molecule 3N – 5 vibrations
- Nonlinear molecules3 translations (along x, y, z)
3 rotations (around x, y, z)3N – 6 vibrations 102
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Selection Rules: Infrared and Raman Spectroscopy
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Water molecule (C2V)
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Reducible Representations
105 106
Reducible to Irreducible representation
9x9 vector
The other entries for can also be found without the matrices
E: all 9 vectors are unchanged--> 9C2: H atoms change position in C2 rotation, so all vectors have zero contribution to the character. O atom vectors in x and y are reversed, each contributing -1 and in z direction is the same, contributing 1. --> (C2) = (-1)+(-1)+1 = -1v(xz): reflection in the plane of the molecule changes the direction of all the y vector, the x and z are unchanged. ---> 3-3+3 = 3.v(yz): reflection perpendicular to the plane of the molecule changes the position of H atoms so their contribution is zero, the x vector on O changes direction, the y and z are unchanged. ---> -1+1+1 = 1
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Reducible representation of H2O
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Reducible to Irreducible representation
ap= nR(R)p(R)
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Reducible to Irreducible representation
H2O
3A1 + A2 + 3B1 + 2B2
ap= nR(R)p(R)
B1
B2
irrep.
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111 112
IR and Raman Active
113 114
B2
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A1
B1
B2
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B2 1
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117 118
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Vibrational modes of SO2 (C2v)
121 122
BCl3
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