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CHEM 515 Spectroscopy. Lecture # 8 Molecular Symmetry. Molecular Symmetry. Group theory is an important aspect for spectroscopy. It is used to explain in details the symmetry of molecules. Group theory is used to:
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CHEM 515Spectroscopy Lecture # 8 Molecular Symmetry
Molecular Symmetry • Group theory is an important aspect for spectroscopy. It is used to explain in details the symmetry of molecules. • Group theory is used to: • label and classify molecule’s energy levels / molecular orbitals (electronic, vibrational and rotational) • look up the possibility of molecular and electronic transitions between energy levels / molecular orbitals.
Symmetry Operations • A symmetry operation is geometrical action that leaves the nuclei in a molecule in equivalent positions. (leaves them indistinguishable). • Five main classes of symmetry operations: • Reflections (σ). • Rotation (Cn). • Rotation-reflection “Improper rotation” (Sn). • Inversion (i). • Identity (E). “do nothing”
Operator Algebra • Operator algebra is similar in many aspects to ordinary algebra. • For: Af1 f2 , operator A is said to transform functions f1 to f2 by a sort of operation. • Addition of operators: Cf = (A + B)f = Af + Bf or C = (A + B) = A + B
Operator Algebra • Multiplication of operators: Cf = (AB)f = A(Bf) or C = (AB) = AB However, it is important to note that: A(Bf) is not necessarily equivalent to B(Af). We say operators A and B don’t necessarily commute.
Operator Algebra • Example: For x = x andD = d/dx ,does Dx = xD ? • Associative law and distributive laws both hold for operators “see book”.
Identity Operator (E) • The identity operator leaves a molecule unchanged. It is applied for all molecule with any degree of symmetry or asymmetry. • It is important not by itself but for specific operator algebra as going to be discussed later.
Rotation Operator (Cn) • Cn rotates a molecule by an angle of 2π/n radians in a clockwise direction about a Cn axis. • If a rotation of 2π/n leaves out the molecule indistinguishable, the molecule is said to have an n-fold axis of rotation. C2 Rotation by 2π/2 radians 1 2 2 1
Rotation Operator (Cn) • When a molecule has several rotational axes of symmetry, the one with the largest value of n is called the principle axis. Example: Trifluoroborane
Rotation Operator (Cn) • Successive Rotations (Cnk). Cnk = Cn Cn … Cn(k times) Also: Cnn = E • Example: BF3 1 3 2 2 1 3 Rotation by 4π/3 radians C32 Rotation by 2π/3 radians C3 3 2 1 Rotation by - 2π/3 radians C3-1
Reflection Operator (σ) • σ reflects a molecule through a plane passing through the center of the molecule. The molecule is said to have a plane of symmetry. C2 σv Reflection through σv plane 1 2 2 1
Reflection Operator (σ) • σ reflects a molecule through a plane passing through the center of the molecule. The molecule is said to have a plane of symmetry.
Reflection Operator (σ) • There are three types of mirror planes: • σv vertical mirror plane which contains the principle axis. • σh horizontal mirror plane which is perpendicular to the principle axis. • σd dihedral mirror plane which is vertical and bisects the angle between two adjacent C2 axes that are perpendicular to the principle axis.
Improper Rotation Operator (Sn) • This operator applies a clockwise rotation on the molecule followed by a reflection in a plane perpendicular to that axis of rotation. Sn = σhCn • Example: Methane σh C4 S4
Inversion Operator (i) • This operator inverts all atoms through a point called “center of inversion” or “center of symmetry”. i (x,y,z) (-x,-y,-z)
Symmetry Operator Algebra • Symmetry operators can be applied successively to a molecule to produce new operators. σv’’’ = σv’’ C3 σv’ = C3 σv’’
Group Multiplication Tables • A group multiplication must satisfy the following conditions in regard with the group’s elements: 1- Closure: If P and Q are elements of a group and PQ = R , then R must be also an element of that group. 2- Associative Law: The order of multiplication is not important. (PQ)R = P(QR). 3- Identity Element: There must be an identity element (E) in the group so that: RE = ER = R.
Group Multiplication Tables 1- Closure: If P and Q are elements of a group and PQ = R , then R must be also an element of that group. 2- Associative Law: The order of multiplication is not important. (PQ)R = P(QR). 3- Identity Element: There must be an identity element (E) in the group so that: RE = ER = R. 4- Inverse: Every element has an inverse in the group so that: RR-1 = R-1R = E .
Group Multiplication Tables 5- If the group elements commute, i.e. PQ = QP, then the group is said to be “Abelian group”. For “point symmetry groups”, we have non-Abelian groups. • “Point groups” retain the center of mass of the molecules under all symmetry operations unchanged and all of the symmetry elements meet at this point
Point Group for Ammonia • The ammonia molecule has six symmetry operators. E , C3, C3-1 (or C32), σv’ , σv’’ and σv’’’
Multiplication Table for NH3 Notice that: • Each operator appears just once in a given row or column in the table but in a different position.
Classes • The members of a group can be divided into classes. The members of a class within a group have a certain type of a geometrical relationship. For ammonia with the C3v symmetry, the three classes are: E , C3 and σv • The point group C3v will contain E , 2C3 and 3σvelements.
Point Groups and Symmetry for Various Molecules C1 Symmetry (only E) Cs Symmetry
