1 The next two parts of the course are closely related, though at first it may not seem so.

What we hope to answer How to recognize if a molecule is chiral What vibrational modes in a molecule are IR or Raman active (visible) How to predict the molecular orbital energy levels based on atomic orbitals, without doing a lot of math 2

Which is more symmetrical? pyramidal trigonal planar Square planar octahedral Chapter 6 “ Molecular Symmetry ”

…….although you might have an intuitive sense as to which of these molecules is most symmetrical, the answer can be succinctly described using formalized rules, by determining the molecular point group…... Knowledge of a molecule ’ s point group, i.e., symmetry, allows you to: construct a molecular orbital (MO) diagram predict the number of unpaired electrons predict the number of IR stretches ………..etc.

Which is more symmetrical? C 3v (5 symmetry elements) D 3h (10 symmetry elements) D 4h (14 symmetry elements) OhOh (31 symmetry elements) Point group associated symmetry elements the more symmetrical molecules possess more symmetry elements

6 Group theory is a mathematical analytic method that describes the relationship between geometrical objects and operations performed on them. A mathematical group consists of a set of operations that, in combination on an object, do not result in changes to the object. Thus these operations can be referred to as symmetry elements.

In order to be able to do this, we ’ ll need to know how to determine a molecule ’ s symmetry properties A molecule ’ s symmetry properties can be summarized by one symbol called the “ Point Group ” symbol Once we know a molecule ’ s “ Point Group ” then we know the symmetry of all of its orbitals Orbitals can overlap (bond) only if they have identical symmetry

What is a symmetry element? Symmetry elements are used to perform operations that relate equivalent nuclei, or transform a molecule into itself There are four distinct symmetry elements: 1. mirror planes (  ) 2. rotation axes (C n ) 3. inversion centers (i) 4. improper rotation axes (S n = C n +  )

SymmetrySymbolOperation Element 1. Identity E nothing! 2. Rotation Axis C n rotation 4. Inversion center i reflection thru a point C 2 = 180˚ 360/n˚ C 3 = 120˚ C 4 = 90˚ C 5 = 72˚ C 6 = 60˚ 3. Mirror Planes  reflection  v,  h,  d

Rotation about an axis 360/n˚, n= 2 CnCn example labeled atom has moved molecule looks the same (ie it is in the exact same orientation)

Rotation about an axis 360/n˚, n= 3 CnCn example labeled atom has moved

Rotation about a Rotation Axis C n = 360˚/n Note: the symmetry axis goes through the molecule CnCn example

Rotation about a Rotation Axis C n = 360˚/n CnCn example

Rotation about a Rotation Axis C n = 360˚/n CnCn example L5L5 L6L6 L4L4 L1L1 L2L2 L3L3 L6L6 L4L4 L5L5 L2L2 L3L3 L1L1

SymmetrySymbolOperation Element 1. Identity E nothing! 2. Rotation Axis C n rotation 3. Mirror Planes  reflection C 2 = 180˚ 360/n˚ C 3 = 120˚ C 4 = 90˚ C 5 = 72˚ C 6 = 60˚  v,  h,  d ⊥ highest-fold axis contain highest-fold axis

reflection thru a mirror plane σnσn example labeled atom has moved all atoms lie in this mirror plane (differs from that shown above), therefore nothing is transformed by the operation hh plane bisects molecular angle

Reflection through a Mirror plane σnσn example

SymmetrySymbolOperation Element 1. Identity E nothing! 2. Rotation Axis C n rotation 3. Mirror Planes  reflection 4. Inversion center i reflection thru a point C 2 = 180˚ 360/n˚ C 3 = 120˚ C 4 = 90˚ C 5 = 72˚ C 6 = 60˚  v,  h,  d

reflection thru a point i example (x, y, z)(–x, –y, –z)

A molecule which does NOT possess an inversion center (i) molecule is in a different orientation

Distinguishing two different operations note how the color labels (used for book-keeping purposes to track the atoms) differ for the two operations

5. Improper Rotation S n rotation C n (rotation axis followed by + mirror plane) reflection thru the   to C n SymmetrySymbolOperation Element

Improper Rotation S 4 rotation 360/n˚, followed by reflection thru a mirror plane perpendicular to C n example SnSn molecule is in a different orientation. Therefore, the molecule does not possess an indpendent C 4 axis molecule is now in the original orientation, thus C 4 in combination with perpendicular mirror= symm element

rotation 360/n˚, followed by reflection thru a mirror plane perpendicular to C n example SnSn C4C4 C4C4 note how this plane is perpendicular ( ⊥ ) to the C 4 axis note how this plane is perpendicular ( ⊥ ) to the C 4 axis

C 2v = point group key symmetry elements: C 2 axis C 2 axis 2 mirror planes 2 mirror planes Symmetry Properties of H 2 O molecule

a closed mathematical group which describes the symmetry of an object What is a Point Group ? e.g., T d = symbol which summarizes all of the symmetry elements associated with molecules such as CH 4. there are 47 point groups, which can be divided into 14 distinct classes (listed by character table in your text pp )

How do we identify a molecule ’ s point group? Identify all of the symmetry elements. Identify just the key symmetry elements. Or With the aid of a set of character tables With the aid of a flow chart Or

Flow chart used for assign a molecule to a point group

Character Table describing C 2v Point Group

symmetry label orbital function

31 For this section of the course, you ’ ll find the textbook website ’ s “ 3D rotatable molecules ” link useful, as well as the “ Tables for Group Theory ” Shriver & Atkins textbook web site

point group symmetry elements Character Table describing C 3v Point Group

33 point group more symmetry elements Character Table describing D 3h Point Group

34 even more symmetry elements Character Table describing D 4h Point Group point group

35 point group even more symmetry elements Character Table describing O h Point Group

n=2 point group n=2, 3, 4, 5, 6