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PG510

Symmetry and Molecular Spectroscopy

Lecture no. 1

Group Theory:

Definitions and Theorems

Giuseppe Pileio

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Introduction to the course

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About Me

Name: Giuseppe Surname: Pileio

Nickname: Peppe

Location: 30/3047, School of Chemistry

E-mail: g.pileio@soton.ac.uk Phone: 023 80 59 4146

Web: http://www.mhl.soton.ac.uk/~peppe

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About You

! Fill Form A with your details and past experiences

! Use Form B throughout the course to annotate crucial sentences

! Use Form C throughout the course to give me comments and feedbacks on your understanding level

! Keep the molecular model in the envelope for the full duration of the course. It will help you in understanding

! Finally, note that to get the 12 credits only attendance to all the lectures and workshops is needed. No exams required!

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Course plan: Overview

Group Theory

Molecular Spectroscopy

Symmetry Groups

Symmetry Operations

Vibrational IR Raman IR and MW

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Course plan details: Part I

Part I: Group Theory (5 lectures + 1 workshop)

1- Group Theory

2- Molecular Symmetry

3- Point Groups

4- Representations of groups

5- Character tables

Workshop 1 (group theory and molecular symmetry)

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Part II: Molecular Spectroscopy (7 lectures + 2 workshops)

6- Introduction to spectroscopy

7- Rotational, Vibrational and Vibro-rotational Spectroscopy

8- Vibrational Spectroscopy of polyatomic molecules

9- Raman Spectroscopy

Workshop 2 (about Rotational, Vibrational and Raman Spectroscopy)

10-12 Electronic Spectroscopy

Workshop 3 (about MO and Electronic Spectroscopy)

Course plan details: Part II

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Plan of action

Lectures are introductory to the subject and medium-level. Only basic math is required. Quantum mechanics concepts have been avoided as much as possible and when present a deep understanding is not required

A molecular model is given to you to be used throughout the course (bring it to all lectures and workshops). Very simple tasks on that will be asked as homework

Workshops are intended as an application of these concepts. Most likely an exercise will be done by me on the board and some others by you (working in groups and with my help). Those papers will not be marked although I will correct them

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Crucial concepts will be highlight and collected by you in Form B as an useful reminder (bring it to all lectures and workshops)

Form C (one for each lecture) is intended to help me to figure out your understanding level of the concepts discussed so to, eventually, come back to these concept in a clearer way or address them in other circumstances

Form A is intended to help me in understanding your starting level and your knowledge on the subject in order to better meet your requirements

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textbooks

Introductive:

Molecular Spectroscopy (Oxford Chemistry Primers), J. M. Brown, ISBN 978-0198557852

Foundation of Spectroscopy (Oxford Chemistry Primers), S. Duckett and B. Gilbert, ISBN 978-0198503354

Fundamental:

Chemical Applications of Group Theory, F. A. Cotton (3rd ed.), ISBN 9971-51-267-X

Deeper Understanding:

Symmetry and Spectroscopy D. C. Harris and M. D. Bertolucci, ISBN 0-486-66144-X

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Overall Learning Outcomes

By the end of this course you will be able to:

!! Understand the fundamentals of Group Theory and apply it to molecular spectroscopy

!! Understand the link between molecular spectroscopy, symmetry and information content of molecular spectra

!! Calculate/predict energy levels and spectral features using symmetry as a simplification tool

!! Use symmetry arguments to possibly solve molecular problems

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Lecture no. 1

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Groups

What is a group?

A Group is a collections of elements that follow these rules:

1.! The combination of any two elements of the group is itself a member of the group

2.! One element must commute with all the others leaving them unchanged

A * B = C and C belongs to the same group as A and B

E * A = A * E = A

This element, E is usually called identity element

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4.! Every element must have a reciprocal which is still an element of the same group

3.! Combination must be associative

A * (B *C) = (A * B) * C

A * X = X * A = E

! A, X and E belong to the same group (rule 1) ! If X is the reciprocal of A then A is the reciprocal of X ! E is the reciprocal of itself i.e. E * E = E ! (A * B ** Z)-1 = Z-1 ** B-1 * A-1

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Abelian Groups

If a 5th rule is also fulfilled we call the group Abelian Group or Commutative Group:

5.! The combination of two element is commutative

A * B = B * A

Note: Although very common in many everyday mathematical applications this rule generally does not hold in Group Theory!

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Properties of Groups

! The number of element in the group is called order and is indicated by h

In the simplest h=3 group (G3):

A A*A=B A*A*A=A*B=E

! A group is called Cyclic if it is made by only one element and its h powers

! Smaller group found in a group are called subgroups and their order is indicated with g so that h/g=k with k an integer

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Similarity Transformations

A similarity transformation is present if A and X are both elements of the group G and

B = X-1 * A * X

with B being another element of G (rule 1). In words we say:

B is the similar transform of A by X

or

A and B are conjugate

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Properties of conjugation

! Every element is conjugate with itself

! If A is conjugate with B then B is conjugate with A

! If A is conjugate with B and C then B and C are also conjugate with each other

A = X-1 * A * X

A = X-1 * B * X

B = Y-1 * A * Y with Y = X-1 and Y-1 =X

if A = X-1 * B * X and A = Y-1 * C * Y

Then B = Z-1 * C * Z

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Classes of elements

A complete set of elements that are conjugate to one another is called a class of the group

The order of each class must be integer factor of h and their sum must be h

To figure out all the classes within a group we have to take every element of the group and work out the similarity transform with all the other element. The elements conjugate each other will form a class

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Examples of Groups: The Integers with respect to +

Lets take the collection of all the integers and lets choose the sum (+) as a rule for the combination of these elements. Do they form a group?

The group is also Abelian since the sum is commutative

Rule 4 is satisfied if we choose k as the reciprocal of any element k since: k + (-k) = 0

Rule 3 holds since the sum is obviously associative

Rule 2 is satisfied if we choose 0 as identity: 0 + k = k + 0 = k (with k any integer)

Rule 1 is satisfied since the sum of any two integers give another integer i.e. another element of the group

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Examples of Groups: The Integers with respect to x

Lets take the integers again but choose the simple product (x) as a rule for the combination. Do they still form a group?

The group is also Abelian since x is commutative

Rule 4 is satisfied if we choose 1/k as the reciprocal of any element k since: k x (1/k) = 1

Rule 3 holds since the product is obviously associative

Rule 2 is satisfied if we choose 1 as identity: 1 x k = k x 1 = k (with k any integer)

Rule 1 is satisfied since the product of any two integers give another integer i.e. another element of the group

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What did we learn in this lecture?

! The concept of group

! The rules that make a collection of elements be a group

! Some properties of groups

! The concept of similarity transform and its properties

! The concept of class

! Examples of groups