MOLECULAR SYMMETRY, GROUP THEORY, &

MOLECULAR SYMMETRY, GROUP THEORY, & APPLICATIONS
Lecturer: Claire Vallance (CRL office G9, phone 75179, e-mail [e-mail])
These are the lecture notes for the second year general chemistry course named ‘Symmetry I’ in the course outline. They contain everything in the lecture slides, along with some additional information. You should, of course, feel free to make your own notes during the lectures if you want to as well. If anyone would desperately like a copy of the lecture slides, e-mail me at the end of the course and I’ll send you one (the file is about 2MB in size).
At some point after each lecture and before the next, I STRONGLY recommend that you read the relevant sections of the lecture handout in order to consolidate the material from the previous lecture and refresh your memory. Most people (including me!) find group theory quite challenging the first time they encounter it, and you will probably find it difficult to absorb everything on the first go in the lectures without doing any additional reading. The good news is that a little extra effort on your part as we go along should easily prevent you from getting hopelessly lost!
If you have questions at any point, please feel free to ask them either during or after the lectures, or contact me by e-mail or in the department (contact details above).
Below is a (by no means comprehensive) list of some textbooks you may find useful for the course. If none of these appeal, have a look in your college library, the Hooke library or the RSL until you find one that suits you.
Atkins - Physical Chemistry Atkins - Molecular Quantum Mechanics Ogden – Introduction to Molecular Symmetry (Oxford Chemistry Primer) Cotton – Chemical Applications of Group Theory Davidson – Group Theory for Chemists Kettle – Symmetry and Structure Shriver, Atkins and Langford – Inorganic Chemistry Alan Vincent – Molecular Symmetry and Group Theory (Wiley)
Also, to get you started, here are a few useful websites. I’m sure there are many more, and if you find any others you think I should include, please e-mail me and let me know so I can alert future generations of second years.
http://www.reciprocalnet.org/edumodules/symmetry/intro.html (a good tutorial on point groups and some aspects of symmetry and group theory, with lots of 3D molecular structures for you to play with)
http://www.chemistry.nmsu.edu/studntres/chem639/symmetry/group.html (a helpful applet providing character tables and reduction of representations, which you’ll know all about by about lecture 5 of this course)
NOTE: A PROBLEM SHEET IS ATTACHED TO THE END OF THIS HANDOUT
Contents
1.
Introduction
2.
Symmetry operations and symmetry elements
3.
Symmetry classification of molecules – point groups
4.
Symmetry and physical properties
4.1.
Polarity
4.2.
Chirality
5.
Combining symmetry operations: ‘group multiplication’
6.
Constructing higher groups from simpler groups
7.
Mathematical definition of a group
8.
Review of Matrices
8.1.
Definitions
8.2.
Matrix algebra
8.3
Direct products
8.4.
Inverse matrices and determinants
9.
Transformation matrices
10.
Matrix representations of groups
10.1.
Example: a matrix representation of the C3v point group (the ammonia molecule)
10.2.
Example: a matrix representation of the C2v point group (the allyl radical)
11.
Properties of matrix representations
11.1.
Similarity transforms
11.2.
Characters of representations
12.
Reduction of representations I
13.
Irreducible representations and symmetry species
14.
Character tables
15.
Reduction of representations II
15.1
General concepts of orthogonality
15.2
Orthogonality relationships in group theory
15.3
Using the LOT to determine the irreps spanned by a basis
16.
Symmetry adapted linear combinations
17.
Determining whether an integral can be non-zero
18.
Bonding in diatomics
19.
Bonding in polyatomics - constructing molecular orbitals from SALCs
20.
Calculating the orbital energies and expansion coefficients
21.
Solving the secular equations
21.1
Matrix formulation of a set of linear equations
21.2
Solving for the orbital energies and expansion coefficients
22.
Summary of the steps involved in constructing molecular orbitals
23.
A more complicated bonding example – the molecular orbitals of H2O
23.1
Matrix representation, characters and SALCs
24.
Molecular vibrations
24.1
Molecular degrees of freedom – determining the number of normal vibrational modes
24.2
Determining the symmetries of molecular motions
24.3
Atomic displacements using the 3N Cartesian basis
24.4
Molecular vibrations using internal coordinates
25.
Summary of applying group theory to molecular motions
26.
Group theory and molecular electronic states
27.
Spectroscopy – interaction of atoms and molecules with light
27.1
Electronic transitions in molecules
27.2
Vibrational transitions in molecules
27.3
Raman scattering
28.
Summary
29.
Appendix A – a few proofs for the mathematically inclined
30.
Appendix B – Character tables and direct product tables
Problem sheet
3

1.
Introduction
You will already be familiar with the concept of symmetry in an everyday sense. If we say something is ‘symmetrical’, we usually mean it has mirror symmetry, or ‘left-right’ symmetry, and would look the same if viewed in a mirror. Symmetry is also very important in chemistry. Some molecules are clearly ‘more symmetrical’ than others, but what consequences does this have, if any? The aim of this course is to provide a systematic treatment of symmetry in chemical systems within the mathematical framework known as group theory(the reason for the name will become apparent later on). Once we have classified the symmetry of a molecule, group theory provides a powerful set of tools that provide us with considerable insight into many of its chemical and physical properties. Some applications of group theory that will be covered in this course include:


i) Predicting whether a given molecule will be chiral, or polar.
ii) Examining chemical bonding and visualising molecular orbitals.
iii) Predicting whether a molecule may absorb light of a given polarisation, and which spectroscopic
transitions may be excited if it does.
iv) Investigating the vibrational motions of the molecule.


You may well meet some of these topics again, possibly in more detail, in later courses (notably Symmetry II, and for the more mathematically inclined amongst you, Supplementary Quantum Mechanics). However, they will be introduced here to give you a fairly broad introduction to the capabilities and applications of group theory once we have worked through the basic principles and ‘machinery’ of the theory.
2.
Symmetry operations and symmetry elements
A symmetry operation is an action that leaves an object looking the same after it has been carried out. For example, if we take a molecule of water and rotate it by 180° about an axis passing through the central O atom (between the two H atoms) it will look the same as before. It will also look the same if we reflect it through either of two mirror planes, as shown in the figure below.
reflection
rotation
reflection
(operation) (operation) (operation)
axis of symmetry
(element) mirror plane
(element) mirror plane(element)
Each symmetry operation has a corresponding symmetry element, which is the axis, plane, line or point with respect to which the symmetry operation is carried out. The symmetry element consists of all the points that stay in the same place when the symmetry operation is performed. In a rotation, the line of points that stay in the same place constitute a symmetry axis; in a reflection the points that remain unchanged make up a plane of symmetry. The symmetry elements that a molecule may possess are:
1.
E - the identity. The identity operation consists of doing nothing, and the corresponding symmetry element is the entire molecule. Every molecule has at least this element.
2.
Cn - an n-fold axis of rotation. Rotation by 360°/n leaves the molecule unchanged. The H2O molecule above has a C2 axis. Some molecules have more than one Cn axis, in which case the one with the highest value of n is called the principal axis. Note that by convention rotations are counterclockwise about the axis.
3.
σ - a plane of symmetry. Reflection in the plane leaves the molecule looking the same. In a molecule that also has an axis of symmetry, a mirror plane that includes the axis is called a vertical mirror plane and is labelled σv, while one perpendicular to the axis is called a horizontal mirror plane and is labelled σh. A vertical mirror plane that bisects the angle between two C2 axes is called a dihedral mirror plane, σd.
4.
i - a centre of symmetry. Inversion through the centre of symmetry leaves the molecule unchanged. Inversion consists of passing each point through the centre of inversion and out to the same distance on the other side of the molecule. An example of a molecule with a centre of inversion is shown below.
5.
Sn - an n-fold improper rotation axis (also called a rotary-reflection axis). The rotary reflection operation consists of rotating through an angle 360°/n about the axis, followed by reflecting in a plane perpendicular to the axis. Note that S1 is the same as reflection and S2 is the same as inversion. The molecule shown above has two S2 axes.
The identity E and rotations Cn are symmetry operations that could actually be carried out on a molecule. For this reason they are called proper symmetry operations. Reflections, inversions and improper rotations can only be imagined (it is not actually possible to turn a molecule into its mirror image or to invert it without some fairly drastic rearrangement of chemical bonds) and as such, are termed improper symmetry operations.
A note on axis definitions: Conventionally, when imposing a set of Cartesian axes on a molecule (as we will need

النتائج (العربية) 3:

و symmetry , Group theory , & التطبيقات
): واضحة vallance )النطق مكتب g9, والتى 75179, البريد الالكترونى [e-mail] )
These are The قراءة الملاحظات For The General chemistry الثانى year سباق named "symmetry الاول" in The سباق العام. لمدة on الغاية in The قراءة slides, Along additional " Some المعلومات. فيا should, of السباق,Nonwhites Free to make ) عون الحواشى during The القراءات if فيا change to as well. if anyone الواردة desperately like مسبقة وقد OF THE قراءة slides, البريد الالكترونى لى at the الاسبوع OF THE سباق and I اليوم رم " One ( The فيا القيادة is about 2mb in الالكترونى). Some
at each قراءة البند After before and The next,الاول strongly فيا to that " The ضمن الفروع OF THE قراءة handout in Order ( The to consolidate from previous قراءة and The refresh ) Die. Most People ) including لى ولدى find Group Theory quite challenging The First Time لمدة زهاء IT- , and فيا will probably find IT- difficult to absorb الغاية يمكن The First go in The القراءات"الافراط any Reading اضافية.The Good that is News , Little الخارجى ) جهدا من جانب as we go Along should easily prevent فيا from Getting hopelessly Lost! فيا
if اذا المسائل at any البند, اسم Nonwhites to Free بعديد them ان during الذهب After The القراءات الاتصال, بيد اننى by البريد الالكترونى ولكن in The Department )الاتصال details above .
الاطلاع is by (رقم بالمعاهدات Comprehensive) List of some المدرسية فيا ماى find useful For The السباق. لا شيء if Appeal of these, اذا كان in ) انظر كلية Library , The hooke library الذهب The RSL وهو فيا find One that suits فيا.
تقدم - physical chemistry تقدم - و مقدار Mechanics شركة: مقدمة to و symmetry )اكسفورد chemistry يسود) cotton - التطبيقات of chemical Group Theory Davidson - Group for نظرية chemists kettle - symmetry and هيكل shriver, تقدم and langford - : chemistry Alan فنسنت و - symmetry theory and Group (Wiley)
also , to get فيا started,وقد here 'are useful شبكة. اليوم الاول م المنظمة there are more مانى, if and others فيا find any فيا think الاول should " , اسم البريد الالكترونى لى and يوليه اننى اعرف can ومنظمة الاشعار الدولية سو الاول مستقبلا الثانى Years of البشرية.
http://www.reciprocalnet.org/edumodules/symmetry/intro.html Good - ), ان البند groups and some of جوانب symmetry theory and Group ," 3 D شحنات of و الهياكل for فيا to Play " (
http://www.chemistry.nmsu.edu/studntres/chem639/symmetry/group.html helpful applet ), انظر الجداول and Providing reduction of representations , which فيا اليوم رم know about all by about قراءة 5 of this سباق)
ملاحظة: Sheet is a problem attached to End The of this handout
المحتويات
.
مقدمة 1 2
.
symmetry Operations and symmetry عناصر
3.
symmetry تصنيف of molecules - البند 4 Groups
.
symmetry and physical Properties
4.1.
polarity
4.2.
chirality
5.
combining symmetry Operations : "زيادة" Group
6.
Constructing العالى Groups from تبسيط Groups
.
mathematical 7 , Group of Definition
8.
Review of المصفوفات
8.1.
التعاريف
8.2.
ماتريكس algebra
8.3
المباشر Products
8.4.
العكس المصفوفات and Determinants
9.
تحويل المصفوفات
10.
ماتريكس representations Groups of
10.1.
example ماتريكس التمثيل: OF THE c3v البند Group ( The مع molecule)
10.2. وقد
example : ماتريكس التمثيل OF THE c2v البند Group (allyl جذرى ( The
11 .
Properties of ماتريكس representations
11.1.
similarity transforms
11.2.
characters of representations
12.
reduction of representations الاول
13 .
irreducible representations and symmetry
( 14 . انظر الجداول

15 .
reduction of representations الثانى
15.1
General of مفاهيم orthogonality
15.2
orthogonality اجرى in theory Group
15.3
)The الكمية to يمس The irreps spanned by on
16 الف.
symmetry adapted ( combinations
.
17determining whether عام Integral can Be عدم Zero
18 .
bonding in diatomics
19 .
bonding in polyatomics - Constructing و orbitals from salcs
20 . The المدارى
calculating الطاقات and التوسع عوامل
21 . The
solving secular equations
21.1
ماتريكس صياغة of ( وقد of ( equations
21.2
solving for and The المدارى الطاقات التوسع عوامل
22 .
Summary of The steps ? in Constructing orbitals
و 23 . وقد
more [bonding example , و The orbitals of h2o
23.1
ماتريكس التمثيل, characters and salcs
24 . و
الذبذبات
24.1
و degrees of freedom - determining The number of العادى vibrational انماط
24.2
determining symmetries OF THE و نقاط
24.3
The الذرية displacements )3n cartesian on
24.4
و الذبذبات )Internal coordinates
25 .
Summary of / Group Theory to و نقاط
26 .
Group theory and electronic States
و 27 .
spectroscopy - التفاعل of atoms and Light
molecules " Electronic 27.1
التحولات in molecules
27.2
vibrational التحولات in molecules
27.3
والمعارف scattering
28 .موجز

29.
التذييل الف - 'proofs FOR THE mathematically inclined
30 .
التذييل باء - انظر الجداول and المباشر ومراقبة المائدة
problem Sheet
3

1.
مقدمة
فيا will nouveau بى"" The مفهوم of symmetry in عام Everyday sense . if we ساى something is symmetrical' 'مؤتمر we usually IT- هاس Mirror symmetry, ولكن "left - Right' symmetry,and الواردة انظر The سامى if viewed Mirror , in. symmetry is also in chemistry very هامة. Some molecules are more clearly "symmetrical' ثان others , Consequences does What الهدف this اذا, if any?The سباق الاسلحة of this is to provide والرصد وقد symmetry Treatment of chemical Systems in The داخل mathematical framework known as Group ( The Theory reason for The name will يصبحوا الظاهر بعد ذلك). يعيد we اذا classified symmetry OF THE molecule وقد theory , Group , provides powerful ( of Tools that " provide US الاعوام Insight into Many of its Chemical and physical properties .Some التطبيقات Group of نظرية that will Be covered " in this سباق:


(ط) predicting whether molecule will مع ا بى, ولكن كما القطبية.
الثانى) examining Chemical and bonding visualising و orbitals. ' "'
predicting whether molecule ماى absorb وقد Light of مع ا ستقطاب, and which spectroscopic
التحولات ماى Be excited if IT- does.
' "' investigating The vibrational نقاط OF THE molecule.


فيا ماى well meet some of these Topics Again, possibly in more detail , in بعد سباقات )notably symmetry الثانى , and for more The mathematically inclined amongst فيا, تكميلية مقدار Mechanics). However ,Be introduced لمدة will here to واتضح فى المستقبل فيا عليها مقدمة TO AND THE تعزيز التطبيقات Theory of Group يعيد we اذا worked through The Basic Principles and "مع" The Theory of. 2
.
symmetry Operations and symmetry عناصر
symmetry عملية is فى سنة العمل that leaves عام عولجت Looking The سامى After IT- هاس عدد انه out. For example ,if we take molecule ا of Water and rotate رقم IT- by 180 about عام"العبور through The المركزى atom س )between The Two )ح( atoms IT- will انظر The سامى as before. IT- will also انظر The سامى if we reflect IT- through ان of two Mirror -, as shown in The الشكل الاطلاع. )

التناوب
)
(عملية )عملية )عملية)
"of symmetry
(اساسى( Mirror تحوم
)اساسى( Mirror تحوم ( اساسى)
each symmetry عملية corresponding symmetry هاس, اساسى , which is", The line تحوم, ولكن البند " The احترام to which is symmetry عملية انه out. The symmetry اساسى consists of all The البنود that in " the " The انشاء 'symmetry عملية is performed. وقد التناوب, inThe line of البنود that " in The 'انشاء symmetry" (, وقد ); in The البنود that (unchanged make up of symmetry تحوم فى. The symmetry عناصر that molecule ماى قد possess are :
.
1 )ه( - The والهوية. The identity عملية consists of الافراط Nothing , and The corresponding symmetry اساسى is The ليس molecule. every molecule هاس at least this اساسى.
2 .
CN - عام رقم fold"of التناوب. تناوب by 360°/ن leaves The molecule unchanged. The h2o molecule above هاس جيم 2 ,". Some molecules اذا more than one CN", in The case which " One " The value of The رقم % is الرئيسى". مذكرة that are by الاتفاقية تناوب counterclockwise about The".
3 .
σ - تتعرض of symmetry.), in The تحوم leaves The molecule Looking The سامى. وقد molecule that in also هاس عام", of symmetry Mirror تحوم that is تشمل The", راسى % Mirror تحوم and is labelled σv, while One perpendicular to The is", الافقى % Mirror تحوم and is labelled σh. وقد راسى Mirror تحوم that bisects The منظور between two جيم 2 % محاور is dihedral Mirror , يخيم,σd.
4 .
الاول - مركز of symmetry. The انعكاس through مركز OF THE symmetry leaves molecule unchanged. انعكاس consists of العبور each البند through The مركز out of انعكاس and to The سامى بعد ذلك The Other الغربى OF THE molecule. السنة example , molecule of " فى مركز of انعكاس is shown الاطلاع.
5 .
SN - عام رقم fold improper التناوب") also % - وقد منظمة الروتارى" ).The الروتارى )عملية consists of rotating through سنة حيث 360°/ن about The", followed by قد تتعرض reflecting in The perpendicular to". مذكرة that is s1 The سامى as )and 2 is as انعكاس The سامى. The molecule shown above هاس two 2 محاور. )ه(
and The identity تناوب CN symmetry Operations that are could actually Be out انه تم molecule.For this reason are لمدة % "( symmetry Operations. , and on والتقلبات improper تناوب can only )بى Imagined IT- is Not actually ممكن to Turn molecule into its وقد Mirror صورة غير to invert IT-"some واتضح drastic rearrangement of Chemical and هائلة) as such, are termed improper symmetry Operations. مذكرة من ا
التعاريف": conventionally," ( of : A cartesian محاور وقد molecule (as we need will