Symmetries and group theory

Most of the following slides are from an old PowerPoint I used in an undergraduate quantum mechanics class.

© 2017, U. Burghaus

Piece of cake for you (?)

This class will be rather short.

• Lorentzian line shape function • Gaussian line shape • Voight line shape

• Natural line width • Pressure broadening • Doppler broadening • Transit-time broadening • Power broadening

What is the origin of a Lorentzian line shape? Remember?

Covers about the same, but is somewhat more detailed/advanced • Spectra of Atoms and Molecules, 3rd Ed., Peter F. Bernath,

Oxford University Press, Chapters 2 – 3

I mostly used these to prepare this PowerPoint • Physical Chemistry, Engel/Reid (2nd edition), Chapter 27 • Physical Chemistry, I.N. Levine, Chapter 21.17 (p. 804)

Group theory - motivation

• How to design molecular orbitals? Making LCAO’s correctly.

• Selection rules. • Normal modes Vibrations

• Moment of inertia Rotations

• Predicting dipole moments • Quantum mechanics & group theory particle physics

Group theory is a very very general approach since symmetries are everywhere.

LCAO: linear combination of orbitals

Remember this story … details later

Raman

Motivation - molecular symmetry in microwave spectroscopy

A molecule has a given symmetry if the symmetry operation leads to a configuration of the molecule that is physically indistinguishable from the original one.

Def.:

Erot = Erot(moment of inertia) Moment of inertia depends on symmetry of the molecule. 1)

Why is that important? Molecular rotation of more complicated molecules

Consider

Symmetry is closely related to the possibility to excite a rotation. Electric dipole moment depends on symmetry of the molecule.

Selection rules are related to the electric dipole moment.

2)

and

Motivation -molecular rotation – more complicated molecules

Classical mechanics: Erot = Erot(moment of inertia)

Idea in Q.M. basically identical with classical concept.

Erot I I Ia b c∝ + +1 1 1

Look at the principal axes of symmetry of the molecule. Those determine the moment of inertia and hence Erot.

Example: Principal axes of symmetry

Motivation - molecular rotation

E Erot rot I I Ia b c= ∝ + +( )moment of inertia 1 1 1

A symmetric top is a molecule in which two moments of inertia are the same.

Molecular symmetry – symmetry elements

A molecule has a given symmetry if the symmetry operation leads to a configuration of the molecule that is physically indistinguishable from the original one.

Def.:

Cn symmetry := rotation by 360/n

Cn

water ammonia

Rotation axis of highest symmetry is oriented along the z axis.

Molecular symmetry – symmetry elements

σ

Mirror planes containing the principal rotation axis σV: vertical mirror plane Mirror planes perpendicular to the principal rotation axis: σH: horizontal mirror plane σV’(σV): molecule in (out) of the plane

σ symmetry := reflection on plane mirror planes

Molecular symmetry -- symmetry elements

rotation & reflection

Sn symmetry := rotation & reflection

Molecular symmetry -- symmetry elements

i

i symmetry :=“center of symmetry”, inversion x, y, z -x, -y, -z

no dipole moment

i

σV

σHσ’V

σ’H

Molecular symmetry

There are only five different types of symmetry elements:

From Engel’s book.

https://en.wikipedia.org/wiki/Point_group

Symmetry group - definition

Engel p. 659; Levine p. 805

1) Closure 2) Associative 3) Existence of identity 4) Existence of inverse

Example: Definition of the system -) all integers: positive, negative, zero (A = 1; B = 7; C = 15; D = 0) -) combination rule is “addition”

All integers form a group with respect to “addition”.

Verification 1) Closure: A+B=8, 8 is an integer 2) Associative: (A+B)+C = A+(B+C) 3) Identity: B+0=0+B=B, identity is zero I=0 4) Inverse: inverse of B is –B B+(-B)=(-B)+B=0=I

Finding the group of a molecule.

Example Nitrogen trifluoride

C3v group

Engel page 659

Follow red line

What are point groups?

A symmetry operation moves each point in space to a new location such that the point that is the molecular center of mass is unmoved.

Definition

In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. https://en.wikipedia.org/wiki/Point_group

Multiplication tables

Engel’s book Levine’s book

Different nomenclature but the same thing.

Multiplication tables - example

How to read this table?

Matrix representations of a symmetry operation

These symmetry operations form a group (e.g. C2v) and the matrices are the matrix representation of this group.

A set of nonnull square matrices that multiply the same way the corresponding members of a point group multiply form the representation of the group. The order of the matrices is the dimension of the representation.

Next • We need to remember how to use a matrix • Example for using the matrix representation

Matrix algebra

(a)(b) = (a*b) Matrices of order one multiply as numbers.

matrix * vector = vector

Matrix algebra - example

Example: Effect of C2v group operations on a vector (x y z)

180º rotation

reflection at orange mirror plane

reflection at red mirror plane

Z-component is Not affected

Matrix representations of a symmetry operation

Rotation about the z-axis:

Example

The 3D matrix representation obeys the multiplication table.

Matrix representations of a symmetry operation

Many different representations of a group can be found.

These matrices are in diagonal form. Therefore, a simpler representation can be found.

Irreducible representation – 1D matrices

The irreducible representations are the matrices of smallest dimension that obey the multiplication table of the group.

Enge

l’s b

ook

Levi

n’s

book

(1) (-1) (-1) (1)

Irreducible representation - example

Matrices of order one multiply as numbers.

(This is the old example above.)

(-1) (-1) = (1) (-1) (1) = (-1)

The 1D representation obeys the multiplication table.

What can we do with this kind of table ???

?

Irreducible representations in quantum mechanics

Consider the 2p orbitals of an atom.

How do the orbitals transform under symmetry operations?

For example effect of C4(z), i.e., 90º rotation about z-axis

C4(z)

2px 2py

Symmetry of the transformation

“+1” sign if lobe is unchanged “-” sign if lobe is changed

Effect of transformation:

Chapter 28.4 in Engel’s book

Symmetry of transformations

2px 2py 2pz

E, C2, σv, σ’v

sign of lobes unchanged

+1 +1 +1 +1

E, C2, σv, σ’v E, C2, σv, σ’v

+1 -1 +1 -1 +1 -1 -1 +1

What about the effect of the symmetry operation of the C2v group on p-orbitals?

p orbitals in a water molecule

Group theory bla bla… 3x3 matrices, set of +1 and -1, and 2p orbitals form a basis of the representation.

AOs are basis functions

Set of coordinates are basis functions

Coordinates and rotations are basis functions

What can we do with this table? Irreducible representations in quantum mechanics

What can we do with this table? Irreducible representations in quantum mechanics

A: symmetric B: antisymmetric with respect to rotation about principal axis

subscript 1: symmetric 2: antisymmetric

with respect to rotation about axis perpendicular to principal axis

Symmetry-adapted MOs

Only AO combinations with nonzero overlap integral contribute to MOs.

Example: H2O H AOs H 1sA + H1sB

O AOs

Group theory: The overlap integral of AOs is nonzero only if the combination belongs to the same representation.

Chapter 28.6 – Engel’s book

Symmetry-adapted MOs

Group theory: The overlap integral of AOs is nonzero only if the combination belongs to the same representation.

Example:

A2 B2 Representations:

“A2*B2” = (1x1 1x(-1) (-1)x(-1) (-1)x1) = (1 -1 1 -1) B1

A2*B2 overlap integral is zero

Symmetry-adapted MOs

Only the oxygen 2sAO and 2px AOs belong to the same irreducible representation as H1sA + H1sB AOs

How to read this table?

In-c

lass

ho

mew

ork

Problem 10

)(ˆ)(ˆ)(ˆ 2 yzzCxz σσ =

RESULT

RE

SULT

σ ?? C ?? S rotation-reflections I inversion E ???

11)

In-c

lass

ho

mew

ork

Complete the table Problem 11

σ reflections C rotations S rotation-reflections I inversion E identity

Symmetry operations

• What are the most important (and common) symmetry operations? • How to read and use multiplication tables? • Name & discuss one example of group theory.

Presenter

Presentation Notes

Copyright note:

• Spectra of Atoms and Molecules, 3rd Ed., Peter F. Bernath, Oxford University Press, chapter 5

• Physics of Atoms and Molecules, B.H. Bransden, C.J. Joachain, Wiley,

chapters 3-9 • Foundations of Spectroscopy, S.

Duckett, B. Gilbert, Oxford Chemistry Primers, Vol. 78

chapter 4 • Molecular Spectroscopy, J.M. Brown,

Oxford Chemistry Primers, Vol. 55 chapter 7

Next class: • Atomic spectroscopy

• Absorption spectroscopy • Bohr model • QM of H atom review • Eigenfunctions, orbitals

group theory …&… quantum mechanics misc.

• Spectra of Atoms and Molecules, 3rd Ed., Peter F. Bernath, Oxford University Press, Chapter 4

if we have time left

ψψ EH =ˆ

PChem – Quantum mechanics

What do we get out of this?

wave functions eigenvalues E

ψ

spectroscopy Ei - Ej

0)(82

2

2

2

=−+∂∂ ψπψ VE

hm

x

Eigenvalue equation

https://en.wikipedia.org/wiki/Matrix_mechanics

Schrödinger/Dirac wave formulation of quantum mechanics, 1926

Matrix quantum mechanics by Heisenberg, Born, Jordan in 1925.

Wave mechanics Matrix mechanics

equivalent

More versions do exist: Feynman, density matrix, …

Some colleagues apparently believe that this version is better for

spectroscopy

• Uses differential equations

ψψ EH =ˆ• Uses matrix algebra

>=< jiij fHfH |ˆ|H select a basis set, then

becomes a matrix

A function f is written as |f>

https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation

bra–ket notation

Paul Adrien Maurice Dirac (1902 – 1984) English theoretical physicist

<f| |f>

Expanding wave function using set of basis functions

∑=i

ii fcψ >>=∑i

ii fc ||ψ

ψψ EH =ˆ >>= ψψ ||ˆ EH>=< jiij fHfH |ˆ|

H becomes a matrix Eigenvalue equation

Used in matrix QM

Claude Cohen-Tannoudji, et al., Quantum mechanics, Chapter IIB

Scalar product τφψφψ d∫>=< *|

)(| 321 ccc=<ψ

>=

3

2

1

|ddd

φ

=>=< ∫

3

2

1*3

*2

*1

* )(|ddd

cccdτφψφψ

bra ket )(|| ***

321ccc==<>+ ψψ

Why do we need new quantum numbers for multi-electron systems? (Engel ch. 21.7)

EXAMPLES Good quantum numbers Bad quantum numbers

H-atom (single electron)

Multi electron system Multi electron system

2 2ˆ ˆˆ ˆ, , , commute with Hz zl l s s

ˆˆ ˆ ˆ,S,J commute with HL l s, l, m , mn

Good quantum numbers: Eigenvalues of operators (i.e. the observable) are only independent of time if the operator commutes with the Hamilton operator.

ˆ

ˆ ˆ[ , ] 0 ( , ) ( )

A a

A H a r t a r

ψ ψ=

= → =

https://en.wikipedia.org/wiki/Good_quantum_number https://en.wikipedia.org/wiki/Symmetry_in_quantum_mechanics

Matrices & vectors nomenclature for the spin.

1 0 and

0 1α β = =

Spin eigenfunctions

0 1 0 1 0ˆ ˆ ˆ, ,

1 0 0 0 12 2 2x y z

is s s

i−

= = = −

Spin operators

P21.7) In this problem we represent the spin eigenfunctions and operators as vectors and matrices.

1 0 and

0 1α β = =

and α β

a) The spin eigenfunctions are often represented as the column vectors

Show that are orthogonal using this representation.

Using the rules of matrix multiplication,

( )1

0 1 1 0 0 1 0.0

αβ

= = × + × =

Therefore α and β are orthogonal.

0 1 0 1 0ˆ ˆ ˆ, ,

1 0 0 0 12 2 2x y z

is s s

i−

= = = −

ˆ ˆ ˆ,x y zs s i s =

b) If the spin angular momentum operators are represented by the matrices

, show that the commutation rule holds.

2

2 2

0 1 0 0 0 1ˆ ˆ ˆ ˆ

1 0 0 0 1 02

0 0 1 0 1 0ˆ 2

0 0 0 1 0 12 2 2

x y y x

z

i is s s s

i i

i ii i i s

i i

− − − = − − = − = = = − − −

and α β 2ˆ ˆ and .zs sd) Show that are eigenfunctions of

What are the eigenvalues?

1 0 1 1 1 0 0 0ˆ ˆ and

0 1 0 0 0 1 1 12 2 2 2z zs sα β = = = = − − −

Szα α= 12

12

ˆzS β β= −