4cm Physical Chemistry 5 (Spectroscopy) 1mm A short ... · Our concern: Symmetry operations O^ i such that h O^ i;H^ i = 0!eigenvalues of O^ i can be used to categorize eigenstates

Physical Chemistry 5 (Spectroscopy)

A short introduction to Group Theory

Axel Schild

Axel Schild 2020-04-24/05-08 1

Our concern

Use the symmetry of molecules!

(mostly the symmetry of the equilibrium nuclear conguration)

Axel Schild 2020-04-24/05-08 2

Why group theory?

Our concern: Symmetry operations Oi such that[Oi , H

]= 0

→ eigenvalues of Oi can be used to categorize eigenstates

Useful applications for spectroscopy:

• classify (electronic, vibrational, rotational) eigenfunctions of H

• predict allowed spectroscopic transitions (selection rules)

• predict physical properties (electric dipole moment, optical activity)

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Our plan

molecular symmetry, point groups

↓matrix representations of point groups

↓reducible & irreducible representations

↓character tables

↓applications

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Our literature

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What is a group?

Group G: Set of elements together with an operation (group law) combining two

elements A and B to form another element, A · B ≡ AB = C , with axioms:

• Closure: For all A, B ∈ G , also A · B ∈ G .

• Associativity: For all A, B, C ∈ G , we have (A · B) · C = A · (B · C).

• Identity: There exists a unique element E ∈ G such that A · E = E · A = A.

• Inverse element: For all A ∈ G there exists an element B ∈ G such that

A · B = B · A = E . The inverse element of A is commonly denoted as A−1.

Examples:

• group of integers with addition operation

• group of non-zero real numbers with multiplication operation

• group of translations in a plane

• symmetry groups

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Point groups

If a molecule is energetically constrained to its equlibrium nuclear structure or if it is

rigid within the relevant time scale, its symmetry properties are given by a point group.

Symmetry operations vs. symmetry elements:

Element Operation

E Identity E nothing

Cn Proper axis Cn counterclockwise rotation by 2π/n

σ Plane σ reection at a plane

i Inversion center i inversion through inversion center

Sn Improper axis Sn rotation by 2π/n followed by reection

at plane perpendicular to rotation axis

Note:

• The symmetry operations are the group elements.

• The symmetry elements are closely related to the classes of the group.

• All symmetry elements meet in one point.

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Symmetry element: Identity

operation E : do nothing

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Symmetry element: Proper axis

operation Cn: counterclockwise rotation by 2π/n

• principle axis: Cn with highest n

• if only one principle axis: permanent dipole moment

• if several principle axes: no permant dipole moment

• a Cn axis gives rise to n − 1 symmetry operations:

Cn = rot. by 2πn , C2

n = rot. by 2 2πn , . . . , C n−1

n = rot. by (n − 1) 2πn

examples: water (one C2), NCl3 (one C3), methane (four C3 and three C2)

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Symmetry element: Plane

operation σ: reection at a plane

• σv reection at vertical plane (contains symmetry axis Cn)

• σh reection at horizontal plane (perpendicular to symmetry axis)

• σd reection at diagonal plane (bisects angle between two C2-axes)

• molecule cannot be optically active

examples: H2O (two σv), ethene C2H4 (two σv, one σh), methane CH4 (six σd)

z

y

x

C2

σv'σv

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Symmetry element: Inversion center

operation i : inversion through inversion center


examples: SF6, trans-1,2-Dichloroethene C2H2Cl2

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Symmetry element: Improper axis

operation Sn: rotation by 2π/n, reection at plane perpendicular to rotation axis


• an Sn-axis results in n − 1 symmetry operations Sn, S2n , . . . , S

n−1n

examples: allene (S4), trans-1,2-Dichloroethene C2H2Cl2 (S2)

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Types of point groups

Schönies notation:

• Cn (cyclic): n-fold rotation axis

Cnh: with horizontal mirror plane

Cnv : with n vertical mirror planes

• S2n (Spiegel): 2n-fold rotation-reection axis (NB: Sn = Cnh if n odd)

• Dn (dihedral): n-fold rotation axis and n perpendicular C2-axes

Dnh: with horizontal mirror plane (and n vertical mirror planes)

Dnd : n diagonal mirror planes

• T (tetrahedral): four 3-fold axes and three 2-fold axes

Th: three horizontal mirror planes

Td : has diagonal mirror planes

• O (octahedral): like cube, three 4-fold axes, four 3-fold axes, six diagonal 2-foldaxes

Oh: horizontal (and vertical) mirror planes

• I (icosahedral): six 5-fold axes, ten 3-fold axes, and 15 2-fold axes

Ih: horizontal mirror planes, inversion center, improper rotation axes

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Some common point groups: C1

group symmetry elements example

C1 E no symmetry, e.g. CHClFBr

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Some common point groups: C∞v


C∞v E, C∞, ∞σv, . . . linear, e.g. HCl

NB: Actually, there are innitely many symmetry elements Cn with n = 1, 2, . . . ,∞.

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Some common point groups: D∞h


D∞h E, C∞, ∞σv, i, S∞, ∞C2, . . . linear with inversion center, O2, CO2

NB: Compare with C∞h.

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Some common point groups: C2v


C2v E, C2, σv(xz), σ′v(yz) H2O

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Some common point groups: C3v


C3v E, C3, 3σv CH3Cl, NCl3

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Some common point groups: D2h


D2h E, C2(z), C2(y), C2(x), i, σ(xy), σ(xz), σ(yz) ethylene, pyrazine

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Some common point groups: D6h


D6h E, C6, C3, C2, 3C′2, 3C′′2 , i, S6, S3, σh, 3σd, 3σv benzene

NB: The C3- and C2-axes are contained in the C6-axis; The S3-axis is contained in the

S6-axis; S2 is the same as i.

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Some common point groups: Td


Td E, 4C3, 3C2, 3S4, 6σd methane

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Some common point groups: Oh


Oh E, 3C4, 4C3, 6C2, 3C2, i, 4S6, 3S4, 3σh, 6σd SF6, cubane

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Determining the point group

source: https://commons.wikimedia.org/wiki/File:Pt_Group_chart_2.png

Note: Those are the symmetry elements, not the operations.

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Hs. H. Günthard Master Scholarshipfor students of Chemistry or Interdisciplinary Sciences:

Call for Proposals in the Academic Year 2020/2021

The next period for applications will be May 1 to May 31, 2020.

The Günthard Foundation oers scholarships for Master students

with a proven interest in physical chemistry. Prof. Hs. H. Günthard

was the founder of the Laboratory of Physical Chemistry of ETH

Zurich. The maximum grant contribution will be CHF 22,000 for

the entire Master program.

Requirements for your Application and further dates:

https://www.chab.ethz.ch/studium/guenthard-master-scholarship.html

Contact person: Veronika Sieger, [email protected]

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Remark: CNPI groups

If the nuclear conguration is not rigid on the timescale of the experiment (e.g. due to

tunneling, like for the ammonia molecule), point groups won't work. Then we can use

complete nuclear permuation inversion groups.

• all point groups are contained in the CNPI groups

• CNPI groups can be very large,

CH4: 4! × 2 = 48 operations

C6H6: 6! × 6! × 2 = 1036800 operations

C60: 60 × 2 ≈ 1082 operations

but most of these operations are unfeasible (energetically forbidden)

See the script or one of the two Bunker & Jensen books for more information.

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Goal: Understanding character tables

C2v E C2 σv σ′vA1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy

B1 1 -1 1 -1 Ry , x xz

B2 1 -1 -1 1 Rx , y yz

C3v E 2C3 3σvA1 1 1 1 z x2 + y2, z2

A2 1 1 -1 Rz

E 2 -1 0 (Rx ,Ry ), (x , y) (x2 − y2, xy), (xz, yz)

What is this and what is it good for?

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Concepts: Order of a group

Order h (or g) of a group:

number of group elements = number of symmetry operations

• discrete/nite groups have nite order

• continuous groups have innite order

• example: C2v has order 4 (E , C2, σv, σ′v)

• example: C3v has order 6 (E , 2C3, 3σv)

C3v E 2C3 3σvA1 1 1 1 z x2 + y2, z2

A2 1 1 -1 Rz

E 2 -1 0 (Rx , Ry ), (x, y) (x2 − y2, xy), (xz, yz)

Order k of a group element O is the smallest number k such that Ok = E

NB: all members of a class (see below) have the same order

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Concepts: Product table, abelian

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Concepts: Product table, non-abelian

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Concepts: Similarity transform, conjugate elements, classes

For group elements Oi , Ok , we can make the similarity transform

Ok · Oi · O−1k = Oj

with Ok · O−1k = E . Clearly, Oj is also a member of the group. We call Oi and Oj

conjugated elements. All conjugated elements of a group for a class.

C2v E C2 σv σ′vA1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy

B1 1 -1 1 -1 Ry , x xz

B2 1 -1 -1 1 Rx , y yz

C3v E 2C3 3σvA1 1 1 1 z x2 + y2, z2

A2 1 1 -1 Rz


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Concepts: Classes

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Concepts: Representations

C2v E C2 σv σ′vE E C2 σv σ′vC2 C2 E σ′v σvσv σv σ′v E C2

σ′v σ′v σv C2 E

C2v E A B C

E E A B C

A A E C B

B B C E A

C C B A E

Whatever the elements and operation of a group, if it has the same multiplication

table it is the same thing. . .

. . . , well, almost:

• isomorphism: one-to-one correspondence between group elements

• homomorphism: same group structure, but elements may be the same

We look for a set of matrices that represent the symmetry operation in the sense that

they are homomorphic. Those are a representation of the point group.

geometry of symmetry operations ↔ matrix algebra

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Concepts: Representations

Construct representations Γ from position vectors, basis vectors, function spaces. . .

. . . we use vectors

• symmetry operation O → n × n matrix B

• n is dimensionality of representation

• basis e, matrix representation B(e), some vector ~x :

~y = B(e)~x

• dierent basis e′ = Se (with S being the transfomation matrix):

~y ′ = S~y = SB(e)~x = SB(e)S−1~x ′ = B(e′)~x ′

→ B(e′) is an equivalent representation of O (similarity transform)

• similarity transforms preserve the trace of the matrix: Tr B(e′) = Tr B(e)

The trace of the representation is called the character χ of the operation.

All elements of a class have the same character.

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Representation Γ of a point groupset of matrices for all group elements / symmetry operations

Character χtrace of a matrix in some representation Γ

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Character table C2v , again

C2v E C2 σv σ′vA1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy

B1 1 -1 1 -1 Ry , x xz

B2 1 -1 -1 1 Rx , y yz

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Example 1: Some one-dimensional representations of C2v

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Example 2: Some more one-dimensional representations of C2v

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Example 3: Yet another one-dimensional representation of C2v

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Example 4: A two-dimensional representation of C2v

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Example 5: A two-dimensional representation of C3v

C3v E 2C3 3σvA1 1 1 1 z x2 + y2, z2

A2 1 1 -1 Rz


C3 rotation around the z-axis

x

y

x

y

C3

(x

y

)=

(cos(2π/3) sin(2π/3)

− sin(2π/3) cos(2π/3)

)(x

y

)=

(−1/2

√3/2

−√3/2 −1/2

)(x

y

)Character χ(C3) = −1/2− 1/2 = −1

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Concepts: Reducible and irreducible representations

If all matrices of a representation Γ can be brought into identical block-diagonal form

by a similarity transformation, Γ is reducible. Otherwise, Γ is irreducible.

(Somewhat artical) example:

If the two matrices of the 5-dimensional representation Γ can be brought into the form

A =

A11 A12 0 0 0

A21 A22 0 0 0

0 0 A11 A12 0

0 0 A21 A22 0

0 0 0 0 A33

, B =

B11 B12 0 0 0

B21 B22 0 0 0

0 0 B11 B12 0

0 0 B21 B22 0

0 0 0 0 B33

,

Γ is reducible and is composed of the representations

Γ1 : A1 =(A11 A12A21 A22

), B1 =

(B11 B12B21 B22

)and

Γ2 : A2 = A33,B2 = B33.

We write Γ = 2Γ1 ⊕ Γ2. If the two matrices of Γ1 cannot simultaneously be

diagonalized, Γ1 (and the one-dimensional representation Γ2) are irreducible.

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Concepts: Character tables

contains characters χ of the IRREPS Γ

Γ

(x

y

)is reducible, Γ

(x

y

)= B1 ⊕ B2

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Concepts: Systematic reduction

∑O

χ(i)(O)× χ(j)(O) = hδij

Reduction:

Γred =∑k

credk Γk

credk :=1

h

∑O

χred(O)× χ(k)(O)

with

O . . . all group elements/symmetry operations

(sum is over all operations, not only the classes!)

h . . . order of the group

Γk . . . IRREP

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Concepts: Systematic reduction example

c(1s)A1

=1

4(2× 1 + 0× 1 + 0× 1 + 2× 1) = 1

c(1s)A2

=1

4(2× 1 + 0× 1− 0× 1− 2× 1) = 0

c(1s)B1

=1

4(2× 1− 0× 1 + 0× 1− 2× 1) = 0

c(1s)B2

=1

4(2× 1− 0× 1− 0× 1 + 2× 1) = 1 ⇒ Γ(1s) = A1 ⊕ B2

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Summary: Character tables

C2v E C2 σv σ′vA1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy

B1 1 -1 1 -1 Ry , x xz

B2 1 -1 -1 1 Rx , y yz

C3v E 2C3 3σvA1 1 1 1 z x2 + y2, z2

A2 1 1 -1 Rz


What is this?

A table showing the characters of all symmetry operations

of a group for all irreducible representations (IRREPS).

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Summary: Character tables

C2v E C2 σv σ′vA1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy

B1 1 -1 1 -1 Ry , x xz

B2 1 -1 -1 1 Rx , y yz

C3v E 2C3 3σvA1 1 1 1 z x2 + y2, z2

A2 1 1 -1 Rz


• also included: rotations and monomials of up to second order of the

x , y , z-coordinates (e.g. for p- and d-orbitals)

• characters of operations in the same class are identical

• characters of IRREPS are orthonormal,∑

O χ(i)(O)× χ(j)(O) = hδij

• columns are also orthonormal (w.r.t. classes)

NB: The sums run over all symmetry operations, not only over the classes.

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Labels of IRREPS

C3v E 2C3 3σvA1 1 1 1 z x2 + y2, z2

A2 1 1 -1 Rz


• one-dimensional representations are A or B

• A/B if character of rotation around highest symmetry axis is +1/− 1

• two-dimensional representations are E

• three-dimensional representations are T or F

• for inversion: add subscript g or u

• special rules for innite groups

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Clarication: Symmetry elements vs. symmetry operations

D6h E 2C6 2C3 C2 3C ′2 3C ′′2 i 2S3 2S6 σh 3σd 3σvA1g +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1

A2g +1 +1 +1 +1 -1 -1 +1 +1 +1 +1 -1 -1

B1g +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1

B2g +1 -1 +1 -1 -1 +1 +1 -1 +1 -1 -1 +1

E1g +2 +1 -1 -2 0 0 +2 +1 -1 -2 0 0

E2g +2 -1 -1 +2 0 0 +2 -1 -1 +2 0 0

A1u +1 +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 -1

A2u +1 +1 +1 +1 -1 -1 -1 -1 -1 -1 +1 +1

B1u +1 -1 +1 -1 +1 -1 -1 +1 -1 +1 -1 +1

B2u +1 -1 +1 -1 -1 +1 -1 +1 -1 +1 +1 -1

E1u +2 +1 -1 -2 0 0 -2 -1 +1 +2 0 0

E2u +2 -1 -1 +2 0 0 -2 +1 +1 -2 0 0

The C6-axis has the symmetry operations:

• C6 and C56 = C−16 → class 2C6

• C26 = C3 and C4

6 = C23 = C−13 → class 2C3

• C36 = C2 → class C2

The S6-axis has the symmetry operations:

• S6 and S56 = S−16 → class 2S6

• S26 = S3 and S4

6 = S23 = S−13 → class 2S3

• S36 = S2 = i → class i

Character tables contain the

symmetry operations, sorted in

classes. Symmetry elements are

convenient to determine the point

group but are not needed

otherwise.

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Application: Symmetrized LCAO

Projector on an IRREP Γ:

PΓ =1

h

∑O

χ(Γ)(O)× O

e.g. our 1s H-orbitals of H2O:

PA1 =1

4(1× E + 1× C z

2 + 1× σxz + 1× σyz ) , PA11s(1) =1

2(1s(1) + 1s(2))

PA2 =1

4(1× E + 1× C z

2 − 1× σxz − 1× σyz ) , PA21s(1) = 0

PB1 =1

4(1× E − 1× C z

2 + 1× σxz − 1× σyz ) , PB11s(1) = 0

PB2 =1

4(1× E − 1× C z

2 − 1× σxz + 1× σyz ) , PB21s(1) =1

2(1s(1)− 1s(2))

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Application: Symmetrized LCAO

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Application: Symmetry of normal modes

What are normal modes?

• Take your favorite Born-Oppenheimer energy surface (the electronic state) V

• V (X ) depends on the coordinates X = (x1, y1, . . . , zN) of the N nuclei

• Go to the minimum Xeq (equilibrium conguration) and expand to 2nd order in

displacement coordinates (X1,X2, . . . ,X3N) = Xeq − X :

V (X ) ≈∑i

∑k

∂2V

∂Xi∂Xj

∣∣∣∣X=Xeq

XiXj

• Diagonalize:

V (X ) ≈∑i

Ωiq2i

• normal mode coordinates: qi

• nuclear wavefunction: ψnuc ≈∏3N

i=1 φi (qi )

where each φi is an eigenfunction of the harmonic oscillator with potential Ωiq2i

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3N nuclear coordinates for N nuclei:

• 3 for translation

• 3 (or 2) for rotation

• 3N − 6 (or 3N − 5) for vibration

Γ3N = Γt ⊕ Γr ⊕ Γv

example: H2O with x−, y−, z−coordinates at each nucleus

With the basis x1, y1, z1, . . . , z3(which is representation Γ9), the

C2v-symmetry operations are

represented by 9× 9 matrices

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Exi = xi ; Eyi = yi ; Ezi = zi ⇒ χΓ9 (E) = 9

C z2 x3 = −x3; C z

2 y3 = −y3; C z2 z3 = z3 ⇒ χΓ9 (C z

2 ) = −1

σxzy3 = −y3; σxzx3 = x3; σxzz3 = z3 ⇒ χΓ9 (σxz ) = 1

σyzxi = −xi ; σyzyi = yi ; σyzzi = zi ⇒ χΓ9 (σyz ) = 3

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C2v E C2 σv σ′vA1 1 1 1 1 z, x2, y2, z2

A2 1 1 -1 -1 Rz , xy

B1 1 -1 1 -1 Ry , x , xz

B2 1 -1 -1 1 Rx , y , yz

Γ9 9 -1 1 3

Γ9 = 3A1 ⊕ A2 ⊕ 2B1 ⊕ 3B2

cA1 =1

4(9− 1 + 1 + 3) = 3

cA2 =1

4(9− 1 + 1 + 3) = 1

cB1 =1

4(9− 1 + 1 + 3) = 2

cB2 =1

4(9− 1 + 1 + 3) = 3

vibrational modes: 2A1 ⊕ B2

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vibrational modes: 2A1 ⊕ B2, can be obtained with projector, e.g. B2

PB2

x1y1z1

=1

4(E − C z

2 − σxz + σyz )

x1y1z1

=1

4

x1 + x2 − x2 − x1y1 + y2 + y2 + y1z1 − z2 − z2 + z1

=1

2

0

y1 + y2z1 − z2

PB2

x3y3z3

=1

4(E − C z

2 − σxz + σyz )

x3y3z3

=1

4

x3 + x3 − x3 − x3y3 + y3 + y3 + y3z3 − z3 − z3 + z3

=

0

y30

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Some selection rules (without proof):

A normal mode is IR-active if its symmetry corresponds to one or more of the

translations x , y , z.

A normal mode is Raman-active if its symmetry corresponds to one or more of the

translational products x2, y2, z2, xy , xz, yz.

In a centrosymmetric molecule a mode cannot be both IR- and Raman-active.

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Application: Symmetry of vibrational levels

nuclear vibrational wavefunction in normal mode approximation:

ψnuc ≈ φ(ν1)1 (q1)× φ(ν2)

2 (q2)× · · · × φ(ν3N−6)

3N−6 (q3N−6)

where each φ(νi )i is eigenfunction of harmonic oscillator with potential Vi = Ωiq

2i and

with quantum number νi ⇒ state quantum numbers (ν1, ν2, . . . , ν3N−6)

Total symmetry:

Γvib = (Γν1 )ν1 ⊗ (Γν2 )ν2 ⊗ · · · ×(Γν3N−6

)ν3N−6

Example: (2, 1, 3)-state of H2O (ν1 is symmetric stretch (A1), ν2 is bending (A1), ν3is asymmetric stretch (B2))

Γ(2,1,3)vib

= A1 ⊗ A1 ⊗ A1 ⊗ B2 ⊗ B2 ⊗ B2 = B2

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Application: Symmetry of electronic states

Electronic states, Hartree-Fock:

ψel ≈ Aφ1(x1)× φ2(x2) · · · × φm(xm)

Total symmetry:

Γel = (Γν1 )n1 ⊗ (Γν2 )n2 ⊗ · · · × (Γνm )nm

with

Γνj . . . IRREP of orbital j

nj . . . occupation of orbital j

Note: Totally lled subshells are totally symmetric

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Example:

electronic ground state of H2O, conguration . . . (b2)2(a1)2(b1)2

Γel = B2 ⊗ B2 ⊗ A1 ⊗ A1 ⊗ B1 ⊗ B1 = A1

Label X 1 A1

Example:

electronic ground state of H2O+, conguration . . . (b2)2(a1)2(b1)1

Γel = B2 ⊗ B2 ⊗ A1 ⊗ A1 ⊗ B1 = B1

Label X+ 2B1

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Example: electronic ground state of BH3

Orbitals: B → 2s, 2px , 2py , 2pz , 3×H → 3×1s

where Γ3D is three-dimensional representation spanned by the H 1s-orbitals

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Reduction:

cA′1=

1

12(3 · 1 + 1 · 3 · 1 + 3 · 1 + 1 · 3 · 1) = 1

cA′2=

1

12(3 · 1− 1 · 3 · 1 + 3 · 1− 1 · 3 · 1) = 0

cE ′′ =1

12(3 · 2 + 1 · 0 + 3 · 2 + 1 · 0) = 1

(no need to go further) ⇒ Γ3D = A′1 ⊕ E ′

Axel Schild 2020-04-24/05-08 63


PE ′1s(1) =1

12

(2E − C3 − C2

3 + 2σh − S3 − S23

)1s(1)

=1

12(21s(1)− 1s(2) - 1s(3) + 21s(1)− 1s(2)− 1s(3))

=1

3

(1s(1)−

1

2(1s(2) + 1s(3))

)PE ′1s(2) =

1

3

(1s(2)−

1

2(1s(3) + 1s(1))

)PE ′1s(3) =

1

3

(1s(3)−

1

2(1s(1) + 1s(2))

)→ linear dependence, orthogonalize

Axel Schild 2020-04-24/05-08 64


molecular orbitals:

combine B 2s- & 2p-orbitals with H 1s-orbital combinations of same symmetry

2s(B): A′1 ↔ A′12px (B), 2py (B): E ′1 ↔ E ′12pz (B): A′′2 → non-bonding

electronic conguration: (1a′1)2(2a′1)2(1e′)4(1a′′2 )0 label: X 1A′1

Axel Schild 2020-04-24/05-08 65


oxygen molecule O2

Axel Schild 2020-04-24/05-08 66


oxygen molecule O2

electronic conguration: (1σg )2(1σ∗u )2(2σg )2(2σ∗u )2(3σg )2(1πu)4︸︷︷︸Σ+g

(1π∗g )2(3σ∗u )0

Axel Schild 2020-04-24/05-08 67


oxygen molecule O2

gerade & ungerade: g ⊗ g = u × u = g , g ⊗ u = u ⊗ g = u

electronic conguration: (1σg )2(1σ∗u )2(2σg )2(2σ∗u )2(3σg )2(1πu)4︸︷︷︸Σ+g

(1π∗g )2(3σ∗u )0

Πg ⊗ Πg = Σg ⊕[Σ−g]⊕∆g

Axel Schild 2020-04-24/05-08 68


oxygen molecule O2 electronic conguration:

(1σg )2(1σ∗u )2(2σg )2(2σ∗u )2(3σg )2(1πu)4︸︷︷︸Σ+g

(1π∗g )2(3σ∗u )0

Πg ⊗ Πg = Σ+g ⊕

[Σ−g]⊕∆g

• anti-symmetry spatial part/symmetric spin part: 3Σ−g

• symmetry spatial part/anti-symmetric spin part: 1Σ+g ,

1∆g

Axel Schild 2020-04-24/05-08 69

Application: Pauli principle and allowed states

Molecular wavefunction:

ψtot ≈ ΦelΦvibΦrotΦnsΦes

Γtot = Γel ⊗ Γvib ⊗ Γrot ⊗ Γns ⊗ Γes

ψtot must transform as IRREP with characters

χ(Oj ) =

NF∏i=1

(−1)Pi (Oj )

NF . . . number of types of identical fermions

Pi (Oj ) . . . parity of permutation of ith kind of fermions

• If Oj applies to bosons: χ(Oj ) = +1.

• If Oj applies to fermions:

if Oj is even number of permutations: χ(Oj ) = +1

if Oj is odd number of permutations: χ(Oj ) = −1

Axel Schild 2020-04-24/05-08 70

Application: Pauli principle and allowed states

Example: two protons (fermions) in H2O

Permutation (12) ≡ C z2 : χ((1, 2)) = χ(C z

2 )!

= −1 → IRREPS B1 and B2

Electronic ground state:

Γtot︸︷︷︸B1 or B2

= Γel︸︷︷︸A1

⊗ Γvib︸︷︷︸A1

⊗Γrot ⊗ Γns ⊗ Γes︸︷︷︸A1

→ only some combinations of rotational levels & nuclear spin functions are allowed

Example: two deuteron (bosons) in D2O

Permutation (12) ≡ C z2 : χ((1, 2)) = χ(C z

2 )!

= +1 → IRREPS A1 and A2

Axel Schild 2020-04-24/05-08 71

Application: Selection rules with group theory

Interaction with radiation eld: V = −~µlab · ~E

molecule-xed frame vs. lab frame (dened by radiation eld)

Euler angles:• rotate around Z by ϕ: (x ′, y ′, z ′)• rotate around y ′ by θ: (x ′′, y ′′, z ′′)• rotate around z ′′ by χ: (x , y , z)

Axel Schild 2020-04-24/05-08 72



dipole operator, molecule-xed frame:

~µ =

µxµyµz

=N∑i=1

qi

xiyizi

but radiation eld ~E is in the space-xed frame X ,Y ,Z , so we need

~µlab =

µXµYµZ

= λ~µ

with transformation matrix λ.

Axel Schild 2020-04-24/05-08 73



Assume ~E = (0, 0,EZ )T , then:

〈ψf |V |ψi〉 = 〈ψf |~µZ |ψi〉EZ =

⟨ψf

∣∣∣∣∣∣∑

α=x,y,z

λZαµα

∣∣∣∣∣∣ψi⟩

EZ

initial state: ψi ≈ Φ′′elΦ′′vibΦ′′rotΦ

′′nsΦ′′es

nal state: ψf ≈ Φ′elΦ′vibΦ′rotΦ

′nsΦ′es

Selection rules from⟨Φ′elΦ

′vibΦ′rot

∣∣∣∣∣∣∑

α=x,y,z

λZαµα

∣∣∣∣∣∣Φ′′elΦ′′vibΦ′′rot

⟩〈Φ′es|Φ′′es〉〈Φ′ns|Φ′′ns〉

Axel Schild 2020-04-24/05-08 74


Selection rules from⟨Φ′elΦ

′vibΦ′rot

∣∣∣∣∣∣∑

α=x,y,z

λZαµα

∣∣∣∣∣∣Φ′′elΦ′′vibΦ′′rot

⟩〈Φ′es|Φ′′es〉〈Φ′ns|Φ′′ns〉

• Φ′′es!

= Φ′es: ∆S = 0

• Φ′′ns!

= Φ′ns: ∆I = 0

Note that φel, φvib, µα do not depend on Euler angles:∑α=x,y,z

⟨Φ′rot

∣∣λZα ∣∣Φ′′rot⟩ ⟨

Φ′elΦ′vib

∣∣µα ∣∣Φ′′elΦ′′vib

⟩

Axel Schild 2020-04-24/05-08 75


∑α=x,y,z

⟨Φ′rot


Φ′elΦ′vib


⟩From 〈Φ′rot |λZα |Φ′′rot〉 come angular momentum selection rules:

• ∆J = 0,±1; 0 = 0

• polarization along Z , ∆M = 0

• polarization along X/Y , ∆M = ±1• (diatomics) dipole along z-axis: parallel transition, ∆Λ = 0

• (diatomics) dipole along x , y -axis: perpendicular transition, ∆Λ = ±1

NB: M is projection of J on Z -axis

Axel Schild 2020-04-24/05-08 76


∑α=x,y,z

⟨Φ′rot


Φ′elΦ′vib


⟩Vanishing integral theorem:

〈ψf |O|ψi〉 6= 0⇒ Γ(ψ∗f )⊗ Γ(O)⊗ Γ(ψi) ⊃ Γsym

product needs to contain the totally symmetric IRREP of the group

Axel Schild 2020-04-24/05-08 77


Vanishing integral theorem:

〈ψf |O|ψi〉 6= 0⇒ Γ(ψ∗f )⊗ Γ(O)⊗ Γ(ψi) ⊃ Γsym

Rotational spectroscopy, Φ′el = Φ′′el, Φ′vib = Φ′′vib• ⟨Φ′elΦ′vib ∣∣µα ∣∣Φ′elΦ

′vib

⟩is expectation value of µα

• molecule needs permanent dipole moment

Axel Schild 2020-04-24/05-08 78


Vibrational spectroscopy, Φ′el = Φ′′el, Φ′vib 6= Φ′′vib• ⟨Φ′elΦ′vib ∣∣µα ∣∣Φ′elΦ

′′vib

⟩=⟨Φ′vib

∣∣µel,α ∣∣Φ′′vib⟩

• we need Γ′vib ⊗ Γα ⊗ Γ′′vib ⊃ Γsym

Example H2O, excitation of symmetric stretch (0, 0, 0)→ (1, 0, 0):

Γ′vib ⊗ Γα ⊗ Γ′′vib = A1 ⊗B1

B2

A1

⊗ A1 =

B1

B2

A1

→ allowed transition comes from change of z-component of dipole

Example H2O, excitation of asymmetric stretch (0, 0, 0)→ (0, 0, 1):

Γ′vib ⊗ Γα ⊗ Γ′′vib = B2 ⊗B1

B2

A1

⊗ A1 =

A2

A1

B2

→ allowed transition comes from change of y -component of dipole

Axel Schild 2020-04-24/05-08 79


Electronic spectroscopy, Φ′el 6= Φ′′el

• ⟨Φ′elΦ′vib ∣∣µα ∣∣Φ′′elΦ′′vib

⟩=⟨

Φ′vib

∣∣∣µel,α ∣∣∣Φ′′vib

⟩• we need Γ′el ⊗ Γα ⊗ Γ′′el ⊃ Γsym to have µel,α 6= 0

Example H2O, excitation from electronic ground state X 1A1:

Γ′el ⊗ Γα ⊗ Γ′′el = Γ′el ⊗B1

B2

A1

⊗ A1!

=

A1

A1

A1

→ transitions to B1,B2,A1 are allowed, those to A2 are forbidden

Axel Schild 2020-04-24/05-08 80


Electronic spectroscopy, Φ′el 6= Φ′′el

• ⟨Φ′elΦ′vib ∣∣µα ∣∣Φ′′elΦ′′vib

⟩=⟨

Φ′vib


⟩• we need Γ′el ⊗ Γα ⊗ Γ′′el ⊃ Γsym to have µel,α 6= 0

Expand electronic transition dipole element along nuclear coordinates:

µel,α = µ,eqel,α +

3N−6∑j=1

(∂µel,α

∂Qj

)eq

Qj + . . .

Zero-th order: ⟨Φ′vib


⟩≈⟨Φ′vib

∣∣Φ′′vib⟩µ,eqel,α

The Franck-Condon factor |⟨Φ′vib

∣∣Φ′′vib⟩|2 is proportional to the intensity of the

transition → if transition is electronically allowed, we have the vibrational selection

rule Γ′vib ⊗ Γ′′vib ⊃ Γsym.

Axel Schild 2020-04-24/05-08 81


The Franck-Condon factor |⟨Φ′vib

∣∣Φ′′vib⟩|2 is proportional to the intensity of the

transition → if transition is electronically allowed, we have the vibrational selection

rule Γ′vib ⊗ Γ′′vib ⊃ Γsym.

Example: H2O X 1A1(0, 0, 0)→ C1B1

excitation from electronic & vibrational ground state to an electronically excited state

Γ′′vib = A1 ⇒ Γ′vib!

= A1

• symmetric stretch ν1: A1, allowed

• bending mode ν2: A1, allowed

• asymmetric stretch ν3: B2, forbidden if ν3 odd

Axel Schild 2020-04-24/05-08 82


Separation of electronic and vibrational wavefunction is only an approximation, in

general

Γ′vib ⊗ Γ′el ⊗ Γα ⊗ Γ′′vib ⊗ Γ′′el ⊃ Γsym

→ transitions may be weakly allowed

Example: H2O X 1A1(0, 0, 0)→ A2(0, 0, 1)

• X 1A1(0, 0, 0): Γ′′el = A1, Γ′′vib = A1, Γ′′el ⊗ Γ′′vib = A1

• A2(0, 0, 1): Γ′′el = A2, Γ′′vib = B2, Γ′′el ⊗ Γ′′vib = B1

Γ′vib ⊗ Γ′el ⊗ Γα ⊗ Γ′′vib ⊗ Γ′′el = B1 ⊗B1

B2

A1

⊗ A1 =

A1

A2

B1

→ vibronically allowed due to change in x-component of dipole

Axel Schild 2020-04-24/05-08 83

Documents

4cm Physical Chemistry 5 (Spectroscopy) 1mm A short ... · Our concern: Symmetry operations O^ i such that h O^ i;H^ i = 0!eigenvalues of O^ i can be used to categorize eigenstates