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Group theory, representations and their applications in solid state

TMS 2018

1

Outline of the course

1. Introduction: Symmetries, degeneracies and representations.

2. Irreducible representations as building blocks. Application to molecular vibrations.

3. Operations with representations: Physical properties and spectra.

4. Spin and double valued representations. Splitting of atomic orbitals in crystals.

5. Representation theory and electronic bands.

2

Molecular polarizability

From vectors to tensors

To lowest order, the induced dipole moment of a molecule is proportional to the applied electric field

~P = ↵~E

Pi = ↵ijEj

is the polarizability tensor. In components ↵

The transformation properties of the polarizability are those of a tensor

8g 2 G : ↵ij ! Vik(g)Vjl(g)↵kl = V (g)↵V t(g)<latexit sha1_base64="dOxiLjmdvaUOwv+zDZp9e47Z+wo=">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</latexit><latexit sha1_base64="dOxiLjmdvaUOwv+zDZp9e47Z+wo=">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</latexit><latexit sha1_base64="dOxiLjmdvaUOwv+zDZp9e47Z+wo=">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</latexit><latexit sha1_base64="dOxiLjmdvaUOwv+zDZp9e47Z+wo=">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</latexit>

3



The polarizability describes a propertity of the molecule in equilibrium and must be invariant under all its symmetries

We will use group theory to find the most general form of the polarizability compatible with the symmetry constraints

This is easier if we consider the electrostatic energy

E ⇠ ~P · ~E ⇠ ↵ijEiEj

↵g = V (g)↵V (g)t = ↵ ) [↵, V (g)] = 0, 8g 2 G<latexit sha1_base64="xpTEHjnBnVp9yaM66OtL/Dago9Q=">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</latexit><latexit sha1_base64="xpTEHjnBnVp9yaM66OtL/Dago9Q=">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</latexit><latexit sha1_base64="xpTEHjnBnVp9yaM66OtL/Dago9Q=">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</latexit><latexit sha1_base64="xpTEHjnBnVp9yaM66OtL/Dago9Q=">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</latexit>

4



The energy is quadratic in the electric field.

All the representations in this equation are unitary: norms and scalar products are invariant under the symmetry group

The energy must be a linear combination of quadratic invariants in the electric field.

The electric field belongs to the vector representation. For the group

V (Ex

, Ey

, Ez

) = A1(Ez

) + E(Ex

, Ey

)

C3v

5


A1(Ez

)⇥A1(Ez

) ! E2z

E(Ex

, Ey

)⇥ E(Ex

, Ey

) ! E2x

+ E2y

This implies↵ij

Ei

Ej

= a(E2x

+ E2y

) + bE2z

or

↵ =

0

@a 0 00 a 00 0 b

1

A

C3vThis is valid for any molecule with symmetry group

V (Ex

, Ey

, Ez

) = A1(Ez

) + E(Ex

, Ey

)<latexit sha1_base64="GL4eUHQ48lx/gpXohZQqG6dG7dQ=">AAACBXicbVDLSgMxAMzWV62vVY9eQovQopRd8dWDUJWCxwr2Ae2yZNNsG5p9kGTFdelZf0ZPot78AX/AvzFbW/A1EJjMTCAzTsiokIbxoWVmZufmF7KLuaXlldU1fX2jKYKIY9LAAQt420GCMOqThqSSkXbICfIcRlrO8Dz1W9eECxr4VzIOieWhvk9dipFUkq3nm8WafbNbs2N1bksnp7ZZTMlObaqXbL1glI0xYEoOKocVaE6VKSmACeq2/t7tBTjyiC8xQ0J0TCOUVoK4pJiRUa4bCRIiPER90lHURx4RVjLuMoLbbsChHBA4vn/PJsgTIvYclfGQHIjfXir+53Ui6R5bCfXDSBIfq4jy3IhBGcB0EtijnGDJYkUQ5lT9EuIB4ghLNVxO1f9T9i9p7pVNo2xe7heqZ5MhsmAL5EERmOAIVMEFqIMGwOAePIIX8KrdaQ/ak/b8Fc1okzeb4Ae0t08ZX5Wr</latexit><latexit sha1_base64="GL4eUHQ48lx/gpXohZQqG6dG7dQ=">AAACBXicbVDLSgMxAMzWV62vVY9eQovQopRd8dWDUJWCxwr2Ae2yZNNsG5p9kGTFdelZf0ZPot78AX/AvzFbW/A1EJjMTCAzTsiokIbxoWVmZufmF7KLuaXlldU1fX2jKYKIY9LAAQt420GCMOqThqSSkXbICfIcRlrO8Dz1W9eECxr4VzIOieWhvk9dipFUkq3nm8WafbNbs2N1bksnp7ZZTMlObaqXbL1glI0xYEoOKocVaE6VKSmACeq2/t7tBTjyiC8xQ0J0TCOUVoK4pJiRUa4bCRIiPER90lHURx4RVjLuMoLbbsChHBA4vn/PJsgTIvYclfGQHIjfXir+53Ui6R5bCfXDSBIfq4jy3IhBGcB0EtijnGDJYkUQ5lT9EuIB4ghLNVxO1f9T9i9p7pVNo2xe7heqZ5MhsmAL5EERmOAIVMEFqIMGwOAePIIX8KrdaQ/ak/b8Fc1okzeb4Ae0t08ZX5Wr</latexit><latexit sha1_base64="GL4eUHQ48lx/gpXohZQqG6dG7dQ=">AAACBXicbVDLSgMxAMzWV62vVY9eQovQopRd8dWDUJWCxwr2Ae2yZNNsG5p9kGTFdelZf0ZPot78AX/AvzFbW/A1EJjMTCAzTsiokIbxoWVmZufmF7KLuaXlldU1fX2jKYKIY9LAAQt420GCMOqThqSSkXbICfIcRlrO8Dz1W9eECxr4VzIOieWhvk9dipFUkq3nm8WafbNbs2N1bksnp7ZZTMlObaqXbL1glI0xYEoOKocVaE6VKSmACeq2/t7tBTjyiC8xQ0J0TCOUVoK4pJiRUa4bCRIiPER90lHURx4RVjLuMoLbbsChHBA4vn/PJsgTIvYclfGQHIjfXir+53Ui6R5bCfXDSBIfq4jy3IhBGcB0EtijnGDJYkUQ5lT9EuIB4ghLNVxO1f9T9i9p7pVNo2xe7heqZ5MhsmAL5EERmOAIVMEFqIMGwOAePIIX8KrdaQ/ak/b8Fc1okzeb4Ae0t08ZX5Wr</latexit><latexit sha1_base64="GL4eUHQ48lx/gpXohZQqG6dG7dQ=">AAACBXicbVDLSgMxAMzWV62vVY9eQovQopRd8dWDUJWCxwr2Ae2yZNNsG5p9kGTFdelZf0ZPot78AX/AvzFbW/A1EJjMTCAzTsiokIbxoWVmZufmF7KLuaXlldU1fX2jKYKIY9LAAQt420GCMOqThqSSkXbICfIcRlrO8Dz1W9eECxr4VzIOieWhvk9dipFUkq3nm8WafbNbs2N1bksnp7ZZTMlObaqXbL1glI0xYEoOKocVaE6VKSmACeq2/t7tBTjyiC8xQ0J0TCOUVoK4pJiRUa4bCRIiPER90lHURx4RVjLuMoLbbsChHBA4vn/PJsgTIvYclfGQHIjfXir+53Ui6R5bCfXDSBIfq4jy3IhBGcB0EtijnGDJYkUQ5lT9EuIB4ghLNVxO1f9T9i9p7pVNo2xe7heqZ5MhsmAL5EERmOAIVMEFqIMGwOAePIIX8KrdaQ/ak/b8Fc1okzeb4Ae0t08ZX5Wr</latexit>

6



V (Ex

, Ey

, Ez

) = T2(Ex

, Ey

, Ez

)

the only quadratic invariant is E2x

+ E2y

+ E2z

This implies an isotropic polarizability

↵ =

0

@a 0 00 a 00 0 a

1

A

For any molecule with symmetry Td

7

Molecular vibrationsIf the molecule has A atoms, its configuration is described by a 3A dimensional vector withu = (~u1, ~u2, . . . , ~uA)

The kinetic energy is given by

where M is the diagonal mass matrix

T =AX

↵=1

1

2m↵~u

2↵ ⌘ 1

2utMu

M = dm1,m1,m1, . . . ,mA,mA,mAc

~r↵ = ~R↵ + ~u↵

8

Molecular vibrationsThe potential energy is expanded about the equilibrium configuration keeping only second order contributions

where U is the symmetric, but non-diagonal potential matrix

Uij =@2U

@ui@uj

U(ui) '3AX

i,j=1

1

2Uijuiuj =

1

2utUu

The lagrangian is thus given by

L =1

2utMu� 1

2utUu

9

Molecular vibrationsThis lagrangian gives rise to a set of coupled linear differential equations

The normal modes and frequencies are found by making the ansatz which givesu(t) = nei!t

(U �M!2)n = 0

This is not a standard eigenvalue problem due to the presence of the matrix M. To solve it, we make the change of coordinates .u = M1/2u

10

Molecular vibrationsThe lagrangian in the new variables

L =1

2˙ut ˙u� 1

2utU u

with gives rise to an ordinary eigenvalue problem.

U = M�1/2UM�1/2

(U � !2)n = 0

Solving the eigenvalue problem yields 3A frequencies .The corresponding normal modes satisfy .nt

↵n� = �↵�

!2↵

Reverting to the original coordinates shows that normal modes satisfy

nt↵Mn� = �↵�

11

Molecular vibrationsPURE VIBRATIONS The mechanical representation acts on

LM

= Ltrans

� Lrot

� LV ib

Ltrans : u = (~t, . . . ,~t )

is defined by the constraints LV ib

AX

↵=1

m↵~u↵ = 0 ,AX

↵=1

m↵(~R↵ ⇥ ~u↵) = 0

that imply with respect to the mass metric

LV ib

? Ltrans

and LV ib

? Lrot

(u1, u2) = ut1Mu2

Lrot

: u = (⇥� ⇥ ⇥R1, . . . , ⇥� ⇥ ⇥RA

)

12

Molecular vibrations

Assume we have obtained symmetry-adapted coordinates for the IRs in the vibrational representation

Consider the methane molecule with symmetry .Td

V ib = A1(q1) + E(q2, q3) + T2(q4, q5, q6) + T2(q7, q8, q9)

The normal modes and frequencies are obtained by diagonalizing the potential matrix U = M�1/2UM�1/2

(U � !2)n = 0


13

Molecular vibrationsThe potential matrix has to be invariant under all the elements of the group . Td

[U ,M(g)] = 0, 8g 2 Td

These constrains can be solved more easily by considering the invariance of the potential energy

EP =1

2utU u

This must be a linear combination of quadratic invariants in the symmetric coordinates {q1, q2, . . . , q9}


14


This shows that the most general potential energy for the methane molecule depends only on 5 free parameters.

A1(q1)⇥A1(q1) ! q21

E(q2, q3)⇥ E(q2, q3) ! q22 + q23

T2(q4, q5, q6)⇥ T2(q4, q5, q6) ! q24 + q25 + q26

T2(q7, q8, q9)⇥ T2(q7, q8, q9) ! q27 + q28 + q29T2(q4, q5, q6)⇥ T2(q7, q8, q9) ! q4q7 + q5q8 + q6q9


15


or


16


Reorder the coordinates


17



18


Reorder the coordinates


19


In general, for each IR with dimension d and multiplicity m in the decomposition of Vib, there are d identical mxm matrices in the diagonal of the potential matrix.

Upon diagonalization, they give rise to m different d-degenerated frequencies.


20

Given two representations and acting on representation spaces of dimensions and , the tensors in belong to the product representation

T1 T2

d1 d2L1 ⌦ L2

T3 = T1 ⇥ T2

with�3(g) = �1(g)�2(g)

General representations

Rank-2 tensors live in the 9-dimensional space and transform like the productsR3 ⌦ R3

{x1x2, y1y2, z1z2, x1y2, y1x2, y1z2, z1y2, z1x2, x1z2}

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Rank-2 tensors belong to the product representationV ⇥ V ⌘ V 2

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Appendix: The product representation

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Number of bilinear invariantsGiven two real IRs and acting on and , bilinears in their coordinates belong to the product representation .

⌧1 ⌧2

⌧3 = ⌧1 ⇥ ⌧2

L1(x1, . . . , xd1)L2(y1, . . . , yd2)

There are no invariants if . And for equivalent IRs, the only bilinear invariant is given by

⌧1 6⌘⌧2

x1y1 + . . . xdyd

The number of bilinear invariants is equal to the number of times the identity representation is contained in ⌧3 = ⌧1 ⇥ ⌧2

m1 =1

N

X

g2G

�⇤3(g) · 1 =

1

N

X

g2G

�1(g)�2(g) = (�1,�2) = �1,2

This form of the invariant is valid only if the matrices of the two equivalent IRs are identical.

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