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
Molecular polarizability
From vectors to tensors
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
Molecular polarizability
From vectors to tensors
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
From vectors to tensors
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
Molecular polarizability
From vectors to tensors
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
Molecular vibrations
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}
Molecular vibrations
14
Molecular vibrations
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
Molecular vibrations
15
Molecular vibrations
or
Molecular vibrations
16
Molecular vibrations
Reorder the coordinates
Molecular vibrations
17
Molecular vibrations
Molecular vibrations
18
Molecular vibrations
Reorder the coordinates
Molecular vibrations
19
Molecular vibrations
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.
Molecular vibrations
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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