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Translation symmetry, Space groups,

Bloch functions, Fermi energy

Roberto Orlando

Dipartimento di Chimica

Università di Torino

Via Pietro Giuria 5, 10126 Torino, Italy

roberto.orlando@unito.it

Outline

The crystallographic model of a crystal:

• translation invariance and lattice

• point symmetry and crystallographic point groups

• Bravais lattices and space groups

Our model of a perfect crystal

The Pack-Monkhorst net

Local basis sets and Bloch functions

The calculation of the Crystalline Orbitals with a local basis set

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Quartz structure

Quartz crystals

Si

O

Crystal structures

How do we generate a crystal from a structural motif?

The infinite crystal of quartz can be generated from two atoms, Si—O,

by the application of symmetry operations

a2

a1

2a1 + 3a2

-a1 + 2a2

The lattice

Every lattice vector is a linear combination of the basis vectors

A lattice is translation invariant: translations along lattice vectors

are symmetry operations

a1, a2 : basis vectors

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a

2

a3

a

b

g

a1

Bravais lattice

Unit cell: a volume in space that fills

space entirely when translated by all

lattice vectors.

The obvious choice:

a parallelepiped defined by a1, a2, a3,

three basis vectors with

the best a1, a2, a3 are as orthogonal

as possible

the cell is as symmetric as possible

(14 types)

A unit cell containing one lattice point is called primitive cell.

Point symmetry operations

Rotation around an axis:

2

n = 360°/

axis order 3

When a point symmetry operation is applied to a lattice, the

transformed unit cell is indistiguishable from the original unit cell.

During the action of a point symmetry operation at least one point stays put

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1 3 4 62

....

Rotations

m Reflexion 1Inversion

Rotation with inversion

Combining rotation axes

Combining rotation axes and inversion

1, 2, 3, 4, 6

3

4

6

1 c

2 m a l’eixto the axis

32 crystallographic point goups

Triclinic Monoclinic Orthorhombic Trigonal Tetragonal Hexagonal Cubic

1 2 3 4 6

1 2 m 3 4 6

2/ m 4 / m 6 / m 3m

2mm 3m 4mm 6mm

222 322 422 622 23

432

3m 42m 6 2m 43m

mmm 4/ mmm 6/ mmm 3m m

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Triclinic: 1, 1

P

P

P

P

P

P

I F

C

C F

I

I

R

Monoclinic: 2, m, 2/m

Orthorhombic: 222,

mm2, mmm

Tetragonal: 4, 4, 422,

42m, 4mm, 4/mmm, 4/m

Cubic: 23, 432, 43m,

m3, m3m

Trigonal: 3, 32, 3, 3m

Hexagonal: 6, 622, 6mm,

6, 6/m, 6/mmm, 6m2

The 7 Crystal Systems

Primitive and centered lattices

Primitive lattice

P

a b

c

181 8 m a

b

Centered lattices:

Body-centered

I

ab

c

t

2 1 8

1 8 m

c 21 b

21 a

21 t

2

1 ,2

1 ,2

1 (0,0,0)

2

1 z ,2

1 y 2

1 x z) y, (x, ,

a

b

½

Face-centered

F, A, B, C

I

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Triclinic: 1, 1

P

P

P

P

P

P

I F

C

C F

I

I

R

Monoclinic: 2, m, 2/m

Orthorhombic: 222,

mm2, mmm

Tetragonal: 4, 4, 422,

42m, 4mm, 4/mmm, 4/m

Cubic: 23, 432, 43m,

m3, m3m

Trigonal: 3, 32, 3, 3m

Hexagonal: 6, 622, 6mm,

6, 6/m, 6/mmm, 6m2

The 14 Bravais Lattices

Screw axes

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b4

a4

d

n

b

a

(001) types

a2

b2

a2

a b 001

}b2

c4

a4

c2

a2

c2a

2 }a c n d

(001) types

100 010c2

c2

b2

c4

b4

dn

cb} types parallel

to (001)

SIMBOLS

PROJ. I I PROJ.

a, b

c

n

d

b2

c

Glide planes

The 230 Space Groups

All possible combinations of symmetry operations compatible with

the 14 Bravais lattices originate the 230 space groups

Classification and details about space groups can be found in the

International Tables for Crystallography vol. A

ed. Theo Hahn, Kluwer Academic Publishers

Hermann-Mauguin symbols

diagrams of the space groups

symmetry operations

general and special positions

subgroups

supergroups

reflection conditions

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Defining the crystal structure: α-quartz

Space group: P 32 2 1 (No. 154)

Unit cell parameters (γ=120°):

a = 4.916 Å c = 5.4054 Å

Formula unit: SiO2

No. of formula units per cell: 3 SiO2

Fractional coordinates of the atoms in the asymmetric unit:

Si

O

0.4697x

a

0.4135x

a

0y

a 0

z

c

0.2669y

a 0.1191

z

c

(x, y, z – atom coordinates in terms of the basis vectors - in Å)

Crystal system: trigonal

Hermann-Mauguin symbol

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Generating the unit cell of α-quartz

0 1 0 0

1 1 0 0

0 0 1 2 / 3

+3

1 1 0 0

1 0 0 0

0 0 1 1/ 3

-3

1 1 0 0

0 1 0 0

0 0 1 1/ 3

2

1 0 0 0

1 1 0 0

0 0 1 2 / 3

2

Symmetry operations:

0.4697 0.0000 0.0000

0.0000 0.4697 0.3333

0.4697 0.4697 0.3333

0.4135 0.2669 0.1191

0.2669 0.1466 0.2142

0.1466 0.4135 0.4524

0.2669 0.4135 0.1191

0.1466 0.4135 0.2142

0.4135 0.1466 0.4524

Si fractional coordinates O fractional coordinates

0 1 0

1 0 0

0 0 1

2

1 0 0

0 1 0

0 0 1

1

b

a

along the two-fold axes (special position)

(general position)

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The Wigner-Seitz cell

Wigner-Seitz cell: the portion of

space which is closer to one

lattice point than to anyone else.

A Wigner-Seitz cell is primitive by

construction.

There are 14 different types of

Wigner-Seitz cells.

A reciprocal lattice with lattice basis vectors b1, b2, b3 corresponds

to every real (or direct) lattice with lattice basis vectors a1, a2, a3.

Basis vectors obey the following orthogonality rules:

The reciprocal lattice

b1·a1 = 2π b2·a2 = 2π b3·a3 =2π

b1·a2 = 0 b2·a3 = 0 b3·a1 = 0

b1·a3 = 0 b2·a1 = 0 b3·a2 = 0

or, equivalently,

b1 = 2π/V a2Λa3 b2 = 2π/V a3Λa1 b3 = 2π/V a1Λa2

V = a1·a2Λa3 V* = (2π)3/V

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The first Brillouin zone

Unit cells for face-

centered-cubic crystals

First Brillouin zone

for a fcc lattice

Wigner-Seitz cell for

body-centered-cubic

crystals

transformation to the

reciprocal space

Brillouin zone: Wigner-Seitz cell in the

reciprocal space.

Special points in the Brillouin zone have

been classified and labelled with letters,

used in the specification of paths.

The model of a perfect crystal

A real crystal: a macroscopic finite array of a very large number n of atoms/ions

with surfaces

the fraction of atoms at the surface is proportional to n -1/3 (very small)

if the surface is neutral, the perturbation due to the boundary is limited to a few

surface layers

a real crystal mostly exhibits bulk features and properties

A macro-lattice of N unit cells is good model of such a crystal

as N is very large and surface effects are negligible in the bulk, the macro-lattice

can be repeated periodically under the action of translation vectors N1 a1, N2 a2,

N3 a3 without affecting its properties

thus, our model of a perfect crystal coincides with the crystallographic model of an

infinite array of cells containing the same group of atoms

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i

i i i

N

N i a aT AT

Commutative group of N

translation operators Ti

The macro-lattice

A1

A2

a1

a2

Our model of a crystal consists of an infinite array of macro-lattices.

Every macro-lattice contains N unit cells

Every function or operator defined for the crystal then admits the following

boundary conditions:

f (r + Niai) = f (r) for i = 1, 2, 3 translation

symmetry

a new metric is defined

Born-Von Kárman boundary conditions

Ai = Ni ai for i = 1, 2, 3

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A1

A2

a1

a2

b1

b2

B1

B2

From real to reciprocal space: metrics

Real

space

Reciprocal

space

Real space lattice:

spanned by a1, a2

Real space macro-lattice:

spanned by A1, A2

Reciprocal space lattice:

spanned by b1, b2

Reciprocal space micro-lattice:

spanned by B1, B2

A reciprocal lattice can be associated with the macro-lattice with metric A:

the reciprocal micro-lattice with metric B = N-1 b

this micro-lattice is much denser than the original reciprocal lattice

with metric b

the unit cell of the original reciprocal lattice with metric b can be

partitioned into N micro-cells, each located by a vector (wave vector)

km:

the Pack-Monkhorst net

Integers Ni are called the shrinking factors

When referred to the 1st Brillouin zone, the complete set of the km vectors form

31 21 2 3 1 2 3

1 2 3

, , integersm

mm mm m m

N N N k b b b

Pack-Monkhorst net

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0

0

mk

mk

jl

jl…………………… ……………………

1

1

1

1 1…………………… ……………………

exp( )m ji k l exp( )m ji k l

exp( )m ji k l exp( )m ji k l

……………

……………

……

……

…………

……

…………

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N translation operations N classes N one-dimensional irreducible

representations

associated with

N km vectors of

Brillouin Zone

N

m m j mm

j 1

exp i N δ

k k l1

exp ( )N

m j j jj

m

i N

k l l

Due to the

orthogonality

between rows

and columns

&

……

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Character Table of the translation group

the local basis set

M×N atomic orbitals

Choice of a set of local functions L.C.A.O method

(Linear Combination of Atomic Orbitals)

M atomic orbitals per cell

( )j r - l

*( , ) ( ) ( ) ( - ) j jj j

V

f f d l l r - l r r l r

representation of

operator f

( , )j jf l l

( ) r

Atomic orbital basis set

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*( , ) ( ) ( ) ( ) j jj j

V

f f d l l r l r r l r

( )j r l r

( ) ( )jff r r l

, , ,j jj j jf f f 0l l l l 0 l

but, for translation invariance and if f(r) is periodic with the same

periodicity of the direct lattice,

j j j j r l r l l r l

the origin is translated by lj’

If in

*, ( ) ( ) j jj j j

V

f f d

0 l l r r l r l l r

so that matrix elements are also translation invariant

Translation invariance of matrix elements

Consequences of translation invariance:

Square Matrix: N2

blocks of size M

lj

lj’

0

( , ) ( , )j j j jf f l l 0 l l

Representation of operators in the AO basis

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The local basis

M×N atomic orbitals

( )j r - l

Bloch function basis basis for the irreducible representation

of the translations group

( , )m k r

1

1( , ) exp( ) ( )

N

m m j j

j

iN

k r k l r l

*

, 1

1( , ) ( ) m j m j

Ni

m m j j

j j V

f e f dN

k l k l

k k r l r r l r

In this basis set the

matrix element fμν

depends on two new

indices: km and km’

which are vectors of

the reciprocal space:

Bloch functions

1

( , ) e ( )m j

m m

Ni

m m j

j

f f

k l

k kk k l

for translation invariance and after multiplying and dividing by

1 1

1( , ) , m j m j

N Ni i

m m j j

j j

f e e fN

k l k l

k k l l

1 1

1( , ) , m j jm m j

N Nii

m m j j

j j

f e e fN

k l lk k l

k k 0 l l

m jie k l

but, as the sum over j extends to all direct lattice vectors, it is equivalent

to summing over all vectors lj” = lj – lj’

1 1

1( , ) ,m m j m j

N Ni i

m m j

j j

f e e fN

k k l k l

k k 0 l

and recalling the character orthogonality relation for rows, we get

Bloch theorem

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the AO basis

( )j r - l

Bloch function basis

( , )m k r

0

N blocks of size M associated with km

0

0

Consequences of Bloch theorem

Representation of one-electron operator F in the basis of the AOs :

Calculation of (M 2 × N) elements of F: ( )F l

At every km, the eigenvalue equation of size M is solved:

Representation in the basis of Bloch functions:

1

) exp( ) ( )N

m m j j

j

F i F

(k k l l

At every point km of Pack-Monkhorst net:

M crystalline orbitals associated with ( , )m k r ( )m k

1

( , ) ( ) ( , )M

m m mC

k r k k r

F( ) C( ) S( )C( )E( )m m m m mk k k k k

Calculation of the Crystalline Orbitals with

the Self-consistent-field approximation

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Aufbau principle in populating electronic bands

Fermi Energy F

Calculation of density matrix elements in the AO basis from occupied CO

ready for the calculation of F in the AO basis during the next iterative step

*

m m m

( ) 2 ( ) ( ) )M

total

occupied

P C C

k k k

in reciprocal space

m j m

1

1) exp( ) ( )

Ntotal total

m

P i PN

j(l k l k

in direct space

Calculation of the Crystalline Orbitals with

the Self-consistent-field approximation

Band structure representation: α-quartz

1sSi

1sO

2sSi 2pSi

}

k path

En

erg

y (

ha

rtre

e)

valence bands

conduction band

2sO

3pSi

2pO

top valence band

gaps

Fermi energy

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1 1

2 ( ( )) /M N

states F

m

n N

mk

θ(x)=0 if x < 0 and

θ(x)=1 if x ≥ 0

with

1

12 ( ( ))

M

states F

BZ BZ

n d

k k

definition of a new function :

1

1( ) 2 ( ( ))

BZ

M

states

BZ

n d

k k

iterative evaluation of F (nstates()-nstates)

Fermi energy

Beryllium

Fermi

energy

1

1( ) 2 ( , ) ( ( ))

Mtotal

F

BZ BZ

P P d

l k l k k

*( , ) ( ) ( )expP C C i

k l k k k l

Definition of the density matrix