Bloch functions, crystalline orbitals and

Fermi energy

Roberto Orlando

Dipartimento di Chimica

Università di Torino

Via Pietro Giuria 5, 10125 Torino, Italy

roberto.orlando@unito.it

Outline

The crystallographic model of a crystal

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

a2

a3

!"

#

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.

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·a

1 = 2! b

2·a

2 = 2! b

3·a

3 =2!

b1·a

2 = 0 b

2·a

3 = 0 b

3·a

1 = 0

b1·a

3 = 0 b

2·a

1 = 0 b

3·a

2 = 0

or, equivalently,

b1 = 2!/V a

2"a

3 b

2 = 2!/V a

3"a

1 b

3 = 2!/V a

1"a

2

V = a1·a

2"a

3 V* = (2!)3/V

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

( )T T Ti

i i i

N

N= =

a a 0

Commutative group of N

translation operators Tl

The macro-lattice

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, 3translation

symmetrya new metric is defined

Born-Von Kárman boundary conditions

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

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

0

0

mk

m!k

jl

j!l…………………… ……………………

1

1

1

1 1…………………… ……………………

exp( )m j

i !k l exp( )m j

i!

"k l

exp( )m j

i!"k l exp( )

m ji

! !"k l

……………

……………

……

……

…………

……

…………

•

•

•

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 ä! !=

" #$ % =& '( k k l

1

exp ( )N

m j j jj

m

i N !" "=

# $% & =' () k l l

Due to the

orthogonality

between rows

and columns

&

……

•

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•

•

•

•

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Character Table of the translation group

( )1

exp ( )N

m j j jj

m

i N!" "

=

# $ =% k l l

one of the two points (any) is assigned to

the reduced Pack-Monkhorst net (type C).

the point is assigned to the reduced Pack-Monkhorst

net (type R); it can only be km = (0 0 0) or km = (! !

!)

*

m m!k k"

( )'

1

cos (N

m m j j jj

m

w !" "

=

# $ =% k l l

*

1(2 )

m mmw

N!= "k k

*

( )m m m= !k K kIf vector belongs to Pack-Monkhorst net, it may happen:

The reduced Pack-Monkhorst net

mk

*

mk

*

m m=k k#

KK

mk

*

mk

K

*

mk

the local basis set

M"N atomic orbitals

Choice of a set of local functionsL.C.A.O method

(Linear Combination of Atomic Orbitals)

$ M atomic orbitals per cell

( )jµ! r - l

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

V

f f dµµ! !" "# #= $l l r - l r r l r

representation of

operator f

( , )j jfµ! "l l

( )µ! r

Atomic orbital basis set

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

V

f f dµµ! !" "# #= $ $%l l r l r r l r

( ) ( )jµ µ! !"# =r l r

( ) ( )jff!

= +r r l

( ) ( ) ( ), , ,j jj j jf f fµ! µ! µ!" " ""#= =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 dµµ! !" "# # #$ %& &' (= + &)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

l

l’

0

( , ) ( , )j j j jf fµ! µ!" "= #l l 0 l l

Representation of operators in the AO basis

The local basis

M"N atomic orbitals

( )jµ! r - l

Bloch function basisbasis for the irreducible representation

of the translations group

( , )mµ! k r

1

1( , ) exp( ) ( )

N

m m j j

j

iN

µ µ! "=

= # $%k r k l r l

( ) ( ) ( )*

, 1

1( , ) ( ) m j m j

Ni

m m j j

j j V

f e f dN

µ! µ !" "# #$ % $# #

#=

= % %& 'k l k l

k k r l r r l r

In this basis set the

matrix element fµ!

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 fµ! µ!"#

$

#

=

= %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

µ! µ!

" "# $ $

" "

"= =

= % %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

µ! µ!

"" " # $$ #

" "

"= =

= $% %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

µ! µ!

" " ""# $ $

" ""

" ""= =

= % %k k l k l

k k 0 l

and recalling the character orthogonality relation for rows, we get

Bloch theorem

the AO basis

( )jµ! r - l

Bloch function basis( , )

mµ! 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: ( )Fµ! 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 Fµ! µ!

=

= "#(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! µ! µ

µ

"=

# =$k r k k r

F( )C( ) S( )C( )E( )m m m m m

=k k k k k

Calculation of the Crystalline Orbitals with

the Self-consistent-field approximation

% 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 Cµ! µ" !""

# $= % &

' ()k k k

& in reciprocal space

m j m

1

1) exp( ) ( )

N

total total

m

P i PN

µ! µ!

=

= " #$j(l k l k

& in direct space

Calculation of the Crystalline Orbitals with

the Self-consistent-field approximation

Band structure representation: !-quartz

1sSi

1sO

2sSi2pSi

}

k path

En

erg

y (

ha

rtre

e)

valence bands

conduction band

2sO

3pSi

2pO

top valence band

gaps

Fermi energy

1 1

2 ( ( )) /M N

states F

m

n N!!

" # #= =

$ %= &' (

) *++ m

k!(x)=0 if x < 0 and

!(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 ( , ) ( ( ))

M

total

F

BZ BZ

P P d!

µ" µ" !!

# $ $=

% &= '( )*+ ,

- .l k l k k

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

µ" µ! "!= # $k l k k k l

Definition of the density matrix

Fourier expansion of %t(k):

1

( ) ( )fN

j j

j

D f!

!"

=

#$k k

( ) ( )! !" "+ =k K k

( ) ( )! !" "= #k k

Analytic expression of even periodic function %"(k) in reciprocal space

1

2( ) cos( )

(1 )

j

j j

n

j jl

lj

fn ! =

" #= $% &

+ ' ()

l l

k k l

'

m m

1

( ) ( )N

j m j

m

D w f!

!"

=

=# k k

Fourier analysis of energy bands

Definition of a denser meshdenser mesh than Pack-Monkorst in BZ

1 2 3

1 2 3

1 2 3

n n nn

p p p

S S S= + +k b b b 0 1

ni ip S! ! "

3( )

1

( ) ( ) ( ) ( )i

n n i in

i

p p! ! !" " "

=

# + $%k k k

Linear expansion of %$(k) in the domain of point kn

The first BZ is partitioned into S = S1 S2 S3 domains centred at kn

1 2 3

1 2 3

1/ 2 1/ 2 1/ 2

electrons 1 2 31/ 2 1/ 2 1/ 2

1 1

2( ) ( ( ))

n n n

n n n

M S p p p

p p pn

n dp dp dpS

!!

" # " "+ + +

$ $ $= =

% &= $' (

) *++, , , k

Gilat net

1

1( ) 2 ( , ) ( ( ))

BZ

M

total

F

BZ

P P d!

µ" µ" !!

# $ $= %

& '= () *

%) *+ ,- .l k l k k

1

( ) ( ) ( )fN

j j

j

P D f! !

µ" µ"

=

#$k l k Fourier expansion

'

1 1

2( ) ( , )

M Ntotal

m m

mrecip

P W P! !µ" µ"

! = =

# $% & '( ) *

++l k l

Calculating the density matrix: quadrature

1

( )fN

m m i i m

i

W w Q f! !

=

= " k

1( ) ( ( ))

BZ

j j F

BZ

Q f d!!" # #

$

= %$ & k k k

full bands: m mW w

!=

In Gilat net:

partially occupied bands:m

1

( )fN

m m i i

i

W w Q f! !

=

= " k

Calculation of the quadrature coefficients

1 2 3

1 2 3

' 1/ 2 1/ 2 1/ 2

1 2 31/ 2 1/ 2 1/ 2

1

( ( )) n n n

n n n

S p p p

j n nj Fp p p

n

Q v f dp dp dp!!" # #

+ + +

$ $ $=

% &= $' () *+ , , , k