PHY 752 Solid State Physics
11-11:50 AM MWF Olin 107
Plan for Lecture 9:
Reading: Chapter 8 in MPM; Electronic Structure
1. Linear combination of atomic orbital (LCAO)
method
2. Slater and Koster analysis
3. Wannier representation
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Linear combinations of atomic orbitals (LCAO) methods
for analyzing electronic structure
a
R
Bloch wave:
 nk  r   e ik  r u nk  r 
periodic function
Let R a  τ a  T
lattice translation
basis vector
Bloch wave identity:
LCAO basis functions with Bloch symmetry:
 nk  r  T   e

ik  T
 nk  r 
a nlm
k
(r )   e

ik  τ a  T
 a
nlm  r  τ a  T 
T
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LCAO methods -- continued
LCAO basis functions with Bloch symmetry:
a
 ka nlm (r )   eik Tnlm
 r  τ a  T  Example for nlm (r)  100 (r)  Ce r
T
k=0
k=p/2a
k=p/a
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LCAO methods -- continued – angular variation
http://winter.group.shef.ac.uk/orbitron/
l=0
l=1
l=2
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LCAO methods -- continued – angular variation
While, for atoms the “z” axis is an arbitrary direction,
for diatomic molecules and for describing bonds, it is
convenient to take the “z” axis as the bond direction.
Atom
l=0
l=1
l=2
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Bond
symbol
m=0
s
m=0
p
m= ±1
p
m=0
d
m= ± 1
d
m= ± 2
d
l=0
l=1
l=2
PHY 752 Spring 2015 -- Lecture 9
l=0
l=0
l=1
l=0
l=1
l= 2
symbol
s
s
p
s
p
d
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LCAO methods -- continued – bond types
pps
sss
ppp
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ddp
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LCAO methods -- continued – Slater-Koster analysis
LCAO basis functions with Bloch symmetry:

a nlm
k
(r )   e

ik  τ a  T
 a
nlm  r  τ a  T 
T
Approximate Bloch wavefunction:
 k (r ) 

X aknlm  ka nlm (r )
a nlm
In this basis, we can estimate the electron energy by variationally
computing the expectation value of the Hamiltonian:
 k H  k
E k 
 k  k
Terms in this expansion have the form:
e

ik  τ a  τ b  T

na' l ' m '  r  τ b  H nla m  r  τ a  T 
T
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LCAO methods – Slater-Koster analysis -- continued
e

ik  τ a  τ b  T

a
a
na' l ' m '  r  τ b  H nlm
r

τ
 T

T
Notation in Slater-Koster paper
k   τ a  τ b  T   l  m  n
na' l ' m '  r  τ b  H nalm  r  τ a  T  :
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LCAO methods – Slater-Koster analysis -- continued
Simple cubic lattice
on site NN
( s / s )  Es , s (000)  2 Es , s (100)  cos   cos  cos  
NNN
+4Es , s (110)  cos  cos  cos cos 
τ
a
l
 cos  cos    ...
 τ b  T   apxˆ  aqyˆ  razˆ
p
p2  q2  r 2
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m
q
p2  q2  r 2
n
r
p2  q2  r 2
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LCAO methods – summary
LCAO basis functions with Bloch symmetry:

a nlm
k
(r )   e

ik  τ a  T
 a
nlm  r  τ a  T 
T
Approximate Bloch wavefunction:
 k (r ) 
 X
a nlm
k
 ka nlm (r )
a nlm
In this basis, we can estimate the electron energy by variationally
computing the expectation value of the Hamiltonian:
E k 
 k H  k
 k  k
Terms in this expansion have the form:
e

ik  τ a  τ b  T

a
na' l ' m '  r  τ b  H nlm
r  τa  T
T
and also
e

ik  τ a  τ b  T

a
na' l ' m '  r  τ b  nlm
r  τa  T
T
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LCAO methods – summary
LCAO basis functions with Bloch symmetry:

a nlm
k
(r )   e

ik  τ a  T
 a
nlm  r  τ a  T 
T
Is there a “best choice” for atom-centered functions?
Introduction to the Wannier representation
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Wannier representation of electronic states
Note: This formulation is based on the relationship between
the Bloch and Wannier representations and does not
necessarily imply an independent computational method.
Bloch wave:
 nk  r   e
Bloch wave identity:
ik  r
u nk  r 
 nk  r  T   e ik  T  nk  r 
Orthonormality:
 nk  r   n ' k '  r   d nn d 3  k  k '
Wannier function in lattice cell T, associated with band n is given by
Wn (r  T) 
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V
 2p 
3
 ik  T
d
k
e
 nk (r )
3 
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Wannier representation of electronic states -- continued
Wannier function in lattice cell T, associated with band n is given by:
Wn (r  T) 
V
 2p 
3
 ik  T
d
k
e
 nk (r )
3 
Note that:
Wn (r  T) Wn ' (r  T ')  d nn d TT '
Comment: Wannier functions are not unique since the
the Bloch function may be multiplied by a k-dependent
phase, which may generate a different function Wn(r-T).
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Example from RMP 84, 1419 (2012) by Mazari, Mostofi,
Yates, Souza, and Vanderbilt
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