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1
Electronic States in a Crystal
~10
23
interacting electrons and ion cores – a big computational problem.
22
2
2
2
,,
'
,
spinorbitinteraction
soi
11
2
2
...
...
in
n
n
n
T
i
i j
n
nn
ni
n
i
n
ij
el
core
el core
pp
Z
Z
e
Z
e
e
H
mM
HH
H
H
H
RR
r
R
rr
Frozen core approximation
:
cores at equilibrium positions; no phonons; no electron
phonon interactions, etc.
Ignore spinorbitinteraction.
Te
le
l
c
o
r
e
HHH
, still cannot be solved exactly even for two electrons because of cross
terms.
Independent
e
approximation
: assuming
Ti
(no cross terms), and
i
independent of
i
(all electrons treated equally). A good approximation if
i
is chosen
wisely.
Oneelectron problem (singleparticle approximation):
2
2
2
2
HU
m
r
U
=
crystal potential
Hartree approx:
U
r
= potential from ion cores + average potential from all other
electrons
HartreeFock approx
Hartree
exchange
UU
U
r
(exchange from exclusion principle)
Generally,
Hartree
exchange
correlation
U
U
r
(correlation = whatever is left)
For now, consider
U
as given (as a parameterized function, for example).
Translational symmetry:
+
R
Bloch theorem:
wave function is of the form
i
eu
kr
k
,
where
uu
kk
R
= a periodic function.
k
= wave vector = quantum number related to translation
k
crystal momentum
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Translating the wave function by
R
:
i
eu
krR
k
rR
Equivalent form of Bloch theorem
i
e
kR
r
Proof:
Define translation operator
T
Rr
r
R
.
Explicit form of
T
R
from Taylor expansion:
00
11
exp
!!
n
n
nn
i
i
pp
R
r
R
r
R
r
So,
exp
Ti
p
RR
Translation symmetry:
1
HT
H
T
H
R r
R
r
;
,0
TH
R
.
Also,
TT
.
H
and all
T
R
form a commuting set.
Can choose
to be a simultaneous eigenfunction.
H
;
Tc
.
4
Since
T
R
R
,
cc
c
R
R
.
Let
2
i
ix
i
ce
a
;
i
x
= complex in general;
i
=
1, 2, 3.
3
12
2 2
3 3
3
2
2
22
i
ii
n
in
x
n
x
n
x
x
xx
i
n
e
e
e
e
Ra
Let
i
x
kb
i
R
r
R
r
i
Te
.
QED
Note
k
is complex in general.
Alternate proof based on group theory:
T
R
form a 3d cyclic group of
order.
All
representations are 1D.
Choose the 1D matrix to be
2
x
i
a
. Etc.
Another quick proof:
Because of translation symmetry, if
r
is a solution,
is also a solution. Choose
the basis set such that
c
.
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This note was uploaded on 03/21/2012 for the course PHYS 560 taught by Professor Flynn during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Flynn
 Physics

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