Computational Solid
State Physics
計算物性学特論 第６回
6. Pseudopotential
Potential energy in crystals
V (r  R )  V (r )
:periodic potential
R  a n  b l  cm
a,b,c: primitive vectors of the crystal
n,l,m: integers
V (r )   vG expiG  r 
G
Fourier transform
of the periodic
potential energy
G: reciprocal lattice vectors
Summation over ionic potentials
V (r ) 
 v (r  an  b l  cm τ )
j
j
n ,l ,m, j
2
Z
e
1
j
v j (r )  
40 r
Zj: atomic number
vG   exp(iG τ
 j ) v j (r ) exp(iG  r )dr
j
an  bn  cl   j
:position of j-th atom in
(n,l,m) unit cell
Bragg reflection
vG   exp(iG τ
 j ) v j (r ) exp(iG  r )dr
j
 j : position of the j-th atom in a unit cell
Assume all the atoms in a unit cell are the same kind.
vG   exp(iG τ
 j )   v(r ) exp(iG  r )dr
j
 SG   v(r ) exp(iG  r )dr
SG   exp(iG τ
 j)
j
:structure factor
The Bragg reflection disappears
when SG vanishes.
Valence states
We are interested in behavior of valence electrons,
since it determines main electronic properties of solids.
Valence states must be orthogonal to core states.
Core states are localized near atoms in crystals and
they are described well by the tight-binding approximation.
Which kinds of base set is appropriate to
describe the valence state?
Orthogonalized Plane Wave
(OPW)
OPW
 kG (r )  k G (r )  [   ck * (r ' )k G (r ' )dr' ] ck (r )
c
1
k G (r ) 
exp[i(k  G)  r ]
V
 ck (r )

ck
: core Bloch function
* (r )  kG (r )dr  0
: plane wave
Core Bloch function
・Tight-binding approximation
1
 ck (r ) 
N

c
 exp(ik  R ) (r  R )
n
c
n
* (r  Rn )c ' (r  Rn ' )dr   cc' nn '
H ck (r )   c  ck (r )
n
Inner product of OPW
 kG (r )  k G (r )  [ ck * (r ' )k G (r ' )dr' ]ck (r )
c
  kG' |  kG   GG '    k G ' | ck ck | k G 
c
Expansion of valence state
by OPW
k (r )   aG (k )  kG (r )
G
2
2

*
(
r
)(
H


)

(
r
)
dr

[
|
k

G
|
 ] GG '  VGG '  (VR ) GG '
kG '
 kG
2m
VR (r , r ' )    ck (r )(   c ) ck * (r ' ) :Extra term due to
c

k
OPW base set
* (r ) ck (r )dr  0
orthogonalization of valence Bloch
functions to core functions
Pseudo-potential:
OPW method
VR (r, r ' )   ck (r )(   c )ck * (r ' )
c
Fc(r’)
generalized pseudo-potential
VR (r, r ' )   ck (r ) Fc (r ' )
c
Generalized
pseudopotential
VR (r, r ' )   ck (r ) Fc (r ' )
c
2 2
(
  V  VR )   '  :pseudo wave function
2m
2 2
:real wave function
(
  V )  
2m
2 2
  * (r )( 2m   V  VR )dr   '   * (r )(r )dr
2 2
  * (r )( 2m   V )dr   '   * (r )(r )dr
   * ( r ) ck (r ) dr  0
  '
Empty core model
Fc (r ' )   ck * (r ' )V (r ' )
VR (r )    ck (r ) ck * (r ' )V (r ' )
c
Core region

ck
(r )ck * (r ' )   (r  r ' ) completeness
c
V (r )  VR (r )  0
Empty core pseudopotential
v ps (r )  0
ze
v ps (r )  
r
(r<rc)
2
(r>rc)
2
1 i ( k G )r
4

e
z
i ( k G ')r 3
e
v ps (r )e
d r 
cosqrc
2


q
q | G  G' |
Ω: volume of a unit cell
Screening effect by free electrons
qTF
 (q)  1  2
q
qTF 
2
6e n
2
dielectric susceptibility for metals
2
F
n: free electron concentration
εF: Fermi energy
1 i ( k G )r
4e z
i ( k G ')r 3
e
v ps (r )e
d r 
cos qrc
2


 (q)q
2
Screening effect by free electrons
1 i ( k G )r
4e z
i ( k G ')r 3
e
v ps (r )e
d r
cos qrc
2

2

(q  qTF )
2
q | G  G' |
・screening length in
metals
・Debye
screening length
in semiconductors
lTF
1
F


qTF
6e2 n
kT
lD  2
en
Empty core pseudopotential and
screened empty core pseudopotential
Brillouin zone for fcc lattice
Pseudopotential for Al
Energy band structure of metals
Merits of pseudopotential
 The valence states become orthogonal to the
core states.
 The singularity of the Coulomb potential
disappears for pseudopotential.
 Pseudopotential changes smoothly and the
Fourier transform approaches to zero more
rapidly at large wave vectors.
The first-principles norm-conserving
pseudopotential (1)
 d r  (r)   d r  (r)
2
3
r  rc

rc
0
3
r rc
ps
drrRl (r;  )
2
t
2
: Norm conservation
2
rc Rl (rc ;  )2  Dl (rc ;  )

2m

logarithmic derivative
Rl (r;  )
Dl (rc ;  ) 
/ Rl (r;  ) |r r
r
c
First order energy
dependence of the
scattering
The first-principle normconserving pseudopotential (2)
( H   )  0

(H   )  




  Rl (r )Ylm ( ,  )
Ylm ( ,  )
: spherical harmonics
The first-principle norm
conserving pseudo-potential(3)
rc
R
l
r dr 
2 2

  * ( H   ) dr
|r | rc
0



2
   ( H   ) * dr 
( *      *)dS

2m |r| rc
|r | rc
2


2
d Rl
dRl
 2
 2 2 d 1 dRl

rc ( Rl
 Rl
)
rc Rl
(
) |r  rc
2m
dr
dr
2m
d Rl dr
The first-principles normconserving pseudopotential (4)
 Pseudo wave function has no nodes, while
the true wave function has nodes within core
region.
 Pseudo wave function coincides with the true
wave function beyond core region.
 Pseudo wave function has the same energy
eigenvalue and the same first energy
derivative of the logarithmic derivative as the
true wave function.
Flow chart describing the construction
of an ionic pseudopotential
First-principles pseudopotential
and pseudo wave function
Pseudopotential of Au
Pseudopotential of Si
Pseudo wave function of Si(1)
Pseudo wave function of Si(2)
Siの各種定数
計算値
実験値
計算値と実験値の
ずれ
格子定数
5.4515[Å]
5.429 [Å]
+0.42%
凝集エネルギー
5.3495[eV/atom]
4.63[eV/atom]
+15.5399%
体積弾性率
0.925[Mbar]
0.99 [Mbar]
-7.1%
エネルギーギャップ
0.665[eV]
1.12[eV]
-40.625%
凝集エネルギー=Total Energy－2×EXC(非線形内殻補正による分)－2×ATOM Energy－(ゼロ点振動エネルギー)
Total Energy = -0.891698734009E+01 [HR]
EXC = -0.497155935945E+00 [HR]
ATOM TOTAL = -3.76224991 [HT]
Siのゼロ点振動エネルギー = 0.068 [eV]
Lattice constant vs.
total energy of Si
Energy band of Si
Problems 6
 Calculate Fourier transform of Coulomb
potential and obtain inverse Fourier
transform of the screened Coulomb potential.
 Calculate both the Bloch functions and the
energies of the first and second bands of Al
crystal at X point in the Brillouin zone,
considering the Bragg reflection for free
electrons.
 Calculate the structure factor SG for silicon
and show which Bragg reflections disappear.

Geometry I