PHY 752 Fall 2015 -- Lecture 16 19/30/2015
PHY 752 Solid State Physics11-11:50 AM MWF Olin 103
Plan for Lecture 16:Reading: Chapter 5 in GGGPP
Ingredients of electronic structure calculations1. Plane wave basis sets2. Construction of pseudopotentials3. Projector Augmented Wave (PAW) method
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Some practical considerations in electronic structure calculations
( ) ( ) ( )Plane wav
Bloch
e representation
( )
theorem
( )
i in n n
in n
e e u
C e
k T k rk k k
k G rk k
G
r T r r
r G
22
In practice, summation is truncated:
2 cutEm
k G
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Kohn-Sham equations (assuming “local” potential)
2 2
2
( )
nk nk nk
ieff
eff
eff V e
V Em
V
G r
G
r r r
r G
3
max
( )
Convenient representation provided that
( ) for
ieff
e
f
f
ef
f
V d r eV
V G
G rG
G G
r
òStrong motivation for the development of pseudopotentials
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How can we construct a pseudopotential?
Norm-conserving pseudopotentials
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PHY 752 Fall 2015 -- Lecture 16 99/30/2015r (Bohr)
rV(r
) (B
ohr *
Ry
Self-consistent full potential for C
Local pseudopotential for C
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Some details of pseudopotential construction from Troullier and Martins
Atomic Kohn-Sham equation for all-electron potential (atomic (Hartree) units
AEAEAE
Corresponding atomic Kohn-Sham equation for pseudoptential
PP PPPP
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Conditions of AE and PP functions:
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Benefit of these conditionsPseudo-wavefunction accurately represents correct functional form in valence regionKohn-Sham equations correctly solved at energy of reference state and its neighborhood
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Troullier-Martins recipe
Conditions to determine coefficients cn
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Conditions on coeffients, continued:
PHY 752 Fall 2015 -- Lecture 16 159/30/2015r (Bohr)
rV(r
) (B
ohr *
Ry
Self-consistent full potential for C
Local pseudopotential for C
Example of Troullier-Martin pseudopotential for C
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Properties of pseudopotential and corresponding pseudo-wavefunctions
Correct logarithmic derivatives at energy E and at nearby energies
(1( ln ((
)) ))
c c
cr
nln
c rl nl
nl
dP rdL P r P rdr P r dr
Pseudopotential form in terms of polynomial p(r):
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Construction energyE (Ry)
L[P
]
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2
2 2
2
2 2
( 1) ( ) ( ) ( )
Formally take energy
Useful relationship in Rydberg uni
derivative:
( )( 1) ( ) ( )
ts
i i i i i i
i i
i i i i
i i
ieff l l l
lief
in n n
nin n
nf l l
l
l ld V r P r E P rdr r
dP rl ld V r E P rd Er dr
2 2
0
( ) ( )] ( )[c
i i i i i i
i ic
r
n l n l n ln l r
r r ddP L P PdE
r r The construction ensures that
PS and AE have same log derivatives near E
What about other partial waves? (Non-local contributions to pseudopotential)
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PAW representation of Kohn-Sham orbitals
Evaluation of the energy of the system
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Example of atomic basis and projector functions
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Clever secret of PAW method; partitioning of planewave and one center contributions
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Construction of atom centered basis and projector functions: KS( ) ( ) 0a a
i i r rH
rc
2s2ss
s
r
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Construction of atom centered basis and projector functions – continued (scheme developed by David Vanderbilt for ultra-soft pseudopotentials; for each l channel at at time):
41 2
1( )(
Let )
i ml
cmmi
i c
rr C r r
rr
r
r
KS
1
( ) ( )
Calculate overlap mat
Construct auxiliary function:
Form projector function:
rix:
( ) ( )
i i i
ij i j
i j jij
r r
B
rp r
B
H
This construction ensur
(
es that
)
a aj i ijP r
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Example for C
2s2s 2sp
r
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Example for C -- continued
s
s
sp
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Example for C -- continued log derivatives for l=0
E
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Example for C -- continued log derivatives for l=1
E
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Example for C -- continued log derivatives for l=2
E