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Electronic structure of solids
Sergey V. Levchenko
Fritz Haber Institute of the Max Planck Society, Berlin, Germany
Extended (periodic) systems
There are 1020 electrons per 1 mm3 of bulk Cu
x
y
z
a1
a2
a3
Position of every atom in the crystal (Bravais lattice):
332211321 )0,0,0(),,( aaarr nnnnnn
lattice vector:
,2,1,0,,
),,(
321
332211321
nnn
nnnnnn aaaR
Example: two-dimensional Bravais lattice
1a
2a
primitive unit cells 12 23 aa
The form of the primitive unit cell is not unique
From molecules to solids
Electronic bands as limit of bonding and anti-bonding combinations of
atomic orbitals:
electronic
band
band
gap
Adapted from: Roald Hoffmann, Angew. Chem. Int. Ed. Engl. 26, 846 (1987)
Bloch’s theorem
Rr
r
R
0
)()exp()( rkRRr i
In an infinite periodic solid, the solutions of the
one-particle Schrödinger equations must
behave like
)()( rRr UU Periodic potential
(translational symmetry)
Consequently:
)()(),()exp()( rRrrkrr uuui
332211 aaaR nnn
Index k is a vector in reciprocal space
332211 gggk xxx ijji 2ag
1g
2g
nm
laa
g 2 – reciprocal lattice vectors
The meaning of k
chain of hydrogen atoms
j
sjk ajikx )()exp( 1
Adapted from: Roald Hoffmann, Angew. Chem. Int. Ed. Engl. 26, 846 (1987)
k shows the phase with which the orbitals are combined:
k = 0: )2()()()0exp( 1110 aaaj ss
j
s
k = : )3()2()()()exp( 11110 aaaajji sss
j
s π a
a a a a
k is a symmetry label and a node counter, and also
represents electron momentum
Bloch’s theorem: consequences
kGkGk krGrkr nn uiiui ~)exp()]exp()[exp(
a Bloch state
at k+G with
index n
a Bloch state at k with
a different index n’
kkk nnnh ˆ
In a periodic system, the solutions of the Schrödinger equations are
characterized by an integer number n (called band index) and a vector k:
For any reciprocal lattice vector
332211 gggG nnn
a lattice-periodic
function u~
1g
2g
21 ggG
Can choose to consider only k within single primitive
unit cell in reciprocal space
)()(),()exp()( rRrrkrr kkkk nnnn uuui
Brillouin zones
A conventional choice for the reciprocal lattice unit cell
For a square lattice
For a hexagonal lattice
Wigner-Seitz cell
Wigner-Seitz cell
In three dimensions:
Face-centered
cubic (fcc) lattice Body-centered
cubic (bcc) lattice
Time-reversal symmetry
For Hermitian , can be chosen to be real h kn
)()(ˆ)()(ˆ **rrrr kkkkkk nnnnnn hh
From Bloch’s theorem:
)()exp()()()exp()( **rkRRrrkRRr kkkk nnnn ii
)(*
kk nn kk nn )(
Electronic states at k and –k are at least doubly degenerate
(in the absence of magnetic field)
Electronic band structure
kn
k0 π/a -π/a
For a periodic (infinite) crystal, there is an infinite number
of states for each band index n, differing by the value of k
Band structure represents dependence of on k
)(k
Electronic band structure in three dimensions
z
Γ Λ
L U
X W K
Δ Σ y
x
Brillouin zone of the fcc lattice
By convention, are measured (angular-resolved
photoemission spectroscopy, ARPES) and calculated
along lines in k-space connecting points of high symmetry
kn
εn(k
), e
V
Al band structure (DFT-PBE)
Finite k-point mesh
kk nn , – smooth functions of k
can use a finite mesh, and then
interpolate and/or use perturbation
theory to calculate integrals
Charge densities and other quantities are
represented by Brillouin zone integrals:
occ
BZ
32
BZ
)()(j
j
kdn rr k 8x4 Monkhorst-Pack grid
occ
1
2kpt
)()(j
N
m
jm mwn rr k
xk
yk 1b
2b
H.J. Monkhorst and J.D. Pack, Phys. Rev.B 13, 5188 (1976); Phys. Rev. B 16, 1748 (1977)
Band gap and band width (dispersion)
a a a … 0.8 Å – hydrogen molecule chain
(DFT-PBE)
a = 10.0 Å
ε(k
)
a = 5.0 Å
ε(k
)
0 a
a
0
a
a
a = 2.0 Å
ε(k
)
a = 1.6 Å
ε(k
)
Overlap between interacting orbitals determines band gap and band width
Insulators, semiconductors, and metals
Eg>>kBT
Insulators (MgO, NaCl,
ZnO,…)
Eg~kBT
Semiconductors (Si,
Ge,…)
Eg=0
Metals (Cu, Al, Fe,…)
k k
εF
In a metal, some (at least one) energy bands are only partially occupied
The Fermi energy εF separates the highest occupied states from lowest
unoccupied
Density Of States (DOS)
n
N
j
jnj
n
n wkdgkpt
BZ1
3 ))(())(()( kk
Number of states in energy interval dε per unit volume,
d
d1
1
Atom-Projected Density Of States (APDOS)
Decomposition of DOS into contributions from different atomic
functions : i
n
nnii kdrdgBZ
32
3 ))(()()()( krr k
Recovery of the chemical interpretation in terms of orbitals
Qualitative analysis tool; ambiguities must be resolved by truncating
the r-integral or by Löwdin orthogonalization of i
O(2s)
O(2p)
Mg(3s)
Mg(3p) Mg(3d)
O(3d)
Periodic and cluster models of defects
Embedded cluster model Periodic model
+ Higher-level ab initio methods
can be applied
+/- Defects in dilute limit
- Effect of embedding on the
electronic structure and Fermi
level – ?
+ Robust boundary conditions
+ Higher defect concentrations
+/- Higher defect concentrations
- Artificial defect-defect
interactions
DFT and wave-function methods for solids
})({extiV r
Ground-state electronic structure problem
}{},{ ii r – many-body wave function, depends on spatial ( )
and spin ( ) coordinates of particles (also on nuclear
coordinates ( ) and )
iri
JR
}{},{
}{},{})({1
2
1 ext
2
2
ii
ii
i i J ji
i
jiJi
J
i
E
VZ
r
rrrrRrr
• already include approximations (Born-Oppenheimer,
non-relativistic, no spin-orbit coupling, no magnetic field)
• wave function depends on 4N variables (spatial + spin)
• electrons interact via Coulomb forces
Non-interacting fermions
i i J ji
i
jiJi
J
i
VZ
H })({1
2
1ˆ ext
2
2
rrrRrr
i
ii vV )(})({extrr
),()()(2
12
2
rrrRrr
nnn
J J
Jv
Z
),(,),,(det!
1,, 11111 NNNNN
N rrrr
one-particle states
Non-interacting fermions – periodic system
)()()(2
12
2
rrrRRrr
kkk
R
nnn
J J
Jv
Z
n – band index,
k – k-point
41 ,,,1 kkk n
41 ,,,2 kkk n
41 ,,,3 kkk n
Born-von Karman periodic boundary conditions
finite number of k-points, infinite (macroscopic) system as physical
limit
)()( Rrr kk nn
),(,),,(det!
1,, 11111 NNNNN
N rrrr
Interacting fermions (electrons)
i i J ji jiJi
J
i
ZH
rrRrr
1
2
1ˆ2
2
0ˆ
ˆmin
i
HH
i
Introduce basis set: ,
variational principle
j
jiji cs )()(),( rr 0ˆ
ijc
H
Approximations to the electronic problem: Basis set
Widely used basis sets:
gaussians (localized, analytic integrals)
plane waves (delocalized, analytic integrals)
Slater-type (localized, nuclear cusp)
grid-based (localized, analytic integrals)
)exp( 2rzyx kji
)exp( rk i
)exp( rzyx kji
)( irr
Core electrons are often treated separately
(pseudopotentials, plane-wave + localized basis)
Introduce basis set: , j
jiji cs )()(),( rr 0ˆ
ijc
H
k
kkkR
k
R rr )(e1
)( )(m
m
nmi
n UN
w
From reciprocal to real space
Wannier functions: not unique, can
be chosen more or less localized An arbitrary unitary
transformation within
occupied and virtual
subspaces
)()(det!
111 11 NNww
Nrr RR
Wannier functions in each
unit cell
The Hartree-Fock (HF) approximation
),(,),,(det!
1,, 11111 NNNNN
N rrrr
rrddnnZZ
hEM
I
M
J JI
JIN
n
nn
33
1 11
tot
)()(
2
1
2
1ˆrr
rr
RR
N
nm
mnnmddrrdd
1,
33** ),(),(),(),(
2
1
rr
rrrr
HF (exact) exchange energy
• No self-interaction
• Coulomb mean-field no dynamic correlation, single determinant
no static correlation
The Hartree-Fock (HF) approximation
Dynamic correlation:
versus
Coulomb
potential
Non-dynamic (static) correlation:
versus +
HF approximation ≥90% of total energy, overestimates ionicity
smaller e density, large r /
(quasi)degenerate HOMO-LUMO)
Density functional theory
Density functional theory: Hohenberg-Kohn theorem
rn
H NN rr ,,11
totE
– many-body Hamiltonian
– many-body wave function
– total energy
][)()(
2
1
2
1)(][ XC
33
1 1
3
1
tot nErrddnnZZ
rdn
ZnTEM
I
M
J JI
JIM
I I
I
rr
rr
RRRr
r
Approximations to : Local density approximation (LDA),
generalized gradient approximation (GGA), meta-GGA
][XC nE
Standard DFT and the self-interaction error
][)()(
2
1
2
1)(][ XC
33
1 1
3
1
tot nErrddnnZZ
rdn
ZnTEM
I
M
J JI
JIM
I I
I
rr
rr
RRRr
r
)(rn -- electron density
LDA, GGA, meta-GGA: ][][][ loc
C
loc
XXC nEnEnE
Standard DFT: (Semi)local XC operator low computational cost
(includes self-
interaction)
exchange-
correlation
(XC) energy
Removing self-interaction + preserving fundamental properties (e.g.,
invariance with respect to subspace rotations) is non-trivial
residual self-interaction (error) in standard DFT
Consequences of self-interaction (no cancellation of errors):
localization/delocalization errors, incorrect level alignment (charge
transfer, reactivity, etc.)
Hybrid DFT
][][)α1(}][{α}][{ loc
C
loc
X
HF
XXC nEnEEE
-- easy in Kohn-Sham formalism ( ) n
nnfn2
Perdew, Ernzerhof, Burke (J. Chem. Phys. 105, 9982 (1996)): α = 1/N
MP4 N = 4, but “An ideal hybrid would be sophisticated enough to
optimize N for each system and property.”
PBEC
PBEX
HFX
PBE0XC )α1(α EEEE
Hartree-Fock exchange – the problem
rrddDDE
ljki
lkji
jkil33
,,,
HFX
)()()()(
2
1
rr
rrrr
)(ri
)(rk
)'(rj
)'(rl
rr
1
electron repulsion integrals
Lots of integrals, naïve implementation N4 scaling (storage
impractical for N > 500 basis functions)
• need fast evaluation
• need efficient use of sparsity (screening)
Hartree-Fock exchange in extended systems
',,
33*
''*
HFX
),(),(),(),(
2
1
kk
kkkk
rr
rrrr
nm
mnnmddrrddE
rrdd
DDE
ljki
lkji
jkil
33
,,,,,
HFX
)()()()(
)()(2
1
rr
RRrRrRrr
RRRR
RRR
R
kRkk Rrr
,
e)()(),(i
iimim cs
Electron repulsion integrals in periodic systems
)(ri
)( Rr j
Q Q
Q
RQQRQRRRRR
μ ν
ν
,)(μνμ,, lkjilkij CVCI
RI-LVL: Sparse, easier to calculate, linear scaling is possible!
RI-V is impractical for extended systems
)(μ Qr P
0μ, Q
RjiC
Q
QR QrRrr
μ
μμ, )()()( PC jiji
Hybrid functionals -- scaling with system size
Periodic GaAs, HSE06 hybrid functional
Hybrid functionals -- CPU scaling
Periodic GaAs, HSE06 hybrid functional
Beyond mean-field for extended systems
The concept of mixing excitations
iii EH 00ˆ – non-interacting effective particles (HF, DFT, etc.)
}{ i – a basis set for N-electron wave functions
j
j
ijij
j
ijj
j
iji cEcHHHc )]ˆˆ(ˆ[, 00
ikki
j
jkij cEEHc )(ˆ 0
Project onto equations for : k ijc
configuration interaction
,,, 210
Configuration interaction – matrix diagonalization
abij
abij
abij
ai
ai
ai cc
,,
0
},{},{,0abij
ai DS
00ˆ H0
0ˆ HSS
D
T
Q
0
0ˆ HD
0
0
S D T Q
SH0
SHS ˆ
SHD ˆ
SHT ˆ
0
DH0
DHS ˆ
DHD ˆ
DHT ˆ
DHQ ˆ
0 0
THS ˆ
THD ˆ
THT ˆ
THQ ˆ
0
QHD ˆ
QHT ˆ
QHQ ˆ
!)!(
!
nnM
M
n-tuple
excitations
Configuration interaction in periodic systems
n – band index,
k – k-point
41 ,,,1 kkk n
41 ,,,2 kkk n
41 ,,,3 kkk n
0
rqpk
rqpkrpqk
rqpk
qk
qkqk
qk
baji
bjai
baji
ai
ai
ai cc ,
,
,,
,,
,
0
excitations can change not only n but also k-point
momentum conservation
rqpk
qk
baji
ai H ,
,ˆ
RI is a must!
Hierarchies of GS wave-function methods
Møller-Plesset
perturbation theory
(MP2, MP3, MP4,…)
Coupled-cluster
(CCD,CCSD,CCSDT,…)
0)0()0(
0
2
0)2(
0
ˆ
i i
i
EE
HE
MPm: ~nNm+2
0
ˆ}{
0 e mi
iTm
95
100
CID
CISD
CISDT
CISDTQ
MP2 MP3
MP4 MP5 MP6
CCD
CCSD
CCSD(T) CCSDT
CCSDTQ
R.J. Bartlett, private communication
Truncated CI
(CISD, CISDT,…)
0}{
0ˆ
mi
im T
CI{m}: ~nmNm+2
90
85
80
75
70
CC{m}: ~nmNm+2
% c
orr
ela
tion e
nerg
y
Ripples in the charged Fermi liquid
0000ˆ EH VHH λˆˆ
00
λ0
1
tadiabatic switching
Long-range correlation (screening) contributions can be
summed up to infinite order (EX+cRPA, rPT2, GW, BSE)
- - - - - - - - -
- - - - -
+ + + + + + + + +
+ + + +
fluctuating charge density
polarization (screening)
i i
i
EE
VE
)0()0(0
2
0)2(0
λ
Random-phase approximation
Second-order energy correction (MP2, HF orbitals):
rr
rrrr )()()()()|(
**bjai
jbia
abij bajii i
i jaibjbiajbia
EE
VE
,)0()0(
0
2
0)2(0
εεεε
))|()|)((|(2
Random-phase approximation
correlation exchange
Second-order energy correction (MP2, HF orbitals):
rr
rrrr )()()()()|(
**bjai
jbia
abij bajii i
i jaibjbiajbia
EE
VE
,)0()0(
0
2
0)2(0
εεεε
))|()|)((|(2
Random-phase approximation
correlation exchange
Second-order energy correction (MP2, HF orbitals):
abij bajii i
i jaibjbiajbia
EE
VE
,)0()0(
0
2
0)2(0
εεεε
))|()|)((|(2
rr
rrrr )()()()()|(
**bjai
jbia
- - - - - - - - -
- - - - -
+ + + + + + + + +
+ + + + cE
kcjbia cakibaji
iakckcjbjbia
,, )εεεε)(εεε(ε
)|)(|)(|(
Ren, Rinke, Joas, and Scheffler, Invited Review, Mater. Sci. 47, 21 (2012)
Random-phase approximation and beyond
RPAcE
0
μν0μν0RPAc ω)}ω)(χ({Tr))}ω)(χ{1ln(det( dVVE
.c.c)εε(ω
)(ψ)(ψ)(ψ)(ψω),,(χ
occ unocc **
0
i a ia
iaai
i
rrrrrr
rPT2cE
Kohn-Sham polarizability:
Renormalized second-order perturbation theory (rPT2):
single excitations second-order screened exchange RPA
Conclusions
• Periodic models can be efficiently used to study
concentration/coverage dependence, including infinitely dilute limit
(low-dimensional systems, defects, etc.)
• Practically all beyond-DFT methods involve calculation of the
electron repulsion integrals (ERIs)
• ERIs can be evaluated efficiently by using resolution of identity
(density fitting) and taking into account sparsity
• Wave function methods slowly but steadily enter the arena of
condensed matter physics
• Exploration of the connection between wave-function methods
(coupled-cluster) and many-body perturbation theory (beyond RPA)
continues
Acknowledgements
Volker Blum
Patrick Rinke
Rainer Johanni
Jürgen Wieferink
Xinguo Ren
Igor Ying Zhang
Matthias Scheffler