The electronic structure of materials 2 - DFT - Quantum...
Transcript of The electronic structure of materials 2 - DFT - Quantum...
Contents Density functional theory (DFT) Literature
The electronic structure of materials 2 - DFTQuantum mechanics 2 - Lecture 9
Igor Lukacevic
UJJS, Dept. of Physics, Osijek
December 19, 2012
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
1 Density functional theory (DFT)
2 Literature
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Contents
1 Density functional theory (DFT)
2 Literature
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Historical background
The beginnings:
L. de Broglie (1923)
E. Schrodinger (1925)
W. Pauli (1925)
L. Thomas & E. Fermi(1927)
P. A. M. Dirac (1928)
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Historical background
Quantum mechanics 99K technology transition:
F. Bloch (1928)
R. Wilson (1931) - implications of band theory - insulators/metals
J. Slater (1934-1937) - bands of Na
E. Wigner & F. Seitz (1935) - quantitative calcs on Na
J. Bardeen (1935) - Fermi surface
first understanding of semiconductors (1930’s)
transistor (1940’s)
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Historical background
Electronic structure methods:
E. Hylleraas (1929) - numerically exact solution for H2
J. Slater (1937) - augmented plane waves (APW)
C. Herring (1940) - orthogonalized plane waves (OPW)
S. F. Boys (1950’s) - Gaussian orbitals
J. C. Phillips & L. Kleinman (1950’s) - pseudopotentials
O. K. Andersen (1975) - linear muffin tin orbitals (LMTO)
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Historical background
Density functional theory:
P. Hohenberg & W. Kohn (1964)
W. Kohn & L. J. Sham (1965)
R. Car & M. Parrinello (1985) - CPMD
improved approximations for functionals
evolution of computer power
W, Kohn (1998) - Nobel Prize for Chemistry
widely used codes - abinit, quantum espresso, vaps, castep, wien2k, cpmd,fhi98md, siesta, crystal, fplo, elk,...
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Historical background
From Physics Today, June (2005).
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Hohenberg-Kohn ansatz
Starting ideas of Hohenberg & Kohn
create an exact many-body theory around the electronic ground-statedensity n(r)
start from the system of N interacting particles (B-O approx.):
Helec = −N∑
i=1
1
2∆i −
N∑i=1
M∑A=1
ZA
riA+
N∑i=1
N∑j>i
1
rij
A question
What do you think, why did they want n(r) as the main variable?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Hohenberg-Kohn ansatz
Reduction of the number of variables
Example: system of 100 particles
positions (r1, . . . , r100) , rk = (xk , yk , zk )
Schrodinger theory: ψ = ψ (r1, . . . , r100)
Questions
1 ψ (r1, . . . , r100) depends on how many variables?
2 if you want to calculate the energy from the variational principle, andguess the initial w.f. with p = 3 parameters, on how many variables willthe minimization of energy depend?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Hohenberg-Kohn ansatz
Reduction of the number of variables
Example: system of 100 particles
positions (r1, . . . , r100) , rk = (xk , yk , zk )
Schrodinger theory: ψ = ψ (r1, . . . , r100)
Questions
1 ψ (r1, . . . , r100) depends on how many variables? 300
2 if you want to calculate the energy from the variational principle, andguess the initial w.f. with p = 3 parameters, on how many variables willthe minimization of energy depend? 3300 ≈ 10150!
Exponential wall!
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Hohenberg-Kohn ansatz
Reduction of the number of variables
Example: system of 100 particles
positions (r1, . . . , r100) , rk = (xk , yk , zk )
Schrodinger theory: ψ = ψ (r1, . . . , r100)
DFT: ψ = ψ [n (r)] , n = n (nx , ny , nz )
Questions
1 ψ [n (r)] depends on how many variables?
2 on how many variables now does the minimization of energyE0 = minn(r) E [n (r)] depend?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Hohenberg-Kohn ansatz
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Hohenberg-Kohn ansatz
1st HK theorem
The ground-state density n(r) of a bound system of interacting electrons insome external potential Vext(r) determines this potential uniquely.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Hohenberg-Kohn ansatz
2nd HK theorem
A universal functional E [n] can be defined for any external potential Vext(r).The exact ground state energy of the system is the global minimum of thisfunctional, and the density n(r) that minimizes the functional is the exactground state density n0(r).
E0 = E [n0(r)] = minn(r)
E [n (r)]
E [n (r)] =
∫Vext(r)n (r)d3r + F [n]
F [n] = T [n] + Eint [n]
A question
What about F [n]? Do we know it?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Hohenberg-Kohn ansatz
HK DFT pros
an exact theory for ineractingparticle systems
provides a fundamentalunderstanding of physicalquantities like electron density andresponse functions
adds a practical contribution vialowering the number of variables
HK DFT cons
we don’t know F [n]
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Kohn-Sham ansatz
Motivation Hartree’s single particle self consistent equations
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Kohn-Sham ansatz
Motivation Hartree’s single particle self consistent equations
=⇒ ignore interaction: Vint = 0
=⇒ E [n (r)] =∫Vext(r)n (r)d3r + Ts [n]
Ts [n] = kinetic energy of the ground state of noninteracting electronswith density n (r)
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Kohn-Sham ansatz
From this assumption, they got:(−1
2∆ + Vext(r)− εj
)ϕj (r) = 0 ,
E =N∑
j=1
εj ,
n (r) =N∑
j=1
|ϕj (r)|2
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Kohn-Sham ansatz
To obtain the connection between HK and KS ansatz, rewrite:
EHK =
∫Vext(r)n (r) d3r + T [n] + Eint [n]︸ ︷︷ ︸
EKS =
∫Vext(r)n (r) d3r + Ts [n] + EHartree [n] + Exc [n]︸ ︷︷ ︸
Definition of exchange-correlation energy
Exc [n] = T [n] + Eint [n]− (Ts [n] + EHartree [n])
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Kohn-Sham ansatz
Rewriting EKS as
EKS =
∫Veff (r)d3r + Ts [n(r)]
gives the equations(−1
2∆ + Veff (r)− εj
)ϕj (r) = 0 ,
n (r) =N∑
j=1
|ϕj (r)|2 ,
Veff = Vext(r) + VHartree(r) + Vxc (r)
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Kohn-Sham ansatz
Noninteracting system Interacting system(−1
2∆ + Vext(r)− εj
)ϕj (r) = 0 ⇔
(−1
2∆ + Veff (r)− εj
)ϕj (r) = 0
↓single particle equations for thesystem of interacting particles= noninteracting particles inan effective potential
A question
What about Veff ? (hint: it depends on Vxc )
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Kohn-Sham ansatz
KS DFT pros
one N-particle problem None-particle problems
if Exc was known =⇒ exact E0
and n0
KS DFT cons
an exact form of Exc is unknown
A question
What do you think is the meaning of:
a) ϕj (use an analogy with Schrodinger theory)?
b) εj ?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Kohn-Sham ansatz
KS DFT pros
one N-particle problem None-particle problems
if Exc was known =⇒ exact E0
and n0
KS DFT cons
an exact form of Exc is unknown
A question
What do you think is the meaning of:
a) ϕj ? n (r) =∑N
j=1 |ϕj (r)|2
b) εj ? εhighestj = −Eionization. Starting approx. for more precise calculations.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Kohn-Sham ansatz
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Approximation to Exc - LDA
LDA = Local Density Approximation
E LDAxc =
∫n(r) εhom
xc [n(r)]︸ ︷︷ ︸ d3r
↓xc density at each point same asthat in a homogenous electron gaswith that density
A question
Where do you think LDA performs well and where not?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Approximation to Exc - LDA
LDA precision
Quantity Deviation from exp.
atomic & molecular ground state energies < 0.5%molecular equilibrium distances < 5%band structures of metals few %band gap < 100%lattice constants < 2%Ex O(10%)Ec ×2E atom
ionization & Edissoc & Ecoh 10− 20%
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Approximation to Exc - GGA
GGA = General Gradient Approximation
E (0)xc =
∫εhom
xc [n(r)]n(r)d3r (LDA)
E (1)xc =
∫f (1)[n(r), |∇n(r)|]n(r)d3r (GGA)
E (2)xc =
∫f (2)[n(r), |∇n(r)|]∇2n(r)d3r
A question
What do you think why GGA opened DFT to chemists?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Approximation to Exc - GGA
GGA precision
Quantity Deviation wrt LDA
B underestimatevalence bandwidth narrowslattice constants corrects or overcorrectsTO(Γ) νphon underestimateEbinding corrects the overestimationE atom
ionization & Etot corrects
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Plane waves
core states
strongly bound to nuclei
atomic-like
valence states
changes in the materials
bonding, electric & opticproperties, magnetism,...
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
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Plane waves
A question
What would be the most appropriate basis for representing the wave functionsin solids?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Plane waves
A question
What would be the most appropriate basis for representing the wave functionsin solids?
Possibilities:
1 plane waves e ikr
2 localized orbitals (gaussians,...)
3 augmented functions (LAPW, PAW,...)
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Plane waves
Period solid simulation
Bloch’s theorem:
ψm,k(r + ai ) = e ik·ai um,k(r)
Born-von Karman boundarycondition:
ψm,k(r + Ni ai ) = ψm,k(r)
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Plane waves
How to make a plane wave basis?
Plane wave kin. energy
E pwkin = −1
2∆
|k + G|2
2
Plane wave sphere|k + G|2
2< Ecut
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Plane waves
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Plane waves
But, there are problems!
Si core and valence electron w.f.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Pseudopotentials
Basic assumption
ψcore are the same in atomic or bounding conditions
=⇒ n(r) = ncore(r) + nval (r)
Atomic Si electron energy levels and core w.f.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Pseudopotentials
Basic assumption
ψcore are the same in atomic or bounding conditions
=⇒ n(r) = ncore(r) + nval (r)
A question
What do you think, is this core/valence partitioning obvious for all elements?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Pseudopotentials
Basic assumption
ψcore are the same in atomic or bounding conditions
=⇒ n(r) = ncore(r) + nval (r)
Examples
F atom: (1s)2 + (2s)2(2p)5
IP 1 keV 10 − 100 eV E expion ≈ 18 eV
Ti atom: (1s)2(2s)2(2p)6 + (3s)2(3p)6(4s)2(3d)2 small core
IP 99.2 eV E expion ≈ 7 eV
(1s)2(2s)2(2p)6(3s)2(3p)6 + (4s)2(3d)2 large core
IP 43.3 eV
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Pseudopotentials
Basic idea
“Freeze” core electrons =⇒ effective potential
Si 3s and 3s with frozen 3s electron w.f.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Pseudopotentials
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Pseudopotentials
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Pseudopotentials
Sometimes the core correction is needed. (a) Na as semi-metal (no core correction). (b) Na as insulator (with core correction).
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Brillouin zone integration
In order to calculate something in a period case, you have to:
1 sum over bands
2 integrate over the BZ
Expressions for T and n
T =∑
n
1
Ω0k
∫Ω0k
(εF − εnk) 〈ψnk| −1
2∆|ψnk〉dk
n(r) =∑
n
1
Ω0k
∫Ω0k
(εF − εnk)ψ∗nk(r)ψnk(r)dk
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Brillouin zone integration
How to calculate an integral over the BZ?
1
Ω0k
∫Ω0k
Xkdk 7→∑k
wkXk
How to choose k andwk?
Monkhorst-Pack grids
tetrahedron methods...
Homogenous sampling of the BZ.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Contents
1 Density functional theory (DFT)
2 Literature
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT
Contents Density functional theory (DFT) Literature
Literature
1 R. M. Martin, “Electronic Structure - Basic Theory and PracticalMethods”, Cambridge University Press, Cambridge, 2004.
2 W. Kohn “Electronic structure of matter” - Nobel Lecture
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The electronic structure of materials 2 - DFT