Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b
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© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Periodic Boundary Methods and Applications: Ab-initioQuantum Mechanics for Band Structures
CH121b
Jamil Tahir-KheliMSC, Caltech
May 4, 2011
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Outlines
(1) What is different about crystalline solids?(2) Bloch theorem(3) First Brillouin zone(4) Reciprocal space sampling(5) Plane wave, APW, Gaussian basis sets(6) SeqQuest (7) Crystal06
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What is different about solids?
H H H HH HH
a
Infinite repeating pattern of atoms with translational symmetry
Even if you have 1 basis function per atom, there is still an infinite number of atoms leading to diagonalization of an infinite matrix!
This implies we can never solve crystals
By exploiting the translational symmetry of the crystal, we can find a way to break the problem into finite pieces that approximate the solution
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Bloch Theorem (simplification due to translation symmetry)K-Vectors
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Bloch Theorem (example: one dimensional hydrogen chain)
H H H HH HH
a
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Bloch Theorem (example: one dimensional hydrogen chain)band structure
k = 0 k = /a
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Density of States
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Bloch Theorem (example: two dimensional hydrogen surface)
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The First Brillouin zone
The first Brillouin zone contains all possible interactions between two adjacent unit cells.
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Hartree-Fock-Roothaan Equation in periodic systems
Finite diagonalizations
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We can solve for each k-point, but there are an infinite number of them
By evaluating each k-point at the first Brillouin zone and summing them together, we can obtain the properties such as total energy or electron density of the system
In practice, the only computationally feasible approach is to approximate the full BZ integral with summation over a finite set of k-points.
Impossible !!!
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Reciprocal Space Sampling (Monkhost-Pack grids)
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Differences between Molecularand Periodic Codes
There is an infinity far away from the moleculewhere the density decays to zero as an exponential.
The exponent is the ionization potential (up to a factor)and can be shown to equal the HOMO eigenvalue.
.)(
),.(2 , as )(
,|)(|)(
2
2
2
r
IPr
HOMO
n
er
uaErer
rr
DFT obtains exact density and thus IP.
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There is no vacuum away from infinite crystal where wecan define the zero of the electrostatic potential.
No physical significance can be attached to the Kohn-Shameigenvalues for solid calculations.
Empirically, we do it anyway.
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Orbital energies are arbitrary up to a constant.
To obtain the work functions, you need to knowthe surface charge distribution of a finite sample.
Ionization potential
Fermi level
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Plane Wave Augmented Plane Waves
Gaussian Orbitals
Ewald (CRYSTAL)
Reference Density(SeqQuest)
Ab-Initio Methods
FLAPW, Wien2kVASP
“Exact” GW
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Numerical BasisSets
DMOL3
SIESTA
Green’s Function (GW)
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Plane Waves
Basis functions for each k in Brillouin Zone,
rG)i(k2/1 e)( rGk
where G is a reciprocal lattice vector.
Solve for wavefunctions and energies,
G
GkGnk rar )()(
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Practically, to obtain a finite set of states, the basis functions are cutoff,
cutGG
The cutoff is quoted as an energy,
,2
22
mG
E cutcut
or as a cutoff wavelength,
.2
cutcut G
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Assembling the Fock matrix to diagonalize is easy withPlane waves.
),(2
)(|
21
|22
2 GGm
GkGkGk
2)(
41|
1|
GGr GkGk
212
41|
1|
qr kkqkqk
Kinetic
Nuclear
Coulomb +Exchange
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Problem: cutoff G must be chosen extremely large to capturevariation of wavefunction near nuclei.
Fock matrix to diagonalize cheap to assemble, but large.
Diagonalization becomes time consuming.
CASSTEP is a pure plane wave code.
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Augmented Plane Wave codes try to reduce the number ofbasis functions of pure plane wave by using atomic orbitalsin the vicinity of nuclei that are smoothly joined to planewaves in the interstitial region.
Self-Consistent spherical potential inside spheres
Constant potential in interstitial regions
Wavefunctions in two regions are smoothly joined
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APW works well for computing band structures, but has threedrawbacks:
1.) There are no standard basis functions. This makes it difficult to visualize the wavefunction in terms of atomic orbitals. Mulliken populations are hard to quantify.
2.) Exact exchange is hard to compute. Thus, modern hybrid functionals that include Hartree-Fock exchange are not presently available with this approach.
3.) There is a certain arbitrariness to the choice of sphere radii.
Wien2k and FLAPW
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GW Method
Feynman diagram method
= +
= + +
+ …..
Gives good bandgaps and excitations, but computationallyvery very expensive. Not competitive with DFT.
Poles of propagator are physical excitation energies.
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Gaussian Orbitals
Trial wavefunctions for crystal momentum k are built upfrom linear combinations of localized atomic Gaussian orbitals.
R
ikRk Rre
Nr )(
1)( Atomic Gaussian
localized at R
)()()( rkcr knnk
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Advantages:
1.) Fewer basis functions needed to solve problem. 2.) Intuitive wavefunctions that are easily visulalized.
3.) Mulliken populations4.) Can do surface problems
Disadvantage:
1.) Much harder to calculate elements in Fock matrix.
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SeqQuest(Sequential QUantum Electronic STructure)
][
||)(
21 2
exnuc Vrr
rrdVH
||)]()([
||)(
||)( 00
rrrr
rdrrr
rdrr
rrd
Worked out once Varies slowly so solve in Fourier space using Poisson equation,
][4 02 V
Can obtain linear scaling!!
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The linear scaling method does not lend itself to an easyway to compute exact Hartree-Fock exchange.
HF exchange requires brute force calculation taking the scaling back to O(N^3).
In fact, no one has found a fast way to compute exact exchange for periodic systems.
If you can, PUBLISH!
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do setupdo itersno forceno relaxsetup datatitle2 GaN bulk wurzite: a=6.13 bohr, c/a=1.630714474, GaN(Z)=3.755109729 Example: change functional to PBE flavor of GGA functional LDA-SPspin polarization 1.0000 dimension of system (0=cluster ... 3=bulk) 3primitive lattice vectors 5.308735725 -3.0650 0.000000000 0.000000000 6.1300 0.000000000 0.000000000 0.0000 9.996279726grid dimensions 24 24 36atom types 2atom file n.atmatom file ga.atm
GaN Quest Input Deck
Bohr
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number of atoms in unit cell 4atom, type, position vector 1 1 3.539157150 0.0000 0.025788046 2 2 1.769578575 3.0650 1.268818179 3 1 1.769578575 3.0650 5.023927909 4 2 3.539157150 0.0000 6.266958042kgrid2 4 4 2end setup phase datarun phase input dataconvergence criterion 0.000500end run phase data
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http://www.cs.sandia.gov/~paschul/Quest/
Online manual for Quest
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CRYSTAL: A Gaussian CodeInput Structure of CRYSTAL
Structure
Basis set(atomic orbital)
Method (HF or DFT)SCF control
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Input Structure of CRYSTAL (example)Your personal note about this calculation
“crystal” “slab” “polymer” “molecule”
Space group sequence numberCell parameters
Number of non-equivent atoms
Atomic coordiantes
Basis set
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Input Structure of CRYSTAL (Basis set)
atomic numberFor example:C: 6O: 8Ni: 28Ni: 228
number of shells
all electron basis set
effective core potential
1st shell
2nd shell
3th shell
4th shell
End of basis set section
basis set type0: input by hand1: STO-nG2: 3(or 6)-21G
shell (orbital) type0: s orbital1: s+p orbital2: p orbital3: d orbital4: f orbital
number of Gaussian functions
number of electrons at this shell
scale factor
Si (1s22s22p63s23p2) 14 electronsSi ((function)3s23p2) 4 electrons
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Crystal06 Input (basis set)http://www.crystal.unito.it/Basis_Sets/Ptable.html
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Crystal06 Input (SCF control)
k-point net
for insulator: n nfor metal n 2n
maximum SCF iterations
mixing control 30% P0 + 70% P1 for second step