Download - Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b
![Page 1: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/1.jpg)
© 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
![Page 2: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/2.jpg)
© 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
![Page 3: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/3.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 4: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/4.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Bloch Theorem (simplification due to translation symmetry)K-Vectors
![Page 5: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/5.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Bloch Theorem (example: one dimensional hydrogen chain)
H H H HH HH
a
![Page 6: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/6.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Bloch Theorem (example: one dimensional hydrogen chain)band structure
k = 0 k = /a
![Page 7: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/7.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Density of States
![Page 8: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/8.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Bloch Theorem (example: two dimensional hydrogen surface)
![Page 9: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/9.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
The First Brillouin zone
The first Brillouin zone contains all possible interactions between two adjacent unit cells.
![Page 10: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/10.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Hartree-Fock-Roothaan Equation in periodic systems
Finite diagonalizations
![Page 11: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/11.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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 !!!
![Page 12: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/12.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Reciprocal Space Sampling (Monkhost-Pack grids)
![Page 13: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/13.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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.
![Page 14: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/14.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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.
![Page 15: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/15.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 16: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/16.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Plane Wave Augmented Plane Waves
Gaussian Orbitals
Ewald (CRYSTAL)
Reference Density(SeqQuest)
Ab-Initio Methods
FLAPW, Wien2kVASP
“Exact” GW
![Page 17: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/17.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Numerical BasisSets
DMOL3
SIESTA
Green’s Function (GW)
![Page 18: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/18.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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 )()(
![Page 19: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/19.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 20: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/20.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 21: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/21.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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.
![Page 22: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/22.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 23: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/23.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 24: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/24.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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.
![Page 25: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/25.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 26: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/26.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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.
![Page 27: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/27.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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!!
![Page 28: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/28.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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!
![Page 29: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/29.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 30: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/30.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 31: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/31.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
http://www.cs.sandia.gov/~paschul/Quest/
Online manual for Quest
![Page 32: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/32.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
CRYSTAL: A Gaussian CodeInput Structure of CRYSTAL
Structure
Basis set(atomic orbital)
Method (HF or DFT)SCF control
![Page 33: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/33.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 34: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/34.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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
![Page 35: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/35.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
Crystal06 Input (basis set)http://www.crystal.unito.it/Basis_Sets/Ptable.html
![Page 36: Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b](https://reader035.vdocuments.us/reader035/viewer/2022081603/56813abb550346895da2c4fd/html5/thumbnails/36.jpg)
© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14
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