Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b

© 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


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


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


Bloch Theorem (simplification due to translation symmetry)K-Vectors


Bloch Theorem (example: one dimensional hydrogen chain)

H H H HH HH

a


Bloch Theorem (example: one dimensional hydrogen chain)band structure

k = 0 k = /a


Density of States


Bloch Theorem (example: two dimensional hydrogen surface)


The First Brillouin zone

The first Brillouin zone contains all possible interactions between two adjacent unit cells.


Hartree-Fock-Roothaan Equation in periodic systems

Finite diagonalizations


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 !!!


Reciprocal Space Sampling (Monkhost-Pack grids)


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.


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.


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


Plane Wave Augmented Plane Waves

Gaussian Orbitals

Ewald (CRYSTAL)

Reference Density(SeqQuest)

Ab-Initio Methods

FLAPW, Wien2kVASP

“Exact” GW


Numerical BasisSets

DMOL3

SIESTA

Green’s Function (GW)


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 )()(


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


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


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.


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


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


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.


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


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.


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!!


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!


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


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


http://www.cs.sandia.gov/~paschul/Quest/

Online manual for Quest


CRYSTAL: A Gaussian CodeInput Structure of CRYSTAL

Structure

Basis set(atomic orbital)

Method (HF or DFT)SCF control


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


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


Crystal06 Input (basis set)http://www.crystal.unito.it/Basis_Sets/Ptable.html


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

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Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b

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