05 Basis sets - Harvard University

Basis sets forcrystals and molecules

Lecture 5 2/12/18

1Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky

Matrix formulation

How to discretize the problem for a numerical solution?

Expand a wavefunction as a linear combination of basis vectors

2

EHrErH ˆ)()(ˆ

functions orthogonalk ..1

nkn

nnc

mm EH ˆmn

knmn EcHc

ˆ..1

Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky

Linear algebra: diagonalization

3

mnmkn

n EcHc

ˆ,1

mkn

nmn EccH ,1

kkkkk

k

c

c

E

c

c

HH

HH

.

.

.

.

.

.

............

...... 11

1

111

HH T*


Material dimensionality

4

1 dim

2 dim 3 dim

0 dim


Basis Set Approximation

For numerical solutions, molecular orbitals are written as linear combinations of basis functionsLinear Combination of Atomic Orbitals (LCAO)Schrodinger equation becomes a linear algebra diagonalization problem

5

• Molecular orbital coefficients, cki, are determined in numerical procedure• The basis functions, i, are atom‐centered functions that mimic solutions

of the H‐atom (s orbitals, p orbitals,...)

i ckikk

M


Basis set for quantum chemistry

“Slater type” orbitals (STO) are exponential and are exact solutions to the radial part of the simple Hydrogen atom

Slater orbitals are accurate but not often used because multi‐center integrals are very difficult to computeInstead, what is almost exclusively done is to use Gaussian type orbitals (GTO), of varying spatial spreads, to approximate STO.


Basis set for quantum chemistry

Product of two Gaussians is a (shifted) Gaussian, so the two‐electron integrals are much simpler to evaluate

7

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

GTOSTO


Available basis sets

Many basis sets have been made for different elementsBest resource: https://bse.pnl.gov/bse/portal


Double‐zeta: Two basis functions for each atomic orbital Triple‐zeta: Three …

Basis set convergence

The complete basis set limit can be approximated by taking basis sets of different size and extrapolating the fitted trend

Additional basis functions added: diffuse, polarization…


Any periodic function s(x) = s(x+mP) can be expressed as a Fourier series

coefficients live on the “reciprocal lattice”

Periodic functions

10

P


Reciprocal lattice

11

For any point on the reciprocal lattice and all translations T=mP

Systematically better representation of s(x) is obtained by including more reciprocal lattice points


Bravais lattice

12

1a2a

3a

),,( 321

332211

nnnR

anananR

Infinite array of points with a periodic arrangement having discrete translational symmetry, defined in terms of its primitive lattice vectors


Reciprocal lattice in 3D


j i ijG a

Crystal = Lattice + Basis


Bravais lattices in 3D

14 Bravais lattices exist in 3 dimensions


Properties of reciprocal lattice

The volume of the unit cell in reciprocal space is inversely proportional to the volume of the unit cell in direct space

Direct lattice is the reciprocal of its own reciprocal

16

Direct lattice Reciprocal lattice

Simple cubic Simple cubicFCC BCCBCC FCCOrthorhombic Orthorhombic


Periodic potentials


Electrons in a periodic potential

Hamiltonian is “periodic”: commutes with translation operator

Eigenfunctions can have any periodicity but should have a special symmetry

Density should be periodic

18

0ˆˆˆˆ]ˆ,ˆ[ HTTHTH RRR

)()()()(ˆ rRcRrrTR

)'()(ˆ)'(ˆ'' RcRcTTRRcT RRRR

ikReRc )()'()()'( RcRcRRc


Bloch theorem

For any eigenstate of H there exists a vector k such that translation by a lattice vector is equivalent to multiplying by the phase factor

Every solution to the Schrodinger equation in a periodic potential is a plane wave times some function with the periodicity of the lattice


The k label is not unique

20

k1 and k2 can differ by a reciprocal wave vector G

identical solutions


The Brillouin Zone removes ambiguity

21

BZ is a Wigner‐Seitz cell in the reciprocal lattice


Brillouin zone of fcc lattice


Electronic spectrum in solids

For each value of k, there are multiple solutions to the Schrödinger equation labelled by n, the band index, which simply numbers the energy “bands”. Each of these states evolves continuously with k, forming a smooth band.


Schrodinger equation in momentum space

Operators become matrices in plane wave basis

24

)(21ˆ 2 rVH

GG

rGiiGr GeedrGG ,222

21

21

21

'( ) ( )iGr iG rG GG V r G dr e V r e V


Schrodinger equation in momentum space

Equation couples only k+G and k+G’ componentsAll values of k (in the BZ) are independentThis means k is a good label for eigenfunctions

25

( ) n iGrnk k G

Gu r c e ikr

nknk erur )()(

22 0

2n n nc V c

m

k k G G G k GG

k G


Band structure of a free electron

An equivalent solution is found by shifting by a G vectorSeveral bands in the BZ is equivalent to a single parabolaThis is called zone folding and depends on the choice of the periodic unit cell


Adding a potential distorts the bands

Weak potential acts at BZ boundaryCertain energies become forbidden

No solutions exist with these eigenvalues

Two standing waves are not degenerateThis is called a band gap


Free electron in fcc and Silicon


Band structure


From atoms to solids


Density of States

The “density of states” is a measure of the number of states in a given energy interval that are available to be occupied.The density of states can be computed from the band structure.


Band structure classification


Basis size: energy cutoff (G‐vectors)

Size of plane‐wave basis‐set limited by the kinetic‐energy cut‐offThis controls the resolution of the Fourier representation

Only include G’s that satisfy

33

2 22 2

22

2 2

2

i

i

c em m

c em

k G rk k G

G

k G rk G

G

r

k G

22

cutoff2E

m k G


Brillouin zone integration (k‐points)

We also need to integrate over all “k‐points” in the BZ to get densities and energies

34

G

iGrGknnk ecru ,)(

ikrnknk erur )()(

kn

nkk

rN

r,

2)(1)(


Exploiting symmetry

Save time by computing only k‐points in the “irreducible” portion of the Brillouin zone.

35

2

,,

1, ,

22

, ,, , ,

21

,, ,

( )

( )irr

irr

irrirr

n kn k

n k n Sk

n k n Skn k n S k

n kn S k

r r

S r r

r r r

S r


The plane wave basis set

Systematic improvement of completeness/resolutionMay need a large number of basis elements

calculations only possible because of soft pseudopotentials

Allows for easy evaluation of gradients and LaplacianBasis set does not depend on atomic positions

This is more complex in Gaussian basis sets


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05 Basis sets - Harvard University