8/11/2019 Finding the Electronic Ground State

1/50

Castep Workshop '07 Finding the Groundstate 1

CASTEP Workshop 2006

University of York

16th-21stSeptember 2007

Finding the Kohn-Sham

Groundstate

8/11/2019 Finding the Electronic Ground State

2/50

Castep Workshop '07 Finding the Groundstate 2

Overview

The FT S!hr"dinger e#uation

$inimising the KS energ%

$eta&s and inite temperatures ensit% mi(ing and )nsem*&e FT

8/11/2019 Finding the Electronic Ground State

3/50

Castep Workshop '07 Finding the Groundstate +

The o*,e!t is to ind the sing&e-parti!&eso&utions to the S!hr"dinger e#uation

where i&a*e&s the dierent eigenstates and theami&tonian is given *%

The FT ami&tonian

Hksi=ii

Hks=1

22VHVxcVps

8/11/2019 Finding the Electronic Ground State

4/50

Castep Workshop '07 Finding the Groundstate .

We are using a p&ane-wave *asis andsamp&ing the /ri&&ouin one at a inite num*er

o k-points1

where the sum is over a&& re!ipro!a& &atti!eve!tors *e&ow some !ut-owaveve!tor

The waveun!tions at dierent k-points on&%intera!t via the densit%1 so ea!h k-point %ie&dsa separate Kohn-Sham e#uation

=G

ck,Ge

i kG . r

8/11/2019 Finding the Electronic Ground State

5/50

8/11/2019 Finding the Electronic Ground State

6/50

Castep Workshop '07 Finding the Groundstate

Se&-Consisten!%

The artree potentia& represents the Cou&om*repu&sion *etween the dierent parti!&es and isgiven *%8

where n is the !harge densit%

nr=i

f ii r2

VHr= nr 'rr 'd

3r '

8/11/2019 Finding the Electronic Ground State

7/50

Castep Workshop '07 Finding the Groundstate 7

9oti!e that the artree potentia& depends onthe densit%1 and the densit% depends on the

waveun!tions1 we no &onger have a i(edami&tonian to so&ve

The e(!hange-!orre&ation potentia& a&sodepends on the densit% : the ami&tonian

!hanges as the waveun!tions !hange; We need to ind a set o waveun!tions that

&ead to a ami&tonian whose so&utions are the

waveun!tions we started with This is !a&&ed self-consistency

8/11/2019 Finding the Electronic Ground State

8/50

Castep Workshop '07 Finding the Groundstate