-
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
