Overview of OpenMX: Implementation and practical aspects · PDF fileIntroduction 2....
Transcript of Overview of OpenMX: Implementation and practical aspects · PDF fileIntroduction 2....
Open
MX
/QM
AS W
S08, 21th
April 2008 in JA
IST
Tai
suke
Oza
ki (J
AIS
T)
1.Introduction
2.Im
plemen
tation
Overview of OpenMX:
Implementation and practical aspects
3.Ongoing and planned works
4.Summary
Tota
l en
ergy, fo
rces
, num
eric
al inte
gra
tions,
bas
is funct
ions, g
etting S
CF, geo
met
ry o
ptim
izat
ion,
par
alle
liza
tion, co
des
.
Short history of OpenMX
Improvement of
geometry optimization
The
dev
elopm
ent of th
e co
de
has
bee
n sta
rted
from
the
mid
dle
of 2000.
The
firs
t public
rele
ase
was
done
at 2
003 Jan
uar
y, an
d
thirte
en rel
ease
sw
ere
mad
e
until now
.
The
code
has
bee
n ste
adily
dev
eloped
as sh
ow
n the
figure
,
and the
com
munity itsel
f has
bee
n a
lso g
row
ing.
The number of lines of the code
Large im
provement
of parallelization
NC-D
FT
LDA+U
Electric polarization
Transport
Contributors
•T. O
zaki (J
AIS
T)
•H
. K
ino (N
IMS)
•J. Y
u (SN
U)
•M
. J. H
an(U
C,D
avis
)
•N
. K
obay
ashi (T
sukuba
Univ
.)
•M
. O
hfu
ti (Fujitsu)
•F. Is
hii (K
anaz
awa
Univ
.)
•T. O
hw
aki (N
issa
n)
•H
. W
eng (JA
IST)
•K
. Ter
akura
(JA
IST)
Open
MX
Open
source package for Material eX
plorer
The m
emory requirementscales as O(N
),
and the computational costis proportinal to
the third power of number of atoms or linearly.
OpenMXis based on LDA, GGA,
LDA+U, local pseudo-atomic basis
functions, and norm-conserving
peudopotentials.
Implementation of OpenMX
•Total energy
•Numerical integration
•Forces
•Basis functions
•Getting SCF
•Geometry optimization
•Parallelization
•Handling of the code
Total energy No.1
The total energy is given by that of the conventional DFT. The reorganization
of the Coulomb energiesis a key
for the accurate implementation
Total energy No.2
The
reorg
aniz
atio
n o
f Coulo
mb e
ner
gie
s giv
esth
ree
new
ener
gy ter
ms.
The
neu
tral
ato
m e
ner
gy
Diffe
rence
char
ge
Har
tree
ener
gy
Scr
eened
core
-core
rep
uls
ion e
ner
gy Neu
tral
ato
m p
ote
ntial
Diffe
rence
char
ge
Total energy No.3
So, th
e to
tal en
ergy is giv
en b
y
} }
Eac
h ter
m is ev
aluat
ed b
y u
sing a
diffe
rent num
eric
al g
rid.
Spher
ical
coord
inat
e in
mom
entu
msp
ace
Rea
lsp
ace
regula
r m
esh
Rea
l sp
ace
fine
mes
h
Projector expansion of V
naNo.1
wher
e a
set of ra
dia
l fu
nct
ions {R
lζ} is
an
orthonorm
al
set def
ined
by a
norm
∫r2drRV
na,kR′
for ra
dia
l fu
nct
ions R a
nd R′
, an
d is ca
lcula
ted b
y the
follow
ing G
ram
-Sch
mid
torthogonal
izat
ion:
Vnate
nds to
be
ver
y d
eep, le
adin
g
to a
ser
ious num
eric
al p
roble
m.
Vnaca
n b
e ex
pan
ded
by p
roje
ctors
:
Thre
e ce
nte
r in
tegra
ls w
ith V
naca
n b
e tran
sform
ed to p
roduct
s
of tw
o c
ente
r in
tegra
ls b
y the
pro
ject
or ex
pan
sion m
ethod.
TO
and H
.Kin
o, PR
B 7
2, 045121 (2005)
Projector expansion of V
naNo.2
Conver
gen
ce p
roper
ties
Com
par
ison b
etw
een n
on-p
roje
ctor
and p
roje
ctor m
ethods
Forces
Easy calc.
See the left
Forc
es a
re a
lway
s an
alytic
at a
ny g
rid
finen
ess an
d a
t ze
ro tem
per
ature
, ev
en if
num
eric
al b
asis
funct
ions an
d n
um
eric
al g
rids.
Basis functions in O
penMX
1) Primitive functions
gen
erat
ed b
y a
confinem
ent sc
hem
e
2) Optimized functions
by a
n o
rbital
optim
izat
ion m
ethod
Primitive basis functions
1. Solv
e an
ato
mic
Kohn-S
ham
eq.
under
a c
onfinem
ent pote
ntial
:
2. C
onst
ruct
the
norm
-conse
rvin
g
pse
udopote
ntial
s.
3. Solv
e gro
und a
nd e
xci
ted sta
tes fo
r th
e
the
peu
dopote
ntial
s.
s-orb
ital
of oxygen
In m
ost c
ases
, th
e ac
cura
cy a
nd e
ffic
iency
can
be
controlled
by
Cutoff radius
Number of orbitals
PR
B 6
7, 155108 (2003)
PR
B 6
9, 195113 (2004)
Convergence with respect to basis functions
mole
cule
bulk
The
two p
aram
eter
s ca
n b
e re
gar
ded
as var
iational
par
amet
ers.
Benchmark of primitive basis functions
Gro
und sta
te c
alcu
lations of dim
er u
sing p
rim
itiv
e bas
is funct
ions
All the
succ
esse
s an
d fai
lure
s by the
LD
A a
re rep
roduce
d
by the
modes
t size
of bas
is funct
ions (D
NP in m
ost
cas
es)
Optimization of basis functions
Pra
ctic
ally
, th
e ac
cura
cy a
nd e
ffic
iency
can
be
controlled
by
Num
ber
of bas
is funct
ions
Cuto
ff rad
ius
But, ther
e is
anoth
er o
ne
var
iational
par
amet
er:
Rad
ial sh
ape
If the
radia
l sh
ape
can b
e optim
ized
, it is ex
pec
ted that
the
hig
h a
ccura
cy w
ill be
atta
inab
le w
ith a
sm
all num
ber
of bas
is funct
ions.
Variationaloptimization of basis functions No.1
One-
par
ticl
e w
ave
funct
ions
Contrac
ted o
rbital
s
The
var
iation o
f E w
ith res
pec
t to
cw
ith fix
ed a
giv
es
Reg
ardin
g c
as dep
enden
t var
iable
s on a
and a
ssum
ing K
S
eq. is
solv
ed sel
f-co
nsisten
tly w
ith res
pec
t to
c, w
e hav
e
PR
B 6
7, 155108 (2003)
Variationaloptimization of basis functions No.2
The
bas
is funct
ions ca
n b
e optim
ized
in the
sam
e
pro
cedure
as fo
r th
e geo
met
ry o
ptim
izat
ion.
Primitive vs. O
ptimized
Energy convergence
Radial shape of carbon atom
Sin
ce the
fluorine
attrac
ts e
lect
rons sittin
g p
-orb
ital
s of ca
rbon,
the
p-o
rbital
larg
ely shrinks.
Notes in LCPAO
•It is not a complete basis
•A m
odest accuracy is attainable in practical calculations
•The double valence plus a single polarization functions
are an optimum choice.
•The use of many basis functions for dense bulk system
s
tends to be problematic due to the overcompleteness.
RMM-D
IIS for obtaining SCF
In m
ost c
ases
, th
e Res
idual
Min
imiz
atio
n M
ethod in the
direc
t In
ver
sion
of Iter
ativ
e su
bsp
ace
(RM
M-D
IIS) in
mom
entu
m spac
e w
ork
s w
ell.
Res
idual
vec
tors
Ker
ker
met
ric
with the
Ker
ker
fac
tor
Let
us as
sum
e th
e re
sidual
vec
tor at
the
nex
t step
is e
xpre
ssed
by
Min
imiz
e
with res
pec
t to
α
optim
um
αs
Assume an optimum charge is given by
G.K
ress
e an
d J. Furthm
euller
, PR
B 5
4, 11169 (1996).
Comparison of mixing m
ethods
Anderson m
ixing
→ eq
uiv
alen
t to
RM
M-D
IIS
Broyden m
ixing
→
RMM-D
IIS, Anderson, Broyden m
ethods are all equivalentfrom
the mathem
atical point of view and based on a quasi-N
ewton m
ethod.
→ eq
uiv
alen
t to
RM
M-D
IIS
V. Eyer
t, J. C
om
p.P
hys. 1
24, 271 (1996)
A way for im
proving the SCF convergence
→
Broyden m
ethod
If G
can
be
store
d, th
e Bro
yden
met
hod m
ay b
e th
e bes
t m
ethod
among them
. H
ow
ever
, G
is to
o lar
ge
to b
e store
d. Thus, fro
m
the
theo
retica
l poin
t of vie
w a
rea
sonab
le im
pro
vem
entof th
e
conver
gen
ce c
an b
e obta
ined
by incr
easing th
e num
ber
of of th
e
pre
vio
us step
s.
In fac
t, the
conver
gen
t re
sults w
ere
obta
ined
using 3
0-5
0 p
revio
us
step
s in
the
RM
M-D
IIS for 20 d
ifficu
lt system
s th
at the
SCF is har
dly
obta
ined
using a
sm
alle
r num
ber
of pre
vio
us step
s.
The
resu
lts ca
n b
e fo
und in h
ttp://w
ww
.jai
st.a
c.jp
/~t-oza
ki/la
rge_
exam
ple
.tar
.gz
Geometry optimization
The
geo
met
ry o
ptim
izat
ion in O
pen
MX
is bas
ed o
n q
uas
i N
ewto
n
type
optim
izat
ion m
ethods . F
our kin
d o
f m
ethods ar
e av
aila
ble
.
Bro
yden
-Fle
tcher
-Gold
farb
-Shan
no
(BFG
S) m
ethod
Treatm
ent of H
DII
S BFG
S RF
(rat
ional
funct
ion)
EF
(eig
envec
tor fo
llow
ing)
H=I
BFG
SBFG
S+RF
BFG
S p
lus m
onitoring
of ei
gen
val
ues
of H
If the
red p
art is
positive,
the
positive
def
initiv
enes
s of H
iskep
t.
Mole
cule
sBulk
s
The comparison of four quasi Newton m
ethods
It turn
ed o
ut th
at the
EF m
ethod is ro
bust
and e
ffic
ient, w
hile
the
RF a
lso show
s co
mpar
able
per
form
ance
.
The
input file
s an
d o
ut file
s use
d in the
calc
ula
tions sh
ow
n in the
figure
can
be
found in "open
mx3.3
/work
/geo
opt_
exam
ple
".
Parallelization
•The
par
alle
liza
tion is bas
ical
ly d
one
by a
1D
-dom
ain d
ecom
position.
•A
lso a
diffe
rent par
alle
liza
tion
schem
e is c
onsider
ed d
epen
din
g o
n
the
dat
a stru
cture
in e
ach subro
utine.
•The
dynam
ic load
bal
anci
ng is at
tem
pte
d a
t ev
ery M
D ste
p.
1-D
domain decomposition
Dynamic load balancing
Parallel efficiency
(a)
Dia
mond (512 a
tom
s)
(b)SM
M (148 a
tom
s)
(c)
Dia
mond (64 a
tom
s,
k-p
oin
ts=3x3x3)
Cra
y-X
T3
2.4
GH
z
Inte
rconnec
t
actu
al p
erfo
rman
ce 1
.0G
B/s
The code
is w
ritten
by a
sta
ndar
d C
, an
d c
onsist
s of 800 subro
utines
,
about 210000 lin
es, an
d is nee
ded
to incl
ude
lapac
k, bla
s, fftw
.
Mai
n c
ode:
open
mx
VPS a
nd P
AO
gen
erat
or: a
dpac
k
About te
n p
ost
pro
cess
ing c
odes
:
ban
dgnu13: b
and d
isper
ion
dosM
ain: D
OS
jx: ex
chan
ge
coupling c
onst
ant
polB
: m
acro
scopic
pola
riza
tion
esp: an
alysi
s of ch
arge
stat
e
……
How to handle the large software ?
The number of combination of parameters: ∝ ∝∝∝ 2N
Thus, it is
not ea
sy to a
ssure
the
reliab
ility o
f th
e co
de
as the
code
bec
om
es c
om
plica
ted.
In o
rder
to c
hec
k the
reliab
ility o
f th
e m
ost
funct
ional
itie
s,
sever
al a
uto
mat
ic tes
ting syst
ems hav
e bee
n d
evel
oped
:
runte
st:
a te
ster
of in
stal
lation a
nd M
PI
mltes
t:
a te
ster
of m
emory
lea
k
forc
etes
t:
a te
ster
of an
alytic
forc
es
filA
3:
a ch
ecker
of dynam
ic a
lloca
tion
Ongoing and planned works
1.
Hybrid p
aral
leliza
tion u
sing M
PI an
d O
pen
MP b
y T
O
2.
Monkhors
t-Pac
k k
-poin
ts b
y H
.Wen
g
3.
Cal
cula
tion o
f W
annie
r fu
nct
ions by H
.Wen
g
4.
Impro
vem
ent of num
eric
al inte
gra
tions by T
O
5.
Rel
ease
of N
EG
F b
y T
O
1.
Imple
men
tation o
f D
MFT b
y J.Y
u’s
gro
up
2.
Cal
cula
tion o
f an
om
alous H
all ef
fect
by S
. O
noda
3.
Imple
men
tation o
f st
ress
by T
O
4.
Ref
inem
ent of pse
udopote
ntial
s by T
O
5.
Imple
men
tation o
f hybrid funct
ional
by M
. Toyoda
Ongoing
Planned
Summary
OpenMX is a state-of-the art software package based on the local
PAO basis functions and norm
-conserving pseudopotentials.
The method can provide reasonable computational accuracy and
effciency
in a balanced way for a wide variety of system
s including
solid state and m
olecular system
as with careful consideration for
the basis functions and pseudopotentials.
To improve the reliability and efficiency, and to add m
ore
functionalities, the code is still under development.