Invariant MD w/ Variable Cell ShapeR. Wentzcovitch

U. MinnesotaVlab Tutorial

-Simulate solids at high PTs

-Useful for structural optimizations

-Useful for structural search (shake and bake)

-Various fictitious Lagrangian formulations

Fictitious molecular dynamicsH. C. Andersen (1978)

I II

dUm R

dR

(N,E,V) (N,H,P)

( )

I I II

ext

dUm R f

dR

d U P VWV F

dV

Variable Cell Shape MD

h1

h2

)(thiji=vector indexj=cart. index

( )

I I II

extij ij

dUm R f

dR

d U P VWh F

dV

Anderson’s Fictious MD (HPN ensemble)

Anderson’s variable volume fixed shape constant pressure MD (Anderson, J. Chem. Phys 72,2384(1980))

L K U E K U cte

2/3 2

1 1 1

.2 2

N N NTi

And i i ij iji i i j i

m WL V s s r V PV

" "at cell at cellE K K U U cte H

1/3i ir V s

1/3i ir V s

( )V t

The ensemble (trajectory) averages produce the HPN ensemble averages

Cell volume

atK cellK cellUatU

Fictious MD (continue…)

Parrinello/Rahman variable cell shape MD (Parrinello and Rahman, J. Appl. Phys 52, 7182 (1981))

1 1 1

.2 2

N N NT Ti

PR i i ij iji i i

m WL s gs r Tr h h PV

i ir hs

" "i i iv r hs

, ,h a b c

Tg h h

1

1

1 ij ij

i i j iii ij

rs s s g gs

m r

1h P

W

, ,b c c a a b

1 1

1 1N NijT T

i i i ij iji i j i ij

rm v v r r

V V r

2T Vh I

12 T

hV

L L

t q q

Applying Lagrange’s equation

- in PR-VCSMD is not uniquely definedKLatt

The trajectory is not uniquely defined.It does not depend only on the initial conditions.

0

0

ah

a

0

a ah

a

0

0

vh

v

0

v vh

v

2LattK Wv 23

2LattK Wv

aa

a a

2

equivalent

equivalent

Solution: use strain ε instead of h as dynamical variable

1 oh h

(1 )i ir q

ε is strain

, ,o o o oh a b c

1 1

.2 2

N N NT Ti

Inv i i ij iji i j i

m WL q dq r Tr PV

1 1T

d

1

1TV

PW

1

1

1 Nij ij

i i j iii ij

rq q q d dq

m r

Invariant dynamics I

Wentzcovitch, PRB 44,2358 (1991)

Alternative form of LInv-I in terms of h and s:

1 1

.2 2

N N NT Ti

Inv i i ij ij oi i j i

m WL s gs r Tr h f h PV

( )o

To of

1

1

1 ij ij

i i j iii ij

rs s s g gs

m r

1o

Vh P f

W

, ,o o o o o o ob c c a a b

with

Final observation:Inv I AndLatt cellK K

2~2 2

Inv II TLatt

W WK Tr hfh V

In the limit of variable V-only

Solution: ( )Tf with

Eq. of motion given by Eq. 9 in PRB 44, 2358(1991)

a

a

2a

Fluctuations in the cell edges lengths of fcc X-tal of Ar initially placed away from Veq.

Beeman integration algorithm

dt= 10 fmt (1 a.u. = 2.5 x 10-17s (in Ry))Mi = 39 mp

W= 35 mp in (a); W= 0.0007 mp/ao3 in (b)

Rc= 10 ao

0 0

0 0

0 0 2

a

h a

a

Wentzcovitch, PRB 44,2358 (1991)WV

B~

01

02

0

a a

h a a

a a

fcc

bccscθ

d

dd

2

ad

bccfcc

sc

fcc

bcc

Potential energy isosurfaces

a

b

c

Basins of attraction if weuse andin the MD

a

b

c

Basins of attraction if weuse and

in the PR-MD

a

c a b c

b

Wentzcovitch, PRB 44,2358 (1991)

Typical Computational Experiment

Damped dynamics (Wentzcovitch, 1991)

)(~ PI),(~ int rffr

P = 150 GPa

(Wentzcovitch, Martins, and Price, PRL 1993)

hcp to bcc transition in Mg(Wentzcovitch, Phys Rev. B 50, 10358 (1994))

(0001)(110)

Distortion of the (0001) plane of the hcp structure into the (110) plane of the bcc structure. Arrows indicate atomic displacements.

Atoms at1

( )4

ua ub c

u=1/6 or 1/3 u=1/4

Enthalpy barrier separating the hcp from the bcc phases at P=35 GPa at T=0K.u=1/6 ↔ hcp u=1/4 ↔ bcc

Ideal phase boundary (solid)and blurry cause by hysteresis (dashed). Phase transitions willbe simulated at the pointsmarked by dots and error bars (undertainties in P and T).

Exp. PT = 45-55 GPaat 300 K

~150 K

hcp to bcc transition

Time evolution of the internal parametersu’s, and angles and lengths of simulationcell vectors.

Simulation w/ 16 atoms only T = 700 KP = 72 GPadt = 6 ftsW=0.02mat=24.3 mp

Θab = 70.53o

Θab = 60o

u=1/6 u=1/4

u=1/6

u=1/4

u=1/4

bcc to hcp transition

Time evolution of the internal parametersu’s, and angles and lengths of simulationcell vectors.

Simulation w/ 16 atoms only T = 500 KP = 12 GPadt = 6 ftsW=0.02mat=24.3 mp

u=1/6

u=1/6

u=1/4

u=1/4

u=1/4

Θab = 70.53o

Θab = 60o

MgSiO3 Perovskite----- Most abundant constituent in the Earth’s lower mantle----- Orthorhombic distorted perovskite structure (Pbnm, Z=4)----- Its stability is important for understanding deep mantle (D” layer)

b

ca

Lattice system: Bace-centered orthorhombicSpace group: CmcmFormula unit [Z]: 4 (4)Lattice parameters [Å] a: 2.462 (4.286)[120 GPa] b: 8.053 (4.575)

c: 6.108 (6.286)Volume [120 GPa] [Å3]: 121.1 (123.3) ( )…perovskite

6 8 10 12 14 16

2 theta (deg)

Inte

nsi

ty (

arb

itra

ry u

nit)

= 0.4134 Å

120 GPaExp

Calc

020

021

002

022

110

111

040

041

023/

130

131

042

132

113

004

Pt

Crystal structure of post-perovskiteTsuchiya, Tsuchiya, Umemoto, Wentzcovitch, EPSL, 2004

Ab initio exploration of post-perovskite phase in MgSiO3

Perovskite

SiO3 layer

SiO3

MgSiO3

MgSiO3

MgSiO3

- Reasonable polyhedra type and connectivity under ultra high pressure -

SiO4 chain

Post-perovskite

c’a’

b’

Structural relation between Pv and Post-pv

Deformation of perovskite under shear strain ε6

a

b

c

Perovskite θTsuchiya, Tsuchiya, Umemoto, Wentzcovitch, EPSL, 2004

Conclusions

-VCSMD is very useful for structural optimizations whenthe dynamics has the correct symmetry properties (invariant dynamics)

- It is capable of simulating a phase transition whenone knows how the transformation occurs

- There is unavoidable hysteresis associated withthe simulation, which makes the simulation often unfeasible

-Alternative approaches for obtaining phase boundaries by computations will be discussed throughout the course

Practice(Go to http://www.msi.umn.edu and navigate to the tutorial web site… …to … software. You will use VCSMD today. Click and download program, Input, and instruction.)

Some Instructions for Lind24-Lab

1) OpenDX is a visualization software you may use. To enable access to OpenDX:

module load soft/opendxmodule initadd soft/opendx

The first line enables the software for the current session, the second for every future session. Every user will need to type those two lines, but once they do, the software will be permanently enabled for your individual accounts.To launch the software, type 'dx'.

2) xmgr is a basic plotting software available in Linux. To launch it type ‘xmgr'.

3) The command for compiling fortran a code is 'f77'. It's part of the GCC 3.3.5 package built into Linux.

4) You can SSH to MSI machines. They are on a different network and use a different account, so you will need to incorporate that into the command. For example, if your username is 'user' and the computer is 'altix.msi.umn.edu', you would need to type ‘ssh user@altix.msi.umn.edu'.

5) They machines called lind24-01.itlabs.umn.edu, lind24-02.itlabs.umn.edu, etc, all the way up to lind24-40.itlabs.umn.edu. Both OpenDX and Xmgr are graphical, so you'll need to enable X Forwarding for the SSH connection if you're logging in remotely. Usually this can be done by adding the '-XY' flag to your SSH command in Unix.

Run1 Test: md of Ar atom in fcc cell (title)nd (calc) s n (ic,iio) 11.000000 (alatt) 1 1 1 (nsc) 1.000000 0.000000 0.000000 (avec) 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 0.00100 0.00000 (cmass, press) 1 (ntype) 4 Ar 40.00000 (natom,nameat,atmass) 0.000000 0.000000 0.000000 (rat) 0.500000 0.500000 0.000000 0.000000 0.500000 0.500000 0.500000 0.000000 0.500000 40.000000 (rcut) 5 5 5 (ncell) 1000 1110 10 (nstep,ntcheck,ntimes) 000.00000 0.00100 200.00000 (temp,ttol,dt)

~

Run2

Decrease step size by ½ and increase# of steps by 2

Run3

Test: md of Ar atom in fcc cell (title)nd (calc) s n (ic,iio) 11.000000 (alatt) 1 1 1 (nsc) 0.500000 0.500000 0.000000 (avec) 0.000000 0.500000 0.500000 0.500000 0.000000 0.500000 0.00100 0.00000 (cmass, press) 1 (ntype) 1 Ar 40.00000 (natom,nameat,atmass) 0.000000 0.000000 0.000000 (rat)40.000000 (rcut) 9 9 9 (ncell) 2000 2110 10 (nstep,ntcheck,ntimes) 000.00000 0.00100 100.00000 (temp,ttol,dt)

~

Run4

Adjust cell mass to get sameperiod of oscillation

Run5

Test: Optimization under pressure (fcc) (title)nm (calc) s n (ic,iio) 11.000000 (alatt) 1 1 1 (nsc) 1.000000 0.000000 0.000000 (avec) 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 0.00100 0.00000 (cmass, press) 1 (ntype) 4 Ar 40.00000 (natom,nameat,atmass) 0.000000 0.000000 0.000000 (rat) 0.500000 0.500000 0.000000 0.000000 0.500000 0.500000 0.500000 0.000000 0.500000 40.000000 (rcut) 6 6 6 (ncell) 100 1110 10 (nstep,ntcheck,ntimes) 000.00000 0.00100 500.00000 (temp,ttol,dt)

~

Run6

Test: Optimization under pressure (hcp) (title)nm (calc) s n (ic,iio) 9.000000 (alatt) 1 1 1 (nsc) 1.000000 0.000000 0.000000 (avec) 0.500000 s 0.750000 0.000000 0.000000 0.000000 1.633000 0.00100 0.00000 (cmass, press) 1 (ntype) 2 Ar 40.00000 (natom,nameat,atmass) 0.000000 0.000000 0.000000 (rat) t 1.000000 t 1.000000 0.50000040.000000 (rcut) 9 9 9 (ncell) 100 1110 10 (nstep,ntcheck,ntimes) 000.00000 0.00100 500.00000 (temp,ttol,dt)

~

Run7 Test: MD of 32 atoms at 200K (title)md (calc) s n (ic,iio) 10.000000 (alatt) 2 2 2 (nsc) 1.000000 0.000000 0.000000 (avec) 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 0.00100 0.00000 (cmass, press) 1 (ntype) 4 Ar 40.00000 (natom,nameat,atmass) 0.000000 0.000000 0.000000 (rat) 0.500000 0.500000 0.000000 0.000000 0.500000 0.500000 0.500000 0.000000 0.500000 40.000000 (rcut) 3 3 3 (ncell) 1000 100 10 (nstep,ntcheck,ntimes) 200.00000 0.00100 200.00000 (temp,ttol,dt)~

Run8

Test: MD of 32 atoms at 2000K (title)md (calc) s n (ic,iio) 10.000000 (alatt) 2 2 2 (nsc) 1.000000 0.000000 0.000000 (avec) 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 0.00100 0.00000 (cmass, press) 1 (ntype) 4 Ar 40.00000 (natom,nameat,atmass) 0.000000 0.000000 0.000000 (rat) 0.500000 0.500000 0.000000 0.000000 0.500000 0.500000 0.500000 0.000000 0.500000 40.000000 (rcut) 3 3 3 (ncell) 1000 100 10 (nstep,ntcheck,ntimes) 2000.00000 0.00100 100.00000 (temp,ttol,dt)~