1. Invariant MD w/ Variable Cell Shape R. Wentzcovitch U. Minnesota Vlab Tutorial -Simulate solids at high PTs -Useful for structural optimizations -Useful for structural search (shake and bake) -Various fictitious Lagrangian formulations

2. Fictitious molecular dynamics H. C. Andersen (1978) (N,E,V) (N,H,P)

3. Variable Cell Shape MD i=vector index j=cart. index

4. Anderson’s Fictious MD (HPN ensemble) Anderson’s variable volume fixed shape constant pressure MD (Anderson, J. Chem. Phys 72,2384(1980)) The ensemble (trajectory) averages produce the HPN ensemble averages Cell volume

5. Fictious MD (continue…) Parrinello/Rahman variable cell shape MD (Parrinello and Rahman, J. Appl. Phys 52, 7182 (1981)) Applying Lagrange’s equation

6. - in PR-VCSMD is not uniquely definedK Latt The trajectory is not uniquely defined. It does not depend only on the initial conditions. a a aa equivalent

7. Solution: use strain ε instead of h as dynamical variable ε is strain Invariant dynamics I Wentzcovitch, PRB 44,2358 (1991)

8. Alternative form of L Inv -I in terms of h and s: with Final observation: In the limit of variable V-only Solution: with Eq. of motion given by Eq. 9 in PRB 44, 2358(1991)

9. a a 2a Fluctuations in the cell edges lengths of fcc X-tal of Ar initially placed away from V eq. Beeman integration algorithm dt= 10 fmt (1 a.u. = 2.5 x 10 -17 s (in Ry)) M i = 39 m p W= 35 m p in (a); W= 0.0007 m p /a o 3 in (b) R c = 10 a o Wentzcovitch, PRB 44,2358 (1991)

10. fcc bcc sc θ d d d bcc fcc sc fcc bcc Potential energy isosurfaces Basins of attraction if we use and in the MD Basins of attraction if we use and in the PR-MD Wentzcovitch, PRB 44,2358 (1991)

11. Typical Computational Experiment Damped dynamics (Wentzcovitch, 1991) P = 150 GPa (Wentzcovitch, Martins, and Price, PRL 1993)

12. 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 at u=1/6 or 1/3 u=1/4

13. 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 will be simulated at the points marked by dots and error bars (undertainties in P and T). Exp. P T = 45-55 GPa at 300 K ~150 K

14. hcp to bcc transition Time evolution of the internal parameters u’s, and angles and lengths of simulation cell vectors. Simulation w/ 16 atoms only T = 700 K P = 72 GPa dt = 6 fts W=0.02 m at =24.3 m p Θ ab = 70.53 o Θ ab = 60 o u=1/6 u=1/4 u=1/6 u=1/4

15. bcc to hcp transition Time evolution of the internal parameters u’s, and angles and lengths of simulation cell vectors. Simulation w/ 16 atoms only T = 500 K P = 12 GPa dt = 6 fts W=0.02 m at =24.3 m p u=1/6 u=1/4 Θ ab = 70.53 o Θ ab = 60 o

16. MgSiO 3 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)

17. b c a Lattice system: Bace-centered orthorhombic Space group: Cmcm Formula 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] [Å 3 ]: 121.1(123.3)( )…perovskite Pt Crystal structure of post-perovskite Tsuchiya, Tsuchiya, Umemoto, Wentzcovitch, EPSL, 2004

18. Ab initio exploration of post-perovskite phase in MgSiO 3 Perovskite SiO 3 layer SiO 3 Mg SiO 3 Mg SiO 3 MgSiO 3 - Reasonable polyhedra type and connectivity under ultra high pressure - SiO 4 chain

19. Post-perovskite c’ a’ b’ Structural relation between Pv and Post-pv Deformation of perovskite under shear strain ε 6 a b c Perovskite θ Tsuchiya, Tsuchiya, Umemoto, Wentzcovitch, EPSL, 2004

20. Conclusions -VCSMD is very useful for structural optimizations when the dynamics has the correct symmetry properties (invariant dynamics) - It is capable of simulating a phase transition when one knows how the transformation occurs - There is unavoidable hysteresis associated with the simulation, which makes the simulation often unfeasible -Alternative approaches for obtaining phase boundaries by computations will be discussed throughout the course

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

22. Some Instructions for Lind24-Lab 1)OpenDX is a visualization software you may use. To enable access to OpenDX: module load soft/opendx module 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'.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.

23. 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) ~

24. Run2 Decrease step size by ½ and increase # of steps by 2

25. 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) ~

26. Run4 Adjust cell mass to get same period of oscillation

27. 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) ~

28. 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 0.500000 40.000000 (rcut) 9 9 9 (ncell) 100 1110 10 (nstep,ntcheck,ntimes) 000.00000 0.00100 500.00000 (temp,ttol,dt) ~

29. 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) ~

30. 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) ~