1. PWscf, FPMD and CP (Democritos Package) Tutorial
Carlo Cavazzoni (High Performance Computing Group) CINECA
16/11/2018

2. Outline ab-initio simulations introduction
PWSCF and Democritos package
Installation
Functionality
Simple examples
16/11/2018

3. The scope of computer simulation
Measure theories. Solve equations which could not be solved otherwise. “Get numbers out of theories” much in the same way as experiments “get numbers out of a natural process”
Do virtual experiments where experimental conditions can be controlled down to the atomic scale in ways which would not be possible in the lab

4. The Born-Oppenheimer approximation (M>>m)
Ab initio simulations
The Born-Oppenheimer approximation (M>>m)

5. Density functional theory
Kohn-Sham Hamiltonian

6. Kohn-Sham equations from functional minimization
Helmann & Feynman
Kohn & Sham

7. The tricks of the trade
Expanding the Kohn-Sham orbitals into a suitable basis set turns DFT into a multi-variate minimization problem, and the KS equation into a non-linear matrix eigenvalue problem
The use of pseudo-potentials allows to ignore chemically inert core states and to use plane waves (the name of the game!)

8. The tricks of the trade (II)
Plane waves are orthogonal and the matrix elements are usually easy to calculate; the effective completeness of the basis is easy to check
Plane-waves allow to calculate efficiently matrix-vector products and to solve the Poisson equation using FFT’s
Supercells for treating finite (or semi-infinite) systems
Iterative diagonalization vs. global minimization

9. The tricks of the trade (III)
Summing over occupied states: special-point and Gaussian-smearing techniques
Non-linear extrapolation for SCF acceleration and density prediction in MD
Choice of fictitious masses in CP dynamics

10. Which algorithm shall I use?
Electronic structure: SCF diagonalization vs. energy minimization
Geometry optimization: standard DFT
Lattice vibrations, static response functions: DF perturbation theory
Dynamics: Car-Parrinello vs. Born-Oppenheimer
Slow kinetics and rare events: path sampling vs. Parrinello-Laio metadynamics
Optical properties, excited states: Time-dependent DFT & many-body perturbation theory

11. PWSCF and Democritos (www.democritos.it)
(DEmocritos MOdeling Center for Research in aTOmistic Simulatios)
Democritos is a National Simulation Center of the Italian
Istituto Nazionale per la Fisica della Materia (INFM).
Hosted by SISSA in Trieste
PWSCF is promoted, maintained and developed
by Democritos
CINECA is a Partner of Democritos.
16/11/2018

12. Democritos and Scientific Software
GOAL:
Development of software for next generation parallel computers (N>10000 processors)
Production of high-quality software package for atomistic simulations based on density-functional theory (DFT), plane wave (PW) and pseudopotentials (PP)
16/11/2018

13. PWSCF is also part of a merge effort of pre-existing scientific software
PWscf package: electronic structure, structural optimization, molecular dynamics, vibrational and dielectric properties. Developed by S. Baroni, S. de Gironcoli, A. Dal Corso (SISSA, Trieste), P. Giannozzi (Scuola Normale, Pisa) and others. See for more information and downloads.
CP code: Car-Parrinello variable-cell molecular dynamics. Developed by A. Pasquarello (IRRMA, Lausanne), K. Laasonen (Oulu), A. Trave (UCBerkeley), R. Car (Princeton), P. Giannozzi and others. Based on the original code written by R. Car and A. Pasquarello. Download CP (initial public release)
FPMD code: Car-Parrinello variable-cell molecular dynamics. Developed by C. Cavazzoni (CINECA, Bologna), S. Scandolo (ICTP, Trieste), G. Chiarotti (SISSA, Trieste), P. Focher, G. Ballabio and others. Based on the original code written by R. Car and M. Parrinello. Download FPMD (initial public release)
GNU License
16/11/2018

14. Present Package Status
The Package is developed in Fortran90
The codes in the Package share:
installation
most parts of the basic code
input format
PP format
a graphical GUI is being developed
output format (CP/FPMD fully compatible/restartable)
16/11/2018

15. Basic Data Type Charge density 3D arrays Wave functions 1D arrays
Real space
Charge density
3D arrays
Wave functions
1D arrays
Reciprocal space
16/11/2018

16. Reciprocal Space Representation
Wave
Functions
To truncate the infinite sum
Charge
Density
To retain the same accurancy
as the wave function
16/11/2018

17. DFT Functional
16/11/2018

18. FFTs Reciprocal Space FFT Real Space |G|2/2< Ecut |G|2/2< 4Ecut
16/11/2018

19. Parallelization Reciprocal Space distribution
PWSCF parallelize also over K-points
16/11/2018

20. Parallelization FFT algorithm
16/11/2018

21. SP4, optimization issues
use essl
use non-blocking communications
use mass
modify strides to reduce TLB misses
use block-algorithm whenever possible ATLAS strategy
16/11/2018

22. Case study: matrix transposition
do ib = 1, nb
ioff = (ib-1) * bsiz
do jb = 1, mb
joff = (jb-1) * bsiz
do j = 1, bsiz
do i = 1, bsiz
buf(i,j) = x(i+ioff, j+joff)
enddo
do i = 1, j-1
bswp = buf(i,j)
buf(i,j) = buf(j,i)
buf(j,i) = bswp
do i=1,bsiz
do j=1,bsiz
y(j+joff, i+ioff) = buf(j,i)
do i=1,n
do j=1,m
y(j,i) = x(i,j)
enddo
block-algorithm
bsiz = block size
nb = n / bsiz
mb = n / bsiz
more code for reminder:
MOD(n / bsiz) /= 0 OR
MOD(m / bsiz) /= 0
16/11/2018

23. Block algorithm for matrix transposition80 70 60 50
Mips 40 30 20 10 20 40 60 80
100
120
block dimension
16/11/2018

24. CP code Verlet dynamics with mass preconditioning
Developed by A. Pasquarello (IRRMA, Lausanne), K. Laasonen (Oulu), A. Trave (UCBerkeley), R. Car (Princeton), P. Giannozzi and others. Based on the original code written by R. Car and A. Pasquarello.
Verlet dynamics with mass preconditioning
Temperature control: Nose’ thermostat, velocity rescaling
Metallic systems: Nose’ thermostat for both electrons and ions
electronic and ionic minimization via damped dynamics
Modified kinetic functional for costant-pressure calculations
“grid box” for fast treatment of augmentation terms in Ultrasoft PP’s
Nudget Elastic Band (NEB) scheme for transition paths and energy barriers
Limitations:
no k-points
no fancier minimization chemes
no constraints
16/11/2018

25. FPMD code Verlet dynamics with mass preconditioning
Developed by C. Cavazzoni (CINECA, Bologna), S. Scandolo (ICTP, Trieste), G. Chiarotti (SISSA, Trieste), P. Focher, G. Ballabio and others. Based on the original code written by R. Car and M. Parrinello.
Verlet dynamics with mass preconditioning
Temperature control: Nose’ thermostat, velocity rescaling
Metallic systems: Nose’ thermostat for both electrons and ions
Various electronic and ionic minimization schemes
Modified kinetic functional for costant-pressure calculations
Macroscopic polarization via Berry Phase
Nudget Elastic Band (NEB) scheme for transition paths and energy barriers
Constrained dynamics
Limitations:
no Ultrasoft PP’s
16/11/2018

26. PWSCF code Self-consistent ground-state energy and Kohn-Sham orbitals
Developed by S. Baroni, S. de Gironcoli, A. Dal Corso (SISSA, Trieste), P. Giannozzi (Scuola Normale, Pisa) and others.
Self-consistent ground-state energy and Kohn-Sham orbitals
Structural optimization
Molecular dynamics on the ground-state Born-Oppenheimer surface
Variable-cell molecular dynamics
Phonon frequencies and eigenvectors at a generic wave vector
Effective charges and dielectric tensors
Electron-phonon interaction coefficients for metals
Third-order anharmonic phonon lifetimes
Macroscopic polarization via Berry Phase
Nudget Elastic Band (NEB) scheme for transition paths and energy barriers
Limitations:
no Car-Parrinello dynamics
very limited constrained minimization and dynamics
16/11/2018

27. Access the anonymous CVS of the Democritos PW-PP-DFT Package
anonymous CVS contains the whole package,
with all the codes
(tcsh/csh):
setenv CVS_RSH ssh
setenv CVSROOT
(sh/bash):
export CVS_RSH=ssh
export
Then:
cvs login
(password: cvsanon).
For the first code download:
cvs co O-sesame
Open-Sesame: Open-Source Scalable Electronic Structure and Atomistic Modeling Environment
16/11/2018

28. Installation ./configure make target 16/11/2018
where target is one of the following:
pw basic code for scf, structure optimization, MD
fpmd FPMD code for Car-Parrinello MD
cp CP code: CP MD with ultrasoft pseudopotentials
ph phonon code
pp postprocessing programs
gamma Gamma-only version of phonon code
nc non collinear magnetic version of pw code
pwcond ballistic conductance
d third-order derivatives
tools misc tools for data analysis
upf utilities for pseudopotential conversion
pwall same as "make pw ph pp gamma nc pwcond d3 tools"
all same as "make pwall fpmd cp upf"
links creates links to executables in bin/
clean remove executables and objects
veryclean revert distribution to the original status
tar create a tarball of the source tree
tar-gui create a tarball of the GUI sources
16/11/2018

29. Installation (DEMO)
how to configure configure?
16/11/2018

30. Directories tree I/II include Modules clib flib upftools CPV FPMD PW
includes files
Modules
clib
flib
upftools
common modules, subroutine and tools
CPV
FPMD PW
main codes directories D3 PH PP
pw post-processing codes
16/11/2018

31. Directories tree II/II
pwdocs
cpdocs
documentation
pseudo
pseudopotential library
install
preconfigured installation parameter files
cp_examples
pw_examples
run examples
16/11/2018

32. Input files PWscf, CP and FPMD share the same input files format
All codes read input parameters from
standard input
Codes read from files PP data,
all codes read the same PP files
In directory upftools there are conversion
tools for PP formats (there is one also for CPMD)
16/11/2018

33. Standard Input Layout
The standard input is divided in two main section
the first contains fortran namelists
the seconds contains optional input “CARDS”
all input parameter are described in cpdocs and
pwdocs file.
all input parameter for all codes are contained
in file Modules/input_parameters.f90
16/11/2018

34. layout namelists cards the namelist order is fixed, but within a
&CONTROL
control_parameter_1, control_parameter_2, /
&SYSTEM
sistem_parameter_1, sistem_parameter_2,
&ELECTRONS
electrons_parameter_1, electrons_parameter_2,
&IONS
ions_parameter_1, ions_parameter_2,
&CELL
cell_parameter_1, cell_parameter_2,
‘ATOMIC_SPECIES’
label_1 mass_1 pseudo_file_1
label_2 mass_2 pseudo_file_2
.....
‘ATOMIC_POSITIONS’
label_1 px_1 py_1 pz_1
label_2 px_2 py_2 pz_2
the namelist order is
fixed, but within a
namelist the parameter
sequence is irrelevant
namelists
card order is irrelevant,
but within a card the
parameter layout is fixed
cards
16/11/2018

35. Input Namelists
&CONTROL input variables that control the flux of teh calculation and the amount of I/O on disk and on the screen
&SYSTEM input variables that specify the system under study
&ELECTRONS input variables that control the algorithms used to reach the self-consistent solution of KS equations for the electrons, and the electrons dynamics (for CP and FPMD)
&IONS input variables that control ionic motion in molecular dynamics run or structural relaxation
&CELL input variables that control cell-shape evolution in a variable-cell MD run or structural relaxation
16/11/2018

36. Input Cards
ATOMIC_SPECIES name, mass and pseudopotential used for each atomic species present in the system
ATOMIC_POSITIONS type and coordinate of each atom in the unit cell
K_POINTS coordinate and weights of the k-points used for BZ integration
CELL_PARAMETERS …
OCCUPATIONS …
CLINBING_IMAGES …
16/11/2018

37. Silicon Band Structure (www.cineca.it/~acv0/Public/PWSCF_Tutorial.tgz)
Select the appropriate unit cell
Input the atomic coordinate
Choose the pseudopotentials
Determine k-point sampling
Select the size of the basis set (Ecut)
16/11/2018

38. Select the appropriate unit cell
ibrav
bravais lattice
celldm(.)
cell parameters
celldm(1) = a,
celldm(2) = b/a,
celldm(3) = c/a
celldm(4) = cos(bc)
celldm(5) = cos(ac)
celldm(6) = cos(ab)
ibrav structure celldm(2)-celldm(6)
"free", see above not used
cubic P (sc) not used
cubic F (fcc) not used
cubic I (bcc) not used
Hexagonal and Trigonal P celldm(3)=c/a
Trigonal R celldm(4)=cos(aalpha)
Tetragonal P (st) celldm(3)=c/a
Tetragonal I (bct) celldm(3)=c/a
Orthorhombic P celldm(2)=b/a,
celldm(3)=c/a
Orthorhombic base-centered(bco) celldm(2)=b/a,
Orthorhombic face-centered celldm(2)=b/a,
Orthorhombic body-centered celldm(2)=b/a,
Monoclinic P celldm(2)=b/a,
celldm(3)=c/a,
celldm(4)=cos(ab)
Monoclinic base-centered celldm(2)=b/a
Triclinic P celldm(2)= b/a,
celldm(3)= c/a,
celldm(4)= cos(bc),
celldm(5)= cos(ac),
celldm(6)= cos(ab)
16/11/2018

39. Unit cell
ibrav = 2, cubic fcc
ibrav = 1, simple cubic
16/11/2018

40. Unit cell (supercells)
ibrav = 1, simple cubic
(crystal with defects)
ibrav = 6, tetragonal
(surfaces)
16/11/2018

41. Silicon Crystal unit cell
&control
prefix='silicon',
pseudo_dir = '$PSEUDO_DIR/',
outdir='$TMP_DIR/' /
&system
ibrav= 2, celldm(1) =10.20, nat= 2, ntyp= 1,
ecutwfc =12.0,
&electrons
empty namelist, keep default values
for Self-Consistent Field iterations
16/11/2018

42. Atomic Positions ATOMIC_POSITION (units)
units: if units is not specified, unit of “celldm(1)” is assumed
units = bohr position in Bohr radius
units = angstrom position in Angstrom
units = crystal position in crystal coord.
ATOMIC_SPECIES
Si Si.vbc.UPF
ATOMIC_POSITIONS Si Si
16/11/2018

43. K-points set Automatic User specified K_POINTS (automatic) 2 2 2 1 1 1
k-points grid
grid off-set
K_POINTS 2
User specified
number of k-points
k-points weight
k-points coordinate
(crystal coord.)
start with coarse grid, then try finer ones to check convergence
16/11/2018

44. Input for Silicon Crystal SCF calculation
&control
prefix='silicon',
pseudo_dir = '$PSEUDO_DIR/',
outdir='$TMP_DIR/' /
&system
ibrav= 2, celldm(1) =10.20, nat= 2, ntyp= 1,
ecutwfc =12.0,
&electrons
ATOMIC_SPECIES
Si Si.vbc.UPF
ATOMIC_POSITIONS Si Si
K_POINTS 2
pw.x < inputfile
16/11/2018

45. Non SCF calculation To compute band structure, we need the value of
eigenvectors on all k-point of the bands.
If the computation is converged with respect to
the k-points grid, we do not need to repeat the
SCF calculation on all k-points of the bands, but
we can diagonalize the Hamiltonian using the SCF
Kohn-Sham potential.
16/11/2018

46. Input for Non SCF calculation
&control
calculation='nscf'
pseudo_dir = '$PSEUDO_DIR/',
outdir='$TMP_DIR/',
prefix='silicon' /
&system
ibrav= 2, celldm(1) =10.20, nat= 2, ntyp= 1,
ecutwfc =12.0, nbnd = 8,
&electrons
ATOMIC_SPECIES
Si Si.vbc.UPF
ATOMIC_POSITIONS Si Si
K_POINTS …
16/11/2018

47. K-points set along high symmetry lines
# band structure calculation along delta, sigma and lambda lines
K_POINTS 36
16/11/2018

48. Post Processing bands.x < inputfile
&inputpp
prefix = 'silicon'
outdir = '$TMP_DIR/'
filband = 'sibands.dat' /
bands.x < inputfile
prints the bands to graphical files
sibands.dat
sibands.xmgr
sibands.ps
6.3369
plotband.x < inputfile
16/11/2018

49. the silicon bands!
16/11/2018

50. How to find equilibrium lattice parameters
different
scf computation,
changing cut-off
and lattice
parameter
16/11/2018

51. Plot Charge Density 1) 2) 3)
# post-processing for charge density
cat > si.pp_rho.in << EOF
&inputpp
prefix = 'silicon'
outdir = '$TMP_DIR/'
filplot = 'sicharge'
plot_num= 0 /
EOF
./pp.x < si.pp_rho.in > si.pp_rho.out 1)
# chdens
cat > si.chdens.in << EOF
&input
nfile = 1
filepp(1) = 'sicharge'
weight(1) = 1.0
iflag = 2
plot_out = 1
output_format = 2
fileout = 'si.rho.dat'
e1(1) =1.0, e1(2)=1.0, e1(3) = 0.0,
e2(1) =0.0, e2(2)=0.0, e2(3) = 1.0,
nx=56, ny=40 /
EOF
./chdens.x < si.chdens.in > si.chdens.out 2)
cat > si.plotrho.in << EOF
si.rho.dat
si.rho.ps n
EOF
./plotrho.x < si.plotrho.in > si.plotrho.out 3)
16/11/2018

52. Charge Density
16/11/2018

53. MD - Molecular Dynamics
Stop
accumulate statistics
(RDF, Energies, MSD, ecc..)
Initial configuration
calculate forces
update atomic positions
16/11/2018

54. DFT - Solution of the KS equation.
trial n(r)
build VKS
solve [-1/22+VKS]i = ii
|nnew-nold|<nthr ?
groud state
calculate nnew(r) = ∑fi|i|2
nold = nnew
yes no
16/11/2018

55. AIMD ab-Initio Molecular Dynamics Born-Oppenheimer MD
Stop
accumulate statistics
Initial configuration
Solve the DFTproblem Egs[{RI}]
update atomic positions
calculate forces FI=∂Egs[{RI}] /∂RI
16/11/2018

56. CPMD Car-Parrinello Molecular Dynamics
Stop
accumulate statistics
Initial configuration
Solve the DFTproblem Egs[{RI}, {i}]
update wave functions i
ionic forces FI = - ∂Etot[{RI}] / ∂RI
update atomic positions RI
electronic forces F i = - ∂Etot[{RI}, {i}] / ∂i*
16/11/2018

57. Car-Parrinello Lagrangian
(deformable cell extension)
Density Functionl Energy
Diagonal in Fourier space
Diagonal in Real space
Computationally it is convenient to calculate quantities
in the space where they are diagonal
16/11/2018

58. Equations of motion
Electrons treated as classical scalar fields, follow
adiabatically the ions remaining close to the ground state.
Ions in scaled coordinates, move along the Born Oppenheimer surface
Cell shape is driven by the imbalance of internal
stress  and external pressure p.
16/11/2018

59. CINECA/Democritos CP Code
FORM
NLRH
Pseudopotential Form factors
RHOOFR
(r) =  |(r)|2
VOFRHO
PRESS
V(R, ) = VrDFT(R, (r)) + VGDFT(R, (G))
FORCE
Forces on the electrons: F
ORTHO
Orthogonalize wave functions: 
16/11/2018

60. A CP simulation require usually many RUNs and JOBs
Minimize the Electronic degrees of freedom ( d.o.f. )
Minimize the ionic d.o.f.
Randomize ionic d.o.f.
Re-minimize Electronic d.o.f
Move Electronic and Ionic d.o.f. using Verlet to integrate the equations of motion
Change temperature using thermostat
Accumulate statistics for few picoseconds of simulated time ( – time steps )
16/11/2018

61. Electronic Minimization I/II
&CONTROL
title = ' Water Molecule ',
calculation = 'cp',
restart_mode = 'from_scratch',
prefix = 'h2o_mol‘
nstep = 50,
dt = 5.0d0,
etot_conv_thr = 1.d-9,
ekin_conv_thr = 1.d-4, /
&SYSTEM
ibrav = 1,
celldm(1) = 10.0,
nat = 3, ntyp = 2,
nbnd = 4, nelec = 8,
ecutwfc = 70.0,
xc_type = 'BLYP'
CP time step (in a.u.)
number of CP step
convergence threshold,
effective only for minimization
Simulation Cell (in a.u.)
number of bands, and
electrons
number of atom,
and species
Energy cut-off,
in Rydberg
Exchange and correlation
functional
16/11/2018

62. Electronic Minimization II/II
&ELECTRONS
emass = 400.d0,
emass_cutoff = 2.5d0,
orthogonalization = 'ortho',
electron_dynamics = 'sd', /
&IONS
ion_dynamics = 'none',
ATOMIC_SPECIES
O 16.0d0 O.BLYP.UPF 4
H 1.00d0 H.fpmd.UPF 4
ATOMIC_POSITIONS (bohr) O H H
Fictitious electron mass (a.u.)
m parameter in CP dynamic
orthogonalization
algorithm
‘ortho’ or 'Gram-Schmidt'
PW Energy cut-off (in Rydberg)
for Fourier Acceleration
electron dynamics
sd -> steepest descend
cg -> conjugate gradient
damp -> damped
verlet -> Verlet
Label Mass (uma) PP
ion dynamics
none -> kept fixed
sd -> steepest descend
cg -> conjugate gradient
damp -> damped
verlet -> Verlet
Label x y z
16/11/2018

63. CP – Dynamics I/II verlet should be used for ions and electrons
&CONTROL
title = ' Water Molecule ',
calculation = 'cp',
restart_mode = ‘restart',
nstep = 50,
dt = 5.0d0,
prefix = 'h2o_mol' /
&SYSTEM
ibrav = 1,
celldm(1) = 10.0,
nat = 3, ntyp = 2,
nbnd = 4, nelec = 8,
ecutwfc = 70.0,
xc_type = 'BLYP'
&ELECTRONS
emass = 400.d0,
emass_cutoff = 2.5d0,
orthogonalization = 'ortho',
electron_dynamics = ‘verlet',
electron_velocities = ‘zero’
verlet should be used
for ions and electrons
&IONS
ion_dynamics = ‘verlet',
ion_velocities = ‘zero’ /
ATOMIC_SPECIES
O 16.0d0 O.BLYP.UPF 4
H 1.00d0 H.fpmd.UPF 4
ATOMIC_POSITIONS (bohr) O H H
in the very first run of
dynamics velocities
should be set to zero
16/11/2018

64. CP – Dynamics II/II ( thermostat )
&CONTROL
title = ' Water Molecule ',
calculation = 'cp',
restart_mode = ‘restart',
nstep = 50,
dt = 5.0d0,
prefix = 'h2o_mol' /
&SYSTEM
ibrav = 1,
celldm(1) = 10.0,
nat = 3, ntyp = 2,
nbnd = 4, nelec = 8,
ecutwfc = 70.0,
xc_type = 'BLYP'
&ELECTRONS
emass = 400.d0,
emass_cutoff = 2.5d0,
orthogonalization = 'ortho',
electron_dynamics = ‘verlet',
Termostat temperature and frequency
tempw ( Kelvin )
fnosep ( THz )
&IONS
ion_dynamics = ‘verlet',
ion_temperature = ‘nose’
tempw = 300,
fnosep = 70.0 /
ATOMIC_SPECIES
O 16.0d0 O.BLYP.UPF 4
H 1.00d0 H.fpmd.UPF 4
ATOMIC_POSITIONS (bohr) O H H
16/11/2018

65. Parrinello-Rahman Dynamics
&CONTROL
title = ' Water Molecule ',
calculation = ‘vc-cp',
restart_mode = ‘restart',
nstep = 50,
dt = 5.0d0,
prefix = 'h2o_mol' /
&SYSTEM
ibrav = 1,
celldm(1) = 10.0,
nat = 3, ntyp = 2,
nbnd = 4, nelec = 8,
ecutwfc = 70.0,
xc_type = 'BLYP'
&ELECTRONS
emass = 400.d0,
emass_cutoff = 2.5d0,
orthogonalization = 'ortho',
electron_dynamics = ‘verlet',
change the calculation
type (variable cell cp)
&IONS
ion_dynamics = ‘verlet', /
&CELL
cell_dynamics = ‘pr’,
press = 0.0
ATOMIC_SPECIES
O 16.0d0 O.BLYP.UPF 4
H 1.00d0 H.fpmd.UPF 4
ATOMIC_POSITIONS (bohr) O H H
Add the CELL namelists,
with cell dynamics:
pr -> parrinello rahman
sd -> steepest descent
damp -> damped
external
pressure
(Kbar)
16/11/2018

66. high pressure – high temperature water phase diagram
C.Cavazzoni, G.L. Chiarotti, S.Scandolo, E.Tosatti, M.Bernasconi, and M. Parrinello
“Superprotonic and metallic states of water and ammonia at giant planet conditions”,
Science.283, 44 (1999).
16/11/2018

67. ionic trajectory at finite temperature
H trajectories in the superionic phase
16/11/2018

68. Run on SP4 - Water128
16/11/2018

69. Run on SP4 - Water32
16/11/2018