On the role of quantum mechanical simulation in materials science.

Page 1

João Pessoa, 2014

Roberto Dovesi

On the role of quantum mechanical

simulation in materials science

Page 2

João Pessoa, 2014

Is simulation useful?

Does it produce reasonable numbers?

Or can only try to reproduce the experiments?

Connected question:

Is simulation expensive?

Page 3

João Pessoa, 2014

In the year 1960-1990, calculations for the structural properties of

periodic compounds ( oxides, halides, ..) were performed at the

semi-classical or force-field level (Catlow, Gale, Macrodt and others)

The first quantum mechanical ab initio calculations of periodic

systems date back to 1979-1981 (diamond, silicon, cubic BN: band

structure, total energy, charge density maps)

The first periodic code publicly available to the scientific community

is released in 1988 (CRYSTAL through QCPE, Quantum Chemistry

Program Exchange)…

Afterwards………..very quick evolution

Page 4

João Pessoa, 2014

How many transistors on a chip?

Intel i7 Sandy bridge 32 nm

2.27 billions of transistors 434 mm2

GPU NVIDIA GK110 28 nm

7.1 billions of transistors

Gordon Moore

The number of transistors per chip doubles

every 18 months

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João Pessoa, 2014

Performance of HPC

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João Pessoa, 2014

DFT & Kohn-Sham

• “Density Functional Theory (DFT)

is an incredible success story” *

• DFT has enable to tackle complex

problems with an accuracy

unobtainable by any other

approach

• DFT methods has now been

applied to chemistry, materials

science, solid-state physics, but

also geology, mineralogy and

biology.

• Kohn-Sham formalism

* from K. Burke Perspective on Density Functional

Theory JCP 136 (2012) 150901

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João Pessoa, 2014

Is simulation expensive?

The last computer we bought….

Server Supermicro 64 CORE OPTERON euros 6.490 ,00

1 x Chassis 2U - 6 x SATA/SAS - 1400W

4 x CPU AMD Opteron 16-Core 6272 2,1Ghz 115W

8 x RAM 8 GB DDR3-1333 ECC Reg. (1GB/core)

1 x Backplane SAS/SATA 6 disks

1 x HDD SATAII 500 GB 7.200 RPM hot-swap

1 x SVGA Matrox G200eW 16MB

2 x LAN interface 1 Gbit

1 x Management IPMI 2.0

Cheap… but 64 cores- Parallel computing

Much less than most of the experimental equipments

64 cores enough for large calculation……..

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João Pessoa, 2014

At the other extreme: SUPERCOMPUTERS

Available, but:

a) They are fragile

b) Not so much standard (compiler, libreries)

c) The software (that is always late with respect to hardware) MUST BE

ABLE TO EXPLOIT this huge power

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João Pessoa, 2014

The PRACE Tier-0 Resources

HORNET (HLRS, DE)

Cray XC30 system - 94,656 cores CURIE (GENCI, FR)

BULL x86 system – 80,640 cores (thin nodes)

FERMI (CINECA, IT)

BlueGene Q system – 163,840 cores

SUPERMUC (LRZ, DE)

IBM System x iDataPlex

system– 155,656 cores

MARENOSTRUM (BSC, SP)



JUQUEEN (JÜLICH, DE)


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João Pessoa, 2014

CRYSTAL parallel versions: MPPcrystal

MPPcrystal– Distributed data

– Each processor hold only a part of each of the matrices used

in the linear algebra

– Most but not all of CRYSTAL implemented

– Will fail quickly and cleanly if requested feature not

implemented

– Good for large problems on large processor counts

– For large systems can scale well, but not so good for

small to medium size ones

– Size of linear algebra matrices is, at present, not an

issue given enough processors

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João Pessoa, 2014

The software must be

a) Easy to use (freindly)

b) Robust,

c) Protected

d) Documented

e) General as much as possible

f) Transferable

g) Parallel

h) ………..

I few axamples referring to the CRYSTA14 code, that uses a

guassian basis set.

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João Pessoa, 2014

One of the specific features of solids are the

TENSORIAL PROPERTIES

that in the liquid or gas phase can be known (measured or

calculated ) only as mean values (invariants of the tensor)

Many of them can be computed

• Thttt

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João Pessoa, 2014

Tensorial Properties of Crystals

Second order Third order Fourth order

✔ Dielectric✔ Polarizability

✔ Piezoelectric✔ First hyperpolarizability

✔ Elastic✔ Photoelastic✔ Second hyperpolarizability

Maximum number of independent elements according to crystal symmetry:

6 18 21

Minimum number of independent elements according to crystal symmetry:

1 1 3

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João Pessoa, 2014

Effect of the Crystal Symmetry on Tensors

CubicTriclinic

Third Order Tensors:

Fourth Order Tensors:

Cubic

Hexagonal

Triclinic Hexagonal

J. F. Nye, Oxford University Press, (1985)

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João Pessoa, 2014

Tensorial Properties Related to Crystal Strain

Elastic Tensor Piezoelectric Tensor Photoelastic Tensor

Order of the Tensors

First derivative of the inverse dielectric tensor (difference

with respect to the unstrained configuration)

with respect to strain

First derivative of the polarization P (computed through the Berry phase

approach) with respect to the strain

Second derivatives of the total energy E with respect

to a pair of strains, for a 3D crystal

Voigt’s notation is used according to v, u = 1, . . . 6 (1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 =xz, 6 = xy) and i,j =1,2, 3 (1 = x , 2 = y, 3 = z).

4 3 4

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João Pessoa, 2014

Geometry definitionELASTCON

[Optional keywords]END

ENDBasis set definitionENDComput. ParametersEND

Tensorial Properties Related to Crystal Strain

Elastic Tensor Piezoelectric Tensor Photoelastic Tensor

Geometry definitionPIEZOCON



Geometry definitionPHOTOELA



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João Pessoa, 2014

Geometry optimization and calculation of the cell gradients of the reference structure

Full symmetry analysis and definition of minimal set of strains

Application of each strain and calculation of cell gradients of strained configurations,

for different strain amplitudes

CRYSTAL14: Elastic Properties – The Algorithm

Numerical fitting of analytical gradients with respect to strain and calculation of

elastic constants

From a posteriori calculations: seismic wave velocities (through Christoffel's

equation), bulk, shear and Young moduli.

Page 23

João Pessoa, 2014

Six Silicate Garnets

✔Garnets constitute a large class of materials of great geological and technological interest

✔Silicate garnets are among the most important rock-forming minerals

✔Earth’s lower crust, upper mantle and transition zone

✔Interest in discussion of different models for Earth's interior

✔Characterized by a cubic structurewith space group Ia3d

✔80 atoms per unit cell

Pyraspite

Mg3Al

2(SiO

4)3

Pyrope

Fe3Al

2(SiO

4)3

Almandine

Mn3Al

2(SiO

4)3

Spessartine

Grossular

Ca3Al

2(SiO

4)3

Ca3Fe

2(SiO

4)3

Andradite

Ca3Cr

2(SiO

4)3

Uvarovite

Ugrandite

X3Y

2(SiO

4)

3

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João Pessoa, 2014

Mg

Al

O

Si

O

O

•Cubic Ia-3d

•160 atoms in the UC (80 in the primitive)

•O general position (48 equivalent)•Mn (24e) Al (16a) Si (24d) site positions

distorted

dodecahedra

tetrahedra

octahedra

Structure of pyrope: Mg3Al2(SiO4)3

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João Pessoa, 2014

CRYSTAL14: Elastic Properties

Pyrope-Mg3Al

2(SiO

4)

3

A. Erba, A. Mahmoud, R. Orlando and R. Dovesi, Phys. Chem. Minerals (2013) DOI 10.1007/s00269-013-0630-4

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João Pessoa, 2014



Almandine

Spessartine Grossular

Andradite Uvarovite

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João Pessoa, 2014


From the elastic constants, through Christoffel's equation, seismic wave velocities can be computed:

Some elastic properties of an isotropic polycrystalline aggregate can be computed from the elastic and compliance constants defined above via the Voigt-Reuss-Hill averaging scheme:

Bulk modulus

Shear modulus

Young modulus Poisson's ratio Anisotropy index

The average values of transverse (shear), vs, and longitudinal, vp, seismic wave velocities, for an isotropic polycrystalline aggregate, can be computed

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João Pessoa, 2014


Voigt-Reuss-Hill averaging scheme



Andradite Uvarovite

AlmandinePyrope

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João Pessoa, 2014



Andradite-Ca

3Fe

2(SiO

4)3

Directional seismic wave velocities of an andradite single-crystal, as computed ab initio in the present study (continuous lines) and as measured by Brillouin scattering at ambient pressure by Jiang et al (2004) (black symbols). Seismic wave velocities are reported along an azimuthal angle θ defined in the inset. Computed values are down shifted by 0.1 km/s.

Vp

Vs2

Vs1

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João Pessoa, 2014




Andradite Uvarovite

AlmandinePyrope

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João Pessoa, 2014




Andradite Uvarovite

AlmandinePyrope

Elastic Anisotropy

Seismic wave velocity

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João Pessoa, 2014

Are those calculations expensive?

✔ 80 atoms✔ 1488 atomic orbitals✔ 800 electrons✔ 48 symmetry operators

✔ Geometry optimization + cell gradients✔ 2 active deformation (compression, expansion), two geometry optimization + cell gradients each one (cubic crystal symmetry)

✔ CPU time: 18146.938 s ≈ 5 h on 256 processors (elastic properties of Pyrope)

Per unit cellPyrope

Reference structure

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João Pessoa, 2014



Application of each strain and calculation of cell gradients and Berry phase of

strained configurations, for different strain amplitudes

Piezoelectric Properties – The Algorithm

Berry phase calculation

Piezoelectric constants are obtained by numerical fitting with respect to the

strain

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João Pessoa, 2014



Application of each strain and calculation of cell gradients and the dielectric tensor

of strained configurations, for different strain amplitudes

Photoelastic Properties – The Algorithm

Dielectric tensor calculation through CPHF/KS

Photoelastic constants are obtained by numerical fitting with respect to the

strain

Page 35

João Pessoa, 2014

CRYSTAL14: Piezoelectric and Dielectric Properties

393 K 278 K 183 KTemperature

✔ BaTiO3

prototypical ferroelectric oxide✔ ABO

3-type perovskite crystal structure

✔ Advanced technological applications:✔ capacitor ✔ component of non-linear optical, piezoelectric and energy/data-storage devices.

Cubic

Tetragonal Orthorhombic Rhomohedral

✔ Upon cooling, three consecutive ferroelectric transitions occur starting from the cubic structure, due to the displacement of Ti ions along different crystallographic directions✔ The resulting macroscopic polarization of thematerial is always parallel to this displacement

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João Pessoa, 2014


A. Mahmoud, A. Erba, Kh. E. El-Kelany, M. Rérat and R. Orlando, Phys. Rev. B (2013)

✔ Two independent dielectric tensor component: є

11and є

33

✔ Computed as a function of the electric field wavelength λ with four different one-electron Hamiltonians✔ Experimental values at λ = 514.5 nm

✔ (є11

= 6.19 and є33

= 5.88)

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João Pessoa, 2014



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João Pessoa, 2014

CRYSTAL14: Photoelastic Properties


✔ Elasto-optic constants here refer to the λ → ∞ limit✔ No experimental data are currently available to compare with✔ From previous studies, we expect the hybrid PBE0 scheme to give the best description of elastic properties and the PBE functional the best description of photoelastic properties✔ Electronic “clamped-ion” and total “nuclear-relaxed” values are reported

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João Pessoa, 2014

CRYSTAL14: Photoelastic Properties

A. Erba and R. Dovesi, Phys. Rev. B 88, 045121 (2013)

✔ The three independent elasto-optic constants of MgO, computed at PBE level, as a function of the electric field wavelength λ ✔ p44 is almost wavelength independent✔ p11 and p12 show a clear dependence from λ✔ Dashed vertical lines in the figure identify the experimental range ofadopted electric field wavelengths

Page 40

João Pessoa, 2014

IR and RAMAN spectra

Wavenumbers and intensities

Page 41

João Pessoa, 2014

Reflectivity is calculated from dielectric constant by means of:

(θ is the beam incident angle)

The dielectric function is obtained with the classical dispersion relation

(damped harmonic oscillator):

IR reflectance spectrum

Page 42

João Pessoa, 2014

Garnets: X3Y2(SiO4)3

Space Group: Ia-3d

80 atoms in the primitive cell (240 modes)

Γrid = 3A1g + 5A2g + 8Eg + 14 F1g + 14 F2g + 5A1u + 5 A2u+ 10Eu + 18F1u + 16F2u

17 IR (F1u) and 25 RAMAN (A1g, Eg, F2g) active modes

X Y Name

Mg Al Pyrope

Ca Al Grossular

Fe Al Almandine

Mn Al Spessartine

Ca Fe Andradite

Ca Cr Uvarovite

Page 43

João Pessoa, 2014

25 modes

The RAMAN spectrum of Pyrope:

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João Pessoa, 2014

From A1g+Eg wavenumbers...

Ours Hofmeister Chopelas Kolesov

Sym M υ (cm-1) υ (cm-1) Δυ (cm-1) υ (cm-1) Δυ (cm-1) υ (cm-1) Δυ (cm-1)

1 352.5 362 -10 362 -10 364 -12

A1g 2 564.8 562 3 562 3 563 2

3 926.0 925 1 925 1 928 -2

4 209.2 203 6 203 6 211 -2

5 308.5 309 -1 284 25

6 336.5 342 -6 344 -8

7 376.9 365 12 379 -2 375 2

Eg A 439 439

8 526.6 524 3 524 3 525 2

9 636.0 626 10 626 10 626 10

10 864.4 867 -3

B 911

11 937.4 938 -1 938 -1 945 -8

Frequency differences are

evaluated with respect to

calculated data.

Hofmeister: Hofmeister &

Chopelas, Phys. Chem.

Min., 1991

Chopelas: Chaplin & Price

& Ross, Am. Mineral.,

1998

Kolesov: Kolesov &

Geiger, Phys. Chem. Min.,

1998

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João Pessoa, 2014

... to RAMAN spectra!

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João Pessoa, 2014

And now F2g wavenumbers...Ours Hofmeister Chopelas Kolesov

Sym. M υ (cm-1) υ (cm-1) Δυ (cm-1) υ (cm-1) Δυ (cm-1) υ (cm-1) Δυ (cm-1)

12 97.9 - - - - 135 -37

13 170.1 - - - - - -

14 203.7 208 -4 208 -4 212 -8

C 230 230

15 266.9 272 -5 272 -5 - -

D 285

16 319 318 1 318 1 322 -3

F2g E 342

17 350.6 350 1 350 1 353 -2

18 381.9 379 3 379 3 383 -1

19 492.6 490 3 490 3 492 1

20 513.5 510 4 510 4 512 2

21 605.9 598 8 598 8 598 8

22 655.3 648 7 648 7 650 5

23 861 866 -5 866 -5 871 -10

24 896.7 899 -2 899 -2 902 -5

25 1068.4 1062 6 1062 6 1066 2

Frequency differences are

evaluated with respect to

calculated data.

Hofmeister: Hofmeister &

Chopelas, Phys. Chem.

Min., 1991

Chopelas: Chaplin & Price

& Ross, Am. Mineral.,

1998

Kolesov: Kolesov &

Geiger, Phys. Chem. Min.,

1998

B3LYP overstimates

the lattice parameter!

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João Pessoa, 2014

... and the RAMAN spectra!

A1g peaks also in F2g spectrum caused by the presence of different crystal orientations

and/or rotation of the polarized light.

Page 48

João Pessoa, 2014

Grossular

LM, R. Demichelis, R. Orlando, M. De La Pierre, A. Mahmoud, R. Dovesi, J. Raman Spectrosc., in press

Page 49

João Pessoa, 2014

A couple of other examples of

RAMAN SPECTRA

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João Pessoa, 2014

Jadeite

Experimental spectrum from rruff database

M. Prencipe, LM, B. Kirtman, S. Salustro, A. Erba, R. Dovesi J. Raman Spectrosc., in press

Page 51

João Pessoa, 2014

Raman Spectrum of UiO-66 Metal-Organic Framework

Theory

Experiment

Exp. spectra from S. Bordiga and collaborators

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João Pessoa, 2014

Reflectivity is calculated from dielectric constant by means of:


The dielectric function is obtained with the classical dispersion relation

(damped harmonic oscillator):


Page 56

João Pessoa, 2014


Reflectivity is calculated from

dielectric constant by means of:


The dielectric function is obtained

with the classical dispersion relation:Comparison of computed and experimental IR reflectance spectra

for garnets: a) pyrope b) grossular c) almandine .

Page 57

João Pessoa, 2014

IR reflectance spectrum of grossular

Computed and experimental IR reflectance spectra of grossular garnet, plus imaginary parts of ε and 1/ε.

Page 58

João Pessoa, 2014

High frequency modes

Dependence on lattice parameter

Isotopic substitution on X and Y

cations: small dependence

Graphical analysis of

eigenvectors:

• modes 11-14: bending

• modes 15-17: stretching

Garnets: compositional trends

Page 59

João Pessoa, 2014

• Changing the mass of one atomic species at a time

– Natural isotopic masses

– Percentage mass variations

– Infinite mass

• Hessian re-diagonalization not required (zero

computational cost)

• Tool for the assignment of the modes and the

interpretation of the spectrum

The isotopic substitution

Page 60

João Pessoa, 2014

(cm-1)

(cm-1)100 350

Pyrope : 24Mg → 26Mg

Isotopic shift on the vibrational frequencies of pyrope when 26Mg is substituted for 24Mg.

Page 61

João Pessoa, 2014

Isotopic shift on the vibrational frequencies of pyrope when 29Al is substituted for 27Al.

(cm-1)

(cm-1)300 700

Pyrope : 27Al → 29Al

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João Pessoa, 2014

Isotopic shift on the vibrational frequencies of pyrope when 30Si is substituted for 28Si.

(cm-1)

(cm-1)

850 1050

Pyrope : 28Si → 30Si

250 700

Low ν : rotations and

bending of tetrahedra and

octahedra (involving by

connectivity also Si)

High ν: stretching of

tetrahedra

Page 63

João Pessoa, 2014

The PRACE Tier-0 Resources

HORNET (HLRS, DE)

Cray XC30 system - 94,656 cores CURIE (GENCI, FR)

BULL x86 system – 80,640 cores (thin nodes)

FERMI (CINECA, IT)


SUPERMUC (LRZ, DE)



MARENOSTRUM (BSC, SP)



JUQUEEN (JÜLICH, DE)


Page 64

João Pessoa, 2014

A model for the MCM-41 mesoporous silica material

OSiH

Cell: 41x41x12 Å

579 atoms in the unit cell (Si142O335H102)

Ordered arrangement of cylindrical pores

Pores: mesoporous size (2-10 nm)

High surface area: up to 1000 m2g-1

FunctionalizableAPPLICATIONS

Separation - Catalysis – Sensors – Drug Delivery

Page 65

João Pessoa, 2014

B3LYP/6-31G(d,p)

579 atoms in the UC, 7756 AO

Standard tolerances

41 Å

T-CPU(64) SCF+G 9000 s

For diagonalization the empirical

rule is N-AO/60 N-cores

Massive parallel performances

MCM-41

IBM Power PC 970MP 2.3 GHz BSC MN

Page 66

João Pessoa, 2014

MPPCRYSTAL: Memory Usage

Memory occupation peak in the SCF calculation of different supercells of the

mesoporous silica MCM-41, with a 6-31G** basis set and B3LYP functional.

The single unit cell (X1) contains 579 atoms and 7756 atomic orbitals.

The largest cell (X12) contains 6948 atoms and 93072 atomic orbitals.

X1

X12

X8

X4

X2

Page 68

João Pessoa, 2014

MPPCRYSTAL: Time Scaling

•Scaling of computational time required for a complete SCF (13 cycles) with

the size of the MCM-41 supercell,

•on 1024 processors at SUPERMUC (Munich).

•X1

•X8

•X4•X2

•X12

Page 71

João Pessoa, 2014

CRAMBIN

Crambin is a small seed storage protein from the Abyssinian

cabbage. It belongs to thionins. It has 46 aminoacids (642

atoms).

Primary structure:

Secondary structure:

N-termC-term

α-HELIX A

α-HELIX B

β-SHEET

RANDOM COIL

Page 72

João Pessoa, 2014

AB-INITIO PROTEIN OPTIMIZATION

Geometry FULLY optimized at the B3LYP-D*/6-31d level of theory with CRYSTAL14.

B3LYP-D*Experimental

RMSD (backbone)0.668 Å

Notes:

- Crystallographic structure has a

30% solvent content (v/v).

- Nakata et al., who optimized

crambin using the Fragment

Molecular Orbital method (HF/6-

31d) with the polarizable

continuum model, report a RMSD

of 0.525 Å with respect to PDB

structure 1CRN.

AVERAGE

OPTIMIZATION STEP

ON 640 CPUs*

323 seconds

*SuperMUC (LRZ, Munich)

Page 73

João Pessoa, 2014

AB-INITIO PROTEIN INFRARED SPECTRUM

The FULL vibrational spectrum is computed at the B3LYP-D*/6-31d level of theory

3500 3000 2500 2000 1500 1000 500 0

Wavenumber (cm-1)

1900 1850 1800 1750 1700 1650 1600 1550 1500 1450 1400

Wavenumber (cm-1)

AMIDE I AMIDE II

AMIDE I: C=O stretching (backbone)

AMIDE II: N-H bending and C-N stretching (backbone)

TOTAL TIME ON 1024 CPUs*

222 hours *SuperMUC (LRZ, Munich)

Page 74

João Pessoa, 2014

ELECTROSTATIC POTENTIAL MAPPED ON THE B3LYP DENSITY

Isovalue: 10-4 e200x200x200 grid

TOTAL TIME ON 256 CPUs* < 1 minute *SuperMUC (LRZ, Munich)

Page 75

João Pessoa, 2014

AB-INITIO PROTEIN OPTIMIZATION – CRYSTAL STRUCTURE

FULL optimization(B3LYP-D*/6-31d)

**Crystallographic experimental

structure has a 30% solvent content

(v/v). Here water was removed.

AVERAGE OPTIMIZATION STEP ON 640 CPUs*1064 seconds

*SuperMUC (LRZ, Munich)

CELL VOLUME:

-10% with respect to the

experimental structure**

P21 - 1284 total atoms / 642 irreducible atoms

Page 76

João Pessoa, 2014

Ab initio modelling of giant MOFs: when the size matters

MIL-100(M)

MOF-5

Comparison between the crystallographic unit cells of the giant MIL-100 and MOF-5

PRACE Grant:

Project 2013081680

M204X68O68[(C6H3)-(CO2)3]204

2788 atoms (primitive u.c.)

M= Al, Sc, Cr, Fe

106 atoms (primitive u.c.)

(Zn4O)2[(C6H4)-(CO2)2]6

Page 77

João Pessoa, 2014

Running time scaling with the number of computing cores for

MIL-100(Al)-N (2720 atoms) on the SuperMUC HPC system.

Timings on 1024 cores:

• one SCF cycle = 767 sec

• Gradient (atoms) = 1801 sec

MIL-100(Al)-N is a model system in

which a N atom substitutes the O at the

center of the inorganic unit. It consists

of a primitive unit cell containing 2720

atoms without symmetry.

MIL-100(Al)-N: MPP-CRYSTAL Scaling

B3LYP calculation with

44606 AOs in the unit cell.

Speedup=T1024

/TnCPUs

94%

86%

PRACE Grant: Project 2013081680

Calculations run on SUPERMUC at LRZ:

HPC IBM System x iDataPlex powered

by 16 Intel cores per node running at 2.7

GHz, with 2 GB/core

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On the role of quantum mechanical simulation in materials science.

Science