On the role of quantum mechanical simulation in materials science.
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Transcript of On the role of quantum mechanical simulation in materials science.
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João Pessoa, 2014
Roberto Dovesi
On the role of quantum mechanical
simulation in materials science
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João Pessoa, 2014
Is simulation useful?
Does it produce reasonable numbers?
Or can only try to reproduce the experiments?
Connected question:
Is simulation expensive?
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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
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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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Performance of HPC
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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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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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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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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)
IBM System x iDataPlex
system– 48,448 cores
JUQUEEN (JÜLICH, DE)
BlueGene Q system – 458,752 cores
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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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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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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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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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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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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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Geometry definitionELASTCON
[Optional keywords]END
ENDBasis set definitionENDComput. ParametersEND
Tensorial Properties Related to Crystal Strain
Elastic Tensor Piezoelectric Tensor Photoelastic Tensor
Geometry definitionPIEZOCON
[Optional keywords]END
ENDBasis set definitionENDComput. ParametersEND
Geometry definitionPHOTOELA
[Optional keywords]END
ENDBasis set definitionENDComput. ParametersEND
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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.
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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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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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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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A. Erba, A. Mahmoud, R. Orlando and R. Dovesi, Phys. Chem. Minerals (2013) DOI 10.1007/s00269-013-0630-4
CRYSTAL14: Elastic Properties
Almandine
Spessartine Grossular
Andradite Uvarovite
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CRYSTAL14: Elastic Properties
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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A. Erba, A. Mahmoud, R. Orlando and R. Dovesi, Phys. Chem. Minerals (2013) DOI 10.1007/s00269-013-0630-4
Voigt-Reuss-Hill averaging scheme
CRYSTAL14: Elastic Properties
Spessartine Grossular
Andradite Uvarovite
AlmandinePyrope
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A. Erba, A. Mahmoud, R. Orlando and R. Dovesi, Phys. Chem. Minerals (2013) DOI 10.1007/s00269-013-0630-4
CRYSTAL14: Elastic Properties
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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A. Erba, A. Mahmoud, R. Orlando and R. Dovesi, Phys. Chem. Minerals (2013) DOI 10.1007/s00269-013-0630-4
CRYSTAL14: Elastic Properties
Spessartine Grossular
Andradite Uvarovite
AlmandinePyrope
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A. Erba, A. Mahmoud, R. Orlando and R. Dovesi, Phys. Chem. Minerals (2013) DOI 10.1007/s00269-013-0630-4
CRYSTAL14: Elastic Properties
Spessartine Grossular
Andradite Uvarovite
AlmandinePyrope
Elastic Anisotropy
Seismic wave velocity
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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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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 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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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 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
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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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CRYSTAL14: Piezoelectric and Dielectric Properties
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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CRYSTAL14: Piezoelectric and Dielectric Properties
A. Mahmoud, A. Erba, Kh. E. El-Kelany, M. Rérat and R. Orlando, Phys. Rev. B (2013)
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CRYSTAL14: Photoelastic Properties
A. Mahmoud, A. Erba, Kh. E. El-Kelany, M. Rérat and R. Orlando, Phys. Rev. B (2013)
✔ 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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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
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IR and RAMAN spectra
Wavenumbers and intensities
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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
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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
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25 modes
The RAMAN spectrum of Pyrope:
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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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... to RAMAN spectra!
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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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... 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.
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Grossular
LM, R. Demichelis, R. Orlando, M. De La Pierre, A. Mahmoud, R. Dovesi, J. Raman Spectrosc., in press
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A couple of other examples of
RAMAN SPECTRA
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Jadeite
Experimental spectrum from rruff database
M. Prencipe, LM, B. Kirtman, S. Salustro, A. Erba, R. Dovesi J. Raman Spectrosc., in press
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Raman Spectrum of UiO-66 Metal-Organic Framework
Theory
Experiment
Exp. spectra from S. Bordiga and collaborators
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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
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IR reflectance spectrum
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:Comparison of computed and experimental IR reflectance spectra
for garnets: a) pyrope b) grossular c) almandine .
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IR reflectance spectrum of grossular
Computed and experimental IR reflectance spectra of grossular garnet, plus imaginary parts of ε and 1/ε.
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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
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• 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
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(cm-1)
(cm-1)100 350
Pyrope : 24Mg → 26Mg
Isotopic shift on the vibrational frequencies of pyrope when 26Mg is substituted for 24Mg.
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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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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
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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)
IBM System x iDataPlex
system– 48,448 cores
JUQUEEN (JÜLICH, DE)
BlueGene Q system – 458,752 cores
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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
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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
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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
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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
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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
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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)
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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)
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ELECTROSTATIC POTENTIAL MAPPED ON THE B3LYP DENSITY
Isovalue: 10-4 e200x200x200 grid
TOTAL TIME ON 256 CPUs* < 1 minute *SuperMUC (LRZ, Munich)
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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
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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
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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