MP2 Description of Solids: The CRYSCOR Project

MP2 Description of Solids: The
CRYSCOR Project
Lorenzo Maschio
[email protected]
Dipartimento di Chimica and NIS centre,
University of Torino, Italy
The Regensburg group
C. Pisani
S. Casassa
L. Maschio
M. Schütz
D. Usvyat
Lorenzo Maschio
The CRYSCOR Project
2
1. Local correlation methods
If the doors of perception were cleansed, every
thing would appear to man as it is, infinite.
W. Blake
Local correlation methods
Occupied and virtual space in post-HF methods are truncated according to distance
criteria.
In the original formulation,[1,2] the virtual space is represented by
Projected Atomic Orbitals (PAOs)
P. Pulay, Chem. Phys. Lett. 100,151 (1983)
M. Schutz, G.Hetzer, H.-J. Werner, JCP 111, 5691 (1999)
1
Local correlation methods
Occupied and virtual space in post-HF methods are truncated according to distance
criteria.
i
occupied
orbital
In the original formulation,[1,2] the virtual space is represented by
Projected Atomic Orbitals (PAOs)
P. Pulay, Chem. Phys. Lett. 100,151 (1983)
M. Schutz, G.Hetzer, H.-J. Werner, JCP 111, 5691 (1999)
1
Local correlation methods
Occupied and virtual space in post-HF methods are truncated according to distance
criteria.
virtual space
i
occupied
orbital
In the original formulation,[1,2] the virtual space is represented by
Projected Atomic Orbitals (PAOs)
P. Pulay, Chem. Phys. Lett. 100,151 (1983)
M. Schutz, G.Hetzer, H.-J. Werner, JCP 111, 5691 (1999)
1
Local correlation methods
Occupied and virtual space in post-HF methods are truncated according to distance
criteria.
virtual space
i
occupied
orbital
i
virtual space
occupied
orbital
In the original formulation,[1,2] the virtual space is represented by
Projected Atomic Orbitals (PAOs)
P. Pulay, Chem. Phys. Lett. 100,151 (1983)
M. Schutz, G.Hetzer, H.-J. Werner, JCP 111, 5691 (1999)
1
Local correlation methods
Occupied and virtual space in post-HF methods are truncated according to distance
criteria.
i
In the original formulation,[1,2] the virtual space is represented by
Projected Atomic Orbitals (PAOs)
P. Pulay, Chem. Phys. Lett. 100,151 (1983)
M. Schutz, G.Hetzer, H.-J. Werner, JCP 111, 5691 (1999)
1
Local correlation methods
Occupied and virtual space in post-HF methods are truncated according to distance
criteria.
j
i
In the original formulation,[1,2] the virtual space is represented by
Projected Atomic Orbitals (PAOs)
P. Pulay, Chem. Phys. Lett. 100,151 (1983)
M. Schutz, G.Hetzer, H.-J. Werner, JCP 111, 5691 (1999)
1
2. Local correlation methods for
periodic systems
Tyger! Tyger! Burning bright
In the forests of the night:
What immortal hand or eye
Could frame thy fearful symmetry?
W. Blake
In Solids
In a crystalline solid everything is infinite.
2
In Solids
In a crystalline solid everything is infinite. But we can refer to the unit cell.
2
In Solids
The reference occupied orbital is in the unit cell…
i
2
In Solids
And other local objects are selected in its neighborhood.
i
2
In Solids
And other local objects are selected in its neighborhood.
j
i
2
Some equations
The MP2 is an energy per cell, including orbital pairs near that cell
We have to minimize the “residuals”
3
Some equations
In the PAO basis residuals take a more complicated form
4
Local Correlation in Crystals
C. Pisani, S. Casassa, L. Maschio
M. Schütz, D. Usvyat [1-3]
Implements LMP2, F12-LMP2,[4] LCIS[5]
[1] C. Pisani, M. Busso, G. Capecchi, S. Casassa, R. Dovesi, LM, C. Zicovich-Wilson, M. Schütz JCP 122, 094113 (2005)
[2] C. Pisani, LM, S. Casassa, M. Halo, M. Schütz, D. Usvyat, JCC 29 (2008) 2113.
[3] C. Pisani, M. Schütz, S. Casassa, D. Usvyat, LM, M. Lorenz, A. Erba, PCCP, 14, 7615-7628 (2012)
[4] D. Usvyat, JCP 139, 194101 (2013)
[5] M. Lorenz, LM, M. Schütz, D. Usvyat, JCP137, 204119 (2012)
5
Local Correlation in Crystals
C. Pisani, S. Casassa, L. Maschio
M. Schütz, D. Usvyat [1-3]
R. Dovesi et al. [6]
Implements LMP2, F12-LMP2,[4] LCIS[5]
HF solution from
[1] C. Pisani, M. Busso, G. Capecchi, S. Casassa, R. Dovesi, LM, C. Zicovich-Wilson, M. Schütz JCP 122, 094113 (2005)
[2] C. Pisani, LM, S. Casassa, M. Halo, M. Schütz, D. Usvyat, JCC 29 (2008) 2113.
[3] C. Pisani, M. Schütz, S. Casassa, D. Usvyat, LM, M. Lorenz, A. Erba, PCCP, 14, 7615-7628 (2012)
[4] D. Usvyat, JCP 139, 194101 (2013)
[5] M. Lorenz, LM, M. Schütz, D. Usvyat, JCP137, 204119 (2012)
[6] R. Dovesi, R. Orlando, A. Erba, C. M. Zicovich-Wilson, B. Civalleri, S. Casassa, LM, M. Ferrabone,
M. De La Pierre, P. D'Arco, Y. Noël, M. Causà, M. Rérat, B. Kirtman, IJQC, 114 (2014) 1287-1317.
6
Rare Gas Crystals: Argon
Basis set: [4s4p3d1f(2f)]
Casassa, Halo, Maschio, Journal of Physics: Conference Series 117 (2008)
M.Halo, S.Casassa, L.Maschio and C.Pisani, Chem. Phys. Lett. 467 (2009) 294–298
M.Halo, S.Casassa, L.Maschio and C.Pisani, PCCP 11( 2009) 568–574
7
Molecular Crystals
CO2
L. Maschio, D. Usvyat, M. Schuetz, B. Civalleri, J. Chem. Phys. 132, 134706 (2010)
L. Maschio, D. Usvyat and B. Civalleri CrystEngComm, 12, 2429 (2010)
L. Maschio, B. Civalleri, P. Ugliengo and A. Gavezzotti, J. Phys. Chem. A, 115 (41),
11179–11186 (2011)
NH3
8
Strategy
•
Use Gaussian functions as a basis set
•
Aim at general, black-box, robust methods and clean
implementation.
•
Implement local correlation methods for periodic
systems.
Lorenzo Maschio
The CRYSCOR Project
2
9
Strategy
•
Use Gaussian functions as a basis set
•
Aim at general, black-box, robust methods and clean
implementation.
•
Implement local correlation methods for periodic
systems. With Density Fitting.
LM, D. Usvyat, C. Pisani, F. Manby, S. Casassa, M. Schütz, PRB. 76, 075101 (2007)
LM, D. Usvyat, PRB 78, 073102 (2008)
M. Schütz, D. Usvyat, M. Lorenz, C. Pisani, LM, S. Casassa, M. Halo, In: Accurate CondensedPhase Quantum Chemistry. p. 29-55, New York:CRC press - Taylor and Francis Group (2010)
Lorenzo Maschio
The CRYSCOR Project
10
2
Applications of CRYSCOR
■
■
■
■
■
■
Rare Gas Crystals.[1]
Molecular crystals. [2,3,4]
Molecules and atoms on surfaces [5,6,7,8]
Relative stability of crystalline phases[9,10]
Compton scattering phenomena in crystalline systems[11,12]
Nanostructures and nanotubes
[1] M. Halo, S. Casassa, LM, C. Pisani, CPL 467, 294-298 (2009)
[2] LM, B. Civalleri, P. Ugliengo, A. Gavezzotti, JPCA, 115, 11179 (2011)
[3] LM; Usvyat, D.; Civalleri, B., CrystEngComm, 12, 2429–2435 (2010).
[4] LM; Usvyat, D.; Schutz, M.; Civalleri, B. JCP 132, 134706 (2011)
[5] Z. Huesges, C. Mueller, B. Paulus, LM Surf. Sci. 627, 11-15 (2014),
[6] D.Usvyat, K. Sadeghian, LM, M. Schütz PRB 86, 045412 (2012)
[7] Halo, M.; Casassa, S.; LM; Pisani, C.; Dovesi, R.; Ehinon, D.; Baraille, I.; Rèrat, M.; Usvyat, D., PCCP 13, 4434 (2011).
[8] Martinez-Casado, R.; Mallia, G.; Usvyat, D.; LM; Casassa, S.; Schuetz, M.; Harrison, N. M. , JCP 134, 014706 (2011).
[9] Halo, M.; Pisani, C.; LM; Casassa, S.; Schuetz, M.; Usvyat, D., PRB 83, 035117 (2011).
[10] Erba, A.; LM; Salustro, S.; Casassa, S. , JCP 134, 074502 (2011).
[11] Erba, A.; LM; Casassa, S.; Pisani, C., PRB 84, 012101 (2011)
[12] Erba, A.; Itou, M.; Sakurai, Y.; Ito, M.; Casassa, S.; LM; Terentjevs, A.; Pisani, C., PRB 83, 125208 (2011).
11
The problem of integrals evaluation
The bottleneck of the whole Local-MP2 method is represented by the calculation of 4 index
integrals, which constitutes by itself about 99% of the whole time needed for a calculation.
The computational time amounts to several days even for the simplest system.
This is due to the four index contraction over atomic orbitals needed to compute this integral
between composite objects.
In order to be able to study systems of general interest it is mandatory to implement fast
techniques to approximate these integrals.
12
The problem of integrals evaluation
The bottleneck of the whole Local-MP2 method is represented by the calculation of 4 index
integrals, which constitutes by itself about 99% of the whole time needed for a calculation.
The computational time amounts to several days even for the simplest system.
This is due to the four index contraction over atomic orbitals needed to compute this integral
between composite objects.
Virtuals
Occupied
In order to be able to study systems of general interest it is mandatory to implement fast
techniques to approximate these integrals.
13
Multipolar integrals
The product distribution between Wannier function i and PAO a can be
represented by a set of electric multipoles,
, of maximum order l and
centered in C.
Pisani, Capecchi, Casassa and Maschio, Mol. Phys 103-18(2005)
2527-2536
If the two distributions are sufficiently apart to be entirely contained in separate
spheres, their electrostatic interaction can be approximated by a sum of
interactions between the respective multipoles :
Due to translational invariance only the multipoles in the zero cell need to be
computed, all the others being the same.
Multipoles can be safely used for distances larger than 6 Å. Attention must be paid in
the case of extended (augmented) basis sets and/or very large excitation domains.
A hierarchy of integrals
Extrapolation to infinity
Multipolar expansion
Density Fitting
Density fitting
Schuetz, Manby, PCCP (2003) 5, 3349-3358
Density Fitting is a powerful technique for integral evaluation that has
widely proved its efficiency in molecular codes like MOLPRO.
The basic idea is to expand product distributions in an Auxiliary Basis (e.g.
gaussian functions
). We will from now on refer to these functions
with latin capital letters P,Q.
Fitting coefficients are determined by minimizing the error functional
leading to the linear equation system
14
Density fitting
So that the 4-index integrals are finally approximated as :
Only 3-center Coulomb integrals are
needed.
where:
The inverse matrix of
2-center coulomb
integrals has the size of
the number of auxiliary
functions in the system.
and:
15
Performance of Density Fitting
Basis set
Time LMP2
Time DF-LMP2
% Error on
Energy
6-21G*
16 days
4 min.
0.03%
Ice
6-311G**
4 days
15 min.
0.01%
CO2
6-311G(3d)
2 days
30 min.
0.01%
MgO
8-511G*/8-411G*
19 h.
8 min.
0.01%
Ar
ECP/[4s4p3d2f]
10 days
5 min.
0.02%
System
Diamond
L. Maschio, D. Usvyat, C. Pisani, F. Manby, S. Casassa, M. Schütz, Phys. Rev. B 76, 075101 (2007).
D. Usvyat, L. Maschio, F. Manby, M. Schütz, S. Casassa, C. Pisani, Phys. Rev. B 76, 075102 (2007).
L. Maschio, and D. Usvyat, Phys. Rev. B 78, 073102 (2008).
16
3. Parallelization
In Xanadu did Kubla Khan
A stately pleasure-dome decree:
Where Alph, the sacred river, ran
Through caverns measureless to man
Down to a sunless sea.
S. T. Coleridge
Parallel LMP2
Lorenzo Maschio
The CRYSCOR Project
17
2
Parallel LMP2
18
Parallel LMP2
Double-walled fully
Hydrogenated nanotube.
144 atoms per unit cell
1944 basis functions per cell
basis:
aug(p,d)-cc-pVDZ
L. Maschio, JCTC, 7, 2818 (2011).
J. Tanskanen, L. Maschio, A. Karttunen, M. Linnolahti, T.
Pakkanen, ChemPhysChem, 13, 2361 (2012)
Lorenzo Maschio
The CRYSCOR Project
19
2
Parallel LMP2
Running on 48 processors
Total time: 28 hours
Aspirin molecule adsorbed on a 10 Angstrom thick SiO2 slab
141 atoms per unit cell, 2465 basis functions per cell
basis: 88-31G* on Si and O atoms in the slab, TZP on the molecule’s atoms
Lorenzo Maschio
The CRYSCOR Project
20
2
Parallel LMP2
MOF-5 3D framework
106 atoms per unit cell,
2884 basis functions per
cell,
532 correlated electrons
basis: cc-pVTZ with ECPs
on Zn atom
Running on 54 processors
Total time: 23 hours
Lorenzo Maschio
The CRYSCOR Project
21
2
Partial summary
We have efficient and general density-fitted PAObased LMP2, F12-LMP2 and LCIS methods
•
•
Fairly large unit cells can be tackled.
However, it is not simple to deal with such systems
due to the need for a definition on truncation
thresholds and excitation domains.
Lorenzo Maschio
The CRYSCOR Project
22
2
4. Towards black-box methods:
the pair screening
Through the window I see no star:
Something more near
Though deeper within darkness
Is entering the loneliness.
T. Hughes
The problem of pair truncation
How to choose a safe threshold for
integrals?
j
i
23
The problem of pair truncation
How to choose a safe threshold for
integrals?
j
i
24
The problem of pair truncation
How to choose a safe threshold for
integrals?
Neglect
Weak pairs
Distant pairs
j
i
25
CO2 crystal Vs Ge crystal
How to choose a safe threshold?
10 -3
∆ Ecorr / hartree
Ge
CO2
10 -4
N DF =2136, N
Mult
=21604
N DF =1460, N
10 -5
N DF =19524, N
10 -6
3
4
5
6
7
8
9
10
11
Mult
=1056
Mult
=4216
12
13
Rdist / Angstrom
Lorenzo Maschio
The CRYSCOR Project
26
2
A simple quasi-population criterion
D. Kats, JCP 141, 244101 (2014)
The mutual penetration of the WFs i and j (and thus the densities φiφa
and φjφb) can be estimated by the products of the atomic populations
of the individual WF densities:
Lorenzo Maschio
The CRYSCOR Project
27
2
CO2 crystal Vs Ge crystal
-10 -6
N DF =1608, N
Mult
=22132
∆ Ecorr / hartree
-10 -5
-10
N DF =628, N
-4
Mult
Ge
CO2
-10 -3
-10 -2
=1888
10 -5
10 -6
10 -7
WFs tolerance
Lorenzo Maschio
The CRYSCOR Project
28
2
4. Towards black-box methods:
Orbital-Specific Virtuals
Water, water, everywhere,
Nor any drop to drink.
S. T. Coleridge
A dinner for a
couple of friends:
PAO domains are
affordable
29
A dinner for a
couple of friends:
PAO domains are
affordable
A dinner for a
hundred (or more)
guests: the OSV (or
PNO) buffet
30
PNOs and OSVs
Pair Natural Orbitals:
diagonalization of the
MP2-like density matrix
Orbital Specific Virtuals:
diagonalization of the
diagonal MP2 amplitudes
•
•
Faster MP2 (fewer virtuals)
Quasi-Black Box
P. Pulay (unpublished)
J. Yang, Y. Kurashige, F. R. Manby, G. K. L. Chan, JCP. 134 (2011) 044123.
J. Yang, G. K. Chan, F. R. Manby, M. Schutz, H.-J. Werner, JCP 136 (2012) 144105.
Lorenzo Maschio
The CRYSCOR Project
31
2
PNOs and OSVs
Pair Natural Orbitals:
pair-dependent
Orbital Specific Virtuals:
occupied orbital—dependent
Projected Atomic Orbitals:
pair/orbital—independent
P. Pulay (unpublished)
J. Yang, Y. Kurashige, F. R. Manby, G. K. L. Chan, JCP. 134 (2011) 044123.
J. Yang, G. K. Chan, F. R. Manby, M. Schutz, H.-J. Werner, JCP 136 (2012) 144105.
Lorenzo Maschio
The CRYSCOR Project
32
2
OSVs in CRYSCOR
D.Usvyat, L. Maschio, M. Schütz, JCP. 143 (2015) 102805
Results converge fast to most
accurate PAO calculation with big
domains…
…but with a considerably reduced
number of virtuals!
33
OSVs in CRYSCOR: smooth potential surfaces
Black Phosphorus
LMP2
p_tzvpp2(3d1f)[2d1f]
1) lattice parameters
bfixed=10.42 [Å]
b
a
fit
c
a=3.27 [Å]
c=4.36 [Å]
34
smooth potential surfaces=geometry optimisation?
Geometry optimisation
LMP2
p_tzvpp2(3d1f)[2d1f]
2) internal coordinates
x/afixed=0.
1 irreducible atom
for symmetry
y/b=0.10339
fit
z/c=0.07863
35
Conclusions
-
Cryscor allows for Local MP2, CIS and F12-MP2
calculations starting from the CRYSTAL wavefunction
-
Can be done either on top of HF or DFT (doublehybrids)
-
New developments move towards better performance,
“black box” use and wider applicability
36
…coming soon…
New release hopefully by the end of
2015
37
(Further) Acknowledgments
PhD students in Torino
Simone
Salustro
Giuseppe
Sansone
Other collaborators (in random order)
-
Nic Harrison, Giuseppe Mallia (London, UK)
Ruth Martinez-Casado (Madrid, Spain)
Antti Karttunen (Aalto University, Finland)
Andreas Savin, Julien Toulouse (Paris, France)
Beate Paulus (Berlin, Germany)
Maosheng Miao (CSU Northridge, US)
Bernard Kirtman (Santa Barbara, US)
Alfonso Pedone, Davide Presti (Modena, Italy)
38
Thank you for your attention
[email protected]