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    • Abstract: Local Molecular Orbitals. Symmetry Adapted Wannier Functions (SAWF) Local, non-orthonormal, redundant representation of virtual manifold. Projected Atomic Orbitals (PAO) ... Bloch functions as Linear Combination of Atomic Orbitals (contractions of Hermite Gaussian Functions) ...

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THE “CRYSCOR” PROJECT : STATUS AND PROSPECTS
(#) Cesare PISANI, Massimo BUSSO, Gabriella CAPECCHI,
Silvia CASASSA, Roberto DOVESI, Lorenzo MASCHIO
(*) Claudio ZICOVICH-WILSON
(°) Vic R. SAUNDERS
(&) Martin SCHÜTZ
(#) Università di Torino (Italy)
(*) Universidad de Cuernavaca (Mexico)
(°) CRLC Laboratories, Daresbury (GB)
(&) Universität Regensburg (Germany)
Cesare Pisani - LCC2004 1
THE STRATEGY
Combining two well assessed, robust, compatible technologies
Geometrical and structural
analysis of periodic
On Local Correlation (LC)
system theory and techniques at
various levels of
Accurate HF solution in sophistication
terms of AOs
Meyer-Pulay-Werner …
Local representation of
occupied manifold (WF)
CRYSTAL ( MOLPRO )
Cesare Pisani - LCC2004 2
Local-correlation approach for crystals
MEYER, PULAY, SAEBØ, …,WERNER, KNOWLES, HETZER, MANBY, SCHÜTZ , …
(Arkansas, Stuttgart, Birmingham, Bristol, ...)
Local, orthonormal representation of occupied HF manifold
Local Molecular Orbitals ↔ Symmetry Adapted Wannier Functions (SAWF)
Local, non-orthonormal, redundant representation of virtual manifold
Projected Atomic Orbitals (PAO) ↔ (PAO)
Reformulation of standard approaches ( MPn, CCSD(T), … )
TRUNCATION STRATEGY
Dynamical correlation effects are short-ranged:
Ignore excitations from very distant pairs of SAWFs
Exploit translational (and point) symmetry of SAWFs and PAOs
N--SCALING → n--SCALING
(n is the size of the irreducible part of the crystalline cell)
Explore the role of computational parameters
Cesare Pisani - LCC2004 3
THE OBJECTIVE
To produce (in a reasonably short time) a public domain ab initio code
for estimating electron correlation effects in non-conducting crystals,
characterized by:
generality
simplicity
robustness
acceptable efficiency,
and open to improvements
As a first step, an MP2 code has been prepared and is here presented.
Cesare Pisani - LCC2004 4
Is the objective worth the effort?
Is the strategy appropriate?
Cesare Pisani - LCC2004 5
Is the objective worth the effort?
In our opinion it is.
A number of ab-initio periodic correlation codes are ready, in
preparation or in project (Stoll, Bartlett, Scuseria, Birkenheuer,
Saunders-Orlando, Malrieu-Evangelisti, Monkhorst ….).
However, a conceptually simple, reliable, easily accessible code,
could serve the purpose of providing reference data, exploring
the role of computational parameters, checking the
usefulness of alternative techniques, etc.
Cesare Pisani - LCC2004 6
Is the strategy appropriate?
With respect to other approaches in this area,
CRYSCOR is characterized by its being strictly
founded on the HF program CRYSTAL (language,
basic techniques, etc.), so as to be fully compatible
with it …
… in a sense, CRYSCOR is the post-HF option of
CRYSTAL.
Disadvantages
Advantages
Cesare Pisani - LCC2004 7
Disadvantages
CRYSTAL’s computational technology is efficient but
somewhat rigid and outdated.
Inserting new parts of code may require a lot of work and
attention.
Cesare Pisani - LCC2004 8
Advantages
Full space group symmetry of system and basis functions
are fully and efficiently provided by CRYSTAL.
Quasi-HF periodic solutions are obtained, and their
quality can be easily assessed.
Efficient and well tested techniques for generating
localized and symmetry adapted WFs are available.
Fourier transformation techniques are available (for
instance, for calculating the inverse of quasi-diagonal
translationally invariant matrices).
Cesare Pisani - LCC2004 9
CRYSTAL 2003 : Main Features
VR Saunders, R Dovesi, C Roetti, R Orlando, CM Zicovich-Wilson, NM Harrison, K Doll, B Civalleri, IJ
Bush, Ph D’Arco, M Llunell
Distributed starting 9/03 . Info : http://www.crystal.unito.it (Rev. Computational Chemistry, in press)
The code
• FORTRAN90, fully parallelized, dynamic memory allocation
The periodic model
• Consistent treatment of periodicity : 3D, 2D, 1D, 0D
• Ewald techniques for lattice sums (specific for 1D, 2D, 3D)
• Full exploitation of point symmetry in direct and reciprocal space
Basis set
• Bloch functions as Linear Combination of Atomic Orbitals (contractions of Hermite Gaussian Functions)
• All-Electron or Valence-only-plus-Pseudopotential basis set
Hamiltonians
• RHF, UHF
• Kohn-Sham techniques with Local and Gradient-corrected
exchange and correlation functionals
• Hybrid DFT-HF exchange functionals
Energy derivatives
• Automated geometry optimisation based on analytical gradient
Wave function analysis and manipulation
• Band structure, PDOS, Charge, spin, electron momentum density, Structure factors, Compton profiles
• Elastic, dielectric constant, piezoelectric, hyperfine and nuclear quadrupole coupling tensors, …
• [ Vibrational frequencies, based on analytical gradient ]
• Localized Wannier Functions [Symmetry adapted]
Cesare Pisani - LCC2004 10
A pre-requisite for Local correlation methods in crystals:
Efficient generation of Wannier functions (WF) to span the occupied HF space
{Ψn(k)}occ → FT +
Ψ )} localization criterion →{ wsg }occ
Edmiston-Ruedenberg (1965) : Maximum intra-LO repulsion (N5)
Boys (1966) : Maximum distance between LO centroids (N3)
Pipek-Mezey (1989) : Minimum number of atoms per LO (N3)
What does efficient mean?
Computationally inexpensive
Well localized WFs
Strictly orthonormal WFs
Zicovich, Dovesi & Saunders, J. Chem. Phys. 115, 9708 (2001)
Symmetry adapted WFs
Zicovich, Casassa (2004)
Cesare Pisani - LCC2004 11
CRYSCOR work to date and scheme of presentation
1. Generalization of the LOCALI part of CRYSTAL03, to produce
symmetry adapted Wannier Functions (see Casassa-Zicovich
poster)
2. Reformulation of LC-MP2 equations, so as to exploit translational
and point symmetry
3. CRYSCOR code preparation (from CRYSTAL output to final
results) (see Casassa et al. poster)
4. Refinement work on the integral part of the code to obtain 2-el
integrals (i a | j b ) either exactly or in a multipolar
approximation (see Capecchi-Maschio poster)
5. Test of computational parameters (molecular cases + Diamond,
Silicon, SiC, BN, BeS)
Cesare Pisani - LCC2004 12
LC Reformulation of LMP2 and use of Symmetry
Rijab = 0 = Kijab + Σcd [ fac Tijcd Sdb + Sac Tijcd fdc ] +
− Σcd [ Sac Σk (f ik Tkjcd + Tikcd f kj ) Scb ]
E2 = Σ(ij) Σab ∈(ij) (Kijab + Rijab ) (2 Tijab − Tijba ) ( a sum of pair energies)
Ψ(1)〉 = Σ(ij) Σab ∈(ij) ( 2 Tijab − Tijba ) 2 Φijab − Φijba 〉
Kijab = ( i a | j b) = Σµρνσ cWFiµ cPAOaρ cWFjν cPAObσ ( µ ρ | ν σ )
Symmetry exploitation
Translational : The “ first WF index ” (i) is always confined in the zero cell
Rotational :
• Only “irreducible WF pairs” (i j) need to be considered
• Irreducible pairs may have a residual symmetry which can be exploited
in the K integral evaluation (Capecchi-Maschio poster).
Cesare Pisani - LCC2004 13
Use of Symmetry in the Update step
Rijab = … − Σcd [ Sac Σk (f ik Tkjcd + Tikcd f kj ) Scb ]
= … − Σcd [ Sac βijcd Scb ]
∆Tijab = Rijab / ∆εijab
To update amplitudes of irreducible WF pairs, we need those of all pairs.
For each irreducible pair ij do
Update Tij
For each symmetry operator of i j (n) do
Obtain amplitude of “rotated pair” T injn
nn
For each irreducible pair kl do
If (k ≈ in •and• l (close-to) jn ) then βkl = βkl + Tinjn f jn l
nn
If (l ≈ jn •and• k (close-to) in ) then βkl = βkl + f k in Tinjn
nn
enddo
enddo
enddo
Cesare Pisani - LCC2004 14
BLOCK DIAGRAM OF CRYSCOR
CRYSTAL
(on disk)
CRYSCOR
symmetry;
Crystal type and symmetry;
AO, SAOg , FAOg , WF (coefficients & symmetry)
MP2MAIN
Computational parameters from input cards
Construct PAOs ,
ORIQAO
Calculate FPAOg , SPAOg , FWFg
SYMPAIR wp- bi-
Recognize irreducible ww pairs and wp-Wp bi-pairs
Boughton-Pulay),
Domains (Boughton-Pulay),
DOMAINS
Pair domains …… and their classification:
classification:
MULTIPOLES WF-
Calculate multipolar expansion of WF-PAO products
MP2INT el-
Calculate 2-el-integrals ( exact or multipolar )
(see Capecchi-Maschio poster)
MP2CORE
Solve LMP2 equations (next dia)
MP2LOOP
Cesare Pisani - LCC2004 15
SOLUTION OF LMP2 EQUATIONS
MP2MAIN MP2CORE
For each irreducible WW pair, do
LON lon)
Construct local orthonormal orbitals (lon)
Organize quantities for iterative loop
BSETR
close- pairs,
(close-by pairs, etc.)
Calculate local pseudo-canonical virtual orbitals
pseudo-
FWPRIM
Provide initial guess for amplitudes T and for
IGUESS
“coupling” β quantities
coupling”
Enddo
MP2LOOP
For each irreducible WW pair, do
NO
ECALC Read K, T, β, Update T ,Update E2
End
YES Update β
conv? UPDATE
Enddo
Cesare Pisani - LCC2004 16
Main computational parameters in LMP2 for crystals
Rijab = Kijab + Σcd [ fac Tijcd Sdb + Sac Tijcd fdc ] − Σcd [ Sac βijcd Scb ]
E2 = Σ(ij) Σab ∈(ij) (Kijab + Rijab ) (2 Tijab − Tijba )
Kijab = ( i a | j b) = Σµρνσ cWFiµ cPAOaρ cWFjν cPAObσ ( µ ρ | ν σ )
• µ
Basis set {µ} (representation of WFs and PAOs)
• Truncation of WF and PAO tails (|cWFiµ| > tow ; |cPAOaρ| > toq)
• Exact / multipolar treatment of K integrals (Capecchi-Maschio )
• Prescreening of exact K integrals (Schwarz+density screening)
• Size of WF domains (range of occ. to virt. excitations in Σab ∈(ij) )
• Maximum WF-WF „distance“ (range of Σ(ij) )
Cesare Pisani - LCC2004 17
C Shell exponent s coefficient p coefficient d coefficient
6-21G*
3048 0.001826
456.4 0.01406
1s
103.7 0.06876
29.23 0.2304
9.349 0.4685
3.189 0.3628
2sp 3.665 -0.3959 0.2365
0.7705 1.216 0.8606
3sp 0.26 1 1
d 0.8 1
Cesare Pisani - LCC2004 18
Si Shell exponent s coefficient p coefficient d coefficient
16120 0.001959
6-21G* 2426 0.01493
1s
553.9 0.07285
156.3 0.2461
50.07 0.4859
17.02 0.3250
292.7 -0.002781 0.004438
2sp 69.87 -0.03571 0.03267
22.34 -0.1150 0.1347
8.150 0.09356 0.3287
3.135 0.603 0.4496
1.225 0.419 0.2614
3sp 1.079 -0.3761 0.0671
0.3024 1.252 0.9569
4sp 0.1233 1 1
d 0.5 1
Cesare Pisani - LCC2004 19
Main computational parameters in LMP2 for crystals
Rijab = Kijab + Σcd [ fac Tijcd Sdb + Sac Tijcd fdc ] − Σcd [ Sac βijcd Scb ]
E2 = Σ(ij) Σab ∈(ij) (Kijab + Rijab ) (2 Tijab − Tijba )
Kijab = ( i a | j b) = Σµρνσ cWFiµ cPAOaρ cWFjν cPAObσ ( µ ρ | ν σ )
• µ
Basis set {µ} (representation of WFs and PAOs)
• Truncation of WF and PAO tails (|cWFiµ| > tow ; |cPAOaρ| > toq)
• Exact / multipolar treatment of K integrals (Capecchi-Maschio )
• Prescreening of exact K integrals (Schwarz+density screening)
• Size of WF domains (range of occ. to virt. excitations in Σab ∈(ij) )
• Maximum WF-WF „distance“ (range of Σ(ij) )
Cesare Pisani - LCC2004 20
PAOs and WFs (SiC and BeS)
Cesare Pisani - LCC2004 21
Main computational parameters in LMP2 for crystals
Rijab = Kijab + Σcd [ fac Tijcd Sdb + Sac Tijcd fdc ] − Σcd [ Sac βijcd Scb ]
E2 = Σ(ij) Σab ∈(ij) (Kijab + Rijab ) (2 Tijab − Tijba )
Kijab = ( i a | j b) = Σµρνσ cWFiµ cPAOaρ cWFjν cPAObσ ( µ ρ | ν σ )
• µ
Basis set {µ} (representation of WFs and PAOs)
• Truncation of WF and PAO tails (|cWFiµ| > tow ; |cPAOaρ| > toq)
• Exact / multipolar treatment of K integrals (Capecchi-Maschio )
• Prescreening of exact K integrals (Schwarz+density screening)
• Size of WF domains (range of occ. to virt. excitations in Σab ∈(ij) )
• Maximum WF-WF „distance“ (range of Σ(ij) )
Cesare Pisani - LCC2004 22
Influence of Computational parameters on Diamond results
Basis set: 6-21G* ; valence-only MP2 calculations
Standard parameter settings are in bold letters (outlined in yellow in tables)
Parameters
WF Size: There is only one type of WF (C-C bond localized orbital). The corresponding
domain comprises 2 (C—C) or 8 ( C3 C—C C3 ) atoms
Tow : threshold of cWFiµ coefficients ( 0.01, 0.005, 0.001, 0.0005, 0.001)
Toq : threshold of cPAOaρ coefficients ( 0.01, 0.005, 0.001, 0.0005, 0.001)
Pairs: n. of irreducible pairs considered [exact/multipolar] (4,19 / 0,154,301,407)
The threshold for disregarding 2-el integrals according to Schwarz criterion has been
set to 10-8, except in the cases indicated with an asterisk, where it is set to 10-10
Results
[bb] : contribution to -E2 from [C’C|C’C] pair (microHartree)
[bCb’] : contribution to -E2 from [C’C|CC’’] (microHartree)
-E2 : absolute value of MP2 energy per cell (microHartree)
Time: CPU time/103 sec on AMD Athlon, 1.6 GHz
Cesare Pisani - LCC2004 23
Influence of Computational parameters on Diamond results
Domain size and number of pairs
All energies in microHartrees, times in 103 sec on AMD Athlon 1.6 GHz
Size Tow Toq Pairs [bb] [bCb’] -E2 Time
2 0.001 0.005 4/0 19357 4813 219053 18
19/0 19389 4842 236091 90
2 0.001 0.005 (+2363)
19/407 19390 4842 238454 90+4
8 0.001 0.005 4/0 19814 4973 225545 26
19/0 19846 4996 247393 131
8 0.001 0.005 (+3861)
19/407 19848 4996 251254 131+5
Cesare Pisani - LCC2004 24
Influence of Computational parameters on Diamond results
Truncation of tails and number of pairs
Size Tow Toq Pairs [bb] [bCb’] -E2 Time
8 0.01 0.01 4/0 20176 5526 249782 10
8 0.001 0.01 4/0 19625 4900 220237 21
8 0.001 0.005 4/0 19814 4973 225545 26
8 0.001 0.001 4/0 19822 4978 225858 293
8 0.0001 0.0001 4/0 19828 4980 225969 385
8 0.01 0.01 19/0 20241 5598 285496 81
8 0.001 0.01 19/0 19655 4917 237687 99
8 0.001 0.005 19/0 19846 4996 247393 131
8 0.001 0.005 19/407 19848 4996 251254 131+5
8 0.001 0.001 19/0 19856 5001 248205 * 1588
8 0.001 0.001 19/154 19856 5001 251497 * 1588+5
8 0.001 0.001 19/301 19856 5001 252279 * 1588+5
8 0.001 0.001 19/407 19856 5001 252355 *1588+5
Cesare Pisani - LCC2004 25
Comparison between 6-21G* results for diamond-like molecules
(MOLPRO) and Diamond (CRYSCOR, standard setting)
[bb] [bCb’] [b||b’] ∨
[b∨b’]
System - E2
4 C-C Neopentane
12 C-H C5H12 20331 5050 --- --- 752436
12 C-C Adamantane
16 C-H C10H16 19814 5155 --- 407 1432646
…. 20107 5138
C35H36 20135 5050 445 405 4852832
4854
Diamond m=4 m=24 m=24 m=48 247401
C∞ 19846 4996 393 373 251551
All energies in µHartree; the best fit estimate of diamond E2 “energy per cell” from
present molecular results is -259780 µHartree
Cesare Pisani - LCC2004 26
Cesare Pisani - LCC2004 27
Comparison between 6-21G* results for diamond-like molecules
(MOLPRO) and Diamond (CRYSCOR, standard setting)
[bb] [bCb’] [b||b’] ∨
[b∨b’]
System - E2
Neopentane
C5H12 20331 5050 --- --- 752436
Adamantane
C10H16 19814 5155 --- 407 1432646
…. 20107 5138
52 C-C
C35H36 20135 5050 445 405 4852832
36 C-H
4854
Diamond m=4 m=24 m=24 m=48 247401
C∞ 19846 4996 393 373 251551
All energies in µHartree; the best fit estimate of diamond E2 “energy per cell” from
present molecular results is -259780 µHartree
Cesare Pisani - LCC2004 28
Cesare Pisani - LCC2004 29
Comparison between 6-21G* results for diamond-like molecules
(MOLPRO) and Diamond (CRYSCOR, standard setting)
[bb] [bCb’] [b||b’] ∨
[b∨b’]
System - E2
4 C-C Neopentane
12 C-H C5H12 20331 5050 --- --- 752436
12 C-C Adamantane
16 C-H C10H16 19814 5155 --- 407 1432646
…. 20107 5138
52 C-C
C35H36 20135 5050 445 405 4852832
36 C-H
4854
2 C-C Diamond m=4 m=24 m=24 m=48 247401
per cell C∞ 19846 4996 393 373 251551
All energies in µHartree; the best fit estimate of diamond E2 “energy per cell” from
present molecular results is -259780 µHartree
Cesare Pisani - LCC2004 30
Summary of MP2 valence-only results (6-21 G* basis set)
BEHF EMP2-EatMP2 BETOT BEINCR BEEXP
Diamond 0.407 0.134 0.541 0.528 0.555
Silicon 0.227 0.091 0.318 * 0.325 0.345
SiC 0.263 0.121 0.384 * 0.440 0.475
BN 0.335 0.117 0.452 * 0.455 0.498
BeS 0.260 0.042 0.302 * -- ≈ 0.320
* Still not at convergence
Cesare Pisani - LCC2004 31
STATE OF PROJECT AND PROSPECTS
Current work :
Refinement and standardization of “basic” program
Higher efficiency, generalization of WF symmetry treatment
Tests
Basis set quality, different systems (MgO, Polymers, Rare gases…)
Open problems and lines of development (Schütz, Usvjat, …):
Efficient calculation and transformation of K integrals (Density fitting, etc.)
Auxiliary basis set (to complement the HF set)
Extension to other local correlation schemes,
either within the MOLPRO framework (CCSD, MP4, .. …),
or different (IEPA + NO ; MP2-R12 (Kutzelnigg, Klopper, Manby) ; …?)
Cesare Pisani - LCC2004 32


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