p. pulay - unito.itp. pulay department of chemistry and biochemistry fulbright college of arts and...
Post on 31-May-2020
6 Views
Preview:
TRANSCRIPT
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 1
Toward full accuracy local correlation methods
P. PulayDepartment of Chemistry and Biochemistry
Fulbright College of Arts and SciencesUniversity of Arkansas, Fayetteville
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 2
Susscrofa
Razorback
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 3
ThanksCoworkers
Prof. Svein Saebo (Mississippi State University, Starkville)Dr. Jon Baker (PQS, LLC, Fayetteville, AR)Prof. K. Wolinski (University of Lublin, Poland)Prof. Amy Apon (U.of Arkansas, Computer Science and Eng.)Prof. Shigeru Nagase, Institute for Molecular Science, Japan Mr. Alan Ford (University of Arkansas)Mr. Youyou ZhaoMs. Yuriko Yara
FundingThe U.S. National Science Foundation (2 grants)NSF SBIR to PQS, LLC
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 4
Motivation and general considerations
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 5
Topics
• An efficient canonical MP2• Parallel MP2 on a Linux cluster• Application to π stacking energies• Dual-basis MP2• Full-accuracy local MP2• AO formulation of the MP2 gradient• Array files (Amy Apon & her coworkers)
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 6
An efficient canonical MP2
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 7
Motivation• Needed canonical to check approximate methods (local
MP2, FMO=Fragment Molecular Orbital, Many-Body Expansion, Density Fitting) against exact results
• Commonly used programs failed or were too expensive• Local electron correlation becomes less efficient for
systems with aromatic delocalization (porphyrins, fullerenes, graphenes).
• There are excellent local correlation implementations (the Saebo-Pulay technique is implemented in MOLPRO and JAGUAR; TRIM in QChem, Scuseria has a method)
• An efficient integral transformation is necessary for higher configuration-based correlation methods
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 8
Four-index transformation: theory• Essentially all effort in canonical MP2 goes into the four-
index integral transformation (i,j occupied, a,b virtual) (ai|bj)=∑ pqrs (pq|rs)CpaCqiCrbCsj ,
usually broken up to four quarter transformations• Formal scaling of the first quarter transformation,
(pi|rs) = ∑ q (pq|rs)Cqi, ( i=1,…n)is O(nN4), where n is the number of correlated occupied orbitals, and N is the number of AOs
• Subsequents steps scale as O(n2N3). As N>>n (for better basis sets N≈ 6-10n), the first transformation is expected to dominate
• Traditional: Saunders and Van Lenthe, Werner and Meyer
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 9
Integral transformation for large calculations: memory limitations
Head-Gordon, Pople, Frisch, Chem. Phys. Lett. 153 (1988) 503
Cubic memory demand. Calculate in batches
(Gets expensive if there is insufficient memory)
For a comparison of methods, and some new algorithms (two methods that require O(N) memory)
see: M. Schütz, R. Lindh and H.-J. Werner, Mol. Phys. 96 (1999) 719
Saebo and Almlöf, Chem. Phys. Lett. 154 (1989) 83Quadratic memory; only one permutation used (integrals calculated 4x)
do pdo r
calculate all (pq|rs)=Xqs
Y=CTXC (matrix mult.)store Yij=(pi|rj) on disk
end do rend do p
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 10
Prescreening
• Efficient prescreening in the integral evaluation phase based on local correlation ideas (Chem. Phys. Lett. 2001, 344, 543)
• Basic idea: an integral which is negligible in local MP2 is also negligible in the canonical method
• Pair correlation amplitudes between well-separated electron pairs decreases as R-3
• In large, well-localized molecules, only a small fraction of the AO integrals needs to be calculated and transformed
• Dilemma: dense matrix multiplication is very fast but scales steeply; sparse matrix multiplication has good scaling but is slow. Solution: compact the matrices before transformation
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 11
Effect of the threshold on the accuracy of the calculated MP2 energy
Molecule Hexapep (glycine)10
Basis 6-311G(dp) 6-31G(dp)
N 672 734
T1=3.16×10-9 -4.516233 -6.202953
T1=10-9 -4.516178 -6.202887
T1=3.16×10-10 -4.516177 -6.202881
Hexapep=N-formyl pentalanine amide
For well-conditioned basis sets, the default threshold guarantees accuracy to a few µµµµEh.
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 12
Timings for calix[4]arenea,b, C60a and C74(C1)c
Formula C32H32O4 C32H32O4 C60 C74Symmetry C2h C2h D2h C1Basis cc-pVDZ cc-pVTZ cc-pVTZ 6-31G(d)N 664 1528 1800 10362n 184 184 240 296V 536 1400 1620 814P (%) 24.8 14.0 30.8 23.6Tint/min 71.4 795.7 2233.9 239.7c
T1/min 58.8 880.4 1905.2 283.4c
Tsort/min 9.9 138.0 639.5c 14.8c
T2/min 111.4 1644.5 3336.9 120.7c
TMP2/min 252.9 3466.6 8132.8 658.7c
TSCF/min 149.0 4××××480.8 3××××1644.0 616.3c
Disk use/GB 3.8 11.6 11.6 <120E(SCF)/Eh -1529.889518 -1530.271932 -1530.271932 -2801.946722E(MP2)/Eh -5.022596 -6.127940 -9.224801 (-36.539500)
a In minutes on an 800 MHz Pentium III bTetramethoxy-calix[4]arenec In minutes on a 3 GHz Pentium 4 Xeon
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 13
Scaling
• The ultimate scaling is determined by the second half transformation which is performed just like in traditional MP2
• Routine calculations on a single processor are possible for molecules with ~1000 basis functions and ~300 correlated electrons in the absence of symmetry
• Larger calculations are possible for symmetrical molecules
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 14
Parallel MP2 on a Linux cluster(Jon Baker; prelim. work Matt Shirel)
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 15
Parallel four-index transformation:mostly aimed at supercomputers
R. A. Whiteside, J. S. Binkley, M. E. Colvin, H. F. Schaefer, IIIJ. Chem. Phys. 1987, 86, 2185
J. D. Watts and M. Dupuis, J. Comput. Chem. 1988, 9, 158T. L. Windus, M. E. Schmidt, and M. S. Gordon, Theor. Chim.
Acta 1994, 89, 77 (for MC-SCF)A. M. Márquez and M. Dupuis, J. Comput. Chem. 1995, 16, 395I. M. B Nielsen and E. T. Seidl, J Comput Chem 1995, 16, 1301 A. T. Wong, R. J. Harrison, and A. P. Rendell, Theor. Chim.
Acta 1996, 93, 317M. Schütz and R. Lindh, Theor. Chim. Acta 1997, 95, 13A. C. Limaye, J. Comput. Chem. 1997, 18, 552C. P. Sosa, J. Ochterski, J. Carpenter, M. J. Frisch, J. Comput.
Chem. 1998, 19, 1053
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 16
Parallelization
• First half transformation: The Saebo-Almlöf algorithm naturally parallelizes over two fixed AO indices p and r
• Yoshimine bin sort. Each node owns a subset of the pair indices (i,j). The parallel sort sends the half-transformed integrals to the appropriate node. Synchronization delays are avoided by spawning a set of independent bin receive/write daemon processes
• The second half transformation naturally parallelizes over the orbital pair indices (i,j)
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 17
Parallel Yoshimine bin sort
Slave 1Read half-transformed integrals from disk and sort them in bins by i,j. Send full bins to the appropriate node red or blue
Master: spawn slaves and bin writer daemons
Bin writer 1 (daemon)Receive sorted bins and store them on disk
Slave 2Read half-transformed integrals from disk and sort them in bins by i,j. Send full bins to the appropriate node red or blue
Bin writer 2 (daemon)Receive sorted bins and store them on disk
spawn send
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 18
Possible improvements
• Half-transformed integrals can be sorted into bins directly ⇒ avoid intermediate storage of half-transformed integrals (implemented recently but there is an additional memory demand)
• This allows the overlap of sort and computation, and renders the sorting step computationally insignificant
• Integral evaluation often dominates the first half transformation: the program would profit from an improved integral code and more efficient use of symmetry
• The second half transformation is done fully in canonical orbitals and dominates for very large calculations because of its fifth-order scaling
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 19
Parallel scaling: dual CPUs
“Hexapep”:For-Ala5-NH26-311G**
Elapsed time on 50 1 GHz Pentium IIIs: 10.9 min for MP2
(672 BF, no symmetry)
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 20
Parallel scaling: single CPUs
Calix[4]arene:cc-pVTZ basis,1528 BF, C2h symmetry
Elapsed time on 24 1 GHz Pentium IIIs: 150.4 min for MP2
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 21
Application to π stacking energies
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 22
General
• Large planar aromatic systems (graphene sheets, porphyrins, phthalocyanines, DNA bases) are attracted by a considerable dispersion force.
• Dispersion forces are also important in bucky onions and carbon nanotubes
• The true dimerization energy (heat of vaporization) is difficult to measure because of decomposition
• In the limit of large parallel sheets, the dispersion force diminishes as 1/d4, not as 1/d6
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 23
Coronene dimer(parallel displaced)• Counterpoise correction is important• At the 6-311G* level (936 BF, C2h)
– Energy minimum at 3.27 Å (graphite 3.35 Å room temp.)
– Minimum energy = -0.0442 Eh = -27.7 kcal/mol– At 3.35 Å: SCF=+22.1, corr. -49.6, B3LYP=+15.7
kcal/mol – Elapsed time on 4 dual-processor computers: 2hrs 1 min
• At the 6-311G(2d,f) level (1512 BF):– Energy at 3.35 Å (close to minimum) is -0.0556 Eh=-35
kcal– Elapsed time on 10 800 MHz P III processors: 7.1 hrs
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 24
Potential curve for (C24H12)2a,b
aCounterpoise
corrected
bDistance of sheets in graphite = 3.35 Å
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 25
Porphine dimer
• Porphyrins show a tendency to dimerize in solution
• Porphyrin dimer: C40H28N8, 228 correlated electrons
• 6-311G* basis: 948 basis functions 73 min total on 8 proc.
• Dimerization energy at 3.35 Å is comparable to coronene with the same basis, -0.043 Eh = -26.9 kcal, SCF=+20.5, correlation -47.4
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 26
Coronene dimer
-37
-32
-27
-22
-17
-12
-7
-2
3
8
3 3.5 4 4.5 5 5.5 6Rz Displacement (Å)
Inte
ract
ion
Ener
gy (k
cal/m
ol)
MP2 6-311G(0.25 d) Rx=0, Ry=0
MP2 6-311G(d) Rx=1.42, Ry=0
MP2 6-311G(0.25 d) Rx=1.42, Ry=0
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 27
Circumcoronene dimer (C108H36)
• Two circumcoronene molecules separated by 3.35 Å• 6-31G* binding energy = 0.138 Eh = 86.3 kcal/mol; with
soft d functions (P. Hobza) 0.157 Eh=98.5 kcal/mol• Per carbon atom = 0.80 kcal/mol (significantly more than
in coronene, 0.57 kcal/mol (0.91 with soft d functions)
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 28
Circumcoronene dimer 6-31G(0.25d)
-110
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
C:/documents and settings/pulay/My Documents/Papers04/GRAPHITE/cir2-vertical.txt" using 1:($2*627.51)"C:/documents and settings/pulay/My documents/Papers04/GRAPHITE/cir2-vertical.txt" u 1:($3*627.51)"C:/documents and settings/pulay/My documents/Papers04/GRAPHITE/cir2-vertical.txt" u 1:($4*627.51)"C:/Documents and settings/pulay/My documents/Papers04/GRAPHITE/cir2-vertical.txt" u 1:($5*627.51)
Sandwich
P.D. 1/4
Parallel Displaced 1/2P.D. 3/4
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 29
Circumcoronene dimer: the effect of horizontal displacement
-96
-94
-92
-90
-88
-86
-84
-82
-80
-78
-76
-74
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
"D:/cir2-horizontal.txt" u 1:($2*627.51)
Vertical distance: 3.4 Å
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 30
Spin-component scaled MP2
• MP2 overestimates π-stacking energies, e.g. in benzene. The well depth is the result of a delicate balance between the Pauli repulsion and dispersion attraction. Errors in the latter are magnified.
• S. Grimme (JCP, 2003). An empirical correction: increase slightly the opposite spin contribution, diminish strongly the parallel spin part. Note that these are NOT identical with singlet and triplet pairs.
• Diminishes the absolute value of the dispersion interaction. Yields good values for the benzene dimer
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 31
Circumcoronene dimer: MP2 versus SCS-MP2
-120
-100
-80
-60
-40
-20
0
20
2.5 3 3.5 4 4.5 5
"D:/cir2-scs-2.txt" u 1:($2*627.51)"D:/cir2-scs-2.txt" u 1:($3*627.51)
MP2 (CP corrected)
SCS-MP2 (CP corr.)
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 32
Applications to water clusters
• MP2 is excellent for water clusters, see S. Xantheas, C. J. Burnham, and R. J. Harrison, J. Chem. Phys. 116, 1493 (2002)
• (H2O)20 aug-cc-pVTZ 1840 BF, 80 correlatedorbitals
• Diffuse functions diminish the efficiency• 4 Pentium 4 processors:
SCF 982 min elapsed timeMP2 2603 min elapsed time
• E(SCF) = -1521.395430 Eh• E(MP2)= -5.517590 Eh
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 33
Pyrazine in a Cram carceplex (courtesy Dr. Suyong Re)
C72H52N2O24
6-31G* basis, 1476 basis functions, D2 symm, 8 2.4 GHz Pentium Xeon processors.
Elapsed times:SCF: 36.9 minFirst ½ tr. 51.0 minSort and Second ½ tr. 295.9 minTotal job 383.9 min
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 34
N-methyl pyrrolidone in a Cram carceplex(courtesy Dr. Suyong Re)
C73H57NO25
6-31G* basis, 1500 basis functions, no symm, 4 or 48 3 GHz processors.
Elapsed times in MP2 (in min):# processors 4 48 ratio
Total MP2 1923 159 12.1
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 35
Can (semi)local DFT describe dispersion?
A thought experiment shows that (semi)local DFT cannot describe genuine dispersion, even if there is an artificial attraction resembling dispersion
e-
Infinite potential barrier
Electromagnetic forces movefreely through the barrier
Electrons can’t penetrate
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 36
Full-accuracy local MP2(Svein Saebo)
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 37
New local correlation program
• S. Saebo, P. Pulay, J. Chem. Phys. 2001, 115, 3975• Simultaneous transformation of 2 indices (P. R. Taylor,
Int. J. Quantum Chem. 1987, 31, 521). This is not the essence of the method but makes it very memory-efficient
• Memory and disk space requirements are minimal (E.g. S. Sabo evaluated the MP2 energy of (glycine)50, 6-31G(d): 3118 BF on a single PC. Similarly, (glycine)30, 6-311G**: 2638 BF. However, distant pairs were approximated in these calculations
• With distant pairs added, the results are essentially identical with canonical theory
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 38
0200400600800
100012001400
gly10 gly15 gly20
NbasisNorbT(SCF)/minAOInt/minT(MP2)/min
SCF: N2.02
MP2:N1.33
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 39
Problems with the current code
• Pair domains are large. If distant pairs are included, computational times approach the traditional timings for medium-sized molecules. The major computational task is MP2 iteration
• Even weak correlation requires a considerable number of AOs for full description. The current program is wasteful because it uses the same local basis for all pairs: strong, weak, distant. A version which uses a smaller local basis for distant pairs would be much more efficient.
• A more fundamental approach is to switch to a molecular orbital representation (W. Meyer)
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 40
MO treatment of weak correlation
By changing to an MO basis, a much smaller basis is sufficient.
Consider the pair correlation between two localized orbitals|i> and |j> (use canonical virtuals)
Tijab = -(ia|jb)/(εa + εb -εi - εj)
Invoke the multipole expansion but not for approximating integrals: 1/r12 = R-3[r1⋅r2 -3R-2(r1⋅R)(r2⋅R) + …]
Substitute into the pair wavefunction Ψij = ∑a,b TijabΨij
ab
If the energy denominator is assumed constant then, in the dipole approximation, three virtual orbitals on each center,
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 41
∑=a
av ari ϕϕ ηη || bb
w brj ϕϕ ζζ ∑= ||
are sufficient to account for all correlation. η,ζ=x,y,z; rη is the position vector relative to the centroid of ϕ i in the ηdirection, rη=η-Riη ; rζ is analogous for ϕj
Of course, the orbital energies are not constant. Numerical experiments by W. Meyer suggest that 2-3 denominator shifts are sufficient to generate essentially all the virtual space
∑−∆+=
aaav ari ϕεϕ ηη
1)(||
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 42
Moreover, the dipole approximation holds only for very distant pairs; at shorter distances quadrupole, possibly octupole terms may also contribute.
These considerations suggest that a small molecular orbitalbasis, 10-20 orbitals, determined individually for each localized internal orbital, is capable of describing essentially all dispersion interaction in the virtual space.
The MOs belonging to different orbitals are not orthogonal. Note that this is essentially the pseudonatural orbital method of Meyer.
It is best to introduce a pseudocanonical basis in the small subspace, separately for each localized occupied orbital, by diagonalizing the Fock matrix in the subspace
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 43
Natural Orbitals of Distant Pairs
A typical distribution of the natural orbital occupation numbers for a weak pair (in gly5) is shown here:
Natural orbital occupation numbers for pair 9 80.00030674 0.00017801 0.00004943 0.00002241 0.000015010.00001092 0.00000940 0.00000651 0.00000341 0.000002730.00000193 0.00000167 0.00000127 0.00000086 0.000000600.00000045 0.00000043 0.00000040 0.00000027 0.000000200.00000017 0.00000016 0.00000014 0.00000011 0.00000009
Sum of all eigenvalues: 0.0006141Sum of the 8 largest eigenvalues: 0.0006019
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 44
-7
-6.5
-6
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
-2
2 3 4 5 6 7 8 9
Log pair energy versus localized orbital separation in GLY8
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 45
-7
-6.5
-6
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
-2
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Pair energy versus localized orbital separation in GLY8, LOG-LOG plot-6*x-1.1
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 46
An efficient atomic-orbital based MP2 gradient program
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 47
Efficient AO formulation of MP2 gradientsJ. Chem. Phys., to be submitted
Based on the orbital-invariant form of the MP2 energyP. Pulay and S. Saebo, Orbital-invariant formulation
and gradient evaluation in Møller-Plesset Perturbation Theory, Theor. Chim. Acta 69, 357 (1986).
Also used to derived the equations for local MP2 gradient:
A. El-Azhary, G. Rauhut, P. Pulay and H.-J. Werner, Analytical Energy Gradients for Local Second-Order Moller-Plesset Perturbation Theory, J. Chem. Phys., 108, 5185 (1998).
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 48
Generator State Formulation
baji
baji
baji
abij
abij Φ+Φ+Φ+Φ=Ψ �
Normalized to 4 (a≠b) or to 2 (a=b), and pairwise non-orthogonal,
babaij
abij ≠=⟩ΨΨ⟨ if 2|
they allow a substantial simplification of the MPn, CI and Coupled Cluster equations, see
P. Pulay, S. Saebo, and W. Meyer: An Efficient Reformulation of the Closed-Shell Self-Consistent Electron Pair Theory, J. Chem. Phys. 81, 1901 (1984).
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 49
The Self-Consistent Electron Pair TheoryW. Meyer, J. Chem. Phys. 64, 2901 (1976).
An efficient matrix formulation of the singles and doubles CI problem and related methods. Both the amplitudes T
abij
jiΨ+Ψ=Ψ+Ψ=Ψ ∑
≥
ijab010 T
and the integrals, e.g. the exchange and Coulomb integrals)|( );|( abijbjai ij
abijab == JK
are collected in matrices. It is useful to introduce the contravariant amplitude matrices (Tji is the transpose of Tij)
)2)(2(~ jiijij
ij TTT −−= δ jiijij TTT −= 2
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 50
SCEP in AO basis
The canonical virtual orbitals are replaced by AOs:
This complicates the formalism but is essential in local correlation, and is also advantageous in gradient evaluation. E.g. no problem with frozen cores. With a little extra work, “QM/MM” is possible: QM=MP2, “MM”=Hartree-Fock
µνµν ij
jiΨ+Ψ=Ψ+Ψ=Ψ ∑
≥
ij010 T
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 51
SCEP, cont.
For canonical occupied orbitals this reduces to
In a canonical MO basis for virtual orbitals (S=1; F=diag{ε}) it becomes the usual canonical formula
0SSTFSTSFTKR =+−++= ijji
ijijijij )( εε
0)()|( =−−++ ijabjibabjai Tεεεε
1))(|( −−−+−= jibaijab bjai εεεεT
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 52
Orbital-invariant form of MP2
It can be derived in several ways. For gradient evaluation the Hylleraas form is particularly useful:
Ec = 2⟨Ψ1|H-E0|Ψ0⟩ - ⟨Ψ1|H0-E0|Ψ1⟩ = minimum The MP2 energy can be expressed as Ec=∑ij eij
(S is the overlap matrix, F is the Fock matrix).Differentiating the pair energy eij with respect to the (contra-
variant) amplitudes gives the following equation
]~~[~~~2 jiikkj
jikj
kik
ijjijiijijijij FFe TSSTTSSTFSTTTSFTTK +−++= ∑
0][ =+−++= ∑ STTSFSTSFTKR ikkjkj
kik
ijijijij FF
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 53
Timing benchmarks (minutes)on a 2.8 GHz Pentium 4 Xeon processor
930
389
560
282
230
694
285
346
Nbf
21055056-311G(d,p)C20H32N10O11Glycine10
167.936.421.56-31G*C12H22O11Sucrose
665*122118cc-pVTZC8H10N4O2Caffeine , Cs
78.4*10.310.36-311G*C8H10N4O2Caffeine , Cs
47.4*5.85.66-31G*C8H10N4O2Caffeine, Cs
11351451886-311+G(2df,2pd)C12H18N+Retinal Schiff b.
44.55.95.76-31G**C12H18N+Retinal Schiff b.
177.736.739.36-311G(df,p)C10H16αααα-Pinene
GradMP2SCFBasisFormulaMolecule
*No symmetry
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 54
Cavitand with 2-oxo-1-butanol*
C32H32O10, 6-31G*, 652 BFA single 2.8 GHz P 4 Xeonprocessor (parallelization in progress). Elapsed times:SCF = 99.6 minMP2* = 309.0 minMP2 gradient = 883.5 min* incl. storing the amplitudes*Courtesy Dr. Re Suyong
defaults used
point 0
MOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDEN
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 55
Nucleic acid bases
• In connection with a recent plane-wave based DFT paper (M. Preuss et al., J. Comput. Chem. 2004, 25, 112) which predicts essentially planar geometries for these molecules, explicit correlation methods predict significant non-planarity for the amino (-NH2) group.
• We have optimized the geometries of adenine, cytosine, guanine and thymine at the MP2 level using the cc-pVTZ, aug-cc-pvTZ and cc-pVQZ basis sets; the latter has the composition of 5s4p3d2f1g (755 basis functions for guanine). All three large basis sets give very similar geometries, showing that the results are converged. These are among the largest MP2 optimizations performed.
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 56
Guanine, optimized
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 57
Array files (with Amy Apon)
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 58
Middleware for large-scale data-intensive applications
• Both disk storage capacities and random access memories have increased by about 2 orders of magnitude in the past 10 years: a new paradigm is needed
• Global Arrays were conceived more that 10 years ago for small memory machines. They assume that a matrix does not fit into memory, and distribute it across nodes. This is inefficient, not necessary any more.
• Needed: transparent distributed parallel storage. Disk reading rates and network transport rates are comparable.
• First version ready, and was tested over both NSF and PVFS (Parallel Virtual File System); the production version uses PVFS with moderately large stripe size
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 59
Array Files design
• An array file is a collection of variable-length records of the same type. Records are assumed to be relatively large (unlike in transaction processing), e.g. an 500×500 double precision array occupies ~2 MB.
• Records are currently indexed by integer counters. The record size is fixed at the time of the first writing and cannot be increased in the present version
• Records are NOT separate files created by the operating system. Records are locked during write but otherwise it is the user’s responsibility to ensure data coherence.
• Blocking and nonblocking read is available.• We are currently recoding MP2 and MP2 gradients using
Array Files under PVFS.
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 60
Summary
• Canonical MP2 is possible for large systems, by using natural localization
• Parallel implementation can handle >2000 basis functions: π stacking
• Full-accuracy local MP2 has been implemented• MP2 gradients up to ~atoms and ~1000 basis functions are
feasible on a single PC• Array Files have been implemented; will be released soon.
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 61
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 62
The end
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 63
Scaling of configuration-based correlation methods
• Traditional correlation theories scale steeply and this is not reduced automatically by sparsity
• This steep scaling is an artifact of using delocalized canonical MOs, as dynamical correlation is very local
• To get rid of it, correlation theories must be formulated in terms of localized quantities: either localized molecular orbitals or atomic orbitals (basis functions)
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 64
Advantages of localized orbitals
Two sources of savings• Distant localized orbitals interact weakly (~ R-6): neglect• Correlation orbitals must have nodes dissecting an
occupied orbital. Only the basis functions localized in the neighborhood of the occupied orbital are effective: truncate the correlation space
Left-right correlation Up-down correlation
+-+-
Sept. 9-11, 2004 Local Correlation Methods, Torino, Italy 65
Saebo-Pulay local correlation
• Recent very efficient implementation by H.-J. Werner and his coworkers (M. Schütz, G. Rauhut) in MOLPRO
• Generalization to Coupled Cluster methods (Werner group)
• Other major local correlation methods (still in the formative stage):– Scuseria group (Laplace transform)– Head-Gordon group (diatomics in molecules, triatomics in
molecules)• Still occasional difficulties with localization artifacts• We decided to develop a full accuracy MP2
top related