p. pulay - unito.itp. pulay department of chemistry and biochemistry fulbright college of arts and...

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


Susscrofa

Razorback


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


Motivation and general considerations


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)


An efficient canonical MP2


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


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


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


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


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.


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


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


Parallel MP2 on a Linux cluster(Jon Baker; prelim. work Matt Shirel)


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


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)


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


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


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)


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


Application to π stacking energies


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


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


Potential curve for (C24H12)2a,b

aCounterpoise

corrected

bDistance of sheets in graphite = 3.35 Å


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


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


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)


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


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


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


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.)


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


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


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


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


Full-accuracy local MP2(Svein Saebo)


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


0200400600800

100012001400

gly10 gly15 gly20

NbasisNorbT(SCF)/minAOInt/minT(MP2)/min

SCF: N2.02

MP2:N1.33


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)


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,


∑=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)(||


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


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


-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


-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


An efficient atomic-orbital based MP2 gradient program


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).


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).


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


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


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


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


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


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


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.


Guanine, optimized


Array files (with Amy Apon)


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


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.


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.


The end


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)


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

+-+-


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