2011 10 月 6 日 Selected Chapters Budapest Fall 20111 Major methods The J matrix engine (Ahmadi &...

2011 　 10 　月 6 日 Selected Chapters Budapest Fall 2011 1

Major methods

• The J matrix engine (Ahmadi & Almlöf, 1995; White & Head-Gordon, 1996; Shao &, Head-Gordon, 2000)

• Density fitting (RI) (Dunlap & Rösch, Ahlrichs)

• Numerical solution of the Poisson equation (Delley, Becke, Termath & Handy, Baker, Nemeth & Pulay, unpublished)

• Gaussian fast multipoles and related methods (White, Johnson, Gill, Head-Gordon, 1994; Strain, Scuseria and Frisch, 1996; Challacombe & Schwegler, 1997)

• Pseudospectral methods (Friesner)

• Intermediate Plane Wave expansion (GAPW and the Fourier Transform Coulomb method)


Plane wave basis: widely used in solid-state physics

• The Coulomb operator is diagonal in momentum space:

|r1-r2|-1 = (1/22)dk k-2 exp[ik( r1 - r2)]

• Include the molecule in a sufficiently large box (for simplicity, it is treated as a cube with sides L)

• Plane wave basis exp(iagr); g integer |g|< gmax; a=2/L

• Introduce a real-space grid with spacing h = L/2gmax

• Functions which can be represented by the plane wave basis are exactly determined by the real-space grid values

• Calculate each quantity in the appropriate (momentum or real space) representation, using fast Fourier transformation to switch between representations


Plane wave vs. Gaussians basis sets

• Plane wave basis sets have important advantages: flexibility, efficient calculation of forces (Hellmann-Feynman), extensible to crystals, no BSSE, no near-singularities due to overcompleteness

• They also have problems: periodic ghost images, divergences for charged systems, cannot describe compact orbitals, very large basis sets are needed poor scaling, exact exchange (for hybrid DFT) is extremely costly


The Fourier Transform Coulomb method: The best of both worlds


Calculating the Coulomb operator in a Gaussian basis by plane-wave expansion

ADVANTAGES vs. PURE PLANE WAVES

• No new model chemistry: strictly comparable to existing Gaussian-based codes

• Conventional (pseudo)diagonalization is applicable

• Can be combined with the conventional calculation of the exchange operator - hybrid DFT is possible

DISADVANTAGES

• The extra flexibility of the plane wave basis is lost

• Calculation of the forces is more expensive

• BSSE reappears

• For large diffuse basis sets, Gaussians are ill-conditioned


The GAPW method of Parrinello et al.

The application of plane wave expansion for the solution of the Coulomb problem in Gaussian basis was pioneered by the Parrinello group:

• G. Lippert, J. Hutter and M. Parinello, Theor. Chem. Acc. 103 (1999) 124. (Pseudopotentials; Blöchl’s PAW)

• M. Krack and M. Parrinello, Phys. Chem. Chem. Phys. 2 (2000) 2105. (All electron: for the first time comparison of absolute energies with Gaussian results)

• Our motivations and goals are similar and both are based on plane waves. However, the methods are very different technically. In particular, we aim at much higher precision


Calculation of the Coulomb operator in plane wave basis

Let us assume first that the whole density can be represented by plane waves

• Calculate the density (r) on the real space rectangular grid (asymptotically O(N))

• Fast Fourier Transform to k space: (k) O(NlogN)

• Calculate the potential V(k)=C(k)/k2 O(N)

• Reverse FFT yields V(r): the Coulomb potential on the grid

• Add VXC(r) (currently done separately)

• Calculate the Coulomb matrix elements J by quadrature (exact if the basis functions can be represented on the PW basis), asymptotically O(N)


Difficulties I: Ghost Images


Eliminating ghost images• Periodic ghost images use a truncated Coulomb operator

• f(r) = 1/|r| if r<D, = 0 if r>D (D=diameter of the molecule)

• analytical Fourier transform (4/k2)(1-cos(kD))

• contrary to accepted wisdom, its cost is negligibl

• Can be generalized to 1, 2, 3-dimensional periodic systems

ghostmolecule

aa+D

L extended

L+D 2L

ghost...


Generalization to systems periodic in 1, 2 or 3 dimensions

• This concept can be generalized to polymers, layers or crystals by using an appropriately modified form of the Coulomb operator.

• For 1-D: f(r)=1/r if x2+y2<(D/2)2, 0 otherwise• For 2-D: f(r)=1/r if |z|<D/2, 0 otherwise• For 3-D” f(r)=1/r• Exact Fourier transforms:

– {1+D[kJ1(Dk )K0(Dk║) - k║J0(Dk )K1(Dk║)]}4k-2

– {1-sign(k )exp(-kD/2)} 4k-2

– 4k-2

(J0, J1: Bessel functions of the first kind; K0,K1: of the 2nd kind)


Treatment of compact orbitals

• They are necessary also for the inner valence shells (O,F…)

• At first we have retained the usual Gaussian Electron Repulsion Integral (ERI) evaluation for integrals which cannot be calculated in plane wave basis

• Some integrals involving compact charge distributions can be calculated by a plane-wave expansion. E.g. contributions arising from (c|d) [c=compact, d=diffuse charge distribution (AO product)] can be evaluated because the Coulomb operator is diagonal in k-space. However, the subsequent numerical quadrature requires that |c> is subjected to a low-pass filter operation.

• Currently: multipole approximation for (c|c’) contributions


Characteristics of the FTC method

• Results are essentially exact with a sufficient PW basis

• Asymptotic scaling with mol. size (at constant basis quality) is Nlog N

• Scaling with the basis set size at constant mol. size is O(N2)

• The onset point is favorable for moderate (6-31G*) and larger basis sets; already for a small molecule like aspirin and the 6-31G** basis, the calculation is faster than the conventional one

• Without the multipoles no savings for the 3-21G basis in systems with ~100 atoms

• Exact exchange is calculated by Electron Repulsion Integrals


Accuracy comparison with GAPW (error relative to accurate Gaussian calculations for small molecules)

Molecule GAPW/Eh* FTC/ Eh

H2 -50 0.00

H2O -220 0.00

CO2 -730 0.01

SF6 2670 0.57

*M. Krack and M. Parrinello, Phys. Chem. Chem. Phys. 2000, 2, 2105


Examples


AspirinC9H8O4


Timings for aspirin (B-LYP)a

Basis set 6-31G(d,p)-dc*

(NBF=295)6-311G(2df,2pd)-dc*

(NBF=555)

FTC ERI FTC ERI

Coulomb 4.09 8.47 9.54 116.9

XC 4.32 5.36 17.51 17.11

Matrix 0.10 0.07 0.43 0.36

Total 8.51 14.08 27.53 134.92

Energy -648.526478 -648.526478 -648.719612 -648.719612

a In minutes on a 2.4 GHz Pentium 4 *Slightly decontracted


Sucrose, C12H22O11


Timings for sucrose (B-LYP)a

Basis set 6-31G(d,p)-dc(NBF=569)

6-311G(2df,2pd)-dc (NBF=1080)

FTC ERI FTC ERI

Coulomb 9.09 36.34 28.2 483.4

XC 12.48 14.23 71.9 69.0

Matrix 0.48 0.36 3.2 2.0

Total 22.1 51.1 103.5 554.8

Energy -1297.606133 -1297.606137 -1298.000428 -1298.000425

a In minutes on a 2.4 GHz Pentium 4


Taxol C47H51NO14


Timings for taxol (B-LYP)a

Basis set 6-31G(d,p)-dc(NBF=1484)

6-311G(2df,2pd)-dc (NBF=2860)

FTC ERI FTC ERI

Coulomb 60.3 516.6 197.3 6694

XC 102.3 117.3 470.3 501

Matrix 8.5 7.3 30.3 41

Total 172.5 642.8 722.6 7240

Energy+ 2928

-0.615109 -0.615114 -0.442403 -0.442390

a In minutes on a 2.4 GHz Pentium 4


Scaling with respect to molecular size(alanine)n, n=1-15, 6-311G(2df,2pd)

0100200300400500600700

300 540 1260 2460 3660

Number of basis functions

Tim

e in

min

ute

s/it

er

FTC PW

FTC ERI

Full ERI

627 min

30.2 min

5.8 min


Scaling with respect to molecular size - (alanine)n, n=2-15 log-log plot

-1

0

1

2

3

2.73 3.1 3.39 3.56

log Nbasis

log

tim

e/m

in

FTC PW

FTC ERI

Full ERI


Scaling with respect to basis set size - (alanine)5,

0

100

200

300

400

500

41

9

57

5

65

6

76

1

97

2

12

89

15

00

Number of basis functions

Tim

e p

er

ite

rati

on

(m

in)

FTC PW

FTC ERI

Full ERI

*1010


Average computational costs with and without multipole expansion in FTC

6-311G(2df,2pd)

min/SCF iteration

Alanine5 Alanine10 Alanine15

No. of contracted basis functions

1260 2460 3660

FTC plane wave + multipoles

1.22 3.01 5.18

FTC (analytical ERI part)

2.42 10.12 23.82

Exchange-correlation 6.44 18.31 32.89

Diagonalization+DIIS

0.35 2.45 8.15

FTC total energy(deviation)

-1313.042279 (0.079 µEh)

-2549.614340(0.062 µEh)

-3786.171392 (0.066 µEh)


Conclusions

• The FTC method, i.e. the calculation of the Coulomb operator by plane-wave expansion of the valence AOs has significant advantages, particularly for large basis sets:– Accurate

– Low scaling with molecular size

– Low scaling with basis set size

– Early onset

• Main projected applications: DFT calculations – on large clusters

– of NMR chemical shifts

– direct molecular dynamics

A new project: Genuine Mixed Gaussian and Plane Wave Basis Sets

Both the Basis Set Overcompleteness (BSO) and the Basis Set Superposition Error (BSSE) are caused by diffuse functions

Diffuse functions are essential for intermolecular (van der Waals) forces, for negative ions and negatively charged fragments, for excited states, polarizabilities, hyperpolarizabilities…

Replace the diffuse basis functions (and only these) by a plane wave basis is a box

The number of plane waves remains manageable. The truncated Coulomb operator trick eliminates periodic ghost images. However: new kinds of integrals are needed


Problems with atomic basis sets 1• Small and medium-sized atomic basis sets are probably the

best choice for routine molecular calculations. Large basis sets, particularly if augmented with diffuse functions, have serious problems in larger 2-D and 3-D systems– In principle, AO expansions around one center are complete if

all states (including continuum states) are included. Therefore, a full AO expansion around all centers is overcomplete: it contains redundant functions

– For finite basis sets, the basis becomes nearly linearly dependent. There is a linear combination of basis functions with normalized coefficients (i |cik|2 = 1) which has a very low norm in physical space (ij cik

*Sij cjk = k << 1); Sij = <i |j >

This leads to large MO coefficients (up to 102 – 103).


Problems with atomic basis sets 2• Explicitly (in wavefunction-based correlation theories), or

implicitly (in SCF), we always transform to MOs: this transformation becomes numerically unstable in the presence of large, cancelling MO coefficients. This is always the case for diffuse basis functions and larger systems which became accessible recently. Particularly bad for wavefunction-based correlation theories (virtuals!).

• Possible remedies:– Eliminate linear combinations of basis functions which

belong to low eigenvalues of the overlap (Gram) matrix. This may cause discontinuities on potential energy surfaces and does not help in really serious cases. Penalty function?

– Use an orthogonal basis set


An exampleConsider a fragment of a crystal lattice with 1 Å lattice constant and

diffuse Gaussians (=0.036). Overlap of neighbors S=0.93774

Number of atoms (k3) 8 27 64

Lowest eigenvalue of S 2.41E-4 1.45E-7 2.02E-10

min(1-S)2k asymptotically

Even in a 1-D ring, if only nearest neighbors overlap, the overlap matrix is singular for even n if S0.5.

Most atomic basis sets, and all diffuse basis sets become singular in the limit, and have very low eigenvalues for larger systems.

Overlap of two 1s Slater functions (=1.24) is 0.66 at R=0.74 Å


Problems with atomic basis sets 3

Another ubiquitous problem with atomic basis sets is the basis set superposition error (BSSE). Particularly significant for weak intermolecular interactions but also present within the molecule and is nearly impossible to eliminate.

BSSE: Artificial attraction between molecular fragments due to the improved description of the orbitals as two fragments approach each other

BSSE is usually corrected for by the Boys-Bernardi Counterpoise correction (CP). There are strong theoretical arguments for the validity of the original CP methodology (J. van Lenthe, van Duijneveldt). However, in practice CP overcorrects.



The 3a1 MO in water along a line passing through the symmetry axis. Red= Gaussian basis, Green=small PW

How many plane waves are needed?


• To replace diffuse AOs, a plane wave density of about 0.4-1 PW/Å is needed. Not economical for small systems because even a hydrogen atom needs a sizable box. However, the box size does not grow proportionately with the molecular size

• We don’t have to be as accurate as in FTC where the plane waves are used to evaluate integrals over Gaussian orbitals. Approximating orbitals means changing the Hamiltonian the error is first-order. Changing a (nearly optimum) basis function introduces only a second-order error in the energy.

A diffuse Gaussian and its PW simulation


Gaussian with exponent 0.036(recommended for H diffuseS orbital by Radom, P. Schleyer)Largest PW density 0.2 PW/a0

distance/a0

Using PWs as basis functions is much less demanding than reproducing a Gaussian


Largest PW density0.133 PW/a0

Largest PW density0.167 PW/a0 (0.3/Å

Previous work: Scattering

Integrals for electron-molecule scattering cross sections– McWeeny, Acta Cryst. 1953; Miller and Krauss, JCP 1967,

Stewart, JCP 1969; Liu, JCP, 1973; Ostlund CPL 1975; Pulay et al. JCP 1983; Kohl and Pulay, J. Mol. Struct. 1984 (elastic)

• Polašek and Čarsky Int. J. Quantum Chem. 2007, J. Phys. B. 2010 (Accuracy is OK for scattering, not sufficient for electronic structure; Kuchitsu, Okuda, Tachikawa Int. J. Quantum Chem. 2009 (exchange included)

• More general integrals: Čarsky and Polašek, J. Comput. Phys. 1998. The accuracy of the routine used is sufficient for scattering but unfortunately not sufficient for molecular calculations


PreviousWork - GIAOsGauge-Including Atomic Orbitals (GIAOs)

Polynomial(x,y,z) exp[-a(r – A)2] exp[(-i/2c) BAr]

• Usually, only the derivatives of the GIAOs with respect to the magnetic induction B are needed. In this case, no complex arithmetic is necessary. Ditchfield, Mol. Phys.1974

• In our case, the full complex integrals are needed. GIAO Integrals: Ishida JCP 2003. Useful for very strong magnetic fields (cannot be created in the lab)

• Also H. Fukui, 1998, 1999

• Genuine mixed basis proposal: Colle, Fortunelli, Simonucci, Nuovo Cimento 1987, 1988 (no continuation)

• Problem: probably divergent terms in the expansion


Integral typesThe product of a Gaussian and a plane wave gives a Gaussian

with complex center coordinates (G. G. Hall; also Gauge-Including Atomic Orbitals needed for magnetic properties)

Integral types (g = Gaussian, w = plane wave):

(gg|gg) : use traditional Gaussian techniques (Rys polynomial, Obara-Saiko, McMurchie-Davidson etc.) for complex coord

(gw|g’g”), (gw|g’w’): traditional integral methods extended to complex origins/GIAOs (we use the Rys polynomial technique of H. King and M. Dupuis). Need complex Fm(t).

(gg’|w’w”),(gw|w’w”): expand |g> in a plane wave basis. Compact functions are OK: Only components of |g> with |k|<|k’- k”| are needed. Use the truncated Coulomb operator. Same for <ww’|w”w’”>: only diagonal terms


Integral evaluation: Rys quadrature

2011 　 10 　月6 日

Selected Chapters Budapest Fall 2011 41

H. F. King, M. Dupuis, J. Rys, 1976-

The Gaussian transform of the inverse distance factorizes

The product of 2 Gaussians or a PW and a Gaussian is a Gaussian (possibly with complex center coordinates) which factorizes to x,y,z factors:

P, Q are Gaussians (exponents p,q) with polynomial factors in x,y,z; Ix, Iy, Iz are 2-D integrals (simple but numerous)

Change of variable: t2 = u2/(+u2), = pq/(p+q)

])(exp[])(exp[])(exp[/2

])(exp[/2||

221

2221

222

0

12

22

0

121

211

12

zzuyyuxxudu

duur

rrrr

zyx IIIQruP ]|)exp(|[ 2

122


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) 83

Quadratic memory; only one permutation used (integrals calculated 4x)

do p

do r

calculate all (pq|rs)=Xqs

Y=CTXC (matrix mult.)

store Yij=(pi|rj) on disk

end do r

end 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=3.1610-9 -4.516233 -6.202953

T1=10-9 -4.516178 -6.202887

T1=3.1610-10 -4.516177 -6.202881

Hexapep=N-formyl pentalanine amide

For well-conditioned basis sets, the default threshold guarantees accuracy to a few Eh.


Timings for calix[4]arenea,b, C60a and C74(C1)c

Formula C32H32O4 C32H32O4 C60 C74 Symmetry C2h C2h D2h C1

Basis cc-pVDZ cc-pVTZ cc-pVTZ 6-31G(d) N 664 1528 1800 1036 2n 184 184 240 296 V 536 1400 1620 814 P (%) 24.8 14.0 30.8 23.6 Tint/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 4480.8 31644.0 616.3c

Disk use/GB 3.8 11.6 11.6 <120 E(SCF)/Eh -1529.889518 -1530.271932 -1530.271932 -2801.946722 E(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


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


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 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 is difficult to measure (heat of vaporization would give a clue)

• 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 6

Rz Displacement (Å)

Inte

rac

tio

n E

ne

rgy

(k

ca

l/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 -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.)


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)


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 correlationUp-down correlation

+-+-


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


0200400600

800100012001400

gly10 gly15 gly20

Nbasis

Norb

T(SCF)/min

AOInt/min

T(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)/(a + b -i - j)

Invoke the multipole expansion but not for approximating integrals: 1/r12 = R-3[r1r2 -3R-2(r1R)(r2R) + …]

Substitute into the pair wavefunction ij = a,b Tijabij

ab

If the energy denominator is assumed constant then, in the dipole approximation, three virtual orbitals on each center,


a

av ari || bb

w brj ||

are sufficient to account for all correlation. ,=x,y,z; r is the position vector relative to the centroid of i in the direction, r=-Ri ; r is analogous for j

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

a

aav ari 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 orbital basis, 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 hexane:

Eigenvalues of S1/2T†STS1/2 for pair 13 4 0.000000259 0.000000063 0.000000045 0.000000008 0.000000002 0.000000001 0.000000001 0.000000001 0.000000001 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 Sum of all eigenvalues 0.000000385 Sum of the 3 largest eigenvalues 0.000000368 Sum of the 8 largest eigenvalues 0.000000381


-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 gradients J. Chem. Phys. 120, 11423 (2004).

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

Normalized to 4 (ab) or to 2 (a=b), and pairwise non-orthogonal,

babaij

abij 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

ji

ijab010 T

and the integrals, e.g. the exchange and Coulomb integrals

)|( );|( abijbjai ijab

ijab JK

are collected in matrices. It is useful to introduce the contravariant amplitude matrices (Tji is the transpose of Tij)

)2)(2(~ jiij

ijij TTT 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

ij

ji

ij010 T


SCEP, cont.

For canonical occupied orbitals this reduces to

In a canonical MO basis for virtual orbitals (S=1; F=diag{}) it becomes the usual canonical formula

0SSTFSTSFTKR ijji

ijijijij )(

0)()|( ijabjibabjai T

1))(|( jibaijab bjai T


Orbital-invariant form of MP2

It can be derived in several ways. For gradient evaluation the Hylleraas form is particularly useful:

Ec = 21|H-E0|0 - 1|H0-E0|1 = 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

Molecule Formula Basis Nbf SCF MP2 Grad

-Pinene C10H16 6-311G(df,p) 346 39.3 36.7 177.7

Retinal Schiff b. C12H18N+ 6-31G** 285 5.7 5.9 44.5

Retinal Schiff b. C12H18N+ 6-311+G(2df,2pd) 694 188 145 1135

Caffeine, Cs C8H10N4O2 6-31G* 230 5.6 5.8 47.4*

Caffeine , Cs C8H10N4O2 6-311G* 282 10.3 10.3 78.4*

Caffeine , Cs C8H10N4O2 cc-pVTZ 560 118 122 665*

Sucrose C12H22O11 6-31G* 389 21.5 36.4 167.9

Glycine10C20H32N10O11 6-311G(d,p) 930 505 2105

*No symmetry


Cavitand with 2-oxo-1-butanol*

C32H32O10, 6-31G*, 652 BF

A single 2.8 GHz P 4 Xeon

processor (parallelization in progress).

Elapsed times:SCF = 99.6 minMP2* = 309.0 minMP2 gradient = 883.5 min* incl. storing the amplitudes

*Courtesy Dr. Re Suyong


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


Summary

• Canonical MP2 is possible for large systems, by using natural localization

• Parallel implementation can handle >2000 basis functions: 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


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


The end

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