Fred Manby - University of · PDF fileFred Manby Centre for ... Quantum dynamics of nuclei ... • Approaches MP2 basis set limit amazingly rapidly

Converged quantum chemistry for large systems

Fred Manby

Centre for Computational Chemistry, School of Chemistry

University of Bristol

Seminar — University College London, 15/03/2006

Outline

Three problems in quantum chemistry

Three solutions

Converged quantum chemistry for large molecules

Quantum chemistry

HΨ = EΨ

(One possible) hierarchy of methods

HF −→ MP2 −→ CCSD −→ CCSD(T) −→ CCSDT −→ · · · −→ FCI

(One possible) hierarchy of basis sets

VDZ −→ VTZ −→ VQZ −→ V5Z −→ · · · −→ ∞

Chemical accuracy ∼ a few kJ mol−1


Benchmarking standard methods

• Set of simple reactions (from Helgaker, Jørgensen and Olsen)

CO2 + 4H2 −→ CH4 + 2H2O N2 + 3H2 −→ 2NH3

C2H2 + H2 −→ C2H4 CO + H2 −→ CH2O

CH2O + 2H2 −→ CH4 + H2O F2 + H2 −→ 2HF

HCN + 3H2 −→ CH4 + NH3 O3 + 3H2 −→ 3H2O

C2H2 + 3H2 −→ 2CH4 CH2 + H2 −→ CH4

CO + 3H2 −→ CH4 + H2O 2CH2 −→ C2H4

HNO + 2H2 −→ H2O + NH3

• Plot normal distributions of errors (in kJ mol−1)


Errors for benchmark reaction energies in kJ mol−1

CCSD(T)

−80 80 −80 80 −80 80 −80 80

CCSD

−80 80 −80 80 −80 80 −80 80

MP2

−80 80 −80 80 −80 80 −80 80

HF

−80 80 −80 80 −80 80 −80 80

VDZ VTZ VQZ V5Z



Steep scaling of effort wrt molecular size

Steep scaling of effort wrt basis set size

Slow convergence wrt basis set size

Beyond CCSD(T)

Multireference methods

Excited states

Quantum dynamics of nuclei

Chemistry in solution

. . .



(H2O)n

2 3 4 5 6n

0

500

1000

1500tim

e/s

CCSD(T)CCSDMP2HF



H2O with MP2/cc-pVnZ

VDZ VTZ VQZ V5Z V6Zbasis set

0

50

100tim

e/s


Slow convergence in quantum chemistry

error ∝ [basis size]−1

time ∝ [basis size]4

⇒ error ∝ time−1/4

time0

erro

r10 000-fold improvement in CPU speed gives only one order of magnitude







Canonical and local orbitals

Indinavir



Indinavir — a canonical orbital



Indinavir — a local orbital


Rapid decay of correlation energy

0 5 10 15 20 25 30Interorbital Distance [bohr]

1e-06

0.0001

0.01

1

Incr

emen

tal

Cor

rela

tio

n E

nerg

y [

Har

tree

]

1 kcal/mol

Indinavir

rh

ob

72.

33


Local correlation theories

• Use localized orbitals

• Take advantage of the short-ranged nature of correlation

• Pioneered by Pulay and Saebø

• Linear scaling implementation by Werner and Schutz


LMP2/VDZ on (Gly)n: Werner et al.

2 4 6 8 12 16 20n

0

50

100

150

CP

U-t

ime

/ min

ute

standard MP2local MP2


Local coupled cluster (Schutz and Werner)

Glyn with LCCSD/VDZ and LCCSD(T)/VDZ

0 2 4 6 8 10 12 14 16n

0

5000

10000

15000

20000

25000

30000

35000

40000C

PU ti

me

/ s(T) CCSD Iteration

(~N )

L(T) (~ N)

(~N )7 6

LCCSD Iteration(~N)



Steep scaling of effort wrt molecular size — local methods




Density fitting

All ab initio quantum chemistry needs 2-electron integrals∫d~r1

∫d~r2 ψ

∗p(~r1)ψq(~r1)

1

r12ψ∗r(~r2)ψs(~r2)

|pq) |rs)Idea is to fit these orbital product densities in a set of functions

|pq) =∑A

DpqA |A)


Efficiency of density fitting

H2O with MP2/cc-pVnZ

VDZ VTZ VQZ V5Z V6Zbasis set

0

100

200

300

400

500

600

700tim

e/s MP2

DF-MP2


Impact of density fitting

• Reduces severity of moving to larger basis sets

• Introduces only tiny errors (numbers later)

• Ideal for large basis sets and medium molecular size

• But still has poor scaling with system size


Combining local and density fitting methods

• Use localized orbitals Werner, FRM, Knowles, JCP 118 8149 (2003)

• Perform density fitting

• Fit products of local orbitals in localized fitting expansions

|ia) ≈∑

A near i,a

DiaA |A)


DF-LMP2 performance

Indinavir in cc-pVTZ (2008 bf)

CPU time/second

LMP2 DF-MP2 DF-LMP2

Integrals 25540 2992 2816

Transformation 56620 4795 970

Solve 0 3364 362

Assemble 0 82663 38

Iteration 3772 0 3775

Total MP2 86177 93914 8247

Werner, FRM, Knowles, JCP 118 8149 (2003)


DF-LMP2/VDZ on (Gly)n

2 4 6 8 12 16 20n0

50

100

150C

PU

-tim

e / m

inut

e LMP2DF-MP2DF-LMP2


DF-LMP2 accuracy

Comparison of MP2 and DF-LMP2 reaction energies (kcal/mol)

VTZ VQZ

MP2 DF-LMP2 MP2 DF-LMP2

I −16.28 −16.29 −15.24 −15.24

II −51.50 −51.49 −50.83 −50.79

III −151.58 −151.58 −156.27 −156.27

I HF + 2-butene → 2-fluorobutane

II

III THF + 2 H2O2 → γ-butyrolactone + 3 H2O


An application in enzyme catalysis

PHBH enzyme

Walter Thiel (Mulheim), Ricardo Mata,

Hans-Joachim Werner (Stuttgart)

QM region: 49 atoms

Chorismate mutase

Fred Claeyssens, Adrian Mulholland,

Jeremy Harvey

QM region: 24 atoms


An application in enzyme catalysis

Does transition state theory work for enzymes?

• Protein bulk treated by molecular mechanics (QM/MM)

• Typically DFT or semi-empirical theory used for active site

• Aim to converge the quantum chemistry for these systems

• We have done DF-LMP2/aug-cc-pV5Z and DF-LCCSD(T)/(aug)-cc-pVTZ


Barrier heights

Method CM PHBH

HF 28.3 36.7

B3LYP 10.2 8.4

LMP2 9.5 10.7

LCCSD(T0) 13.1 13.3

Experiment 12.7a 12.0b

14.6c

aKast et al., Tet. Lett. 37 2691 (1996) bvan Berkel et al., Eur. J. Biochem. 179 307 (1989) cOrtiz-

Maldonado et al., Biochemistry 43 1569 (2004) and refs. therein; includes a computed entropic correction




Steep scaling of effort wrt basis set size — density fitting



Slow convergence

• Orbital expansions are not very good for describing correlation

• Orbitals expanded about nuclei, not about other electrons

• Hylleraas in 1929 saw the simple solution:

include terms that depend on interelectronic distances (r12) in the wavefunction


Explicitly correlated theories

• Putting r12 in wavefunction leads to excellent convergence. . .

. . . but also to many-electron integrals

• Resolution of the identity (Kutzelnigg and Klopper)

• Let P =∑

p |p〉〈p| ≈ 1 then

〈ijk|r12r−123 |klm〉 ≈ 〈ijk|r12P2r

−123 |klm〉

=∑p

〈ij|r12|kp〉〈pk|r−112 |lm〉


MP2-R12 theory

• Standard MP2 ansatz

|uij〉 =∑ijab

T ijab|ab〉

• Augment the standard virtual space with explicitly correlated terms

|uij〉 =∑ijab

T ijab|ab〉 +∑ijkl

tijklQ12r12|kl〉

◦ operator Q12 a technical detail

• All many-electron integrals computed by resolution of identity


Improving on MP2-R12

• Replace r12 with f12, a short-range correlation factor (Ten-no; Manby; Klopper)

• Use density fitting FRM, JCP 119 4607 (2003)

• Make local approximations Werner, FRM, JCP, 2005–06

Gives the snappily named DF-LMP2-F12/2*A(loc) method

• Approaches MP2 basis set limit amazingly rapidly

• Errors in MP2 correlation energies typically an order of magnitude smaller than

Hartree-Fock errors


Performance of DF-LMP2-F12/2*A(loc)

• MP2-F12 is fast

• HF+MP2-F12 in aug-cc-pVTZ is faster than HF+MP2 in aug-cc-VQZ

• And much more accurate

Test set: H2 CH4 NH3 H2O C2H2 C2H4 C2H6 CO H2CO CH3OH H2O2 H2CCO

C2H4O CH3CHO C2H5OH HNCO HCONH2 CO2 HCOOH NH2CONH2 HCOOCH3


Performance of DF-LMP2-F12/2*A(loc)

2 4 6 8 10 12 14 16 18 20Molecule

-20

0

20

40

60

80

100E

rror

/ m

illih

artr

ee

MP2/AVTZMP2/AVQZMP2/AV5ZMP2-F12/AVTZHF/AVTZ



. . . and three solutions


Steep scaling of effort wrt basis set size — density fitting

Slow convergence wrt basis set size — explicit correlation


Conclusions

• Chemical accuracy can now be achieved for large molecules

• New methods combine three key technologies

◦ Local description of correlation

◦ Density fitting

◦ Explicit correlation

• No large effects beyond TST are at work in CM and PHBH


Acknowledgements

Hans-Joachim Werner (Stuttgart)

Martin Schutz (Regensburg)

Ricardo Mata (Stuttgart)

Peter Knowles (Cardiff)

Andrew May (Bristol)

Ed Valeev (Georgia, USA)

Seiichiro Ten-no (Nagoya)

Fred Claeyssens (Bristol)

Jeremy Harvey (Bristol)

Adrian Mulholland (Bristol)

Walter Thiel (Mulheim)


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Fred Manby - University of · PDF fileFred Manby Centre for ... Quantum dynamics of nuclei ... • Approaches MP2 basis set limit amazingly rapidly