Accurate Quantum Chemistry for Large · PDF fileAccurate quantum chemistry for large molecules. ... time/s MP2 DF-MP2 Accurate quantum chemistry for large molecules. ... Quantum mechanics

Accurate Quantum Chemistry forLarge Molecules

Fred Manby

Centre for Computational Chemistry, School of Chemistry

University of Bristol

Outline

Three problems in quantum chemistry

Three solutions

Accurate 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

Excited states

Quantum dynamics of nuclei

Solvation

Multireference methods



(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

• Current implementation by Werner and Schutz

• Achieves linear scaling with respect to system size


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)


Density fitting

• First used by Boys and Shavitt for H3 (1959)

• Used by Baerends and Dunlap in density functional theory (1970s)

• Recently used in ab initio methods: Ahlrichs, Feyereisen, Komornicki

FRM, Knowles, Lloyd PRL 87 163001 (2001); JCP 115 9144 (2001)

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

FRM, JCP 119 4607 (2003); Ten-no, FRM, JCP 119 5358 (2003)

Schutz, FRM, PCCP 5 3349 (2003)

Schutz, Werner, Lindh, FRM, JCP 121 737 (2004)

May, FRM, JCP 121 4479 (2004)


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


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

150

CP

U-ti

me

/ min

ute LMP2

DF-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


Other DF-local theories

• DF-HF Polly, Werner, FRM, Knowles, Mol. Phys. in press (2004)

◦ Needed since the DF-LMP2 program was so fast

• DF-LCCSD(T) Schutz and FRM, PCCP 5 3349 (2003)

◦ In progress, but indicates 100-fold improvement in speed over LCCSD(T)

• DF-LMP2 gradients Schutz, Werner, Lindh, FRM, JCP 121 737 (2004)




Steep scaling of effort wrt basis set size — density fitting



The origin of slow convergence

• H only has terms like 1/r12 and −12∇

2

• 1/r12 blows up when electrons coalesce

• HΨ(r12)/Ψ(r12) = E does not

• Cancellation of divergence must be from KE

• −12∇

2Ψ(r12) = − 1

r12for small r12

• −12∇

2r12 = − 1

r12


Slow convergence in the helium atom


Slow convergence

• Orbital expansions are not very good for describing correlation

• Orbitals expanded about nuclei, not about other electrons

• Simple solution: include terms that depend on r12 in the wavefunction


Explicitly correlated theories

Hylleraas’ pioneering calculations on helium — 1929

0 100 200 300 400 500 600 700number of terms

-9

-8

-7

-6

-5

-4lo

g(er

ror)

orbital-based wavefunctionsHylleraas wavefunctions


DF-MP2-R12 theory

• Based on MP2-R12 theory (Kutzelnigg, Klopper et al.)

• Uses density fitting FRM, JCP 119 4607 (2004)

• Uses an optimal correlation factor Andy May, FRM, JCP 121 4479 (2004)

• Local versions under development (FRM and H-J Werner)


Benchmarking explicitly correlated theories

1 CO + SO3 → CO2 + SO2 2 C2H2 + H2O → CH3CHO

3 CO + Cl2 → COCl2 4 furan + H2S → thiophene + H2O

5 CO + CH3OH → HCOOCH3 6 CO + NH3 → HCONH2

7 CO + H2O → CO2 + H2 8 CS2 + 2 H2O → CO2 + 2 H2S

9 H2CCO + HCHO → C2H4O + CO 10 H2O2 + H2 → 2 H2O

11 C2H2 + H2 → C2H4 12 C2H4 + H2 → C2H6

13 HCHO + H2 → CH3OH 14 C2H6 + H2 → 2 CH4

15 CO + H2 → HCHO 16 CH4 + 4 H2O2 → CO2 + 6 H2O

17 NH3 + 4 H2O2 → HNO3 + 5 H2O 18 CO + H2O2 → CO2 + H2O

19 SO2 + H2O2 → SO3 + H2O 20 HNCO + NH3 → NH2CONH2


Benchmarking explicitly correlated theories

0 2 4 6 8 10 12 14 16 18 20Reaction number

-2

-1

0

1

2

3

4E

nerg

y di

ffer

ence

(kca

l/mol

)

AVTZAVQZAVTZ+R12

CCSD(T) correlation energy contributions relative to AVQZ+R12



. . . and three solutions


Steep scaling of effort wrt basis set size — density fitting

Slow convergence wrt basis set size — explicit correlation


Some applications

DNA intercalators:

J Platts (Cardiff)

PHBH enzyme: W Thiel (Mulheim),

R Mata, H-J Werner (Stuttgart)

Gas-phase conformation of enkephalins:

J Hirst (Nottingham)

Chorismate mutase:

Fred Claeyssens, AJM, JNH


Chorismate mutase

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

• Calculations by Fred Claeyssens


Chorismate mutase

-2 -1 0 1 2reaction coordinate / angstrom

-30

-20

-10

0

10

20

30

40en

ergy

/ kc

al m

ol-1

HFMP2CCSD(T)


Future work

• Method development

◦ Correlated alternatives to Hartree-Fock theory

◦ Optimal control, interaction of light and matter (with GGBK)

◦ Quantum mechanics of light nuclei

◦ Combined molecular mechanics and quantum mechanics (with JNH, AJM)

• Applications

◦ High-level quantum chemistry of enzymes (with AJM, JNH)

◦ Quantum chemistry at interfaces (with NLA, JPR)


Conclusions

• Chemical accuracy can now be achieved for large molecules

• New methods combine three key developments

◦ Local description of correlation

◦ Density fitting

◦ Explicit correlation

• Methods available in Molpro quantum chemistry package

◦ Distributed to 300 research groups across the world


Acknowledgements

Hans-Joachim Werner (Stuttgart)

Martin Schutz (Regensburg)

Ricardo Mata (Stuttgart)

Peter Knowles (Cardiff)

Andrew May (Bristol)

Seiichiro Ten-no (Nagoya)

Fred Claeyssens (Bristol)

Jeremy Harvey (Bristol)

Adrian Mulholland (Bristol)


Documents

Accurate Quantum Chemistry for Large · PDF fileAccurate quantum chemistry for large molecules. ... time/s MP2 DF-MP2 Accurate quantum chemistry for large molecules. ... Quantum mechanics