Post on 23-Feb-2018
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
Converged quantum chemistry for large molecules
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)
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
Three problems in quantum chemistry
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
. . .
Converged quantum chemistry for large molecules
Steep scaling of effort wrt molecular size
(H2O)n
2 3 4 5 6n
0
500
1000
1500tim
e/s
CCSD(T)CCSDMP2HF
Converged quantum chemistry for large molecules
Steep scaling of effort wrt basis set size
H2O with MP2/cc-pVnZ
VDZ VTZ VQZ V5Z V6Zbasis set
0
50
100tim
e/s
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
Three problems in quantum chemistry
Steep scaling of effort wrt molecular size
Steep scaling of effort wrt basis set size
Slow convergence wrt basis set size
Converged quantum chemistry for large molecules
Canonical and local orbitals
Indinavir — a canonical orbital
Converged quantum chemistry for large molecules
Canonical and local orbitals
Indinavir — a local orbital
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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)
Converged quantum chemistry for large molecules
Three problems in quantum chemistry
Steep scaling of effort wrt molecular size — local methods
Steep scaling of effort wrt basis set size
Slow convergence wrt basis set size
Converged quantum chemistry for large molecules
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)
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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)
Converged quantum chemistry for large molecules
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)
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
Three problems in quantum chemistry
Steep scaling of effort wrt molecular size — local methods
Steep scaling of effort wrt basis set size — density fitting
Slow convergence wrt basis set size
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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〉
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
Three problems in quantum chemistry
. . . and three solutions
Steep scaling of effort wrt molecular size — local methods
Steep scaling of effort wrt basis set size — density fitting
Slow convergence wrt basis set size — explicit correlation
Converged quantum chemistry for large molecules
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
Converged quantum chemistry for large molecules
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)
Converged quantum chemistry for large molecules