1. The Effective Fragment Molecular Orbital
Method
Casper Steinmann1 Dmitri G. Fedorov2 Jan H. Jensen1
1
Department of Chemistry, University of Copenhagen, Denmark
2
AIST, Umezono, Tsukuba, Ibaraki, Japan
September 14th, 2011

2. Outline
1 Motivation
2 The Fragment Molecular Orbital Method
3 The Effective Fragment Potential Method
4 The Effective Fragment Molecular Orbital Method
5 Results
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3. Motivation
Treatment of Very Large Systems
• We want quantum mechanics (QM) to do chemistry.
• We want the speed of force-fields (MM) to treat large systems.
Usually done via hybrid QM/MM methods
We propose a fragment based, on-the-fly parameterless polarizable
force-field.
• A merger between FMO and EFP.
• FMO: Faster FMO by the use of classical approximations
• EFP: Flexible EFP’s.
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4. FMO
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5. FMO2 Method
The two-body FMO2 method on a system of N fragments
N N
FMO
E = EI + (EIJ − EI − EJ ),
I I>J
with fragment energies obtained as
ˆ
EX = ΨX |HX |ΨX
where
 
all all
ˆ
HX = − 1 2
i +
−ZC
+
1
+
ρK (r )
dr 
2 |ri − RC | j>i
|ri − rj | |ri − r |
i∈X C K∈X
NR
+ EX
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6. FMO2 Method
Using a single Slater determinant to represent |ΨX , we obtain
ˆ
f X φX = X X
k k φk .
Here,
ˆ ˆ ˆ ˜
f X = hX + V X + g X = hX + g X ,
ˆ ˆ
where
all all
ˆ −ZC ρK (r )
VX = + dr
|r1 − RC | |r1 − r |
C∈X K∈X
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7. FMO2 Method
expanding our molecular orbitals φ in a basis set
φX =
k
X
Cµk χµ
µ
we obtain, the Fock matrix elements of V X
all all
ˆ
Vµν = µ|V X |ν =
X
uK +
µν
K
υµν
K∈X K∈X
which are given as
−ZC
uK = µ|
µν |ν ,
|r1 − RC |
C∈K
and
K K
υµν = Dλσ (µν|λσ).
λσ∈K
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8. FMO approximations
FMO2 formally scales as O(N 2 ), wants to be O(N ). We need
distance-based approximations:
|ri − rj |
RI,J = min vdw vdw
i∈I,j∈J ri + rj
• 1) Approximate ESP (Resppc ):
QC
uK =
µν
K
Dλσ (µν|λσ) → µ| |ν
|r1 − RC |
λσ∈K C∈K
• 2) Approximate dimer interaction (Resdim ):
EIJ ≈ EI +EJ +Tr DI uJ +Tr DJ uI + I J
Dµν Dλσ (µν|λσ)
Usually, Resdim and Resppc are equal (2.0)
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9. A couple of pictures to help
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10. A couple of pictures to help
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11. Covalent Bonds in FMO
In FMO, bonds are detatched instead of capped.
BDA|-BAA
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12. HOP vs. AFO
Two methods in FMO: hybrid orbital projection (HOP) and adaptive
frozen orbitals (AFO)
Both modifies the Fock-operator
ˆ ˜
f X = hX + g X +
ˆ Bk |φ φ|
k
• HOP: External model system generated and used
• AFO: Generated on the fly automatically:
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13. AFO
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14. AFO
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15. AFO
We shall return to AFO later ...
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16. EFP
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17. EFP
EFP is an approximation to the RHF interaction energy, E int
E int = E RHF − EI ≈ E EFP
0
I
The EFP energy
E EFP = EFP ind
∆EIJ + Etotal
I>J
EFP es xr ct
∆EIJ = EIJ + (EIJ + EIJ )
es
EIJ using distributed multipoles.
ind
Etotal using induced dipoles based on distributed polarizabilities.
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18. EFP
EFP is an approximation to the RHF interaction energy, E int
E int = E RHF − EI ≈ E EFP
0
I
The EFP energy
E EFP = EFP ind
∆EIJ + Etotal
I>J
EFP es xr ct
∆EIJ = EIJ + (EIJ + EIJ )
es
EIJ using distributed multipoles.
ind
Etotal using induced dipoles based on distributed polarizabilities.
• The internal geometry is fixed.
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19. EFP
EFP is an approximation to the RHF interaction energy, E int
E int = E RHF − EI ≈ E EFP
0
I
The EFP energy
E EFP = EFP ind
∆EIJ + Etotal
I>J
EFP es xr ct
∆EIJ = EIJ + (EIJ + EIJ )
es
EIJ using distributed multipoles.
ind
Etotal using induced dipoles based on distributed polarizabilities.
• The internal geometry is fixed.
• You need to construct the EFP’s before you can use them
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20. EFMO
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21. What is the EFMO method?
You start with FMO ...
• Remove the ESP
Now you have N gas phase calculations.
Then you mix in some EFP
• Use EFP to describe many-body interactions
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22. EFMO RHF Energy
The two-body FMO2 method on a system of N fragments
E EFMO = 0
EI −→ do MAKEFP
I
Resdim ≥RI,J
0 0 0 ind
+ EIJ − EI − EJ − EIJ
IJ
Resdim <RI,J
es
+ EIJ
IJ
ind
+ Etot ,
• QM: Gas phase RHF (and MP2) calculations
• MM: Interaction energies by Effective Fragment Potentials
0
* obtain EI , q, µ and Ω and α from RHF via a fake MAKEFP run.
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23. A couple of pictures to help
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24. A couple of pictures to help
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25. Correlation in EFMO
Correlation as in FMO
E = E EFMO + E COR .
Here E COR is
N RI,J <Rcor
E COR = COR
EI + COR COR
EIJ − EI COR
− EJ .
I IJ
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26. EFMO vs. EFP vs. FMO
EFMO vs. EFP
• EFMO energy includes internal energy, i.e. total energy can be
obtained.
• Short range interactions are computed using QM.
we assume E ex and E ct are negligible when RI,J > Resdim .
EFMO vs. FMO
• No ESP, i.e. one SCC iteration.
• Many-body interactions are entirely classical.
General EFMO considerations
• Calculation of classical parameters on-the-fly.
• Every EFMO calculation requires re-evaluation of EFP
parameters.
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27. Rigorous Analysis of Small Water Clusters
• Water trimer: Estimate lower bound to energy error
• Water pentamer: Estimate upper bound to energy error
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28. Rigorous Analysis of Small Water Trimer
Kitaura-Morokuma energy analysis
∆3 E int 6-31G(d) 6-31++G(d)
-1.9 kcal/mol -1.45 kcal/mol
Many-body terms ∆3 E ind +∆3 E xr + ∆3 E ct + ∆3 E MIX
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29. Rigorous Analysis of Small Water Trimer
Kitaura-Morokuma energy analysis
∆3 E int 6-31G(d) 6-31++G(d)
-1.9 kcal/mol -1.45 kcal/mol
Many-body terms ∆3 E ind +∆3 E xr + ∆3 E ct + ∆3 E MIX
∆3 E ind 6-31G(d) 6-31++G(d)
-0.9 kcal/mol -1.67 kcal/mol
EFMO error 1.0 kcal/mol -0.22 kcal/mol
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30. Rigorous Analysis of Small Water Trimer
Kitaura-Morokuma energy analysis
∆3 E int 6-31G(d) 6-31++G(d)
-1.9 kcal/mol -1.45 kcal/mol
Many-body terms ∆3 E ind +∆3 E xr + ∆3 E ct + ∆3 E MIX
∆3 E ind 6-31G(d) 6-31++G(d)
-0.9 kcal/mol -1.67 kcal/mol
EFMO error 1.0 kcal/mol -0.22 kcal/mol
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31. Rigorous Analysis of Small Water Trimer
Kitaura-Morokuma energy analysis
∆3 E int 6-31G(d) 6-31++G(d)
-1.9 kcal/mol -1.45 kcal/mol
Many-body terms ∆3 E ind +∆3 E xr + ∆3 E ct + ∆3 E MIX
∆3 E ind 6-31G(d) 6-31++G(d)
-0.9 kcal/mol -1.67 kcal/mol
EFMO error 1.0 kcal/mol -0.22 kcal/mol
6-31G(d):
• Lower bound to the error in energy is ≈ 0.33 kcal/mol/HB
6-31++G(d):
• Lower bound to the error in energy is ≈ -0.07 kcal/mol/HB
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32. Rigorous Analysis of Small Water Pentamer
Kitaura-Morokuma energy analysis
∆n E int 6-31G(d) 6-31++G(d)
-6.10 kcal/mol -5.14 kcal/mol
Many-body terms ∆n E ind +∆n E EX + ∆n E CT + ∆n E MIX
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33. Rigorous Analysis of Small Water Pentamer
Kitaura-Morokuma energy analysis
∆n E int 6-31G(d) 6-31++G(d)
-6.10 kcal/mol -5.14 kcal/mol
Many-body terms ∆n E ind +∆n E EX + ∆n E CT + ∆n E MIX
∆n E ind 6-31G(d) 6-31++G(d)
-1.97 kcal/mol -3.68 kcal/mol
EFMO error -4.13 kcal/mol -1.46 kcal/mol
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34. Rigorous Analysis of Small Water Pentamer
Kitaura-Morokuma energy analysis
∆n E int 6-31G(d) 6-31++G(d)
-6.10 kcal/mol -5.14 kcal/mol
Many-body terms ∆n E ind +∆n E EX + ∆n E CT + ∆n E MIX
∆n E ind 6-31G(d) 6-31++G(d)
-1.97 kcal/mol -3.68 kcal/mol
EFMO error -4.13 kcal/mol -1.46 kcal/mol
6-31G(d):
• Upper bound to the error in energy is ≈ 1.03 kcal/mol/HB
6-31++G(d):
• Upper bound to the error in energy is ≈ 0.37 kcal/mol/HB
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35. Rigorous analysis of Small Water Clusters
6-31G(d):
• Lower bound to the error in energy is ≈ 0.33 kcal/mol/HB
• Upper bound to the error in energy is ≈ 1.03 kcal/mol/HB
6-31++G(d):
• Lower bound to the error in energy is ≈ -0.07 kcal/mol/HB
• Upper bound to the error in energy is ≈ 0.37 kcal/mol/HB
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36. 20 water molecule clusters
6-31G(d) ∆E [kcal/mol/HB] [kcal/mol] Resdim = 2.0
EFMO 0.53 15.8
FMO2 -0.39 -11.6
6-31++G(d)
EFMO -0.08 -2.5
FMO2 -0.76 -21.8
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37. EFMO Gradient
∂E EFMO ∂ 0
= E
∂xI ∂xI I
I
RI,J ≤Rcut
∂ 0 ∂ ind
+ ∆EIJ − E
∂xI ∂xI IJ
I>J
RI,J >Rcut
∂ es ∂ ind M
+ E + E + TxI
∂xI IJ ∂xI total
I>J
TM is the contribution to the gradient on atom I due to torques
I
arising from nearby atoms.
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38. EFMO Gradient
• For water clusters, EFMO
XI Ea
EFMO
− En ≈ 10−4 Hartree / Bohr
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39. EFMO Gradient
• For water clusters, EFMO
XI Ea
EFMO
− En ≈ 10−4 Hartree / Bohr
"Those are not good gradients."
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40. Wait until you see the covalently bonded systems then.
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41. EFMO for Covalent Systems
Covalent systems pose a problem in EFMO because ...
• ... Inherent close (and even overlapping) electrostatics.
• ... Inherent close position of polarizable points and nearby
electrostatics.
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42. Back to the drawing board
a) b) c)
H H H H
H H
C C C5 + C1 C
H H H
H H H
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43. Back to the drawing board
• 1) The frozen orbital (during the SCF) is allowed to mix during
Foster-Boys localization.
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44. Back to the drawing board
• 1) The frozen orbital (during the SCF) is allowed to mix during
Foster-Boys localization.
• 1) Crap. Errors ≈ 50 kcal/mol. FMO2: ≈ 5 kcal/mol
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45. Back to the drawing board
• 1) The frozen orbital (during the SCF) is allowed to mix during
Foster-Boys localization.
• 1) Crap. Errors ≈ 50 kcal/mol. FMO2: ≈ 5 kcal/mol
• 2) The localized orbital is kept frozen, i.e. as it is during the SCF.
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46. Back to the drawing board
• 1) The frozen orbital (during the SCF) is allowed to mix during
Foster-Boys localization.
• 1) Crap. Errors ≈ 50 kcal/mol. FMO2: ≈ 5 kcal/mol
• 2) The localized orbital is kept frozen, i.e. as it is during the SCF.
• 2) Works, Errors grows with system size and are on-par with
FMO2. Requires much more screening.
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47. Energies for Conformers of Polypeptides
k(R, α, β) = 1 − exp − αβ|R|2 1+ αβ|R|2
N
1
AM,X = M X
EI − EI .
N
I
Peptide MAD for EFMO/2 vs. Screening Parameter
12
10
AEFMO,X [kcal/mol]
8
6
P1(RHF)
P1(MP2)
P2(RHF)
4 P2(MP2)
P3(RHF)
P3(MP2)
2
0.1 0.2 0.3 0.4 0.5 0.6
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48. Energies for Conformers of Polypeptides
k(R, α, β) = 1 − exp − αβ|R|2 1+ αβ|R|2
N
1
AM,X = M X
EI − EI .
N
I
Peptide MAD for 2 Residues per Fragment
8
FMO2-RHF/HOP
7 FMO2-MP2/HOP
FMO2-RHF/AFO
6 FMO2-MP2/AFO
EFMO-RHF
5 EFMO-MP2
AM,X [kcal/mol]
4
3
2
1
0 P1 P2 P3
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49. Energies for Proteins
Table: Energy Error of EFMO and FMO2/AFO compared to ab initio
calculations on proteins using two residues per fragment.
Nres EFMO FMO2/AFO
Rcut = 2.0 Rcut = 2.0
RHF MP2 RHF MP2
1L2Y 20 3.2 -4.3 1.7 6.4
1UAO 10 1.8 1.5 0.4 1.4
• Timings: 5 times faster than FMO2.
• Requires lots of screening.
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50. Back to the drawing board
• The backbone is the main problem.
• Errors around 10−3 (10−4 ) Hartree / Bohr
• Timings: 1.5 times faster than FMO2-MP2
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51. Timings
Gain-Factor in CPU walltime for increasing CPU count
40
35
30
Gain-Factor in CPU walltime
25
20
15
10
5
5 10 15 20 25 30 35 40
Total CPU count
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52. Summary
• Successful merger of the FMO and EFP method
• For molecular clusters, it performer pretty good.
• For systems with covalent bonds, work is needed.
• Faster than FMO2, roughly same accuracy.
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53. Outlook
• EFMO-PCM (meeting with Hui Li tomorrow)
• EFMO QM/MM (Based on FMO/FD)
• More EFP, less QM (Spencer Pruitt)
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54. Acknowledgements
Jan H. Jensen
Dmitri G. Fedorov
Bad Boys of Quantum Chemistry:
Anders Christensen
Mikael W. Ibsen (FragIt)
Luca De Vico (FragIt)
Kasper Thofte
$$ - Insilico Rational Engineering of Novel Enzymes (IRENE)
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55. Thank you for your attention
proteinsandwavefunctions.blogspot.com
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56. Gradient Contribution
K dwA = −dum − dvA
m
A
m
m m m
J duA = wA (τA · vA ) + uA × wA (τA · wA )
m m m
r
dvA = −wA (τA · uA )+vA ×wA (τA · wA )
1
r2
duI
dvI
uI
vI
I dwI
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