The document describes the Effective Fragment Molecular Orbital (EFMO) method, which combines the Fragment Molecular Orbital (FMO) method with the Effective Fragment Potential (EFP) method. EFMO treats large systems by dividing them into fragments and using quantum mechanics (QM) to calculate the gas phase energies of each fragment and effective fragment potentials (EFP) to describe many-body interactions between fragments. This allows EFMO to achieve a balance between the accuracy of QM methods and the efficiency of forcefield methods for treating large molecular systems.

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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