Download - The Effective Fragment Molecular Orbital Method

The Effective Fragment Molecular OrbitalMethod

Casper Steinmann1 Dmitri G. Fedorov2 Jan H. Jensen1

1Department of Chemistry, University of Copenhagen, Denmark

2AIST, Umezono, Tsukuba, Ibaraki, Japan

September 14th, 2011

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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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 polarizableforce-field.

• A merger between FMO and EFP.

• FMO: Faster FMO by the use of classical approximations

• EFP: Flexible EFP’s.

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FMO

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FMO2 MethodUsing a single Slater determinant to represent |ΨX〉, we obtain

fXφXk = ǫXk φX

k .

Here,fX = hX + V X + gX = hX + gX ,

where

V X =

all∑

C∈X

−ZC

|r1 −RC |+

all∑

K 6∈X

∫

ρK(r′)

|r1 − r′|dr′

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FMO2 Methodexpanding our molecular orbitals φ in a basis set

φXk =

∑

µ

CXµkχµ

we obtain, the Fock matrix elements of V X

V Xµν = 〈µ|V X |ν〉 =

all∑

K 6∈X

uKµν +

all∑

K 6∈X

υKµν

which are given as

uKµν = 〈µ|

∑

C∈K

−ZC

|r1 −RC ||ν〉,

andυKµν =

∑

λσ∈K

DKλσ(µν|λσ).

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FMO approximationsFMO2 formally scales as O(N2), wants to be O(N). We needdistance-based approximations:

RI,J = mini∈I,j∈J

|ri − rj |

rvdwi + rvdwj

• 1) Approximate ESP (Resppc):

uKµν =

∑

λσ∈K

DKλσ(µν|λσ) →

∑

C∈K

〈µ|QC

|r1 −RC ||ν〉

• 2) Approximate dimer interaction (Resdim):

EIJ ≈ EI+EJ+Tr(

DIuJ)

+Tr(

DJuI)

+∑∑

DIµνD

Jλσ(µν|λσ)

Usually, Resdim and Resppc are equal (2.0)

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A couple of pictures to help

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Covalent Bonds in FMOIn FMO, bonds are detatched instead of capped.

BDA|-BAA

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HOP vs. AFOTwo methods in FMO: hybrid orbital projection (HOP) and adaptivefrozen orbitals (AFO)

Both modifies the Fock-operator

fX = hX + gX +∑

k

Bk|φ〉〈φ|

• HOP: External model system generated and used

• AFO: Generated on the fly automatically:

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AFO

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AFO

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AFO

We shall return to AFO later ...

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EFP

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EFPEFP is an approximation to the RHF interaction energy, Eint

Eint = ERHF −∑

I

E0I ≈ EEFP

The EFP energy

EEFP =∑

I>J

∆EEFPIJ + Eind

total

∆EEFPIJ = Ees

IJ + (ExrIJ + Ect

IJ )

EesIJ using distributed multipoles.

Eindtotal using induced dipoles based on distributed polarizabilities.

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Eint = ERHF −∑

I

E0I ≈ EEFP

The EFP energy

EEFP =∑

I>J

∆EEFPIJ + Eind

total

∆EEFPIJ = Ees

IJ + (ExrIJ + Ect

IJ )



• The internal geometry is fixed.

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Eint = ERHF −∑

I

E0I ≈ EEFP

The EFP energy

EEFP =∑

I>J

∆EEFPIJ + Eind

total

∆EEFPIJ = Ees

IJ + (ExrIJ + Ect

IJ )



• The internal geometry is fixed.

• You need to construct the EFP’s before you can use them

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EFMO

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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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EFMO RHF Energy

The two-body FMO2 method on a system of N fragments

EEFMO =∑

I

E0I −→ do MAKEFP

+

Resdim≥RI,J∑

IJ

(

E0IJ − E0

I − E0J − Eind

IJ

)

+

Resdim<RI,J∑

IJ

EesIJ

+ Eindtot ,

• QM: Gas phase RHF (and MP2) calculations

• MM: Interaction energies by Effective Fragment Potentials

* obtain E0I , q, µ and Ω and α from RHF via a fake MAKEFP run.

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Correlation in EFMOCorrelation as in FMO

E = EEFMO + ECOR.

Here ECOR is

ECOR =

N∑

I

ECORI +

RI,J<Rcor∑

IJ

(

ECORIJ − ECOR

I − ECORJ

)

.

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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 Eex and Ect 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 EFPparameters.

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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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Rigorous Analysis of Small Water TrimerKitaura-Morokuma energy analysis

∆3Eint 6-31G(d) 6-31++G(d)-1.9 kcal/mol -1.45 kcal/mol

Many-body terms ∆3Eind +∆3Exr +∆3Ect +∆3EMIX

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∆3Eind 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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∆3Eind 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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Rigorous Analysis of Small Water PentamerKitaura-Morokuma energy analysis

∆nEint 6-31G(d) 6-31++G(d)-6.10 kcal/mol -5.14 kcal/mol

Many-body terms ∆nEind +∆nEEX +∆nECT +∆nEMIX

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∆nEind 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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∆nEind 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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Rigorous analysis of Small Water Clusters6-31G(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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20 water molecule clusters

6-31G(d) ∆E [kcal/mol/HB] [kcal/mol] Resdim = 2.0EFMO 0.53 15.8FMO2 -0.39 -11.6

6-31++G(d)EFMO -0.08 -2.5FMO2 -0.76 -21.8

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EFMO Gradient

∂EEFMO

∂xI

=∑

I

∂

∂xI

E0I

+

RI,J≤Rcut∑

I>J

(

∂

∂xI

∆E0IJ −

∂

∂xI

EindIJ

)

+

RI,J>Rcut∑

I>J

∂

∂xI

EesIJ +

∂

∂xI

Eindtotal + TM

xI

TMI is the contribution to the gradient on atom I due to torques

arising from nearby atoms.

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EFMO Gradient• For water clusters, ∇XI

EEFMOa − EEFMO

n ≈ 10−4 Hartree / Bohr

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EFMO Gradient• For water clusters, ∇XI

EEFMOa − EEFMO

n ≈ 10−4 Hartree / Bohr

"Those are not good gradients."

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Wait until you see the covalently bonded systems then.

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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 nearbyelectrostatics.

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Back to the drawing board

C1 C

H

H

H

H

C5

H

H

C C

H

HHH

HH

+

a) b) c)

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Back to the drawing board• 1) The frozen orbital (during the SCF) is allowed to mix during

Foster-Boys localization.

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Foster-Boys localization.• 1) Crap. Errors ≈ 50 kcal/mol. FMO2: ≈ 5 kcal/mol

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• 2) The localized orbital is kept frozen, i.e. as it is during the SCF.

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• 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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Energies for Conformers of Polypeptides

k(R, α, β) = 1− exp(

−√

αβ|R|2)(

1 +√

αβ|R|2)

AM,X =1

N

N∑

I

∣

∣EMI − EX

I

∣

∣ .

0.1 0.2 0.3 0.4 0.5 0.6

2

4

6

8

10

12

AEFMO,X

[kca

l/mol

]

Peptide MAD for EFMO/2 vs. Screening Parameter

P1(RHF)P1(MP2)P2(RHF)P2(MP2)P3(RHF)P3(MP2)

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Energies for Conformers of Polypeptides

k(R, α, β) = 1− exp(

−√

αβ|R|2)(

1 +√

αβ|R|2)

AM,X =1

N

N∑

I

∣

∣EMI − EX

I

∣

∣ .

P1 P2 P30

1

2

3

4

5

6

7

8

AM,X

[kca

l/mol

]

Peptide MAD for 2 Residues per Fragment

FMO2-RHF/HOPFMO2-MP2/HOPFMO2-RHF/AFOFMO2-MP2/AFOEFMO-RHFEFMO-MP2

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Energies for Proteins

Table: Energy Error of EFMO and FMO2/AFO compared to ab initiocalculations 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 6.41UAO 10 1.8 1.5 0.4 1.4

• Timings: 5 times faster than FMO2.

• Requires lots of screening.

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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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Timings

5 10 15 20 25 30 35 40Total CPU count

5

10

15

20

25

30

35

40

Gain

-Fac

tor i

n CP

U w

allti

me

Gain-Factor in CPU walltime for increasing CPU count

EFMO, FMO2 40 / 45

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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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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AcknowledgementsJan H. JensenDmitri G. Fedorov

Bad Boys of Quantum Chemistry:Anders ChristensenMikael W. Ibsen (FragIt)Luca De Vico (FragIt)Kasper Thofte

$$ - Insilico Rational Engineering of Novel Enzymes (IRENE)

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Thank you for your attention

proteinsandwavefunctions.blogspot.com

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Gradient Contribution

JK

I

dvdu

dw

IuvII

II

r

r1

2

dwmA = −dum

A − dvmA

dumA = wA (τmA · vA)+uA×wA (τmA ·wA)

dvmA = −wA (τmA · uA)+vA×wA (τmA ·wA)

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