1. The Effective Fragment Molecular Orbital MethodCasper Steinmann1 Dmitri G. Fedorov2 Jan H. Jensen11Department of Chemistry, University of Copenhagen, Denmark2AIST, Umezono, Tsukuba, Ibaraki, JapanSeptember 14th, 2011

2. Outline1 Motivation2 The Fragment Molecular Orbital Method3 The Effective Fragment Potential Method4 The Effective Fragment Molecular Orbital Method5 Results2 / 45 3. MotivationTreatment of Very Large Systems We want quantum mechanics (QM) to do chemistry. We want the speed of force-elds (MM) to treat large systems.Usually done via hybrid QM/MM methodsWe propose a fragment based, on-the-y parameterless polarizableforce-eld. A merger between FMO and EFP. FMO: Faster FMO by the use of classical approximations EFP: Flexible EFPs.3 / 45 4. FMO4 / 45 5. FMO2 MethodThe two-body FMO2 method on a system of N fragmentsNNFMOE = EI + (EIJ EI EJ ),II>Jwith fragment energies obtained as EX = X |HX |Xwhereall allHX = 12i + ZC + 1+ K (r )dr 2|ri RC | j>i |ri rj | |ri r |iX C KX NR +EX 5 / 45 6. FMO2 MethodUsing a single Slater determinant to represent |X , we obtainf X X = X X k k k .Here, f X = hX + V X + g X = hX + g X ,whereall all ZC K (r ) VX =+dr|r1 RC ||r1 r |CXKX 6 / 45 7. FMO2 Methodexpanding our molecular orbitals in a basis setX = kX Ck we obtain, the Fock matrix elements of V Xallall V = |V X | =XuK + K KX KXwhich are given as ZC uK = | | ,|r1 RC | CKand KK =D (|).K 7 / 45 8. FMO approximationsFMO2 formally scales as O(N 2 ), wants to be O(N ). We needdistance-based approximations: |ri rj |RI,J = min vdwvdwiI,jJ ri + rj 1) Approximate ESP (Resppc ):QC uK =K D (|) |||r1 RC |K CK 2) Approximate dimer interaction (Resdim ):EIJ EI +EJ +Tr DI uJ +Tr DJ uI +I J D D (|)Usually, Resdim and Resppc are equal (2.0) 8 / 45 9. A couple of pictures to help 9 / 45 10. A couple of pictures to help 10 / 45 11. Covalent Bonds in FMOIn FMO, bonds are detatched instead of capped. BDA|-BAA 11 / 45 12. HOP vs. AFOTwo methods in FMO: hybrid orbital projection (HOP) and adaptivefrozen orbitals (AFO)Both modies the Fock-operator f X = hX + g X + Bk | |k HOP: External model system generated and used AFO: Generated on the y automatically: 12 / 45 13. AFO13 / 45 14. AFO14 / 45 15. AFOWe shall return to AFO later ... 15 / 45 16. EFP16 / 45 17. EFPEFP is an approximation to the RHF interaction energy, E intE int = E RHF EI E EFP0 IThe EFP energy E EFP = EFP ind EIJ + Etotal I>JEFP es xrctEIJ = EIJ + (EIJ + EIJ ) esEIJ using distributed multipoles. indEtotal using induced dipoles based on distributed polarizabilities.17 / 45 18. EFPEFP is an approximation to the RHF interaction energy, E intE int = E RHF EI E EFP0 IThe EFP energy E EFP = EFP ind EIJ + Etotal I>JEFP es xrctEIJ = EIJ + (EIJ + EIJ ) esEIJ using distributed multipoles. indEtotal using induced dipoles based on distributed polarizabilities. The internal geometry is xed.17 / 45 19. EFPEFP is an approximation to the RHF interaction energy, E intE int = E RHF EI E EFP0 IThe EFP energy E EFP = EFP ind EIJ + Etotal I>JEFP es xrctEIJ = EIJ + (EIJ + EIJ ) esEIJ using distributed multipoles. indEtotal using induced dipoles based on distributed polarizabilities. The internal geometry is xed. You need to construct the EFPs before you can use them17 / 45 20. EFMO 18 / 45 21. What is the EFMO method?You start with FMO ... Remove the ESPNow you have N gas phase calculations.Then you mix in some EFP Use EFP to describe many-body interactions 19 / 45 22. EFMO RHF EnergyThe two-body FMO2 method on a system of N fragments E EFMO =0EI do MAKEFPI Resdim RI,J 0 00ind +EIJ EI EJ EIJIJ Resdim 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-y. Every EFMO calculation requires re-evaluation of EFPparameters. 24 / 45 27. Rigorous Analysis of Small Water Clusters Water trimer: Estimate lower bound to energy error Water pentamer: Estimate upper bound to energy error25 / 45 28. Rigorous Analysis of Small Water TrimerKitaura-Morokuma energy analysis3 E int 6-31G(d)6-31++G(d) -1.9 kcal/mol -1.45 kcal/molMany-body terms 3 E ind +3 E xr + 3 E ct + 3 E MIX26 / 45 29. Rigorous Analysis of Small Water TrimerKitaura-Morokuma energy analysis3 E int 6-31G(d)6-31++G(d) -1.9 kcal/mol -1.45 kcal/molMany-body terms 3 E ind +3 E xr + 3 E ct + 3 E MIX3 E ind 6-31G(d)6-31++G(d) -0.9 kcal/mol -1.67 kcal/molEFMO error1.0 kcal/mol -0.22 kcal/mol26 / 45 30. Rigorous Analysis of Small Water TrimerKitaura-Morokuma energy analysis3 E int 6-31G(d)6-31++G(d) -1.9 kcal/mol -1.45 kcal/molMany-body terms 3 E ind +3 E xr + 3 E ct + 3 E MIX3 E ind 6-31G(d)6-31++G(d) -0.9 kcal/mol -1.67 kcal/molEFMO error1.0 kcal/mol -0.22 kcal/mol26 / 45 31. Rigorous Analysis of Small Water TrimerKitaura-Morokuma energy analysis3 E int 6-31G(d)6-31++G(d) -1.9 kcal/mol -1.45 kcal/molMany-body terms 3 E ind +3 E xr + 3 E ct + 3 E MIX3 E ind 6-31G(d)6-31++G(d) -0.9 kcal/mol -1.67 kcal/molEFMO error1.0 kcal/mol -0.22 kcal/mol6-31G(d): Lower bound to the error in energy is 0.33 kcal/mol/HB6-31++G(d): Lower bound to the error in energy is -0.07 kcal/mol/HB26 / 45 32. Rigorous Analysis of Small Water PentamerKitaura-Morokuma energy analysis n E int6-31G(d)6-31++G(d)-6.10 kcal/mol -5.14 kcal/molMany-body terms n E ind +n E EX + n E CT + n E MIX27 / 45 33. Rigorous Analysis of Small Water PentamerKitaura-Morokuma energy analysis n E int6-31G(d)6-31++G(d)-6.10 kcal/mol -5.14 kcal/molMany-body terms n E ind +n E EX + n E CT + n E MIX n E ind6-31G(d)6-31++G(d)-1.97 kcal/mol -3.68 kcal/mol EFMO error -4.13 kcal/mol -1.46 kcal/mol27 / 45 34. Rigorous Analysis of Small Water PentamerKitaura-Morokuma energy analysisn E int6-31G(d)6-31++G(d) -6.10 kcal/mol -5.14 kcal/molMany-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/mol6-31G(d): Upper bound to the error in energy is 1.03 kcal/mol/HB6-31++G(d): Upper bound to the error in energy is 0.37 kcal/mol/HB 27 / 45 35. 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/HB6-31++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/HB28 / 45 36. 20 water molecule clusters6-31G(d)E [kcal/mol/HB] [kcal/mol] Resdim = 2.0 EFMO0.53 15.8 FMO2-0.39 -11.66-31++G(d) EFMO-0.08-2.5 FMO2-0.76 -21.8 29 / 45 37. EFMO GradientE EFMO 0=ExI xI I IRI,J Rcut 0 ind+ EIJ ExIxI IJ I>JRI,J >Rcut es ind M+ E +E+ TxI xI IJ xI total I>JTM is the contribution to the gradient on atom I due to torquesIarising from nearby atoms.30 / 45 38. EFMO Gradient For water clusters, EFMO XI Ea EFMO En 104 Hartree / Bohr31 / 45 39. EFMO Gradient For water clusters, EFMO XI Ea EFMO En 104 Hartree / Bohr "Those are not good gradients."31 / 45 40. Wait until you see the covalently bonded systems then. 32 / 45 41. EFMO for Covalent SystemsCovalent systems pose a problem in EFMO because ... ... Inherent close (and even overlapping) electrostatics. ... Inherent close position of polarizable points and nearby electrostatics.33 / 45 42. Back to the drawing boarda)b) c)HH HHH H C C C5 + C1CHHH H HH34 / 45 43. Back to the drawing board 1) The frozen orbital (during the SCF) is allowed to mix during Foster-Boys localization. 35 / 45 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 35 / 45 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.35 / 45 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 withFMO2. Requires much more screening.35 / 45 47. Energies for Conformers of Polypeptidesk(R, , ) = 1 exp |R|21+ |R|2N1AM,X = MXEI EI .NIPeptide MAD for EFMO/2 vs. Screening Parameter1210 AEFMO,X [kcal/mol]86P1(RHF)P1(MP2)P2(RHF)4 P2(MP2)P3(RHF)P3(MP2)20.1 0.2 0.30.40.5 0.636 / 45 48. Energies for Conformers of Polypeptidesk(R, , ) = 1 exp |R|2 1+ |R|2 N1 AM,X = MX EI EI .NI 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 37 / 45 49. Energies for ProteinsTable: Energy Error of EFMO and FMO2/AFO compared to ab initiocalculations on proteins using two residues per fragment.NresEFMO FMO2/AFORcut = 2.0 Rcut = 2.0 RHF MP2 RHFMP2 1L2Y203.2-4.31.76.4 1UAO101.8 1.50.41.4 Timings: 5 times faster than FMO2. Requires lots of screening. 38 / 45 50. Back to the drawing board The backbone is the main problem. Errors around 103 (104 ) Hartree / Bohr Timings: 1.5 times faster than FMO2-MP2 39 / 45 51. TimingsGain-Factor in CPU walltime for increasing CPU count 40 35 30Gain-Factor in CPU walltime 25 2015105 51015 2025 30 3540 Total CPU count40 / 45 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. 41 / 45 53. Outlook EFMO-PCM (meeting with Hui Li tomorrow) EFMO QM/MM (Based on FMO/FD) More EFP, less QM (Spencer Pruitt) 42 / 45 54. AcknowledgementsJan H. JensenDmitri G. FedorovBad Boys of Quantum Chemistry:Anders ChristensenMikael W. Ibsen (FragIt)Luca De Vico (FragIt)Kasper Thofte$$ - Insilico Rational Engineering of Novel Enzymes (IRENE)43 / 45 55. Thank you for your attentionproteinsandwavefunctions.blogspot.com44 / 45 56. Gradient Contribution KdwA = dum dvAm A m mmmJduA = wA (A vA ) + uA wA (A wA ) m m mr dvA = wA (A uA )+vA wA (A wA )1r2duIdvIuI vII dwI45 / 45