Constructing a rigorous fluctuating-
charge model for molecular mechanics
             -           -
                      ...
Molecular mechanics is useful



           water flow in aquaporins1                     mechanical deformation in ceramic...
Molecular mechanics
• Classical energy function with bonded and
  nonbonded terms




                                Van ...
MM leaves out something
                  time 0                time
• Ab initio molecular dynamics (MD)
  nuclear forces ...
QEq1, a fluctuating charge model
    • Given geometry, find charge distribution
                     energy to charge atom  ...
Physical interpretation of QEq
• In equilibrium:
  – each atom i has the same chemical potential μ
  – μ uniquely determin...
Physical interpretation of QEq
     • Three-point approximation for derivatives


                                        ...
Why QEq is bad
• Wrong asymptotic charges predicted
     1.2
           q/e
                                   equilibrium...
New charge model: Desiderata
•       Transferable parameters
    –     Generic, application-independent
    –     No atom ...
QTPIE: charge transfer with
 polarization current equilibration
• Shift focus to charge transfer variables pji:
   – Charg...
NaCl asymptote correct
• QTPIE prediction improved over QEq, even without
  reoptimized parameters
   1.2
   q/e




   1....
Water fragments correctly
       • Asymmetric dissociation: correct asymptotics, charge
         transfer on OH fragment r...
Water parameters transferable
• Parameters transferable across geometries
1.0
          q/e
0.8
                          ...
Water parameters transferable
• Parameters transferable across geometries
1.0
          q/e
0.8
                          ...
Water parameters transferable
• Parameters transferable across geometries
1.0
          q/e
0.8
                          ...
Water parameters transferable
• Parameters transferable across geometries
1.0
          q/e
0.8
                          ...
Water parameters transferable
1.0       • Parameters transferable across geometries
          q/e                         ...
Dipole polarizability of phenol
    • Response of dipole moment to external electric
      field



    • QTPIE: overestima...
QTPIE = coarse-grained ab initio?
• Reparameterizing with ab initio (MP2/aug-cc-
  pVDZ) IPs and EAs improves agreement of...
Dealing with charged systems I
• Constrained minimization with Lagrange
  multipliers


  – Problem 1: Cannot be enforced ...
Dealing with charged systems II
     • Redefine atoms with formal charges

                               E                ...
Test case - water : phenol : sodium -stack
 • Chemically “obvious”
   localized charge
 • Reparameterization
   appears to...
Outlook
• QTPIE is a promising new charge model
  – Implement scalable solution algorithm
  – Interface with MD code
  – C...
Conclusions
• Focus on charge transfer and including distance penalty
  improves description of atomic charges
           ...
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Constructing a rigorous fluctuating-charge model for molecular mechanics

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Constructing a rigorous fluctuating-charge model for molecular mechanics

  1. 1. Constructing a rigorous fluctuating- charge model for molecular mechanics - - + - + + + + + - - -!+ - + + + + + + +!- +!+ Funding Acknowledgments NSF DMR-03 25939 ITR •Todd Martínez DOE DE-FG02-05ER46260 •Martínez Group members, esp. Ben Levine Jiahao Chen September 19, 2006
  2. 2. Molecular mechanics is useful water flow in aquaporins1 mechanical deformation in ceramics2 • Since atomic nuclei behave mostly classically, molecular mechanics (MM) is a useful method for doing dynamics • In MM, classical electrostatic effects are important, including polarization 1. E. Tajkhorshid et. al., Science 296 (2002), 525-530. 2. P. S. Branicio, R. K. Kalia, A. Nakano, P. Vashishta, Phys. Rev. Lett. 96 (2006), art. no. 065502.
  3. 3. Molecular mechanics • Classical energy function with bonded and nonbonded terms Van der Waals interactions Molecular electrostatics • Nuclear motions propagated using classical equations of motion
  4. 4. MM leaves out something time 0 time • Ab initio molecular dynamics (MD) nuclear forces from wavefunction • MM/MD nuclear forces from fixed charge distribution - - + - + + + + + specified • MM/MD cannot describe chemical reactions
  5. 5. QEq1, a fluctuating charge model • Given geometry, find charge distribution energy to charge atom Coulomb interaction q1 q2 q3 • Minimization with fixed total charge q4 q5 defines Lagrange multiplier μ 1. A. K. Rappe, W. A. Goddard III, J. Phys. Chem. 95 (1991) 3358-3363.
  6. 6. Physical interpretation of QEq • In equilibrium: – each atom i has the same chemical potential μ – μ uniquely determines the atomic charges qi • Atoms interpreted as subsystems in equilibrium molecule i atom N, V, T Energy derivatives: chemical potential μ, hardness
  7. 7. Physical interpretation of QEq • Three-point approximation for derivatives Mulliken1 E Parr-Pearson2 IP EA N N0-1 N0 N0+1 1. R. S. Mulliken, J. Chem. Phys. 2 (1934) 782-793. 2. R. G. Parr, R. G. Pearson, J. Am. Chem. Soc. 105 (1983) 7512-7516.
  8. 8. Why QEq is bad • Wrong asymptotic charges predicted 1.2 q/e equilibrium geometry 1.0 0.8 0.6 QEq Mulliken 0.4 ab initio DMA charges 0.2 Ideal dipole 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 R/Å 8.0 • No penalty for long-range charge transfer • Overestimates molecular electrostatic properties • Especially bad far from equilibrium
  9. 9. New charge model: Desiderata • Transferable parameters – Generic, application-independent – No atom typing • Accurate – Able to describe polarization and charge transfer – Correct asymptotic charge distributions – Predicts electrostatic properties accurately • Flexible – Able to handle arbitrary total charge – Able to describe electronic excited states • Rigorous – Well-defined coarse-graining picture from conventional electronic structure methods • Practical to compute – O(N ) or better – Faster than conventional electronic structure methods
  10. 10. QTPIE: charge transfer with polarization current equilibration • Shift focus to charge transfer variables pji: – Charge accounting: where it came from, where it’s going p 12 p23 p34 p45 – Explicitly penalize long-distance charge transfer
  11. 11. NaCl asymptote correct • QTPIE prediction improved over QEq, even without reoptimized parameters 1.2 q/e 1.0 equilibrium geometry 0.8 0.6 QEq 0.4 QTPIE 0.2 DMA 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 R / Å8.0 • Slope wrong: cannot capture nonadiabatic effects
  12. 12. Water fragments correctly • Asymmetric dissociation: correct asymptotics, charge transfer on OH fragment retained 1.0 q/e equilibrium geometry R 0.5 R/Å 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 -0.5 -1.0
  13. 13. Water parameters transferable • Parameters transferable across geometries 1.0 q/e 0.8 O H 0.6 H 0.4 DMA 0.2 QEq 0.0 QTPIE R/Å QTPIE -0.20.5 1.5 2.5 3.5 4.5 DMA -0.4 -0.6 QEq -0.8 -1.0
  14. 14. Water parameters transferable • Parameters transferable across geometries 1.0 q/e 0.8 O H 0.6 0.4 DMA H 0.2 QEq 0.0 QTPIE R/Å -0.20.5 1.5 2.5 3.5 4.5 QTPIE DMA -0.4 -0.6 QEq -0.8 -1.0
  15. 15. Water parameters transferable • Parameters transferable across geometries 1.0 q/e 0.8 O H 0.6 0.4 H DMA 0.2 QEq 0.0 QTPIE R / Å QTPIE -0.20.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 DMA -0.4 -0.6 QEq -0.8 -1.0
  16. 16. Water parameters transferable • Parameters transferable across geometries 1.0 q/e 0.8 O H 0.6 H 0.4 DMA 0.2 QEq 0.0 QTPIE R / Å QTPIE -0.20.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 DMA -0.4 -0.6 QEq -0.8 -1.0
  17. 17. Water parameters transferable 1.0 • Parameters transferable across geometries q/e 1.0 q/e 0.8 O H 0.8 0.6 O H 0.6 H 0.4 0.4 H DMA 0.2 0.2 DMA 0.0 QEq QEq R/Å QTPIE 0.0 R / Å QTPIE -0.20.5 1.5 2.5 3.5 4.5 QTPIE-0.20.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 QTPIE -0.4 DMA -0.4 DMA -0.6 -0.6 -0.8 QEq -0.8 QEq -1.0 -1.0 1.0 1.0 q/e q/e 0.8 0.8 O H O H 0.6 0.6 H 0.4 H 0.4 0.2 DMA 0.2 DMA 0.0 QEq 0.0 QEq R/Å R / Å QTPIE QTPIE -0.20.5 1.5 2.5 3.5 4.5 -0.20.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 QTPIE QTPIE -0.4 DMA -0.4 DMA -0.6 -0.6 -0.8 QEq -0.8 QEq -1.0 -1.0
  18. 18. Dipole polarizability of phenol • Response of dipole moment to external electric field • QTPIE: overestimates less than QEq QEq/STO QTPIE/STO MP2/STO- MP2/aug-cc- 3G pVDZ x 24.6244 13.0298 8.4240 13.6758 y 20.3270 10.7566 7.0488 12.3621 z 0.0000 0.0000 0.8595 6.9981 (Å ) • Out-of-plane component missing in QEq, QTPIE • MP2/STO-3G suggests this is largely because of inflexible basis set
  19. 19. QTPIE = coarse-grained ab initio? • Reparameterizing with ab initio (MP2/aug-cc- pVDZ) IPs and EAs improves agreement of in- plane polarizabilities at same level of theory (eV) Original ab initio Eigenvalues of dipole IP(H) 11.473 13.588 polarizability tensor/Å IP(C) 10.406 9.607 Old QTPIE New QTPIE ab initio IP(O) 15.423 14.565 13.0298 13.4285 13.6758 EA(H) -2.417 -0.068 10.7566 11.1316 12.3621 EA(C) 0.280 1.000 0.0000 0.0000 6.9981 EA(O) 2.059 3.127 • Similar results for other ab initio methods, e.g. FCI/STO-3G, RHF/aug-cc-pVDZ…
  20. 20. Dealing with charged systems I • Constrained minimization with Lagrange multipliers – Problem 1: Cannot be enforced for diatomic molecule and – Problem 2: Generalizing to non-zero diagonal charge transfer variables destroys asymptotic property – Model has insufficient constraints at large bond lengths to guarantee integer charges
  21. 21. Dealing with charged systems II • Redefine atoms with formal charges E E IP+1 IP - e- EA+1 EA N N N0-1 N0 N0+1 N0-2 N0-1 N0 • Problem: must account for multiple references IP0, EA0 IP+1, EA+1 IP0, EA0 - e- + + +… IP0, EA0 IP0, EA0 + IP0, EA0 IP0, EA0 IP0, EA0 IP+1, EA+1
  22. 22. Test case - water : phenol : sodium -stack • Chemically “obvious” localized charge • Reparameterization appears to work well for QTPIE • Need to figure out extension to general systems qNa/e QEq QTPIE Lagrange 0.6177 0.1876 reparam. 0.4798 0.8648 Mulliken/MP2/cc-pVDZ charge: 0.7394
  23. 23. Outlook • QTPIE is a promising new charge model – Implement scalable solution algorithm – Interface with MD code – Chemical applications, e.g. enzyme-substrate docking, electrochemistry • Many open theoretical questions, e.g.: – How to account for out-of-plane polarizabilities? – When does a molecule stop being a molecule? – What is the quantum-mechanical analogue of charge transfer variables? – How to deal with excited states?
  24. 24. Conclusions • Focus on charge transfer and including distance penalty improves description of atomic charges Fluctuating-charge model QEq QTPIE (now) Transferable parameters Yes Yes Correct asymptotics No Yes Correct molecular electrostatics No Almost! Established Arbitrary total charge Yes* No New result Coarse-graining picture Yes* Some evidence In progress Practical scaling Yes, O(N2) No, O(N4) Need ideas Excited states No No *with caveats
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