Methods for computing partial charges

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Methods for computing partial charges

  1. 1. Models for computing partial charges Jiahao Chen Martínez Group Meeting September 27, 2005
  2. 2. Outline• An atom-site charge model: QEq – Results for amino acids – NaCl dissociation – Reparameterization study• A minimal bond-space model – Study of NaCl.6H2O dissociation• Quantum mechanical analogs – Derivative discontinuities
  3. 3. Molecular charge distributions• Molecules as clusters of point charges• Electrostatics in the classical limit• Useful for molecular modeling
  4. 4. Point charge models • Key atomic parameters: – Electronegativity – Hardness • Mulliken definitions – Ionization potential – Electron affinity • Sanderson electronegativity equilibrationIczkowsky, R. P.; Margrave, J. L., J. Am. Chem. Soc. 83, 1961, 3547-3553.
  5. 5. QEq: Rappé and Goddard, 1991 • Parameters: Mulliken electronegativities and hardnesses internal energy Coulomb interactionRappé, A. K.; Goddard, W. A. III, J. Phys. Chem. 95, 1991, 3358-3363.
  6. 6. QEq (continued)• Screened Coulomb interaction: two- electron integrals over ns-ms STOs• Sanderson electronegativity equalization principle• Linear system of simultaneous equations
  7. 7. QEq: Electrical interpretation• Molecules as classical Circuit QEq circuits element Atom Capacitor + Resistor Bond (Ideal) Wire
  8. 8. QEq on equilibrium geometries• Compare QEq results with ab initio calculations for ground state geometries• Molecules: 20 naturally occurring amino acids• Ab initio method: – MP2 geometry optimization – DMA0 (distributed multipole analysis) charges: 0th order = monopoles
  9. 9. QEq v. DMA0 on MP2/6-31G*1.00.80.6 C0.4 CHx0.2 NH20.0 N, NH-0.2 OH S-0.4 O-0.6-0.8-1.0 -1.0 -0.5 0.0 0.5 1.0 1.5
  10. 10. QEq v. DMA0 on MP2/cc- 1.0 QEq pVDZ 0.8 0.6 0.4 C CHx 0.2 0.0 NH2 N, NH-0.2 OH S-0.4 O-0.6-0.8-1.0 ab initio -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
  11. 11. QEq v. DMA0 on MP2: Results• Only singly bonded atoms have good agreement (Δq<0.1) – Deviations: 1° > 2° > aromatic > 3° – N termini – Hydrocarbons – Carboxyls, imines…• Higher correlation between QEq and DMA0 on MP2/cc-pVDZ
  12. 12. Does QEq neglect polarizability?• 6-31G v. 6-31G* on Cys: very similar 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.0 -0.5 0.0 0.5 1.0 1.5
  13. 13. QEq on Diatomics• Compare QEq results with experimental results for diatomics• Molecule: NaCl (g)• Dipole moments from experimental literature• Given bond length, can QEq predict the dipole moment ?• QEq parameters derived from fit to experimental dipole moments
  14. 14. QEq results: NaCl dissociation0.9 qN a _ _ R eq_ R0.80.7 Too slow!0.6 Not zero!0.5 qN a _ _ R0.4 qN a _R ! 1 _ _ __ :___ 6 _ _0.30.2 Â_ _ R0.10.0 0 2 4 6 8 10 12 14 16 18 20
  15. 15. QEq: What is Missing?• No HOMO-LUMO band gap! – All bonding is completely metallic• Wrong asymptotic limit of quantum statistical mechanics – Have: No Fermi gap => T  ∞ limit – Need: Ground state only => Want T  0 limit!• No notion of bond length and bond order – All atoms are pairwise “σ”–bonded together!• No out-of-plane polarizability
  16. 16. QEq: Parameterization • Can reparameterizing QEq improve its accuracy? +q -q • Molecules: 94 diatomics r • Benchmark: experimental (and high- precision computational) dipole moments • Partial charges from ideal dipole model • χ² goodness-of-fit minimizationHuber, K. P.; Herzberg, G. Constants of Diatomic Molecules, VanNostrand Reinhold, 1978, New York, NY.
  17. 17. 5.0 QEq: Original parameters QEq CsI4.5 CsBr RbI KI CsCl RbBr KBr RbCl4.0 KCl LiI LiBr LiCl NaI NaBr3.5 NaCl3.0 CsF RbF KF2.5 NaF LiF2.0 SiO1.5 CF HF HCl OH1.0 CO HBr ICl BrF HI SH IBr ClF SO PN0.5 NO BrCl Expt. ClO NS0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
  18. 18. 5.0 QEq: Optimized parameters QEq CsI RbI4.5 KIRbBr CsBr KBr RbCl CsCl KCl4.0 NaI NaBr NaCl3.5 KF RbF NaF CsF3.0 LiI LiBr LiCl2.5 LiF2.01.5 SiO PN1.0 NS HClBrF HF SO OH ClO ICl0.5 SH CF ClF HBr IBr BrCl Expt. NOHI CO0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
  19. 19. 1.0 QEq: New parameters on aa’s0.80.6 Worse than before!0.40.20.0 1.0-0.2 0.8 0.6-0.4 0.4 0.2 0.0-0.6 -0.2 -0.4 -0.6-0.8 -0.8 -1.0 -1.0 -0.5 0.0 0.5 1.0 1.5-1.0 -1.0 -0.5 0.0 0.5 1.0 1.5
  20. 20. QEq Reparamet.: Conclusions• Optimization procedure is insufficient to improve parameter quality beyond the standard values.• Lack of sufficient data, esp. for radicals and ions.• Published parameters likely to be optimal, despite physical difficulty in interpretation e.g. EA(H) <0
  21. 21. Outline• An atom-site charge model: QEq – Results for amino acids – NaCl dissociation – Reparameterization study• A minimal bond-space model – Study of NaCl.6H2O dissociation
  22. 22. Electronegativity, revisited• Many definitions and scales – Pauling, Mulliken – Different dimensionalities!• Intrinsic chemical potential for electrons• Substantial empirical evidence for variations depending on context, e.g., C-C v. C=C• Electronegativity a characteristic of bonds, rather than atoms?
  23. 23. Charge-transfer model• “Derivation” – Replace electronegativity by distance- dependant electronegativity – Replace charges by charge-transfer variables – Impose detailed balance• Sum over CTs are deviations from reference charge, not actual charge per se
  24. 24. nQEq: Formulation• Linear system of simultaneous equations
  25. 25. Computation• Cast system into matrix problem• Degenerate system of equations – Singular value decomposition – Generalized Moore-Penrose inverse (psuedoinverse) O+2δ O η ηH+δ H+δ H H
  26. 26. Theoretical Results• Singular values/zero eigenvalues correspond to closed loops of circulation – Faraday’s Law – Linear responses• N-1 nonzero eigenvalues/singular values – N-1 linearly independent flow variables – Minimum spanning tree for N nodes has N-1 edges
  27. 27. Results: NaCl0.40 0.9 0.80.35 0.7 0.6 0.50.30 0.4 0.30.25 0.2 0.1 0.00.20 0 2 4 6 8 10 12 14 16 18 200.150.10 Correct asymptotic limit!0.050.00 0 2 4 6 8 10 12 14 16 18 20
  28. 28. 0.2 Results: H2O0.10.0-0.1-0.2-0.3-0.4 0 2 4 6 8 10 12 14 16 18 20
  29. 29. Solvation of salt in H2O 6-mer• 6-mer known to be smallest cluster needed to fully solvate NaCl• Sudden limit of dissociation dynamics: no solvent reorganization
  30. 30. q/e Results: NaCl.6H2O0.5 0.70 0.650.4 0.60 0.550.3 0.50 0.450.2 0.40 0.350.1 0.30 0 2 4 6 8 10 12 14 16 18 200.0-0.1 nonvanishing residue-0.2-0.3 R(Na-Cl)/Å 0 2 4 6 8 10 12 14 16 18 20
  31. 31. Representations in Bond-space• How to describe molecule in bond space? – Bonds : Adjacency matrices – Atoms and Bond lengths: Metrized graphs• How to solve for electrostatic equilibrium? – Topological/geometric properties – Cutoffs for Coulomb interactions (optional)
  32. 32. Numerical Issues• For large systems, algorithm does not find correct dissociation limits• Large residual found• Low condition number• What’s going on?
  33. 33. Future work• Look at adiabatic limit of NaCl.6H2O dissociation – Need ab initio equilibrium geometries• Computation of molecular properties – Dipole moments – Polarizabilities – pKa?• More efficient algorithm for solving model – Graph/network flow algorithms?
  34. 34. Outline• An atom-site charge model: QEq – Results for amino acids – NaCl dissociation – Reparameterization study• A minimal bond-space model – Study of NaCl.6H2O dissociation• Quantum mechanical analogs – Derivative discontinuities
  35. 35. Quantum Analogues?• Quantum analogue of partial charges? – Spin-statistics theorem – Anyons• QEq analog: Heisenberg spin magnet
  36. 36. Janak’s Theorem • Kohn-Sham one-particle orbital energies dictate change in total energy • Implies discontinuities as a function of particle number at integers:Janak, J. F.; Phys. Rev. B, 18, 1978, 7165-7168.
  37. 37. Origin of discontinuity• Which term in universal functional contributes the most? – Coulomb exchange – Kinetic: Pauli exclusion principle – Unsolved question!
  38. 38. Future work• Notion of generating density matrices compatible with a given Hamiltonian
  39. 39. Derivative discontinuities and ionization potentials • Implementing discontinuities improve estimates of ionization potentials • “Double knee” feature in laser-induced ionization of helium atoms • Model discontinuity in correlation potential needed to obtain correct limitLein, M.; Kümmel, S. Phys. Rev. Lett., 94, 2005, 143003.
  40. 40. Acknowledgments

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