Theory and applications of
fluctuating charge models
                  Jiahao Chen
                 Martínez Group

    Dep...
Acknowledgments
                                     Committee
                                 Prof. Nancy Makri
        ...
“The supreme goal of all theory is to
make the irreducible basic elements as
simple and as few as possible without
   havi...
Electronic structure and dynamics
ˆ                              What is the charge distribution?
HΨ = EΨ

   direct      ...
Electronic structure and dynamics
ˆ                              What is the charge distribution?
HΨ = EΨ

   direct      ...
Molecular models/force fields
Typical energy function


E = covalent bond effects
                          +


           ...
Molecular models/force fields
Typical energy function


             kb (rb − rb )2 +              κa (θa − θa )2 +
 E=    ...
Molecular models/force fields
Typical energy function


             kb (rb − rb )2 +              κa (θa − θa )2 +
 E=    ...
Why care about polarization
  and charge transfer?

  They are important in
 condensed phases, where
   most chemistry and...
Polarization in chemistry
• Ex. 1: Stabilizes carbonium in lysozyme
      carbonium
        forms
                        ...
Fluctuating charges
                                      -0.3

charge transfer = 1.1e                              charge...
Charge formation vs.
    charge-charge interactions
  Electronic     Coulomb
energy of atom interactions
                 ...
Electronegativity

                  IP + EA
               χ=
                     2
                             R. S. M...
A quantitative definition
           IP + EA
    =
χ
              2
           E(N − 1) − E(N + 1)
    =
                 ...
Chemical hardness

                = IP − EA
     η
                    2
                  ∂E
                =     2
   ...
QEq, a fluctuating-
  charge model
                 1
E=       qi χi +                                qi qj Jij
           ...
Principle of electronegativity
          equalization
                                                     1
           Mi...
Physical interpretation
In equilibrium:
 o each atom i has the same chemical potential µ
 o µ uniquely determines the ato...
QEq, a fluctuating-
  charge model
                 1
E=       qi χi +                                qi qj Jij
           ...
QEq has wrong asymptotics
1.0
        q/e

                                      Na               Cl
                     ...
Fluctuating-charge models map
molecules onto electrical circuits

                                    screened
           ...
Fluctuating-charge models map
molecules onto electrical circuits

                                       screened
        ...
Fluctuating-charge models map
molecules onto electrical circuits

                                           screened
    ...
QEq has wrong asymptotics
1.0
        q/e

                                            Na                   Cl
           ...
In fluctuating-charge
  models like QEq, all
molecules are metallic
Problems due to metallicity
Fractional charge distributions predicted
 for dissociated systems
Overestimates charge tran...
QTPIE, our new charge model
    Charge-transfer with polarization current
     equilibration
    Voltage attenuates with...
QTPIE, our new charge model
    Charge-transfer with polarization current
     equilibration
    Voltage attenuates with...
Making QTPIE (Step 1)
To make the proposed change, first change variables
                            qi =          pji
   ...
Making QTPIE (Step 2)
atomic electronegativities become bond electronegativities

                                        ...
QTPIE has correct limit
1.0
        q/e

                                      Na                Cl
                      ...
However...
                                1
              =         qi χi +
        QEq
    E                            ...
Origin of rank deficiency
     Charge transfer variables are massively
     redundant due to Kirchhoff’s voltage law

     ...
Reverting to atomic charges
                             qi =       pji        q1
     p12                            j
  ...
Reverting to charge variables
                            qi =         pji              q4
             p14
              ...
Reverting to charge variables
                                       qi =                 pji                            q...
Reverting to charge variables
                                       qi =                 pji                            q...
Execution times
                                   TImes to solve the QTPIE model
                       4
               ...
Atom-space QTPIE vs QEq
                                         1
                     =           qi χi +
              ...
Cooperative
     polarization in water
        +                  −→
• Dipole moment of water increases from 1.854
  Debye...
Polarization in water chains
  • Use parameters from gas phase data to
    model chains of waters


  • Compare QTPIE with...
The flexible SPC model
                                      2
    =                                         bond stretch
 ...
Our new water model
                                      2
    =                                         bond stretch
   ...
Our new water model
                                       reparameterized
                                     2
    =   ...
Parameterization
1 230 monomers sampled by systematic variation of coords.
890 dimers sampled from flexible SPC at 30 000 ...
Dipole moment per water
                                     2.6
Dipole moment per molecule (Debye)

                     ...
Polarizability per water
Longitudinal polarizability per molecule (Å!)
                                                5.0...
Polarizability per water
Transverse polarizability per molecule (Å!)
                                              3.5


 ...
Polarizability per water
Out of plane polarizability per molecule (Å!)
                                                1.5...
Charge transfer in 15 waters
                     .20



                     .10
  Molecular charge




                 ...
Summary

• Polarization and charge transfer are important
  effects usually neglected in classical MD
• Our new charge mod...
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Theory and application of fluctuating-charge models

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  • Thus the chemical potential is the key concept underlying the workings of the QEq model. We can consider individual atoms as subsystems on which we can define atomic chemical potentials. Then in equilibrium, the QEq model postulates that the chemical potential on each atom is equal, and this therefore defines a unique atomic charge for each atom.


























































  • Theory and application of fluctuating-charge models

    1. 1. Theory and applications of fluctuating charge models Jiahao Chen Martínez Group Dept. of Chemistry, Frederick Seitz Materials Research Laboratory and the Beckman Institute University of Illinois at Urbana-Champaign Stanford Linear Accelerator Center Dept. of Chemistry and Dept. of Photon Sciences Stanford University
    2. 2. Acknowledgments Committee Prof. Nancy Makri Prof. Duane Johnson Prof. Dirk Hundertmark Discussions Prof. Susan Atlas (UNM) Dr. Ben Levine (UPenn) Prof. Todd J. Martínez Dr. Steve Valone (LANL) Martínez Group and friends Prof. Troy van Voorhis (MIT) $: DOE
    3. 3. “The supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” Albert Einstein, “On the Method of Theoretical Physics”, Phil. Sci. 1 (1934), 163-9.
    4. 4. Electronic structure and dynamics ˆ What is the charge distribution? HΨ = EΨ direct density coarse- semiempirical molecular continuum ab initio numerical functional grained theories methods models (MM) electrostatics quadrature theory models numerical quadrature classical coarse- finite element ab initio molecular dynamics path integrals molecular grained methods emiclassical dynamics dynamics dynamics ˆ ˙ What does the system do? HΨ = iΨ
    5. 5. Electronic structure and dynamics ˆ What is the charge distribution? HΨ = EΨ direct density coarse- semiempirical molecular continuum ab initio numerical functional grained theories methods models (MM) electrostatics quadrature theory models numerical quadrature classical coarse- finite element ab initio molecular dynamics path integrals molecular grained methods emiclassical dynamics dynamics dynamics ˆ ˙ What does the system do? HΨ = iΨ
    6. 6. Molecular models/force fields Typical energy function E = covalent bond effects + noncovalent interactions
    7. 7. Molecular models/force fields Typical energy function kb (rb − rb )2 + κa (θa − θa )2 + E= ldn cos(nπ) 0 0 d∈dihedrals n a∈angles b∈bonds bond stretch angle torsion dihedrals + - 12 6 σij σij qi qj + 4 − + + ij rij rij rij i<j∈atoms i<j∈atoms dispersion electrostatics Usually fixed charges
    8. 8. Molecular models/force fields Typical energy function kb (rb − rb )2 + κa (θa − θa )2 + E= ldn cos(nπ) 0 0 d∈dihedrals n a∈angles b∈bonds bond stretch angle torsion dihedrals + - 12 6 σij σij qi qj + 4 − + + ij rij rij rij i<j∈atoms i<j∈atoms dispersion electrostatics Usually fixed charges
    9. 9. Why care about polarization and charge transfer? They are important in condensed phases, where most chemistry and biology happens
    10. 10. Polarization in chemistry • Ex. 1: Stabilizes carbonium in lysozyme carbonium forms sugar bond cleaved • Ex. 2: Hydrates chloride in water clusters TIP4P/FQ OPLS/AA non-polarizable polarizable force field force field 1. A Warshel and M Levitt J. Mol. Biol. 103 (1976), 227-249. 2. SJ Stuart and BJ Berne J. Phys. Chem. 100 (1996), 11934 -11943.
    11. 11. Fluctuating charges -0.3 charge transfer = 1.1e charge transfer = 0.2 e -0.5 χ2 , η2 +0.8 charge transfer = 1.3 e χ3 , η3 Response = change in atomic charges Review: H Yu and WF van Gunsteren Comput. Phys. Commun. 172 (2005), 69-85.
    12. 12. Charge formation vs. charge-charge interactions Electronic Coulomb energy of atom interactions 1 = Eat (qi ) + E qi qj Jij 2 i i=j 12 1 ∂2E ∂E = + + ··· + qi qi qi qj Jij 2 2 2 ∂qi ∂qi qi =0 qi =0 i i i=j 12 1 = qi χi + qi ηi + · · · + qi qj Jij 2 2 i i i=j chemical hardness electronegativity R. P. Iczkowski and J. L. Margrave J. Am. Chem. Soc. 83:(1961), 3547–3551
    13. 13. Electronegativity IP + EA χ= 2 R. S. Mulliken J. Chem. Phys 2:(1934), 782–793 Electronegativity: “Concept introduced by L. Pauling as the power of an atom to attract electrons to itself.” IUPAC Compendium of Chemical Terminology, aka “The Gold Book”, goldbook.iupac.org
    14. 14. A quantitative definition IP + EA = χ 2 E(N − 1) − E(N + 1) = 2 ∂E ∼ ∂N R. S. Mulliken J. Chem. Phys 2:(1934), 782–793 R. P. Iczkowski and J. L. Margrave J. Am. Chem. Soc. 83:(1961), 3547–3551 R. G. Parr, R. A. Donnelly, M. Levy and W. E. Palke J. Chem. Phys. 68:(1978), 3801–3807
    15. 15. Chemical hardness = IP − EA η 2 ∂E = 2 ∂N R. G. Parr, R. G. Pearson J. Am. Chem. Soc. 105:(1983), 7512–7516
    16. 16. QEq, a fluctuating- charge model 1 E= qi χi + qi qj Jij 2 atomic screened i ij Coulomb electronegativities “voltages” interactions φ2 (r1 )φ2 (r2 ) i j Jij = dr1 dr2 |r1 − r2 | R3×2 ni −1 −ζi |r−Ri | φi (r) = Ni |r − R| e AK Rappé and WA Goddard III J. Phys. Chem. 95 (1991), 3358-3363.
    17. 17. Principle of electronegativity equalization 1 Minimize energy E= qi χi + qi qj Jij 2 i ij qi = Q subject to charge constraint i Using the method of Lagrange multipliers, reduces to solving the linear equation J 1 q −χ = 0 0 1 T µ (electronic) chemical potential
    18. 18. Physical interpretation In equilibrium: o each atom i has the same chemical potential µ o µ uniquely determines the atomic charges qi Atoms are subsystems in equilibrium molecule Ω Ωi atom N, V, T Energy derivatives: chemical potential µ, hardness η
    19. 19. QEq, a fluctuating- charge model 1 E= qi χi + qi qj Jij 2 atomic screened i ij Coulomb electronegativities “voltages” interactions φ2 (r1 )φ2 (r2 ) i j Jij = dr1 dr2 |r1 − r2 | R3×2 ni −1 −ζi |r−Ri | φi (r) = Ni |r − R| e AK Rappé and WA Goddard III J. Phys. Chem. 95 (1991), 3358-3363.
    20. 20. QEq has wrong asymptotics 1.0 q/e Na Cl R 0.8 χ1 − χ2 q= J11 + J22 − J12 0.6 QEq asymptote ~ 0.43 ≠ 0 0.4 0.2 ab initio R/Å 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
    21. 21. Fluctuating-charge models map molecules onto electrical circuits screened electro- chemical Coulomb molecule negativity hardness interaction
    22. 22. Fluctuating-charge models map molecules onto electrical circuits screened electro- chemical Coulomb molecule negativity hardness interaction electric (inverse) Coulomb electrical potential capacitance interaction circuits
    23. 23. Fluctuating-charge models map molecules onto electrical circuits screened electro- chemical Coulomb molecule negativity hardness interaction electric (inverse) Coulomb electrical potential capacitance interaction circuits More electropositive χ - Voltage + η 1 1 χ η 2 0V 2 More electronegative
    24. 24. QEq has wrong asymptotics 1.0 q/e Na Cl R 0.8 + + χ1 − χ2 - - q= J11 + J22 − J12 0.6 QEq asymptote ~ 0.43 ≠ 0 0.4 + + J12 → 0 - - 0.2 ab initio R/Å 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
    25. 25. In fluctuating-charge models like QEq, all molecules are metallic
    26. 26. Problems due to metallicity Fractional charge distributions predicted for dissociated systems Overestimates charge transfer for stretched / reactive geometries In practice, existing models must introduce ad hoc cutoffs on charge flows Polarizabilities are not size-extensive
    27. 27. QTPIE, our new charge model Charge-transfer with polarization current equilibration Voltage attenuates with increasing distance voltage η 2 distance J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.
    28. 28. QTPIE, our new charge model Charge-transfer with polarization current equilibration Voltage attenuates with increasing distance voltage η 2 η 2 distance J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.
    29. 29. Making QTPIE (Step 1) To make the proposed change, first change variables qi = pji p12 j Charge transfer variables quantify how much charge went from one atom to another, and are p23 indexed over pairs 1 p34 E= qi χi + qi qj Jij p45 2 ij i Still QEq! 1 Same model, = pji χi + pki plj Jij 2 new representation ij ijkl J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.
    30. 30. Making QTPIE (Step 2) atomic electronegativities become bond electronegativities 1 = pji χi + QEq E pki plj Jij 2 ij ijkl 1 = pji χi kij Sij + QT P IE E pki plj Jij 2 ij ijkl Sij = φi (r)φj (r)dr R3 J Chen and T J Martínez, Chem. Phys. Lett. 438 (2007), 315-320.
    31. 31. QTPIE has correct limit 1.0 q/e Na Cl R 0.8 χ1 − χ2 q= J11 + J22 − J12 0.6 QEq (χ1 − χ2 )S12 0.4 q= J11 + J22 − J12 QTPIE 0.2 ab initio R/Å 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
    32. 32. However... 1 = qi χi + QEq E qi qj Jij 2 i ij 1 = pji χi kij Sij + QT P IE E pki plj Jij 2 ij ijkl N times as many variables as before - costly! Equations are rank deficient - need SVD
    33. 33. Origin of rank deficiency Charge transfer variables are massively redundant due to Kirchhoff’s voltage law p12 p31 p23 p12 + p13 + p31 = 0 only N-1 of these variables are linearly independent! Therefore, charge transfer variables contain exactly the same amount of information as atomic charges
    34. 34. Reverting to atomic charges qi = pji q1 p12 j p31 p23 q2 q3 ? Topological analysis of the relationship between charges and charge transfer variables allows the reverse transformation to be derived as qi − qj = pji N
    35. 35. Reverting to charge variables qi = pji q4 p14 j q1 p24 p34 p12 p13 q2 q3 ? p23   p12       0 0 0 −1 −1 −1 q1 p13    q2   1 0   0 0 −1 −1 p14  =   −1     q3   0 1 0 1 0 p23     0 0 1 0 1 1 q4 p24 p34 Adjacency matrix of an oriented complete graph with 4 vertices
    36. 36. Reverting to charge variables qi = pji q4 p14 j q1 p24 p34 p12 p13 q2 q3 ? p  23  p12  +    p13  0 0 0 −1 −1 −1 q1    p14  1 0   q2  0 0 −1 −1      =  p23  0 −1   q3  1 0 1 0    p24  0 0 1 0 1 1 q4 p34   1 0 0 −1     0 1 0 −1 q1   1  q2  0 0 1 −1    = 4  q3  0 1 0 −1     0 0 1 −1 q4 0 0 1 −1 Inverse transformation is determined by pseudoinverse of adjacency matrix
    37. 37. Reverting to charge variables qi = pji q4 p14 j q1 p24 p34 p12 p13 qi − qj q2 q3 pji = p  23  N p12  +    p13  0 0 0 −1 −1 −1 q1    p14  1 0   q2  0 0 −1 −1      =  p23  0 −1   q3  1 0 1 0    p24  0 0 1 0 1 1 q4 p34   1 0 0 −1     0 1 0 −1 q1   1  q2  0 0 1 −1    = 4  q3  0 1 0 −1     0 0 1 −1 q4 0 0 1 −1 Inverse transformation is determined by pseudoinverse of adjacency matrix
    38. 38. Execution times TImes to solve the QTPIE model 4 10 N6.20 N1.81 1000 100 Solution time (s) 10 1 Bond-space SVD 0.1 Bond-space COF Atom-space iterative solver Atom-space direct solver 0.01 4 5 10 100 1000 10 10 N Number of atoms
    39. 39. Atom-space QTPIE vs QEq 1 = qi χi + QEq E qi qj Jij 2 i ij 1 = qi χi + ¯ QT P IE E qi qj Jij 2 i ij A charge model with bond electronegativities is equivalent to one with renormalized atomic electronegativities kij Sij (χi − χj ) kij Sij kij Sij χj χ= ¯ = χi − N N N j j j
    40. 40. Cooperative polarization in water + −→ • Dipole moment of water increases from 1.854 Debye1 in gas phase to 2.95±0.20 Debye2 at r.t.p. (liquid phase) • Polarization enhances dipole moments • Missing in models with implicit or no polarization, e.g. Bernal-Fowler, SPC, TIPnP... 1. D R Lide, CRC Handbook of Chemistry and Physics, 73rd ed., 1992. 2. AV Gubskaya and PG Kusalik J. Chem. Phys. 117 (2002) 5290-5302.
    41. 41. Polarization in water chains • Use parameters from gas phase data to model chains of waters • Compare QTPIE with: QEq and reparameterized QEq ๏ ˆ Ab initio DF-LMP2/aug-cc-pVTZ ๏ HΨ = EΨ AMOEBA2, an inducible dipole model ๏ 1. WF Murphy J. Chem. Phys. 67 (1977), 5877-5882. 2. P Ren and JW Ponder J. Phys. Chem. B 107 (2003), 5933-5947.
    42. 42. The flexible SPC model 2 = bond stretch 0 kO–H RO–H − E RO–H O–H Urey-Bradley 2 + UB 0 RH—H − kH—H RH—H 1,3 term H—H angle torsion 2 + 0 κ∠HOH θ∠HOH − θ∠HOH ∠HOH 12 6 σO—H σO—H + 4 − O—H RO—H RO—H O—H,nonbonded dispersion qi qj + Rij ij,nonbonded electrostatics LX Dang and BM Pettitt J. Phys. Chem. 91 (1987) 3349-3354.
    43. 43. Our new water model 2 = bond stretch 0 kO–H RO–H − E RO–H O–H Urey-Bradley 2 + UB 0 RH—H − kH—H RH—H 1,3 term H—H angle torsion 2 + 0 κ∠HOH θ∠HOH − θ∠HOH ∠HOH 12 6 σO—H σO—H + 4 − O—H RO—H RO—H O—H,nonbonded dispersion qi qj + EQTPIE Rij ij,nonbonded electrostatics LX Dang and BM Pettitt J. Phys. Chem. 91 (1987) 3349-3354.
    44. 44. Our new water model reparameterized 2 = 0 kO–H RO–H − E RO–H to ab initio (DF- O–H LMP2/aug-cc-pVTZ) 2 + UB 0 kH—H RH—H − RH—H energies, dipoles H—H 2 and polarizabilities + 0 κ∠HOH θ∠HOH − θ∠HOH of sampled ∠HOH 12 monomer and 6 σO—H σO—H + 4 O—H − geometries dimer RO—H RO—H O—H,nonbonded qi qj + EQTPIE Rij ij,nonbonded
    45. 45. Parameterization 1 230 monomers sampled by systematic variation of coords. 890 dimers sampled from flexible SPC at 30 000 K Step 1: Fit electrostatics to dipoles and polarizabilities Step 2: Fit non-electrostatic parameters with ab initio energies Parameter flexible SPC This work Parameter/eV QEq New QEq QTPIE LJ radius of OH/Å 3.1656 1.7055 H electronegativity 4.528 3.678 4.528 LJ well depth/kcm 0.1554 0.2798 H hardness 13.89 18.448 11.774 bond stretch 527.2 226 O electronegativity 8.741 9.591 7.651 eq. bond length /Å 1 1.118 O hardness 13.364 17.448 13.364 angle stretch 37.95 40.81 eq. angle/deg. 109.47 111.48 UB stretch 39.9 54.32 UB eq. length/Å 1.633 1.518
    46. 46. Dipole moment per water 2.6 Dipole moment per molecule (Debye) DF-LMP2/aug-cc-PVTZ AMOEBA 2.5 QTPIE 2.4 2.3 QEq (reparameterized) 2.2 2.1 2.0 1.9 QEq 1.8 0 5 10 15 20 25 Number of molecules
    47. 47. Polarizability per water Longitudinal polarizability per molecule (Å!) 5.0 QEq QEq (reparameterized) 4.0 3.0 2.0 AMOEBA DF-LMP2/aug-cc-PVTZ QTPIE 1.0 .0 0 5 10 15 20 25 Number of molecules
    48. 48. Polarizability per water Transverse polarizability per molecule (Å!) 3.5 3.0 QEq 2.5 2.0 QTPIE AMOEBA 1.5 QEq (reparameterized) DF-LMP2/aug-cc-PVTZ 1.0 0 5 10 15 20 25 Number of molecules
    49. 49. Polarizability per water Out of plane polarizability per molecule (Å!) 1.5 DF-LMP2/aug-cc-PVTZ AMOEBA 1.0 .5 QTPIE, QEq (reparameterized) and QEq .0 -.5 0 5 10 15 20 25 Number of molecules
    50. 50. Charge transfer in 15 waters .20 .10 Molecular charge .00 QEq -.10 QEq (reparameterized) QTPIE DMA Charges -.20 -.30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Index of water molecule
    51. 51. Summary • Polarization and charge transfer are important effects usually neglected in classical MD • Our new charge model corrects deficiencies in existing fluctuating-charge models at similar computational cost • We obtain quantitative polarization and qualitative charge transfer trends in linear water chains
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