Scientific Computing Partial Differential Equations Explicit Solution of Wave Equation.
Methods for computing partial charges
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Models for computing partialcharges
Jiahao ChenMartínez Group Meeting
September 27, 2005
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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
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Molecular charge distributions
• Molecules as clusters of point charges• Electrostatics in the classical limit• Useful for molecular modeling
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Point charge models
• Key atomic parameters:– Electronegativity– Hardness
• Mulliken definitions– Ionization potential– Electron affinity
• Sanderson electronegativity equilibration
Iczkowsky, R. P.; Margrave, J. L., J. Am. Chem. Soc. 83, 1961, 3547-3553.
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QEq: Rappé and Goddard, 1991
• Parameters: Mulliken electronegativitiesand hardnesses
internal energy Coulombinteraction
Rappé, A. K.; Goddard, W. A. III, J. Phys. Chem. 95, 1991, 3358-3363.
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QEq (continued)• Screened Coulomb interaction: two-
electron integrals over ns-ms STOs
• Sanderson electronegativity equalizationprinciple
• Linear system of simultaneous equations
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QEq: Electrical interpretation• Molecules as classical
circuits
(Ideal)Wire
Bond
Capacitor+ Resistor
Atom
QEqCircuitelement
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QEq on equilibrium geometries
• Compare QEq results with ab initiocalculations for ground state geometries
• Molecules: 20 naturally occurring aminoacids
• Ab initio method:– MP2 geometry optimization– DMA0 (distributed multipole analysis)
charges: 0th order = monopoles
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-1.0 -0.5 0.0 0.5 1.0 1.5
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
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0.6
0.8
1.0
QEq v. DMA0 on MP2/6-31G*
S
C
CHx
NH2
N, NH
OHO
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QEq v. DMA0 on MP2/cc-pVDZ
ab initio
QEq
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
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1.0
S
CCHx
NH2
N, NH
OHO
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QEq v. DMA0 on MP2: Results
• Only singly bonded atoms have goodagreement (Δq<0.1)– Deviations: 1° > 2° > aromatic > 3°– N termini– Hydrocarbons– Carboxyls, imines…
• Higher correlation between QEq andDMA0 on MP2/cc-pVDZ
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Does QEq neglect polarizability?
• 6-31G v. 6-31G* on Cys: very similar
-1.0 -0.5 0.0 0.5 1.0 1.5
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-0.2
0.0
0.2
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1.0
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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
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0 2 4 6 8 10 12 14 16 18 20
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QEq results: NaCl dissociation
qN a_R ! 1 __ __ :___6_ _
Â_R_
qNa _R _
qNa _R _ R eq_
Too slow!
Not zero!
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QEq: What is Missing?
• No HOMO-LUMO band gap!– All bonding is completely metallic
• Wrong asymptotic limit of quantumstatistical 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
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QEq: Parameterization
• Can reparameterizing QEq improve itsaccuracy?
• Molecules: 94 diatomics• Benchmark: experimental (and high-
precision computational) dipole moments• Partial charges from ideal dipole model
• χ² goodness-of-fit minimization
+q -qr
Huber, K. P.; Herzberg, G. Constants of Diatomic Molecules, VanNostrand Reinhold, 1978, New York, NY.
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NaClNaBr
NaI
KCl
KBr
KI
RbCl
RbBr
RbI
CsCl
CsBr
CsI
LiF
LiCl
LiBrLiI
NaF
KFRbF
CsF
CO
CF
NO
NS
OH
SiO
PNSO
SH
ClF
ClO
BrCl
BrF
IBr
ICl
HI
HBr
HCl
HF
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
QEq: Original parameters
Expt.
QEq
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NaClNaBrNaI
KCl
KBr
KI
RbCl
RbBr
RbI
CsCl
CsBr
CsI
LiF
LiClLiBrLiI
NaF
KFRbF
CsF
CO
CF
NO
NSOH
SiO
PN
SO
SHClF
ClO
BrCl
BrF
IBr
ICl
HI
HBr
HClHF
NaClNaBrNaI
KCl
KBr
KI
RbCl
RbBr
RbI
CsCl
CsBr
CsI
LiF
LiClLiBrLiI
NaF
KFRbF
CsF
CO
CF
NO
NSOH
SiO
PN
SO
SHClF
ClO
BrCl
BrF
IBr
ICl
HI
HBr
HClHF
NaClNaBrNaI
KCl
KBr
KI
RbCl
RbBr
RbI
CsCl
CsBr
CsI
LiF
LiClLiBrLiI
NaF
KFRbF
CsF
CO
CF
NO
NSOH
SiO
PN
SO
SHClF
ClO
BrCl
BrF
IBr
ICl
HI
HBr
HClHF
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
QEq: Optimized parameters
Expt.
QEq
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-1.0 -0.5 0.0 0.5 1.0 1.5
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1.0
QEq: New parameters on aa’s
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0.0
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1.0
Worse than before!
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QEq Reparamet.: Conclusions
• Optimization procedure is insufficient toimprove parameter quality beyond thestandard values.
• Lack of sufficient data, esp. for radicalsand ions.
• Published parameters likely to be optimal,despite physical difficulty in interpretatione.g. EA(H) <0
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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
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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-Cv. C=C
• Electronegativity a characteristic of bonds,rather than atoms?
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Charge-transfer model
• “Derivation”– Replace electronegativity by distance-
dependant electronegativity– Replace charges by charge-transfer variables
– Impose detailed balance
• Sum over CTs are deviations fromreference charge, not actual charge per se
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nQEq: Formulation
• Linear system of simultaneous equations
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Computation
• Cast system into matrix problem• Degenerate system of equations
– Singular value decomposition– Generalized Moore-Penrose inverse
(psuedoinverse)
O+2δ
H+δ H+δ
O
H H
η η
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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
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0 2 4 6 8 10 12 14 16 18 20
0.00
0.05
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0.15
0.20
0.25
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0.35
0.40
Results: NaCl
Correct asymptotic limit!
0 2 4 6 8 10 12 14 16 18 20
0.0
0.1
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0.9
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0 2 4 6 8 10 12 14 16 18 20
-0.4
-0.3
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-0.1
0.0
0.1
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Results: H2O
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Solvation of salt in H2O 6-mer• 6-mer known to be
smallest clusterneeded to fullysolvate NaCl
• Sudden limit ofdissociationdynamics: no solventreorganization
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R(Na-Cl)/Å
q/e
0 2 4 6 8 10 12 14 16 18 20
-0.3
-0.2
-0.1
0.0
0.1
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Results: NaCl.6H2O
nonvanishing residue
0 2 4 6 8 10 12 14 16 18 20
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
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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)
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Numerical Issues
• For large systems, algorithm does not findcorrect dissociation limits
• Large residual found• Low condition number• What’s going on?
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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?
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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
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Quantum Analogues?
• Quantum analogue of partial charges?– Spin-statistics theorem– Anyons
• QEq analog: Heisenberg spin magnet
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Janak’s Theorem
• Kohn-Sham one-particle orbital energiesdictate change in total energy
• Implies discontinuities as a function ofparticle number at integers:
Janak, J. F.; Phys. Rev. B, 18, 1978, 7165-7168.
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Origin of discontinuity
• Which term in universal functionalcontributes the most?– Coulomb exchange– Kinetic: Pauli exclusion principle– Unsolved question!
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Future work
• Notion of generating density matricescompatible with a given Hamiltonian
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Derivative discontinuities andionization potentials
• Implementingdiscontinuities improveestimates of ionizationpotentials
• “Double knee” feature inlaser-induced ionizationof helium atoms
• Model discontinuity incorrelation potentialneeded to obtain correctlimit
Lein, M.; Kümmel, S. Phys. Rev. Lett., 94, 2005, 143003.
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Acknowledgments