Slides for group meeting (Fall 2005).

1. Models for computing partial
charges
Jiahao Chen
Martínez Group Meeting
September 27, 2005

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. Molecular charge distributions
• Molecules as clusters of point charges
• Electrostatics in the classical limit
• Useful for molecular modeling

4. 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.

5. QEq: Rappé and Goddard, 1991
• Parameters: Mulliken electronegativities
and hardnesses
internal energy Coulomb
interaction
Rappé, A. K.; Goddard, W. A. III, J. Phys. Chem. 95, 1991, 3358-3363.

6. QEq (continued)
• Screened Coulomb interaction: two-
electron integrals over ns-ms STOs
• Sanderson electronegativity equalization
principle
• Linear system of simultaneous equations

7. QEq: Electrical interpretation
• Molecules as classical Circuit QEq
circuits element
Atom Capacitor
+ Resistor
Bond (Ideal)
Wire

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. QEq v. DMA0 on MP2/6-31G*
1.0
0.8
0.6
C
0.4
CHx
0.2
NH2
0.0
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. 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. 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. 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. 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. QEq results: NaCl dissociation
0.9
qN a _ _ R eq_
R
0.8
0.7
Too slow!
0.6
Not zero!
0.5 qN a _ _
R
0.4 qN a _R ! 1 _ _ __ :___ 6 _
_
0.3
0.2 Â_ _
R
0.1
0.0
0 2 4 6 8 10 12 14 16 18 20

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. 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 minimization
Huber, K. P.; Herzberg, G. Constants of Diatomic Molecules, Van
Nostrand Reinhold, 1978, New York, NY.

17. 5.0
QEq: Original parameters
QEq CsI
4.5 CsBr RbI
KI
CsCl RbBr
KBr
RbCl
4.0 KCl
LiI
LiBr
LiCl NaI
NaBr
3.5 NaCl
3.0 CsF
RbF
KF
2.5 NaF
LiF
2.0
SiO
1.5
CF
HF
HCl OH
1.0 CO HBr ICl
BrF
HI SH
IBr
ClF SO PN
0.5 NO
BrCl
Expt.
ClO NS
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

18. 5.0
QEq: Optimized parameters
QEq CsI
RbI
4.5
KIRbBr
CsBr
KBr
RbCl
CsCl
KCl
4.0
NaI
NaBr
NaCl
3.5 KF
RbF
NaF
CsF
3.0 LiI
LiBr
LiCl
2.5 LiF
2.0
1.5
SiO
PN
1.0
NS
HClBrF HF
SO
OH
ClO
ICl
0.5
SH
CF
ClF
HBr
IBr
BrCl Expt.
NOHI
CO
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

19. 1.0
QEq: New parameters on aa’s
0.8
0.6
Worse than before!
0.4
0.2
0.0
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. 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. 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. 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. 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. nQEq: Formulation
• Linear system of simultaneous equations

25. Computation
• Cast system into matrix problem
• Degenerate system of equations
– Singular value decomposition
– Generalized Moore-Penrose inverse
(psuedoinverse)
O+2δ O
η η
H+δ H+δ H H

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. Results: NaCl
0.40
0.9
0.8
0.35 0.7
0.6
0.5
0.30
0.4
0.3
0.25 0.2
0.1
0.0
0.20 0 2 4 6 8 10 12 14 16 18 20
0.15
0.10
Correct asymptotic limit!
0.05
0.00
0 2 4 6 8 10 12 14 16 18 20

28. 0.2
Results: H2O
0.1
0.0
-0.1
-0.2
-0.3
-0.4
0 2 4 6 8 10 12 14 16 18 20

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. q/e Results: NaCl.6H2O
0.5
0.70
0.65
0.4 0.60
0.55
0.3 0.50
0.45
0.2 0.40
0.35
0.1 0.30
0 2 4 6 8 10 12 14 16 18 20
0.0
-0.1 nonvanishing residue
-0.2
-0.3 R(Na-Cl)/Å
0 2 4 6 8 10 12 14 16 18 20

31. 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. Numerical Issues
• For large systems, algorithm does not find
correct dissociation limits
• Large residual found
• Low condition number
• What’s going on?

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. 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. Quantum Analogues?
• Quantum analogue of partial charges?
– Spin-statistics theorem
– Anyons
• QEq analog: Heisenberg spin magnet

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. Origin of discontinuity
• Which term in universal functional
contributes the most?
– Coulomb exchange
– Kinetic: Pauli exclusion principle
– Unsolved question!

38. Future work
• Notion of generating density matrices
compatible with a given Hamiltonian

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
limit
Lein, M.; Kümmel, S. Phys. Rev. Lett., 94, 2005, 143003.

40. Acknowledgments