Multiscale Methods for Electrostatics and Electronic Structure

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Multiscale Methods for Electrostatics and
Electronic Structure
Thomas L. Beck Department of Chemistry University
of Cincinnati thomas.beck_at_uc.edu
Acknowledgments
NSF AFOSR DoD/MURI
People
Michael Merrick, Karthik Iyer, Jian Wang, Anping
Liu, Nimal Wijesekera, Guogang Feng, Rob Coalson,
Achi Brandt, Shlomo Taasan, Jian Yin, Zhifeng
Kuang, Nobu Matsuno, Uma Mahankali
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Outline

• Physical problems multiple length and time
  scales
• Multiscale methods PDEs and molecular
  simulations
• Use of multiscale methods in our research
  Kohn-Sham, electron transport, Poisson problems,
  Poisson-Boltzmann, Poisson-Nernst-Planck
  electrodiffusion, configurational bias simulation
  of polyelectrolytes
• Electronic structure eigenvalue, Poisson,
  pseudopotentials, generalized Ritz, numerical
  results
• Monte Carlo simulations of flexible
  polyelectrolytes
• Ion channels Cl channels, PNP permeation
  calculations, homology models of human Cl channels

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Applications
Molecular Electronics Diodes, Wires, Transistors?
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I
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Au
Au
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Polyelectrolytes
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Transport through membrane channels, gating
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Multiscale Methods

• Solve PDEs with multigrid methods
• Simulate large amplitude, long time scale motions
  in molecular systems
• In both cases, utilize information from wide
  range of length scales to accelerate convergence
  the long wavelength modes are the problem

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PDEs to Solve
electron transport

variable dielectric Poisson
steady-state diffusion (PNP)
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Electronic Structure, Kohn-Sham Equations
Nonlinear!
(Goedecker, et al pseudopotentials incorporated)
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Finite-difference representation (Poisson)
(Results from Taylor series expansion of the
function)
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2-d 4th order Laplacian
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Order and accuracy H atom
Eigenvalues
Virial Ratio
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Finite element discretization

• Localized polynomial basis set
• Variational theorem obeyed
• For example, if piecewise linear functions used
  as basis, problem looks very much like (2nd
  order) FD discretization
• More adaptable
• Number of terms along row of matrix proportional
  to p3, vs. 3p1 for FD (p is the order)

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Development of real-space methods

• Poisson problems and biophysical electrostatics
• Finite differences and finite elements
• Mainly single-level solvers (Honig, McCammon,
  Sharp, Nichols)
• More recently application of MG techniques
  Holst, Coalson, Beck, Tomac, Graslund)
• Mesh refinements Bai and Brandt, Beck, Goedecker
• Electronic structure reviews
• Goedecker (linear scaling) Rev. Mod. Phys. 71,
  1085 (1999).
• Arias (wavelets) Rev. Mod. Phys. 71, 267
  (1999).
• Beck (finite differences, finite elements, and
  multiscale Rev. Mod. Phys. 72, 1041 (2000).
• Eigenvalue problems, fixed potential Rabitz,
  Seitsonen, Kolb, Ackerman, Ferrarri, Pask.
• Self-consistent problems Chelikowsky(FD), Hoshi
  and Fujiwara(FD).
• Multigrid White, Wilkins, and Teter Bernholc
  Ancilotto Davstad Beck Martin Thijssen,
  Chang, Heiskanen.

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• 6. Mesh refinement Gygi and Galli Kaxiras
  Fattebert Ono and Hirose, Beck.
• Finite elements Heinemann, Kopylow, White,
  Wilkins, Teter, Gillan, Batcho, Tsuchida, Tsukada
• Time-dependent DFT for excitations Yabana,
  Bertsch, Chelikowsky

Typical of real-space solvers without MG
acceleration (or linearized MG) is requirement of
20-50 or more self-consistency iterations to
reach the ground state. Optimal plane-wave
solvers require 5-10 iterations (Kresse and
Furthmuller).
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Advantages of Real-Space

• All iterative steps near-local in space, parallel
  algorithms
• Linear scaling algorithms and localized orbitals
• Multiscale acceleration
• More natural for finite systems
• Local mesh refinements

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Iterative relaxation efficiency
Eigenvalues of update matrix (weighted Jacobi)
Longest wavelength modes
Critical slowing down
MG
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Multigrid V-cycle
MG accelerates convergence by decimating error
components with all wavelengths!
2 relaxations per level
Correct, relax
Restrict, relax
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Alternative cycles
V-cycle
Full multigrid (FMG) good preconditioning
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Full Approximation Scheme (FAS) (For nonlinear
problems)
Coarse-grid equation
Restriction
Defect correction
Correction step
Gauss-Seidel, SOR, or Kaczmarz Relax (2 steps)
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Finite-difference representation (eigenvalue)
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FAS Eigenvalue modifications
Brandt, McCormick, and Ruge (1983)
Coarse-grid orthonormality constraint
Coarse-grid eigenvalues (same as fine grid)
Fine-grid Ritz projection preceded by
Gram-Schmidt orthogonalization this is costly
q2Ng step (see below for algorithmic
improvements) Only once per V-cycle (once per 4-6
relaxations)
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Self consistency
Ritz and update of Veff
C
Constraints
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Efficiency
All electron
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Glycine (15 states)
Benzene dithiol (21 states)
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Pseudopotentials (Goedecker et al.)
Local Part
Nonlocal Part
Projectors
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Total pseudopotential
Application of pseudopotential
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Benzene dithiol convergence
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Benzene dithiol electron density map
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Algorithm scaling

• q orbitals and Ng fine grid-points (NgH
  coarse-grid points)
• Relaxation of orbitals qNg
• Relaxation of potential Ng
• Gram-Schmidt on fine grid q2Ng
• Ritz projection on fine grid q2Ng to construct
  matrix and q3 to solve
• Computation of eigenvalues on coarse grid qNgH
• Solution of constraint equations on coarse grid
  q2NgH to construct matrix and q3 to solve
• Costiner and Taasan (1995) have developed an
  algorithm which moves the Ritz projection to
  coarse levels. Effective scaling reduced to qNg
  which is for relaxation on fine grid for orbitals
  which span the whole domain. Linear scaling if
  localized orbitals.
• Pseudopotential application qNnucNg,loc

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Generalized Ritz/backrotation (Costiner and
Taasan)
Goal is to move expensive q2Ng Ritz operation to
coarse levels using FAS strategy

• Generalized Ritz (on coarse grid) generalized
  eigenvalue problem

• Backrotation Z modified. Prevents rotations
  in degenerate subspaces, permutations,
  rescalings, and sign changes of solutions during
  MG corrections

Update
No orthogonalization required on fine grid
(except within degenerate clusters) only
relaxation
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3D Harmonic Oscillator Convergence 10 states
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Hyrogen Atom Convergence 14 states
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Neon Atom (pseudopotential)
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CO Molecule Convergence (PP)
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Generalized Ritz/Backrotation, Large Molecules?

• Converges for fixed potential problems with
  well-defined eigenvalue cluster structure
• Converges for small to medium sized molecule
  self-consistent pseudopotential calculations
• Stalls for larger systems, problem appears
  related to mixing of states in the backrotation
  step
• Solution? Update fine grid functions with
  generalized Ritz Z matrix before correction step,
  eliminates backrotation (in progress, so far have
  shown converges for fixed potential and small
  molecule cases)

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High-order mesh refinements (Beck, 1999)
Flux conservation at patch boundaries (on coarse
grids)? Lack of conservation due to defect
correction tH. Both Poisson and eigenvalue
solvers developed.
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Polyelectrolyte simulation configuration bias
MC with multigrid Poisson-Boltzmann solution for
each configuration 20000 PB solves required
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PNP Continuum Theory
Simple Moderate Complex

• PNP Theory Ion Transport

200mV
?(e(r) ??(r) ) -4??(r) Szeci J(r,t)
-D?ci(r,t) b ?V(r)ci(r,t)
At steady-state dc(r,t)/dt 0, 0 ?.J
?.?ci(r) b ?V(r)ci(r) where VU zi e? i
1,2.. N
static charge
mobile charge
Upon Slotboom transformation, 0 ?.eeff(r) (? ?
(r)) where eeff exp(- bV) and ?i exp (bV) ci
Laplace Equation
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Human ClC-2 in DMPC membrane
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Conclusions

• Discretization scheme is simple
• Rapid convergence of solvers efficient
  algorithm requires less than 10 self-consistency
  cycles in electronic structure calculations
• Time per self-consistency step comparable to (or
  slightly less) than plane-wave codes on uniform
  grids
• Optimal algorithmic scaling if localized orbitals
• With high-order methods, accuracies comparable to
  or better than plane-wave methods without
  complications of supercells
• Adaptive discretization possible
• Locality of updates and parallel algorithms
• Convergence to exact result controlled only by
  order of representation and grid spacing
• New eigenvalue algorithms reduce overhead of
  wavefunction orthogonalization and subspace
  diagonalization

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Future work

• Large systems FAS generalized Ritz with
  pseudopotentials, efficiency
• Updates of the effective potential on coarse
  levels during self-consistency iterations can
  the ground state be obtained in one
  self-consistency cycle?
• Forces
• Mesh refinements for Poisson and eigenvalue
  problems
• Transport algorithm (steady-state) Kosov
  (2002), novel current constraint method.
  Applications in molecular electronics.
• Biological channel ion transport
  Poisson-Nernst-Planck theory requires solution of
  coupled Poisson and drift/diffusion equations
  (Laplace equation with variable dielectric)