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Multiscale Methods for Electrostatics and Electronic Structure

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Title: Multiscale Methods for Electrostatics and Electronic Structure


1
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
2
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

3
Applications
Molecular Electronics Diodes, Wires, Transistors?
DV
I
e-
Au
Au
S
S
4
Polyelectrolytes
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5
Transport through membrane channels, gating
W

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P
M
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6
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

7
PDEs to Solve
electron transport

variable dielectric Poisson
steady-state diffusion (PNP)
8
Electronic Structure, Kohn-Sham Equations
Nonlinear!
(Goedecker, et al pseudopotentials incorporated)
9
Finite-difference representation (Poisson)
(Results from Taylor series expansion of the
function)
10
2-d 4th order Laplacian
11
Order and accuracy H atom
Eigenvalues
Virial Ratio
12
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)

13
(No Transcript)
14
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.


15
  • 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).
16
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

17
Iterative relaxation efficiency
Eigenvalues of update matrix (weighted Jacobi)
Longest wavelength modes
Critical slowing down
MG
18
Multigrid V-cycle
MG accelerates convergence by decimating error
components with all wavelengths!
2 relaxations per level
Correct, relax
Restrict, relax
19
Alternative cycles
V-cycle
Full multigrid (FMG) good preconditioning
20
Full Approximation Scheme (FAS) (For nonlinear
problems)
Coarse-grid equation
Restriction
Defect correction
Correction step
Gauss-Seidel, SOR, or Kaczmarz Relax (2 steps)
21
Finite-difference representation (eigenvalue)
22
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)
23
Self consistency
Ritz and update of Veff
C
Constraints
24
Efficiency
All electron
25
Glycine (15 states)
Benzene dithiol (21 states)
26
Pseudopotentials (Goedecker et al.)
Local Part
Nonlocal Part
Projectors
27
Total pseudopotential
Application of pseudopotential
28
Benzene dithiol convergence
29
Benzene dithiol electron density map
30
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

31
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
32
3D Harmonic Oscillator Convergence 10 states
33
Hyrogen Atom Convergence 14 states
34
Neon Atom (pseudopotential)
35
CO Molecule Convergence (PP)
36
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)

37
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.
38
Polyelectrolyte simulation configuration bias
MC with multigrid Poisson-Boltzmann solution for
each configuration 20000 PB solves required
39
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
40
Human ClC-2 in DMPC membrane
41
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

42
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)
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