Multiscale Methods for Electronic Structure and Membrane Channels

1
Multiscale Methods for Electronic Structure and
Membrane Channels
Thomas L. Beck Department of Chemistry University
of Cincinnati thomas.beck_at_uc.edu
Acknowledgments
NSF DoD/MURI
People
Anping Liu, Nimal Wijesekera, Guogang Feng, Jian
Yin, Zhifeng Kuang, Uma Mahankali, Rob Coalson,
Achi Brandt
2
Applications
Molecular Electronics Diodes, Wires, Transistors?
DV
I
Benzene dithiol
e-
Au
Au
S
S
3
Transport through membrane channels, gating
Regulation
W

-
P
M
-

Gate

Selectivity
4
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)
6
Electronic Structure, Kohn-Sham Equations
Nonlinear!
(Goedecker, et al analytic pseudopotentials)
7
Finite-difference representation (eigenvalue)
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2-d 4th order Laplacian (12th order used here in
electronic structure calculations)
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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 nonlinear methods (FAS)
• 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
FMG
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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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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
Potential Updates?
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Efficiency
All electron
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Glycine (15 states)
Benzene dithiol (BDT, 21 states)
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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
Orthogonalization of full subspace not required
on fine grid
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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? Perform Ritz projection on overlapping
  eigenvalue clusters on the fine scale ? restores
  convergence

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• S simultaneous
• C cluster
• R Ritz

Glycine
S, CR
S, R
CR
R
21

• S simultaneous
• C cluster
• R Ritz

Benzene dithiol (BDT)
S, CR
R
CR
S, R
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Transport in molecular devices
Applied potential e ? I-V (?)
BDT
Au
Au
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Constrained current formulation (Kosov, 2002)

• Impose the current
• Solve a modified Schrodinger equation
• Finite system, localized eigenfunctions (no
  difficulties with scattering boundary
  conditions).
• No E contour integration to get the density
  matrix
• Problem becomes one of obtaining the potential
  that drives the current
• Solution nonorthogonal localized orbital method
  (Fattebert, Bernholc, and Gygi) ? obtain
  transmission function T(E) from NEGF method ?
  potential using Landauer theory. FAS solver
  developed, linear scaling

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NEGF method (nonorthogonal localized orbitals)
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(Au Benzene dithiol Au) T(E)
(LDA level)
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LUMO
Conducting ?





HOMO
Stokbro, et al.
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I-V for BDT
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Conductance for BDT
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(GGA level, equilibrium)
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(No Transcript)
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Proteins
Bacterial Cl channel mutant (open)
Biased MC for locating ion transit pathways
TransPath. Uses MG methods to obtain the
potential.
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1OTS closed structure (gating and proton access?)
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Conclusions

• Nonlinear multigrid methods (FAS) provide
  efficient solution of the Kohn-Sham equations ?
  competitive with best plane-wave methods for
  convergence
• Further exploitation of the FAS ideas can move
  costly orthogonalization operations to the coarse
  level while maintaining efficiency
• FAS nonorthogonal localized orbital method ?
  linear scaling
• Above methods applied to constrained current/NEGF
  formalism to produce ab initio I-V curves for
  molecular electronic devices
• MG methods can be applied to study ion transport
  through biological ion channels