Minimization of the Kohn-Sham Energy with a Localized, Projected Search Direction

Transcription

1 Minimization of the Kohn-Sham Energy with a Localized, Projected Search Direction Courant Institute of Mathematical Sciences, NYU Lawrence Berkeley National Lab 29 October 2009

2 Joint work with Michael Overton NYU Juan Meza LBNL Chao Yang LBNL October SIAM ALA, Monterey CA

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4 In the past Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry. If mathematical analysis should ever hold a prominent place in chemistry - an aberration which is happily almost impossible - it would occasion a rapid and widespread degeneration of that science. Auguste Comte, 1830

5 Today Advances in density functional theory coupled with multinode computational clusters now enable accurate simulation of the behavior of multithousand atom complexes that mediate the electronic and ionic transfers of solar energy conversion. These new and emerging nanoscience capabilities bring a fundamental understanding of the atomic and molecular processes of solar energy utilization within reach. Basic Research Needs for Solar Energy Utilization Report of the BES Workshop on Solar Energy Utilization April 18-21, 2005

6 In the future Whereas computational methods nowadays mainly supplement experimentally obtained information, they are expected increasingly to supersede this information. Stephen Goedecker, 1999

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8 CdSe Solar Cell CMOS electrons > 1,000,000 electrons > 50,000 electrons

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10 The Problem is solved...in the Schrödinger equation we very nearly have the mathematical foundation for the solution of the whole problem of atomic and molecular structure G. N. Lewis, J. Chem. Phys. 1, 17 (1933)

11 Many-body Schrödinger Equation H Ψ ( r 1, r 2,..., r ne ) = λψ ( r 1, r 2,..., r ne ) If the n u nuclei are fixed in space at positions ˆr j, n e H = - 1 Δ ri 2 1 n u j=1 n e i=1 z j r i ˆr j + 1 r 1 i,j n i r j e Ψ contains all the information in the system and satisfies orthogonality and antisymmetry properties Ψ ( r 1, r 2,..., r ne ) 2 dr 1... dr ne gives probability density λ represents the quantized energy

12 The problem is solved almost...in the Schrödinger equation we very nearly have the mathematical foundation for the solution of the whole problem of atomic and molecular structure but the problem of the many bodies contained in the atom and the molecule cannot be completely solved without a great further development in mathematical technique. G. N. Lewis, J. Chem. Phys. 1, 17 (1933)

13 Many-body Schrödinger Equation (the but ) H Ψ ( r 1, r 2,..., r ne ) = λψ ( r 1, r 2,..., r ne ) If the n u nuclei are fixed in space at positions ˆr j, n e H = - 1 Δ ri 2 1 n u j=1 n e i=1 z j r i ˆr j + 1 r 1 i,j n i r j e Ψ contains all the information in the system and satisfies orthogonality and antisymmetry properties Ψ ( r 1, r 2,..., r ne ) 2 dr 1... dr ne gives probability density λ represents the quantized energy If we attempt to solve on a 32 x 32 x 32 grid, and n e = 5, H has dimension

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15 Density Functional Theory Hohenberg and Kohn 1964: Proved that at ground state, total energy is function of charge density ρ ( r) = n e Ψ* ( r, r 2,..., r ne ) Ψ ( r, r 2,..., r ne ) dr 2 dr 3... dr ne Kohn and Sham 1965: Proposed a practical formulation, using n e single-particle, orthogonal wavefunctions ψ j that do not interact. Electron-electron interactions modeled by exchange-correlation energy.

16 The Kohn-Sham Equations Charge Density: n e ρ (r) = ψ* i (r ) ψ i ( r ) Exchange-correlation energy: E xc (ρ) = ρ (r ) ε xc [ρ ( r )]dr 1 Kohn-Sham energy: n e E KS [{ψ j }] = ψ j ( Δψ j ) dr + V ion ( r ) ρ ( r ) dr ρ ( r ) ρ ( r ) r r drdr + E xc ( ρ ).

17 Kohn-Sham Equations continued KKT Conditions: min E KS ( ψ j ) s.t. ψj * ψ j = δ i,j for i = 1, 2,..., n e H KS ( ρ ) ψ i = λ i ψ i, ψ j * ψ j = δ i,j, for i = 1, 2,..., n e where H KS = Δ + V ion ( r ) +ρ ( r ) * 1 r + V xc ( ρ) = V ( ρ ( r ) )

18 Non-orthogonal Kohn-Sham Equations Overlap matrix: S jk = ψ* j ψ k dr Charge-density: ρ (r) = j ψ* i (r ) (S -1 ) jk ψ i ( r) k Kohn-Sham Energy: E KS ( {ψ j }) = ( S -1 ) jk j k ψ j ( Δψ k ) dr + V ion ( r ) ρ ( r ) dr ρ ( r ) ρ ( r ) r r drdr + E xc ( ρ) This energy is independent of basis

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20 Numerical Methods Step 1: Choose a discretization Plane-waves basis: ψ ( r) = k α k e ig k r 1 Advantages: ρ ( r ) * r Matrix Multiplication (FFT) Disadvantages: Inherently nonlocal Finite-differences: ψ ( r j ) = 1 h [ψ ( r j + h ) ψ ( r j h)] Advantages: Localized, highly sparse 1 Disadvantages:ρ ( r ) * r requires solution to a Poisson problem

21 Discretized Kohn-Sham Equations ( ( E (X) = trace (X*X ) -1 1 X* 2 L + α ) ) 1 2 V (r) X + α 2 4 ρt L -1 ρ [ ρ = diag X ( X*X ) -1 ] X* Look Familiar? x*ax ( x*x trace (X*X ) -1 ) X*AX Observe: E (XG) = E (X)

22 Numerical Methods Step 1: Choose a discretization Step 2: Compute a solution SCF: Solve the nonlinear Kohn-Sham equations CG: Directly minimize the Kohn-Sham Energy

23 Self-Consistent Field Iteration O ( n 3 e ) Convergence slow, non-monotonic Charge-mixing to accelerate convergence

24 CG (Fletcher-Reeves) 1. RM = - X E ( X 0 ) 2. PM = RM 3. XM = X 0 4. repeat until convergence criteria are satisfied 5. Choose αm from a line search on E ( XM + αpm) 6. X k+1 = XM + αmpm 7. R k+1 = - X E ( X k+1 ) 8. β k+1 = (rk+1 ) T r k+1 (r k ) T r k 9. P k+1 = R k+1 + β k+1 ( PM) 10. k = k end loop

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26 Two Equal Solutions Orthogonal Solution A Localized Solution Student Version of MATLAB Student Version of MATLAB

27 Sparsity Pattern nz = 150 Sparsity structure of X: Support region size 30, n = 100, n e = 5

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29 Maintaining Localization/sparsity An iterative algorithm need not maintain the sparsity pattern. Naive approach: Truncate, i.e. set X k+1 = (X k+1 ) T E and Gao approach: Minimize truncation error (localize) Compute invertible G which minimizes XG (XG) T F Remember E (XG) = E (X) for any invertible G

30 Sparse Subspace (XMG) T E(XMG) = E(XM) XM

31 CG + Localization (CG L by E and Gao) 1. RM = - X E ( X 0 ) 2. PM = RM 3. XM = X 0 4. repeat until convergence criteria are satisfied 5. Choose αm from a line search on E ( XM + αpm) 6. X k+1 = XM + αmpm 7. G = arg min G X k+1 G (X k+1 G ) T 8. X k+1 = (X k+1 G) T 9. R k+1 = - X E ( X k+1 ) 10. β k+1 = (rk+1 ) T r k+1 (r k ) T r k 11. P k+1 = R k+1 + β k+1 ( PMG) T 12. k = k end loop

32 Avoiding Local Minima: CG L versus truncation n = 100, n e = localized truncated

33 Our Approach: Localize the search direction { Sparse Dense { Pk { Sparse + Pk Localization (( + Pk)Gk) T

34 Localized Search Direction 1. RM = - X E ( X 0 ) 2. PM = RM 3. XM = X 0 4. repeat until convergence criteria are satisfied 5. G = arg min G ( XM + PM ) G (( XM + PM ) G ) T 6. PM = (( XM + PM ) G ) T XM 7. Choose αm from a line search on E ( XM + α PM) 8. X k+1 = XM + αm PM 9. R k+1 = - X E ( X k+1 ) 10. β k+1 = (rk+1 ) T r k+1 (r k ) T r k 11. P k+1 = R k+1 + β k+1 PM 12. k = k end loop

35 Avoiding Local Minima: Us vs: CG L!0.625! E and Gao o - Our method!0.635!0.64!0.645!0.65! Tolerance = 10 ⁷; n e = 5, n = 100

36 Monotonic Convergence !1 - E and Gao - Our method log(e - E*) 10!2 10!3 10!4 10!5 10!6 10!7 10!8 10! n e = 5, n = 100

37 Some timings on a bigger problem Method Mean(sec) Std. Dev. #evals time/eval Truncation E and Gao Proposed Method runs, n=5000, n e =10, Support =850

38 Where are the savings? The Gradient! S = X*X E = -XS -1 X* ( L + V ) XS -1 + (L + V ) XS -1 +α (Diag ( L -1 ρ ) XS -1 XS -1 X*Diag ( L -1 ρ ) XS -1)

39 The Overlap Matrix x S S -1 Student Version of MATLAB Student Version of MATLAB

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41 Deeper Analysis: Maintaining Descent ( trace E T P ) < 0 ( trace ( E ) T ( )) T ( XM + PM ) G < 0.

42 Integration with BFGS 10 2 CG !2 BFGS 10!4 10!6 10!8 10!10 10!12 10!14 10! n e = 5, n = 100. Bad Search Directions: CG (0), BFGS (7)

43 Solve real problems by Chelikowsky, Troullier, and Saad Open-source Fortran code In real-space, using finite-differences Exploits parallel linear algebra routines and MPI for parallelization

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