Algorithms and Computational Aspects of DFT Calculations

Transcription

1 Algorithms and Computational Aspects of DFT Calculations Part I Juan Meza and Chao Yang High Performance Computing Research Lawrence Berkeley National Laboratory IMA Tutorial Mathematical and Computational Approaches to Quantum Chemistry Institute for Mathematics and its Applications, University of Minnesota September 26-27, 2008 Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

2 Outline 1 Preliminaries 2 Density Functional Theory 3 Pseudopotentials 4 Bloch s Theorem 5 Diagonalization / Minimization 6 Improving Convergence 7 Summary Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

3 Goals 1 Brief introduction to Schrödinger s equation and Density Functional Theory 2 Overview of most commonly used approximations 3 Description of the Self-Consistent Field Iteration 4 Overview of major algorithmic components Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

4 References M. C. Payne, M. P. Teter, D. C. Allen, T. A. Arias, J. D. Joannopoulos, Iterative minimization techniques for ab initio total energy calculation: Molecular dynamics and conjugate gradients, Reviews of Modern Physics, Vol. 64, Number 4, pp (1992). Christopher J. Cramer, Essentials of Computational Chemistry, John Wiley and Sons (2003). Richard M. Martin, Electronic Structure Basic Theory and Practical Methods, Cambridge University Press (2005). F. Nogueira, A. Castro, A.L. Marques, A Tutorial on Density Functional Theory, Chapter 6, pp , A Primer in Density Functional Theory, Springer-Verlag (2002). J.M. Thijssen, Computational Physics, Cambridge University Press (2003). Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

5 Many-body electronic Schrödinger equation H Ψ k (r 1, r 2,..., r N ) = E k Ψ(r 1, r 2,..., r N ) (1) N N H = 2 2 i + V ext (r i ) + 1 e 2 (2) 2m 2 r i r j i=1 i=1 Ψ k contains all the information needed to study a system Ψ k 2 probability of finding an electron at a certain state V ext represents an external potential, e.g. Coulomb attraction by nuclei E k quantized energy Ψ k is a function of 3N variables; the electron positions, r 1,..., r N. Computational work grows like O(10 3N ) i j Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

6 Approximations commonly used Born Oppenheimer Also called adiabatic approximation Due to large difference in mass between electrons and nuclei Take nuclear positions as fixed Density Functional Theory for modeling electron-electron interactions Local Density Approximation (LDA) Pseudopotentials for handling electron-ion interactions Supercells to model systems with aperiodic geometries Methods for minimizing total energy functional Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

7 Density Functional Theory The unknown is very simple, i.e. the electron density, ρ(r) Hohenberg-KohnTheory There is a unique mapping between the ground state energy, E 0, and the ground state density, ρ 0 Exact form of the functional unknown and probably unknowable Independent particle model Electrons move independently in an average effective potential field Must add correction for exchange and correlation terms Good compromise between accuracy and feasibility Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

8 Kohn-Sham Total Energy Kohn and Sham proposed using n e noninteracting electrons moving in an effective potential due to the other electrons Replace many-particle wavefunctions with single-particle wavefunctions Kohn-Sham Total Energy n e ψ i 2 + V ext ρ + Ω E total [{ψ i }] = 1 2 i=1 Ω 1 ρ(r)ρ(r ) 2 Ω r r drdr + E xc [ρ(r)], where ρ(r) = n e i=1 ψ i(r) 2, Ω ψ iψ j = δ i,j, n e is the number of electrons, and E xc [ρ(r)], denotes the exchange correlation functional Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

9 Kohn-Sham Equations Goal is to find the ground state energy by minimizing the Kohn-Sham total energy, E total Leads to: Kohn-Sham equations Hψ i = ɛ i ψ i, i = 1, 2,..., n [ e H = 1 ] V (ρ(r)), ρ V (ρ(r)) = V ext (r) + r r + V xc(ρ) Nonlinear eigenvalue problem since the Hamiltonian, H, depends on ψ through the charge density, ρ V xc (ρ) = E xc (ρ(r))/ ρ Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

10 Pseudopotentials Interaction between electrons and the nucleus creates a problem; one needs to deal with a singularity near the atomic core, specifically the 1/r term in computation of V ext (r) Pseudopotentials are based on idea that most chemistry is dependent on valence electrons rather than core electrons Therefore we replace the core electrons (and the ionic potential) with a weaker pseudopotential Using pseudopotentials reduces the number of electrons that we need to consider, as well as the number of plane waves needed to accurately represent the wavefucntions, thereby reducing the computational cost Both empirical and ab initio forms available. Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

11 Exchange Correlation Functional Most of the complexity of DFT is hidden in the exchange correlation functional Exchange arises from antisymmetry due to the Pauli exclusion principle Correlation accounts for other many-body effects missing from single-particle approximation, e.g. K.E. not covered by first term of Hamiltonian No systematic way to improve the exchange correlation functional Local Density Approximation (LDA) Simplest approximation to exchange correlation term Assumes energy is equal to energy from a homogeneous electron gas Purely local, yet remarkably successful Known limitations Literally hundreds of functionals proposed. For an interesting historical perspective see In Pursuit of the Divine Functional, A.E. Mattsson, Science, Vol. 298, No. 5594, pp (2002). Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

12 Discretization Options Finite difference ψ (r j ) [ψ(r j + h) ψ(r j h)]/h Finite elements ψ(r) n α j φ j (r), j φ j (r) nice functions with local support Local orbital method (good for molecules) Choose φ j(r) as Gaussian or other nice functions Planewave expansion Choose φ j(r) as e ig j r Useful for modeling solids with a periodic structure Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

13 Blochs Theorem and Periodic Supercells Bloch s Theorem: In a periodic solid each electronic wave function can be expressed as the product of a periodic function φ and exp(ik r), where k is a wavevector, i.e. ψ(r) = e (ik r) φ(r) Can expand φ(r) in a set of plane waves so that ψ(r) is a sum of plane waves (more in a minute) Bloch s Theorem allows us to express the electronic wavefunctions in terms of a discrete set of plane waves Can model large periodic systems by focusing on a smaller primary cell Can also be used to model nonperiodic systems, like molecules Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

14 Plane-wave Basis Set Write wavefunction as: ψ i (r) = e ik r g j α j e igj r (3) In principle, you need an infinite plane-wave basis set In practice, you introduce an energy cutoff to truncate the basis set All terms for which the kinetic energy is bigger than the cutoff are ignored Pseudopotentials also allow us to use a much smaller number of plane-wave basis thereby reducing the computational cost As a bonus, the kinetic energy term of Hamiltonian is diagonal (in Fourier space) when using a plane-wave basis set Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

15 Finite Dimensional Problem Recall we want to min E[{ψ i}] = 1 2 Xn e i=1 Substituting (3) and after some algebra we have Z Ω Z ψ i 2 + V extρ + 1 Z ρ(r)ρ(r ) Ω 2 Ω r r drdr + E xc(ρ) min E KS (X) E kinetic (X) + E ext (X) + E Hartree (X) + E xc (X), X X=I ne E kinetic = 1 2 trace(x LX) where E ionic = trace(x V ext X) E Hartree = 1 2 ρ(x)t L ρ(x) E xc = ρ(x) T (µ xc [ρ(x)]) ρ(x) = diag(xx ) X N n e matrix Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

16 Minimizing the Total Energy KKT conditions X L(X, Λ) = 0, X X = I ne. Discretized Kohn-Sham equations can now be written as: H(X)X = XΛ, X X = I ne. Kohn-Sham Hamiltonian given by: H(X) = 1 2 L + V (X), V (X) = V ext + Diag (L ρ(x)) + Diag g xc (ρ(x)) Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

17 Approaches for Solving the Kohn-Sham Equations Work with the KS equations indirectly Self-Consistent Field Iteration View as solving a sequence of linear eigenvalue problems Need to precondition Need other acceleration techniques to improve convergence Minimize the total energy directly Direct Constrained Minimization Constrained optimization problem Also requires globalization techniques In general more robust Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

18 The SCF Iteration [ ] V (ρ(r)) ψ i = E i ψ i {ψ i } i=1,...,ne ρ(r) = n e i ψ i (r) 2 V (ρ(r)) Most of the work is in solving the linear eigenvalue problem Orthogonality constraint for the wavefunctions must be enforced explicitly If using reciprocal (Fourier) space, then you also have many 3D FFTs For large systems, the calculation of nonlocal potentials can also be expensive SCF does NOT decrease the energy monotonically Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

19 Checking for Self-consistency Convergence is usually checked by computing the change in total energy or density between iterations Recall that neither quantity is guaranteed to decrease monotonically Sometimes difficult to decide when self-consistency is reached Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

20 SCF Convergence Properties Surprisingly few results A good starting point: E. Cancès and C. Le Bris, Can we outperform the DIIS approach for electronic structure calculations? Intl. J. Quantum Chem. 79 (200), E(x) may not monotonically decrease between SCF iterations SCF does not always converge; lim i E(x (i+1) ) E(x (i) ) 0, or lim i ρ(x (i+1) ) ρ(x (i) ) 0 For some problems, one can show subsequence convergence; lim i ρ(x(i+1) ) ρ(x (i 1) ) = 0 Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

21 Example L = E(x) = 1 2 xt Lx + α 4 ρ(x)t L 1 ρ(x) ( ) ( ) ( x1 x 2, x =, ρ(x) = 1 x 2 x 2 2 min E(x) s.t. x x 2 2 = 1 [ ] L + αdiag(l 1 ρ(x)) x = λ 1 x ) Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

22 SCF Converges When α = 1.0 ρ (i) = ρ (i) ρ (i 1) Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

23 SCF Fails When α = 12.0 Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

24 Subsequence Convergence odd subsequence even subsequence Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

25 Why Does SCF Fail? SCF is attempting to minimize a sequence of surrogate models Objective: E(x) = 1 2 xt Lx + α 4 ρ(x)t L 1 ρ(x) E sur(x) = 1 2 (xt H(x (i) )x), Gradient: E(x) = H(x)x E sur(x) = H(x (i) )x Gradients match at x (i) E(x (i) ) = E sur (x (i) ) Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

26 SCF Step is Too Long! Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

27 Improving SCF Construct better surrogate Cannot afford to use local quadratic approximation (Hessian too expensive) Charge mixing to improve convergence (heuristic) Trust Region to restrict the update of the x in a small neighborhood of the gradient matching point, e.g. TRSCF Thogersen, Olsen, Yeager & Jorgensen (2004) Direct Constrained Minimization Yang, Meza & Wang (2006) 1 See talk by Chao Yang, Friday, Oct. 4, C. Yang, J. Meza, L. Wang, A Constrained Optimization Algorithm for Total Energy Minimization in Electronic Structure Calculation, J. Comp. Phy., (2006) Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

28 Mixing Linear mixing New ρ is a linear combination of the previous value and the quantity computed from the solution of the linear eigenvalue problem, i.e. ρ i+1 = β ˆρ + (1 β)ρ i Anderson extrapolation Broyden and Modified Broyden Mixing DIIS (Direct Iterative Inversion Subspace) All methods are some form of an acceleration technique for a nonlinear iteration Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

29 Trust Region Subproblem Solve min s.t. x T x = 1, E sur (x) xx T x (i) (x (i) ) T 2 F trust region constraint Equivalent to solving [H(x (i) ) σx (i) (x (i) ) T ] x = λ 1 x x T x = 1 σ is a penalty parameter (Lagrange multiplier for the trust region constraint) Need heuristic for choosing σ Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

30 SCF + Charge Mixing Improves Convergence E(x (i) ) = E(x (i) ) E min α = 12, n = 10, n e = 2 Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

31 TRSCF Further Improves Convergence How should we choose σ? Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32

32 Summary Reviewed basic approximations used in DFT Introduced the major algorithmic components Discussed methods for improving SCF convergence Introduced trust region ideas Part II of this talk will discuss many of the computational issues Juan Meza (LBNL) Algorithms and Computational Aspects of DFT Calculations September 26, / 32