Solving Many-Body Schrödinger Equation Using Density Functional Theory and Finite Elements

Transcription

1 Solving Many-Body Schrödinger Equation Using Density Functional Theory and Finite Elements Institute of Physics, Academy of Sciences of the Czech Republic June 21, 2008

2 Introduction Contens Density Functional Theory Spherically symmetric problems, examples Mixing schemes Non symmetric 3D problems, Finite Element Method

3 Schrödinger equation where Ĥ Ψ = ( ˆT + Û + ˆV ) Ψ = E Ψ Û = i<j ˆT = N i i U(r i, r j ) = 1 U(r i, r j ) 2 U(r i, r j ) = U(r j, r i ) = ˆV = i,j N v(r i ) i 1 r i r j v(r i ) = k Z k r i R k

4 Density Functional Theory We solve the Kohn-Sham equations: ( V H (r) + V xc (r) + v(r))ψ i (r) = ɛ i ψ(r) that yield the orbitals ψ i that reproduce the density n(r) of the original interacting system n(r) = V H (r) = δe H δn(r) = 1 2 N ψ i (r) 2 i n(r ) r r d3 r 2 V H = n(r) E xc [n] = (T + U)[n] E H [n] T S [n] V xc (r) = δe xc[n] δn(r) v(r) = k Z k r R k

5 Atomic wavefunctions Spherically symmetric potential: Radial Schrödinger equation: V (x) = V (r) ψ nlm (x) = R nl (r) Y lm ( x r R nl + 2 r R nl + 2M 2 (E V )R l(l + 1) nl r 2 R nl = 0 )

6 Relativistic atomic wavefunctions Dirac equation: (icγ µ D µ mc 2 )ψ = 0 D µ = µ + iea µ Radial Dirac equation: ( ) [ 2 (E g κ + g κ+ ) κ(κ + 1) V r + V 2Mc 2 Radial Schrödinger equation: R + 2 r R + f κ = g κ 2Mc + κ + 1 g κ r 2Mc R 2 = f 2 + g 2 [ (E V ) 2Mr 2 + κ + 1 4M 2 c 2 r V ] l(l + 1) 2Mr 2 2MR = 0 ] 2Mg κ = 0

7 Code (300 lines in Python, 800 in Fortran) from atom import atom, show import radial import utils def do(z): R = radial.create_log_grid(z) s = atom(z,alpha=0.3,iter=20,relat=0,grid=r) n5_lda = radial.ks_construct_density(s,r,z)*r*r s = atom(z,alpha=0.3,iter=20,relat=2,grid=r) n5_rlda = radial.ks_construct_density(s,r,z)*r*r utils.makeplot(r,[ (n5_lda,"b-","non-relativistic"), (n5_rlda,"g-","full relativistic"), ],title="z=%d"%(z),xleg="r", yleg="radial density * R^2") do(5) #do(82)

8 Boron

9 Lead

10 Lead - nonrelativistic calculation Iterations: 20 F (x) = Agrees with NIST: 1s( 2): s( 2): p( 6): s( 2): p( 6): d(10): s( 2): p( 6): d(10): f(14): s( 2): p( 6): d(10): s( 2): p( 2):

11 Lead - relativistic calculation Iterations: 20 F (x) = s( 2) j=l+1/2: s( 2) j=l+1/2: p( 6) j=l-1/2: p( 6) j=l+1/2: s( 2) j=l+1/2: p( 6) j=l-1/2: p( 6) j=l+1/2: d(10) j=l-1/2: d(10) j=l+1/2: s( 2) j=l+1/2: p( 6) j=l-1/2: p( 6) j=l+1/2: d(10) j=l-1/2: d(10) j=l+1/2: s( 2) j=l+1/2: f(14) j=l-1/2: f(14) j=l+1/2: p( 6) j=l-1/2: p( 6) j=l+1/2: d(10) j=l-1/2: d(10) j=l+1/2: s( 2) j=l+1/2: p( 2) j=l-1/2:

12 Iteration to self-consistency The problem: F(x) = x equivalently R(x) = 0 for R(x) = F(x) x. We approximate R(x M+1 ) R(x M ) J (x M+1 x M ) with the Jacobian We want R(x M+1 ) = 0: J ij = R i x j x M+1 x M J 1 R(x M ) J is approximated by a sequence of J 0, J 1, J 2,...

13 Linear mixing with so x M+1 x M J 1 M R(x M) J 1 M = α1 x M+1 = x M + αr(x M ) = x M + α(f(x M ) x M ) SciPy from scipy.optimize.nonlin import linearmixing

14 exciting mixing Used in the FP-LAPW DFT code ( with x M+1 x M J 1 M R(x M) J 1 M = diag(β 1, β 2, β 3,... ) start with β 1 = β 2 = β 3 = = α and at every iteration adjust the parameters β i according to this very simple algorithm: if R i (x M 1 )R i (x M ) > 0 then increase β i by α otherwise set β i = α (if β i > α max, set β i = α max ). SciPy from scipy.optimize.nonlin import excitingmixing

15 Broyden update The first Broyden method: J M+1 = J M ( R(x M) + J M x M ) x T M x M 2 The second Broyden method: J 1 M+1 = J 1 M + ( x M J 1 M starting with the linear mixing: J 1 0 = α1 R(x M)) R(x M ) T R(x M ) 2 SciPy from scipy.optimize import broyden1, broyden2

16 low memory second Broyden update The second Broyden method (J 1 M+1 = J 1 M with + ( x M J 1 M R(x M)) R(x M ) T R(x M ) 2 J 1 M+1 = J 1 M + uvt u = x M J 1 M R(x M) v = R(x M) R(x M ) 2 so the whole inverse Jacobian can be written as ) can be written as J 1 M = α1 + u 1v T 1 + u 2 v T 2 + u 3 v T 3 + J 1 M y = αy + u 1(v T 1 y) + u 2 (v T 2 y) + u 3 (v T 3 y) + SciPy from scipy.optimize import broyden3

17 Other methods The generalized Broyden method (modified Broyden method): M 1 p=m k (1 + ω 2 0δ pn ) R(x n ) T R(x p )γ p = R(x n ) T R(x M ) x M+1 = x M + β M R(x M ) M 1 p=m k other methods: Anderson, extended Anderson γ p ( x p + β M R(x p )) SciPy from scipy.optimize import broyden generalized, anderson, anderson2

18 Finite element formulation One particle Schrödinger equation: ) ( 2 2m 2 + V ψ = Eψ. FEM: (K ij + V ij ) q j = EM ij q j + F i, V ij = φ i V φ j dv, M ij = φ i φ j dv, K ij = 2 φ i φ j dv, 2m F i = 2 2m dψ dn φ i ds. Usually we set F i = 0.

19 SfePy SfePy = general finite element analysis software BSD open-source license available at (developers) mailing lists, issue (bug) tracking we encourage and support everyone who joins! (project information) selected applications: homogenization of porous media (parallel flows in a deformable porous medium) acoustic band gaps (homogenization of a strongly heterogenous elastic structure: phononic materials) shape optimization in incompressible flow problems

20 Particle in the box V (x) = Analytic solution: { 0, inside the box a a a, outside E n1 n 2 n 3 = π2 ( n 2 2a n2 2 + n3 2 ) where n i = 1, 2, 3,... are independent quantum numbers. We chose a = 1, i.e.: E 111 = , E 211 = E 121 = E 112 = , E 122 = E 212 = E 221 = , E 311 = E 131 = E 113 = E 222 = , E 123 = Eperm. = Numerical solution (a = 1, nodes): E theory FEM

21 3D Harmonic oscillator V (r) = { 1 2 ω2 r 2, inside the box a a a, outside Analytic solution in the limit a : ( E nl = 2n + l + 3 ) ω 2 where n, l = 0, 1, 2,.... Degeneracy is 2l + 1, so: E 00 = 3 2, triple E 01 = 5 2, E 10 = 7 2, quintuple E 02 = 7 2 triple E 11 = 9 2, quintuple E 12 = 11 2 : Numerical solution (a = 15, ω = 1, nodes): E theory FEM

22 3D Harmonic oscillator Eigenvectors: 0th 12th 10th 12th

23 Hydrogen atom V (r) = Analytic solution in the limit a : { 1 r, inside the box a a a, outside E n = 1 2n 2 where n = 1, 2, 3,.... Degeneracy is n 2, so: E 1 = 1 2 = 0.5, E 2 = 1 8 = 0.125, E 3 = 1 18 = 0.055, E 4 = 1 32 = Numerical solution (a = 15, nodes): E theory FEM

24 Hydrogen atom 11th eigenvalue (calculated: , exact: ), on the mesh with tetrahedrons and nodes, for the hydrogen atom (V=-1/r).

25 FEM+DFT We solve the Kohn-Sham equations using FEM: ( V H (r) + V xc (r) + v(r))ψ i (r) = ɛ i ψ(r) that yield the orbitals ψ i that reproduce the density n(r) of the original interacting system n(r) = N ψ i (r) 2 i 2 V H = n(r) v(r) = k Z k r R k

26 Boron Current status (uniform tetrahedral mesh, nodes): radial FEM Bad convergence should greatly improve with a better mesh 2th eigenvector (contours and a slice), 3th eigenvector (slice)

27 Conclusion, Future work automatic mesh generation and refining use pseudopotentials (reduces the number of electrons to solve for) only depend on open source (free software) solutions Acknowledgements This research was partly supported by the LC06040 research center project and the GACR grant no. IAA