On the Numerics of 3D Kohn-Sham System

Transcription

1 On the of 3D Kohn-Sham System Weierstrass Institute for Applied Analysis and Stochastics Workshop: Mathematical Models for Transport in Macroscopic and Mesoscopic Systems, Feb On the of 3D Kohn-Sham System

2 Outline 1 The Kohn-Sham System 2 Existence and Uniqueness of Solutions 3 On the of 3D Kohn-Sham System

3 Outline The Kohn-Sham System 1 The Kohn-Sham System 2 Existence and Uniqueness of Solutions 3 On the of 3D Kohn-Sham System

4 The System The Kohn-Sham System Poisson equation: (ɛ ϕ) = q (N A N D + ) e ξ u ξ on Ω + mixed b.c. Schrödinger equation: [ 2 2 (m 1 ξ ) + V ξ ] ψ l,ξ = E l,ξ ψ l,ξ on Ω + hom. Dirichl. b.c. with carrier density u = (u 1,..., u σ ), σ N u ξ (x) = N l,ξ (V ξ ) ψ l,ξ (V ξ )(x) 2 and effective potential l=1 V ξ (u) = e ξ E ξ + V xc,ξ (u) + e ξ qϕ(u) On the of 3D Kohn-Sham System

5 The System The Kohn-Sham System Poisson equation: (ɛ ϕ) = q (N A N D + ) e ξ u ξ on Ω + mixed b.c. Schrödinger equation: [ 2 2 (m 1 ξ ) + V ξ ] ψ l,ξ = E l,ξ ψ l,ξ on Ω + hom. Dirichl. b.c. with carrier density u = (u 1,..., u σ ), σ N u ξ (x) = N l,ξ (V ξ ) ψ l,ξ (V ξ )(x) 2 and effective potential l=1 V ξ (u) = e ξ E ξ + V xc,ξ (u) + e ξ qϕ(u) On the of 3D Kohn-Sham System

6 The System The Kohn-Sham System Poisson equation: (ɛ ϕ) = q (N A N D + ) e ξ u ξ on Ω + mixed b.c. Schrödinger equation: [ 2 2 (m 1 ξ ) + V ξ ] ψ l,ξ = E l,ξ ψ l,ξ on Ω + hom. Dirichl. b.c. with carrier density u = (u 1,..., u σ ), σ N u ξ (x) = N l,ξ (V ξ ) ψ l,ξ (V ξ )(x) 2 and effective potential l=1 V ξ (u) = e ξ E ξ + V xc,ξ (u) + e ξ qϕ(u) On the of 3D Kohn-Sham System

7 The System The Kohn-Sham System Poisson equation: (ɛ ϕ) = q (N A N D + ) e ξ u ξ on Ω + mixed b.c. Schrödinger equation: [ 2 2 (m 1 ξ ) + V ξ ] ψ l,ξ = E l,ξ ψ l,ξ on Ω + hom. Dirichl. b.c. with carrier density u = (u 1,..., u σ ), σ N u ξ (x) = N l,ξ (V ξ ) ψ l,ξ (V ξ )(x) 2 and effective potential l=1 V ξ (u) = e ξ E ξ + V xc,ξ (u) + e ξ qϕ(u) On the of 3D Kohn-Sham System

8 The Kohn-Sham System quasi Fermi-Level and Fermi-Function occupation factor N l,ξ (V ξ ) given by N l,ξ (V ξ ) = f ξ (E l,ξ (V ξ ) E F,ξ (V ξ )) f ξ a distribution function, i.e. Fermi s function (3D) f (s) = e s/k BT and quasi Fermi-level E F,ξ (V eff,ξ ) defined Ω u ξ (V eff,ξ (x))dx = N l,ξ (V eff,ξ ) = N ξ l=1 N ξ being the fixed total number of ξ-type carriers. On the of 3D Kohn-Sham System

9 The Kohn-Sham System quasi Fermi-Level and Fermi-Function occupation factor N l,ξ (V ξ ) given by N l,ξ (V ξ ) = f ξ (E l,ξ (V ξ ) E F,ξ (V ξ )) f ξ a distribution function, i.e. Fermi s function (3D) f (s) = e s/k BT and quasi Fermi-level E F,ξ (V eff,ξ ) defined Ω u ξ (V eff,ξ (x))dx = N l,ξ (V eff,ξ ) = N ξ l=1 N ξ being the fixed total number of ξ-type carriers. On the of 3D Kohn-Sham System

10 Outline Existence and Uniqueness of Solutions 1 The Kohn-Sham System 2 Existence and Uniqueness of Solutions 3 On the of 3D Kohn-Sham System

11 The Particle Density Operator Existence and Uniqueness of Solutions Definition Define the carrier density operator corresponding to f and m by N (V )(x) = f (E l (V ) E F (V )) ψ l (V )(x) 2, V L 2 (Ω) x Ω. l=1 E l (V ) and ψ l (V ) are EV and L 2 -normalized EF of H V E F (V ) defined by N (V)dx = f (E l (V ) E F (V )) = N. eigenvlue asymptotics of H V and properties of f ensure well-definedness of E F right-hand side series absolutely converges On the of 3D Kohn-Sham System

12 The Particle Density Operator Existence and Uniqueness of Solutions Definition Define the carrier density operator corresponding to f and m by N (V )(x) = f (E l (V ) E F (V )) ψ l (V )(x) 2, V L 2 (Ω) x Ω. l=1 E l (V ) and ψ l (V ) are EV and L 2 -normalized EF of H V E F (V ) defined by N (V)dx = f (E l (V ) E F (V )) = N. eigenvlue asymptotics of H V and properties of f ensure well-definedness of E F right-hand side series absolutely converges On the of 3D Kohn-Sham System

13 The Particle Density Operator Existence and Uniqueness of Solutions Definition Define the carrier density operator corresponding to f and m by N (V )(x) = f (E l (V ) E F (V )) ψ l (V )(x) 2, V L 2 (Ω) x Ω. l=1 E l (V ) and ψ l (V ) are EV and L 2 -normalized EF of H V E F (V ) defined by N (V)dx = f (E l (V ) E F (V )) = N. eigenvlue asymptotics of H V and properties of f ensure well-definedness of E F right-hand side series absolutely converges On the of 3D Kohn-Sham System

14 Solution of the Kohn-Sham System Existence and Uniqueness of Solutions Definition Suppose N A N D W 1,2 Γ (Ω), E ξ L 2 (Ω), ξ {1,..., σ}. Let ɛ,m 1,..., m σ, f 1,..., f σ be from L (Ω, B(R 3, R 3 )) and ϕ Γ given. Define the external potentials V ξ and the effective doping D by D = q(n A N D ) ϕ Γ, V ξ = e ξ qϕ Γ eξ E ξ, ξ {1,..., σ}. (V, u 1,..., u σ ) W 1,2 Γ (L 2 (Ω) σ ) is a solution of the Kohn-Sham system, if AV = D + q e ξ u ξ, ξ u ξ = N ξ (V ξ + V xc,ξ (u) + e ξ qv ). On the of 3D Kohn-Sham System

15 Existence and Uniqueness of Solutions Existence and Uniqueness of Solution without V xc Monotonicity and Lipschitz continuity of the Operator A yield the result: Theorem The Schrödinger-Poisson system without exchange-correlation potential has the unique solution (V, N 1 (V 1 + V ),..., N σ (V σ + V )). the operator assigning the solution V to V = (V 1,..., V σ ) is L : (L 2 (Ω)) σ W 1,2 Γ (Ω), L(V) = V L is boundedly Lipschitz continuous On the of 3D Kohn-Sham System

16 Existence and Uniqueness of Solutions Existence and Uniqueness of Solution without V xc Monotonicity and Lipschitz continuity of the Operator A yield the result: Theorem The Schrödinger-Poisson system without exchange-correlation potential has the unique solution (V, N 1 (V 1 + V ),..., N σ (V σ + V )). the operator assigning the solution V to V = (V 1,..., V σ ) is L : (L 2 (Ω)) σ W 1,2 Γ (Ω), L(V) = V L is boundedly Lipschitz continuous On the of 3D Kohn-Sham System

17 Existence and Uniqueness of Solutions Existence of Solution to the Kohn-Sham System Definition (Fixed Point Mapping) Let V = (V 1,..., V σ ) (L 2 (Ω)) σ be a given tupel of external potentials and N 1..., N σ the fixed number of carriers. Define L 1 N = {u = (u 1,..., u σ ) : u ξ 0, u ξ (x)dx = N ξ } and Φ : L 1 N L1 N as Φ ξ (u) = N ξ ( Vξ + V xc,ξ (u) + e ξ ql(v 1 + V xc,1 (u),..., V σ + V xc,σ (u)) ) Theorem (Existence of Fixed Point) If V xc,ξ is for any ξ {1,..., σ} a bounded and continuous mapping from (L 1 (Ω)) σ into L 2 (Ω), then the mapping Φ has a fixed point. On the of 3D Kohn-Sham System

18 Existence and Uniqueness of Solutions Existence of Solution to the Kohn-Sham System Definition (Fixed Point Mapping) Let V = (V 1,..., V σ ) (L 2 (Ω)) σ be a given tupel of external potentials and N 1..., N σ the fixed number of carriers. Define L 1 N = {u = (u 1,..., u σ ) : u ξ 0, u ξ (x)dx = N ξ } and Φ : L 1 N L1 N as Φ ξ (u) = N ξ ( Vξ + V xc,ξ (u) + e ξ ql(v 1 + V xc,1 (u),..., V σ + V xc,σ (u)) ) Theorem (Existence of Fixed Point) If V xc,ξ is for any ξ {1,..., σ} a bounded and continuous mapping from (L 1 (Ω)) σ into L 2 (Ω), then the mapping Φ has a fixed point. On the of 3D Kohn-Sham System

19 Existence and Uniqueness of Solutions Existence of Solution to the Kohn-Sham System Theorem (Equivalence of Solutions) u = (u 1,..., u σ ) is a fixed point of Φ if and only if (V, u 1,..., u σ ) = (A ( 1 D + q ) ) e ξ u ξ, u 1,..., u σ is a solution of the Kohn-Sham system. the Kohn-Sham system always admits a solution. On the of 3D Kohn-Sham System

20 Outline 1 The Kohn-Sham System 2 Existence and Uniqueness of Solutions 3 On the of 3D Kohn-Sham System

21 The Mapping T (η) η η = (η 1,..., η σ ) u = e η δ, δ > 0 constant solve Poisson s equation for potential ϕ(u, N A N D ) obtain V ξ (u) = e ξ E ξ + V xc,ξ (u) + e ξ qϕ solve EVP for Schrödinger s equation [ ( 2 /2) (1/m ξ ) + V eff,ξ ]ψ l,ξ = E l,ξ ψ l,ξ compute carrier densities u ξ (x) = l N l,ξ ψ l,ξ (x) 2 η = log(u + δ) On the of 3D Kohn-Sham System

22 The Mapping T (η) η η = (η 1,..., η σ ) u = e η δ, δ > 0 constant solve Poisson s equation for potential ϕ(u, N A N D ) obtain V ξ (u) = e ξ E ξ + V xc,ξ (u) + e ξ qϕ solve EVP for Schrödinger s equation [ ( 2 /2) (1/m ξ ) + V eff,ξ ]ψ l,ξ = E l,ξ ψ l,ξ compute carrier densities u ξ (x) = l N l,ξ ψ l,ξ (x) 2 η = log(u + δ) On the of 3D Kohn-Sham System

23 The Mapping T (η) η η = (η 1,..., η σ ) u = e η δ, δ > 0 constant solve Poisson s equation for potential ϕ(u, N A N D ) obtain V ξ (u) = e ξ E ξ + V xc,ξ (u) + e ξ qϕ solve EVP for Schrödinger s equation [ ( 2 /2) (1/m ξ ) + V eff,ξ ]ψ l,ξ = E l,ξ ψ l,ξ compute carrier densities u ξ (x) = l N l,ξ ψ l,ξ (x) 2 η = log(u + δ) On the of 3D Kohn-Sham System

24 The Mapping T (η) η η = (η 1,..., η σ ) u = e η δ, δ > 0 constant solve Poisson s equation for potential ϕ(u, N A N D ) obtain V ξ (u) = e ξ E ξ + V xc,ξ (u) + e ξ qϕ solve EVP for Schrödinger s equation [ ( 2 /2) (1/m ξ ) + V eff,ξ ]ψ l,ξ = E l,ξ ψ l,ξ compute carrier densities u ξ (x) = l N l,ξ ψ l,ξ (x) 2 η = log(u + δ) On the of 3D Kohn-Sham System

25 Properties of T (η) Solution of Kohn-Sham system is a fixed point of T (η) δ = added to avoid singularity of logarithm at zero additional smoothness of logarithm improves convergence pure iteration scheme may or may not converge stabilization and acceleration needed On the of 3D Kohn-Sham System

26 Properties of T (η) Solution of Kohn-Sham system is a fixed point of T (η) δ = added to avoid singularity of logarithm at zero additional smoothness of logarithm improves convergence pure iteration scheme may or may not converge stabilization and acceleration needed On the of 3D Kohn-Sham System

27 Properties of T (η) Solution of Kohn-Sham system is a fixed point of T (η) δ = added to avoid singularity of logarithm at zero additional smoothness of logarithm improves convergence pure iteration scheme may or may not converge stabilization and acceleration needed On the of 3D Kohn-Sham System

28 Heuristic Motivation via Gummel s Method Gummel s method originally for the drift-diffusion model Kerkhoven analyzed qualitative behavior converges while sufficiently far away from solution slows down when approaching to the solution opposite to Newton s method this behavior is due to the ellipticity of the involved equations true for the quantum-mechanical system as well On the of 3D Kohn-Sham System

29 Heuristic Motivation via Gummel s Method Gummel s method originally for the drift-diffusion model Kerkhoven analyzed qualitative behavior converges while sufficiently far away from solution slows down when approaching to the solution opposite to Newton s method this behavior is due to the ellipticity of the involved equations true for the quantum-mechanical system as well On the of 3D Kohn-Sham System

30 Heuristic Motivation via Gummel s Method Gummel s method originally for the drift-diffusion model Kerkhoven analyzed qualitative behavior converges while sufficiently far away from solution slows down when approaching to the solution opposite to Newton s method this behavior is due to the ellipticity of the involved equations true for the quantum-mechanical system as well On the of 3D Kohn-Sham System

31 Stabilization and Acceleration of T (η) Stabilization: pure appliance of T (η) causes convergence instabilities stabilize through adaptive underrelaxation fixed point iteration T (η) = η until close to the solution Acceleration: accelerate convergence by employing Newton s method root-finding problem T (η) η = 0 Jacobian-free version based on GMRES until convergence On the of 3D Kohn-Sham System

32 Stabilization and Acceleration of T (η) Stabilization: pure appliance of T (η) causes convergence instabilities stabilize through adaptive underrelaxation fixed point iteration T (η) = η until close to the solution Acceleration: accelerate convergence by employing Newton s method root-finding problem T (η) η = 0 Jacobian-free version based on GMRES until convergence On the of 3D Kohn-Sham System

33 Adaptive Underrelaxation initialize: ω = 1, choose η 0, set η 1 = 0 and η 2 = η 0 Iterate on i: if T (η i ) η i T (η i 1 ) η i 1 > T (η i 1) η i 1 T (η i 2 ) η i 2 then ω := ω 0.8, ω := min(ω, T (η i 1) η i 1 ) T (η i ) η i η i+1 = ω T (η i ) + (1 ω )η i until convergence or ω decreases 5 times in a row or ω remains constant 10 times in a row On the of 3D Kohn-Sham System

34 Adaptive Underrelaxation initialize: ω = 1, choose η 0, set η 1 = 0 and η 2 = η 0 Iterate on i: if T (η i ) η i T (η i 1 ) η i 1 > T (η i 1) η i 1 T (η i 2 ) η i 2 then ω := ω 0.8, ω := min(ω, T (η i 1) η i 1 ) T (η i ) η i η i+1 = ω T (η i ) + (1 ω )η i until convergence or ω decreases 5 times in a row or ω remains constant 10 times in a row On the of 3D Kohn-Sham System

35 Adaptive Underrelaxation initialize: ω = 1, choose η 0, set η 1 = 0 and η 2 = η 0 Iterate on i: if T (η i ) η i T (η i 1 ) η i 1 > T (η i 1) η i 1 T (η i 2 ) η i 2 then ω := ω 0.8, ω := min(ω, T (η i 1) η i 1 ) T (η i ) η i η i+1 = ω T (η i ) + (1 ω )η i until convergence or ω decreases 5 times in a row or ω remains constant 10 times in a row On the of 3D Kohn-Sham System

36 Acceleration by Newton s Method Reformulation: η i+1 = T (η i ) η T (η) = 0 Newton: requires solution of linear system [I η T (η i )]dη = [η i T (η i )] η T (η i ) is the Jacobian matrix of T at η i not known explicitly solve system without generating the Jacobian nonlinear version of GMRES (NLGMR) On the of 3D Kohn-Sham System

37 Acceleration by Newton s Method Reformulation: η i+1 = T (η i ) η T (η) = 0 Newton: requires solution of linear system [I η T (η i )]dη = [η i T (η i )] η T (η i ) is the Jacobian matrix of T at η i not known explicitly solve system without generating the Jacobian nonlinear version of GMRES (NLGMR) On the of 3D Kohn-Sham System

38 Acceleration by Newton s Method Reformulation: η i+1 = T (η i ) η T (η) = 0 Newton: requires solution of linear system [I η T (η i )]dη = [η i T (η i )] η T (η i ) is the Jacobian matrix of T at η i not known explicitly solve system without generating the Jacobian nonlinear version of GMRES (NLGMR) On the of 3D Kohn-Sham System

39 Derivative-free GMRES Solution of Newton s equation equivalent to minimization over dη of (I T )(η i ) + [I η T (η i )]dη 2 GMRES: find approximate solution in Krylov subspace K m = span{v 1, [I η T (η i )]v 1,..., [I η T (η i )] m 1 v 1 } ONB of K m easily gained by Arnoldi process, provided v [I η T (η i )]v is available η T (η i ) never needed explicitly only matrix-vector multiplication η T (η i )v approximate by: η T (η i )v T (η i + hv) T (η i ) h On the of 3D Kohn-Sham System

40 Derivative-free GMRES Solution of Newton s equation equivalent to minimization over dη of (I T )(η i ) + [I η T (η i )]dη 2 GMRES: find approximate solution in Krylov subspace K m = span{v 1, [I η T (η i )]v 1,..., [I η T (η i )] m 1 v 1 } ONB of K m easily gained by Arnoldi process, provided v [I η T (η i )]v is available η T (η i ) never needed explicitly only matrix-vector multiplication η T (η i )v approximate by: η T (η i )v T (η i + hv) T (η i ) h On the of 3D Kohn-Sham System

41 Derivative-free GMRES Solution of Newton s equation equivalent to minimization over dη of (I T )(η i ) + [I η T (η i )]dη 2 GMRES: find approximate solution in Krylov subspace K m = span{v 1, [I η T (η i )]v 1,..., [I η T (η i )] m 1 v 1 } ONB of K m easily gained by Arnoldi process, provided v [I η T (η i )]v is available η T (η i ) never needed explicitly only matrix-vector multiplication η T (η i )v approximate by: η T (η i )v T (η i + hv) T (η i ) h On the of 3D Kohn-Sham System

42 Adaption of NLGMR adjust accuracy of solution to Newton s method adaptively vary number m of steps in NLGMR η 0 : current approximate solution to η T (η) = 0 η m : solution after m steps of GMRES nonlinear residual: linear residual: res nl = η m T (η m ) res lin = η 0 T (η 0 ) + [I η T (η 0 )](η m η 0 ) nonlinearity mild res nl res lin res nl res lin linearized model not good accurate solution of Newton s method wasteful On the of 3D Kohn-Sham System

43 Adaption of NLGMR adjust accuracy of solution to Newton s method adaptively vary number m of steps in NLGMR η 0 : current approximate solution to η T (η) = 0 η m : solution after m steps of GMRES nonlinear residual: linear residual: res nl = η m T (η m ) res lin = η 0 T (η 0 ) + [I η T (η 0 )](η m η 0 ) nonlinearity mild res nl res lin res nl res lin linearized model not good accurate solution of Newton s method wasteful On the of 3D Kohn-Sham System

44 Adaption of NLGMR adjust accuracy of solution to Newton s method adaptively vary number m of steps in NLGMR η 0 : current approximate solution to η T (η) = 0 η m : solution after m steps of GMRES nonlinear residual: linear residual: res nl = η m T (η m ) res lin = η 0 T (η 0 ) + [I η T (η 0 )](η m η 0 ) nonlinearity mild res nl res lin res nl res lin linearized model not good accurate solution of Newton s method wasteful On the of 3D Kohn-Sham System

45 Adaption of NLGMR adjust accuracy of solution to Newton s method adaptively vary number m of steps in NLGMR η 0 : current approximate solution to η T (η) = 0 η m : solution after m steps of GMRES nonlinear residual: linear residual: res nl = η m T (η m ) res lin = η 0 T (η 0 ) + [I η T (η 0 )](η m η 0 ) nonlinearity mild res nl res lin res nl res lin linearized model not good accurate solution of Newton s method wasteful On the of 3D Kohn-Sham System

46 Resulting NLGMR Iteration set m = 2 get initial guess η 0 for carrier density employ m steps of GMRES: yields η m adapt m: 2 3 res nl / res lin 3 2 m := min(2m, 25) 3 2 res nl / res lin 5 m := m else m := max(2, m/2) perform linesearch for stepsize τ guarantee decrease of (η 0 + τdη) T (η 0 + τdη) until convergence form of Newton s method quadratic rate of convergence On the of 3D Kohn-Sham System

47 Resulting NLGMR Iteration set m = 2 get initial guess η 0 for carrier density employ m steps of GMRES: yields η m adapt m: 2 3 res nl / res lin 3 2 m := min(2m, 25) 3 2 res nl / res lin 5 m := m else m := max(2, m/2) perform linesearch for stepsize τ guarantee decrease of (η 0 + τdη) T (η 0 + τdη) until convergence form of Newton s method quadratic rate of convergence On the of 3D Kohn-Sham System

48 Resulting NLGMR Iteration set m = 2 get initial guess η 0 for carrier density employ m steps of GMRES: yields η m adapt m: 2 3 res nl / res lin 3 2 m := min(2m, 25) 3 2 res nl / res lin 5 m := m else m := max(2, m/2) perform linesearch for stepsize τ guarantee decrease of (η 0 + τdη) T (η 0 + τdη) until convergence form of Newton s method quadratic rate of convergence On the of 3D Kohn-Sham System

49 Resulting NLGMR Iteration set m = 2 get initial guess η 0 for carrier density employ m steps of GMRES: yields η m adapt m: 2 3 res nl / res lin 3 2 m := min(2m, 25) 3 2 res nl / res lin 5 m := m else m := max(2, m/2) perform linesearch for stepsize τ guarantee decrease of (η 0 + τdη) T (η 0 + τdη) until convergence form of Newton s method quadratic rate of convergence On the of 3D Kohn-Sham System

50 Resulting NLGMR Iteration set m = 2 get initial guess η 0 for carrier density employ m steps of GMRES: yields η m adapt m: 2 3 res nl / res lin 3 2 m := min(2m, 25) 3 2 res nl / res lin 5 m := m else m := max(2, m/2) perform linesearch for stepsize τ guarantee decrease of (η 0 + τdη) T (η 0 + τdη) until convergence form of Newton s method quadratic rate of convergence On the of 3D Kohn-Sham System

51 Summary of the Algorithm take initial guess perform adaptive underrelaxation until close to the solution perform NLGMR method Newton s method; derivative-free GMRES until convergence On the of 3D Kohn-Sham System

52 Summary of the Algorithm take initial guess perform adaptive underrelaxation until close to the solution perform NLGMR method Newton s method; derivative-free GMRES until convergence On the of 3D Kohn-Sham System

53 Summary of the Algorithm take initial guess perform adaptive underrelaxation until close to the solution perform NLGMR method Newton s method; derivative-free GMRES until convergence On the of 3D Kohn-Sham System

54 Outline 1 The Kohn-Sham System 2 Existence and Uniqueness of Solutions 3 On the of 3D Kohn-Sham System

55 General Implementation: in the Framework of WIAS-pdelib2 (C++) Discretization: Finite Volume Method TetGen: Tetrahedral Mesh Generator and 3D Delaunay Triangulator (tetgen.berlios.de) Eigenvalues: ARPACK: Large Scale Eigenvalue Solver On the of 3D Kohn-Sham System

56 Grid 4 regions, cells, 6 bregions 9905 bfaces Input: 320 points, 172 faces, 4 regions, 6 bregions Output: 4 regions, points, cells, 6 bregions, 9905 bfaces On the of 3D Kohn-Sham System

57 Single Electron f f e e-26 cpu seconds steps total underrelaxed Newton On the of 3D Kohn-Sham System

58 Residual Evolution residual iteration steps On the of 3D Kohn-Sham System

59 A.T. Galick, T. Kerkhoven, U. Ravaioli, J.H. Arends, Y. Saad Efficient numerical simulation of electron states in quantum wires. J. Appl. Phys., 68(7): , 1990 On the of 3D Kohn-Sham System