Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

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Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

Eric Cancès, Rachida Chakir, Yvon Maday

To cite this version:

Eric Cancès, Rachida Chakir, Yvon Maday. Numerical analysis of the planewave discretization of

some orbital-free and Kohn-Sham models. ESAIM: Mathematical Modelling and Numerical Analysis,

EDP Sciences, 2012, 46 (02), pp.341-388. �10.1051/m2an/2011038�. �hal-00471938�

Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

Eric Canc` es

, Rachida Chakir

and Yvon Maday

April 8, 2010

Abstract

We providea priorierror estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas- Fermi-von Weizs¨acker (TFW) model and for the spectral discretization of the Kohn-Sham model, within the local density approximation (LDA).

These models allow to compute approximations of the ground state energy and density of molecular systems in the condensed phase. The TFW model is stricly convex with respect to the electronic density, and allows for a comprehensive analysis. This is not the case for the Kohn-Sham LDA model, for which the uniqueness of the ground state electronic density is not guaranteed. Under a coercivity assumption on the second order optimality condition, we prove that for large enough energy cut-offs, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of any Kohn-Sham ground state, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.

1 Introduction

Density Functional Theory (DFT) is a powerful method for computing ground state electronic energies and densities in quantum chemistry, materials science, molecular biology and nanosciences. The models originating from DFT can be classified into two categories: the orbital-free models and the Kohn-Sham models. The Thomas-Fermi-von Weizs¨acker (TFW) model falls into the first

∗Universit´e Paris-Est, CERMICS, Project-team Micmac, INRIA-Ecole des Ponts, 6 & 8 avenue Blaise Pascal, 77455 Marne-la-Vall´ee Cedex 2, France.

†UPMC Univ Paris 06, UMR 7598 LJLL, Paris, F-75005 France ; CNRS, UMR 7598 LJLL, Paris, F-75005 France

‡Division of Applied Mathematics, Brown University, Providence, RI, USA

category. It is not very much used in practice, but is interesting from a math- ematical viewpoint [1, 7, 12]. It indeed serves as a toy model for the analysis of the more complex electronic structure models routinely used by Physicists and Chemists. At the other extremity of the spectrum, the Kohn-Sham models [8, 11] are among the most widely used models in Physics and Chemistry, but are much more difficult to deal with. We focus here on the numerical analysis of the TFW model on the one hand, and of the Kohn-Sham model, within the local density approximation (LDA), on the other hand. More precisely, we are interested in the spectral and pseudospectral Fourier, more commonly called planewave, discretizations of the periodic versions of these two models. In this context, the simulation domain, sometimes referred to as the supercell, is the unit cell of some periodic lattice ofR³. In the TFW model, periodic boundary conditions (PBC) are imposed to the density; in the Kohn-Sham framework, they are imposed to the Kohn-Sham orbitals (Born-von Karman PBC). Impos- ing PBC at the boundary of the simulation cell is a standard method to compute condensed phase properties with a limited number of atoms in the simulation cell, hence at a moderate computational cost.

This article is organized as follows. In Section 2, we briefly introduce the func- tional setting used in the formulation and the analysis of the planewave dis- cretization of orbital-free and Kohn-Sham models. In Section 3, we provide a priori error estimates for the planewave discretization of the TFW model, in- cluding numerical integration. In Section 4, we deal with the Kohn-Sham LDA model.

2 Basic Fourier analysis for planewave discretiza- tion methods

Throughout this article, we denote by Γ the simulation cell, byRthe periodic lattice, and by R^∗ the dual lattice. For simplicity, we assume that Γ = [0, L)³ (L > 0), in which case R is the cubic lattice LZ³, and R^∗ = ^2π_LZ³. Our arguments can be easily extended to the general case. For k∈ R^∗, we denote byek(x) =|Γ|^−1/2e^ik·x the planewave with wavevectork. The family (ek)_k∈R^∗ forms an orthonormal basis of

L²_#(Γ,C) :=

u∈L²_loc(R³,C)|uR-periodic , and for allu∈L²_#(Γ,C),

u(x) = X

k∈R^∗

b

ukek(x) with buk = (ek, u)_L²

# =|Γ|^−1/2 Z

Γ

u(x)e^−ik·xdx.

In our analysis, we will mainly consider real valued functions. We therefore introduce the Sobolev spaces of real valued R-periodic functions

H_#^s(Γ) :=

(

u(x) = X

k∈R^∗

b

ukek(x)| X

k∈R^∗

(1 +|k|²)^s|buk|²<∞and∀k, bu−k=ub^∗_k )

, s∈R(here and in the sequela^∗ denotes the complex conjugate of the complex number a), endowed with the inner products

(u, v)H^s_#= X

k∈R^∗

(1 +|k|²)^sbu^∗_kbvk. ForNc∈N, we denote by

VNc =





X

k∈R^∗| |k|≤^2π_LNc

ckek| ∀k, c−k=c^∗_k



 (1) (the constraintsc−k=c^∗_k imply that the functions ofVNc are real valued). For all s ∈ R, and each v ∈ H_#^s(Γ), the best approximation of v in VNc for any H_#^r-norm,r≤s, is

ΠNcv= X

k∈R^∗| |k|≤^2π_LNc

b vkek.

The more regular v (the regularity being measured in terms of the Sobolev normsH^r), the faster the convergence of this truncated series tov: for all real numbers randswithr≤s, we have for eachv∈H_#^s(Γ),

kv−ΠNcvk^H#^r = min

v_Nc∈VNckv−vNck^H#^r ≤ L

2π s−r

N_c^−(s−r)kv−ΠNcvk^Hs#

≤ L

2π s−r

N_c^−(s−r)kvk^Hs#. (2) ForNg ∈N\ {0}, we denote byφb^FFT,Ng the discrete Fourier transform on the carterisan gridG^Ng :=_N^L_gZ³ of the functionφ∈C_#⁰(Γ,C), where

C_#⁰(Γ,C) :=

u∈C⁰(R³,C)|uR-periodic . Recall that ifφ=P

k∈R^∗φbkek∈C_#⁰(Γ,C), the discrete Fourier transform ofφ is theNgR^∗-periodic sequenceφb^FFT,Ng = (φb^FFT,N_k ^g)k∈R^∗ where

φb^FFT,N_k ^g = 1 N_g³

X

x∈GNg∩Γ

φ(x)e^−ik·x=|Γ|^−1/2 X

K∈R^∗

φbk+N_gK.

We now introduce the subspaces

W_N^1D_g =

Span

e^ily|l∈ 2π

LZ, |l| ≤ 2π L

Ng−1 2

(Ng odd), Span

e^ily|l∈ 2π

LZ, |l| ≤ 2π L

Ng

2

⊕C(e^iπNgy/L+e^−iπNgy/L) (Ng even),

(W_N^1D_g ∈C_#^∞([0, L),C) and dim(W_N^1D_g) =Ng), andW_N^3D_g =W_N^1D_g ⊗W_N^1D_g ⊗W_N^1D_g. Note thatW_N^3D_g is a subspace ofH_#^s(Γ,C) of dimensionN_g³, for alls∈R, and that ifNg is odd,

W_N^3D_g = Span

ek|k∈ R^∗=2π

LZ³, |k|^∞≤2π L

Ng−1 2

(Ng odd).

It is then possible to define the interpolation projectorI^Ng fromC_#⁰(Γ,C) onto W_N^3D_g by [I^Ng(φ)](x) =φ(x) for all x∈ G^Ng. It holds

∀φ∈C_#⁰(Γ,C), Z

ΓI^Ng(φ) = X

x∈G_Ng∩Γ

L Ng

³

φ(x). (3)

The coefficients of the expansion of I^Ng(φ) in the canonical basis of W_N^3D_g is given by the discrete Fourier transform ofφ. In particular, whenNg is odd, we have the simple relation

I^Ng(φ) =|Γ|^1/2 X

k∈R^∗| |k|∞≤^2π_L“_Ng−1

2

”

φb^FFT,N_k ^gek (Ng odd).

It is easy to check that ifφis real-valued, then so is I^Ng(φ).

We will assume in the sequel that Ng ≥ 4Nc+ 1. We will then have for all v4Nc∈V4Nc,

Z

Γ

v4Nc = X

x∈G_Ng∩Γ

L Ng

3

v4Nc(x) = Z

ΓI^Ng(v4Nc). (4) The following lemma gathers some technical results which will be useful for the numerical analysis of the planewave discretization of orbital-free and Kohn- Sham models.

Lemma 2.1 Let Nc∈N^∗ andNg ∈N^∗ such thatNg≥4Nc+ 1.

1. LetV be a function ofC_#⁰(Γ,C)andvNcandwNcbe two functions ofVNc.

Then Z

ΓI^Ng(V vN_cwN_c) = Z

ΓI^Ng(V)vN_cwN_c; (5)

Z

ΓI^Ng(V|vNc|²)

≤ kVk^L∞kvNck²L²_#. (6) 2. Lets >3/2,0≤r≤s, andV a function ofH_#^s(Γ). Then,

(1− I^Ng)(V)_Hr

# ≤ Cr,sN_g^−(s−r)kVk^H#^s; (7) Π_2Nc(I^Ng(V))

L²_# ≤ Z

ΓI^Ng(|V|²) 1/2

; (8)

Π_2Nc(I^Ng(V))

H_#^s ≤ (1 +Cs,s)kVk^Hs#, (9)

for constantsCr,s independent ofV. Besides if there existsm >3andC∈ R₊ such that|Vbk| ≤C|k|^−m, then there exists a constantCV independent ofNc andNg such that

Π_2Nc(1− I^Ng)(V)_Hr

# ≤ CVN_c^r+3/2N_g^−m. (10) 3. Let φ be a Borel function from R₊ toR such that there exists Cφ ∈R₊

for which|φ(t)| ≤Cφ(1 +t²)for allt∈R₊. Then, for allvNc ∈VNc,

Z

ΓI^Ng(φ(|vN_c|²))

≤ Cφ

|Γ|+kvN_ck⁴L⁴_#

. (11)

Proof Forz2N_c ∈V2N_c, it holds Z

ΓI^Ng(V z2N_c) = X

x∈GNg∩Γ

L Ng

3

V(x)z2N_c(x)

= X

x∈G_Ng∩Γ

L Ng

3

(I^Ng(V))(x)z2Nc(x)

= Z

ΓI^Ng(V)z2Nc (12)

sinceI^Ng(V)z2Nc∈VNg+2Nc⊂V2Ng is exactly integrated. The functionvNcwNc

being inV2Nc, (5) is proved. Moreover, as|vNc|²∈V4Nc, it follows from (4) that

Z

ΓI^Ng(V|vNc|²) =

X

x∈G_Ng∩Γ

L Ng

3

V(x)|vNc(x)|²

≤ kVk^L∞

X

x∈GNg∩Γ

L Ng

3

|vN_c(x)|²

= kVk^L∞ Z

Γ|vN_c|².

Hence (6). The estimate (7) is proved in [6]. To prove (8), we notice that kΠ2N_c(IN_g(V))k²L²_# ≤ kIN_g(V)k²L²_#

= Z

Γ

(IN_g(V))^∗(IN_g(V))

= X

x∈G_Ng∩Γ

L Ng

3

(INg(V))(x)^∗(INg(V))(x)

= X

x∈GNg∩Γ

L Ng

3

|V(x)|²

= Z

Γ

IN_g(|V|²).

The bound (9) is a straightforward consequence of (7):

kΠ2Nc(INg(V))k^H#^s ≤ kINg(V)k^Hs#≤ kVk^H#^s +k(1−INg)(V)k^H#^s ≤(1 +Cs,s)kVk^H#^s. Now, we notice that

Π2N_c(I^Ng(V)) = |Γ|^1/2 X

k∈R^∗| |k|≤^4π_LNc

Vb_k^FFT,Ngek

= X

k∈R^∗| |k|≤^4π_LN_c

X

K∈R^∗

Vbk+NgK

!

ek. (13) From (13), we obtain

Π_2Nc(1− I^Ng)(V)²_Hs

# = X

k∈R^∗| |k|≤^4π_LNc

(1 +|k|²)^s

X

K∈R^∗\{0}

Vbk+NgK

2

≤



 X

k∈R^∗| |k|≤^4π_LN_c

(1 +|k|²)^s



 max

k∈R^∗| |k|≤^4π_LNc

X

K∈R^∗\{0}

Vbk+NgK

2

.

On the one hand, X

k∈R^∗| |k|≤^4π_LN_c

(1 +|k|²)^s ∼

Nc→∞

32π 2s+ 3

4π L

2s

N_c^2s+3, and on the other hand, we have for eachk∈ R^∗ such that|k| ≤ ^4πLNc,

X

K∈R^∗\{0}

Vbk+NgK

≤ C X

K∈R^∗\{0}

1

|k+NgK|^m

≤ C C0

L 2π

m

N_g^−m where

C0= max

y∈R³| |y|≤1/2

X

K∈Z3\{0}

1

|y−K|^m.

The estimate (10) then easily follows. Let us finally prove (11). Using (3) and (4), we have

Z

ΓI^Ng(φ(|vNc|²)) =

X

x∈G_Ng∩Γ

L Ng

3

φ(|vNc(x)|²)

≤ Cφ

X

x∈GNg∩Γ

L Ng

3

(1 +|vNc(x)|⁴)

= Cφ

Z

Γ

(1 +|vN_c|⁴) =Cφ

|Γ|+kvN_ck⁴L⁴_#

.

This completes the proof of Lemma 2.1.

3 Planewave approximation of the TFW model

In the TFW model, as well as in any orbital-free model, the ground state elec- tronic density of the system is obtained by minimizing an explicit functional of the density. Denoting by N the number of electrons in the simulation cell and by

R_N =

ρ≥0|√ρ∈H_#¹(Γ), Z

Γ

ρ=N

the set of admissible densities, the TFW problem reads I^TFW= inf

E^TFW(ρ), ρ∈R_N , (14)

where

E^TFW(ρ) = CW

2 Z

Γ|∇√ρ|²+CTF

Z

Γ

ρ^5/3+ Z

Γ

ρV^ion+1

2DΓ(ρ, ρ).

CWis a positive real number (CW= 1, 1/5 or 1/9 depending on the context [8]), and CTF is the Thomas-Fermi constant: CTF = ¹⁰₃(3π²)^2/3. The last term of the TFW energy models the periodic Coulomb energy: forρandρ^′ inH_#⁻¹(Γ),

DΓ(ρ, ρ^′) := 4π X

k∈R^∗\{0}

|k|⁻²ρb^∗_kρb^′_k.

We finally make the assumption thatV^ion is aR-periodic potential such that

∃m >3, C ≥0 s.t. ∀k∈ R^∗, |Vb_k^ion| ≤C|k|^−m. (15) Note that this implies thatV^ionis inH^m−3/2−ǫ(Γ) for allǫ >0, hence inC_#⁰(Γ) sincem−3/2−ǫ >3/2 forǫsmall enough. It is convenient to reformulate the TFW model in terms of v=√ρ. It can be easily seen that

I^TFW = inf

E^TFW(v), v∈H_#¹(Γ), Z

Γ|v|²=N

, (16)

where

E^TFW(v) = CW

2 Z

Γ|∇v|²+CTF

Z

Γ|v|^10/3+ Z

Γ

V^ion|v|²+1

2DΓ(|v|²,|v|²).

Let F(t) = CTFt^5/3 and f(t) = F^′(t) = ⁵₃CTFt^2/3. The function F is in C¹([0,+∞))∩C^∞((0,+∞)), is strictly convex on [0,+∞), and for all (t1, t2)∈ R₊×R₊,

|f(t²₂)t2−f(t²₁)t2−2f^′(t²₁)t²₁(t2−t1)| ≤ 70

27CTFmax(t^1/3₁ , t^1/3₂ )|t2−t1|². (17)

The first and second derivatives ofE^TFW are respectively given by hE^TFW′(v), wiH_#⁻¹,H_#¹ = 2hH^TFW|v|² v, wi;

hE^TFW′′(v)w1, w2iH_#⁻¹,H_#¹ = 2hH|v|^TFW2 w1, w2i+ 4DΓ(vw1, vw2) + 4 Z

Γ

f^′(|v|²)|v|²w1w2, where we have denoted by H^TFWρ the TFW Hamiltonian associated with the

densityρ

H^TFWρ =−CW

2 ∆ +f(ρ) +V^ion+V_ρ^Coulomb, where

V_ρ^Coulomb(x) = 4π X

k∈R^∗\{0}

|k|⁻²bρkek(x)

is theR-periodic Coulomb potential generated by theR-periodic charge distri- bution ρ. Recall that V_ρ^Coulomb can also be defined as the unique solution in H_#¹(Γ) to









−∆V_ρ^Coulomb= 4π

ρ− |Γ|⁻¹ Z

Γ

ρ Z

Γ

V_ρ^Coulomb= 0.

Let us recall (see [12] and the proof of Lemma 2 in [3]) that

• (14) has a unique minimizerρ⁰, and that the minimizers of (16) areuand

−u, whereu=p ρ⁰;

• uis inH_#^m+1/2−ǫ(Γ) for eachǫ >0 (hence inC_#²(Γ) sincem+1/2−ǫ >7/2 forǫsmall enough);

• u >0 onR³;

• usatisfies the Euler equation H^TFW|u|² (u) =−CW

2 ∆u+ 5

3CTFu^4/3+V^ion+V_u^Coulomb2

u=λu for some λ ∈ R, (the ground state eigenvalue of H^TFWρ⁰ , that is non- degenerate).

The planewave discretization of the TFW model is obtained by choosing 1. an energy cut-off Ec > 0 or, equivalently, a finite dimensional Fourier

spaceVNc, the integerNc being related toEc through the relationNc :=

[√

2EcL/2π];

2. a cartesian grid G^Ng with step size L/Ng where Ng ∈ N^∗ is such that Ng≥4Nc+ 1,

and by considering the finite dimensional minimization problem I_N^TFW_c,Ng = inf

E^TFW_Ng (vN_c), vN_c∈VN_c, Z

Γ|vN_c|²=N

, (18) where

E_N^TFW_g (vN_c) = CW

2 Z

Γ|∇vN_c|²+CTF

Z

ΓI^Ng(|vN_c|^10/3) + Z

ΓI^Ng(V^ion)|vN_c|² +1

2DΓ(|vNc|²,|vNc|²),

I^Ng denoting the interpolation operator introduced in the previous section. The Euler equation associated with (18) can be written as a nonlinear eigenvalue problem

∀vNc∈VNc, h(He^TFW,N|u_Nc,Ng^g|²−λNc,Ng)uNc,Ng, vNciH_#⁻¹,H_#¹ = 0, where we have denoted by

He^TFW,Nρ ^g =−CW

2 ∆ +I^Ng

5

3CTFρ^2/3+V^ion

+V_ρ^Coulomb

the pseudospectral TFW Hamiltonian associated with the density ρ, and by λN_c,N_g the Lagrange multiplier of the constraintR

Γ|vN_c|² =N. We therefore have

−CW

2 ∆uNc,Ng+ΠNc

I^Ng

5

3CTF|uNc,Ng|^4/3+V^ion

+V_|u^Coulomb_Nc,Ng|²

uNc,Ng

=λNc,NguNc,Ng. Under the condition thatNg≥4Nc+ 1, we have for allφ∈C_#⁰(Γ),

∀(k, l)∈ R^∗× R^∗ s.t. |k|,|l| ≤ 2π LNc,

Z

ΓI^Ng(φ)e^∗_kel=φb^FFT_k−l, so that,He^TFWu_Nc,Ng is defined onVNc by the Fourier matrix

[Hb^TFW,N_|uNc,Ng^g_|²]kl = CW

2 |k|²δkl+5

3CTF \

(|uNc,Ng|^4/3)

FFT,Ng

k−l +(V\^ion)^FFT,N_k−l ^g +4π

(|u\N_c,N_g|²)^FFT,N_k−l ^g

|k−l|² (1−δkl),

where, by convention, the last term of the right hand side is equal to zero for k=l.

We also introduce the variational approximation of (16) I_N^TFW_c = inf

E^TFW(vNc), vNc∈VNc, Z

Γ|vNc|²=N

. (19)

Any minimizeruNc to (19) satisfies the elliptic equation

−CW

2 ∆uNc+ ΠNc

5

3CTF|uNc|^4/3uNc+V^ionuNc+V_|u^Coulomb

Nc|² uNc

=λNcuNc, (20) for someλNc ∈R.

The main result of this section is an extension of results previously obtained by A. Zhou [16].

Theorem 3.1 For each Nc ∈ N, we denote byuNc a minimizer to (19) such that (uNc, u)_L²

# ≥ 0 and, for each Nc ∈ N and Ng ≥ 4Nc + 1, we denote by uNc,Ng a minimizer to (18) such that (uNc,Ng, u)_L²

# ≥ 0. Then for Nc large enough,uNc anduNc,Ng are unique, and the following estimates hold true

kuNc−uk^H#^s ≤ Cs,ǫN−(m−s+1/2−ǫ)

c ; (21)

|λNc−λ| ≤ CǫN_c^{−(2m−1−ǫ)}; (22) γkuNc−uk²H_#¹ ≤I_N^TFW_c −I^TFW ≤ CkuNc−uk²H_#¹; (23) kuN_c,N_g−uN_ck^H#^s ≤ CsN_c^3/2+(s−1)+N_g^−m; (24)

|λNc,Ng−λNc| ≤ CN_c^3/2N_g^−m; (25)

|I_N^TFW_c,Ng −I_N^TFW_c | ≤ CN_c^3/2N_g^−m, (26) for all −m+ 3/2 < s < m+ 1/2 and ǫ > 0, and for some constants γ > 0, Cs,ǫ≥0,Cǫ≥0,C≥0 andCs≥0 independent ofNc andNg.

Remark 1 More complex orbital-free models have been proposed in the recent years [15], which are used to perform multimillion atom DFT calculations. Some of these models however are not well posed (the energy functional is not bounded from below [2]), and the others are not well understood from a mathematical point of view. For these reasons, we will not deal with those models in this article.

3.1 A priori estimates for the variational approximation.

In this section, we prove the first part of Theorem 3.1, related to the variational approximation (19). The estimates (21), (22) and (23) originate from arguments already introduced in [3]. For brevity, we only recall the main steps of the proof and leave the details to the reader.

The difference between (16) and the problem dealt with in [3] is the presence of the Coulomb termDΓ(|v|²,|v|²), for which the following estimates are available:

0≤DΓ(ρ, ρ) ≤ Ckρk²L²_#, for allρ∈L²_#(Γ), (27)

|DΓ(uv, uw)| ≤ CkvkL²_#kwkL²_#, for all (v, w)∈(L²_#(Γ))², (28)

|DΓ(ρ, vw)| ≤ CkρkL²_#kvkL²_#kwkL²_#, for all (ρ, v, w)∈(L²_#(Γ))³, (29) kV_ρ^Coulombk^L∞ ≤ CkρkL²_#, for allρ∈L²_#(Γ), (30) kV_ρ^CoulombkH_#^s+2 ≤ Ckρk^H#^s, for allρ∈H_#^s(Γ). (31) Here and in the sequel, C denotes a non-negative constant which may depend on Γ, V^ionandN, but not on the discretization parameters.

Using (27), (28) and the fact thatf^′>0 on (0,+∞), we can then show (see the proof of Lemma 1 in [3]) that there existβ >0,γ >0 andM ≥0 such that for allv∈H_#¹(Γ),

0≤ h(H^TFWρ⁰ −λ)v, viH⁻_#¹,H¹_#≤Mkvk²H_#¹, (32) βkvk²H_#¹ ≤ h(E^TFW′′(u)−2λ)v, viH_#⁻¹,H_#¹ ≤Mkvk²H_#¹, (33) and for allv∈H_#¹(Γ) such thatkvkL²_# =N^1/2 and (v, u)L²_#≥0,

γkv−uk²H_#¹ ≤ h(H^TFWρ⁰ −λ)(v−u),(v−u)iH_#⁻¹,H_#¹. (34) Remarking that

E^TFW(uN_c)−E^TFW(u) = h(Hρ^TFW0 −λ)(uN_c−u),(uN_c−u)iH_#⁻¹,H¹_#

+1

2DΓ(|uNc|²− |u|²,|uNc|²− |u|²) +

Z

Γ

F(|uNc|²)−F(|u|²)−f(|u|²)(|uNc|²− |u|²) (35) and using (34), the positivity of the bilinear formDΓ, and the convexity of the functionF, we obtain that

I_N^TFW_c −I^TFW=E^TFW(uNc)−E^TFW(u)≥γkuNc−uk²H¹_#.

For each Nc ∈ N, euNc = N^1/2ΠNcu/kΠNcukL²_# satisfies (euNc, u)_L²_# ≥ 0 and keuNckL²_# =N^1/2, and the sequence (ueNc)Nc∈Nconverges to uinH_#^m+1/2−ǫ(Γ) for each ǫ >0. As the functionalE^TFW is continuous onH_#¹(Γ), we have

kuN_c−uk²H_#¹ ≤γ⁻¹ I_N^TFW_c −I^TFW

≤γ⁻¹ E^TFW(euN_c)−E^TFW(u)

N−→c→∞0.

Hence, (uNc)Nc∈Nconverges touinH_#¹(Γ), and we also have λNc = N⁻¹

CW

2 Z

Γ|∇uNc|²+ Z

Γ

f(|uNc|²)|uNc|²+ Z

Γ

V^ion|uNc|²+DΓ(|uNc|²,|uNc|²)

N−→c→∞ N⁻¹ CW

2 Z

Γ|∇u|²+ Z

Γ

f(|u|²)|u|²+ Z

Γ

V^ion|u|²+DΓ(|u|²,|u|²)

= λ.

As f(|uNc|²)uNc+V^ionuNc+V_|u^Coulomb

Nc|² uNc is bounded inL²_#(Γ), uniformly in Nc, we deduce from (20) that the sequence (uN_c)N_c∈N is bounded in H_#²(Γ), hence inL^∞(Γ). Now

∆(uNc−u) = 2C_W⁻¹

ΠNc

f(|uNc|²)uNc−f(|u|²)u+V^ion(uNc−u) + V_|u^Coulomb

Nc|² uNc−V_|u|^Coulomb2 u

+ (1−ΠNc)

f(|u|²)u+V^ionu+V_|u|^Coulomb2 u

−λN_c(uN_c−u)−(λN_c−λ)u

.

Observing that the right-hand side goes to zero in L²_#(Γ) when Nc goes to infinity, we obtain that (uNc)Nc∈N converges to u in H_#²(Γ), and therefore in C_#^0,1/2(Γ). In addition, we know from Harnack inequality [10] thatu >0 inR³. Consequently, for Nc large enough, the function uN_c (which is continuous and R-periodic) is bounded away from 0, uniformly inNc. Asf ∈C^∞(0,+∞), one can see by a simple bootstrap argument that the convergence of (uNc)Nc∈Ntou also holds inH_#^m+1/2−ǫ(Γ) for eachǫ >0. The upper bound in (23) is obtained from (35), remarking that

0 ≤

Z

Γ

F(|uN_c|²)−F(|u|²)−f(|u|²)(|uN_c|²− |u|²)

≤ 35 9 CTF

Z

Γ

max(|uNc|^4/3,|u|^4/3)|uNc−u|²

≤ 35 9 CTF

Nmaxc∈NkuNck^{L∞ 4/3}

kuNc−uk²L²_#, and that

0 ≤ DΓ(|uNc|²− |u|²,|uNc|²− |u|²)≤Ck|uNc|²− |u|²k²L²_#

≤ 4C

Nmaxc∈NkuNck^{L∞ 2}

kuNc−uk²L²_#.

The uniqueness ofuNc forNclarge enough can then be checked as follows. First,

(uNc, λNc) satisfies the variational equation

∀vNc∈VNc, h(H^TFW|uNc|²−λNc)uNc, vNciH_#⁻¹,H_#¹ = 0.

Therefore λNc is the variational approximation in VNc of some eigenvalue of H^TFW|uNc|². As (uNc)Nc∈N converges to u in L^∞(Γ), H^TFW|uNc|² − H^TFWρ⁰ converges to 0 in operator norm. Consequently, the n^th eigenvalue of H^TFW|uNc|² converges to then^th eigenvalue ofH^TFWρ⁰ whenNc goes to infinity, the convergence being uniform inn. Together with the fact that the sequence (λNc)Nc∈Nconverges to λ, the non-degenerate ground state eigenvalue of H^TFWρ⁰ , this implies that for Nc large enough,λNc is the ground state eigenvalue of H^TFW|u_Nc|² in VNc and for allvN_c∈VN_c such thatkvN_ckL²_# =N^1/2 and (vN_c, uN_c)L²_# ≥0,

E^TFW(vN_c)−E^TFW(uN_c) = h(H^TFW|uNc|²−λN_c)(vN_c−uN_c),(vN_c−uN_c)iH_#⁻¹,H_#¹

+1

2DΓ(|vNc|²− |uNc|²,|vNc|²− |uNc|²) +

Z

Γ

F(|vN_c|²)−F(|uN_c|²)−f(|uN_c|²)(|vN_c|²− |uN_c|²)

≥ h(H^TFW|uNc|²−λN_c)(vN_c−uN_c),(vN_c−uN_c)iH_#⁻¹,H_#¹

≥ γ

2kvNc−uNck²H_#¹. (36) It easily follows that forNclarge enough, (19) has a unique minimizeruNc such

that (uNc, u)_L²

# ≥0.

Let us now establish the rates of convergence of |λN_c−λ| and kuN_c −uk^H#^s. First,

λNc−λ = N⁻¹

h(H^TFW|u|² −λ)(uNc−u),(uNc−u)iH_#⁻¹,H_#¹

+ Z

Γ

wNc(uNc−u)

(37) with

wNc =f(|uN_c|²)−f(|u|²)

uNc−u |uNc|²+V_|u^Coulomb

Nc|² (uNc+u).

As uNc is bounded away from 0 and f ∈ C^∞((0,+∞)), the function wNc is uniformly bounded inH_#^m−3/2−ǫ(Γ) (at least forNclarge enough). We therefore obtain that for all 0≤r < m−3/2, there exists a constantCr∈R₊ such that for allNc large enough,

|λNc−λ| ≤Cr

kuNc−uk²H_#¹ +kuNc−ukH⁻_#^r

. (38)

In order to evaluate theH_#¹-norm of the error (uNc−u), we first notice that

∀vNc ∈VNc, kuNc−ukH¹_# ≤ kuNc−vNckH_#¹ +kvNc−ukH_#¹, (39)

and that

kuN_c−vN_ck²H_#¹ ≤ β⁻¹h(E^TFW′′(u)−2λ)(uN_c−vN_c),(uN_c−vN_c)iH_#⁻¹,H_#¹

= β⁻¹

h(E^TFW′′(u)−2λ)(uNc−u),(uNc−vNc)iH_#⁻¹,H_#¹

+h(E^TFW′′(u)−2λ)(u−vNc),(uNc−vNc)iH_#⁻¹,H_#¹

.(40) For allzNc∈VNc,

h(E^TFW′′(u)−2λ)(uN_c−u), zN_ciH_#⁻¹,H_#¹

= −2 Z

Γ

[f(|uNc|²)uNc−f(|u|²)uNc−2f^′(|u|²)|u|²(uNc−u)]zNc

−2DΓ((uNc−u)(uNc+u),(uNc−u)zNc)−2DΓ((uNc−u)², uzNc) +2(λNc−λ)

Z

Γ

uNczNc. (41)

On the other hand, we have for allvN_c∈VN_c such thatkvN_ckL²_# =N^1/2, Z

Γ

uNc(uNc−vNc) =N − Z

Γ

uNcvNc =1

2kuNc−vNck²L²_#.

Using (17), (29), (38) with r= 0 and the above equality, we therefore obtain for allvNc ∈VNc such thatkvNckL²_#=N^1/2,

h(E^TFW′′(u)−2λ)(uNc−u),(uNc−vNc)iH_#⁻¹,H_#¹

≤C

kuNc−uk²H_#¹ kuNc−vNckH_#¹

+

kuNc−uk²H_#¹ +kuNc−ukL²_#

kuNc−vNck²L²_#

. (42) Therefore, forNc large enough, we have for allvNc ∈VNc such thatkvNckL²_# = N^1/2,

kuNc−vNckH_#¹ ≤C

kuNc−uk²H_#¹ +kvNc−ukH_#¹

.

Together with (39), this shows that there exists N ∈NandC ∈R₊ such that for allNc≥N,

∀vNc∈VNc s.t. kvNckL²_#=N^1/2, kuNc−ukH_#¹ ≤CkvNc−ukH_#¹. By a classical argument (see e.g. the proof of Theorem 1 in [3]), we deduce from (2) and the above inequality that

kuNc−ukH_#¹ ≤C min

v_Nc∈V_NckvNc−ukH¹_#≤C1,ǫN_c^{−(m−1/2−ǫ)}, (43)

for some constant C1,ǫ independent of Nc. This completes the proof of the estimate in theH_#¹–norm. We proceed with the analysis of theL²_#–norm.

Forw∈L²_#(Γ), we denote byψw the unique solution to the adjoint problem

( findψw∈u^⊥ such that

∀v∈u^⊥, h(E^TFW′′(u)−2λ)ψw, viH_#⁻¹,H_#¹ =hw, viH⁻¹_# ,H_#¹, (44)

where

u^⊥=

v∈H_#¹(Γ)| Z

Γ

uv= 0

.

The functionψw is solution to the elliptic equation

−CW

2 ∆ψw+ V^ion+V_u^Coulomb2 +f(u²) + 2f^′(u²)u²−λ

ψw+ 2V_uψ^Coulomb_w u

= 2 Z

Γ

f^′(u²)u³ψw+DΓ(u², uψw)

u+1 2

w−(w, u)_L²

#u ,

from which we deduce that if w ∈ H_#^r(Γ) for some 0 ≤ r < m−3/2, then ψw∈H_#^r+2(Γ) and

kψwkH^r+2_# ≤Crkwk^H#^r, (45)

for some constantCr independent ofw. Letu^∗_Nc be the orthogonal projection, for the L²_#inner product, ofuNc on the affine spacen

v∈L²_#(Γ)|R

Γuv=No . One has

u^∗_Nc∈H_#¹(Γ), u^∗_Nc−u∈u^⊥, u^∗_Nc−uNc = 1

2NkuNc−uk²L²_#u,