Wavelet approximations of the Hamiltonian operator and computation

Wavelet approximations of the Hamiltonian operator and computation of related energies Claire Chauvin

∗

and Valérie Perrier

†

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February 2, 2008

Abstract Multiresolution analysis in Quantum Chemistry provide ecient computational methods. In this article, we propose several representations of the Hamiltonian operator arising from the Density Functional Theory, based on orthogonal and interpolating scaling function bases. These high order approximations allows to compute the potential and kinetic energies with a linear complexity. Finally numerical examples show the accuracy of the method. Keywords: Density Functional Theory, Orbital energy, Interpolating scaling function, O(N ) method, Stiness Matrix, Harmonic Oscillator, Hydrogen.

Introduction

Ab initio molecular dynamics [25] provides a large class of methods for the Electronic structure calculations in quantum Chemistry. In this framework, we will focus on the Hohenberg-Kohn-Sham density functional theory, which leads to the resolution of nonlinear partial dierential equations. In quantum chemistry two types of methods have been widely used:

on the one

hand, Slater-type or Gaussian-type basis decompositions for the solution use basis functions localized in physical space, generally centered at the center of nuclei and well adapted to the structure of the solution.

On the other

hand, the plane waves method involves functions localized in Fourier space but delocalized in physical space which does not favor the non uniform ∗

Laboratoire

des

Champs

Magnétiques

Intenses,

CNRS

Grenoble,

France

[email protected]

†

Laboratoire

Jean

Kuntzmann,

Grenoble

[email protected]

1

university

and

CNRS,

structure of the solution. The advantage of this second method lies in the easy evaluation of operators and on the low computational complexity provided by the FFT. In between these two methods, wavelets appear as a good compromise between physical and Fourier representations. Like plane waves, wavelet decompositions are computed by a fast algorithm, the fast wavelet transform (FWT) being of linear complexity, and they provide polynomial approximations of arbitrary order [9]. For these reasons, wavelet methods have been introduced for electronic calculations [31, 8, 2, 18, 32, 30].

Moreover, the interpolation property of

some wavelet families seems to be crucial for the treatment of the poten-

inria-00337464, version 1 - 7 Nov 2008

tial operator [15, 23, 1, 14, 16].

Indeed, the potential operator is usually

represented in numerical works by its values on a grid. The objective of this article is to present an ecient and accurate method for the computation of energies related to the Kohn-Sham operator. Classical error estimates are not available here because of the non elliptic form of the Hamiltonian, and because of its nonlinearity. On the other hand error estimates may be done in terms of energies (e.g. eigenvalues). We will focus in this article on numerical methods based on scaling functions, since it is the rst step towards adaptive algorithms based on wavelets. Our method is rstly dened for the Kohn-Sham operator, and is based on the combination of interpolating scaling functions for the decomposition of the potential, and orthonormal scaling functions for the decomposition of the orbitals. Then the accuracy of the method is studied in the linear case, when the potential is reduced to the external potential: we derive error estimates for both the kinetic and potential energies.

Finally, numerical tests on the simple

models of Hydrogen and Harmonic Oscillator illustrate the eciency of this approach.

1

Density Functional Theory

1.1

Kohn-Sham Equations

We present below the set of equations coming from the Density Functional Theory of Hohenberg, Kohn and Sham [20, 22]. Given at

Rα , α = 1, N )

and

2N

electrons occupying

to compute the electron density

ρ=2

No X i=1

ρ,

ni |ui |2 ,

No

K

nuclei (positioned

energy levels, the aim is

dened as:

No X i=1

2

ni = N,

0 6 ni 6 1,

Z where the occupied orbitals

δi,j ,

and are the

N

ui satisfy the orthogonality relation

ui uj dr = Ω

lowest eigenfunctions of the Kohn-Sham operator:

(1) H[ρ] ui = i ui , i = 1, . . . , N, 1 1 H[ρ] = − ∆ + V (r) + VC [ρ] + Vxc [ρ] = − ∆ + VKS (r). (2) 2 2 For simplicity we assume that ni = 1, ∀ i = 1, . . . , No , and that the number of occupied energy levels No is equal to N . The external potential V and the Coulomb potential VC describe respectively the attraction of the electron to

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the nuclei, and the Coulomb interaction between electrons:

V (r) = −

K X α=1

−∆VC

Zα , |r − Rα |

= 4πρ.

The exchange-correlation potential

Vxc

in the Hamiltonian operator (2) is

often evaluated by the Local Density Approximation (LDA) [25, 19]: depends only on the electron density

Vxc

ρ(r).

Several iterative algorithms have been developed to solve this nonlinear system. The simplest one is called the Roothaan algorithm [29], and behaves like a xed-point algorithm on the density main idea is that for a given

ρ,

ρ.

Without going into details, the

one is able to evaluate

solve (1) by determining the lowest eigenvalues functions

ui .

i

VKS ,

and then to

and corresponding eigen-

The existence of solution of such algorithms can be found in

the literature [6, 4, 5, 30].

1.2

The model problem

An important step of the resolution of the self-consistent problem (1)(2) is the determination of the lowest eigenvalues i of a given Hamiltonian H = − 21 ∆ + V , and of the corresponding eigenvectors ui . At rst, V is a function V (r), and we focus on a priori estimates of approximate solutions of the linear problem in specic nite dimensional spaces. Some results on the existence of a discrete spectrum are present in the

V . A rst example is given by R3 ; in this case the spectrum of H is decomposed ([0, +∞[) and a discrete part, which is minored by

literature [12, 28], with some condition on

V (r) =

K |r| with

K= i < ui , v >,

(3)

Ω

< ui , v >=

R

Ω ui

v.

These solutions

ui

will be approximated in

nite dimensional spaces presented in the next section. The convergence of discrete solutions towards continuous ones has been studied in the case of nite element methods, as explained in [28].

H

In the restrictive case where

is an elliptic operator, its approximate eigenvalues

˜i

satisfy the following

estimate:

|i − ˜i | ≤ C h2(m−1) ,

(4)

assuming suitable conditions on the approximation spaces, and that the rst eigenvectors length). When

i

(u1 , · · · , ui )

belong to

H m (Ω) (h

i-

denotes the subdivision

is an eigenvalue of multiplicity 1, we also get:

kui − u ei kH 1 ≤ C hm−1 ,

(5)

which means that the precision order in the eigenvalue approximation is twice better than the one obtained for the eigenvector in energy norm. The aim of this paper is to study

a priori estimates on the evaluation of

the following energies:

< ui , H u i > i = < ui , u i >

where

ekin

Z 1 = ∇ui ∇v+ < V ui , ui > 2 Ω = ekin + ep ,

(6)

stands for the kinetic energy. We also search for a representation

H into suitable nite dimensional spaces, which will improve the eciency ep . The eigenvectors of H are known to have a polynomial decay when |r| → +∞, of

of the calculation of the potential energy

we will thus study them in a domain that contains their support. Moreover, assuming that this domain is big enough,

ui

vanish far from the boundaries,

and we can consider periodic boundary conditions. Numerical advantages of

4

this choice are shown in next section.

Vxc (2). Indeed in Vxc is evaluated from the grid values of ρ. Our choice is to expand ρ and the potential V into an interpolating scaling function basis. On the contrary, the orbitals ui will be expanded into an orthonormal scaling functions basis, The resolution scheme will depend on the representation of

LDA,

to take advantage of the nearly diagonalisation of elliptic operators, provided by the wavelet decomposition. The next section will detail the nite dimensional spaces that will be used, generated by orthogonal or interpolating scaling function bases.

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2

3D periodic Multiresolution Analysis (MRA) ω = R/Z

Let

be the unit torus on

R,

and

Ω = (R/Z)3 .

We rst recall some

basic properties of 3D periodic MRA in a general context.

Then we will

focus on orthonormal and interpolating scaling functions.

2.1

Construction

For more details on practical computations related to these bases, we refer to [11, 24]. A periodic MRA

L2 (Ω) is constructed MRA's {VJ }.

{VJ }

tensor product of one-dimensional

of

by an isotropic

Notation 1 (MRA of L2 (ω)) Let {VJ } and {VeJ }, J > 0 be the ascending

dense sequences of two biorthogonal MRA's of L2 (ω). In the periodic case, the scaling functions φJ,k and φeJ,k which span the spaces VJ = span{φJ,k ; k ∈ ωJ = [0, . . . , 2J − 1]} and VeJ = span{φg J,k ; k ∈ ωJ }

are generated by dilation, shift and periodization of scaling functions φ and φe dened on R: φJ,k (x) =

X

φ(2J (x + r) − k) , φeJ,k (x) = 2J

r∈Z

X

e J (x + r) − k). φ(2

r∈Z

Moreover, they satisfy the biorthogonality relation: Z φJ,k (x) φeJ,l (x) dx = δk,l , ∀J ∈ N, ∀k, l ∈ ωJ , ω

5

where δk,l is the Kronecker delta. The well-known two-scale equations rewrite in the periodic case: φJ,k =

X

hJ+1 (n − 2k) φJ+1,n ,

n∈ωJ+1

φeJ,k =

X

e hJ+1 (n − 2k) φeJ+1,n .

(7)

n∈ωJ+1

where hJ and ehJ are the lters coming from the 2J −periodization of the lters h and h˜ associated to the scaling functions φ and φe. φ and φ˜ have a compact support; this is equivalent to the fact that lters hJ and e hJ have nite length. If the restrictions to [0, 1] of polynomials of degree less or equal than m − 1 are contained in VJ , then the MRA {VJ } is of approximation order m [9]. In this case, if we note PVJ the biorthogonal projection into the space VJ : X ∀u ∈ L2 (ω), PVJ u = < u/φeJ,k > φJ,k ,

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We will assume that the scaling functions

k∈ωJ then the following estimate holds:

ku − PVJ ukH s

.

2−J(m−s) kukH m

, ∀ u ∈ H m (ω)

(8)

Since our objective is the evaluation of the Kohn-Sham operator expanded into a scaling function basis, we will not introduce here wavelets. But the results presented in this article with scaling function bases, are also valid by using wavelets. The construction of three-dimensional MRA is done as follows.

Denition 1 (MRA of L2 (Ω)) A couple of biorthogonal periodic MRA {VJ }

e J } of L2 (Ω) is dened by isotropic tensor products of one-dimensional and {V MRA's {VJ } and {VeJ }: e J = VeJ ⊗ VeJ ⊗ VeJ . VJ = VJ ⊗ VJ ⊗ VJ , and V ΩJ = ωJ3 . A scaling function for each r = (x, y, z) ∈ Ω:

Let writes

of

VJ

indexed by

k = (k1 , k2 , k3 ) ∈ ΩJ

ΦJ,k (r) = φJ,k1 (x) φJ,k2 (y) φJ,k3 (z).

6

In the following, it will be convenient to adopt a vector-type notation for

e J,k }k∈Ω . FJ = {ΦJ,k }k∈ΩJ , FeJ = {Φ J coecients C = {cJ,k }k∈ΩJ will be written as: the basis:

u = C T FJ =

X

A function

u

of

VJ

with

e J,k > ΦJ,k . < u, Φ

k∈ΩJ In the next part we will introduce two specic families of scaling functions, which will be used for the decomposition of the orbitals

ui

and the potential

V.

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2.2

Orthonormal and interpolating MRA's

thonormal m1 .

or-

ui in (1) are expanded into an ⊥ 2 basis [11]. Let {VJ } be an orthonormal MRA of L (Ω), of order

For computational reasons, the orbitals

In numerical tests we will use the Daubechies family and the Coiets

Figure 1: Left: Daubechies scaling functions of order from 2 (black line) to 5 (red dots). Right: Coifman scaling functions of order 2,4 and 6.

[11] (Fig.1). Daubechies wavelets are well suited to Electronic Structure Calculations [31, 17], because of their small compact supports related to their approximation properties. Observe on Fig.1 that Daubechies scaling functions are asymmetric. Coiets are almost symmetric, but their supports are longer. The potential will be expanded into a basis satisfying the interpolation property:

Denition 2 (Interpolating scaling function) A 1D scaling function θ 7

is interpolating if it satises the condition: θ(x − k) = δ0,k , ∀k ∈ Z.

Let denote by VcJ a 3D interpolating MRA, generated by the 3D scaling functions: ΘJ,k (r) = ΘJ,0 (x − k) = θJ,k1 (x1 ) θJ,k2 (x2 ) θJ,k3 (x3 ) X where θJ,k (x) = θ(2J (x+r)−k) has a L∞ -normalization. Therefore ΘJ,k r∈Z

satises inria-00337464, version 1 - 7 Nov 2008

ΘJ,k (

l ) = δk,l . 2J

Such a function is also called Interpolet [13]. The biorthogonal scaling functions are given by will be called

m2 .

e J,k (.) = 23J δ(2J . − k). Θ

The approximation order of

VcJ

Interpolating scaling functions of several orders are shown

on gure Fig.2.

Figure 2: Interpolating scaling functions of orders 4,6,8,10. The oscillations increase with the order.

As explained in section 1, the potential

J points k/2 ,

k ∈ ΩJ .

V

dened on

Ω

is known at grid

(k/2J )}k∈ΩJ be the grid

= {vJ,k }k∈ΩJ = {V VcJ is given by: X X e J,k > ΘJ,k (r) PVJc V (r) = VJ (r) = VT TJ = vJ,k ΘJ,k (r) = < V, Θ Let V

values, the interpolation of

V

into

k∈ΩJ

k∈ΩJ (9)

TJ = {ΘJ,k }k∈ΩJ . The objective now is to apply the operator V to an orbital ui in an ecient way. This means to reduce as far as possible the with

computational cost, which can become huge in three dimensions.

8

3

Representation of the Hamiltonian operator

In this part we will assume that

V

depends only on the space variable

r.

We rst present estimates for the kinetic and potential energies in some cases. Then, we show that in the Galerkin formulation, the application of the potential operator to an orbital can have a high computational cost. One solution for this problem is to consider the projection operators onto the spaces introduced in section 2.

3.1

Galerkin formulation

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We look for solutions coecients

Ci

u ei

V⊥ J , given by their basis FJ = {ΦJ,k }k∈ΩJ :

of system (3) in the space

in an orthonormal scaling function

X

u ˜i (r) = CiT FJ =

(i)

cJ,k ΦJ,k (r),

(10)

k∈ΩJ with

(i)

Ci = {cJ,k }k∈ΩJ .

The system (3) is then reduced to:

 A+ where

G

G

 B Ci = i Ci

A = [Ak,k0 ] and G B = [G Bk,k0 ] are the following matrices (the exponent

stands for Galerkin):

0

∀ k, k ∈ ΩJ

Ak,k0 G

Bk,k0

Z 1 = ∇ΦJ,k ∇ΦJ,k0 , 2 Ω = < ΦJ,k , VJ ΦJ,k0 > .

We will focus now on the evaluation of the energies for one orbital. We thus abandon the index i, and call the approximated solution

u ˜.

The orbitals

T u ˜ are orthonormal in V⊥ ˜ is J , which implies C C = 1. The total energy of u given by e  = eekin + eep , where eekin stands for the kinetic energy and eep for the potential energy.

Kinetic energy: u ˜

the kinetic energy

eekin related to the approximate orbital

is given by:

eekin =

CT A C = C T A C. CT C

(11)

Then we have the well known result [28] in the particular case where the potential

V

is zero:

9

Proposition 1 Let

u be an orbital. Assume that u and all the orbitals of lower energy belong to H s (Ω). Then the error between the kinetic energy ekin , and the computed energy e˜kin behaves like: |ekin − e˜kin | . 2−2J(r1 −1) ,

(12)

with r1 = min(s, m1 ).

Potential energy:

The potential energy related to an orbital

u

is given

by:

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ep (u) =< u, V u > . In the Galerkin formulation, the potential energy of the approximate orbital

u ˜

is:

e˜p =< u ˜,

u ˜ >=

VJ

C T GB C = CT CT C

V is expanded into {ΘJ,k } (see (9)), an

Since the potential scaling function basis matrix

G

B C.

(13)

a nite dimensional interpolating element of the potential Stiness

G B writes:

F or k, k0 ∈ ΩJ ,

G

Bk,k0

= < ΦJ,k , VJ ΦJ,k0 > Z = ΦJ,k (r) VJ (r) ΦJ,k0 (r) dr Ω Z X ΦJ,k (r) ΘJ,m (r) ΦJ,k0 (r) dr = vJ,m Ω

m∈ΩJ

=

X

vJ,m TJ (k − m, k0 − m),

m∈ΩJ where

TJ (k, k0 ) =

Z ΦJ,k (r) ΘJ,0 (r) ΦJ,k0 (r) dr. Ω

are periodic tensor-product functions, each term three cyclic matrices

T,

Since the basis functions

TJ (k, k0 )

is a product of

whose coecients are called connection coecients

[10, 27]: 0

Z

0

φ(x − k) θ(x) φ(x − k ) dx.

T (k, k ) = ω

Details of the calculation can be found in the thesis [7]. The mean idea

ΘJ,m , the calculation G of Bk,k0 is obtained by a bidimensional convolution of coecients V with is that, thanks to the symmetry around the function

elements of the matrix

TJ .

10

Suppose now that the Hamiltonian es

inf r∈Ω V (r) > −1;

H

is elliptic, for instance if

V

satis-

then the exact and approximate orbitals fulll the

approximation error (using (5) and (8)):

ku − u ˜k ≤ C 2−Jr1 kukH r1 with

r1 = min(s, m1 ).

(14)

In this case, we may derive the following error esti-

mate.

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Proposition 2 Let

be an orbital. Assume that u and all the orbitals of lower energy belong to H s (Ω). Suppose also that the potential V ∈ H L (Ω). Then the error between the exact potential energy ep (u) and the approximate potential energy e˜p (˜u) behaves like: u

|ep (u) − e˜p (˜ u)| . C1 2−Jr1 + C2 2−Jr2 ,

(15)

with r1 = min(s, m1 ), r2 = min(L, m2 ). Proof: |ep (u) − e˜p (˜ u)| = |ep (u) − ep (˜ u) + ep (˜ u) − e˜p (˜ u)| Z Z 2 2 2 ˜ ˜ ) + (V − VJ ) u 6 V (u − u Ω

Ω

Using (14) yields:

Z Z 2 2 6 kV kL∞ V (u − u ˜ ) |(u − u ˜)(u + u ˜)| Ω

Ω

. 2 kV kL∞ kuk ku − u ˜k . 2−Jr1 kV kL∞ kuk kukH r1 . The second term can be majored by:

Z 2 2 (V − VJ ) u ˜ . kV − VJ kL∞ kuk Ω

. 2−Jr2 kV kH r2 kuk2 with

r2 = min(L, m2 ),

using classical interpolation error. We thus get the

estimate (15).

11

Table 1: Number of coecients of

m1 = 2

and

m2 = 8.

TJ

For example 9420 is the number of coecients greater

in absolute value than

10−8 ,

|TJ (k, k0 )|

and smaller than

> 10−4 >

#

684

Drawbacks

9420

The evaluation of

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(Interpolet),

10−4 .

> 10−8 >

TJ

cost, as already mentioned in [26].

8

located between two magnitudes, with

> 10−11 14980

is expensive in terms of computational For

m1 = 4

(Daubechies) and

−8 . has almost 30 coecients greater than 10

T

m2 =

In three

dimensions, Table 1 shows the repartition of the coecient size. Obviously, the coecients are localized near the diagonal (for

k

close to

k0 ).

Several work have been done to overcome this complexity, based on quadrature formula [21, 26]. More particularly, Neelov and Goedecker [26] use quadrature formula with an exactness order linked to scaling function order. They showed that the quadratic order for the error convergence was respected both in uniform and non uniform grid, in one dimension.

3.2

Biorthogonal Projection operators

We present now the projectors associated to the three types of scaling function basis (orthonormal basis, interpolation basis and its biorthogonal one).

Notation 2 (Orthogonal projector PVJ⊥ ) Let PVJ⊥ be the orthogonal projection of L2 (Ω) into V⊥J : PV ⊥ : J

   L2 (Ω) u

 

7−→ V⊥ J, −→ PV ⊥ u =

X

J

< u, ΦJ,k > ΦJ,k .

k∈ΩJ

Notation 3 (Interpolation operator PVJc ) Let operator of PVJc :

C 0 (Ω)

   C 0 (Ω)  

u

into

VcJ

:

PVJc

be the interpolation

7−→ VcJ , −→ PVJc u =

X

e J,k > ΘJ,k = < u, Θ

k∈ΩJ

X k∈ΩJ

12

uJ,k ΘJ,k .

Notation 4 (Biorthogonal projection PVeJc ) Let PVeJc be the projector of L2 (Ω)

e c (⊂ H −1 (Ω)): into V J

PVe c : J

   L2 (Ω)  

ec , 7−→ V J J

Following [9] and [3], the projectors (8): if errors

X

−→ PVe c u =

u

e J,k . < u, ΘJ,k > Θ

k∈ΩJ

PV ⊥ J

and

PVJc

satisfy the error estimate

c m1 (resp. m2 ) is the approximation order of V⊥ J (resp. VJ ), then the 2 s in L -norm between a function u ∈ H (Ω) and its projections behave

inria-00337464, version 1 - 7 Nov 2008

as:

ku − PV ⊥ uk . 2−Jr1 kukH r1 ,

r1 = min(m1 , s),

(16)

ku − PVJc uk . 2−Jr2 kukH r2 ,

r2 = min(m2 , s).

(17)

J

Such an estimate is quite dierent in case of the third operator

PVe c .

As

J

ec V J

PVe c u of a given function J −1 u leaves in the Sobolev space H of negative exponent, endowed by the H −1 -norm: is spanned by Delta-functions, the projection

∀ u ∈ H −1 ,

kukH −1 =

| < u, φ >H −1 ,H 1 |.

sup φ∈

H 1 (Ω)

kφkH 1 = 1

?

In this context, the following error estimate can be proved, following [ ]:

Proposition 3 The projection error of a behaves in the space

H −1

as:

L2 -function u,

provided by PVe c ,

ku − PVe c ukH −1 . 2−J(r2 +1) kukH r2 .

J

(18)

J

if r2 ??.

Remark 1 There exists several ways to dene a projector PVJc , since there

e c associated to the primal space Vc . exist several biorthogonal spaces V J J ⊥ Notice also that for u ∈ VJ , PVJ⊥ PVJc u 6= u in general.

13

Practical use of projectors

Numerically, these projectors are applied to

functions of the nite dimensional spaces

V⊥ J

,

VcJ

, or

ec V J

, spanned by the

eJ . These projectors are entirely dened by the respective bases FJ , TJ and T 3J × 23J real following matrices Z , X and Y ∈ MJ , where MJ is the set of 2 valued matrices.

Notation 5 PV ⊥ :

 

VcJ

7−→

V⊥ J,

Z  u = DT TJ −− → PV ⊥ u = (ZD)T FJ = C T FJ , J  Z =< TJ , FJ >= < ΘJ,k , ΦJ,k0 > k,k0 ∈Ω . J   V⊥ 7−→ VcJ , J PVJc : X  u = C T FJ −− → PV c u = (XC)T TJ = DT TJ , J

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where

J

where

n o e J,k0 > X =< FJ , TeJ >= < ΦJ,k , Θ

k,k0 ∈ΩJ

PVe c :

       

V⊥ J

.

ec , 7−→ V J

u = C T FJ

ZT

e T TeJ . −−→ PVe c u = (Z T C)T TeJ = D J

 ec ,  VcJ 7−→ V  J    Y  u = DT TJ − e T TeJ . → PVe c u = (Y D)T TeJ = D J  Y =< TJ , TJ >= < ΘJ,k , ΘJ,k0 > k,k0 ∈Ω . J

where

J

Applying

Z, X

Y

or

3J coecients is easily done by three to a set of 2

successive one-dimensional convolutions [7]. Using such technique in potential energy evaluation can thus improve signicantly the complexity. Indeed, applying the matrix

Z

to a set of

23J

3(l1 + l2 − 1)23J , with functions φ and θ . Applying e is δ0,k . to θ

coecients costs

l1 and l2 the support lengths of the 1D scaling X costs 3 l1 23J , since the lter corresponding

The main dierence with the method used in [26] is that we can approximate the coecients

C

for the expansion in

V⊥ J

with an order independent of

m1 .

< u, ΦJ,k > is 2m1 , whereas VcJ of any order m2 , and then ⊥ project into VJ . Moreover, the cost per grid point is in our case 3(l1 +l2 −1), and in their case 3(2 l1 + 3). If l1 and l2 are equal, then both methods are

In [26], the best order of approximation of in our method we rst project

u ∈ L2 (Ω)

in

similar. The numerical studies of section 4 gives the estimations of a good choice of

m1

and

m2 ,

according to the following approximation inequalities.

14

Approximation error of successive transfers to dierent MRA Lemma 1 Let u ∈ H s (Ω), m1 be the order of V⊥J , and m2 the order of VcJ .

Successive projections onto V⊥J and VcJ of u provide the following estimates: ku − PVJc PV ⊥ ukL2 . 2−Jr1 kukH r1 + 2−Jr2 kukH r2 .

(19)

J

with r1 = min(m1 , m2 ) and r2 = min(s, m1 ). e c and V⊥ leads to: In the same way, the transfer through V J J ku − PVe c PV ⊥ ukH −1 . 2−J(r1 +1) kukH r1 + 2−J(r2 +1) kukH r2 . J

J

(20)

The proof of the estimates (19) and (20) comes simply using the triangular

inria-00337464, version 1 - 7 Nov 2008

inequality and estimations (16), (17) and (18). The following example is an illustration of lemma 1 with a smooth function.

Example 1 Let u be a Gaussian function discretized on an interpolating basis, with coecient vector D. As u ∈ H s (Ω) for all s > 0, only the orders c m1 of V⊥ J and m2 of VJ interfere in convergence rates. (1) Do successive passage to VcJ and V⊥J , by applying the matrix ZX . Then evaluate the error ||D − (XZ)n D||l2 . According to (19), the approximation follows the law 2−J min(m1 , m2 ) . (2) Second, apply n times Z T Z to C , then compute the error ||D−(Z T Z)n D||l2 . The projection of PVe c PVJ⊥ f to VcJ is done by applying the Identity maJ e c are biorthogonal. trix, since VcJ and V J The results are shown in gure Fig.3 and in table TAB[1] . For one couple c (V⊥ J , VJ ), two curves are plotted: they correspond to the convergence order of the curves shown in gure Fig.3, evaluated by a least square procedure. In the method (1), the convergence orders correspond rather good to the competition of r1 and r2 shown in the a priori estimate (19). For the method (2), we have also evaluated the error in L2 norm: the convergence errors are better than in method (1), which is not predicted by (20). 3.3

Application to the computation of energies

The application of the Hamiltonian operator (2) to an orbital

u

has two

parts: 1) apply a derivation lter (kinetic part), 2) multiply by the real function introduce approximations

u ˜

V

(potential part).

of the orbital

15

u

(10).

In this part, we will

Table 2: Error rates for several

D3 I8

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3.03

6.08

D4 I6 3.94

and

m2 .

5.96

3.76

and

D5 I6

7.83

||D − (XZ)n D||l2

5.78

6.33

VcJ

.

D4 C2 3.42

D4 C6

3.87

3.70

7.46

||D − (Z T Z)n D||l2 for several c n = 10. The legend corresponds to couples (V⊥ J , VJ ) of orders m1 D stands for Daubechies, C for Coiet and I for interpolet families.

Figure 3: Errors (1)

2J , with

D4 I10

V⊥ J

and (2)

3.3.1 Kinetic energy To the kinetic energy (11) we associate the relative error

eekin =

τkin :

CT A C |ekin − eekin | , and τkin = . T C C |ekin |

3.3.2 Potential Energy We construct now two representations of the Hamiltonian operator: we start from the potential VJ expanded in the collocation basis

TJ

Eq. (9) and we

will deduce two expressions for the approximate potential energy

eep .

To that

purpose, we will dene two kinds of matrix-matrix product and matrix-vector product:

Notation 6 (Matrix-matrix and matrix-vector products) Let two real 16

matrices A and B ∈ MJ of size 23J × 23J , and two vectors U and V of size 23J be given. The standard matrix-matrix product writes A B , whereas the standard matrix-vector product writes A U . On the other hand, the matrixmatrix product ·, and the matrix-vector product · will denote an element-byelement multiplication, as follows: (U · A)ij

= Ui Aij ,

(U · V )i = ui vi .

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Method 1

The bilinear form associated to VJ is written in

∀ u, v ∈ V⊥ J.

< u, V v > ∼ < u, PV ⊥ (VJ PVJc v) >, J

and its representation in the basis

1

FJ

⊥ V⊥ J × VJ :

yields the Stiness Matrix:

B = Z (V · X).

e= D = X C , that is if D is the collocation vector of u on VcJ , and if D T e Z C , where D corresponds to the coecients of u in the dual interpolating 1 e is approximated by: basis, then the potential energy e p If

1

Method 2

eep =

e T (V · D) D . eT D D

We approximate the bilinear form associated to VJ by:

< u, V v > ∼ < u, PV ⊥ PVJc (VJ PVe c v) >, J

J

Its matrix representation in the basis element

u ∈ V⊥ J

FJ

∀ u, v ∈ V⊥ J

and the potential energy of an

write:

2 J

B = Z (V · Z T ),

2

eep =

e T (V · D) e D . eT D e D

Next section presents the relative errors for the potential energy in the three cases: Galerkin, methods 1 and 2. The relative error writes:

G,1,2

where

ep

τp =

|ep − G,1,2 eep | , |ep |

is the potential energy of the exact orbital

17

(21)

u.

4

Numerical Applications

We have done three-dimensional tests on two linear Hamiltonian operators. 1. The Harmonic Oscillator:

the potential

Vo

uo

and the solution

are

very regular, so that we can observe numerically the error estimates depending only on the basis order.

Vh and the orbital uh are both in 0 C , and leads to more complicated laws for the error behavior.

2. The Hydrogen atom: the potential

Ω: for all r = (X, Y, Z) ∈ Ω, we associate the triplet (x, y, z) such that X = Y = y z c , Z = , with L = 10. The orbital u is expanded into the space VJ of L L interpolating scaling functions of order m2 : X PVJc u = u(xJ,k ) ΘJ,k . (22) In the tests presented here, we have made a dilation of the space

inria-00337464, version 1 - 7 Nov 2008

x L,

k∈ΩJ

eekin

Starting from Eq.(22), we evaluate

{cJ,k }k∈ΩJ

eep .

and

The coecients

C =

are obtained by applying a change of basis:

PV ⊥ u =

X

J

cJ,k ΦJ,k = C T FJ = (ZD)T FJ .

(23)

k∈ΩJ 4.1

Harmonic oscillator

The model for the Harmonic Oscillator is:

A solution

(Eo , uo )

of

1 1 H = − ∆ + |r|2 , r ∈ Ω. 2 2 the eigenproblem Hu = Eu is

uo (r) = CN e−|r|

2 /2

,

known analytically:

∀r∈Ω

Eo = ekin + ep , ekin = ep = CN

< uo | − 12 ∆|uo > = 0.75 a.u., < uo |uo > < uo | 21 |r|2 |uo > = 0.75 a.u.. < uo |uo >

is a normalization factor, based on the

The orbital

uo

L2

(24)

norm of

decays rapidly, so there is no need to have a

example, the orbital

uo

and the potential

Vo

live in

C ∞,

uo : kuo k2 = 1. great L. In this

then the conver-

gence order depends only on the space approximation orders

18

m1

and

m2 .

inria-00337464, version 1 - 7 Nov 2008

Figure 4: Harmonic Oscillator: relative error ferent orders

m1

and

m2

τkin

as a function of

2J .

Dif-

are considered.

4.1.1 Kinetic energy τkin in terms of 2J , on a log/log plot. Two Interpolet orders, m2 = 6 and m2 = 8, with an orthogonal order of m1 = 4, are considered: the dependency on m2 is weak. We obtain the

Fig.4 shows the behavior of the relative error

following behavior for the relative error:

τkin ∼ C 2−2J(m1 −1) , which can be seen as an extension of property (12). Indeed, this property holds for a pure Laplacian operator. Here, we have recovered this estimation for an eigenvector of the Hamiltonian associated to the Harmonic Oscillator, and its kinetic part. Below

10−9 , the machine precision interferes with the values.

The results

are very similar for a Symmlet and a Daubechies scaling function of same order (the Stiness matrices of the Laplacian are identical). We get the same result with Coiets (the error is of order

10−5

for

2J = 32),

but Daubechies

functions have shorter support, and thus have more numerical interest. The error between the exact and the approximated energy is thus quadratic

J and m1 . Remember that if we want to get an energy √ , we only need a precision of  on the orbital.

as a function of some precision

19

with

4.1.2 Potential Energy Galerkin formulation In case of Harmonic Oscillator, it is easy to evaluate the Galerkin Stiness matrix, thanks to the separability properties of the operator and the solution:

1 2 (x + y 2 + z 2 ) = v(x) + v(y) + v(z), 2 2 uo (r) = CN e−|r| /2 = u(x) u(y) u(z),

Vo (r) =

1/3

u(x) = CN e−x

inria-00337464, version 1 - 7 Nov 2008

Let

ω = [0, L[,

2 /2

.

we can express the potential energy in terms of the one

dimensional quantity:

Z

Z

ep =

uo (r) Vo (r) uo (r) dr = 3 Ω

1 v(x) u2 (x) dx = 3 × . 4 ω

D4 I8

D3 I8

D3 I4

32

0.7500070985

0.750043907

0.75055975

64

0.750000030114

0.75000067514

0.750033937

128

0.750000000058755

0.750000010427

0.750000010427

Table 3: Harmonic Oscillator: potential energy

Ge ˜p in case of Galerkin for-

mulation.

Table 3 shows the potential energies

Ge ˜p for dierent couples

(m1 , m2 ).

As the solution is highly regular, the precision only depends on these two orders. The convergence rates are in these three cases 8.5, 6.2 and 4. This suggests an error behavior like:

G

τp ∼ C 2−J

min(2m1 +α, m2 )

A reasonable choice is thus an Interpolet order

Method 1

1 α6 . 2

,

m2 > 2m1 .

1 τ dened p J in Eq.(21) as a function of 2 , for dierent couples (m1 , m2 ). The coecient Figure Fig.5 shows the evolution of the relative error

rate behaves like:

1

τp ∼ C1 2−J 20

min(m1 ,m2 )

.

(25)

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Figure 5: Harmonic Oscillator. Relative error

J lution 2 . In three case,

p as a function of the reso⊥ VJ is spanned by Coiets of order m1 = 2, 4, 6, and

in the last one, by a Daubechies basis

Table 4:

D4.

C2 I8

C4 I8

C6 I8

D4e I10

D4s I8

1.96

3.96

5.86

3.96

3.90

Harmonic Oscillator.

dierent couples

1τ

(m1 , m2 ). s

Convergence rates for the method 1, for

stands for symmetric,

e

for extremal phase.

Table 4 shows the rates corresponding to gure Fig.5. Coiets are slightly better than Daubechies functions, because the Coiet scaling functions have vanishing moments.

Method 2

The rates obtained for the relative error in method 2 are rela-

tively high. As observed on gure Fig.6, the error is quadratic:

2

the factor

C2

τp ∼ C2 2−J min(2m1 ,m2 ) ,

depending on

||u||H m1

and

||u||H m2 .

(26) The energies for the

method 2 (Table 5) are of same order as those obtained in the Galerkin formulation (Table 3).

We can therefore conclude that this evaluation of

the potential energy is optimal: its behavior is as good as the Galerkin one, although its computational complexity is highly reduced.

21

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2τ

Figure 6: Harmonic Oscillator, nal bases

D4I10

2J ,

for dierent orthogo-

Coiets (left) and Daubechies (right). The curves

show the inuence of

Table 5: couples

m1 :

p as a function of

m2

C6I10

J

D3 I8

D4 I8

C4 I8

5

.750043053

.750010176

.7500094682

6

.75000062740

.750000043524

.750000040569

7

.750000009528

.7500000001122

.75000000010048

Harmonic Oscillator.

and

on the convergence.

2e ˜p for dierent

(m1 , m2 ).

22

J

(column) and dierent

4.2

Hydrogen atom

For the Hydrogen atom, the Hamiltonian operator writes with

1 Vh (r) = − |r| .

The ground state

(Eh , uh )

1 He = − ∆ + Vh , 2

is known analytically:

uh (r) = CN e−|r| , ekin = 0.5 a.u., ep = −1. a.u.,

inria-00337464, version 1 - 7 Nov 2008

where

CN

is a normalization constant.

4.2.1 Kinetic energy The evolution of

τkin

depends on the

H s -regularity of the orbital uh .

Indeed,

we observe (for instance on gure Fig.7) the following behavior:

τkin ∼ C 2−2J(min(m1 ,s)−1) , with a prefactor

C

m1 .

increasing with

Numerically, the Sobolev regularity

of the orbital is around 2.4. This value coincides with a theoretical result obtained by H.-J. Flad (MPI, Leipzig), that would be published, showing that the Hydrogen orbital has a regularity smaller than

5 2.

On Fig.7, the Daubechies basis behaves slightly better than Coiets, certainly due to the fact that the Laplacian Stiness matrix in Daubechies basis is the same that the one in the interpolating basis. For physical systems, where the potential is more irregular, it will be thus not useful to take high order

m1 .

4.2.2 Potential Energy The singularity

1 |r| will be numerically avoided by not centering

Vh

at a

discretisation point.

•

At rst, we evaluate the relative error when the orbital is the Gaussian

uo .

Results of convergence are shown in Table 6. As the singularity

does not appear numerically,

Vh -regularity

does not correspond to the

reality. Nevertheless, these convergence rates give information about the quality of the two methods.

Actually, it appears that method 2

is in general better that method 1. compatible with the behavior around

4,

The values obtained for

2−J min(m1 ,m2 ) .

and certainly represents the numerical regularity of

23

C2

are

The convergence rate is

Vh .

inria-00337464, version 1 - 7 Nov 2008

Figure 7: Hydrogen Atom:

C2 I6 2.06

τkin

D3 I6

4.05

3.95

as a function of

C4 I6

4.14

3.97

4.12

2J ,

for dierent

D4 I6 3.93

4.06

m1 , m 2 .

C6 I8 4.23

4.06

Vh and orbital uo . Convergence 1 2 rates of relative errors τp (left) and τp (right) as a function of (m1 , m2 ). Table 6:

•

Hydrogen Atom with potential

For the Hydrogen orbital

uh ,

except for Coiets of order

get always a convergence rate between 1 and 2, and for any

m2 .

m1 = 2,

we

2.89 and 2.98 for both methods

In this case, numerical results tend to conrm

the estimations (19) and (20). For each couple (m1 , m2 ), the error is of order

10−4

for

2J = 128.

4.2.3 Inuence of pseudo-potential Pseudo-potentials have several properties, the main one being to screen nucleus potential by the rst electrons of the atom, to cancel the singularity. In our tests, we use the following one:

Vloc (r) = −

1 erf |r|

 √

   |r| 2 |r| |r| 2 |r| 4 |r| 6 −1( ) c1 + c2 ( +e 2 rloc ) + c3 ( ) + c4 ( ) , rloc rloc rloc 2 rloc

where erf denotes the error function or repartition of the normal law, and

Ci , rloc

are coecients depending on atom characteristics. Numerical values

24

of these parameters can be found in the Physics literature [19]. The second diculty in atomistic simulations is the long range of potentials, behaving

1/|r|.

like

On the torus

Ω, we have to cut it articially, by applying a window.

The Hamiltonian tested here is thus:

1 Vloc (r) 1 . H = − ∆ + Ve = − ∆ + 2 2 1 + eβ (|r|−rc ) The orbital

u1

is not equal to

associated to the ground state of the system described by

uh .

Nevertheless we use the couple

(Ve , uh )

H

in our numerical

V (r) 6 Ve (r) ∀ r ∈ Ω, implying the energy inequality e < uh , V u h > (recall that ep is the theoretical Hydrogen potential ep 6 < uh , u h > e have also to satisfy energy). The ground state orbital u1 for the potential V

inria-00337464, version 1 - 7 Nov 2008

tests, knowing that

the same inequality:

< u1 , Ve u1 > . < u1 , u 1 >

ep 6 Indeed

V:

V

is deeper than

Ve ,

meaning that the electron is more attracted by

this implies that the kinetic energy

increases.

Table 7 and Figure Fig.8 show dierent convergence rates. We get the maximal order

2.95

for

m1 > 4

and

m 2 > 4.

Figure 8: Hydrogen Atom with pseudo-potential. Relative errors ( and

2τ

p (right)) as a function of

2J ,

for several couples

25

(m1 , m2 ).

1 τ (left), p

D3 I8

C2 I8

D4 I6

D4 I10

C4 I4

C4 I8

C6 I8

1.9

2.2

2.9

2.9

2.95

2.95

2.95

Table 7: Hydrogen Atom with pseudo-potential. Convergence rates of the relative error

1 τ , for dierent couples p

m1 , m 2 .

Conclusion After recalling a priori error estimations for the eigenvalues and the eigenvec-

inria-00337464, version 1 - 7 Nov 2008

tors of an elliptic operator, we give error estimations for kinetic and potential energy in specic cases. The approximation is made using Multiresolution Analysis on the tore of

R3 .

More particularly, we have studied the poten-

tial energy, and given dierent methods to compute this energy to improve the computational cost. These methods are based on the expansion of the potential into an interpolating scaling function basis, whereas the orbitals are expanded into an orthonormal scaling function basis. tional costs are optimal, linear in

O(N )

where

N

Their computa-

is the dimension of the

approximation space, and with a very low prefactor. For both methods, numerical tests on Harmonic Oscillator and on the Hydrogen Atom show their eciency and accuracy.

Acknowledgements Authors would like to thank Stefan Goedecker for his rst idea to combine dierent wavelet families in the context of

ab initio

methods. Authors are

also indebted to Reinhold Schneider for many fruitful discussions. This work was partially supported by the CEA-Grenoble, and by the IHP network

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