DENSITY FUNCTIONAL THEORY-BASED NANOSTRUCTURE...

MULTIBODY DYNAMICS 2005, ECCOMAS Thematic ConferenceJ.M. Goicolea, J. Cuadrado, J.C. Garcıa Orden (eds.)

Madrid, Spain, 21–24 June 2005

DENSITY FUNCTIONAL THEORY-BASED NANOSTRUCTUREINVESTIGATION: THEORETICAL CONSIDERATIONS

Dan Negrut†, Mihai Anitescu†, Todd Munson†, and Peter Zapol‡

†Mathematics and Computer Science DivisionArgonne National Laboratory

9700 S.Cass Ave., Argonne, IL60439, USAe-mails:[email protected] ,[email protected] ,[email protected]

‡ Materials Science and Chemistry DivisionArgonne National Laboratory

9700 S.Cass Ave., Argonne, IL60439, USAe-mail: [email protected]

Keywords: electron density computation, electronic structure reconstruction, quasicontinuum,Density Functional Theory (DFT), optimization, nanostructure

Abstract. This work proposes a theoretical framework for the investigation of chemical andmechanical properties of nanostructures. The methodology is based on a two-step approach tocompute the electronic density distribution in and around a nanostructure and then the displace-ment of its nuclei. TheElectronic Problemembeds interpolation and coupled cross-domain op-timization techniques through a process called electronic reconstruction. In the second stage ofthe solution, theIonic Problemdeals with repositioning the nuclei of the nanostructure giventhe electronic density in the domain. It is shown that the new ionic configuration is the solutionof a nonlinear system obtained based on a first-order optimality condition when minimizing thetotal energy associated with the nanostructure. The long-term goal of this work is a substan-tial increase in the dimension of the nanostructures that can be simulated by using approachesthat include accurate DFT computation. The increase in nanostructure size results from thekey observation that during the solution of theElectronic Problemexpensive DFT calcula-tions typically carried out with dedicated third-party software such as NWChem or Gaussian03are limited to a small number of subdomains; the electronic density is then reconstructed else-where. For theIonic Problem, computational gains result from approximating the dislocation ofthe nuclei in terms of a reduced number of representative nuclei following the quasicontinuumparadigm.

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Dan Negrut, Mihai Anitescu, Todd Munson, and Peter Zapol

1 PARADIGM OF THE PROPOSED APPROACH

The intent of this document is to provide the analytical framework for a longer-term projectthat focuses on the investigation of chemical and mechanical properties of nanostructures.

Nanostructures have dimensions in the range of1 ∼ 100 nm and typically contain102 ∼ 108

atoms. Applying the well-established Kohn-Sham DFT method [18] for nonperiodic structuresof 60 atoms has led to simulations that can take up to three months to complete. When longrange interactions are ignored and pseudo-potentials are used,ab-initio simulations have beencarried out for nonmetallic structures with up to 1,500 atoms [25]. The approach that enabledthe increase in the number of atoms belongs to the family of so-calledO(N) methods [10],which scale asN with the dimension of the problem (in this case the number of electrons).

This work is not concerned with fundamental electronic structure computation methods. Ac-knowledging the small-dimension constraint placed on the problem by the existing DensityFunctional Theory (DFT)-based methods, the goal of the proposed work is to use techniquesthat, by closing the spatial scale gap, render electronic structure information at the nanoscale.This electronic structure information is then used to investigate the chemical and mechanicalproperties of the material.

In the context of mechanical analysis of nanostructures, the methodology proposed followsin the steps of the quasi-continuum work proposed in [7, 16, 26]. Specifically, this is an ex-tension of the work in [26, 16], because rather than considering a potential-based interatomicinteraction that has a limited range of validity and is difficult to generalize to inhomogeneousmaterials, the methodology proposed usesab-initio methods to provide for the particle interac-tion. At the same time it is a generalization of the method proposed in [7] because rather thanconsidering each mesh discretization element to be part of a periodic and uniformly deformedinfinite crystal, the proposed method treats in a generic optimization framework any structure(nonperiodic and inhomogeneous) once the electronic density distribution is available.

Two goals are associated with this project; (1) development and software implementation ofa methodology that can substantially increase the dimension attribute of the electronic structureproblem and (2) support for investigation of general nanostructures (metallic and nonmetallic,nonperiodic structures, inhomogeneous materials).

The electronic density reconstruction described is done in reference to a regular lattice ordomain of a regular lattice. Significant computational savings are anticipated to stem fromtwo assumptions: (1) thegeometric assumption, where the premise is that the lattice is onlyminimally deformed and the state variables are nearly periodic in most of the domain (this latterrequirement will be relaxed to allow for localized defects), and (2) theelectronic assumption,where the premise is that for a given ionic distribution, the electronic energy can be expressedas

E(ρ, ρA) =

∫Θ1(ρ, ρA, r)dr +

∫∫Θ2(ρ, ρA, r; ρ, ρA, r′)drdr′ (1)

This representation is commonly used in conjunction with the so-called Orbital-Free DFT(OFDFT) method [29]. HereΘ1,2 are the relevant energy density functions;ρ is the electronicdensity; andρA is the nuclear density, which may include delta functions. The first term typi-cally includes the kinetic energy and an exchange-correlation term, whereas the second integralincludes all pairwise interactions. Details regarding the definition of these terms are providedby several authors [20, 15, 17].

The electronic structure computation is then formulated as an optimization problem [14]:

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find the electronic densityρ that solves the problem

minρ

E[ρ, ρA] (2a)

subject to the charge conservation constraint∫

ρ(r)dr = Ne (2b)

whereNe represents the number of electrons present. The solution to this problem dependsparametrically on the nuclear densityρA, ρ = ρ(ρA), a consequence of the Born-Oppenheimerassumption. Subsequently, the computation of the ground state of the entire system as thesolution of the optimization problem

minρA

E[ρ(ρA), ρA] (3)

provides the nuclei distribution. The latter problem governs the approach to the first question.As indicated above, one of the twocentral assumptionsis that almost everywhere in the

nanostructure the solution to the nuclei distribution problem results in only small deformations.In order to quantify the concept of small deformation, the nanostructure is considered to oc-cupy an initial reference configurationD0 ⊂ R3. The structure undergoes a change of shapedescribed by a deformation mappingΦ(r0, t) ∈ R3. This deformation mapping gives the loca-tion r in the global Cartesian reference frame of each pointr0 represented in the undeformedmaterial frame. As indicated, the mapping might depend on timet. The variablet does notnecessarily represent the time contemporary with the structure under consideration. In fact, ina quasi-state simulation framework, this variable might be an iteration index of an optimizationalgorithm that solves Eq.(3) in the caseρA is made of nuclear point charges.

The components of the deformation gradient are introduced as

FiJ =∂Φi

∂r0J

(4)

where upper-case indices refer to the material frame, and lower-case indices to the Cartesianglobal frame. Thus,F = ∇0 Φ, where∇0 represents the material gradient operator, andtherefore the deformation of an infinitesimal material neighborhood dr0 about a pointr0 of D0

is expressed as

dri = FiJ dr0J (5)

The concept of small distortion is equivalent to requiring that the spectral radius ofF be suffi-ciently small; that is,

||∇0 Φ||2 < K (6)

is expected to hold for almost everywhere in the domainB0, for a suitable chosen value ofK.As a consequence of thegeometric assumption, computational savings are anticipated be-

cause for all the domains that satisfy the condition of Eq.(6) a two-tier interpolation-basedapproach will reduce the dimension of the problem. First, the electronic structure will be re-constructed in some domains by interpolation using adjacent regions in which a DFT-basedapproach has been used to accurately solve the electronic structure problem; we call this pro-cedureelectronic density reconstruction. Second, the position of the nuclei will be expressed

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in terms of the positions of a reduced set of so-called representative nuclei,repnuclei, in an ap-proach similar to the one proposed in [26]. The proposed approach solves only for the positionof theserepnuclei; the position of the rest of the nuclei is then obtained by interpolation.

The remainder of this document is organized as follows. Section2 discusses the strategy forelectronic density reconstruction. The emphasis is placed on how interpolation is used to esti-mate the value of the electronic density in an entire domainD based on information available ina limited set of interior subdomains. The section starts with a simple DFT approach (Thomas-Fermi) that serves as a vehicle for introducing an otherwise abstract methodology. The entiresection draws on the physics of the problem being addressed. In contrast, section3 focuses onthe numerical solution component of the methodology. This section casts the problem in a nu-merical optimization framework and then presents the difficulties associated with the problemand the way they are addressed. With the electronic structure problem solved, the proposedmethodology uses the Born-Oppenheimer assumption to investigate the mechanical propertiesof a nanostructure given a certain electronic distribution. This analysis is discussed in section4.Section5 succinctly presents an outline of the computational flow at the end of which the cou-pled electronic structure and nanostructure shape problems are solved together. A set of openquestions concludes this section. Following the Conclusions section, the Appendix presents amore formal proof for a domain decomposition approach used within the Thomas-Fermi DFTframework.

2 ELECTRONIC DENSITY RECONSTRUCTION

2.1 A simple example: a domain with a gap and Thomas-Fermi DFT

The notation introduced in section1 as well as the proposed methodology is first appliedwhen the Thomas-Fermi functional is used to describe the dependency of the energy on elec-tronic density [27, 8]. The Thomas-Fermi functional has well-known severe accuracy limita-tions. It provides, however, a simple framework in which several key points of the methodologyproposed for electronic density reconstruction are more easily introduced.

2.1.1 The Thomas-Fermi functional

The Thomas-Fermi-based energy functional assumes the form

E [ρ, {RA}] = Ene [ρ, {RA}] + J [ρ] + K [ρ] + T [ρ] + Vnn ({RA}) (7)

where

Ene [ρ, {RA}] = −M∑

A=1

∫ZA ρ(r)

‖RA − r‖ dr (8a)

J [ρ] =1

2

∫ ∫ρ(r) ρ(r′)‖r− r′‖ dr dr′ (8b)

T [ρ] = CF

∫ρ

53 (r) dr (8c)

K [ρ] = −Cx

∫ρ

43 (r) dr (8d)

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Figure 1:Electronic density reconstruction.

Vnn ({RA}) =M∑

A=1

M∑B=A+1

ZA ZB

‖RA −RB‖ (8e)

HereCF = 310

(3π2)2/3, andCx = 34

(3π

)1/3, and the following notation is used:

• Ene - energy corresponding to nucleus-electron interaction

• J - Coulomb energy

• K - exchange energy

• T - kinetic energy

• Vnn - internuclear interaction energy

• ZA - atomic number associated with nucleus A

• ri - global position of electroni

• RA - global position of nucleus of atomA

• ∫(·) without integration limits - an integral over the entire domain.

The expression of the energy functional of Eq.(7) justifies the notation used in Eq.(1): thekinetic, exchange, and nuclear electronic energy are represented through theΘ1 term; theelectron-electron interaction is associated with the termΘ2.

In this simple example assume that there are three identical domainsD1, D2, D3, as in Fig.1.The electronic density in the respective domains is denoted byρ1(r), ρ2(r), ρ3(r):

ρ(r) =

ρ1(r)ρ2(r)ρ3(r)

r ∈ D1

r ∈ D2

r ∈ D3

The definition of the density outside the domainD = D1 ∪D2 ∪D3 is extended by assumingthat its value is zero.

In the Thomas-Fermi case, the optimization problem of Eq.(2) depends parametrically onthe positions of the nuclei:

minρ E(ρ; {RA}) + λ(∫

ρdr −N)

s.t.∫

ρdr −N = 0(9)

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The constrained optimization problem above is formulated for the case in which there is nodeformation in the underlying crystal structure of the material, whose nuclei are at positions{RA}, for A = 1 . . . M . In a direct approach, the dimension of the problem is prohibitive mostof the time; unless simplifying assumptions are taken into account (such as periodic boundaryconditions, local effects (truncation), or pseudo-potentials), systems that contain thousands ofatoms cannot be typically simulated. For domainsDi, i = 1, 2, 3, the energy is defined as

Ei [ρi, λi; ρi, {RA}] = CF

∫

Di

ρ53i (r) dr− Cx

∫

Di

ρ43i (r) dr +

∫

Di

∫

D−Di

ρi(r) ρi(r′)

‖r− r′‖ dr dr′(10)

+1

2

∫

Di

∫

Di

ρi(r) ρi(r′)

‖r− r′‖ dr dr′ −M∑

A=1

∫

Di

ZA ρi(r)

‖RA − r‖ dr + λi

∫

Di

ρi dr

The symbolρi is used to denote the electronic density outside the domainDi, i = 1, 2, 3.The optimality conditions for the optimization problem (9) can now be represented in terms ofsubdomain problems on the domainsDi, i = 1, 2, 3.

∇ρiEi(ρi, λi; ρi, {RA}) = 0, i = 1, 2, 3 (11a)

λ1 = λ2 = λ3 (11b)

∫ρdr−Ne = 0 (11c)

2.1.2 Reconstruction through interpolation

Assuming that the solution is sufficiently close to, but not necessarily, periodic, forr ∈ D2

the density is reconstructed by averaging, that is (see Fig.1)

ρ2(r) ≈ 1

2(ρ1(r− a(1, 0, 0)) + ρ3(r + a(1, 0, 0))) (12)

wherea(1, 0, 0) is a translation vector that indicates that the structure is periodic in the(1, 0, 0)direction. Likewise,a > 0 is a constant scaling factor associated with the underlying struc-ture, much as it is the case with a Bravais lattice but in this case applied for subdomain-typeperiodicity.

This approximation can be improved by using only domains away from the endpoints of theoverall slablike domain. For simplicity, however, in this example the entire domainsD1 andD3 are considered for reconstruction. Based on Eqs.(11) and (12), this leads to the followingcoupled system of nonlinear equations:

∇ρ1E1(ρ1, λ1; ρ3,1

2(ρ1(r− a(1, 0, 0)) + ρ3(r + a(1, 0, 0))) , {RA}) = 0 (13a)

∇ρ3E3(ρ3, λ3; ρ1,1

2(ρ1(r− a(1, 0, 0)) + ρ3(r + a(1, 0, 0))) , {RA}) = 0 (13b)

λ1 = λ3 (13c)

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∫

D1

ρ1dr +

∫

D3

ρ3dr =2

3Ne (13d)

which can be solved without referencing the second domainD2. Note that no assumptionis made about the charge neutrality in the domains; if an external nonsymmetric potential ispresent, the total charge will reflect the nonsymmetry.

The condition in Eq.(13a) leads to the following integral equation, which must hold for anyr ∈ D1:

53CF ρ

231 (r) −4

3Cxρ

131 (r) +

∫

D1

ρ1(r′)K11(r

′, r)dr′ +∫

D3

ρ3(r′)K13(r

′, r)dr′ (14a)

−M∑

A=1

ZA

||r−RA|| + λ1 = 0

where the kernelsK11 andK13 are defined as

K11 =1

||r− r′|| +0.5

||r− (r′ + T)|| K13 =1

||r− r′|| +0.5

||r− (r′ −T)|| (14b)

whereT = a(1, 0, 0). Similarly, writing the optimality condition of Eq.(13b) leads to thefollowing integral equation, which must hold for anyr ∈ D3:

53CF ρ

233 (r) −4

3Cxρ

133 (r) +

∫

D1

ρ1(r′)K31(r

′, r)dr′ +∫

D3

ρ3(r′)K33(r

′, r)dr′ (15a)

−M∑

A=1

ZA

||r−RA|| + λ3 = 0

where the kernelsK31 andK33 for this problem satisfy the condition

K31 = K11 K33 = K31 (15b)

Equations (14a) and (15a) represent a set of nonlinear integral equations that are solvedthrough standard techniques [2]. These equations were derived under the assumption that thereis no deformation of the domains. However, similar equations can be derived from an interpo-lation approach on the underformed crystal, to which the problem on the deformed crystal isreduced by the compositionρ ◦ Φ(r0, t), whereΦ(r0, t) is the deformation mapping. In thatcase, Eq.(12) is replaced by

ρ2(r, t) = ρ2(Φ(r0, t)) ≈ 1

2

(ρ1(Φ(r0 −T, t)) + ρ3(Φ(r0 + T, t))

)(16)

where the deformationΦ that depends on the positionr0, of the point in the undeformed (mate-rial) frame and time is as defined in section1. The optimality condition∇ρ1E1 = 0 leads to thefollowing integral equation:

53CF ρ

231 −4

3Cxρ

131 +

∫

D1

ρ1(r′)

||r− r′||dr′ +

∫

D2

ρ2(r′)

||r− r′||dr′ +

∫

D3

ρ3(r′)

||r− r′||dr′ (17)

−M∑

A=1

ZA

||r− r′|| + λ1 = 0

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A change of integration variable is performed to take the integration back to the undeformeddomains:

∫

D1

ρ1(r′)

||r− r′|| dr′ =

∫

D01

ρ1(Φ(r0′, t))||Φ(r0, t)− Φ(r0′, t)|| |F(r0′, t)| dr0′ (18a)

∫

D2

ρ2(r′)

||r− r′|| dr′ =

∫

D02

ρ2(Φ(r0′, t))||Φ(r0, t)− Φ(r0′, t)|| |F(r0′, t)| dr0′

=

∫

D02

0.5(ρ1(Φ(r0′ −T, t)) + ρ3(Φ(r0′ + T, t)))

||Φ(r0, t)− Φ(r0′, t)|| |F(r0′, t)| dr0′(18b)

∫

D3

ρ3(r′)

||r− r′|| dr′ =

∫

D03

ρ3(Φ(r0′, t))||Φ(r0, t)− Φ(r0′, t)|| |F(r0′, t)| dr0′ (18c)

Therefore, the optimality condition for any pointr0 assumes the form of an integral equation:

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3CF ρ

231 (Φ(r0, t))− 4

3Cxρ

131 (Φ(r0, t)) +

∫

D01

ρ1(Φ(r0′, t))K11(r0′, r0)dr0′ (19a)

+

∫

D03

ρ3(Φ(r0′, t))K13(r0′, r0)dr0′ −

M∑A=1

ZA

||Φ(r0, t)− Φ(R0A, t)|| + λ1 = 0

where the kernelsK11 andK13 are defined as

K11(r0′, r0) =

|F(r0′, t)|||Φ(r0, t)− Φ(r0′, t)|| +

0.5 |F(r0′ + T, t)|||Φ(r0, t)− Φ(r0′ + T, t)|| (19b)

K13(r0′, r0) =

|F(r0′, t)|||Φ(r0, t)− Φ(r0′, t)|| +

0.5 |F(r0′ −T, t)|||Φ(r0, t)− Φ(r0′ −T, t)|| (19c)

Similarly, writing the optimality condition of Eq.(11) for domainD3 leads to the followingintegral equation that must hold for anyr0 in the undeformed domainD0

3:

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3CF ρ

233 (Φ(r0, t))− 4

3Cxρ

133 (Φ(r0, t)) +

∫

D01

ρ1(Φ(r0′, t))K31(r0′, r0)dr0′ (20a)

+

∫

D03

ρ3(Φ(r0′, t))K33(r0′, r0)dr0′ −

M∑A=1

ZA

||Φ(r0, t)− Φ(R0A, t)|| + λ3 = 0

where the kernelsK31 andK33 satisfy

K31(r0′, r0) = K11(r

0′, r0) K33(r0′, r0) = K31(r

0′, r0) (20b)

Note that Eqs.(19) and (20) are similar to Eqs.(14) and (15). The equations correspondingto the undeformed case are obtained by setting|F(r0′, t)| = 1, andΦ(r0, t) = r0 everywhere

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in D0 = D01 ∪D0

2 ∪D03. These two conditions effectively indicate that none of these domains

experiences any deformation.

In setting up and solving the above equations, an appropriate representation forΦ(r0′, t) isnecessary. In this work mesh-based representations are considered. This leads to the followingrepresentation:

Φ(r0′, t) =∑A∈B

ϕ(r0′|R0A)Φ(R0

A, t) (21)

Thus, the deformation needs to be represented only at the pointsR0A, A ∈ B, and is then

reconstructed by interpolation at the other points of the space, by using the shape functionsϕ(·, ·). The pointsR0

A, A ∈ B may or may not coincide with nuclear positions.A difficulty with Eqs.(19a) and (20a) is that the equations are singular whenr0 = R0

A,A = 1, 2, . . . , M , which raises the question whether the equations are well posed. Consid-ering the procedure used to obtain these equations, one can claim only that they are valideverywhere except in small neighborhoods of the nuclear position; that is, not forr0 = R0

A,A = 1, 2, . . . , M . In addition, asymptotic examination (asr0 → R0

A, A = 1, 2, . . . , M ), ofEqs.(19) reveals that they can be asymptotically satisfied, provided that the leading term is

ρ(Φ(r0, t)

) ≈(

3ZA

5CF

) 32

||Φ(r0, t)− Φ(R0A, t)||− 3

2 (22a)

Since the above expression is integrable in three dimensions, it does not pose a problem forevaluating the total charge integral. In addition, this singularity is an artifact of the coordinatesystem used. Using spherical coordinates to represent the density aroundΦ(R0

A, t), the singu-larity is lifted by the determinant of the Jacobian of the coordinate transformation. With respectto those coordinates, the density satisfies

ρ(Φ(r0, t)

) ∼ ||Φ(r0, t)− Φ(R0A, t)|| 12 (22b)

and the singularity is thus removed.Nevertheless, this type of behavior is not easily captured on a computational mesh. A con-

trolled approximation via smoothing of the potential will be introduced in section2.3, and as aresult the expression of the density will be numerically well behaved.

2.2 Analytic foundation of electronic density reconstruction

In this subsection, the process of electronic density reconstruction is referred to as fluctuationreconstruction, in reference to homogenization terminology [4]. The objective is to developefficient tools that compute the solution to the electronic structure problem up to higher-ordertermsO (F)2 + O (∇0 F). This is equivalent to carrying out the first step of the classicalhomogenization technique [4].

For simplicity, assume that two identical rectangular domains of linear sizea are separated bya vectorLa (1, 0, 0), whereL is an integer. Thesereconstruction domainsDr

1 andDrL+1, have

electronic densitiesρ1 andρL+1, respectively, that might be computed by using an elaborateDFT method. The goal is to reconstruct the density in the domain between the two givenrectangular domains (the darker regions in Fig.2). The notation conventionTk = kT is usedbelow.

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Figure 2:Electronic density reconstruction.

The potential generated by the total charge in the system is

V (r) =

∫ρ(r′) + ρA(r′)||r− r′|| dr′

In our computations it is important to consider separately the potential that is generated byelectronic density outside a given domainD, whose complement isD, that is,

V ext(r; D) =

∫

D

ρ(r′) + ρA(r′)||r− r′|| dr′ +

∫

D

ρA(r′)||r− r′||dr

′, r ∈ D (23)

For solving the electronic problem, we may consider that the effect of the nuclei from thedomain is also a part of an “external” potential, which explains the last term in the previousexpression.

Clearly,

V (r) = V ext(r; D) +

∫

D

ρ(r′)||r− r′||dr

′

One consequence of the geometric assumption (see subsection1) is that the external potentialand the electronic density are nearly periodic, at least in the direction or in the region in whichwe do the reconstruction. For that assumption to be reasonable, one may imagine either that thedomain depicted in Fig.2 is embedded in a crystal sufficiently large in the horizontal directionand periodic across or that the end domainsDr

1 andDrL+1 are sufficiently far away from the

boundary of the crystal. In addition, we assume that the “periodicity defect” is slowly varyingin space on the scale of the domains.

This assumption is typical in homogenization theory, and in this context an observableW (x)can be expressed asW (x) = f(x

b, x

a). Heref(y, z) is a function that is periodic inz with

a vector periodT = O(1) and that is well behaved iny, that is, ∂f∂y

= O(1). An exampleof such function isy sin(z). Herea is the characteristic length scale of the fluctuations (the“microscale”, or a measure of the domainDr

1 in our case, such as its diameter), whereasb is the“macroscale” length scale (in our case, the entire crystal or nanoparticle) anda ¿ b.

As a result of our representation ofW (x), we have from the intermediate value theorem thatfor any integerk the following holds:

∣∣∣W (x + kTa)−W (x)∣∣∣ =

∣∣∣∣∣f(

x + kTa

b,x + kTa

a

)− f

(x

b,x

a

)∣∣∣∣∣

=

∣∣∣∣∣f(

x + kTa

b,x

a

)− f

(x

b,x

a

)∣∣∣∣∣ = O

(ka

b

)

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By a similar argument, fork = 1, 2, . . . L + 1, the following also holds for nearly periodicW (x):

(1− k−1

L

)W (x) + k−1

LW

(x + LaT

)−W

(x + (k − 1)aT

)= O(

(La

b

)2)

Therefore, the nearly periodic assumption forV ext(r; D) implies that the external potentialhas the following two properties:

∣∣V ext(Φ(r0 + Tk1−1, t

); Dk1)− V ext

(Φ

(r0 + Tk2−1, t

); Dk2

)∣∣ ≤ O

((k1 − k2)a

b

)

≤ O

(La

b

)∀ r0 ∈ Dr

1, 1 ≤ k1 < k2 < L + 1,

and

(1− k − 1

L)V ext(Φ

(r0, t

); Dr

1) +k − 1

LV ext

(Φ

(r0 + TL, t

); Dr

L+1

)

−V ext(Φ

(r0 + Tk−1, t

); Dk

)= O

((La

b

)2)

k = 1, 2, . . . L + 1, r0 ∈ Dr1

In the following, we assess the error of reconstructing the electronic density in the domainsDr

k, k = 2, . . . , L by interpolation between its values in domainsDr1 andDr

L+1. The reconstruc-tion rule is the following:

ρ(Φ(r0 + Tk−1, t)) ≈ ρ(Φ

(r0 + Tk−1, t

))= (1− k − 1

L)ρ1

(Φ

(r0, t

))

+k − 1

LρL+1

(Φ

(r0 + TL, t

))k = 1, . . . L + 1, r0 ∈ Dr

1

where the notation

ρ1

(Φ

(r0, t

))= ρ

(Φ

(r0, t

)), ρL+1

(Φ

(r0 + TL, t

))= ρ

(Φ

(r0 + TL, t

))r0 ∈ Dr

1

was used to emphasize the fact that the reconstructed density depends only on the density valuesin Dr

1 andDrL+1.

Theorem 1 Assume that the external potential is nearly periodic. Then the error in the opti-mality conditions of the electronic structure problem is

O

((La

b

)2)

+ O(ρ1

(Φ

(r0, t

))− ρL+1

(Φ

(r0 + TL, t

)))2.

Proof:We now assume that the optimality conditions can be expressed as

θ(ρk; V

ext(Φ(r0 + Tk−1, t

); Dk)

)= 0, r0 ∈ Dr

1, k = 1, 2, . . . , L + 1

11


whereθ is an operator that is twice continuously differentiable in the range of approximation.For example, in the case of the Thomas-Fermi approach, Eq.(14) results in the following ap-proach forθ.

θ(ρk; V

ext(Φ

(r0 + Tk−1, t

); Dk

))=

5

3CF ρ

23k

(Φ

(r0 + Tk−1, t

))− 4

3Cxρ

13k

(Φ

(r0 + Tk−1, t

))

+

∫

Dk

ρk(Φ(r0′ , t))

Φ(r0 + Tk−1, t)− Φ (r0′ , t)|F(r0′ , t)|dr0′ + V ext

(Φ

(r0 + Tk−1, t

); Dr

k

)+ λ = 0

wherer0 ∈ Dr1, k = 1, 2, . . . , L + 1. The second-order differentiability ofθ holds as long asρ

is bounded from below.In our approach, the reconstructed density depends only on the the values of the density in

Dr1 andDr

L+1, in which we assume that the optimality conditions are exactly satisfied, that is,

θ(ρ1; V

ext(Φ

(r0, t

); D1

))= 0, θ

(ρL+1; V

ext(Φ

(r0 + TL, t

); DL+1

))= 0, r0 ∈ Dr

1

Using analysis tools in interpolation theory, as well as the assumption that the external po-tential is nearly periodic leads to

θ(

ρ(Φ(r0 + Tk−1, t)), V ext(Φ(r0 + Tk−1, t); Dk))

= θ(

ρk(r0), V ext

k (r0; Dk))

+O

((La

b

)2)

= O(ρ1

(Φ

(r0, t

))− ρL+1

(Φ

(r0 + TL, t

)))2+ O

((La

b

)2)

wherer0 ∈ Dr1, k = 1, 2, . . . , L + 1, and

ρk(r0) = (1− k − 1

L) ρ1(Φ(r0, t)) +

k − 1

LρL+1(Φ(r0 + TL, t))

V extk (r0; Dk) = (1− k − 1

L) V ext(Φ(r0, t); Dr

1) +k − 1

LV ext(Φ(r0 + TL, t); Dr

L+1)

which completes the proof.

The proposed interpolation-based approach has certain limitations and is not expected to al-ways work well. In particular, the reconstructed density might display discontinuities at theinterface between neighboring domains, which may be an issue with von Weizsacker-type ki-netic energy corrections that are encountered in OF-DFT approaches [29]. However, for thepurpose of generating just the field and computing the force on nuclei in the middle of thedomain, the above approach is expected to provide an adequate level of accuracy.

2.3 The optimize-and-interpolate approach

The optimize-and-interpolate approach refers to the case in which the electronic structureis computed, at least in some subdomains, by solving the system of integral equations that thefirst-order optimality condition leads to. In what follows this approach is exemplified for theThomas-Fermi energy functional and applied to a domainD = D1 ∪ . . . ∪ Du that containsthe nanostructure. The optimality conditions of Eq.(11) are first formulated for a set ofp ≤ urepresentative domainsY1 throughYp. In what follows,Y = Y1∪ . . .∪Yp ⊂ D; more precisely,

12


��

��

��

��

��

��

��

��

�

��

��

��

��

��

Figure 3:Nanostructure at end ofPreprocessing.

there is an integer-to-integer mappingχ : {1, . . . , p} → {1, . . . , u} such thatYj = Dχ(j). Thereconstruction method computes the electronic density inY and uses interpolation to recoverρ in D − Y . For instance, in Fig.1 there are three domainsD1, D2, andD3; Y1 = D1, andY2 = D3. Likewise, in Fig.2 there areL + 1 domains, but only two reconstruction domains:Y1 = D1, andY2 = DL+1. Figure3 also shows the partitioning of the domainD in which theelectronic structure computation is carried out, as well as the reconstruction domainsY1 throughY5. The figure presents a two-dimensional case, but the discussion in this section covers boththe two- and the three-dimensional case. By convention, in what follows Greek subscripts areused to index quantities associated with reconstruction domains.

At the core of the optimize-and-interpolate approach stands the optimality condition for ageneric domainYα ∈ Y :

5

3CF ρ

23α(r)− 4

3Cxρ

13α(r) +

u∑i=1

∫

Di

ρi(r′)

||r− r′||dr′ −

M∑A=1

ZA

||r−RA|| + λ = 0 (25)

The first step is to express the densityρi on domainDi in terms of reconstruction densitiesρα ∈ Yα, α ∈ {1, . . . , p}. A set of weightsϑ determined based on the type of interpolationconsidered (linear, quadratic, etc.) is used to this end:

ρi(Φ(r0′, t)) =

p∑α=1

ϑα(i)ρα(Φ(r0′ + Tiα, t)) (26)

where the vectorTiα is the translation vector that based on the periodicity assumption takesthe pointr0′ in domainDi to its image in the domainYα. Note that not all domains need tobe included in the electronic density reconstruction. For example, if a domain has a defect, theelectronic density is severely distorted away from the near-periodicity assumption and shouldnot be included in the reconstruction process in neighboring domains.

Taking into account the deformation of the structure,

13


∫

Di

ρi(r′)

||r− r′||dr′ =

∫

D0i

p∑α=1

ϑα(i)ρα(Φ(r0′ + Tiα, t))

||Φ(r0, t)− Φ(r0′, t)|| |F(r0′, t)| dr0′ (27a)

=

p∑α=1

∫

Y 0α

ρα(Φ(r0′, t))Kiα(r0, r0′) dr0′

Kiα(r0, r0′) =ϑα(i) |F(r0′ −Tiα, t)|

||Φ(r0, t)− Φ(r0′ −Tiα, t)|| (27b)

For anyYα, Eq.(25) is reformulated as

5

3CF ρ

23α(Φ(r0, t))− 4

3Cxρ

13α(Φ(r0, t)) +

u∑i=1

p∑γ=1

∫

Y 0γ

ργ(Φ(r0′, t)) Kiγ(r0, r0′)dr0′

(28)

−M∑

A=1

ZA

||Φ(r0, t)− Φ(R0A, t)|| + λ = 0

Defining forr0 ∈ Y 0α

Kαγ(r0, r0′) =

u∑i=1

Kiγ(r0, r0′) (29a)

and Eq.(28) then yields

5

3CF ρ

23α(Φ(r0, t))− 4

3Cxρ

13α(Φ(r0, t)) +

p∑γ=1

∫

Y 0γ

Kαγ(r0, r0′)ργ(Φ(r0′, t))dr0′ (29b)

−M∑

A=1

ZA

||Φ(r0, t)− Φ(R0A, t)|| + λ = 0

which should hold for anyr0 ∈ Yα. Finally, sinceρ ≥ 0, a new functionη is introduced suchthat

ρ(Φ(r0, t)) = ηs(r0, t) (30a)

wheres ≥ 4 is an even integer. This new function must then satisfy in the subdomainYα thefollowing integral equations:

5

3CF η

2s3

α − 4

3Cxη

s3α +

p∑γ=1

∫

Y 0γ

Kαγ(r0, r0′)ηs

γ(r0′, t)dr0′ −

M∑A=1

ZA

||Φ(r0, t)− Φ(R0A, t)|| + λ = 0

(30b)The algorithm at this point calls for the solution of an nonlinear system of integral equations

in ρα, α = 1, . . . , p. In order to solve this system, the reconstruction domainsYα are meshed by

14


using hexahedrons. These meshes are denoted in what follows byG1 throughGp, and they areassociated withY1 throughYp, respectively.

The direct numerical solution of the nonlinear system of integral equations becomes in-tractable in Cartesian coordinates because of the singularity when the grid points in a meshGα approach a nuclei of locationRA (see Eq.(30b)). When approached in spherical coordi-nates in a three-dimensional representation this apparent singularity is in fact a nonissue (seethe discussion related to Eq.(22)). Below, a potential-smoothing step is introduced to addressthe situation whenr0 → R0

A. Compared to the original term||Φ(r0, t) − Φ(R0A, t)||−1, the

δ-smoothing function

Sδ(r0,R0

A, t) =1− e−

||Φ(r0,t)−Φ(R0A,t)||

δ

||Φ(r0, t)− Φ(R0A, t)|| (31)

behaves similarly for large values of||Φ(r0, t) − Φ(R0A, t)|| and δ small but positive, but it

converges to1δ

rather than going to infinity whenr0 → R0A. Thus, the smoothing process

applied to Eq.(30b) leads to

5

3CF η

2s3

α − 4

3Cxη

s3α +

p∑γ=1

∫

Y 0γ

Kαγ(r0, r0′)ηs

γ(r0′, t)dr0′ −

M∑A=1

ZA Sδ(r0,R0

A, t) + λ = 0 (32)

To make the presentation simpler, the following notation is introduced:

• ηβj – the value ofη at the nodej of grid Gβ

• τ– a generic grid discretization cell of volume||τ ||• V(τ) – the set of vertices associated with cellτ (four for a tetrahedron, eight for an

hexahedron, etc.)

• |Gα| – the number of grid points inGα

• Y 0γ – undeformed reconstruction domain meshed withGγ; Yγ = ∪τ∈Gγτ

After discretization, the integral equation above yields at an arbitrary grid nodei ∈ Gα oflocationr0

i ∈ Yα,

5

3CF η

2s3

αi −4

3Cxη

s3αi +

p∑γ=1

∑

τ∈Gγ

∫

τ

Kαγ(r0i , r

0′) ηsγ(r

0′, t)dr0′

−

M∑A=1

ZA Sδ(r0i ,R

0A, t)+λ = 0

(33)The integral onτ is performed byq-point Gaussian numerical quadrature with weightswl:

∫

τ

Kαγ(r0i , r

0′) ηsγ(r

0′, t)dr0′ ≈ ||τ ||q∑

l=1

wl Kαγ(r0i , r

0l′) ηs

γ(r0l′, t)

Figure3 shows in the two-dimensional case a mesh cell and the quadrature points. As indicatedin this figure,r0

i describes the position of the grid nodes; the interior points (quadrature points)

15


are located atr0l′. The abscissasr0

l′ of the quadrature points are different from the mesh (grid)

points, and the value of the unknown functionη at these abscissas is obtained by interpolation.Interpolation at pointr0

l′ ∈ τ , using a set of shape functionsϕd associated with the nodes

d ∈ V(τ), yieldsηs

γ(r0l′, t) ≈

∑

d∈V(τ)

ηsγd ϕd(r

0l′, t) =

∑

d∈V(τ)

ηsγd ϕl

d

whereϕld are constants that can be precomputed. If one defines forr0 ∈ Yα andr0

l′ ∈ Yγ

kαγd(r0) =

q∑

l=1

wl ϕld Kαγ(r

0, r0l′) , (34a)

the discretized form of the integral equation expressed at grid nodei ∈ Gα of locationr0i ∈ Yα

becomes

5

3CF η

2s3

αi −4

3Cxη

s3αi +

p∑γ=1

∑

τ∈Gγ

||τ ||∑

d∈Vkαγd(r

0i ) ηs

γd

−

M∑A=1

ZA Sδ(r0i ,R

0A, t)+λ = 0 (34b)

By denoting the left side of Eq.(34b) by Pαi(η), whereη = (η11, η12, . . . , ηp1, ηp2, . . .)T , the

nonlinear system of equations that should be solved becomes

Pαi(η) = 0 (35)

for α ∈ {1, 2, . . . , p}, i = 1, . . . , |Gα|.

One additional equation is added to the set of∑p

α=1 |Gα| equations above, and it follows fromthe charge constraint of Eq.(2b). The central idea is again to use the electronic density in thereconstruction domainsYα to express the electronic density in the whole domainD. Skippingthe intermediary steps, this yields

∫

D

ρ(r)dr =

p∑α=1

∫

Y 0α

ηsα(r0, t)Kα(r0, t)dr0 (36a)

Kα(r0, t) =u∑

i=1

ϑα(i) |F(r0 −Tiα, t)| (36b)

For the charge constraint equation, using for the evaluation of the integral on a cell of theundeformed gridY 0

α the same quadrature rule and interpolation method to evaluate the functionat the quadrature points yields:

p∑α=1

∑

τ∈Y 0α

∑

d∈V(τ)

ηsαdkαd

−Ne = 0 (37a)

kαd =

q∑

l=1

wl ϕld Kα(r0

l , t) (37b)

16


If a Newton-type method is considered for the solution of the nonlinear system of Eqs.(35)and (37a), the partials are computed as

∂Pαi

∂ηαi

=10s

9CF η

2s−33

αi − 4s

9Cxη

s−33

αi +∑

τ∈Y 0α

||τ || δατi kααi(r

0i ) ηs−1

αi (38a)

whereδατi = 1 if for τ ∈ Gα, i ∈ V(τ), andδα

τi = 0 otherwise. Wheni 6= j or α 6= β,

∂Pαi

∂ηβj

=∑

τ∈Y 0β

||τ || δβτj kαβj(r

0i ) ηs−1

βj (38b)

Likewise,∂P00

∂ηβj

=

p∑α=1

∑

τ∈Y 0α

s ηs−1αd δα

τd kαd (38c)

where, by convention,P00(η) is a notation for the left side of Eq.(37a).

3 AN OPTIMIZATIONS PERSPECTIVE ON THE DFT WITH DENSITY RECON-STRUCTION

3.1 The optimize-and-interpolate approach revisited: an abstract formulation

By reference to equation (29a), define

Kαα(r0, r0′) =u∑

i=1, i6=χ(α)

Kiα(r0, r0′), r0 ∈ Y 0α (39)

It then follows that the external potential, as defined in (23), can be computed as

V ext(r0, ρ1, ρ2, . . . , ρp; Yα) =

∫

Y 0α

Kαα(r0, r0′)ρα(Φ(r0′, t))dr0′ −M∑

A=1

ZA Sδ(r0,R0

A, t) (40)

+

p∑

γ=1,γ 6=α

∫

Y 0γ

Kαγ(r0, r0′)ργ(Φ(r0′, t))dr0′ α = 1, 2, . . . , p, r0 ∈ Y 0

α .

Define now the following quantity

FTF (ρα; α) =5

3CF η

2s3

α − 4

3Cxη

s3α +

∫

Y 0α

Kχ(α)α(r0, r0′) ρα(Φ(r0′, t))dr0′. (41)

Then the optimality condition (29b) can be written as

FTF (ρα; α) + V ext(r0, ρ1, ρ2, . . . , ρp; Yα) + λ = 0, α = 1, 2, . . . , p, r0 ∈ Y 0α .

The total charge density can also be computed based on (26)∫

ρ(r)dr =u∑

i=1

p∑α=1

ϑα(i)

∫

Y 0α

ρα(Φ(r0′ + Tiα, t))∣∣∣F

(r0′ + Tiα, t

)∣∣∣ dr0′

17


With these notations, we note that our density reconstruction methodology applies irrespec-tive of the particular DFT used. In that case, onlyFKS changes. Therefore, the general elec-tronic density problem with interpolation-based reconstruction becomes forα = 1, 2, . . . , p,andr0 ∈ Y 0

α :

F(ρα; α) + V ext(r0, ρ1, ρ2, . . . , ρp; Yα) + λ = 0 (42a)u∑

i=1

p∑α=1

ϑα(i)

∫

Y 0α

ρα(Φ(r0′ + Tiα, t))∣∣∣F

(r0′ + Tiα, t

)∣∣∣ dr0′ = N (42b)

A useful fact is thatF(ρα; α) is the gradient of the objective function of the electronic struc-ture problem on domainYα, α = 1, 2, . . . , p. We can therefore use, for a given DFT approach,any software that returns the gradient with respect toρ of the energy functional on domainYα,coupled with a quasi-Newton approach to solve the nonlinear system of Eq.(42).

3.2 The IO approach: the interpolate-and-optimize formulation

This section presents an approach in which the interpolation of Eq.(26) is used to create areduced energy functional that depends only on the electron densityρα, in the reconstructiondomainsYα, α = 1, . . . , p. This reduced energy functional is then minimized.

The key terms emerge from the electrostatic potential. We present a succinct derivation ofthe respective equations.

Forα, γ = 1, 2, . . . , p, r0 ∈ Y 0α , r0′ ∈ Y 0

γ , define

Jαγ(r0, r0′) =

u∑i=1

u∑j=1

vα(i)vγ(j)|F (r0 −Tiα, t)| · |F (r0′ −Tjγ, t)|||Φ(r0 −Tiα, t)− Φ(r0′ −Tjγ, t)||

Lα(r0) =M∑

A=1

u∑i=1

vα(i)|F (r0 −Tiα, t)|

||Φ(r0 −Tiα, t)− Φ(R0A, t)||

Mα(r0) =u∑

i=1

vα(i)∣∣F (r0 −Tiα, t)

∣∣

Based on Eq.(26), several of the terms in Eq.(7) become functions of densities in the repre-sentative domains:

J(ρ) =1

2

p∑α=1

p∑γ=1

∫

Y 0α

∫

Y 0γ

Jαγ(r0, r0′)ρα(Φ(r0, t))ργ(Φ(r0′ , t))dr0dr0′

Ene(ρ) = −p∑

α=1

∫

Y 0α

Lα(r0)ρα(Φ(r0, t))dr0,

∫ρdr =

p∑α=1

∫

Y 0α

Mα(r0)ρα(Φ(r0, t))dr0

The difficult part has to do with the kinetic energy and exchange termsT [ρ], K[ρ] whosedependence on the density is not linear and, outside the Thomas-Fermi theory, not even simple

18


to state. Assume that the latter terms are described by a univariate densityΘ1(ρ, r), as indicatedby the first term of Eq.(1). For instance, for the Thomas-Fermi representation,

Θ1(ρ, r) = CF ρ53 (r)− Cxρ

43 (r)

Then, by an argument similar to the one in Theorem1, we obtain that the following approx-imation is accurate, up to termsO((La

b)2).

T [ρ] + K[ρ] ≈p∑

α=1

∫

Y 0α

Mα(r0)Θ1(ρα, Φ(r0, t))dr0

With these approximations and definitions and referring back to Eq.(7), the following elec-tronic structure computation problem is defined:

min EIO(ρ), subject to∫

ρ = N, (43)

where we use the superscript “IO” to denote the “interpolate-and-optimize” approach, andEIO

represents the quantity obtained in Eq.(7) after expressing the energy as a function of electrondensities in the reconstruction subdomains only. This leads to the following optimality condi-tions:

0 = Mα(r0)∇ρΘ1(ρα, Φ(r0, t))− Lα(r0) +

p∑γ=1

∫

Y 0γ

Jαγ(r0, r0′)ργ(Φ(r0′ , t))dr0′ + λMα(r0)

N =

p∑γ=1

∫

Y 0γ

Mα(r0)

Clearly, the optimality conditions of the “interpolate-and-optimize” approach are more com-putationally intensive to set up. Nonetheless, they open the avenue for using special techniques(such as projected gradient) that are available and stable only for optimization formulation.We plan to compare the relative benefits of the “interpolate-and-optimize” approach versus the“optimize-and-interpolate” approach in the near future.

3.3 Nonlinear equations vs. optimization approaches

Reacall that optimality conditions followed by interploation eventually led to the nonlin-ear system of Eq.(42). One can be immediately prove that, in aggregate, this system does notrepresent the first-order conditions of an optimization problem. That issue is a bit unsettlingbecause solving optimization problems is typically a more robust process than is solving equiv-alent nonlinear equations, since any local minimum of the optimization problem satisfies thenonlinear equation of its optimality conditions. When only a nonlinear system is available, alocal minimum of the residual is not necessarily a solution of the nonlinear system. It is there-fore important to assess whether there exists an optimization problem that is equivalent, at leastup to leading order of the homogenization error (La/b), with the nonlinear system.

In an abstract formulation, we have the following problem:

minx1,x2

f(x1, x2)

19


where the variablesx1 correspond to the representative degrees of freedom, whereasx2 corre-spond to the rest of the degrees of freedom. Thus, in the electronic problem with density recon-struction the representative degrees of freedom are the ones used to parametrize the electronicdensity in the representative domainsYα, α = 1, 2, . . . , p; the mappingT (·) is the interpolation-based operator from (26). Likewise, in the quasicontinuum method [26], the representativedegrees of freedom are the positions of the representative nucleirepnuclei(a concept more ex-tensively defined in section4); the location of the remaining nuclei, abstractly denoted byx2, isthen obtained asx2 = T (x1), whereT (x1) is the piecewise linear interpolation mapping withnodes at therepnuclei.

Based on this observation, one can formulate the nonlinear equation

∇x1 f(x1, x2), x2 = T (x1)

which will provide the same solution as the original problem. However, the problem is an equi-librium problem with equilibrium constraints rather than a minimization problem. Furthermore,it immediately results using the chain rule that the optimization problem

minx1

f(x1, T (x1))

has the same solution as the previous two, provided that the reduced Hessian is positive definite,which should be true if the original Hessian was positive definite, and the interpolation mappingis full rank. This observation presents the advantage that one solves an optimization problem asopposed to a system of nonlinear equations, with better global convergence safeguards. Whenthere are many local minima, this should help avoid the points that do not have the correctinertia of the Hessian.

The first approach corresponds to the “optimize-and-interpolate” approach described in sec-tion 2, whereas the second approach is the “interpolate-and-optimize” approach that describedin subsection3.2. The following result settles in the positive the question of whether the twoapproaches are equivalent in the limit of the ansatzx2 = T (x1). The proof technique is similarto the one we used for proving the approximation order of the interpolation approach and isomitted.

Theorem 2 Assume that the solutionx∗ = (x∗1, x∗2) of the original optimization problem satis-

fies‖x∗2 − T (x∗1)‖ ¿ 1; therefore the multiscale ansatz is not perfect but is merely very good.Then the solutionx1 of the nonlinear equation andx1 of the reduced optimization problemsatisfy

‖x∗1 − x1‖ = O(‖x∗2 − T (x∗1)‖2) ‖x∗1 − x1‖ = O(‖x∗2 − T (x∗1)‖2)

4 NANOSTRUCTURE SHAPE INVESTIGATION

The optimization of the geometry of a nanostructure (called hereafter theIonic Problem), tofind the most stable shape reduces to minimizing the total energy given an electronic groundstate configuration of energyEe as a function of the position of the nuclei. More precisely, theequilibrium configuration of a nanostructure is provided by that distribution of the nuclei thatminimizes the energy

Etot = Ee + Enn (44)

whereEnn is the nucleus-nucleus interaction energy and, central to this development,Ee is theelectronic ground-state energy for the considered nuclear distribution.

20


Following the Born-Oppenheimer assumption, the electronic energy depends parametricallyon the positions of the nuclei through the dependence of the electronic density on the nucleipositions. Thus, in a general form (that has the Thomas-Fermi of Eq.(7) as a subcase),

Ee = T [ρ(r)] + EHar[ρ(r)] + Exc[ρ(r)] +

∫ρ(r) Vext(r; {RA}) dr (45)

whereT [ρ(r)] is the kinetic energy functional,EHar[ρ(r)] is the electron-electron Coulombrepulsion energy,Exc[ρ(r)] is the exchange and correlation energy, andVext(r; {RA}) is theionic potential, which parametrically depends on the distribution of the nuclei{RA}. Theexplicit dependence ofT [ρ(r)] andExc[ρ(r)] on the densityρ(r) is typically not available, andconsequently it is approximated in some fashion [22, 29, 23, 13], an issue beyond the scope ofthis document. According to the Hohenberg-Kohn theorem [14], the electronic density is suchthat it minimizesEe subject to the charge conservation constraint of Eq.(2b).

Theorem 3 Consider the optimization problem

min{RA}

Etot = Ee + Enn (46a)

subject to the constraint that for a nuclear configuration{RA} the energyEe is the electronicground-state energy, and the electronic densityρ that realizes this electronic ground energyadditionally satisfies the charge constraint equation of Eq.(2b). Under these assumptions, thefirst-order optimality conditions for the optimization problem of Eq.(46a) lead to

FK =∂Eext

∂RK

+∂Enn

∂RK

= 0 (46b)

whereFK is interpreted as the force acting on nucleusK, and

Eext(r; {RA}) = −M∑

A=1

∫ρ(r) Vext (r; {RA}) dr = −

M∑A=1

∫ZAρ(r)

|r−RA| dr (46c)

Enn =1

2

M∑A=1

M∑B=A+1

ZAZB

RAB

(46d)

Proof:The proof relies on the calculus of variations. Sinceρ(r) is determined to minimize the elec-tronic energy, there is a parametric dependency of this value on the ionic position:ρ(r) =ρ(r; {RA}). After application of the chain rule, the optimality conditions forEtot will read

δEe

δρ

∂ρ

∂RK

+∂Ee

∂RK

+∂Enn

∂RK

= 0 (47)

whereRK is the position of an arbitrary nucleusK.The optimality conditions for minimizing the electronic energy as a functional of the elec-

tronic density lead toδEe

δρ+ λ

δg

δρ= 0 (48)

21


whereλ is the Lagrange multiplier associated with the constraint

g[ρ] = 0 (49)

that the electronic density must satisfy. For the problem at hand the charge conservation equa-tion results ing[ρ] =

∫ρ(r) dr−Ne. Based on Eq.(49), the variation ofρ(r) with respect to

Rk must satisfyδg

δρ

∂ρ

∂RK

= 0

Multiplying Eq.(48) from the right by ∂ρ∂RK

leads toδEe

δρ∂ρ

∂RK= 0, which, substituted back into

(47) yields the optimality condition stated in Eq.(46b) and thus completes the proof.

Therefore, for each nucleusK in the system, Eq.(46b) leads to the condition

∫ρ(r)

r−RK

||r−RK || 32dr +

M∑

A=1,A 6=K

ZARA −RK

||RA −RK || 32= 0 (50)

Remarks:

1. The value of the above theorem is that it allows us to solve the nuclear equilibrium prob-lem by using only the solution of the electronic density problem, and not the values andthe derivatives of the kinetic and exchange energy functionals. Therefore, we can useeven an entirely nontransparent encapsulation of the electronic structure problem, whichallows the proposed approach to work well with legacy codes that do not provide all theneeded derivatives. The key observation is that once the electronic density is available,the equilibrium conditions of Eq.(50) can be imposed right away. Whether the electronicstructure computation is done with third-party software is irrelevant; moreover, there isno need to know the explicit dependence of the energyEe on the electronic densityρ(r).

2. As suggested in [19], the one-atom conditions of Eq.(46b) can be replaced by clusterconditions, an alternative that will be explored in the future.

3. Because of the presence of the electronic densityρ(r) that displays pronounced cusps inthe vicinity of nuclei, the integral in Eq.(50) must be evaluated by using special techniques[3, 28]. This computational aspect is central to the overall algorithm and will be detailedin a separate document.

When a local quasicontinuum approach is used, the condition of Eq.(50) is imposed onlyfor repnuclei; that is, only forK ∈ B (see Eq.(21)). The position of the rest of the atoms inthe system is then expressed in terms of the position of therepnuclei. The repnucleibecomethe nodes of an atomic mesh, and interpolation is used to recover the position of the remainingnuclei. For instance, if the atomic mesh is denoted byM, τ is an arbitrary cell in this mesh,V(τ)represents the set of the nodes associated with cellτ , andϕL is the shape function associatedwith nodeL, then the condition of Eq.(50) is approximated as

∫ρ(r)

r−RK

||r−RK || 32dr +

∑τ∈M

∑A∈τ

ZA

∑L∈V(τ)

RLϕL(RA)−RK

|| ∑L∈V(τ)

RLϕL(RA)−RK || 32= 0 (51)

22


This effectively reduces the dimension of the problem from3 M (the(x, y, z) coordinates ofthe nuclei), to3 Mrep, whereMrep is the number of nodes in the atomic mesh (the number ofrepnuclei). The sum in Eq.(51) is most likely not going to be the simulation bottleneck (solvingthe electronic problem forρ is significantly more demanding), but fast-multipole methods [1,11, 24] can be considered to speed the summation.

Denoting byPi, i = 1, . . . ,Mrep, the position of the representative nucleusni, the set ofnonlinear equations of Eq.(51) can be grouped into a nonlinear system that is solved for therelaxed configuration of the structure.

f1(P1,P2, . . .PMrep) = 0f2(P1,P2, . . .PMrep) = 0· · ·fMrep(P1,P2, . . .PMrep) = 0

(52)

wherefK is the left side of Eq.(51). Finding the solution of this system is done by a Newton-likemethod. Evaluating the Jacobian information is straightforward but not detailed here.

Finally, note that within Eq.(51) a connection is made back to Eq.(21); the position of anarbitrary nucleusA in cell τ is computed based on interpolation using the nodesV(τ), one ofmany alternatives available (one could considerrepnucleifrom neighboring cells for instance).Effectively, this provides in Eq.(21) an expression forΦ(·, t) that depends only onA ∈ V(τ)rather thanA ∈ B.

5 PROPOSED COMPUTATIONAL SETUP

Given a nanostructure of known atomic composition (not necessarily mono-atomic or single-crystal), the goal is to determine the electron density distribution as well as its final configura-tion, that is, the mappingΦ. Because of the assumption that the kinetic energy of the nucleiis zero, the problem corresponds to a zero Kelvin temperature scenario. A methodology thathandles the nonzero temperature case is not addressed here; most likely, it would follow anapproach similar to that of Car-Parrinello [5], or Payne et al. [21].

As indicated in Fig.4, the proposed solution has three principal modules: thePreprocessingstage, theElectronic Problem, and theIonic Problem. Preprocessingis carried out once at thebeginning of the simulation. A suitable chosen domainD is selected to include the nanostruc-ture investigated. The partitioning ofD into u subdomainsDi, i = 1, . . . , u, is done to mirrorthe underlying periodicity of the structure. A set of subdomainsDχ(1) throughDχ(p) is deter-mined to constitute the reconstruction domains, and as in section2.3 they are denoted byY1

throughYp. In thesep subdomains explicit electronic structure computation will be carried outaccurately. A set of values of the electronic density is required at the nodes of the discretizationmesh; the initial guess for the electronic density could be a uniform distribution throughout thenanostructure or, when practical, could be obtained based on a periodic boundary conditionsassumption by computing it in a domainDj and then cloning for the remaining domainsDk.Preprocessingconcludes with the initialization of the deformation mapΦ to be the identitymapping.

The Electronic Problemcan be solved externally or internally. When it is solved exter-nally, a specialized code such as NWChem [12] or Gaussian03 [9] is employed to compute theelectronic density in the reconstruction subdomainsYα, α ∈ {1, . . . , p}. When theElectronicProblemis solved internally (only for qualitative studies, using for instance the Thomas-Fermi

23


��

��

Reevaluate deformation

mapping Φ

Reposition nuclei

based on newρ

IONIC

PROBLEM

PREPROCESSINGPartition in ( 1, , )iD D i u= �

Initialize deformation

mapping to identityΦ

Select reconstruction

domains Y ( 1, , )pα α = �

Mesh domains Y ( 1, , )pα α = �

Provide in Y ( 1, , )init pα αρ α = �

Select repnuclei

ELECTRONIC

PROBLEM

Solve system of Integral

Equations for newρ

init newρ ρ= ?new initρ ρ ε− <

Run external DFT code

in Y ( 1, , )pα α = �

Reconstruct potential

based on initρ

Interpolate from initρ ρ

Figure 4:Computational flow.

24


DFT; see section2.3or 3.2), the computation requires a mesh grid on which the integrals asso-ciated with the formulation are discretized. The algorithm uses three-dimensional interpolationto provide for the density inDj, wherej ∈ {1, . . . , u} − {χ(1), . . . , χ(p)} (the analytical basisof this process is detailed in section2).

Independent of the type of solver invoked (external or internal), with a suitable norm thenew electronic densityρnew is compared toρinit, and the computation restarts theElectronicProblemafter settingρinit = ρnew, unless the corrected and initial values of the electronicdensity are close. This iterative process constitutes the first inner loop of the algorithm. Itsanalytical foundation is discussed in sections2 and3.

The Ionic Problemuses the newly computed electronic density to reposition the nuclei andthus alter the shape of the structure. The nonlinear system of Eq.(52) provides the position oftherepnuclei; the other nuclei are positioned based on the quasi-continuum paradigm discussedin section4. The nonlinear system in Eq.(52) is solved by an iterative method, which leads tothe second inner loop that in turn has four steps:

1. Evaluate the integral of Eq.(51); when necessary, evaluate its partial with respect toPi

2. Evaluate the double sum of Eq.(51), which is based on a partitioning of the structure;when necessary, evaluate its partial with respect to the position of the representative atoms

3. Carry out a quasi-Newton step to update the positionsPi of theMrep representative nu-clei.

4. Go back to1 if not converged

The precision in determining the position of the nuclei is directly influenced by the accuracyof the electronic densityρ(r). Accurately solving theElectronic Problemis computationallyintensive, and thus an important issue not addressed by this work is the sensitivity of the solutionof the nonlinear system of Eq.(52) with respect toρ(r). It remains to determine whether a crudeapproximation of the electronic density suffices for solving theIonic Problemat a satisfactorylevel of accuracy.

After determining the position of the nuclei, the algorithm computes the new deformationmappingΦ according to Eq.(21). If the overall change in the position ofrepnucleiat the endof the Ionic Problemis smaller than a threshold value, the computation stops; otherwise thenew distribution of the nuclei is the input to a newElectronic Problem(second stage of thealgorithm).

In summary, the algorithm passes through thePreprocessingstage once. It then solves theElectronic Problem(the first inner loop) and proceeds to theIonic Problem(the second innerloop). The outer loop (Electronic Problem, followed byIonic Problem) stops when there is nosignificant change in the position of therepnuclei.

6 NUMERICAL RESULTS

For a simple example, this section compares the numerical results obtained with the directminimization approach of the Thomas-Fermi DFT, with the ones produced by the approachdescribed in subsection3.2(the ”interpolate-and-optimize” approach).

The one-dimensional setup is very similar to the one in Fig.2. Eleven equally spaced nucleiwith unit charge,ZA = 1, are considered; the total number of electrons isNe = 11. The locationof the atoms corresponds to the peaks seen in Fig.(5). A mesh is constructed that has 50 nodes

25


−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

20

25

30

35

40

location

Comparison of two Thomas Fermi implementations

Direct SimulationInterpolation reconstruction

1 2 3 4 5 6 7 8 9 10 110.9

0.92

0.94

0.96

0.98

1

1.02

1.04

Index of the domain

Total charge in subdomains

Direct SimulationLinear ReconstructionQuadratic Reconstruction

Figure 5: Thomas-Fermi solutions and total charge. Direct minimization vs. minimization with density recon-struction

per cell, with 30 of them equally spaced on an interval centered at the position of the atom andwhose length is 1/5 of the distance between two atoms. This results in 11 domains,D1, D2, . . .,D11. For discretization of the integral operators the trapezoidal rule was considered. Only thedomainsD2, D6,D10 are used for interpolation in order to avoid the boundary distortion. Eitherpiecewise linear, or quadratic interpolations are employed for the interpolation-based approach.The parameterδ = 10−4 leads to a slightly different regularization from that described in theprevious sections (see Eq.(31)), whereby terms of the type1/|| · || are replaced with1/|| ·+δ||.

The resulting electronic structure optimization problem is solved with the augmented La-grangian software Lancelot [6], which uses an iterative method to solve the bound constrainedsubproblem obtained after penalization of the constraints. When using the interpolation method,the interpolation conditions of Eq.(26) are actually enforced as constraints, rather than substi-tuting them in the functional that describes the problem (seeEIO of Eq.(43)). When efficiencyis a concern, this substitution would be carried out and only the electronic density degrees offreedom inD1, D2,D6,D10,D11 would be considered. The objective here is only to validate theinterpolation-based reconstruction without regard to computational efficiency.

The solution of the direct numerical simulation and of the linear-interpolation-based opti-mization are presented in Fig.5. The results are almost identical, which is an indication that theinterpolation approach was effective in reconstructing the solution in the “gap” domains. Theresults were better yet for the quadratic-interpolation-based reconstruction (not presented in thefigure). Note, however, that the solutions are not identical. This can be seen by computing thetotal charge in the subdomains. The results for the three numerical experiments are presentedin Fig.5, the right panel. This indicates that the quadratic interpolation method produced a verygood fit, with a relative error that is uniformly below2% for domains2 through10.

The solution presents some artifacts at the very end of the domain. That in itself is notsurprising, given the limitations associated with the Thomas-Fermi DFT. The results obtainedare in this sense promising, setting the stage for future tests with more complex structures andmore sophisticated DFT approaches.

26


7 CONCLUSIONS

This paper proposes a theoretical framework for nanostructure optimization. Thegeomet-ric (space periodicity), and theelectronic(energy functional form) assumptions introduced insection1 are at the center of a methodology that uses interpolation and coupled cross-domainoptimization techniques in an effort to increase the size of the problems that rely on a DFT-basedsolution component. For the electron density computation (theElectronic Problem) formal er-ror bounds are provided for the interpolation and cross-domain reconstruction techniques used.The electronic density reconstruction process can be done internally by following an approachsimilar to the one introduced in section2.3 for the Thomas-Fermi DFT; alternatively, it can becarried out by using dedicated third-party software such as NWChem or Gaussian03. In eithercase, the density is reconstructed by solving a cross-domain coupled nonlinear problem. Thelast step of the proposed methodology calls for solving theIonic Problem; that is, repositioningthe nuclei of the structure given the electronic density in the domain. It was shown in section4 that the new ionic configuration is the solution of a nonlinear system obtained based on afirst-order optimality condition. The Jacobian information for this system is readily available,and its solution does not require the explicit dependency of the kinetic and exchange-correlationenergies on the electronic density.

Based on the proposed methodology, a set of simple test cases are currently under investiga-tion: (a) the test case presented in Fig.1 for a mono-atomic Al structure bounded by a surfaceand (b) the electronic structure study of an inner defect in a silicon crystal.

Acknowledgments

This work was supported in part by the Mathematical, Information, and Computational Sci-ences Division subprogram of the Office of Advanced Scientific Computing Research, Officeof Science, U.S. Department of Energy, under Contract W-31-109-ENG-38.

This work was supported in part by the U.S. Department of Energy, Office of Basic EnergySciences-Materials Sciences, under Contract No. W-31-109-ENG-38.

We thank Dr. Anter El-Azab of Florida State University and Dr. Anna Vainchtein of theUniversity of Pittsburgh for many useful discussions that helped shape this document.

27


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