Author's personal copyamasud.web.engr.illinois.edu/Papers/Masud-Kannan...Author's personal copy The Hartree term and the local pseudopotential term are com-bined into a single Poisson

Author's personal copy

B-splines and NURBS based finite element methods for Kohn–Sham equations

Arif Masud ⇑, Raguraman KannanDepartment of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2352, United States

a r t i c l e i n f o

Article history:Received 29 October 2011Received in revised form 11 April 2012Accepted 20 April 2012Available online 3 May 2012

Keywords:Kohn–Sham equationsSchrödinger wave equationQuantum mechanicsB-splinesNURBSFinite elements

a b s t r a c t

This paper presents a B-splines and NURBS based finite element method for self-consistent solution of theKohn–Sham equations [1,2] for electronic structure modeling of semiconducting materials. A Galerkinformulation is developed for the Schrödinger wave equation (SWE) that yields a complex-valued general-ized eigenvalue problem. The nonlinear SWE that is embedded with a non-local potential as well as thenonlinear Hartree and exchange correlation potentials is solved in a self-consistent fashion. In the self-consistent solution procedure, a Poisson problem is integrated and solved as a function of the electrondensity that yields the local pseudopotential (for pseudopotential formulation) and the Hartree potentialfor SWE. Accuracy and convergence properties of the method are assessed through test cases and thesuperior performance of higher-order B-splines and NURBS basis functions as compared to the corre-sponding Lagrange basis functions is highlighted. Self-consistent solutions for semiconducting materials,namely, Gallium Arsenide (GaAs) and graphene are presented and results are validated via comparisonwith the planewave solutions.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

First-principles (ab initio) methods like the density functionaltheory (DFT) [1,2] provide important insights into the electronic,mechanical and chemical properties of materials. Through DFT,the solution of the many-body electronic-structure problem is re-duced to a self-consistent solution of the single Schrödinger equa-tion. The time-independent Schrödinger equation, termed as theSchrödinger wave equation (SWE) is a quantum mechanical equa-tion that is usually employed to determine the electronic structureof periodic solids. The eigen-solutions of the complex valued SWEcorrespond to the different quantum states of the system. In theDFT framework, Schrödinger wave equation and Poisson equationare solved self-consistently until convergence is achieved. Variousnumerical approaches [3–11] have been adopted for the solution ofSWE that include the finite element method [4–7,10,11] and the fi-nite difference method [8,9]. The advantages and utility of finiteelement method relative to other ab initio methods is highlightedin Pask et al. [12].

Traditional numerical techniques for electronic structure calcu-lations employ planewave (PW) basis functions [3,13] that are notlocal in the real space. There are several shortcomings of thesemethods, namely (i) PW functions limit the size of the problemthat can be solved because these are global functions and cannot

take advantage of the linear scaling methods in electronic structurecalculations [14,15] which exploit the locality of the problem todevelop linear scaling algorithms; (ii) PW functions do not possesslocal support which results in dense matrices that are ill-suited foriterative solution procedures; (iii) PW basis have the same resolu-tion everywhere in the real space and thus are inefficient for prob-lems with local inhomogeneities or situations where local electronstates are important; (iv) PW functions are limited to periodicboundary conditions which is of disadvantage with respect to clus-ter and surface calculations; (v) PW basis functions lead to exces-sive communication between processors for the evaluation of thenon-local terms thereby introducing inefficiencies in parallelimplementation of the code; and (vi) Fourier transforms that areinefficient for parallel computing, are required for PW methods.There have been attempts in the literature to address some ofthe technical issues with the PW basis functions [16] by employingvariational framework and finite element method with Lagrangeand Hermite shape functions.

In a relatively recent effort, Pask et al. [12,16] have employed C0

Lagrange basis functions, while Tsuchida and Tsukada [4] haveused Hermite cubic functions with C1 continuity for DFT calcula-tions in the finite element framework. These works have concen-trated on the pseudopotential formulation [17] and provide afinite element method that is based on the Galerkin formulationfor the solution of Kohn–Sham equations in a periodic domain.These works also provide an overview of the advantage of the finiteelement approach over the traditional planewave methods and fi-nite difference schemes.

0045-7825/$ - see front matter � 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cma.2012.04.016

⇑ Corresponding author. Tel.: +1 217 244 2832; fax: +1 217 265 8039.E-mail address: [email protected] (A. Masud).

Comput. Methods Appl. Mech. Engrg. 241-244 (2012) 112–127

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Since SWE has a differential form that involves continuous func-tions of continuous variables, it is therefore suitable for the applica-tion of variational methods to study electronic properties ofmaterials. We are especially interested in discussing the subject inthe context of using pseudopotentials to represent external potentialof nuclei in the Kohn–Sham equations. In this paper we employ ahigher order Galerkin formulation for SWE that employs B-splineand NURBS basis functions in a finite element setting. Since pseudo-potentials are smooth functions [17] they can be adequately repre-sented via B-splines and Non-uniform Rational B-Splines (NURBS)functions. The present work is inspired by the works of Hughes andcoworkers [18] where they have demonstrated the use of B-splineand NURBS in a finite element framework for computational solidmechanics [19,20] and fluid dynamics [21,22]. The advantage of B-spline and NURBS with respect to Lagrange basis functions are: (i)B-splines and NURBS provide Cp�1 or Cp�1�k continuity, where p isthe order of B-spline or NURBS and k refers to the multiplicity of knotvalues in the knot vector. (This aspect will be explained in Section 3.)(ii) Higher order basis functions can accurately represent high gradi-ents in the atomic potentials. (iii) Unlike Lagrange basis functionsthat display Gibbs phenomenon for higher order polynomials, B-spline and NURBS have variation diminishing property [18] thatturns out to be extremely useful in representing high gradients inthe solution, for example all electron potentials with singularities.

B-splines have a long history in electronic and molecular physicscalculations. An extensive review of B-splines in atomic and molecu-lar physics calculations is given by Bachau et al. [23]. The effectivecompleteness of B-spline basis functions is described as a significantadvantage for atomic/molecular physics calculations and discussedin detail by Hansen et al. [24] and Argenti and Colle [25] with corre-sponding numerical examples. Romanowski has used B-splines inthe context of finite element methods for single atom (one-dimen-sional calculations with all-electron potential) [26] and molecular cal-culations (three dimensional pseudopotential calculations withoutnon-local terms) [27]. Hernandez et al. [28] have used B-spline basisfunctions in place of planewave basis in the context of Fourier spacecalculations for molecules. In our work we have used B-spline basisfunctions with non-local pseudopoetntials for bulk semi-conductormaterials that involve three dimensional calculations. We systemati-cally prove effective completeness of the B-spline basis functions forPoisson equation, Schrdinger Wave equation and non-linear Kohn–Sham equations through convergence plots for the L2 norm and thesquare of energy norm for several representative examples. Conver-gence for different problems under different norms shows effectivecompleteness of B-splines and NURBS with respect to those norms.

An outline of the paper is as follows. In Section 2, we present theKohn–Sham equations and associated Galerkin based finite ele-ment method. In Section 3, we present the important attributesof B-spline and NURBS basis functions that make them ideally sui-ted for application to electronic structure calculations. Section 4presents numerical results for both the self-consistent and thenon-self-consistent solution of Schrödinger wave equation. Con-clusions are drawn in Section 5.

2. Kohn–Sham equations

Kohn–Sham equations of DFT [1,2,16] replace an original many-body interacting particle problem in an external potential with anon-interacting particle problem moving in an effective potential.These equations are given as follows:

�12r2/iðxÞ þ Veff/iðxÞ ¼ ei/iðxÞ; ð1Þ

where /iðxÞ and ei are the Kohn–Sham eigenfunctions and eigen-values respectively, and Veff is the effective electronic potential de-fined as follows:

Veff ¼ VL þ VnL þ VH þ VXC ; ð2Þ

VL ¼X

a

VLaðxÞ; ð3Þ

VnL ¼X

a

ZVnL

a ðx; x0Þ/ðx0Þdx0; ð4Þ

VH ¼Z

qeðx0Þx� x0j j dx0; ð5Þ

VXC ¼ VXCðx;qeÞ; ð6Þ

qe ¼X

i

f i /iðxÞj j2: ð7Þ

VLa and VnL

a are the local and non-local terms in the pseudopotentialapproximation [17] for an atom denoted by subscript a. VXC is theexchange correlation potential, VH is the Hartree potential. The formof exchange–correlation potential is determined by the choice ofthe pseudopotential. qe is the electron density, fi is the occupationnumber associated with eigenstate i. The integrals extend over allspace, with summation extending over all atoms.

The Hartree potential (VH) contains 1=r term because of whichthe total number of terms in the summation of integrand (that ex-tends over all space) is quite large. This makes numerical evalua-tion of Eq. (5) computationally inefficient. Therefore instead ofactually evaluating the integral given in Eq. (5) the Hartree termis usually computed by solving an equivalent Poisson problem.By converting the integral into an equivalent Poisson problem,the potential can be computed efficiently. A similar method is usedto convert the long range local pseudopotential into short rangedensities to achieve computational efficiency.

r2VH ¼Zr2 qeðx0Þ

x� x0j j dx0 ¼Z�4pdðx� x0Þqeðx0Þdx0

¼ �4pqeðxÞ; ð8Þ

r2VL ¼X

a

r2VLaðxÞ ¼

Xa

4pqLaðxÞ; ð9Þ

where qLaðxÞ is ionic density.

As an example the local pseudopotential term for Hartwigsen–Goedecker–Hutter (HGH) pseudopotential [29] and its equivalentdensity are shown in Fig. 1 to present a comparison of the longrange behavior of local pseudopotential term and the short rangebehavior of density. Accordingly, the advantage of using equivalentdensity is that one need not consider large number of atoms toachieve accuracy in numerical calculation at a location x.

Fig. 1. Silicon local pseudopotential [12,29] and its corresponding charge density.

A. Masud, R. Kannan / Comput. Methods Appl. Mech. Engrg. 241-244 (2012) 112–127 113


The Hartree term and the local pseudopotential term are com-bined into a single Poisson problem by evaluating the equivalentdensity of local pseudopotential as discussed in Pask et al. [12,16].

r2VC ¼ r2VH þr2VL ¼ �4pqeðxÞ þX

a

4pqLaðxÞ ¼ f ðxÞ; 8x 2 X;

ð10Þ

where VC is the total Coulomb potential.Due to the nonlinearity engendered by Veff , the set of Eqs. (1)

through (7) are solved self-consistently until convergence is at-tained between the newly calculated density and the computeddensity from the previous iteration. The newly calculated densitymodifies the effective potential Veff , especially Hartree and ex-change correlation terms. We use Pulay mixing scheme [30] anda history of five calculated densities from previous iterations toevaluate the new density.

2.1. Solution of periodic systems

The electronic potential for a perfect crystal is assumed periodicand is given by

V xð Þ ¼ V xþ Rð Þ; ð11Þ

where R represents the lattice vectors associated with the primitiveunit cell of the crystal. Bloch’s theorem [31] states that the solutionof Schrödinger wave equation (1) satisfies the following equation:

/ xð Þ ¼ u xð Þeik�x; ð12Þ

where k is wave vector (position vector in reciprocal space) andu xð Þ ¼ u xþ Rð Þ is a complex-valued cell periodic function that sat-isfies the periodic property for all lattice vectors R.

2.2. Schrödinger wave equation

Let X � Rnsd be an open bounded region with piecewise smooth-boundary C. The number of space dimensions, nsd ¼ 3. ApplyingBloch’s theorem to the Schrödinger equation (1) we get

�12DvðxÞ � ik � rvðxÞ þ 1

2k2vðxÞ þ VðxÞvðxÞ ¼ eðkÞvðxÞ; 8x 2 X

ð13Þ

vðxÞ ¼ vðxþ RÞ; 8x 2 C; ð14Þ

n � rvðxÞ ¼ n � rvðxþ RÞ 8x 2 C; ð15Þ

where vðxÞ is the complex valued cell periodic function or the un-known complex scalar field, namely the wave function (eigenfunc-tion), i is the imaginary unit, x represents the position vector, nrepresents the outward unit normal vector to the boundary C of theunit cell, VðxÞ is the electrostatic potential energy of an electron inan electron density qe xð Þ at the position x and is considered periodicover a unit cell. eðkÞ is the eigen-energy associated with the particle asa function of wave vector (position vector in the reciprocal space) k.

In the context of pseudopotential approximation [17] andKohn–Sham framework, the all-electron potential VðxÞ is replacedby Veff (see Eq. (2)).

� 12DvðxÞ � ik � rvðxÞ þ 1

2k2vðxÞ þ VL þ VH þ VXC

� �vðxÞ

þ e�ik�xVnLeik�xvðxÞ ¼ eðkÞvðxÞ; ð16Þ

The Schrödinger wave equation (16) is solved in a periodic and fi-nite domain. However the non-local term VnL involves integrationover entire space and over all atoms. Therefore, this term needs fur-ther consideration. Pask and Sterne [16] proposed a method to re-duce the non-local term integrated over all space to an integral

form defined over a unit cell. To complete the discussion we high-light the significant features of the method that have been exploitedin the present work. Consider a fully separable pseudopotential [17]for an atom denoted by subscript a that usually has the followingform.

VnLa ðx; x0Þ ¼

Xl;m

valmðxÞh

al v

almðx0Þ; ð17Þ

where valmðxÞ is the product of a projector function and spherical

harmonics and hal is a constant.

Because of periodic properties of wave functions, the non-localterm e�ik�x VnL eik�x v xð Þ reduces to

Xa;l;m

e�ik�xX

n

eik�Rn valmðx� sa � RnÞ

� �� ha

l�

RX eik�x0

Xn0

e�ik�Rn0 valmðx0 � sa � Rn0 Þ

� �vðx0Þdx0

!8>>><>>>:

9>>>=>>>;: ð18Þ

For a detailed derivation refer to Pask and Sterne [16].

2.3. The standard weak form for SWE

Let V � H1ðXnsd Þ \ C0ðXnsd Þ denote the space of trial solutionsand weighting functions for the unknown scalar field.

V ¼ v jv 2 H1ðXnsd Þ; vðxÞ ¼ vðxþ RÞ 8x 2 Cn o

: ð19Þ

The standard weak form is

� w�; ik � rvð Þ þ 12rw�;rvð Þ þ 1

2w�; k2v� �

þ w�;Veffvð Þ ¼ w�; evð Þ;

ð20Þ

where w is the weighting function for v ;w� is its complex conjugate,and �; �ð Þ ¼

RX �ð ÞdX, i.e., L2 product of the indicated arguments over

domain X.Let Vh � V denote the finite-dimensional approximation of the

space of trial solutions and weighting functions for the unknownscalar field. The Galerkin form of the problem is

� w�h; ik � rvh� �

þ 12rw�h;rvh� �

þ 12

w�h; k2vh� �

þ w�h;Veffvh� �

¼ w�h; evh� �

: ð21Þ

Let vh ¼Pn

i¼1ciNi and wh ¼Pn

i¼1diNi, where ci, di are complex coef-ficients associated with the corresponding shape functions for thetrial solution and weighting functions, respectively. Since the shapefunctions have local support, the discrete equation takes the follow-ing form.X

j

Kijcj ¼ eX

j

Mijcj; ð22Þ

where

Kij ¼ Anumel

e¼1Ke

ij; Mij ¼ Anumel

e¼1Me

ij; ð23Þ

Keij ¼

ZXe

12rNi � rNj � ik � NirNj þ

12

k2NiNj þ Veff NiNj

� dx;

ð24Þ

Meij ¼

ZXe

NiNjdx; ð25Þ

where A stands for the assembly operation,R

Xe represents inte-gration over an element domain Xe in the finite element mesh,and numel is the total number of elements in the finite elementmesh.

114 A. Masud, R. Kannan / Comput. Methods Appl. Mech. Engrg. 241-244 (2012) 112–127


Remark 1. All the shape functions employed in Eqs. (24) and (25)are real valued, while the coefficients for both weighting functionand wave functions are complex-valued. Hence there is nocomplex conjugation in the integrals.

Remark 2. The non-local term in Veff needs to be handled differ-ently from the conventional element based local evaluation ofthe finite element matrices and vectors. This is described explicitlyin Appendix A.

Remark 3. The effective potential, Veff , is non-linear because of thepresence of VH and VXC (Eq. (2)) that are in turn functions of theelectron density (Eqs. (5) and (6)). Electronic density (qe) is calcu-lated from the eigenfunctions as shown in Eq. (7).

Remark 4. The higher order smoothness facilitated by the B-splines and NURBS basis functions is especially beneficial in accu-rately evaluating the higher eigenvalues and eigenvectors in thesystem.

Remark 5. B-spline control variables have the interpolation prop-erty only at the boundary of the domain due to the repeated knotvalues at the ends of the knot vectors. Periodic boundary conditionis applied by repeating the same control variables at the corre-sponding boundaries.

2.4. The Poisson problem

Let X � Rnsd be an open bounded region with piecewise smoothboundary C. The number of space dimensions, nsd ¼ 3. The Poissonproblem for the total Coulomb potential together with periodicessential and natural boundary conditions is as follows:

r2VC ¼ f xð Þ; 8x 2 X; ð26Þ

VC xð Þ ¼ VC xþ Rð Þ; 8x 2 C; ð27Þ

n � rVC xð Þ ¼ n � rVC xþ Rð Þ; 8x 2 C; ð28Þ

where VC ¼ VH þ VL, and f xð Þ ¼ �4pqeðxÞ þP

a4pqLaðxÞ as pre-

sented in Eq. (10). n represents the outward unit normal vector tothe boundary C of the unit cell X.

The electron densities that are consistent with a derivative peri-odic smoothly varying function are required to be net neutral inthe unit cell, as given in the following equation.Z

Xf xð ÞdX ¼

ZX

DVC xð ÞdX ¼Z

Cn � rVC xð ÞdC

¼Z

Cn � rVC xð Þ � n � rV C xþ Rð Þð ÞdC ¼ 0: ð29Þ

The local pseudopotential term is converted into equivalent densityterm as shown in Eq. (9). These density terms for each nuclei positionare then superimposed at a location x in order to obtain the total den-sity. (See Section 4 of Pask and Sterne [16] for details.) In addition, thedensity term f ðxÞ also includes the electron density as shown in Eqs.(7) and (10). Thus the solution of Poisson equation includes the ef-fects of local pseudopotential term as well as the Hartree potential.

2.5. The standard weak form for the Poisson problem

Let S � H1ðXnsd Þ \ C0ðXnsd Þ denote the space of trial solutionsand weighting functions for the unknown scalar field.

S ¼ VC jVC 2 H1ðXnsd Þ; VC xð Þ ¼ VC xþ Rð Þ 8x 2 Cn o

: ð30Þ

The standard weak form is

� $w;$VCð Þ ¼ w; f xð Þð Þ; ð31Þ

where w is the weighting function for VC .Let Sh � S denote the finite-dimensional approximation of

space of trial solutions and weighting functions for the unknownscalar field. The discretized weak form is:

� $wh;$VhC

� �¼ wh; f xð Þ� �

: ð32Þ

3. B-splines and NURBS

B-splines and NURBS are parametric functions of rational poly-nomials that are typically employed in Computer Aided Design(CAD) to accurately represent complex geometrical shapes withas few parameters as possible. In a series of landmark papersHughes and coworkers [18] have introduced the notion of Isogeo-metric Analysis (IGA) wherein B-splines and NURBS are employedfor modeling and analysis. IGA has been successfully applied tofluid mechanics [21,22] as well as to solid and structural mechan-ics [19,20]. Amongst the main attributes of the B-splines andNURBS based methodology are: (i) the geometrically exact descrip-tion of the domain of computation, (ii) higher order regularity ofthe method due to the notion of k-refinement [18], and (iii) an effi-cient integration of the analysis methods with the CAD based geo-metric modeling methods.

In electronic structure computations, B-splines and NURBSbased methods offer significant advantages over standard La-grange based finite element methods. This is because (i) higherorder inter-element continuity reduces the error for same numberof total degrees of freedom, and (ii) variation diminishing prop-erty helps suppress Gibbs phenomenon, and it is useful in repre-senting all-electron potentials that have singularity near atomicnuclei.

We first present salient features of B-splines and NURBS inone-dimensional context that make them well suited for elec-tronic structure calculations. We then discuss a procedure tobuild multi-dimensional B-spline basis functions in the contextof finite elements, and conclude with a procedure to buildNURBS functions from B-spline functions for one dimensionaland multi-dimensional cases. Interested reader is directed to Pie-gl and Tiller [32], and Hughes et al. [18] for further technicaldetails.

3.1. Knot vectors

Knot vectors define the parametric space for the B-spline func-tions. They are composed of a sequence of non-decreasing realnumbers, N ¼ nif gnþpþ1

i¼1 , where ni 6 niþ1 2 R, i is the knot index, ni

is the ith knot, n is the total number of basis functions, and p isthe polynomial order or the polynomial degree. The interval be-tween two consecutive knots, called knot span ni; niþ1½ Þ, in a knotvector represents an element of the finite element mesh in theparametric space.

Remark 6. Once the degree of polynomial is chosen, the knotvector completely determines all the basis functions. Unlike theLagrange functions, B-spline basis functions are defined in para-metric space and not in the physical space.

Remark 7. Unlike the Lagrange functions, B-splines do not neces-sarily possess the interpolation property corresponding to any knotvalue. This property exists at internal knots if repeated p times andat first and last knots if repeated pþ 1 times.



Remark 8. For any interior knot, a B-spline basis function of orderp is p� k times continuously differentiable, where k is the multi-plicity of the knot. Increasing multiplicity of knots howeverdecreases continuity of B-spline basis functions.

Remark 9. When the first and last knots are repeated pþ 1 times,the knot vector is termed as non-periodic or open or clamped knotvector. This ensures that B-splines have the interpolation propertyat the corresponding extreme points and the application of Dirich-let or periodic boundary conditions become easier. Periodic bound-ary conditions can also be applied by using periodic knot vectorswhich gives rise to periodic basis functions. However in this paperwe limit our discussion to open knot vectors as defined above.

3.2. Definition and properties of B-splines

There are many definitions of B-spline functions in the litera-ture [18,32,33]. The definition employed here is based on recur-rence formula that Hughes et al. [18] have used in the finiteelement context. For knot sequence N ¼ nif gnþpþ1

i¼1 as defined in Sec-tion 3.1, B-spline functions of degree zero, i.e. p ¼ 0 is defined as

Ni;0 nð Þ ¼1 if ni 6 n < niþ1;

0 otherwise

ð33Þ

and for p P 1

Ni;p nð Þ ¼ xi;p nð ÞNi;p�1 nð Þ þ 1�xiþ1;p nð Þ� �

Niþ1;p�1 nð Þ; ð34Þ

where

xi;p nð Þ ¼ n� nið Þ= niþp � ni

� �if ni – niþp

0 otherwise:

(ð35Þ

Remark 10. Ni;p nð Þ ¼ 0 if n is outside the interval ni; niþpþ1� �

, i.e.the B-spline functions have local support property. This also showsthat B-spline basis functions span over a couple of elements offinite element mesh, because knot span ni; niþ1½ Þ represents anelement of a finite element mesh in the parametric space.

Remark 11. In any given knot span, at most only pþ 1 B-splinebasis functions are non-zero.

Remark 12. Ni;p nð ÞP 0 8 i; p and n (non-negativity).

Remark 13.Pi

j¼i�pNj;p nð Þ ¼ 1 8n 2 ni; niþ1½ Þ (partition of unity).

3.3. Curves as B-spline parametric functions

A curve in Rnsd is represented by linear combination of B-splinebasis functions.

C nð Þ ¼Xn

i¼1

BiNi;p nð Þ; ð36Þ

where n is the number of basis functions, Bi 2 Rnsd ; i ¼ 1;2; . . . ; nare the coefficients of B-spline functions Ni;p nð Þ and are called thecontrol points. As discussed in Remark 7, B-splines do not possessinterpolation property where knot values are not repeated. Henceunlike the coefficients of Lagrange basis functions, the controlpoints are not nodal co-ordinates of the physical mesh at these knotvalues. As discussed in Remark 8, repetition of knots decreases thecontinuity of B-spline basis functions, which in turn reduces thecontinuity of B-spline curves.

Remark 14. Equally spaced control points create a non-uniformmesh. By varying control points one can control the geometricshape of the curve and/or the length of mesh elements.

3.4. h-Refinement, p-refinement and k-refinement

In order to perform convergence tests with B-spline basis func-tions, a h-refinement and a p-refinement process is required. Thefollowing procedure employed by Hughes et al. [18] ensures a uni-form mesh for convergence analysis. In addition this process en-sures that the geometry of the curve does not change with h-, p-or k-refinement.

h-refinement is carried out through knot insertion. LetN ¼ nif gnþpþ1

i¼1 be a knot vector with the properties mentioned inSection 3.1. Let Bif gn

i¼1 be the control points associated with a curvefor the knot vector N ¼ nif gnþpþ1

i¼1 . If �n 2 nk; nkþ1½ Þ is the new knotvalue to be inserted, then the new knot vector isN ¼ n1; n2; . . . ; nk; �n; nkþ1; . . . ; nnþpþ1

� �. The corresponding �Bi

� �nþ1i¼1

control points are defined as follows:

�Bi ¼ aiBi þ 1� aið ÞBi�1 ð37Þ

where

ai ¼1; 1 6 i 6 k� p;

�n�niniþp�ni

; k� pþ 1 6 i 6 k;

0; kþ 1 6 i 6 nþ pþ 2:

8><>: ð38Þ

Remark 15. Mesh refinement for each geometry presented in thenumerical results section was carried out by first starting with thelowest possible mesh resolution. In case of cubic domains, thelowest mesh resolution is a single element in the physical domainand the corresponding knot vector in each direction isf0;0; ::;0|fflfflfflfflffl{zfflfflfflfflffl}pþ1; . . . ;1; . . . ;1;1|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

pþ1

g.

The degree elevation or p-refinement can be carried out as athree step process in Piegl and Tiller [32]. In the degree elevationprocess described in [32], say from p to pþ 1, knot values in theinterior of knot vectors corresponding to order p need to be re-peated at least one more time. This however reduces the higher or-der continuity of the function.

In order to overcome the disadvantage, Hughes et al. [18] sug-gest an alternate to p-refinement, which is termed as k-refinement.In this process, we degree elevate the curve with the lowest possi-ble mesh resolution and then apply h-refinement process as dis-cussed in this section. In all convergence studies in Section 4 thatinvolve degree elevation, k-refinement procedure is used.

3.5. Higher dimensional B-splines

Higher dimensional B-splines in Rnsd are obtained by tensorproduct of single dimensional B-splines. Here we show howthree-dimensional basis functions are used to create volumes orsolids, which is relevant to the problems presented in this paper.Just as control points are used in defining one-dimensional curves,we use control nets to define surfaces or solids. Given a control net

Bi;j;k� �

, where i ¼ 1;2; . . . ; n ; j ¼ 1;2; . . . ;m ; k ¼ 1;2; . . . ; l ; and n,m, l, are the number of basis functions in each of the parametricdirections corresponding to knot vectors N ¼ nif gnþpþ1

i¼1 ,H ¼ gif g

mþqþ1i¼1 , L ¼ fif glþrþ1

i¼1 , where p; q; r, are the degrees of theone-dimensional B-spline basis functions in the correspondingparametric directions. The B-spline solid is defined as follows:

S n;g; fð Þ ¼Xn

i¼1

Xm

j¼1

Xl

k¼1

Bi;j;kNi;p nð ÞMj;q gð ÞLk;r fð Þ: ð39Þ



3.6. Rational B-splines

Rational B-splines are rational polynomial functions. Quadraticrational B-spline functions can be used to define exactly geome-tries from conic sections with as few parameters as possible. A ra-tional B-spline curve in Rnsd is defined as follows:

C nð Þ ¼Xn

i¼1

BiRi;p nð Þ; ð40Þ

where

Ri;p nð Þ ¼ Ni;p nð ÞwiPnj¼1Nj;p nð Þwj

ð41Þ

is the ith rational B-spline basis function of degree p correspondingto control point Bi 2 Rnsd and weight wi for a knot vectorN ¼ nif gnþpþ1

i¼1 . Eqs. (37) and (38) for h-refinement or p-refinementshould be applied to weighted control points defined as follows:

Bwi ¼ wiB

1i ;wiB

2i ; ::;wiB

nsdi ;wi

� �; i ¼ 1;2; . . . ;n; ð42Þ

where the superscript j in Bji denotes the components of the vector

Bi. When rational B-splines are defined in a knot vector that is non-uniform and open, they are called non-uniform rational B-splines(NURBS). Higher dimensional NURBS are defined by taking tensorproduct of one-dimensional NURBS similar to the discussion in Sec-tion 3.5. For a detailed description on how to obtain control pointsand control net for different geometric configurations, interestedreader is referred to Hughes et al. [18] and Piegl and Tiller [32].

Remark 16. All the properties associated with B-splines discussedin Section 3 are also associated with NURBS.

Remark 17. The unknown degrees of freedom are called the con-trol variables and have same properties as that of the controlpoints of the control net, depending on the dimension of the phys-ical domain.

Remark 18. B-splines are a special case of NURBS functions wherethe weights are unity.

4. Numerical results

This section presents a series of numerical tests for the varia-tional form of SWE which is cast in the finite element framework.A significant difference from other finite element based methods isthat instead of using the conventional C0 Lagrange functions or C1

Hermite functions, we employ B-spline and NURBS basis functionsthat provide higher order inter-element continuity. The attributesof the higher order smoothness of B-spline and NURBS are exploredvia a set of benchmark problems. The finite element frameworkalso allows implementation of various classes of boundary condi-tions, namely Dirichlet B.C., Neumann B.C., mixed and periodicB.C.’s, that cover a broad class of problems from calculations ofperiodic solids to cluster and surface calculations [34]. For moredetailed discussion on this topic, interested readers are referredto Pask and Sterne [16].

The numerical section is organized as follows. We first investi-gate the consistency of the method via benchmark problems andestablish numerical convergence rates. Then we employ empiricalpseudopotentials to investigate the solution procedure withembedded Poisson equation. This is followed by a self-consistentsolution employing nonlinear iteration procedure and is appliedto study the electronic structure of different semiconductor mate-rials typically used in the electronics industry.

4.1. Kronig–Penney problem (3D case)

The first test case is a convergence study of the 3D generalizedKronig–Penney problem. The domain under consideration is a cubewith electronic potential given by

V rð Þ ¼ V1D xð Þ þ V1D yð Þ þ V1D zð Þ in X ð43Þwhere

V1D sð Þ ¼0 0 6 s < 2 a:u:

6:5 Ry 2 6 s < 3 a:u:

�In order to obtain the convergence plots, uniform meshes for B-splines are obtained through h-refinement process. For all meshes,the interpolation property is enforced at the location of potentialdiscontinuity by repeating corresponding internal knot values ptimes. In the case of NURBS (for all the orders) the following proce-dure is adopted:

1. A 33 mesh of equal length elements is formed with B-splines.Interpolation property is enforced at the location of potentialdiscontinuity.

2. The weight for B-splines is set equal to 1.0 everywhere, exceptfor the B-spline with embedded interpolation property forwhich the weight value used is 2.236. This value is obtainedvia minimizing the error in the eigenvalues.

3. Thereafter h-refinement is performed consistently that resultsin local refinement around the discontinuity.

Since the meshes for NURBS have unequal length elements, theaverage of the largest and the smallest element length for eachmesh is used as ‘h’ parameter in the convergence rate study.

Remark 19. The meshes are chosen in such a way that the elementboundaries are exactly conforming at 2 a:u. in all three directions.

Figs. 2 and 3 present convergence rates for the fractional error inthe first, third and seventh eigenvalues for the Galerkin method withquadratic/cubic NURBS and quadratic/cubic B-spline basis func-tions. The results are compared to corresponding orders Lagrangebasis functions. Theoretical convergence rates for the eigenvaluesis 2p, where p is the order of the interpolation for the complex valuedwavefunction vðxÞ. Computed rates corroborate the theoretical pre-dictions for NURBS [35], B-spline and Lagrange basis functions.

In Figs. 2 and 3, it is important to note that the absolute error forB-splines and NURBS is consistently lower than that of the corre-sponding order Lagrange basis functions for a given number of de-grees of freedom. We have provided Table 1 that presents thenumber of degrees of freedom in all the meshes for B-splines,NURBS and Lagrange polynomials. It can be seen from Table 1 thatthe crudest mesh for cubic Lagrange polynomials has more degreesof freedom than the corresponding order B-splines/NURBS finemesh. For example, in Fig. 3, a 93 cubic Lagrange mesh with 283 de-grees of freedom shows a normalized error of approximately �5.7on the Logarithmic scale, while a 183 mesh of cubic B-splines with233 degrees of freedom gives a normalized error of �7. This isattributed to the Cp continuity of the B-spline and NURBS basisfunctions as opposed to C0 continuity of Lagrange functions.

Fig. 4 represents the energy band diagram for quadratic B-splinefunction of 83 mesh resolution for which the meshes were again cho-sen in a way that the element edges coincide with the location of po-tential discontinuity. Computed eigenvalues are plotted against exacteigenvalues computed from a non-linear analytical function [36].

4.2. Poisson problem

In this section we study the attributes of Poisson problem forvarious potential functions when modeled via B-spline functions.



4.2.1. Triclinic modelThe first test case for the Poisson problem is a model triclinic

charge density. The source term is defined as

f ¼X

G

G2 aG cos G � xþ bG sin G � xð Þ: ð44Þ

The corresponding analytical solution is

V ¼X

G

aG cos G � xþ bG sin G � x; ð45Þ

where G is the reciprocal lattice vector which is defined in terms ofprimitive lattice vectors b1;b2 and b3. For example (1, 0, 2) impliesG ¼ b1 þ 2b3. The reciprocal lattice vectors are defined as follows:

b1 ¼ 2p a2 � a3ð Þ= a1 � a2 � a3ð Þð Þ ¼ 2p;�0:2p;�0:34pð Þ; ð46Þ

b2 ¼ 2p a3 � a1ð Þ= a2 � a3 � a1ð Þð Þ ¼ 0;2p;�0:6pð Þ; ð47Þ

b3 ¼ 2p a1 � a2ð Þ= a3 � a1 � a2ð Þð Þ ¼ 0;0;2pð Þ: ð48Þ

The reciprocal lattice vectors G and constants aG and bG are given inTable 2.

The domain or the unit cell is defined by the following primitivelattice vectors.

a1 ¼ ð1:0;0:0;0:0Þ; a2 ¼ ð0:1;1:0;0:0Þ; a3 ¼ ð0:2;0:3;1:0Þ; ð49Þ

where periodic boundary conditions are applied on the correspond-ing surfaces. Four uniform meshes composed of 43;63;83 and 123 ele-ments for B-spline order 2, 3 and 4 are employed. Mesh refinement(i.e. h-refinement) and degree elevation (i.e. k-refinement) are done

in the same way as discussed in the previous problem and in Sec-tion 3.4. Fig. 5(a)–(c) present plots of the potential along the bodydiagonal. In the legend, the numbers in brackets denote the numberof degrees of freedom per direction along the three lattice vectorsfor the corresponding meshes and polynomial orders. The plots of er-ror in potential along the body diagonal are shown in Fig. 6(a)–(c).There is a reduction in the error by an order of magnitude as we gofrom the coarsest mesh to the next finer mesh. Furthermore, for anygiven mesh the error in the potential reduces with a correspondingelevation in the order of B-splines. Another significant point thatthe error plots highlight is that there are no discontinuities either inthe value of the computed potential or in its derivative at the inter-element boundaries. This is an attribute of the higher inter-elementcontinuity that is facilitated by B-splines. Fig. 7 shows the normalizedL2 norm of the error in the computed potential as a function of themesh refinement. Here normalization is done with respect to the L2

norm of the analytical potential in Eq. (45).Finite element theory predicts convergence at rate pþ 1 where

p is the order of the polynomial employed. We obtain optimal con-vergence rates in each of the cases. The number of degrees of free-dom for various orders of B-splines and corresponding tensorproduct Lagrange polynomials for various meshes are listed inTable 3 for comparison with the available data.

4.2.2. Silicon empirical pseudopotentialIn this section we employ the Silicon empirical pseudopotential

of Cohen and Bergstresser [37] to study the attributes of theB-spline based finite element method for Poisson problem. Themodel Silicon charge density is given by

Fig. 2. Convergence rates for NURBS: (a) p = 2, (b) p = 3.



f ¼X

G

G2SGVGe�iG�x: ð50Þ

The corresponding analytical solution is

V ¼X

G

SGVGe�iG�x; ð51Þ

where G represents the reciprocal lattice vectors given in Table 2,and constants SG and VG are constants that are defined as follows:

SG ¼ cos G � s; ð52Þ

s ¼ 1;1;1ð Þ a8: ð53Þ

VG ¼

�0:21; Gj j2 ¼ 3ð2p=aÞ2

þ0:04; Gj j2 ¼ 8ð2p=aÞ2

þ0:08; Gj j2 ¼ 11ð2p=aÞ2

0; otherwise

8>>>><>>>>:

9>>>>=>>>>;: ð54Þ

The domain or unit cell is defined by the following primitive latticevectors.

a1 ¼ 0; a=2; a=2ð Þ; a2 ¼ a=2;0; a=2ð Þ; a3 ¼ a=2; a=2; 0ð Þ; ð55Þ

where lattice constant a ¼ 10:261 a:u: The reciprocal lattice vectorsare defined as follows:

b1 ¼ 2p a2 � a3ð Þ= a1 � a2 � a3ð Þð Þ¼ �0:1949p; 0:1949p; 0:1949pð Þ; ð56Þ

b2 ¼ 2p a3 � a1ð Þ= a2 � a3 � a1ð Þð Þ¼ 0:1949p; �0:1949p; 0:1949pð Þ; ð57Þ

Fig. 3. Convergence rates for B-splines: (a) p = 2, (b) p = 3.

Table 1Number of degrees of freedom per direction for different meshes and different orders(p) of B-splines, NURBS and tensor product Lagrange polynomials used in convergenceplots for Kronig–Penney problem.

Mesh No of degrees of freedom/direction

NURBS/B-splines Lagrange (tensor product)

p ¼ 2 p ¼ 3 p ¼ 2 p ¼ 3

9 � 9 � 9 12 14 19 2812 � 12 � 12 15 17 25 3715 � 15 � 15 18 20 31 4618 � 18 � 18 21 23 37 55

Fig. 4. Energy band diagram for quadratic B-splines with 83 mesh.



b3 ¼ 2p a1 � a2ð Þ= a3 � a1 � a2ð Þð Þ¼ 0:1949p; 0:1949p; �0:1949pð Þ: ð58Þ

In Eqs. (50) to (52) and (54), the reciprocal lattice vector G is definedin terms of primitive lattice vectors b1, b2 and b3 as follows:

G ¼ lb1 þmb2 þ nb 3; ð59Þ

where l, m, n are integers in the closed interval �2; 2½ �.In this problem the source term is more intricate than in the

previous test case. Accordingly, the computed potential varies rap-idly as a function of the spatial coordinates. Fig. 8(a)–(c) show theplots of potential along the body diagonal for orders p ¼ 2; 3; 4,respectively. These figures show that the coarsest mesh with 43

elements in combination with the second order B-splines is rathertoo coarse a combination to capture the potential accurately.

However, Fig. 8(a) shows that the method rapidly converges ifthe mesh is refined for a given order B-spline. Fig. 9(a)–(c) showthe plots of error in the potential for different order B-spline basisfunctions. Once again there is no discontinuity either in the poten-tial or its derivatives at inter-element boundaries. These figuresshow rapid convergence for a given mesh, if a systematic k-degreeelevation of B-spline is carried out. With the increase in the poly-nomial order, we see a reduction in the error by an order of mag-nitude for meshes with refinement of 63 elements or higher. Thisshows the h-adaptivity and p-adaptivity features of the presentmethod. Convergence rates are plotted in Fig. 10, where normal-ized L2 norm of the error in the computed potential is plotted asa function of mesh refinement. Here normalization is done with re-spect to the L2 norm of the analytical potential in Eq. (52). Onceagain convergence rates measured in the L2 norm corroborate the

Fig. 5. Plot of the potential along the body diagonal for triclinic model: (a) p = 2, (b)p = 3, (c) p = 4.

Fig. 6. Error in the potential along the body diagonal for triclinic model: (a) p = 2,(b) p = 3, (c) p = 4.



theoretically predicted rates. For this test case as well, Table 3 pro-vides the number of degrees of freedom per direction for variousmeshes and orders of polynomials.

4.3. Schrödinger equation: Silicon empirical pseudopotential

This problem is a continuation of the test case in Section 4.2.2.The primitive unit cell (FCC structure [Face Centered Cubic]) forbulk Silicon is described via the primitive lattice vectors definedin Eq. (55) of Section 4.2.2. A typical conventional unit cell and acorresponding primitive unit cell is shown in Fig. 11. Fig. 12 showsthe first Brillouin zone (in the reciprocal space) and its irreduciblewedge for the corresponding primitive unit cell. The atomic posi-tions are shown with each primitive cell containing two Siliconatoms at positions ð0;0;0Þ and ða=4; a=4; a=4Þ.

The high symmetry points of the Brillouin zone are given asfollows:

C ¼ 0;0;0ð Þ; L ¼ 2pa

12;12;12

� ; K ¼ 2p

a34;34;0

� ;

X ¼ 2pa

1;0;0ð Þ; W ¼ 2pa

1;12;0

� ; U ¼ 2p

a1;

14;14

� The solution of the Poisson problem corresponding to 203 meshwith the forcing function given in Eq. (50) is employed to computethe pseudopotential in Eq. (51), that is then employed to drive theSWE. This test case verifies the self-consistent feature of the methodwhere Poisson problem and SWE are solved self-consistently, ratherthan having used the pseudopotential directly in the SWE. Fourmeshes with 43, 63, 83 and 123 elements are employed for the meshsensitivity study and the convergence rate study. We have used

quadratic and cubic B-splines in this study. In each case the under-lying Poisson problem is solved on a 203 mesh. This requires thatthe electronic potential that is computed over the higher densitymesh of the Poisson problem be projected onto lower density meshof the SWE problem. This projection is done in the L2 norm.

Fig. 13(a) and (b) show the band structure for Silicon pseudopo-tential. In Fig. 13(a) it can be seen that as the mesh is refined thecomputed solution converges to the reference planewave solution[16] with the 123 mesh yielding almost identical solution to theplanewave case. A similar trend is observed for the cubic B-splinesin Fig. 13(b) where 83 mesh results in an almost indistinguishablesolution as compared to the planewave case.

Fig. 14 shows the convergence rates for the first five eigenvaluesat a given k point for the quadratic B-splines. We get optimal con-vergence rates for the solution, which is 2p, where p is the order ofthe basis function.

Fig. 7. Convergence rates for the triclinic model.

Table 2The reciprocal lattice vector G is defined in terms of primitive reciprocal latticevectors b1, b2 and b3. Source Pask et al. [16].

G aG bG

(1,0,0) 0.5 0.90(0,1,0) 0.45 0.85(0,0,1) 0.40 0.80(1,1,0) 0.35 0.75(0,1,1) 0.30 0.70(1,0,1) 0.25 0.65(1,1,1) 0.20 0.60(2,1,0) 0.15 0.55(0,2,1) 0.10 0.50(1,0,2) 0.05 0.45

Fig. 8. Plot of the potential along the body diagonal for Silicon empiricalpseudopotential: (a) p ¼ 2, (b) p ¼ 3, (c) p ¼ 4.



4.4. Self-consistent solution procedure

This section employs the self-consistent numerical solutionprocedure presented in Section 2 to develop electronic band dia-grams for GaAs and for an infinite graphene sheet. In all the casesthe HGH pseudopotential [29] is employed. As noted in Section 2,we use the Pulay mixing scheme [30] and a history of five calcu-lated densities from previous iterations to evaluate the new den-sity, and this procedure ensures convergence for fixed pointiteration method. Monkhorst–Pack algorithm [38] is employedfor Brillouin zone integration.

To carry out the h-refinement study we vary the mesh resolu-tion for the Schrödinger equation while keeping fixed the meshfor the Poisson problem. We employ an equally spaced mesh forthe electron density calculation, and this mesh is four times denser

than the corresponding mesh for the Schrödinger equation. Theelectron densities are calculated at nodal points and are then inter-polated at other spatial points using equal order Lagrange basisfunctions. The solution is transferred between meshes via consis-tent projection in the L2 norm. Initial electron density is obtainedby superposition of electron densities of isolated atoms, ensuringfaster convergence for the bulk materials.

Remark 20. Different resolution meshes have been employed forthe solution of the Poisson problem and the Schrödinger waveequation (SWE). The number of elements for Poisson problem is

Fig. 9. Error in the potential along the body diagonal for Silicon empiricalpseudopotential: (a) p ¼ 2, (b) p ¼ 3, (c) p ¼ 4.

Fig. 10. Convergence rates for the Silicon empirical pseudopotential.

Table 3Number of degrees of freedom per direction for different meshes and different orders(p) of B-splines and tensor product Lagrange polynomials used in convergence plots ofPoisson problems in Section 4.2.

Mesh No of degrees of freedom/direction

B-splines Lagrange (tensor product)

p ¼ 2 p ¼ 3 p ¼ 4 p ¼ 2 p ¼ 3 p ¼ 4

4 � 4 � 4 6 7 8 9 13 176 � 6 � 6 8 9 10 13 19 258 � 8 � 8 10 11 12 17 25 3310 � 10 � 10 12 13 14 21 31 4112 � 12 � 12 14 15 16 25 37 4916 � 16 � 16 18 19 20 33 49 6520 � 20 � 20 22 23 24 41 61 8124 � 24 � 24 26 27 28 49 73 9728 � 28 � 28 30 31 32 57 85 11332 � 32 � 32 34 35 36 65 97 12936 � 36 � 36 38 39 40 73 109 14540 � 40 � 40 42 43 44 81 121 161

Fig. 11. Conventional unit cell and Primitive unit cell.



fixed equal to 20 in each direction, while that of SWE is variedsystematically to evaluate numerical convergence rates. Sincedifferent resolution meshes are used for the coupled Poissonproblem and the SWE, the solution from the Poisson (or SWE)problem is projected onto the mesh of SWE (or Poisson) problemby finding equivalent integration point via Newton–Raphsonmethod.

Remark 21. B-spline shape function coefficients (control vari-ables) do not have interpolation property, except for at the bound-aries. Therefore a third mesh is created to find the electron densityvalues at the nodal points for the case where different resolutionmeshes are used for Poisson and SWE problems. Electron densityat any other physical point is determined via Lagrange interpola-tion functions of the same order as the B-spline functions. In thecalculations shown here the density of this mesh is taken to be fourtimes the density of SWE in each direction.

Remark 22. If we use the same resolution mesh for both the Pois-son and the SWE problems, then higher order B-spline or NURBSbasis functions must be employed for Poisson problem in orderto capture high gradients in the potentials of atoms. This strategyhas been used in the calculation of the graphene band diagram.Using same resolution meshes eliminates the need for projectionbetween different resolution meshes and helps optimize the codefor large scale calculations. However in this case the control pointsneed to be chosen in such a manner that yields a mesh where ele-ment edges coincide with the atomic locations. In addition p� 1repeated knots should be used at points that correspond to theatomic locations to ensure that the peaks of atomic potentialsare captured accurately for the eigenvalue and eigenvector compu-tations in SWE.

Remark 23. Mesh refinement is carried out according to the knotinsertion rule as discussed in Section 3.4. This ensures a structuredmesh wherein mesh refinement helps in conducting the conver-gence rate study.

4.4.1. Gallium Arsenide (GaAs)Gallium Arsenide (GaAs) is comprised of the elements Gallium

and Arsenic, and it possesses some electronic properties that aresuperior to those of Silicon. GaAs has a higher saturated electronvelocity and higher electron mobility, thereby allowing transistorsmade from it to function at higher frequencies. Furthermore, unlikeSilicon junctions, GaAs devices are relatively insensitive to heatdue to their higher band gap.

GaAs has a FCC cubic structure similar to that of bulk Silicon(Si). Therefore, the primitive unit cell and Brillouin zone are de-fined in similar manners. The one difference is that the Galliumatom is located at (0,0,0), while the Arsenic atom is located atða=4; a=4; a=4Þ. The lattice constant for GaAs is a ¼ 10:6831 a:u.HGH pseudopotentials [29] and Perdew–Wang exchange–correla-tion potential [39] are used in this study. Monkhorst–Pack algo-rithm [38] is used to numerically integrate the electron densityin the Brillouin zone. Brillouin zone integration was carried outwith 44 integration points in the irreducible part. Computed re-sults are compared with the planewave solutions that are pre-sented in Pask and Sterne [16].

Since the eigenfunctions are functions of the wave vector k,after the application of Bloch’s theorem (Section 2.1) the electronicdensity needs to be evaluated by performing integration in theBrillouin zone. Eq. (7) in Section 2 therefore transforms to thefollowing.

Fig. 12. First Brillouin zone and irreducible wedge.

Fig. 13. Band structure for Silicon pseudopotential: (a) p ¼ 2, (b) p ¼ 3.

Fig. 14. Convergence rates for the first five eigenvalues at a given k point.



nkðxÞ ¼X

i;ei;k<eF

fi;k /i;kðxÞ�� 2;qeðxÞ ¼

1XBZ

ZXBZ

nkðxÞdk ¼X

k

xknkðxÞ;

ð60Þ

where XBZ is the volume of the first Brillouin zone. The total energyis calculated according to the procedure presented in Section 5 ofPask and Sterne [16]. The expression for total energy is as follows:

Etot ¼X

i

fieiþZ

X�1

2qeðxÞþ

Xa

qLaðxÞ

!VCðxÞþqeðxÞ eXCðxÞ�VXCðxÞð Þ

" #dx

þ12

Xa

ZXqL

aðxÞVLaðxÞdx: ð61Þ

Figs. 15 and 16 show the band diagrams along high-symmetrylines of the first Brillouin zone. As shown in these figures the max-imum of the valence band is directly below the minimum of theconduction band for an electron traveling in the direction of C,and therefore GaAs is categorized as direct gap material. PW meth-od predicts a band gap of 0.0173 Ha. For quadratic B-splines, 103

mesh gives a band gap of 0.0234 Ha, while for cubic B-splinesthe computed band gap is 0.0184 Ha. Fig. 17 shows the band dia-gram plot for NURBS functions. The weights of basis functions forelements around atoms are assigned a higher value of 3.0 whileall other basis functions have a weight of unity. Cubic NURBS func-tion with mesh resolution of 83 predicts a band gap of 0.0257 Ha,while cubic B-spline functions with the same mesh resolution pre-dicts a band gap of 0.0258 Ha. The total energy values are tabulatedin Table 4 for the various order NURBS basis functions.

Fig. 18 shows the convergence rates for total energies computedwith quadratic and cubic B-splines. As in the linear SWE problemdescribed in Section 4.3, we get optimal convergence rates of 2p,where p is the order of the basis function.

4.4.2. GrapheneGraphene is a one-atom thick flat layer of sp2-bonded carbon

atoms that is tightly packed into a two-dimensional (2D) honey-comb lattice. Fig. 19(a) shows the 2D primitive unit cell andFig. 19(b) shows the first Brillouin zone for a graphene sheet. Thedomain or unit cell is defined by the following primitive latticevectors.

a1 ¼ a;0;0ð Þ; a2 ¼ a=2;ffiffiffi3p

=2a;0� �

; a3 ¼ 0;0;3að Þ; ð62Þ

aA ¼a1 þ a2

3; aB ¼

2 a1 þ a2ð Þ3

; ð63Þ

where a ¼ffiffiffi3p

aCC ;aA and aB are atomic locations of atoms A and Bshown in Fig. 19(a). The lattice parameter aCC ¼ 2:6834 Ha; is alsoshown in Fig. 19(a).

Fig. 15. Band diagram for Gallium Arsenide (GaAs) for B-spline order p ¼ 2. PWdata from Pask et al. [16].

Fig. 16. Band diagram for Gallium Arsenide (GaAs) for B-spline order p ¼ 3. PWdata from Pask et al. [16].

Fig. 17. Band diagram for Gallium Arsenide (GaAs) for different order NURBS and 83

mesh.

Fig. 18. Convergence rates for Gallium Arsenide (GaAs) total energy.

Table 4Total energy Etð Þ for various orders of NURBS for 83.

NURBS, p Total energy, Et (Ha)

2 �8.54913 �8.5853



For computations of electronic band structure, we consider thethird dimension to be three times the lattice parameter a. Periodicboundary conditions are assumed on the surfaces in all three direc-tions. The reciprocal lattice vectors are defined as follows:

b1 ¼ 2p a2 � a3ð Þ= a1 � a2 � a3ð Þð Þ ¼ 2pa 1;�1=ffiffiffi3p

;0� �

; ð64Þ

b2 ¼ 2p a3 � a1ð Þ= a2 � a3 � a1ð Þð Þ ¼ 2pa 0;2=ffiffiffi3p

;0� �

; ð65Þ

b3 ¼ 2p a1 � a2ð Þ= a3 � a1 � a2ð Þð Þ ¼ 2pa 0;0;1=3ð Þ: ð66Þ

The irreducible Brillouin zone for graphene is 1=12th for the firstBrillouin zone. 110 k points were considered in the irreducible partfor the integration of electron density. The high symmetry points ofthe Brillouin zone are given as follows:

C ¼ 0; 0; 0ð Þ; M ¼ 2pa

0;1ffiffiffi3p ;

16

� ; K ¼ 2p

a13;

1ffiffiffi3p ;

16

� Fig. 20 shows the band diagram along high symmetry points.

Both the r and p bands are accurately captured as shown in thefigure.

Table 5 shows the total energy values calculated for differentorder B-splines and mesh resolutions. The planewave calculationswere done using abinit package [40,41]. Based on the calculationspresented in references [42,43], we performed planewave calcula-tions with 20 � 20 � 1 integration points in the first Brillouin zone(110 k points in irreducible Brollouin zone), and 27 Ha as the en-ergy cutoff. We used Hartwigsen–Goedecker–Hutter (HGH)

pseudopotential [29] and local density approximation exchange–correlation functional that is based on Gœdecker et al. [44]. It givesvalues close to Perdew–Wang exchange–correlation potential [39].The parameters chosen for planewave calculations closely matchthe parameters used in our finite element calculations.

Table 5 shows that for a fixed, but otherwise arbitrary mesh, asthe order of B-splines is increased, the total energy converges tothe calculated planewave energy. For example, for the 6 � 6 � 18mesh the difference in total energy per atom with respect to theplanewave calculations decreases from 0.09156 Ha for p = 3 to0.01866 Ha for p = 5, while the corresponding increase in the de-grees of freedom is only fifty percent. Table 5 also presents two dif-ferent resolution meshes for the cubic B-splines that show that asthe mesh is refined from 6 � 6 � 18 to 9 � 9 � 24, the difference intotal energy per atom reduces from 0.09156 Ha to 0.00136 Ha.These results show the p-refinement and h-refinement featuresof the method that lead to convergence of the computed solutionto the planewave calculations.

5. Conclusions

We have presented a variational framework and a finite ele-ment method for real-space self-consistent solution of Kohn–Shamequations. We have employed B-splines and NURBS basis functionsthat provide higher inter-element continuity as compared to theLagrange basis functions that only possess C0 continuity. Variousattributes of B-splines and NURBS have been exploited in themethod as well as in its numerical implementation and large scaleparallelization. Higher order B-splines and NURBS possess varia-tion diminishing property [18] and therefore they do not displayGibbs phenomenon that is typically encountered when C0 Lagrangebasis functions are employed to capture steep gradients. This as-pect is especially useful in representing the high gradient potentialfunctions (all-electron potentials with singularity near atomic nu-clei) via higher order B-splines and NURBS. In addition real spaceformulation enables various types of boundary conditions to beimplemented with relative ease. Since B-splines and NURBS pro-vide local support they provide significant advantages over plane-wave basis functions in parallel implementation by minimizingcommunication between processors.

The proposed method is tested via a number of benchmarkproblems and the accuracy and convergence of the computed solu-tions is established in comparison with the published results. Spe-cifically the convergence plots for 3D Kronig–Penney problemshow that B-splines and NURBS give better precision with lowerresolution meshes than the corresponding order Lagrange basisfunctions even with high resolution meshes. In the case of Poissonproblem we observe the higher inter-element continuity and a bet-ter representation of the periodic boundary conditions. Pask andSterne [16] have also reported error plots along the body diagonalthat show that C0 Lagrange functions give rise to cusps at inter-ele-ment boundaries. In the present method with B-splines and NURBSthere is a smooth variation in errors across intert-element bound-aries that is due to the higher inter-element continuity property.

Fig. 19. (a) Primitive unit cell and (b) the first Brillouin zone with high-symmetrypoints.

Fig. 20. Band diagram for graphene for various order B-splines for a meshresolution of 6� 6� 18.

Table 5Total energy Etð Þ for various orders of B-spline and mesh resolutions.

B-spline Mesh No. of degrees Total energyorder, p resolution of freedom EFEM

t (Ha)

2 6 � 6 � 18 8 � 8 � 20 (1280) �11.27123 6 � 6 � 18 9 � 9 � 21 (1701) �11.19374 6 � 6 � 18 10 � 10 � 22 (2200) �11.29645 6 � 6 � 18 11 � 11 � 23 (2783) �11.33953 9 � 9 � 24 14 � 14 � 28 (5488) �11.3741

Planewave calculations, total energy ðEPWt Þ �11.37682



The higher accuracy of the method is shown via model problemsand calculations of band diagrams for GaAs and graphene, andvia comparison of the computed solution with the planewavemethods.

Acknowledgment

Authors would like to thank Dr. John E. Pask and Dr. Nahil Sobhfor many helpful discussions on Kohn–Sham equations. We wouldalso like to thank Professor Thomas J.R. Hughes and Dr. J. AustinCottrell for their help in developing the B-spline and NURBS code.This work was supported by the National Academies Grant NAS7251-05-005, which is gratefully acknowledged.

Appendix A. The non-local term

The non-local term embedded within Veff requires furtherexplanation. The non-local term can be expanded as follows usingEq. (20).

ZXe

e�ik�x Ni VnL eik�x Nj

� �dx¼

Xa;l;m

RXe e�ik�x

Xn

eik�Rn valm x�sa�Rnð Þ

� �Ni xð Þdx

!�ha

l

�R

X eik�x0X

n0e�ik�Rn0 va

lm x0 �sa�Rn0ð Þ� �

Nj x0ð Þdx0 !

8>>>>><>>>>>:

9>>>>>=>>>>>;;

ðA:1Þ

whereR

Xe stands for integration over an element domain Xe in thefinite element mesh, while

RX stands for integration over the entire

domain X or the unit cell. In matrix form, the equations are asfollows:

whereZX�ð Þdx0 ¼ m

numel

e¼1

ZXe�ð Þdx0: ðA:3Þ

Remark A.1. Integral 1 in Eq. (A.2) is localized since the integral isover an element domain Xe and the shape functions have localsupport, i.e., only a few shape functions have non-zero values. Thisimplies that the element level non-local term contributes only tothe rows associated with the element e of the global stiffnessmatrix.

Remark A.2. Integral 2 in Eq. (A.2) is nonlocal, because it isdefined over the entire domain or over the unit cell, X. Howeverthe non-local element stiffness matrix contributes to a limitednumber of columns (although greater than that contributed bylocal pseudopotential) in the global stiffness matrix due to theproduct of localized projector functions and the non-localpseudopotentials.

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