VOLUME 78, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 12 MAY 1997

30314

Orthonormal Wavelet Bases for Quantum Molecular Dynamics

C. J. Tymczak and Xiao-Qian WangDepartment of Physics & Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, Georgia

(Received 7 October 1996)

We report on the use of compactly supported, orthonormal wavelet bases for quantum molecular-dynamics (Car-Parrinello) algorithms. A wavelet selection scheme is developed and tested forprototypical problems, such as the three-dimensional harmonic oscillator, the hydrogen atom, andthe local density approximation to atomic and molecular systems. Our method shows systematicconvergence with increased grid size, along with improvement on compression rates, thereby yieldingan optimal grid for self-consistent electronic structure calculations. [S0031-9007(97)03146-3]

PACS numbers: 31.15.–p, 02.70.Rw

st

ept

dur

ictt

n

b

t

c

n

a

nln

cal-

s-ed

agermts.ianof

n-iza-nainns.ts,

helec-

n-uc-beix,w-

these-r-ve

n-is

e

ies

ea-theus-tsth

Recent advances inab initio calculational methodshave experienced a considerable amount of succespredicting ground-state structural and cohesive properof condensed-matter systems [1]. The pioneering woof Car and Parrinello [2], based on dynamical simulatannealing, promoted a new class of approaches [3] apcable to density-functional theory within the local-densiapproximation (LDA) [4]. Density-functional moleculardynamics (Car-Parrinello) [2] and other iterative metho[5] based on plane-wave basis have made such calctions possible for systems consisting of several hundatoms [6]. While these methods have been very sucessful, several difficulties arise when they are extendto systems with large length scales or those containtransition-metal atoms. Several recently proposed teniques such as optimized pseudopotentials, adapcoordinates, and preconditioning combined wiconjugate-gradient techniques have had considerasuccess [5–7]. Yet, these approaches are still constraiby the plane-wave-basis set which relies on the faFourier transform (FFT) to effectively transform betweereal and reciprocal spaces.

Recently, it has been recognized [8–10] that it mayadvantageous to perform electronic structure calculatiousing a wavelet basis that has dual localization characistics in real and reciprocal spaces. One of the impotant features of the wavelet basis is its ability to reduthe number of components needed for solving the KohSham equation [11], reminiscent of compression in aplying wavelets to image processing [12]. Furthermormultiresolution analysis provides automatic preconditioing [6] on all length scales, which increases the rateconvergence of the electronic wave functions.

Arias and coworkers [8] employed a tight frame geneated by the Mexican hat wavelet and applied it to eletronic structure calculations. Their work representsextension of multiresolution analysis to nonorthogonbases. The generalized multiresolution analysis, utilizia Gaussian scaling function and the Mexican hat waveturns out to be a very good approximation in representithe electronic wave functions. However, the correspon

3654 0031-9007y97y78(19)y3654(4)$10.00

iniesrkdli-

y

sla-

edc-edngh-

ivehbleedstn

enser-r-en-p-e,-

of

r-c-n

alg

et,g

d-

ing inverse transform necessary for the self-consistentculations is complicated [8,13].

Wei and Chou [9] have devised a method for uing compactly supported orthogonal wavelets, developby Daubechies (D6) [12], for self-consistent electronicstructure calculations. This approach takes advantof the existence of a fast, discrete wavelet transfo(DWT) [14] associated with the Daubechies waveleThey have demonstrated that the resulting Hamiltonmatrix in wavelet space can be reduced by a factorhundreds, making it feasible for obtaining the eigevalues and eigenfunctions through standard diagonaltion. While the preliminary application of this method iatomic and molecular systems is promising, there remseveral important issues relevant to practical applicatioThese include the selection of the wavelet componenthe effective construction of the matrix elements of tlocal pseudopotential, and the convergence of the etronic wave functions.

The selection of the wavelet components is a key usolved issue in applying wavelet bases to electronic strture calculations. In principle, the selection shouldbased on the eigenfunctions of the Hamiltonian matri.e., the outcome of the diagonalization process. Hoever, the eigenfunctions are not knowna priori. Approxi-mate methods, e.g., using the wavelet transform oflocal potential as a guide [9], may not guarantee thelected wavelet components being optimal. This may futher lead to problems in convergence of the electronic wafunctions. It is also worth pointing out that the represetation of the local pseudopotential in the wavelet spaceof a complicated form and amounts to anOsN2

gd processfor a system ofNg grid points. This severely hampers thpractical applicability of their method [9].

In this Letter we report on the use of the Daubechwavelet bases (D6-D14) in the Car-Parrinello (CP) al-gorithm [2]. Our approach preserves the advanced ftures of the CP method, and the transform betweenreal and wavelet spaces can be efficiently carried outing the DWT. The DWT associated with the waveleof compact support has many similarities to FFT. Bo

© 1997 The American Physical Society

VOLUME 78, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 12 MAY 1997

l

o

me

r

n

h

v

go-

l

ion

n

heg.edc-rs-

e-r-

re

let-

-heingr

etes

are unitary and scale asNg logNg. The primary advan-tage of DWT over FFT is the capability of representinthe wave functions with a minimal number of componenthat scales asNW , logNg. Using the CP method, thecalculations associated with the local pseudopotential—predominant part in the self-consistent calculations—athen OsNgd for each energy level. In addition, the construction of the separable nonlocal pseudopotential mtrix elements of the Kleinman-Bylander form [15] scaleas OsN2

W d, while the matrix elements of the kinetic energy can be obtained through transformation of a bandiagonal matrix into wavelet space [9,16].

Wavelets are very useful in representing data or funtions [13]. An orthonormal wavelet is generated by aanalyzing wavelet and an auxiliary function called the scaing function. The basis is constructed from the analyzinwavelet and scaling function by the operations of transtions and dyadic dilations. Unlike the sines and cosinthat comprise the bases in Fourier analysis, which arecalized in Fourier space but delocalized in real space, bmother wavelet and scaling function are dual localizedreal and reciprocal spaces [12].

The CP method [2] involves a fictitious moleculadynamics solution of electronic wave functions on thBorn-Oppenheimer surface,

mC̈istd ­ 2HCistd 1X

l

LilClstd , (1)

where m is a fictitious mass andLil are the Lagrangeconstraints. The Gram-Schmidt orthogonalization scheis employed to keep the wave functions of the occupistates orthogonal. It is known that the calculation of nolocal pseudopotentials and the orthogonalization proce(using either Gram-Schmidt or iterative solutions) are botlenecks in plane-wave based calculations. The use ofthonormal wavelet bases is capable of reducing the cfor these calculations. In fact, for both types of calculations, the compression associated with the wavelet bais most beneficial in that anOsN2

gd process can be reducedto OsN2

W ).With the use of orthonormal wavelet bases for th

expansion of wave functions, the corresponding matrepresentation of the Hamiltonian consists of kinetic anpotential energy parts. Choosing the three-dimensiowavelet as a combination of three one-dimensionwavelets, the kinetic matrix factors into products of thresubmatrices ofx, y, and z components, which can bereadily calculated and tabulated beforehand [16]. Tconstruction of the potential energy matrix in wavelespace entails (i) calculatingVeffC for each occupied en-ergy level in real space, and (ii) transforming the produback into wavelet space. For electronic structure calcutions, an inverse wavelet transform of the resulting wafunctions are needed to constructVeff of the Kohn-ShamHamiltonian [4].

gts

are-a-s-d-

c-nl-g

a-eslo-th

in

re

ed

n-sst-or-ost-

ses

eixdalale

et

ctla-e

We have developed a “bootstrap” algorithm for selectinthe significant wavelet components. The wavelet compnents of a continuous functionusxd, at scalej, are of theform:

djk ­Z

dx usxd 22jy2cs22jx 2 kd , (2)

where c is the analyzing wavelet. For an orthonormawavelet with M vanishing momentssDMd, the waveletcomponents at scale (j 2 1) are related to those at(coarser) scalej by

dj21,k ø 22sM1 1

2ddj,k , (3)

provided that one can neglect the Taylor series expansof usxd to orders greater thanM. As an example, weshow in Fig. 1 the wavelet coefficients for a Gaussiafunction using theD8 wavelet. The scaling behavior ofthe coefficients is readily observable, which constitutes tbasis of compression in the context of image processin

Taking advantage of the scaling behavior describabove, we start by solving for the (unknown) wave funtion in wavelet space using the coefficients from the coaest scale (J) to scalej. Next, we select the importantwavelet coefficients from this subset of coefficients by rmoving those below the (predetermined) threshold toleance. Then, at scalej 2 1, only those coefficients thatare connected with the surviving components at scalejneed to be considered further for sorting. This proceduis repeated until the finest scale is reached.

A few remarks are in order. In contrast to the simpsorting scheme used by other applications [9], the “boostrap” algorithm is efficient in that any negligible component at a coarse scale results in the elimination of tassociated components at finer scales. It is worth notthat this algorithm provides preconditioning [6] on the fine

FIG. 1. Wavelet components of a Gaussian function (insplot) usingD8. The dashed lines correspond to the boundaribetween each scale (labeled byj).

3655

VOLUME 78, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 12 MAY 1997

-tesnd

t-

r-n

,

-

lnd,-

-h-e.d

r

TABLE I. The kinetic, potential, and total energies (Ekin,Epot, and Etot) of the harmonic oscillator in units ofv0,calculated with different wavelets (D6, D8, and D14) fordifferent grid size (Ng) and the requested wavelet componen(NW ).

Ng Wavelet NW Ekin Epot Etot

163 D6 3101 0.4955596 0.5180191 1.013578D8 3009 0.4986462 0.5031520 1.001798D14 2680 0.4999382 0.5000889 1.000027

323 D6 4724 0.4994644 0.5016613 1.001125D8 3764 0.4998524 0.5002569 1.000109D14 3341 0.4999917 0.5000093 1.000000

643 D6 3996 0.4997000 0.5006009 1.000300D8 3644 0.4998777 0.5002089 1.000086D 14 3584 0.4999938 0.5000062 1.000000

scales. Most importantly, the dynamic simulated anneing procedure for solving the eigenfunctions allows, at thsame time, the selection of important wavelet componewithout the necessity of resorting to approximations.

For purposes of demonstrating our approach, we aply the method to the three-dimensional harmonic oscilltor. Table I shows our results for the system at variogrid sizes, using Daubechies wavelets with varying suports. As seen from Table I, a systematic convergencethe exact result is achieved with increasing grid size anor with the use of smoother wavelets of wider supporalong with the notable improvement of the compressiorate (as measured byNW yNg). For instance, using theD14 wavelet at a grid size of643, the eigenvalue con-vergence is better than10210 with a compression rate ofabout 1%.

An important ramification of these results is the stabity of our algorithm. The systematic convergence indcates that the bootstrap algorithm is a controlled selectprocedure. We have verified the systematic convergenof our algorithm for the harmonic oscillator, the hydrogen atom (see Table II), and the self-consistent electrostructure calculations utilizing pseudopotentials. Forlustration, we list in Tables III and IV results for LDAcalculations with the pseudopotentials for the helium atoand hydrogen dimer. These results demonstrate thatrelatively smooth potentials, our algorithms convergesystematically with increasing grid size, with improvin

TABLE II. The kinetic, potential, and total energies (Ekin,Epot, and Etot) of the hydrogen atom in atomic units and thcompression rate (NW yNg), calculated usingD8 for differentgrid size (Ng).

Ng NW yNg Ekin Epot Etot

163 57.9% 0.40484 21.23225 20.82740323 13.4% 0.58328 21.48928 20.90599643 2.1% 0.71455 21.66780 20.95325

1283 0.3% 0.78551 21.76712 20.98161

3656

ts

726839470

al-e

nts

p-a-usp-tod/

ts,n

il-i-ionce

-nicil-

mfors

g

e

TABLE III. The calculated total energy (in Rydberg) ofatomic helium with different grid size (Ng) and the correspond-ing compression rate (NW yNg).

System Ng NW yNg ELDA (present work) ELDA

Helium 323 17.6% 21.978 22.814643 2.3% 22.400

1283 0.3% 22.635Pseudo 323 21.4% 22.536 22.831

643 2.4% 22.7851283 0.3% 22.792

compression rates (the typical compression rate for a1283

system is about 0.2%).It is worthwhile to compare CP methods using tradi

tional plane-wave and wavelet bases in order to evaluathe efficiency of our approach. For a single iteration, thiamounts to comparing the relative speed between FFT aDWT. Our best optimized DWT is three times slowerthan the FFT. Although there is room for further opti-mizing the DWT, this is not the main thrust of the presenLetter. As a result, for relatively small systems (e.g., harmonic oscillator at size163), the plane-wave code out-performs our algorithm. This is, however, no longer thecase for larger systems (e.g., harmonic oscillator at643)because the wavelet algorithm yields much faster convegence rate, which can be attributed to the drastic reductioin the number of local minima.

For a better understanding of how our method workswe consider all-electron calculations with the Coulombpotential. We employ a simple approximation to the singular Coulomb potential by replacing it with a nonsin-gular one of the formZerfsrysdyr, wheres is chosento cover only the nearest neighbor points. This potentiaguarantees the ground state energy to be an upper bouand a better approximation of the original Coulomb potential with increasing grid size. It is important to pointout that this choice of the potential is not for obtaining accurate energies for systems like the hydrogen atom, whicis well known and can be properly handled by other methods [10]. Rather, we wish to use this exercise to explorthe applicability of our approach to singular potentialsFigure 2 shows the radial part of the calculated groun

TABLE IV. Calculated bond length (d) and vibrational fre-quencies (n) of the diatomic H2 using the Murnaghan equationof state. Experimental results from Ref. [14] are included focomparison.

Grid d (a.u.) n scm21d

Modified Coulomb 643 1.62 3820Modified Coulomb 1283 1.52 4390Pseudopotential 644 1.57 4180Experiment 1.40 4400

VOLUME 78, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 12 MAY 1997

n

ish

itssedt

u-

e,

.e

-t

.

e

,.

FIG. 2. The radial part of the hydrogen atom’s ground staeigenfunction calculated on a grid of163 (¶), 323 (+), 643 (h),and1283 (3) as compared to the analytical solution (solid line

state wave function for the hydrogen atom with respeto different grid sizes, as compared to the exact solutioThe major deviation from the analytical solution occuras expected, around the ionic core region. Our calcution clearly shows that, with an increase of the grid sizthe algorithm picks up more wavelet components arouthe ionic core region, thereby improving the results. Simlar behavior is also observed for the helium atom (see Tble III).

Table IV shows the results for the hydrogen dimerdifferent grid sizes with use of the simplified potentiaand the hydrogen pseudopotential. Figure 3 showsrepresentative curves and data points used to calculate

FIG. 3. Calculated total energy of H2 as a function of bondlength using pseudopotential (+), and all-electron calculatiwith different grid size [(¶) for 643 and (h) for 1283,respectively]. The curves are fits of the Murnaghan equatiof states of the calculated points.

te

).

ctn.

s,la-e,ndi-a-

atl

thethe

on

on

bonding length and vibrational frequency. Our calculatiousing the simplified potential on a1283 grid yieldedreasonably better results, indicating that the wavelet basis capable of handling singular potentials, even witsimple approximations.

In summary, we have presented a method that permus to perform quantum molecular dynamics simulationbased on orthonormal wavelet bases. We have developan efficient scheme for the selection of important wavelecomponents. The advantage of using a dynamical simlated annealing lies in the fact that it is not only makingthe selection of the components a controlled procedurbut also offering a natural way for reducing the cost incomparison with plane-wave based calculations.

Fruitful discussions with M. Y. Chou, C. R. Handy,R. Murenzi, and S. Wei are gratefully acknowledgedThis work was supported in part by the National SciencFoundation under Grant No. HRD9450386, Air ForceOffice of Scientific Research under Grant No. F4962096-1-0211, and Army Research Office under GranNo. DAAH04-95-1-0651.

[1] See, for instance, R. G. Parr and W. Yang,DensityFunctional Theory of Atoms and Molecules(OxfordUniversity Press, Oxford, 1989).

[2] R. Car and M. Parrinello, Phys. Rev. Lett.55, 2471(1985).

[3] M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, andJ. D. Joannopoulos, Rev. Mod. Phys.64, 1045 (1992), andreferences therein.

[4] W. E. Pickett, Comput. Phys. Rep.9, 115 (1989); O. H.Nielsen and R. M. Martin, Phys. Rev. Lett.50, 697 (1983).

[5] D. Vanderbilt, Phys. Rev. B41, 7892 (1990).[6] J. R. Chelikowsky, N. Troullier, and Y. Saad, Phys. Rev

Lett. 72, 1240 (1994).[7] W. Kohn and L. J. Sham, Phys. Rev.140A, 1133 (1965).[8] T. A. Arias, K. J. Cho, P. K. Lam, and M. P. Teter, in

Toward Teraflop Computing and New Grand ChallengApplications, edited by P. K. Kalia and P. Vashishta (NovaScientific Publications, Commack, 1995), p. 23; K. ChoT. A. Arias, J. D. Joannopoulos, and P. K. Lam, Phys. RevLett. 71, 1808 (1993).

[9] S. Wei and M. Y. Chou, Phys. Rev. Lett.76, 2650 (1996).[10] K. Yamaguchi and T. Mukoyama, J. Phys. B29, 4059

(1996).[11] G. Beylkin, SIAM J. Numer. Anal.6, 1716 (1992).[12] I. Daubechies,Ten Lectures on Wavelets(Society for

Industrial and Applied Mathematics, Philadelphia, 1992).[13] J-P. Antoine and R. Murenzi, inSubband and Wavelet

Transforms, edited by A. N. Akansu and Mark J. T. Smith(Kluwer Academic Publishers, 1996).

[14] I. Daubechies, Commun. Pure Appl. Math.41, 909 (1988).[15] L. Kleinman and D. M. Bylander, Phys. Rev. Lett.48,

1425 (1982).[16] G. Beylkin, SIAM J. Numer. Anal.6, 1716 (1992);

G. Beylkin, R. Coifman, and V. Rokhlin, Commun. PureAppl. Math. 44, 141 (1991).

3657