We generalize the Poisson equation to attenuated Newtonian potentials. If the attenuation is at least exponential, the equation provides a local mapping between the density and its potential. We use this to derive several density functionals for the short-range self-interaction energy

Gilbert, Andrew T. B.

Gill, Peter M. W.

Varganov, Sergey A.

The Journal of Chemical Physics

The Australian National University

A generalized Poisson equation and short-range self-interaction energiesSergey A. Varganov, Andrew T. B. Gilbert, and Peter M. W. GillaResearch School of Chemistry, Australian National University, Canberra ACT 0200, AustraliaReceived 18 February 2008; accepted 23 May 2008; published online 24 June 2008We generalize the Poisson equation to attenuated Newtonian potentials. If the attenuation is at leastexponential, the equation provides a local mapping between the density and its potential. We use thisto derive several density functionals for the short-range self-interaction energy. © 2008 AmericanInstitute of Physics. DOI: 10.1063/1.2945298I. INTRODUCTIONThere has been a recent surge of interest1–20 in electronicstructure methods that split the Coulomb operator,u−1  Su + Lu 1where u=r−r into short- and long-range components. Theproblem of computing the Coulomb potential of a densityr then splits into the subproblems of computing its short-range partVr = Surdr 2and the complementary long-range part. The latter can betreated by Fourier or fast multipole methods, but the short-range potential Vr often presents a burdensome computa-tional bottleneck.21In many of these methods,1,3,6–8,11,14,15,19,20 Vr is foundby embedding the generation of attenuated two-electron in-tegrals within a complicated cutoff strategy. We haveshown21–23 how these integrals can be computed when r isexpanded in a Gaussian basis, but if Su is a short-rangedfunction, it is reasonable to ask whether integration over allspace is appropriate and necessary. More specifically, oneshould ask whether there exists a local relationship betweenVr and r.Newtonian potentials can be attenuated in many ways.We will focus here on the Yukawa attenuator24SYu =exp− uu, 3the Ewald attenuator2,3,25,26SEu =erfcuu, 4and the Heaviside attenuatorSHu = 1/u if u  10 if u  1, 5where the parameter  controls the attenuation strength. TheFermi–Dirac, Chakravorty and Clementi,27 Overhauser,28and “optimal”29 attenuators are also interesting, and our ap-proach below is equally amenable to these.In this Communication, we show that the short-rangepotential Vr of a smooth density r is governed by ageneralized Poisson equation GPE and if Su decays atleast exponentially, Vr is locally related to r. We thenillustrate this by constructing several density functionals forthe short-range self-interaction energy. These can be ex-ploited in any field where Newtonian potentials arise, includ-ing electronic structure, molecular dynamics, classical elec-trostatics, fluid mechanics, and cosmology.II. GENERALIZED POISSON EQUATIONBy the convolution theorem, the short-range potentialEq. 2 can be expressed asVr = 4 k̂keik·rdk , 6wherek = 0u2j0kuSudu , 7̂k = 2−3 reik·rdr 8are the Fourier transforms of Su and r, respectively, andj0x is a spherical Bessel function. We note that k is aneven function.If r is analytic, then ̂k decays exponentially,30 andwe can substitute the Laurent expansionk = 	nnkn 9into Eq. 6, swap the integration and summation, and use theFourier derivative theorem to obtainVr = 4 	nnkn̂keik·rdk= 4	nni  nr = 4i  r . 10Equation 10, a GPE, is the key result of this Communica-tion. It provides an explicit relationship between a densityaElectronic mail: peter.gill@anu.edu.au.THE JOURNAL OF CHEMICAL PHYSICS 128, 241101 20080021-9606/2008/12824/241101/4/$23.00 © 2008 American Institute of Physics128, 241101-1Downloaded 27 Jul 2008 to 150.203.35.38. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jspr and its short-range potential Vr, assuming that the den-sity is smooth. Furthermore, we note that Eqs. 2 and 10imply the formal identitySu  4i  u . 11In the unattentuated limit i.e., =0, whereSu = u−1 ⇔ k = k−2, 12the GPE reduces toVr = −42r , 13which is the familiar Poisson equation. In contrast, if Sudecays at least exponentially, we can writek = 0u2j0kuSudu = 	2 −	43!k2 +	65!k4 − ¯ ,14where 	 j is the jth moment of S. If we substitute this into theGPE, we obtain the local expansionVr = 4	2 + 	43! 2 + 	65! 4 + ¯  , 15which, unlike the Poisson equation Eq. 13, expresses thepotential directly as a linear combination of the density andits hyper- Laplacians.An immediate corollary of Eq. 15 is that the short-range self-interaction energy of a density r can be con-verted from the usual double integralE =12  rSurdrdr 16into a density functionalE = 2 	22 + 	43! 2 + 	65! 4 + ¯ dr 17of the gradient expansion type.Equation 17 is related to Eq. C18 in the work ofToulouse et al.,8 but whereas Toulouse et al. stated that theirsis an asymptotic series, we show below that Eq. 17 is aconvergent series if an appropriate attenuator is chosen.III. ATTENUATORS AND EXPANSIONSThe Maclaurin series Eq. 14 is easy to find for manySu and local expansions for Vr and E immediately fol-low. For example, the Yukawa attenuator Eq. 3 yieldsYk =12−k24+k46− ¯ , 18VYr = 4 2+24+46+ ¯  , 19EY = 2  22+24+46+ ¯ dr , 20the Ewald attenuator Eq. 4 yieldsEk =11!22−k22!24+k43!26− ¯ , 21VEr = 4 1!22 + 22!24 + 43!26 + ¯  , 22EE = 2  21!22 + 22!24 + 43!26 + ¯ dr ,23and the Heaviside attenuator Eq. 5 yieldsHk =12!2−k24!4+k46!6− ¯ , 24VHr = 4 2!2 + 24!4 + 46!6 + ¯  , 25EH = 2  22!2 + 24!4 + 46!6 + ¯ dr . 26Each of formulas 20, 23, and 26 expresses theshort-range self-interaction energy as a local density func-tional and, in cases where the function Su is very shortranged compared with the density r, they offer a naturalalternative to the conventional double integral Eq. 16.IV. RANGE OF APPLICABILITYAlthough the expansions above for the potential and en-ergy are formally exact, they are not guaranteed to convergein all situations.First, they will systematically diverge if we unwiselychoose a function Su that is long ranged compared with thedensity. Consider, for example, the Gaussian densityr = 2/3/2 exp− 2r2 . 27It is not difficult to show that 2ndr = − 1n23/22n + 2!n + 1!, 28and the Yukawa, Ewald, and Heaviside expansions areEY =1 2!1!2 − 4!2!4 + 6!3!6 − ¯  , 29EE =13/21!2−5/22!4+7/23!6− ¯  , 30EH =1 11!2 − 12!4 + 13!6 − ¯  , 31respectively. Because the Yukawa attenuator decays only ex-ponentially, it is longer ranged than the density and its seriesdiverges for all . For similar reasons, the Ewald series isconvergent if, and only if, 1. The Heaviside attenuator,in contrast, has compact support, and its series therefore con-verges for any value of .241101-2 Varganov, Gilbert, and Gill J. Chem. Phys. 128, 241101 2008Downloaded 27 Jul 2008 to 150.203.35.38. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jspSecond, if r contains singularities, the swap of inte-gration and summation in Eq. 10 cannot be justified andexpansions 15 and 17 are no longer exact. This is illus-trated by the exact hydrogen atom densityr = 1/exp− 2r , 32which has a cusp singularity at r=0 and for which2 2ndr = 1 − 2n22n−2. 33Using the Heaviside attenuator Eq. 5, the true short-rangeenergy Eq. 16 can be shown to beEH =516−  516+58+122+163exp− 2 , 34which yields the correct convergent series expansionEH =182−1244+1366−83157+1728− ¯ . 35In contrast, Eq. 26 gives the incorrect expansionEH =182−1244−1606−15048−1810010− ¯ .36Because the cusp in r produces an O−6 error, the GPEis probably less useful for such densities.Of course, approximate cuspless hydrogen atom densi-ties present no such problems. If the Heaviside attenuatorwith =1 is applied to the HF/STO-3G density of the Hatom, the sum of the first 16 terms of Eq. 17 is correct toone part in 108.V. LARGER SYSTEMSWe have examined sums of K identical Gaussiansgar = 2/3/2 exp− 2r − Ra2 37arranged in linear d=1, square d=2, and cubic d=3lattices, each with lattice constant c.To find the short-range self-interaction energy of suchdensities via Eq. 17, one needs the one-electron integral gar2ngbrdr = H2n+1R23/2R exp− R2 38between two Gaussians with Ra−Rb=R, where Hnx is aHermite polynomial. We note that Eq. 38 reduces to Eq.28 when R=0.For comparison, the energy can also be found via Eq.16 using the double integralIR =  garSugbrdrdr . 39The standard Fourier transform approach22 yieldsIR =20k2j0kRexp− k2/4kdk , 40which can be evaluated in terms of the error function foreach of the attenuators in Eqs. 3–5.Truncation of Eq. 17 after a finite number N+1 ofterms yields an approximation Eapprox whose relative error depends on the attenuator Su, the attenuation strength ,the number K of Gaussians, the lattice constant c, and thedimensionality d of the lattice.Figure 1 shows the variation of  with N for the threeattenuators for fixed , K, c, and d. For Gaussian densities,the Yukawa expansion Eq. 20 is an asymptotic serieswhich, after initially converging, eventually diverges. In con-trast, the Ewald expansion Eq. 23 exhibits exponentialconvergence and the Heaviside expansion Eq. 26 con-verges superexponentially. Evidently, the attenuators’ conver-gence behaviors for 125 Gaussians follow exactly the pat-terns observed in Eqs. 29–31 for a single Gaussian.Figure 2 shows the variation of  with c for three attenu-ation rates for fixed Su, K, d, and N. We note that varyingc with a fixed Gaussian exponent is equivalent to varying theexponent with fixed c. At the two extremes c=0 where thedensity is a single Gaussian and c= where it is K noninter-acting Gaussians, the same relative error is obtained. For cvalues comparable to the inverse square root of the Gaussianexponent in Eq. 37, the accuracy improves somewhat be-cause the density is approximately uniform.Figure 3 shows the variation of  with K for the threeFIG. 1. Variation of  with N for SY ,=6.0, SE ,=2.0, andSH+,=1.0 using K=125, c=1.0, and d=3.FIG. 2. Variation of  with c for =1.0 , 2.0 , and 3.0  using SE,K=125, d=3, and N=5.241101-3 Generalized Poisson equation J. Chem. Phys. 128, 241101 2008Downloaded 27 Jul 2008 to 150.203.35.38. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsplattices for fixed Su, c, and N. The relative error decreasestoward a limiting value as K increases, which is consistentwith the fact that the short-range energy is an extensivequantity. The limiting error is reached most rapidly in thelow-dimensional systems, where the number of neighboringGaussians is small.VI. CONCLUDING REMARKSWe have introduced a GPE that relates a smooth densityr to its short-range Newtonian potential Vr. In caseswhere the attenuation is exponential or faster, the GPE pro-vides a local mapping between r and Vr and can besolved without integration. Furthermore, provided that theattenuation is short ranged compared with the density, theseries for Vr is convergent.The GPE will be useful in next-generation electronicstructure methods that require the short-range part of theCoulomb potential because it is simpler than previous ap-proaches and its cost is comparable to the calculation of thedensity itself. In nonconducting systems where the densitymatrix is sparse, this cost will scale linearly withsystem size. It may also be helpful in constructingimproved short-range density functionals in the context ofthe various long-range-corrected density functional theorymethods.5,8–10,16–18,20ACKNOWLEDGMENTSS.A.V. thanks ANU’s Research School of Chemistry fora postdoctoral fellowship.1 I. Panas, Chem. Phys. Lett. 245, 171 1995.2 A. Savin, Recent Developments of Modern Density Functional TheoryElsevier, Amsterdam, 1996.3 J. P. Dombroski, S. W. Taylor, and P. M. W. Gill, J. Phys. Chem. 100,6272 1996.4 P. M. W. Gill, R. D. Adamson, and J. A. 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Variation of  with K for d=1 , 2 , and 3  using SE, =2.0, c=1.0, and N=5.241101-4 Varganov, Gilbert, and Gill J. Chem. Phys. 128, 241101 2008Downloaded 27 Jul 2008 to 150.203.35.38. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

A generalized Poisson equation and short-range self-interaction energies

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