The Poisson equation in density fitting for the Kohn-Sham Coulomb problem

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER NOVEMBER 2001 The Poisson equation in density fitting fo the Kohn-Sham Coulomb poblem F. R. Manby a) School of Chemisty, Univesity of Bistol, Cantock s Close, Bistol BS8 1TS, United Kingdom P. J. Knowles b) and A. W. Lloyd School of Chemisty, Univesity of Bimingham, Edgbaston, Bimingham B15 2TT, United Kingdom Received 18 July 2001; accepted 10 Septembe 2001 A new density fitting appoach to the Coulomb poblem in Kohn Sham and Hatee Fock theoy is intoduced. Almost all of the 2- and 3-index epulsion integals become simple ovelap-like integals, without appoximation. The method is tested on numeous benchmak poblems, which eveal that accuacy equal to o bette than standad density fitting can be achieved with the evaluation of aound a tenth of the numbe of Coulomb integals. The scaling popeties of the method ae illustated fo polyalanine helices up to Ala Ameican Institute of Physics. DOI: / I. INTRODUCTION A bottleneck in taditional implementations of Kohn Sham KS theoy 1,2 is the evaluation of the electon epulsion integals ERIs, necessay fo the evaluation of the Coulomb contibution to the Fock matix. The electonic density in KS and Hatee Fock HF theoy is expanded in a poduct basis, *, 1 so the Coulomb enegy, E 1 2 d 1 d 2 1 2, 2 whee 1 2, has to be constucted fom 4-index ERIs. Fomally, this is an O(N 4 ) computational pocess, but since the density matix is spase, with O(N) nonzeo elements fo lage molecules, the numbe of integals to be evaluated actually scales as O(N 2 ). Despite huge advances in integal evaluation technology, these integals ae still time-consuming to compute, and the Coulomb enegy evaluation is the main bottleneck fo lage calculations. It is possible, howeve, to avoid the evaluation of 4-index ERIs altogethe. This is achieved by the intoduction of an auxiliay basis A in which one constucts an appoximate density, ˇ A d A A. 3 Then the Coulomb enegy aising fom the inteaction of ˇ with itself only equies 2-index ERIs, and has the fom whee Ě 1 2 d T Jd, a Electonic mail: Fed.Manby@bistol.ac.uk b Electonic mail: P.J.Knowles@bham.ac.uk 4 J AB d 1 d 2 A 1 B 2, 5 and whee d is the coefficient vecto that minimizes the eo in the auxiliay density. Thee ae seveal ways to measue this eo, geneally having the fom 1 2 d 1 d 2 ˇ 1 Ŵˇ 2, and diffeing though the choice of the weight opeato Ŵ. Least squaes fitting pefomed by choosing Ŵ( ) at fist appeas attactive, since the only 3-index integals it involves ae ovelaps, not Coulomb integals. Howeve, the pefomance of least squaes fitting has been found to be vey unsatisfactoy, both by othes 3 and in ou own investigations. The weight opeato that shows good convegence of enegies and othe popeties with espect to 1 auxiliay basis set is Ŵ leading to the eo expession 3, d 1 d 2 ˇ 1 ˇ 2. 7 The eo is minimized with espect to the coefficients d in the auxiliay basis by setting d 0, and this leads to the linea equations B J AB d B I A,, 8 whee I is a 3-index matix of ERIs of the fom I A, d 1 d 2 A 1 2 * 2. 9 This fitting of the density consideably educes the pefacto in the evaluation of the Coulomb contibution to the Fock matix, but the scaling emains O(N 2 ) owing to the long-anged natue of the Coulomb inteaction. To achieve a /2001/115(20)/9144/5/$ Ameican Institute of Physics

2 J. Chem. Phys., Vol. 115, No. 20, 22 Novembe 2001 Poisson equation in density fitting 9145 linea-scaling method eithe in Coulomb fitting o taditional 4-index integal methods, the asymptotic multipola fom of the Coulomb inteaction between pais of chage distibutions can be exploited. 5 9 The sepaation of long- and shot-ange Coulomb effects can be futhe efined by patitioning the Coulomb potential using an eo function, 10 1 ef efc. 10 These methods ae all based on analytical six-dimensional integals, but an altenative set of methods exist in which the Coulomb potential is built on a quadatue gid, pio to numeical integation with the density to obtain the Coulomb enegy, o with obital pais to obtain the Coulomb contibution to the Fock matix The Coulomb potential is given by the thee-dimensional integal v 1 2 d 2, 11 which has to be pefomed at each gid point 1. The Coulomb potential also satisfies the Poisson equation, Pˆ v, 12 whee Pˆ (4) 1 2, and one can solve this diffeential equation fo v, avoiding the evaluation of any ERIs at all. Howeve, a compaison of methods fo computing the Coulomb potential on a gid evealed that it was moe efficient to evaluate the integal of Eq. 11 than to solve the Poisson equation. 11 Nonetheless in this wok we einvestigate the possibility of using the Poisson equation in Coulomb fitting. It is woth pointing out that although the exact exchange enegy in HF and hybid KS theoies is, like the Coulomb enegy, constucted fom 4-index ERIs, the spasity of the density matix can be exploited to constuct an efficient linea-scaling method. Thus the Coulomb poblem is a bottleneck even in HF theoy, and we expect the cuent wok to be of use in that context as well. II. THEORY In a pevious pape 14 two of us wote down a density fitting method based on the Poisson equation. Hee we efine the method and appoach the poblem in a slightly diffeent way. We stat off fom the integal identity 15 Pˆ f 2 d 2 f 1, which holds exactly fo any f () that vanishes moe quickly than 1 as. Equation 13 can be obtained by inseting the Poisson equation Eq. 12 into the Coulomb potential expession in Eq. 11. Howeve, we meely note hee that Eq. 13 is an exact elation that implies a means of avoiding the toublesome six-dimensional, long-ange Coulomb integals. To apply the identity we will set up a density fitting method by expanding the auxiliay electonic density ˇ in a set of functions of the fom Pˆ A, which we will call Poisson functions, whee A is a Gaussian and theefoe satisfies the equiement of vanishing at long ange faste than 1 ), ˇ A d A Pˆ A. 14 Witing down the Coulomb enegy in exactly the standad way Eq. 2 we have Ě 1 2 AB d A d B d 1 d 2 Pˆ 1 A 1Pˆ 2 B 2, 15 whee Pˆ i acts on functions in the coodinates i. Inseting Eq. 13 and using the Hemiticity of Pˆ, the enegy educes to Ě 1 d 2 A d AB B d 1 A 1Pˆ 1 B This equation is inteesting because it gives an exact expession fo the Coulomb enegy of the density ˇ using only shot-ange thee-dimensional integals, which diffe fom kinetic enegy integals only by a facto of (2) 1. Thee is a catch in all this. Conside a multipole of the density Pˆ A (), q A lm dy lm ˆ l Pˆ A. 17 The function A () vanishes asymptotically so we can use integation by pats to apply 2 to the left, and then since 2 Y lm (ˆ) l 0 we see that q A lm 0 fo all l,m. In othe wods the fitted density in Eq. 14 can have no total chage, no total dipoles, and so on. In ou ealie pape 14 we alleviated the fist poblem that of the vanishing chage by consideing the electons and nuclei of a neutal system simultaneously. Hee we genealize the method by consideing the addition of a small numbe of odinay basis functions to the expansion in Eq. 14. These functions seve to descibe the total chage and highe multipoles of ˇ, and to give some goss appoximation to the density. The Poisson functions seve to move chage aound and poduce an accuate model density. We will now set up density fitting in a basis of m C standad and m P Poisson functions, following Eqs The Coulomb matix J now blocks into thee types of integals see Fig. 1, having in the integand zeo, one and two instances of the Poisson opeato. These integals ae espectively standad Coulomb integals, standad ovelaps, and the scaled kinetic enegy-like integals of Eq. 16. The 3-index integals block into m C m(m1)/2 Coulomb integals and m P m(m1)/2 ovelaps, whee m is the size of the atomic obital basis. Since m C is small, by fa the geate popotion of the 2- and 3-index integals ae shot-anged, and the S and P blocks in Fig. 1 ae spase. Once the Coulomb matices in the mixed basis have been set up, density fitting can be pefomed in exactly the nomal way. III. BASIS SETS AND TEST CASES To test the method, it is necessay to optimize mixed basis sets. Fo the peliminay calculations pesented hee,

3 9146 J. Chem. Phys., Vol. 115, No. 20, 22 Novembe 2001 Manby, Knowles, and Lloyd FIG. 1. Blocking of 2- and 3-index integals in mixed Poisson density fitting. The AO basis is of size m and thee ae m C and m P standad and Poisson fitting functions, espectively. The blocks ae labeled J standad Coulomb integals, S standad ovelap integals, and P scaled kinetic enegy integals. we optimize basis sets fo the elements of the fist ow of the peiodic table. Ou pocedue fo doing so closely follows that of Eichkon et al., 16 and we choose to use thei 3-paamete extension of even-tempeed seies of exponents Eq. 27 of Ref. 16. The optimizations ae pefomed using a Powell minimize fom seveal stating guesses. The egula way in which the optimal paametes vay acoss the peiodic table makes the optimizations inceasingly easy. Ou pocedue fo optimizing a basis fo a given atom was as follows: 1 Optimize a minimal basis of standad s-type functions fo the isolated atom; 2 Optimize Poisson functions s and in some cases d-type functions fo the isolated atom; 3 Simultaneously optimize a single, standad p function and a set of Poisson p, f, and possibly d functions fo the hydide. The size of bases was chosen to obtain eos in the Coulomb enegy below hatee fo atoms and hatee pe atom fo molecules. The basis sets fo C, N, O, and F have 2s 1p functions contacted to 1s 1p fo the standad basis and 6s 3p 6d 1 f fo the Poisson basis. These TABLE I. LDA bond lengths, dipole moments, and hamonic fequencies of gound states of fist-ow diatomics using an exact Coulomb teatment and the appoximation intoduced in this wok. Aveage eos ae povided in the final ow. Diatom /boh z /D /cm 1 Exact This wok Exact This wok Exact This wok H LiH BeH BH CH NH OH HF LiF BeF BN BO CN CO N NO NF O F Avg. eo bases ae somewhat lage than those used in standad density fitting, but the integal evaluation is easie, and, as we shall show, the numbe of integals to compute is much smalle. To test the accuacy of the method, we have computed the bond lengths, dipole moments and hamonic fequencies of the gound states of 19 fist-ow diatomics. This and all tests ae based on compaisons between LDA calculations 17 using the cc-pvdz atomic obital basis sets of Dunning. 18 The esults and aveage eos ae given in Table I. The eos in bond lengths and dipole moments ae consistently extemely small, and the fequencies ae geneally epoduced to within 1 cm 1. Ou second test examines the accuacy of the method in computing enegy diffeences of diffeent magnitudes. We theefoe compute the dissociation enegy using fixed geometies of benzene into thee acetylene molecules, the singlet tiplet splitting of methylene and the otational baie of ethane. The values and eos ae given in Table II. Although the elative eos do incease as the quantity being computed deceases, the smallest enegy the otational baie of ethane is in eo by only 0.8% in the cuent method. A futhe enegy diffeence that povides a test of the method is that between the zwitteionic and neutal foms of glycine. Table III shows esults using an exact Coulomb teatment, standad density fitting with the basis sets of Eichkon et al. 16 and the cuent method. The pefomance is TABLE II. Enegy diffeences of thee diffeent magnitudes computed using an exact Coulomb teatment and this wok. The thee cases ae the dissociation of benzene into thee acetylene molecules, the singlet/tiplet splitting of methylene, and the otational baie of ethane. The eos ange between 0.07% and 0.8% of as the size of the computed quantity deceases. Exact E/hatee This wok Eo/10 3 C 6 H 6 3C 2 H 2 C 6 H C 2 H Dissociation Singlet/tiplet CH 2 Singlet Tiplet Splitting Baie of C 2 H 6 Eclipsed Staggeed Baie hatee

4 J. Chem. Phys., Vol. 115, No. 20, 22 Novembe 2001 Poisson equation in density fitting 9147 TABLE III. Enegies and dipole moments of zwitteionic and neutal foms of glycine using an exact Coulomb teatment, standad density and the method of this wok, along with the eos in computed values. Clealy the enegy diffeence between the two foms is teated moe accuately in the cuent wok than in standad density fitting, and the dipoles in the two appoaches ae oughly equal in accuacy. Exact Standad This wok Zwitteion E/hatee x /boh y /boh z /boh Neutal E/hatee x /boh y /boh z /boh E/kJ mol vey good less than 0.05% eo in the isomeization enegy, and, pehaps fotuitously, significantly bette than that obtained with the standad fitting basis of Ref. 16. The esults also show that molecula popeties such as the dipole moment, which povide a moe stingent test of the faithfulness of the density fitting in egions of space not stongly weighted in the fitting eo functional, can be epoduced with simila accuacy in the cuent and standad methods. Ou final set of tests egad the numbe of integals computed in the cuent method compaed to standad density fitting methods. We conside polyalanine helices with up to 16 amino acids, and count the numbe of pimitive 3-index integals that have to be computed in the standad and cuent density fitting appoaches. Integals ae sceened with a theshold of 10 8 hatee. The esults ae shown in Fig. 2. The numbe of Coulomb integals evaluated in the cuent method emains consistently aound 10% of the numbe in standad Coulomb fitting, and the numbe of 3-index ovelaps ises only linealy. Even fo the smallest 4 peptide molecule, the total numbe of integals is less than with the standad basis, and the vast majoity of these ae of the S type, athe than the moe expensive J. Since the numbe of non-zeo S integals gows only linealy with system size, fo the lagest Ala 16 ) system, the integal evaluation poblem is consideably educed, and we ae still in the egime whee the O(N 2 ) numbe of J integals is less than the O(N) numbe of S integals. IV. DISCUSSION We have intoduced a new density fitting method fo the Coulomb poblem in KS and HF theoy. Most of the functions in the auxiliay basis having the fom Pˆ A (), and the Coulomb integals involving one o two of these functions become exactly equivalent to simple ovelap-like integals. This allows us to educe the numbe of Coulomb integals in density fitting by a facto of aound 10. A small numbe of standad functions is equied in the auxiliay basis since the functions Pˆ A () contain no chage and have vanishing multipoles. They do, howeve seve to move density aound the molecule to povide an accuate auxiliay density. Futhe savings can be made. Since the standad basis in this method is only pesent to give a vey ough appoximation to the density, we will eoptimize the basis sets using the same exponent fo the moe diffuse s function and the p function. Since none of the functions in the standad basis needs to be diffuse, the multipole appoximation will apidly take ove, and a vey small numbe of tue Coulomb integals will have to be evaluated. Fo vey lage systems, the bottleneck in standad density fitting is the dense linea algeba need to fom d. Inthe cuent method only the J-block of the Coulomb matix see Fig. 1 is dense, and this amounts only to aound 1% of the whole matix. Theefoe even fo lage systems, the linea algeba to be pefomed is effectively spase. Natually thee comes a point whee the dense pat of the matix becomes too lage, but the cuent method delays that theshold until the system is oughly ten times lage. Despite the geat savings in tems of the numbe of integals to be computed, and the fact that most of them ae ovelap integals, the method is accuate. Extensive tests on enegies, enegy diffeences, bond lengths, dipoles, and hamonic fequencies eveals that the cuent method is equal o supeio in accuacy to standad density fitting methods. Also accuacy can be inceased by the addition of futhe Poisson functions at elatively little cost, as these only incu the evaluation of ovelap integals, almost all of which vanish. ACKNOWLEDGMENTS The authos would like to thank D. John Wilkie fo poviding the coodinates fo the alanine helices, and one of the authos F.R.M. is gateful to the Royal Society fo financial suppot. FIG. 2. Numbe of 3-index integals in density fitting calculations on polyalanine helical peptides. The lines ae the numbe of Coulomb integals in standad density fitting cicles and in the cuent wok cosses, and the numbe of 3-index ovelap integals in the cuent wok squaes. 1 W. Kohn and L. J. Sham, Phys. Rev. A140, R. G. Pa and W. Yang, Density-Functional Theoy of Atoms and Molecules Oxfod Univesity Pess, Oxfod, O. Vahtas, J. Almlöf, and M. W. Feyeeisen, Chem. Phys. Lett. 213, B. I. Dunlap, Phys. Chem. Chem. Phys. 2, L. Geengad and V. Rokhlin, J. Comput. Phys. 73, L. Geengad, Science 265,

5 9148 J. Chem. Phys., Vol. 115, No. 20, 22 Novembe 2001 Manby, Knowles, and Lloyd 7 C. A. White, B. G. Johnson, P. M. W. Gill, and M. Head-Godon, Chem. Phys. 230, M. C. Stain, G. E. Scuseia, and M. J. Fisch, Science 271, P. M. W. Gill, Chem. Phys. Lett. 270, J. P. Domboski, S. W. Taylo, and P. M. W. Gill, J. Phys. Chem. 100, V. Temath and N. C. Handy, Chem. Phys. Lett. 230, E. J. Baeends, D. E. Ellis, and P. Ros, Chem. Phys. 2, C. Fonseca Guea, J. G. Snijdes, G. te Velde, and E. J. Baeends, Theo. Chem. Acc. 99, F. R. Manby and P. J. Knowles, Phys. Rev. Lett. 87, P. M. Mose and H. Feshbach, Methods of Theoetical Physics McGaw Hill, New Yok, 1953, Sec K. Eichkon, O. Teutle, H. Öhm,M.Häse, and R. Ahlichs, Chem. Phys. Lett. 240, S. J. Vosko, L. Wilk, and M. Nusai, Can. J. Phys. 58, T. H. Dunning, J., J. Chem. Phys. 90,