Continuum states from time-dependent density functional theory

Transcription

1 THE JOURNAL OF CHEMICAL PHYSICS 1, Cotiuum states from time-depedet desity fuctioal theory Adam Wasserma Departmet of Chemistry ad Chemical Biology, Rutgers Uiversity, Piscataway, New Jersey Neepa T. Maitra Departmet of Physics ad Astroomy, Huter College of the City Uiversity of New York, New York 1001 Kiero Burke Departmet of Chemistry ad Chemical Biology, Rutgers Uiversity, Piscataway, New Jersey Received 1 December 004; accepted 31 Jauary 005; published olie 1 April 005 Liear respose time-depedet desity fuctioal theory is used to study low-lyig electroic cotiuum states of targets that ca bid a extra electro. Exact formulas to extract scatterig amplitudes from the susceptibility are derived i oe dimesio. A sigle-pole approximatio for scatterig phase shifts i three dimesios is show to be more accurate tha static exchage for siglet electro-he + scatterig. 005 America Istitute of Physics. DOI: / I. INTRODUCTION Groud-state desity fuctioal theory DFT Refs. 1 ad has become a popular electroic structure method i both quatum chemistry ad solid-state physics, because moder approximatios produce useful accuracy at moderate computatioal cost. 3,4 Now, electroic excitatio eergies of atoms ad molecules are beig calculated usig liear respose time-depedet desity fuctioal theory TDDFT. 5,6 I this scheme, boud boud trasitio eergies are first approximated by the poles of the frequecydepedet Koh Sham KS desity respose fuctio, 6 8 ad the corrected to the poles of the true respose fuctio, i.e., the true excitatios. Boud cotiuum trasitios, however, have ot bee treated i the same way because brach cuts of the KS ad iteractig respose fuctios overlap, ad because it is the phase shifts, rather tha the eergies, that are of iterest i the scatterig regime. Eve though photorespose was addressed i the early days of TDDFT, 9 ad there has bee a log iterest of usig desity fuctioal methods for the scatterig problem e.g., Ref. 10, there is o formal theory based o TDDFT to study electro scatterig. Such a theory might be particularly useful i the emerget field of electro-impact chemistry, 11 i which large targets are struck by low eergy electros, so that boud-free correlatios are sigificat. 1 Several results relevat to this goal are preseted here. First, we provide a proof of priciple: the time-depedet respose of a N-electro groud state cotais the scatterig iformatio for a electro scatterig from the N 1-electro target, ad this is accessible via TDDFT. Secod, we show how this leads to practical ways of calculatig scatterig phase shifts or, i oe dimesio, trasmissio amplitudes. Fially, i the simplest case, siglet scatterig from He +, we fid that TDDFT yields better results tha static exchage, demostratig its higher accuracy at low computatioal cost. Although we are ot presetig a complete theory of electro scatterig withi TDDFT, such a theory ca be built upo the rigorous results preseted here, ad become a competitive alterative to existig techiques for calculatig electro-molecule scatterig cross sectios e.g., Ref. 13. Sice cotiuum states are the curret-carryig states i molecular electroic devices, we also aticipate applicatios of our oe-dimesioal 1D results i the field of electroic trasport through molecular wires. 14 II. EXTRACTING SCATTERING INFORMATION FROM THE SUSCEPTIBILITY A. Theory Our startig poit is the Dyso-like respose equatio that relates the susceptibility r,r; of a system of N iteractig electros with that of its groud-state KS aalog, s r,r;. 6 I operator form idicates spatial ad spi covolutio: = s + s f HXC, where f HXC is the Hartree-exchage-correlatio kerel we use atomic uits throughout: 1 t t f HXC r,r;t t + v XCr,t, r r r,t a fuctioal of the N-electro groud-state desity r. I Eq., v xc r,t is the time-depedet exchage-correlatio potetial iduced whe a time-depedet perturbatio is applied to the N-electro groud state. We write the spidecomposed susceptibility i the Lehma represetatio: /005/114/144103/5/$.50 1, America Istitute of Physics

2 Wasserma, Maitra, ad Burke J. Chem. Phys. 1, with r,r; = F r = 0 ˆ r, * F rf r + i0 + + cc, N ˆ r = r rˆi ˆ i, i=1 where 0 is the groud state of the N-electro system, its th excited state, ad ˆ r is the -spi desity operator. I Eq. 3, is the 0 trasitio frequecy. For the remaider of this sectio, we restrict the aalysis to oe dimesio. Cosider large distaces, where the N-electro groud-state desity is domiated by the decay of the highest occupied KS orbital; 15 the groud-state wave fuctio behaves as 16 0 N 1 0 x,...,x N x x N S 0,,..., N, N 1 where 0 is the groud-state wave fuctio of the N 1-electro system the target, S 0 the spi fuctio of the groud state, ad x the N-electro groud-state desity. Similarly, t x N 1 x,...,x N k x S,,..., N, N where N 1 t is a eigestate of the target labeled by t, S is the spi fuctio of the th excited state, ad k x a oeelectro orbital. We focus o elastic scatterig, so the cotributio to F x from chaels where the target is excited vaishes as x due to orthogoality. Isertig Eqs. 5 ad 6 ito the 1D versio of Eq. 4, ad takig ito accout the atisymmetry of both 0 ad, F x x xk x 0,t S * 0 N S N. 7 N The susceptibility at large distaces is the obtaied by isertig Eq. 7 ito Eq. 3: x,x; = x,x; xx x,x ± k x * k x + i 0, t S0,S + cc. 8 Sice oly scatterig states of the N-electro optical potetial cotribute to the sum i Eq. 8 at large distaces, it becomes a itegral over wave umbers k=, where is the eergy of the projectile electro: k x * k x + i 1 x,x ±0R,L k x k * x k + i dk. I this otatio, the fuctios k are box ormalized ad k x= k x/ L, where L is the legth of the box. The trasitio frequecy =E N E N N 1 0 is ow simply k =E 0 +k / E N 0 =k /+I, where I is the first ioizatio potetial of the N-electro system, ad E M 0 ad E M are the groud ad th excited state eergies of the M-electro system. The subscript R, L implies that the itegral is over both orbitals satisfyig right ad left boudary coditios: R L k x e±ikx + r k e ikx, x t k e ±ikx 10, x ±. Whe x ad x= x the itegral of Eq. 9 is domiated by a term that oscillates i space with wave umber I ad amplitude give by the trasmissio amplitude for spi-coservig collisios t k at that wave umber. Deotig this by osc, we obtai i t = lim x osc x, x; + I. x x 9 11 While this formula also applies to the KS system, its trasmissio t s ca be easily obtaied by solvig a potetial scatterig problem i.e., scatterig off the N-electro groudstate KS potetial. The exact amplitudes t of the maybody problem are formally related to the t s through Eqs. 11 ad 1. This is the mai result of this work: the timedepedet respose of the N-electro groud state cotais the scatterig iformatio, ad is accessible via TDDFT. A potetial scatterig problem is solved first for the N-electro groud-state KS potetial, ad the scatterig amplitudes thus obtaied are further corrected by f HXC to accout for, e.g., polarizatio effects. While Eq. 11 seems impractical as a basis for computatios, it leads to practical approximatios. For example, if Eq. 1 is iterated oce, we fid through Eq. 11 the followig useful distorted-wave-bor-type approximatio for the trasmissio amplitude: t = t s + 1 i HOMO,fˆHXC + IHOMO,, 1 where HOMO, is the product of the highest occupied KS orbital ad the cotiuum KS orbital of eergy. B. Example We illustrate o a simple 1D model of a electro scatterig from a oe-electro atom of uclear charge Z Ref. 17 i the weak iteractio limit:

3 Cotiuum states from time-depedet desity fuctioal theory J. Chem. Phys. 1, We ow apply Eq. 11 to show that the f HXC term of Eq. 1 corrects the t s values to their exact siglet ad triplet amplitudes. We eed f HXC oly to O, f HX x,x; = x x1, 19 where the f HXC of Eq. 1 is give to O by f HX = f H + f X = 1 4 f HX = 1 f H here. Eq. 19 yields x,x; = s x,x; + dx s x,x;x,x;. 0 FIG. 1. Real ad imagiary parts of the KS trasmissio amplitude t s, ad of the iteractig siglet ad triplet amplitudes to first order i, for the model system of Eq. 13. Z= ad =0.5 i this plot. Ĥ = 1 d dx 1 d 1 dx Zx 1 Zx + x 1 x. 13 Electros iteract via a -fuctio repulsio, scaled by. With =0 the groud state desity is a simple expoetial, aalogous to hydrogeic atoms i 3D. Exact solutio i the weak iteractio limit. First, we solve for the exact trasmissio amplitudes to first order i usig the static exchage method. 18 The results for triplet t trip ad siglet t sig scatterig are t trip = t 0, t 0 = ik Z + ik, 14 ik t sig = t 0 +t 1, t 1 = k iz k + iz. Our TDDFT solutio. The groud-state of the N-electro system N= is give to O by 0 x 1 1,x = 1 0 x 1 0 x 1 1, where the orbital 0 x satisfies 19,0 d 1 dx Zx + 0x 0 x = 0 x. To first order i, x = Ze Zx + e 8 3Zx + e Zx 4Zx Z The bare KS trasmissio amplitudes t s characterize the asymptotic behavior of the cotiuum states of v s x = Zx+ 0 x, ad ca be obtaied to O by a distorted-wave Bor approximatio see, e.g., Ref. 1: t s = t 0 + t The result is plotted i Fig. 1, alog with the iteractig siglet ad triplet trasmissio amplitudes, Eqs. 14. Sice the groud state of the N-electro system is a spi siglet, the Kroecker delta S0,S i Eq. 8 implies that oly siglet scatterig iformatio may be extracted from, whereas iformatio about triplet scatterig requires the magetic susceptibility M=, related to the KS susceptibility by spi TDDFT: Mx,x; = s x,x; dx s x,x;mx,x;. 1 For either siglet or triplet case, sice the correctio to s is multiplied by, the leadig correctio to t s is determied by the same quatity, ˆ 0 s ˆ 0 0 s, where ˆ s is the zeroth-order approximatio to the KS susceptibility i.e., with v s x=v 0 s x= Zx. Its oscillatory part at large distaces 3 multiplied by x x/ik, see Eq. 11 is equal to t 1. We the fid through Eqs. 11, 0, ad 1 that t sig = t s + t 1, t trip = t s t 1, i agreemet with Eqs. 14. III. SINGLE POLE APPROXIMATION IN THE CONTINUUM We have yet to prove a aalog of Eq. 11 for Coulomb repulsio i three dimesios. But here we use quatumdefect theory 4 to deduce the result at zero eergy. Cosider the l=0 Rydberg series of boud states covergig to the first ioizatio threshold I of the N-electro system: E E 0 = I 1/, 3 where is the quatum defect of the th excited state. Let = 1/ s, 4 be the KS orbital eergies of that series. The true trasitio frequecies =E E 0 are related through TDDFT to the KS frequecies s, = HOMO. Withi the sigle-pole approximatio SPA Ref. 6: = s, +HOMO,fˆHXC HOMO,. 5 Numerical studies 5 suggest that = s, is a small umber whe. Expadig aroud =0, ad usig I= HOMO,wefid

4 Wasserma, Maitra, ad Burke J. Chem. Phys. 1, FIG.. s-phase shifts as a fuctio of eergy for electro scatterig from He +. Dashed lies: the lie labeled KS correspods to the phase shifts from the exact KS potetial of the He atom; the other dashed lies correspod to the TDDFT siglet ad triplet phase shifts calculated i the preset work accordig to Eq. 8. Solid lies: accurate wave fuctio calculatios of electro-he + scatterig from Ref. 8. The solid lie i the ceter is the average of siglet ad triplet phase shifts. Dotted lies: Static exchage calculatios, from Ref. 9. The asterisks at zero eergy correspod to extrapolatig the boud boud results of Ref. 7. = s, / s, 3. We coclude that, withi the SPA, 6 = s, 3 HOMO,fˆHXC HOMO,. 7 Lettig, Seato s theorem lim = 0 + Ref. 4 implies = s HOMO,fˆHXC + IHOMO,, 8 a relatio for the phase-shifts i terms of the KS phaseshifts s applicable whe 0 +. The factor s, 3 of Eq. 7 gets absorbed ito the eergy-ormalizatio factor of the KS cotiuum states. We illustrate i Fig. the remarkable accuracy of Eq. 8 whe applied to the case of electro scatterig from He +. For this system, a essetially exact groud-state potetial for the N= electro system is kow. This was foud by ivertig the KS equatio usig the groud-state desity of a extremely accurate wave fuctio calculatio of the He atom. 6 We calculated the low-eergy KS s-phase shifts from this potetial, s dashed lie i the ceter, Fig., ad the corrected these phase shifts accordig to Eq. 8 employig a hybrid approximatio to f HXC Ref. 7 adiabatic local desity approximatio for atiparallel cotributio to f HXC ad exchage-oly approximatio for the parallel cotributio. We also plot the results of a recet highly accurate wave fuctio calculatio 8 solid, ad of static-exchage calculatios 9 dotted. The results show that phase shifts from the N-electro groud-state KS potetial s are a excellet approximatio to the average of the true siglet/triplet phase shifts for a electro scatterig from the N 1-electro target, just as i our oe-dimesioal model; they also show that TDDFT, with existig approximatios, works very well to correct scatterig from the KS potetial to the true scatterig phase shifts, at least at low eergies. I fact, for the siglet phase shifts, TDDFT does better tha the computatioally more demadig static exchage method, ad for the triplet case TDDFT does oly slightly worse. Eve though Eq. 8 is, strictly speakig, oly applicable at zero eergy marked with asterisks i Fig., it clearly provides a good descriptio for fiite low eergies. It is remarkable that the atiparallel spi kerel, which is completely local i space ad time, ad whose value at each poit is give by the exchage-correlatio eergy desity of a uiform electro gas evaluated at the groud-state desity at that poit, yields phase shifts for e-he + scatterig with less tha 0% error. Sice a sigature of desity-fuctioal methods is that, with the same fuctioal approximatios, exchage-correlatio effects are ofte better accouted for i larger systems, the preset approach holds promise as a practical method for studyig large targets. IV. CONCLUSION To summarize, we have show how, i oe-dimesio, scatterig amplitudes may be obtaied from TDDFT, ad deduced the results for three dimesios ear zero eergy for Coulombic systems. The ultimate goal is to accurately treat boud-free correlatio for low eergy electro scatterig from polyatomic molecules, with a computatioal cost lower tha that of static exchage. A obvious limitatio of the preset approach is that it ca oly be applied to targets tha bid a extra electro, ad there is much work yet to be doe: geeral proof of priciple i three dimesios, testig of the accuracy of approximate groud-state KS potetials, developig ad testig approximate solutios to the TDDFT Dyso-like equatio, extedig the methodology to cases where the aio has a sharp resoace rather tha a groud state, etc. ACKNOWLEDGMENTS The authors thak Michael Morriso for ispirig discussios. This work was supported by the Petroleum Research Fud uder Grat No AC6, U.S. Departmet of Eergy uder Grat No. DE-FG0-01ER4598, ad NSF uder Grat No. CHE P. Hoheberg ad W. Koh, Phys. Rev. 136, W. Koh ad L. J. Sham, Phys. Rev. 140, A R. M. Dreizler ad E. K. U. Gross, Desity Fuctioal Theory Spriger, Berli, W. Koch ad M. C. Holthause, A Chemist s Guide to Desity Fuctioal Theory Wiley-VCH, Weiheim, E. Ruge ad E. K. U. Gross, Phys. Rev. Lett. 5, M. Petersilka, U. J. Gossma, ad E. K. U. Gross, Phys. Rev. Lett. 76, M. E. Casida, i Recet Developmets ad Applicatios i Desity Fuctioal Theory, edited by J. M. Semiario Elsevier, Amsterdam, H. Appel, E. K. U. Gross, ad K. Burke, Phys. Rev. Lett. 90, A. Zagwill ad P. Sove, Phys. Rev. A 1, S. Tozai ad C. H. Greee, J. Chem. Phys. 1, G. Hael, B. Gstir, S. Deifl, P. Scheier, M. Probst, B. Farizo, E. Illeberger, ad T. D. Märk, Phys. Rev. Lett. 90, R. K. Nesbet, Phys. Rev. A 6, R C. Wistead ad V. McKoy, i Advaces i Chemical Physics, edited by I. Prigogie ad S. A. Rice Wiley, New York, 1996, Vol. XCVI, p. 103.

5 Cotiuum states from time-depedet desity fuctioal theory J. Chem. Phys. 1, M. Di Vetra ad N. D. Lag, Phys. Rev. B 65, J. Katriel ad E. R. Davidso, Proc. Natl. Acad. Sci. U.S.A. 77, M. Erzerhof, K. Burke, ad J. P. Perdew, J. Chem. Phys. 105, C. M. Rosethal, J. Chem. Phys. 55, B. H. Brasde ad C. J. Joachai, Physics of Atoms ad Molecules Logma, New York, E. H. Lieb, J. P. Solovej, ad J. Ygvaso, Phys. Rev. Lett. 69, R. J. Magyar ad K. Burke, Phys. Rev. A 70, H. Friedrich, Theoretical Atomic Physics Spriger, New York, 1991, Sec M. Petersilka ad E. K. U. Gross, It. J. Quatum Chem., Quatum Chem. Symp. 30, N. T. Maitra, A. Wasserma, ad K. Burke, i Electro Correlatios ad Materials Properties, edited by A. Gois, N. Kioussis, ad M. Cifta Kluwer Academic, Dordrecht, M. J. Seato, Mo. Not. R. Astro. Soc. 118, A. I. Al-Sharif, R. Resta, ad C. J. Umrigar, Phys. Rev. A 57, C. J. Umrigar ad X. Goze, Phys. Rev. A 50, K. Burke, M. Petersilka, ad E. K. U. Gross, i Recet Advaces i Desity Fuctioal Methods, edited by P. Fatucci ad A. Becii World Scietific, Sigapore, 000, Vol. III. 8 A. K. Bhatia, Phys. Rev. A 66, R. R. Lucchese ad V. McKoy, Phys. Rev. A 1,