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# 489 Class Note for CHEM C1260 at Purdue

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Date Created: 02/06/15

33 Scattering Amplitudes Al Wasserman and Ki Burke 331 Introduction Electrons are constantly colliding with atoms and molecules in chemical reactions in our atmosphere in stars plasmas in a molecular wire car rying a current or when the tip of a scanning tunneling microscope in jects electrons to probe a surface When the collision occurs at low ener gies the calculations become especially dif cult due to correlation effects between the projectile electron and those of the target These boundfree correlations are very important For example it is due to boundfree corre lations that ultraslow electrons can break up RNA molecules Hanel 2003 causing serious genotoxic damage The accurate description of correlation effects when the targets are so complex is a major challenge Existing approaches based on wavefunction methods developed from the birth of quantum mechanics and perfected since then to reach great sophistication Morrison 1983 Burke 1994 Winstead 1996 cannot overcome the exponen tial barrier resulting from the manybody Schr dinger equation when the number of electrons in the target is larger Wavefunction based methods can still provide invaluable insights in such complex cases provided powerful com puters and smart tricks are employed see eg Grandi 2004 for lowenergy electron scattering from uracil but a truly abinitio approach circumventing the exponential barrier would be most welcome The purpose of this chapter is to describe several results relevant to this goal Imagine a slow electron approaching an atom or molecule that has N electrons and is assumed to be in its ground state with energy Egsi The asymptotic kinetic energy of the incoming electron is 5 so the whole system of target plus electron has a total energy of Egg 5 This is an excited state of the N 1electron system and as such it can be described by the linear response formalism of TDDFT starting from the ground state of the N 1electron systemi We will explain howl The targets we will consider must be able to bind an extra electroni For example take the target to be a positive ion so that the N 1electron system with groundstate energy Eggl is neutral Previous chapters in this book have described how to employ TDDFT to calculate eigi excitation energies corresponding to bound A bound transitions from the ground state However in the scattering situation considered here the excitation energy is A Wasserman and K Burke Scattering Amplitudes Lect Notes Phys 706 4937507 2005 DOI 101007375407354257333 SpringerrVerlag Berlin Heidelberg 2005 494 A Wasserman and K Burke known in advance it is I 5 where I is the rst ionization energy of the N 1system I Egg 7 It is the scattering phase shifts rather than the energies which are of interest in the scattering regime The TDDFT approach to scattering that we are about to discuss Wasserman 2005b is very different from wavefunctionbased methods yet exact in the sense that if the groundstate exchangecorrelation potential um and timedependent exchangecorrelation kernel fxc were known exactly we could then in principle calculate the exact scattering phase shifts for the system of N 1 interacting electronsl Any given approximation to um and fXC leads in turn to de nite predictions for the phase shifts The method involves the following three steps Finding the groundstate KohnSham potential of the N 1electron system vN17 ii Solving a potential seatten39ng problem namely scattering from UK1T and iii Correcting the Kohn Sham scattering phase shifts towards the true ones via linear response TDDFTI We start by reviewing those aspects of the linear response formalism of TDDFT that were introduced in Chap 1 and will be used in the following sec tions We then derive TDDFT equations for onedimensional scattering and work out in detail two simple examples to show how to calculate transmis sion and reflection amplitudes in TDDFTI The discussion is then generalized to three dimensions where we explain how the familiar single pole approxi mation for bound A bound transitions can be continued to describe bound A continuum transitions to get information about scattering states We end with a brief summary and outlook 332 Linear Response for the N 1Electron System For a thorough treatment see Chap 1 Here we only review what will be needed for the following sections The central equation of the linear response formalism of TDDFT is the Dysonlike response equation relating the sus ceptibility XN17 7quotw of a system of interacting electrons with that of its groundstate KohnSham analog leggl 7quotw Petersilka 1996a see 1123 The N 1 superscript was added in order to emphasize that we are going to perturb the groundstate of the N 1electron system where N is the number of electrons of the target In what follows however for notational simplicity the N 1 superscript will be dropped from all quantities We write the spindecomposed susceptibility in the Lehman representation mam quot1331 v lt33 with N1 FwltTgtltwGsmltrgtwzgt mltrgt26lt lgt6m 332 l1 33 Scattering Amplitudes 495 Where was is the ground state of the N lelectron system Ll7i its 2 excited state and 7307 is the aspin density operator In 331 2 is the was A L17 transition frequency The term ccw A 7w stands for the complex conjugate of the rst term With w substituted by 7w The sum in 331 should be understood as a sum over the discrete spectrum and an integral over the continuum All excited states labelled by With non zero FiU 7 contribute to the susceptibility In particular the scattering state discussed in the introduction consisting on a free electron of energy 5 and an Nelectron target contributes too How to extract from the susceptibility the scattering information about this single state The question Will be answered in the following sections starting in one dimension 333 One Dimension 3331 Transmission Amplitudes from the Susceptibility Consider large distances Where the N lelectron groundstate density is dominated by the decay of the highest occupied Kohn Sham orbital Katriel 1980 the groundstate wavefunction behaves as Ernzerhof 1996 Mr N HSGS039703927A A A 70N1 333 N LPGS 10 Gs127 A A A 71N1 Where gs is the groundstate wavefunction of the target SGS the spin func tion of the Nlelectron groundstate and the Nlelectron ground state density Similarly the asymptotic behavior of the ith excited state is zN1q5 f 15iaag UN1 334 Where is an eigenstate of the target labeled by it 539 is the spin function of the ith excited state of the N l7system and 4ka a oneelectron orbital not to be confused with go notation reserved for Kohn Sham orbitals The contribution to Fi0z 332 from channels Where the target is ex cited vanishes as I A gt0 due to orthogonality We therefore focus on elastic scattering only lnserting 333 and 334 into the lD version of 332 and taking into account the antisymmetry of both was and Ll7i Fi0z 10 160 Z 5555 a UN1SZa UN1gt 335 020N1 The susceptibility at large distances is then obtained by inserting 335 into the lD version of 331 496 A Wasserman and K Burke xz 1w Z XUU z z w 1 11 UU ask 90 a X 1 6 6 cc w A 7w 336 Z w79iin 0 SGsSz Since only scattering states of the N lelectron optical potential con tribute to the sum in 336 at large distances it becomes an integral over wavenumbers k v25 where 5 is the energy of the projectile electron SW A 100 MM 33 w79iin 11Hioo RHL w79kin lnthisnotationthefunctionsqbkl are boxnormalized and qka 45k where L A gt0 is the length of the box The transition frequency 21 EiNJrli Eggl is now simply Qk Egg k22 7 Eg1 k22 I where I is the rst ionization potential of the N lelectron system and Egg and are the ground and 2 excited state energies of the M electron system The subscript RHLl indicates that the integral is over both orbitals satisfying ght and left boundary conditions eiik Tke z A 00 R A tkei1m When I A 700 and z 71 the integral of 337 is dominated by a term that oscillates in space with wavenumber 2V k2 7 21 and amplitude given by the transmission amplitude for spinconserving collisions tk at that wavenumber Denoting this by X05 and setting 5 k2 we obtain M272 warmer Therefore in order to extract the transmission amplitude t5 from the sus ceptibility when an electron of energy 5 collides with an Nelectron target in one dimension one should rst construct the groundstate density of the N lelectron system perturb it in the far left with frequency I 5 and then look at the oscillations of the density change in the far right the am plitude of these oscillations ampli ed77 by i n fll is the transmission amplitude t5 see Fig 331 The derivation of 339 does not depend on the interaction between the electrons Therefore the same formula applies to the Kohn Sham system i 25 hm Vx Ad VMMM I In practice the Kohn Sham transmission amplitudes th5 are obtained by solVing a potential scatten39ng problem ie scattering from the N l electron groundstate KS potential 338 zaioo Xoscz7z51 339 700 t5 Ehm 55 1715 I 3310 33 Scattering Amplitudes 497 Density change Ground7state density of N 1 sysmm Perturbation Fig 331 Cartoon of 339 To extract the transmission amplitude for an electron of energy E scattering from an Nelectron target apply a perturbation of frequency E I on the far left of the N lground state system I is its rst ionization energy and look at hoW the density changes oscillate on the far right Once am pli ed the amplitude of these oscillations correspond to tEi Reproduction from the roof of the Sistine Chapel with permission from artist Illustration of 3310 For one electron the susceptibility is given by Maitra 2003b szltzzneIgt vnltzgtnltwgt mm5 gm is 7 21gt 3311 Where the Green s function 9K5 z z 5 has a Fourier transform satisfying 9 l 92 71E 7 vaz ngzz t 7 t 7 7161 7 z 6t 7 t 3312 Let s nd the transmission amplitude for an electron scattering from a double deltafunction well van 7Z16z 7 Z26z 7 a The Green s function can be readily obtained in this case as Z291I7 a91a7 1 3313 1 Z291a7 a ngOmv 91171 7 where 91 is the Green s function for a single deltafunction of strength Z1 at the origin It is given by Szabo 1989 3314 1 Z eik zHiri 91171 g e klmim 1 7 ikZ1 With k Having constructed XKS explicitly application of 3310 yields the correct answer ik Z1 ik t i KS 1 229101709 3315 498 A Wasserman and K Burke 3332 TDDFT Equation for Transmission Amplitudes The exact amplitudes t5 of the manybody problem are formally related to the th 5 through 339 3310 and 123 the timedependent response of the N 1electron groundstate contains the scattering information and this is accessible via TDDFT A potential scattering problem is solved rst for the N1electron groundstate KS potential and the scattering ampli tudes thus obtained tKS are further corrected by fHXC to account for eg polarization effects Even though 339 is impractical as a basis for computations one can rarely obtain the susceptibility with the desired accuracy in the asymptotic re gions as we did in the previous example it leads to practical approximations The simplest of such approximations is obtained by iterating 123 once sub stituting X by XKS in the righthand side of 123 This leads through 339 and 3310 to the following useful distortedwave Born type approximation for the transmission amplitude t5 55 HOMO a foc8 IHOMO 5 3316 1 i25 In 3316 and from now on the doublebracket notation stands for ltltHOMOefoclte Igt1HOMOegtgt dz dz awome mm a Imam 1 In 3317 where HOMO is the highestoccupied molecular orbital of the N1electron system and 013 is the energynormalized scattering orbital of energy 5 satisfying Klboundary conditions see 338 This is reminiscent of the singlepole approximation for excitation energies of bound A bound transi tions 131 Many other possibilities spring to mind for approximate solu tions to 339 3333 A Trivial Example N 0 The method outlined above is valid for any number of particles In particular for the trivial case of N 0 corresponding to potential scattering Consider an electron scattering from a negative deltafunction of strength Z in one dimension Fig 332 The transmission amplitude as a function of 5 is given by see Sect 25 of Griffiths 1995 ik t 39 k 2 3318 a Z ik We lt gt How would TDDFT get this answer Find the groundstate KS potential of the N1 17electron system The external potential admits one bound 33 Scattering Amplitudes 499 2 90 Z 2 Fig 332 Left cartoon of an electron scattering from a negative deltarfunction potential Right cartoon of an electron bound to the same potential the ground state density decays exponentially just as in hydrogenic ions in 3D 2 Fig 333 Left cartoon of an electron scattering from 1DHe Right cartoon of A111 Please PYOVide two electrons bound to the delta function in a singlet state gure citation in text for Fig 333 state of energy 7Z22 The groundstate KS potential is given by vaz vex z 1ch1 but 1ch 0 for one electron7 so vaz vemz 7Z6z ii Solve the groundstate KS equations for positive energies7 to nd th5 ikZ ik iii In this case7 foc 07 so X XKS and t th Notice that approximations that are not selfinteraction corrected to guarantee 1ch 0 would give sizable errors in this simple case 3334 A NonTrivial Example7 N 1 Now consider a simple 1D model of an electron scattering from a oneelectron atom of nuclear charge Z Rosenthal 1971 H7 1 d2 1 d2 za za w 3319 772dz 72dz 7 11 7 I2 17127 The two electrons interact via a deltafunction repulsion7 scaled by A With A 0 the ground state density is a simple exponential7 analogous to hydro genic atoms in 3D 500 A Wasserman and K Burke i Exact solution in the weak interaction limit First we solve for the exact transmission amplitudes to rst order in A using the static exchange method Bransden 1983 The total energy must be stationary with respect to variations of both the bound 451 and scattering orbitals that form the spatial part of the Slater determinant bzl 5zg i bzgq 5zl where the upper sign corresponds to the singlet and the lower sign to the triplet case The staticexchange equations are 2 71mm 7 was Mr mum 3320 where 7 2A for the singlet and 0 for the triplet Thus the triplet transmis sion amplitude is that ofa simple 6function 3318 This can be understood by noting that in the triplet state the Hartree term exactly cancels the ex change the two electrons only interact when they are at the same place but they cannot be at the same place when they have the same spin from Paulils principle The results for triplet tmplet and singlet tsinget scattering are therefore ik Z ik t 7 7ik2 1 k 7 iZ2k 12 ii TDDFT solution We now show step by step the TDDFT procedure yielding the same result 3321 The rst step is nding the groundstate KS potential for two electrons bound by the 6function The groundstate of the N 1electron system N 1 is given to O by ttriplet 750 7 750 E 3321a tsanglet to 2M1 3321b 1 Wcs110171202 E GS11 GSI2l601T6021 501150211 7 3322 where the orbital gogsz satis es Lieb 1992 Magyar 2004b 1d2 75 7 Z6I Alsocsrl2 SDGSI MSDGS1 3323 To rst order in A spasm aw i yum e ZlEl4lel 7 3 3324 87 The bare KS transmission amplitudes tKS5 characterize the asymptotic behavior of the continuum states of vaz 7Z6z Algogsz12 and can be obtained to O by a distortedwave Born approximation see eg Sect 414 of Friedrich 1991 tKS to Atl 3325 33 Scattering Amplitudes 501 39 t c f smgle triplet 3 06 r E k 04 i interacting 02 7 KS 39 Imt Fig 334 Real and imaginary parts of the KS transmission amplitude th and of the interacting singlet and triplet amplitudes for the model system of 3319 Z 2 and A 05 in this plot Reprinted With permission from Wasserman 2005b Copyright 2005 American Institute of Physics The result is plotted in Fig 334 along With the interacting singlet and triplet transmission amplitudes 3321 The quantity Atl is the error of the ground state calculation The interacting problem cannot be reduced to scattering from the N lKS potential but this is certainly a good starting point in this case the KS transmission amplitudes are the exact average of the true singlet and triplet amplitudes compare 3325 With 3321 We now apply 339 to show that the focterm of 123 corrects the th values to their exact singlet and triplet amplitudes The kernel foc is only needed to OM fo UUzzw A6ltI 7 117 600 3326 Where the foc of 123 is given to O by fo fH fx i 200 fo 00 here Equation 3326 yields xzzw XKsI 1w g dz XKszzwxz zwi 3327 Since the ground state of the 2electron system is a spinsinglet the Kronecker delta 65Gssl in 336 implies that only singlet scattering information may be extracted from x Whereas information about triplet scattering requires the magnetic susceptibility M 200 00XUU related to the KS susceptibility by spinTDDFT Petersilka 1996b chw mm m 7 g dz szltzx wgtMltz znwgti 3328 502 A Wasserman and K Burke For either singlet or triplet case since the correction to XKS is multiplied by A the leading correction to th 5 is determined by the same quantity7 gs 96 gs where is the 0ch order approximation to the KS susceptibility ie7 with vaz 1135 7Z6z lts oscillatory part at large distances Maitra 2003b multiplied by nzn7zik see 339 is precisely equal to Atl We then nd through 339 3327 and 3328 that tsinglec th Air 7 ttriplet th A751 3329 in agreement with 3321 The method illustrated in the preceeding example is applicable to any one dimensional scattering problem Equations 339 and 123 provide a way to obtain scattering information for an electron that collides with an Nelectron target entirely from the N 1electron groundstate KS susceptibilty and a given approximation to fxc 334 Three Dimensions 3341 SinglePole Approximation in the Continuum We have yet to prove an analog of 339 for Coulomb repulsion in three dimensions But we can use quantumdefect theory Seaton 1958 to deduce the result at zero energy Consider the l 0 Rydberg series of bound states converging to the rst ionization threshold I of the N lelectron system E 7 Egg I 71 2239 7 ill2 3330 where M is the quantum defect of the ith excited state Let 5i 1 l2 Ksi2l 3331 be the KS orbital energies of that series The true transition frequencies Lul EZ 7 Egg are related through TDDFT to the KS frequencies wKS i 5i 7 EHQMQ where EHOMQ is the HOMO energy Within the singlepole approximation SPA Petersilka 1996a7 applicable to Rydberg excitations according to the criteria of applicability discussed in Appel 2003 w W3 i 2ltltH0Mo zquot focwiH0Mo igtgt 3332 Numerical studies AlSharif 1998 suggest that AM M 7 MKSJ is a small number whenz39 7 gt0 Expanding wi around AM 0 and using I 75HQMO we nd wi wKsi 7 AMn 7 MKS i3 3333 We conclude that7 within the SPA 33 Scattering Amplitudes 503 1 08 c 9 g 06 JES g avar 39g39 w quot 7a m 04 7 y singlet 02 c 0 l l l l 0 02 04 06 08 1 Ene gy H Fig 335 sphase shifts as a function of energy for electron scattering from He Dashed lines the line labeled KS corresponds to the phase shifts from the exact KS potential of the He atom the other dashed lines correspond to the TDDFT singlet and triplet phase shifts calculated in the present Work according to 3335 Solid lines accurate Wavefunction calculations of electronHeJr scattering from Bhatia 2002 The solid line in the center is the aVerage of singlet and triplet phase shifts Dotted lines Static exchange calculations from Lucchese 1980 The asterisks at zero energy correspond to extrapolating the bound gt bound results of Burke 2002 Reprinted With permission from refWasserman 2005b Copyright 20057 American Institute of Physics AM 722 7 MKsi3ltltHOMO 239 focwZHOMO 2 3334 Letting i A gt07 Seatonls theorem 7rlimiH00 M 65 A 0 Seaton 1958 implies 6a 6mg 7 27rltltHOMO efoce 1HOMO 5 3335 a relation for the phaseshifts 6 in terms of the KS phaseshifts 6K3 applicable When 5 A 0 The factor iiuKS i3 of 3334 gets absorbed into the energy normalization factor of the KS continuum states We illustrate in Fig 335 the remarkable accuracy of 3335 When applied to the case of electron scattering from He For this system7 an essentially exact groundstate potential for the N 2 electron system is known This was found by inverting the KS equation using the groundstate density of an extremely accurate wavefunction calculation of the He atom Umrigar 1994 We calculated the lowenergy KS s phase shifts from this potential7 6K5 5 dashed line in the center7 Fig 3357 and then corrected these phase shifts according to 3335 employing the BPG approximation to foc Burke 2002 which amounts to using the adiabatic local density approximation for the 504 A Wasserman and K Burke antiparallel contribution to foc and exchangeonly approximation for the parallel contributionl We also plot the results of a highly accurate wave function calculation Bhatia 2002 solid and of staticexchange calculations Lucchese 1980 dottedl The results show that phase shifts from the N l electron groundstate KS potential 6Ks5 are excellent approximations to the average of the true singlettriplet phase shifts for an electron scattering from the Nelectron target just as in the onedimensional model of the pre vious section they also show that TDDFT with existing approximations works very well to correct scattering from the KS potential to the true scat tering phase shifts at least at low energies In fact for the singlet phase shifts TDDFT does better than the computationally more demanding sta tic exchange method and for the triplet case TDDFT does only slightly worse Even though 3335 is strictly speaking only applicable at zero en ergy marked with asterisks in Fig 335 it clearly provides a good descrip tion for nite low energies It is remarkable that the antiparallel spin ker nel which is completely local in space and time and whose value at each point is given by the exchangecorrelation energy density of a uniform elec tron gas evaluated at the groundstate density at that point yields phase shifts for eHeJr scattering with less than 20 error Since a signature of densityfunctional methods is that with the same functional approximations exchangecorrelation effects are often better accounted for in larger systems the present approach holds promise as a practical method for studying large targetsl 334 2 Part ialWave Analysis For the case of spherically symmetric N lelectron ground states useful expressions can be derived for the transition matrix elements tmatrix in the angular momentum representation For example the matrix elements in the usual de nition Gonis 1992 t E ik l exp7ik6l sin 6 are given by tz ifs 4ltltfocgtgtz 7 3336 where the tfs are the Kohn Sham tmatrix elements and ltltfocgtgtz dndnwmmwhnm w T1T22 X 8 T1 kzr2yfgggomgom3151 3337 with Jig E Ylm 70quot In 3337 the go s are Tadz39alKohn Sham orbitals regular at the origin and the 457s are quasiparticle amplitudes determined by the asymptotic behavior of the interacting radial Green s function see Sectl 232 of Wasserman 2005al These are generally difficult to obtain in practice but approximating them by the corresponding Kohn Sham orbitals 33 Scattering Amplitudes 505 yields a simple prediction for the tmatrix elements Furthermore the single pole approximation of 3335 is obtained from 3336 after expanding it to rst order in 6 7 Slei 335 Summary and Outlook Based on the linear response formalism of TDDFT we have discussed a new way of calculating elastic scattering amplitudes for electrons scattering from targets that can bind an extra electroni In one dimension transmission ampli tudes can be extracted from the Nlelectron groundstate susceptibility as indicated by 339 Since the susceptibility of the interacting system is determined by the KohnSham susceptibility within a given approximation to the exchangecorrelation kernel the transmission amplitudes of the interact ing system can be obtained by appropriately correcting the bare KohnSham scattering amplitudes Equation 3316 reminiscent of the singlepole ap proximation for bound A bound transitions provides the simplest approxi mation to such a correction A similar formula for scattering phase shifts near zero energy 3335 was obtained in three dimensions by applying concepts of quantum defect theoryi These constitute rst steps towards the ultimate goal which is to accu rately treat boundfree correlation for lowenergy electron scattering from polyatomic molecules An obvious limitation of the present approach is that it can only be applied to targets that bind an extra electron because the start ing point is always the N lgroundstate Kohn Sham system which may not exist if the Nelectron target is neutral and certainly does not exist if the target is a negative ion In addition to extending the formalism to treat such cases there is much work yet to be done a general proof of principle in three dimensions testing of the accuracy of approximate groundstate KS poten tials developing and testing approximate solutions to the TDDFT Dysonlike equation extending the formalism to inelastic scattering etcr Thus there is a long and winding road connecting the rst steps presented here with the calculations of accurate cross sections for electron scattering from large tar gets when bound free correlations are important The present results show that this road is promising Of course the road goes ever on and on 77 Baggins 1973 but this section looks worthwhile Acknowledgements The origin of this book is two summer schools on TDDFT that were inde pendently organized one in the USA another in Europe The USA summer school took place in Santa Fe New Mexico June 5710 2004 It was organized by Carsten Ar Ullrich Kieron Burke and Giovanni Vignale supported by a generous grant from the Petroleum Research Fund

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