A critical re-assignment of the Rydberg states of iodomethane based on new polarization data J. Chem. Phys. 138, (2013); /1.

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1 Recent development of self-nteracton-free tme-dependent densty-functonal theory for nonperturbatve treatment of atomc and molecular multphoton processes n ntense laser felds Shh-I Chu Ctaton: The Journal of Chemcal Physcs 123, (2005); do: / Vew onlne: Vew Table of Contents: Publshed by the AIP Publshng Artcles you may be nterested n Strong feld onzaton rates smulated wth tme-dependent confguraton nteracton and an absorbng potental J. Chem. Phys. 140, (2014); / A crtcal re-assgnment of the Rydberg states of odomethane based on new polarzaton data J. Chem. Phys. 138, (2013); / Exact-exchange kernel of tme-dependent densty functonal theory: Frequency dependence and photoabsorpton spectra of atoms J. Chem. Phys. 131, (2009); / A dscrete tme-dependent method for metastable atoms and molecules n ntense felds J. Chem. Phys. 120, (2004); / Self-nteracton-free tme-dependent densty-functonal theory for molecular multphoton processes n ntense laser felds AIP Conf. Proc. 525, 415 (2000); /

2 THE JOURNAL OF CHEMICAL PHYSICS 123, Recent development of self-nteracton-free tme-dependent densty-functonal theory for nonperturbatve treatment of atomc and molecular multphoton processes n ntense laser felds Shh-I Chu a Department of Chemstry, Unversty of Kansas, and Kansas Center for Advanced Scentfc Computng, Lawrence, Kansas Receved 25 August 2004; accepted 17 March 2005; publshed onlne 17 August 2005 In ths paper, we present a short account of some recent developments of self-nteracton-free densty-functonal theory DFT and tme-dependent densty-functonal theory TDDFT for accurate and effcent treatment of the electronc structure, and tme-dependent quantum dynamcs of many-electron atomc and molecular systems. The conventonal DFT calculatons usng approxmate and explct exchange-correlaton energy functonal contan spurous self-nteracton energy and mproper long-range asymptotc potental, preventng relable treatment of the excted, resonance, and contnuum states. We survey some recent developments of DFT/TDDFT wth optmzed effectve potental OEP and self-nteracton correcton SIC for both atomc and molecular systems for overcomng some of the above mentoned dffcultes. These DFT TDDFT/ OEP-SIC approaches allow the use of orbtal-ndependent sngle-partcle local potental whch s self-nteracton free. In addton we dscuss several numercal technques recently developed for effcent and hgh-precson treatment of the self-nteracton-free DFT/TDDFT equatons. The usefulness of these procedures s llustrated by a few case studes of atomc, molecular, and condensed matter processes of current nterests, ncludng a autoonzng resonances, b relatvstc OEP-SIC treatment of atomc structure Z=2 106, c shell-fllng electronc structure n quantum dots, d atomc and molecular processes n ntense laser felds, ncludng multphoton onzaton, and very-hgh-order harmonc generaton, etc. For the tme-dependent processes, an alternatve Floquet formulaton of TDDFT s ntroduced for tme-ndependent treatment of multphoton processes n ntense perodc or quasperodc felds. We conclude ths paper wth some open questons and perspectves of TDDFT Amercan Insttute of Physcs. DOI: / I. INTRODUCTION In recent years, the densty-functonal theory DFT has become a wdely used formalsm for electron structure calculatons of atoms, molecules, and solds. 1 6 The DFT s based on the earler fundamental work of Hohenberg and Kohn 7 and Kohn and Sham. 8 In the Kohn Sham DFT formalsm, 8 the electron densty s decomposed nto a set of orbtals, leadng to a set of one-electron Schrödnger-type equatons to be solved self-consstently. The Kohn Sham equatons are structurally smlar to the Hartree Fock equatons but nclude, n prncple, exactly the many-body effects through a local exchange-correlaton xc potental. Thus DFT s computatonally much less expensve than the tradtonal ab nto many-electron wave-functon approaches and ths accounts for ts great success for large systems. However, the DFT s well developed manly for the ground-state propertes only. The treatment of the excted states wthn the DFT s a more recent development The essental element of DFT s the nput of the exchange-correlaton energy functonal whose exact form s unknown. The smplest approxmaton for the xc-energy a Electronc mal: schu@ku.edu functonal s through the local spn-densty approxmaton 1,16 LSDA of homogeneous electronc gas. A defcency of the LSDA s that the xc potental decays exponentally and does not follow the correct long-range asymptotc Coulombc 1/r behavor. As a result, the LSDA electrons are too weakly bound and for negatve ons even unbound. More accurate forms of the xc-energy functonals are avalable from the generalzed gradent approxmaton GGA, whch takes nto account the gradent of electron densty. However, the xc potentals derved from these GGA energy functonals suffer smlar problems lke n LSDA and do not have the proper long-range asymptotc potental behavor ether. Thus whle the total energes of the ground states predcted by these GGA densty functonals are reasonably accurate, the excted-state energes and the onzaton potentals obtaned from the hghest occuped orbtal energes of atoms and molecules are not satsfactory, typcally 30% 50% too low. 1,21 The problem of the ncorrect long-range behavor of the LSDA and GGA energy functonals can be attrbuted to the exstence of the self-nteracton energy. 1,4,5,21,22 For proper treatment of atomc and molecular dynamcs such as collsons or multphoton onzaton processes, etc., t s necessary that both the onzaton potental and the excted-state propertes be descrbed more accurately. In addton, the /2005/1236/062207/16/$ , Amercan Insttute of Physcs

3 Shh-I Chu J. Chem. Phys. 123, treatment of tme-dependent processes wll requre the use of tme-dependent densty-functonal theory TDDFT. The TDDFT extends the concept of statonary DFT to tme-dependent doman. For any nteractng many-partcle quantum system subject to a gven tme-dependent potental, all physcal observables are unquely determned by knowledge of the tme-dependent densty and the state of the system at any nstant n tme. 23,24 In partcular, f the tmedependent potental s turned on at some tme t 0 and the system has been n ts ground state untl t 0, all observables are unque functonals of the densty only. In ths case the ntal state of the system at tme t 0 wll be a unque functonal of the ground-state densty tself,.e., of the densty at t 0. Ths unque relatonshp allows one to derve a computatonal scheme n whch the effect of the partcle-partcle nteracton s represented by a densty-dependent snglepartcle potental, so that the tme evoluton of an nteractng system can be nvestgated by solvng a tme-dependent auxlary sngle-partcle problem. Addtonal smplfcatons can be obtaned n the lnear response regme In the last several years there s consderable effort and success n the extenson of the weak-feld TDDFT and the use of lnear response theory to the study of exctaton energes, frequency-dependent multpole polarzabltes, 26,32,33 optcal spectra of molecules, clusters, and nanocrystals, 34,35 and autoonzng resonances, 21 etc. The prmary focus of ths paper s to dscuss some of the recent developments and applcatons of self-nteracton-free TDDFT for the study of atomc and molecular multphoton processes n ntense laser felds. The strong-feld atomc and molecular physcs s one of the most actve felds of forefront research n scence and technology. The rapd advent of hghpower and short-pulse laser technology n the last decade has facltated the expermental exploraton of multphoton and very hgh-order 300th order nonlnear optcal processes, leadng to the dscovery of a host of novel strong-feld phenomena, such as multphoton and above-threshold onzaton of atoms, multphoton and above-threshold dssocaton of molecules, multple hgh-order harmonc generaton HHG, chemcal bond softenng and hardenng, Coulomb exploson, and coherent control of chemcal and physcal processes, etc. For the treatment of these strong-feld processes, the conventonal hgh-order perturbaton approach s generally not adequate. On the other hand, nonperturbatve approach usng ab nto wave functons requres the soluton of 3N+1th-order tme-dependent Schrödnger equaton n space and tme, where N s the number of electrons. But ths s well beyond the capablty of current computer technology for N2. Even for the case of N=2, fully ab nto tmedependent study s stll at the begnnng stage. The sngleactve-electron SAE model 36,37 wth frozen core s thus commonly used for descrbng the strong-feld processes. However, wthn the SAE model, mportant physcal phenomena such as excted-state resonances, dynamcal response from ndvdual valence spn orbtal, nner core exctaton, nonsequental onzaton, and dynamcal electron correlatons, etc., cannot be treated. Clearly, a more complete formalsm beyond the SAE and other phenomenologcal models s very desrable at ths tme for more comprehensve and accurate treatment of atomc and molecular physcs and chemcal physcs n strong felds. We note, however, that the conventonal weak-feld TDDFT s not adequate for the treatment of strong-feld processes. Smlar to the statonary DFT case, due to the exstence of the self-nteracton energy, TDDFT calculatons usng adabatc LDA or GGA energy functonals do not have the correct long-range asymptotc Coulombc 1/r potental. Moreover, nonperturbatve framework for TDDFT wll be requred for the treatment of strong feld processes. The recent development of self-nteracton-free DFT and TDDFT removes some of these problems and provdes powerful and practcal nonperturbatve frameworks for quanttatve treatment of hghly excted states and strong-feld processes of many-electron quantum systems. A short account of these new developments and ther applcatons wll be dscussed and examned n ths paper. In the followng, we frst brefly descrbe the selfnteracton-free DFT for more accurate treatment of the electronc structure of atomc, molecular, and quantum dot systems. Ths s followed by a dscusson of the self-nteractonfree TDDFTs and assocated computatonal technques for nonperturbatve treatment of multphoton dynamcs and very-hgh-order nonlnear optcal processes n ntense laser felds. II. DFT WITH OPTIMIZED EFFECTIVE POTENTIAL AND SELF-INTERACTION CORRECTION In the Kohn Sham KS DFT formulaton, 8 one solves the followng set of one-electron Schrödnger-type equatons for N-electron systems n atomc unts: Ĥ KS r = v eff, r r = r, =1,2,,N, where v eff, r s the effectve KS potental and s the spn ndex. The total densty s gven by r = N r 2 = r + r, =1 and the ground-state wave functon s determned by = 1 N! det 1 2 N. The total energy of the ground state s obtaned by the mnmzaton of the Hohenberg Kohn energy functonal 7 E, = T s + J + E xc, + v ext rrdr. Here T s s the nonnteractng KS knetc energy,

4 Self-nteracton-free tme-dependent densty-functonal theory J. Chem. Phys. 123, N T s = 1 =1 2 2, 5 v ext r s the external potental due to the electron-nucleus nteracton, J s the classcal electron-electron repulsve energy, J = 2 1 rr dr dr, 6 r r and E xc, s the xc-energy functonal. Mnmzaton of the total energy functonal, Eq. 4, subject to the constrant rdr = N, 7 gves rse to the KS equatons 1 wth the effectve potental v eff, r = v ext r + J r + E xc, r = v ext r + r r r dr + v xc,r, where v xc, r s the exchange-correlaton potental, v xc, r = E xc,. 9 r The KS equatons are to be solved self-consstently, startng from some ntal estmate of the densty r, untl convergence s reached. In actual calculatons, the KS Hamltonan n Eq. 4 must be fxed by a partcular choce of the xcenergy functonal, E xc,. However, both LSDA and GGA energy functonal forms contan spurous selfnteracton contrbutons. Such self-nteracton contrbuton can be seen from Eq. 4, where the two terms J and E xc, should, n prncple, cancel each other exactly n the lmt of one-electron system, f the exact form for E xc, s used. In practce, E xc, needs to be approxmated, leadng to the self-nteracton energy. The exstence of such self-nteracton energy s the man source of error responsble for the ncorrect long-range behavor of the exchange-correlaton potental v xc, r. Thus the elmnaton of the self-nteracton contrbuton s essental for the proper treatment of the onzaton potentals and excted-state propertes. In the last several years, consderable attenton has been pad to the methodology for the removng of self-nteracton energy. One approach for mprovng E xc, s based on the generalzaton of the so-called optmzed effectve potental OEP formalsm. 38,39 In ths approach, one solves a set of one-electron equatons, smlar to the KS equatons n Eq. 1, Ĥ OEP r = V OEP r r = r, =1,2,,N The optmzed effectve potental V OEP r s obtaned by the requrement that the spn orbtals n Eq. 10 are those that mnmze the total energy functonal E, j, where E OEP, j V OEP =0, 11 r E OEP, j = T s, j + J, j + E xc, j + v ext rrdr. 12 Equaton 11 can be wrtten as, usng the chan rule for functonal dervatve, j dr EOEP, j r jr V OEP + c.c. = 0. r 13 Whle the physcal dea of the OEP method s smple and appealng, Eq. 13 leads to an ntegral equaton whch s computatonally mpractcal to solve. Recently, Kreger, L, and Iafrate 40 KLI have worked out an approxmate, albet accurate, procedure to crcumvent ths dffculty, reducng the determnaton of V OEP to the soluton of smple lnear equatons. In all the earler KLI calculatons so far, 5,40 however, the exchange part of the densty functonal contans Hartree Fock-type potental. Whle such a procedure provdes accurate results for the exchange part of E xc t s computatonally more expensve than the tradtonal DFT calculatons where only sngle-partcle local potental s used. Thus t s desrable to explore an approxmate and yet accurate procedure wthn the KLI framework nvolvng only the use of local potentals. Ths would greatly speed up the computatons of the statc and dynamcal propertes of manyelectron systems. As wll be shown below, the selfnteracton-correcton SIC procedure, smlar to the orgnal KLI method, allows also the constructon of self-nteractonfree effectve potental whch s local and orbtal ndependent Further, the OEP so constructed, denoted by V SIC KLI, r below, has the proper long-range asymptotc 1/r behavor and thus s sutable for the determnaton of both ground- and excted-state propertes of many-electron systems. We shall adopt the followng total energy functonal wth explct SIC form 21,22,41 E OEP SIC, j = E OEP, j J + E xc,0, 14 where E OEP, j s gven n Eq. 12. Extendng the OEP-KLI procedure, we arrve at where V OEP SIC, r = v ex r + r r r dr + E xc, r + V SIC, r, 15 V SIC, r = r r v rv SIC, v, 16

5 Shh-I Chu J. Chem. Phys. 123, TABLE I. The onzaton potentals n atomc unts of ground states of neutral atoms Z18 calculated from the hghest occuped orbtal energes by varous exchange-correlaton energy functonals. Non-KLI-SIC KLI-SIC Atom xlsda BLYP xlsda BLYP Expt. a He L Be B C N O F Ne Na Mg Al S P S Cl Ar a Reference 104. v r = r r r dr E xc,0, 17 r and V SIC, = V SIC, r, 18 long-range behavor rv eff r 0 asymptotcally. Onthe other hand, the correspondng potentals wth KLI-SIC reproduce the correct asymptotc behavor, namely, rv eff r 1. Ths correct long-range behavor s crucal for proper DFT treatment of excted and contnuum states as well as the autoonzng resonances to be descrbed next. v = v r. 19 The set of OEP equatons n Eq. 10, wth V OEP r replaced by local potental V OEP SIC, r n Eq. 15, s to be solved =N self-consstently. Fnally, one can choose V SIC,=v N for the hghest occuped orbtal as suggested by the KLI procedure. The energy of the hghest occuped orbtal provdes an approxmate value for the frst onzaton potental. 42 In the followng we dscuss some recent applcatons of the OEP/ KLI-SIC procedure to the atomc and quantum dot electronc structure calculatons. A. Nonrelatvstc atomc structure calculatons The OEP/KLI-SIC method descrbed above has been recently appled to the calculaton of total energes and onzaton potentals for neutral atoms and negatve ons Z= Table I shows some representatve results of the OEP/KLI-SIC calculatons for the onzaton potentals of neutral atoms. It s seen that whle the onzaton potentals from the DFT calculatons usng the LSDA and Becke-Lee-Yang-Parr BLYP functonals have 30% 50% dscrepancy from the exact results, the correspondng results after the KLI-SIC procedure are markedly mproved to wthn 1% 5% of the exact values. To understand the physcal orgn of such an mprovement, we show n Fg. 1 the effectve potental rv eff r of LSDA and BLYP wth and wthout KLI-SIC for the Ne atom. Notce that both LSDA and BLYP potentals wthout KLI-SIC gve rse to wrong B. Autoonzng resonances Because of the lack of proper long-range nteracton behavor, prevous photoonzaton calculatons of complex atoms usng LSDA or GGA energy functonals fal to exhbt the excted-state structure such as the promnent autoonzng resonances. 25,26 Usng the OEP/KLI-SIC procedure, 21 we have recently performed a calculaton of the photoonzaton spectrum of Ne usng tme-dependent LSDA wthn lnear response theory. It s seen that the tme-dependent LSDA wth KLI-SIC results agree well wth the expermental data n the broad peak regon Fg. 2, followed by a seres of sharp resonances due to 2s np resonance transtons 21 Fg. 3. The calculated lnewdths and resonance lne profle parameters are also n good agreement wth both FIG. 1. The one-electron effectve potentals r V eff r of LSDA and BLYP wth and wthout the KLI-SIC for Ne atom.

6 Self-nteracton-free tme-dependent densty-functonal theory J. Chem. Phys. 123, = N + r r = =1 N r, =1 and the total energy of the ground state s expressed as 21 E = T s + J + E xc, + ext rrdr. 22 Here T s s the knetc energy of the nonnteractng N-electron systems ncludng the rest-mass energy, FIG. 2. The total photoonzaton cross sectons of tme-ndependent and tme-dependent calculatons wth LSDA/KLI-SIC potental. expermental 43 and confguraton-nteracton R-matrx Ref. 44 results. We note that n the photoonzatons, 21 two values of the 2s orbtal energy are used n the lnear response calculatons, one taken drectly from the OEP/KLI-SIC data and the other from the expermental value. Both calculatons gve rse to nearly dentcal autoonzng spectrum except the resonance postons are slghtly shfted. To our knowledge ths s the frst successful DFT calculaton whch produces the fne structure of autoonzng resonances of complex atoms. C. Relatvstc DFT calculatons of atomc structure Z=2 106 The relatvstc densty-functonal theory RDFT s the generalzaton of the nonrelatvstc Hohenberg Kohn Sham densty-functonal formalsm 7,8 to the relatvstc regme. 45,46 When the many-body effects are approxmated locally as beng those of a relatvstc homogeneous electron gas, the relatvstc local densty approxmaton s obtaned. 45,46 In the RDFT, one solves the followng sngle-partcle Drac Fock-type equaton for N-electron atomc systems n atomc unts c p + c 2 + eff, r = r, =1,2,,N 20 where eff, s the effectve one-partcle local potental, s the spn ndex, and are the four-component spnors. The total electron densty s gven by T s = N c p + c 2, =1 23 E xc s the relatvstc counterpart of the exchange-correlaton energy, and v ext s the external potental ncludng the electron-nucleus nteracton. The effectve potental n Eq. 20 s gven by eff, r = v ext r + r r r dr + v xc,r, 24 where xc, s the relatvstc exchange-correlaton xc potental, v xc, r = E xc,. 25 r Smlar to the nonrelatvstc case, the RDFT descrbed above contans the undesrable self-nteracton energy. Thus the relatvstc xc potental does not have the proper long-range behavor ether. To overcome such problems, a self-nteracton-free relatvstc DFT has been developed, based on the extenson of the nonrelatvstc OEP/KLI-SIC formalsm to the relatvstc doman. 48 Usng the relatvstc OEP/KLI-SIC formalsm, a detaled atomc structure calculaton of the orbtal bndng energes and onzaton potentals obtaned from the hghest occuped orbtal energes s performed for the ground states of atoms wth nuclear charge Z= The results are n good agreement wth the expermental data to wthn a few percent Fg. 4 across the perodc table Z= Fgure 4 shows that the ncluson of the relatvstc correlaton energy functonal 48 leads to sgnfcant mprovement of the results, partcularly for the hgh-z atoms. We note that Eq. 1 or Eq. 20 can be solved accurately and effcently by means of the generalzed pseudospectral GPS technque 49,50 whch allows nonunform and optmal spatal dscretzaton of the Kohn Sham Hamltonan wth the use of only a modest number of grd ponts. 21,48 FIG. 3. The photoonzaton cross sectons near the 2s np resonant transtons, showng the autoonzng resonance profles. The results are obtaned by the TDLSDA wth KLI-SIC potental. The expermental value of the 2s orbtal energy s used n the calculaton. D. Electronc structure of quantum dots Recent advances n semconductor technology have led to the fabrcaton of zero-dmensonal structure called quantum dots. 51,52 A recent measurement by Tarucha et al. 53 has probed the electronc structure of quantum dots through sngle-electron tunnelng spectroscopy. An nstructve fndng s the exstence of shell structure of addton energes. The addton energy N s defned to be the energy needed to add an electron nto N 1 electron system, namely, N =E tot N E tot N 1, where, E tot N s the total energy of the

7 Shh-I Chu J. Chem. Phys. 123, N-electron quantum dot. Several recent theoretcal studes have explored the shell-fllng behavor of a few-electron less than 20 quantum dots usng the conventonal LSDA or GGA energy functonal, but no detaled exploraton has been performed on the general shell-fllng behavor of many-electron quantum dots. In a recent study, we extended the OEP/KLI-SIC formalsm to the study of the electronc structure and shell-fllng behavor of quantum dots wth N = Fgure 5 shows the capactve energy, N N 1, as a functon of the electronc number N calculated by the BLYP sold lne and BLYP/KLI-SIC dashed lne procedures, exhbtng the detaled shell and subshell electronc structure of many-electron quantum dots. We note that although numercal values of the capactve energes dffer by a few percent, all the methods LSDA, LSDA/KLI-SIC, BLYP, BLYP/KLI-SIC used n the calculatons 57 lead to the same shell and subshell structure patterns. To understand the orgn of the shell-fllng structure, we frst examne the energy order of ndvdual electron orbtal. Usng ths energy orderng, one can dentfy all the shell structures n Fg. 5. For example, those peak postons marked correspond to the quantum dots wth flled shells or subshells. The most promnent shell structure occurs at the followng magc numbers N=21s 2, 81s 2 sp 6, 201s 2 2p 6 3d 10 2s 2, 401s 2 2p 6 3d 10 2s 2 4f 14 3p 6, etc., correspondng to the fully occuped shells. The smaller peaks marked n Fg. 5 can also be dentfed as those quantum dots wth half-flled subshells. It s nstructve to see that the Hund s rule s also applcable to the quantum dot systems here. 57 FIG. 4. Ionzaton potentals calculated by a nonrelatvstc OEP/KLI-SIC and b relatvstc OEP/KLI-SIC wth exchange only dashed lne and xc sold lne-energy functonals for neutral atoms wth 2Z106. The expermental onzaton potentals are also presented open crcle for comparson. FIG. 5. Capactve energes N N 1 of N-electron quantum dots confned by a sphercal harmonc potental =0.75, exhbtng the shellfllng structure. Both BLYP and BLYP/KLI-SIC procedures are used n the DFT calculaton. Sold lne corresponds to the BLYP results where N=E tot N E tot N 1 s adopted. The dashed lne shows the BLYP/KLI- SIC results where N s taken drectly from the hghest occuped orbtal energy. III. RECENT DEVELOPMENT OF SELF-INTERACTION- FREE TDDFT FOR NONPERTURBATIVE TREATMENT OF ATOMIC MULTIPHOTON PROCESSES IN INTENSE LASER FIELDS The TDDFT as a rgorous formalsm s a more recent development n DFT, although the hstorcal roots date back to the tme-dependent Thomas Ferm model proposed by Bloch n The central result of modern TDDFT s a set of tme-dependent Kohn Sham TDKS equatons whch are structurally smlar to the tme-dependent Hartree Fock TDHF equatons but nclude n prncple, exactly all many-body effects through a local tme-dependent xc potental. 23,24 To date, most applcatons of TDDFT fall n the regme of weak-feld lnear or nonlnear response and the adabatc LSDA energy functonal s often used. 23,25,26,59 Applcatons of the tme-dependent LSDA approach have been made to the photoresponse of atoms, molecules, clusters, nanocrystals, semconductor surfaces, and bulk semconductor n the weak-feld perturbatve regme. 25,26,34,35,59 As ndcated n the Introducton, the conventonal weakfeld TDKS formalsm cannot be drectly appled to the study of multphoton processes n ntense laser felds. In ths secton, we dscuss a TDDFT wth OEP and SIC for nonperturbatve treatment of many-electron quantum systems n ntense laser felds, 60 based on the extenson of the steady-state OEP/KLI-SIC procedure 21 to the tme doman. We note that a related TDOEP-KLI method wthout the use of SIC was proposed by Ullrch and Gross. 61 The latter method provdes an accurate procedure for the calculaton of the exchange part of the tme-dependent potental. But computatonally t can be more tme consumng than the conventonal TDKS approach snce the TDOEP-KLI procedure requres the constructon of Hartree Fock-type potental at each tme step. The advantage of the TDOEP/KLI-SIC approach 60 s that t allows the constructon of self-nteracton-free tmedependent local OEP whch s also orbtal ndependent. Ths greatly facltates the study of tme-dependent processes of many-electron quantum systems n strong felds.

8 Self-nteracton-free tme-dependent densty-functonal theory J. Chem. Phys. 123, A. TDDFT wth OEP/KLI-SIC for atomc multphoton processes n ntense pulsed laser felds The quantum mechancal acton of a many-electron system nteractng wth an external feld can be expressed as 60,61 N t 1 A = dt * r,t t t 1 r,tdr dt r,tv extr,tdr 1 t 1 dt r,t r,t dr dr 2 r r A xc, 26 where r,t are the tme-dependent spn orbtals, N = N s the total number of electrons, v ext r,t s the external potental whch ncludes the electron-nucleus Coulomb nteracton and the couplng of the electron to the external laser felds, r,t= r,t s the total electron densty wth the spn densty N r,t = * r,t r,t, 27 and A xc s the xc acton functonal. The spn orbtals satsfy the one-electron Schrödnger-type equaton, t r,t = V r,t r,t, 28 where V r,t wll be the TDOEP f we choose the set of spn orbtals whch render the total acton functonal A statonary: A V r,t =0. 29 Followng a procedure smlar to the TDOEP/KLI scheme, 61 one obtans the followng general expresson for the tme-dependent xc potental: V xc, r,t = + r,t 1 r,t 2 v r,t + v * r,t r,t r,tv xc, 1 2 v + v + 4 r,t t 2 r,t * dtv t v * t, 30 A xc = t 1 dtexc r,t, r,t N t 1 dtj + E xc,0, =1 32 where E xc s the adabatc tme-dependent xc energy functonal, we obtan v r,t = v * r,t = E SIC xc. 33 Note that v s now a real functon of r and t. Thus the memory term n Eq. 30 vanshes dentcally. Smlar results are obtaned as long as one uses an explct E xc form such as that n LSDA or GGA of energy functonal and the adabatc approxmaton. The use of the SIC form n Eq. 32 removes the spurous self-nteracton terms n conventonal TDDFT and results n a proper long-range asymptotc potental. Another major advantage of ths procedure s that only local potental s requred to construct the orbtal-ndependent OEP. Ths facltates consderably the numercal computaton. By extendng the steady-state OEP/KLI-SIC procedure 21 to the tme-dependent doman, we obtan the tme-dependent TD OEP as where and V r,t = v ext r,t + V SIC, r,t = v r,t = E xc r,t, r,t r,t J r,t + V SIC,r,t, r,t r,t v r,t + V SIC,t v t, J r,t r,t E xc r,t,0, 36 r,t V SIC, t = V SIC, r,t, v t = v r,t Equatons 28 and 34 are to be solved self-consstently. Note that snce the exact form of v xc r,t s unknown, the adabatc approxmaton s often used n the TDDFT calculatons: v xc r,t = v xc =r,t. 39 where the last term contans the memory effect and A xc v r,t = *, v * r,t = A xc *. 31 If we use the followng explct SIC expresson for the xc acton functonal, 60 Fnally Eq. 28 s an ntal value problem and the ntal wave functon can be determned by r,t t=0 = r e t t=0, 40 where r and are the egenfuncton and egenvalue of the tme-ndependent Kohn Sham equaton wth OEP/KLI- SIC for the statc case. 21

9 Shh-I Chu J. Chem. Phys. 123, B. Tme-dependent generalzed pseudospectral method for numercal soluton of self-nteracton-free TDDFT equatons In ths secton we brefly descrbe a numercal procedure recently developed for accurate and effcent soluton of the tme-dependent OEP/SIC equaton, Eq. 28. The commonly used procedures for the tme propagaton of the Schrödnger or TDDFT equaton employ equal-spacng spatal grd dscretzaton For processes such as HHG, accurate tme-dependent wave functons are requred to acheve convergence snce the ntensty of varous harmonc peaks can span a range of many orders of magntude. Hghprecson wave functons are, however, more dffcult to acheve by the conventonal equal-spacng spatal-grddscretzaton tme-dependent technques, due to the Coulomb sngularty at the orgn and the long-range behavor of the Coulomb potental. To acheve more accurate wavefuncton propagaton, a numercal procedure, the tmedependent generalzed pseudospectral TDGPS method 66 has been recently ntroduced. The TDGPS procedure conssts of the followng two basc elements: The GPS technque 49,50 s used for nonunform optmal grd dscretzaton of the radal coordnates and the Hamltonan. It has been shown that the number of grd ponts requred n the GPS procedure can be orders of magntude smaller than those used by the conventonal equal-spacng dscretzaton methods. Yet consderably hgher accuracy n wave functons and therefore HHG spectra can be acheved snce the physcally more mportant short-range regme s more accurately treated by the TDGPS method. 66 A splt-operator technque n the energy representaton s ntroduced for effcent tme propagaton of the wave functons. More detaled dscusson of the TDGPS method s gven n a later molecular secton. C. Multphoton quantum dynamcs of atomc systems n ntense laser felds In ths secton we dscuss several recent applcatons of the TDOEP/KLI-SIC formalsm to the nonperturbatve study of multphoton processes of many-electron atomc systems n ntense laser felds, focused partcularly on the phenomenon of multple HHG n ntense laser felds. The study of the HHG phenomena s one of the most rapdly developng topcs n strong-feld atomc and molecular physcs. 36,67 73 The generaton of harmonc orders well n excess of 100 from noble gas, datomc and polyatomc molecules, and cluster targets has been demonstrated by several recent experments For example, n a recent experment, 71 ultrashort laser pulses wth 26 fs pulse duraton from a T:sapphre laser have been used to generate coherent radaton at wavelengths as short as 2.7 nm 460 ev. These wavelengths are well wthn the water wndow regon of x-ray transmsson. Thus the HHG mechansm provdes a smple and powerful new route to generate coherent x-ray laser source whch s techncally much less demandng and less energy ntensve than current plasma based x-ray schemes. The avalablty of such a compact laboratory table-top system for the generaton of coherent x rays holds promse as a source for bologcal holography and nonlnear optcs n the x-ray regme. Another potental new applcaton of HHG processes s the possblty of generatng laser pulses of ultrashort duraton tens of attoseconds n the near future, leadng a way to perform attosecond spectroscopy and study new dynamcal phenomena wth attosecond tme resoluton. To study HHG, we start from the calculaton of the total nduced dpole moment and dpole acceleraton of N-electron systems whch can be expressed n terms of electron densty as follows: dt = r,tzdr = r,tz r,t, d A t = r,t d2 z dt 2 dr = r,t V eff,;r,t r,t z 41 Et r snt + r,t r,t. 42 z The correspondng HHG power spectrum can now be obtaned by the Fourer transformaton of the respectve tmedependent dpole moment or dpole acceleraton: P = 1 t f dte dt2 t f t t t d 2 43 and P A = 1 t 1 f t f t 2 da te dt2 t t d A We note that an mportant measure of the accuracy of HHG results s that the power spectrum P should be equal to P A f the tme-dependent wave functon calculaton s fully converged. 1. The effect of dynamcal electron correlaton on the HHG of rare gas atoms One recent applcaton of the TDOEP/KLI-SIC formalsm s to study the role of dynamcal electron correlaton on HHG of He atoms n ntense lnearly polarzed LP laser pulses. 60 Of partcular nterest s the study of the mechansm responsble for the producton of the hgher harmoncs observed n the experment 74 whch cannot be explaned by the SAE model. 36,37 Fgure 6 shows that whle the SAE model fals to produce the hgher harmoncs, the TDDFT/KLI-SIC results agree well wth the expermental data n both lower and hgher HHG regmes, ndcatng the mportant role played by the dynamcal electron correlaton. 60 More detaled study of the HHG processes of rare gas He, Ne, Ar atoms has been recently reported Coherent control of HHG of rare gas atoms n twocolor mxed felds The TDOEP/KLI-SIC formalsm was recently extended to the study of coherent control of the producton of HHG of

10 Self-nteracton-free tme-dependent densty-functonal theory J. Chem. Phys. 123, FIG. 6. The HHG spectrum of He obtaned from the all-electron calculaton open crcle and from the SAE model flled trangle. The expermental data wth error bar are also shown for comparson. The HHG yelds are normalzed to the 13th harmonc peak. The laser peak ntensty used n the calculaton s I= W/cm 2 and wavelength =248.6 nm. He atoms by means of the use of two laser felds wth dfferent frequences and polarzaton drectons. 76 It s shown that by mxng a weak fundamental feld 1053 nm wth a strong second-harmonc feld 527 nm, one can produce hgh-order sum and dfference frequency radaton and even harmoncs wth an effcency smlar to that of odd harmoncs. Further the relatve effcency of the HHG can be controlled by manpulatng the relatve polarzaton drecton of the two felds. These predctons are n accord wth the recent expermental data. 77 As a case study, Fg. 7 shows the results sold lnes of the HHG of He atoms n the mxed felds wth the o -feld ntensty I 1 = W/cm 2 and the 2 o -feld ntensty I 2 = W/cm 2. The felds are parallel to each other wth the relatve phase =0. Also shown n Fg. 7 for comparson are the sngle-feld data, o feld alone open crcle and 2 o feld alone flled crcle, both wth laser ntensty W/cm 2. Frst recall that the snglecolor 2 o feld produces 4n+2 harmoncs, n=0,1,2,. By mxng the 2 o feld wth a weak 200 tmes weaker o feld, three addtonal 4n 1th, 4nth, 4n+1th harmoncs appear n between two adjacent harmoncs of the sngle-2 o feld case. The remarkable feature s that the ntenstes of these extra harmoncs are of smlar orders of magntude of the harmoncs produced by the sngle-color 2 o feld flled crcle and are much hgher than those produced by the sngle-color o feld open crcle. Further these 4nth harmoncs are the even harmoncs whch cannot be produced by ether the sngle-color 2 o or the sngle-color o feld alone. The generaton of these extra harmoncs can be explaned by the hgh-order wave mxng mechansm. Thus the leadng channel for producng the 4n+1th harmonc s due to the absorpton of 2n photons from the 2 o feld and one photon from the o feld. Smlarly, the 4n 1th harmonc s produced by the absorpton of 2n photons from the 2 o feld and the emsson of one photon to the o feld. Lkewse, the even 4nth harmoncs can be produced by the absorpton of 2n 1 photons from the 2 o feld and the absorpton of two photons from the o feld. Thus the combnaton of a strong 2 o feld wth a weak o feld can produce both odd and even harmoncs wth relatvely hgh yelds Generaton of crcularly polarzed HHG In all the theoretcal and expermental nvestgatons up to 1998, only LP HHG has been studed. Ths s because the dpole selecton rule excludes the possblty of producng crcularly polarzed CP harmoncs by multphoton mechansm. Recently a feasble scheme has been proposed for the generaton of purely CP hgh harmoncs, usng two-color laser felds. 78 The proposed setup conssts of a CP fundamental laser feld and a LP second-harmonc laser feld 2 n crossed-beam confguraton. The feasblty of such a scheme s confrmed by a three-dmensonal TDOEP/KLI-SIC calculaton of the He system. 78 IV. SELF-INTERACTION-FREE TDDFT FOR MOLECULAR MULTIPHOTON PROCESSES IN INTENSE LASER FIELDS A. TDDFT wth OEP/KLI-SIC for molecular processes n ntense laser felds The TDOEP/KLI-SIC formalsm descrbed n the last secton for atomc systems can be extended to the molecular systems. Consder the soluton of the tme-dependent Kohn Sham-tpye TDKS equaton for N-electron molecular systems under fxed nucle approxmaton n LP laser felds, n atomc unts, t r,t = Ĥr,t r,t FIG. 7. HHG spectra of He atoms n the two-color mxed felds 1 = o, 2 =2 o wth parallel polarzaton and laser ntenstes I 1 = W/cm 2, I 2 = W/cm 2 sold lne. Also shown are the results of HHG spectra n sngle-color o 1053 nm feld open crcle and snglecolor nm feld flled crcle, both wth ntensty at W/cm 2. = v eff, r,t r,t, =1,2,,N 45 where v eff, r,t s the tme-dependent effectve potental dependng upon the total electron densty t and s the spn ndex. Followng the TDOEP/KLI-SIC procedure, we obtan the tme-dependent OEP as 60,79

11 Shh-I Chu J. Chem. Phys. 123, V OEP eff, r,t = v ext r,t + J r,t + V SIC,r,t. 46 Here v ext r,t s the external potental due to the nteracton of the electron wth the external laser feld and the nucle and V SIC, r,t s gven n Eq. 35. For the specal case of homonuclear datomc molecules, the tme-dependent OEP potental, Eq. 46, has the followng explct form: V OEP Z 1 eff, r,t = R 1 r Z 2 R 2 r + d 3 r r,t r r + Et r sn t + V SIC, r,t. 47 FIG. 8. The grd structure of the spatal coordnates of H 2 obtaned by the generalzed pseudospectral dscretzaton technque. Here r s the electronc coordnate, Et the electrc feld ampltude, and R 1 =0,0,a and R 2 =0,0, a are the coordnates of the two nucle n Cartesan coordnates wth nuclear charges Z 1 and Z 2, respectvely. The nternuclear separaton R s equal to 2a. B. Generalzed pseudospectral method for spatal dscretzaton of two-center systems In ths secton, we dscuss a new procedure for the optmal spatal dscretzaton and hgh-precson soluton of feld-free two-center datomc molecular systems. We shall use the prolate spherodal coordnates,,, 0, 0, and 02 for the descrpton of the system: x = a snh sn cos, y = a snh sn sn, z = a cosh cos. Due to the axal symmetry of the datomc systems, the feldfree soluton takes the form m r = e m,, m =0,±1,±2,. In order to symmetrze the Hamltonan matrx, we transform the Kohn Sham dfferental equaton, Eq. 1, nto a varatonal problem that mnmzes the functonal F s = dr + v eff 2 dr. 48 The Coulomb repulsve potental V c =drr/r r satsfyng the Posson equaton, 2 V c = 4, can also be recast nto the followng varatonal form seekng the mnmzaton of F c = 1 2 V c 2 dr 4 V c dr. 49 The GPS technque for one-center atomc system 49,50 can be extended to dscretze the ntegral representaton n two-center systems. 79,80 In the two-center GPS procedure, one expands any spatal wave functon, by N,N,, the polynomals of order N and N n and, respectvely,, N,N, = N,N, j g xg j y, =0,j=0 50 and further requre the approxmaton to be exact at the collocaton ponts,.e., N,N, j =, j j, where x and y j are the two sets of collocaton ponts to be descrbed below. In Eq. 50, g x and g j y are the cardnal functons 49,50,81 defned as 1 1 x 2 P N x g x =, 51 N N +1P N x x x 1 1 y 2 P N y g j y =. 52 N N +1P N y j y y j In the case of the Legendre pseudospectral method, 49,79 81 the collocaton ponts are determned, respectvely, by the roots of the frst dervatve of the Legendre polynomal P N wth respect to x and the frst dervatve of P N wth respect to y, namely, P N x =0, P N y j =0. 53 It follows that the cardnal functons possess the followng unque and desrable propertes g x =,, g j y j = j,j. 54 The mappng relatonshps between and x and between and y can be chosen as 79,80 = L 1+x 1 x, = 1+y, 55 2 where x 1,1, y 1,1, 0,, 0,, and L s a mappng parameter. A more detaled dscusson of the constructon of the dfferentaton matrx and the symmetrzaton of the Hamltonan matrx can be found elsewhere. 79,80 A major advantage of the outlned generalzed pseudospectral method s that t allows for nonunform optmal spatal grd dscretzaton: denser mesh near the nucle and sparser mesh for long-range part of the Coulombc potental. Wth the use of only a modest number of grd ponts, hgh precson egenvalues and egenfunctons can be obtaned. Fgure 8 shows a typcal grd structure for two-center datomc systems. 79,80 As a measure of the accuracy of the GPS procedure, we have frst tested

12 Self-nteracton-free tme-dependent densty-functonal theory J. Chem. Phys. 123, the method for the H 2 + molecule, where exact results are avalable for comparson. Usng only a modest number of grd ponts 20 for the coordnate and 9 for the coordnate, we obtan the ground-state energy to be a.u., n complete agreement wth the exact value of a.u. 82 C. Tme-dependent generalzed pseudospectral method for numercal soluton of self-nteracton-free TDDFT equatons n two-center systems In the followng, we extend the TDGPS procedure to the numercal soluton of the tme-dependent OEP/SIC equatons n two-center systems. Consder the soluton of tmedependent Kohn Sham-type equaton wth OEP/SIC for N-electron datomc molecular systems n LP laser felds, t r,t = Ĥ r,t = Ĥ 0 r + Vˆ r,t r,t, =1,2,,N. 56 Here Ĥ 0 s the tme-ndependent Hamltonan wth OEP/SIC at t=0 and Vˆ ncludes the electron-laser feld nteracton and the resdual tme-dependent OEP/SIC: snh 2 + sn 2 snh snh 1 + snh 2 + sn 2 sn sn Ĥ 0 r = 1 2a V OEP SIC, r,0, 57 Vˆ r,t = Et r sn t + V OEP SIC, r,t V OEP SIC, r,0, 58 where Et s the electrc feld, assumed to be parallel to the nternuclear ẑ axs, and Et=Fft, where ft s the envelope functon of the laser pulse. The second- or hgher order splt-operator technque n prolate spherodal coordnates and n the energy representaton can be extended for accurate and effcent propagaton of the tme-dependent OEP/SIC equatons: 66,79,80 r,t + te Vˆ r,tt/2 e Ĥ 0 rt e Vˆ r,tt/2 r,t + Ot Note that such an expresson s dfferent from the conventonal splt-operator technques, 62,63,83 where Ĥ 0 s usually chosen to be the knetc energy operator and Vˆ the remanng Hamltonan dependng on the spatal coordnates only. The use of the energy representaton n Eq. 59 allows the explct elmnaton of the undesrable fast-oscllatng hghenergy components and speeds up consderably the tme propagaton. 66,79,80 In addton, the symmetry propertes possessed by Ĥ 0 can be used to smplfy and facltate the calculatons. FIG. 9. The HHG power spectrum of H 2 at R=1.4a 0 n a 20 optcal cycle, 1064 nm, sn 2 pulse shape laser felds wth peak ntensty W/cm 2. Both the length form sold lne and acceleraton form dotted lne power spectra are shown for comparson. D. Exploraton of the underlyng mechansms for hgh harmonc generaton H 2 n ntense laser felds In ths secton we show an applcaton of the TDOEP/ KLI-SIC procedure to the study of HHG of H 2 n ntense pulsed laser felds. Frst we dscuss the feld-free electronc structure calculatons usng the steady-state OEP/KLI-SIC procedure, 21,79 and the GPS procedure 79,80 s extended to dscretze the molecular Hamltonan n the prolate spherodal coordnates. For H 2, the calculated ground-state energy s a.u. usng LSDA exchange energy functonal only and a.u. ncludng both LSDA exchange and correlaton energy functonals; the latter s wthn 1% of the exact value of a.u. If the GGA energy functonal such as that of BLYP Ref. 1 s used, the calculated groundstate energy s mproved to a.u. Consder now the nteracton of H 2 molecules wth an ntense LP laser feld wth wavelength 1064 nm, sn 2 pulse shape, and 20 optcal cycles n pulse length. The tmedependent xc potental s constructed by means of the tmedependent OEP/KLI-SIC procedure usng the adabatc LSDA exchange and correlaton energy functonal. We shall assume the electrc feld polarzaton s algned along the nternuclear-axs ẑ drecton. Ths approxmaton s justfed by the expermental observaton that the laser-molecular nteracton tends to force the molecule to algn along the polarzaton axs. In the followng, we shall focus our dscusson on the HHG process of H 2 molecules from the ground vbratonal state wth the nternuclear separaton R fxed at the equlbrum dstance R=R e =1.4a 0. The fxed nucle approxmaton s justfable snce the zero-pont vbraton of H 2 n the ground state s rather small wthn 0.25a 0 of R e and the ncluson of the vbratonal degree of freedom s not expected to alter the man features of the HHG phenomenon, partcularly when the tme duraton of the laser pulse s short. The soluton of the TDOEP/KLI-SIC equaton s performed by means of the TDGPS method descrbed above. As an example of the numercal accuracy of the TDGPS technque, Fg. 9 shows the comparson of the HHG power spectrum of H 2 for the case of laser ntensty I=10 14 W/cm 2

13 Shh-I Chu J. Chem. Phys. 123, FIG. 10. Color. The tme-frequency spectra modulus of H 2 at R=1.4a 0 n 1064 nm, 20 optcal cycle, sn 2 pulse shape laser felds wth peak ntensty W/cm 2. The colors shown are n logarthmc scale n the powers of 10. obtaned by the Fourer transform of the nduced dpole and dpole acceleraton, respectvely. Excellent agreement of the two spectra s obtaned from the lowest harmoncs all the way to the cutoff regme, ndcatng the the full convergence of the tme-dependent wave functons. Fgure 9 shows that those harmonc peaks near the cutoff regme are structureless. However, for harmoncs n the plateau and well below the cutoff, they possess some multplepeak fne structures. To explore the detaled spectral and temporal structure of HHG and the underlyng mechansms n dfferent energy regmes, one can perform the tmefrequency analyss by means of the wavelet transform 84,85 of the nduced dpole or dpole acceleraton, A W t 0, = dtw t0,tdt d t, 60 wth the wavelet kernel W t0,t= Wt t0. For the harmonc emsson, a natural choce of the mother wavelet s gven by the Morlet wavelet 85 Wx = 1/ e x e x2 / Fgure 10 shows the modulus of the tme-frequency profles of H 2 at R=1.4a 0 n 1064 nm, 20 optcal cycle, sn 2 pulse shape, and W/cm 2 laser felds, revealng strkng and vvd detals of the spectral and temporal structures. Several salent features are notced. Frst, for the lowest few harmoncs, the tme profle at a gven frequency shows a smooth functon of the drvng laser pulse. Ths s an ndcaton that the multphoton mechansm domnates ths lower harmonc regme. In ths regme, the probablty of absorbng N photons s roughly proportonal to I N, and I laser ntensty s proportonal to Et 2. Second, the smooth tme profle s gettng shorter n tme duraton and broadened n frequency as the harmonc order s ncreased, as s evdent n Fg. 10 from the frst to the seventh harmoncs. As the harmonc order s further ncreased, the tme profles see partcularly the 11th harmonc n Fg. 10 develop extended fne structures. Ths can be attrbuted to the effect of excted states and the onset of the onzaton threshold. Thrd, for those hgh harmoncs n the plateau regme well above the onzaton threshold, the most promnent feature s the development of fast burst tme profles. At a gven tme, we see that such bursts actually form a contnuous frequency profle n Fg. 10. Ths s clear evdence of the exstence of the bremsstrahlung radaton emtted by each recollson of the electron wave packet wth the parent onc cores. In contrast, we fnd that the multphoton-domnant lowest-order harmoncs form a contnuous tme profle at a gven frequency. In the ntermedate energy regme where both multphoton and tunnelng mechansms contrbute, the tmefrequency profles show a netlke structure. More detaled analyss of the orgn of the power spectrum patterns near and below the cutoff can be pursued by performng the cross secton of the tme-frequency profle of Fg. 10 at a gven harmonc frequency. 79 Fnally t wll be nstructve to explore the orgn of the fne-structure peak splttng of harmoncs n the plateau regme below the cutoff, see for example, the 23rd harmonc n Fg. 9. Fgure 11 shows the tme profles at the three subpeak postons denoted by 1, 2, and 3 wthn the 23rd harmonc. Strkngly, ther tme profles nearly concde. Ths s evdence that all the harmonc subpeaks wthn a gven harmonc are produced by the same mechansm, namely, they are produced by the nterference n tme of all the bremsstrahlung radaton emtted from all the returnng electronc wavepackets wthn the ncdent laser pulse duraton. To our knowledge, ths s the frst ab nto calculaton exhbtng the

14 Self-nteracton-free tme-dependent densty-functonal theory J. Chem. Phys. 123, FIG. 11. The tme profles of the subpeaks of the 23rd harmonc of H 2 R=1.4a 0 n ntense pulsed laser felds. The laser parameters are the same as those n Fg. 3. detals of the tme profles of the subpeak harmoncs for a molecular system. E. Multphoton onzaton and hgh harmonc generaton of N 2 n ntense laser felds In the last secton, we descrbe the TDOEP/KLI-SIC method for the study of multphoton processes of molecular systems n ntense laser felds, takng nto account the correct long-range Coulombc 1/r potental. Here we consder an alternatve and smpler procedure by adoptng the mproved Leeuwen Baerends LB type potental, 86 v LB xc, for the statc xc potental. The correspondng tme-dependent xc potental n the adabatc approxmaton conssts of two emprcal parameters and and possesses the followng explct form, 87 v LB xc r,t = v LSDA x r,t + v LSDA c r,t x 2 r,t 1/3 r,t 1+3x r,tlnx r,t + x 2 r,t +1 1/2. 62 The frst two terms n Eq. 62, v LSDA x and v LSDA c are the LSDA exchange and correlaton potentals whch do not have the correct asymptotc behavor. The last term n Eq. 62 s the nonlocal gradent correcton wth x r = r r 4/3, whch ensures the proper long-range asymptotc behavor v LB xc 1/r as r. For the tmendependent case, ths exchange-correlaton LB potental has been found to be relable for the electronc structure and frequency-dependent hyper polarzablty calculatons of a number of atomc and molecular systems. 86 Fgure 12 presents an example of the tme-dependent sngle-electron populatons of dfferent spn orbtals of N 2 molecule. 87 The slope of the decay of the electron populaton n tme descrbes the onzaton rate. The laser electrc feld wth ntensty W/cm 2 and wavelength 1064 nm s assumed to be parallel to the nternuclear axs and the nternuclear dstance of N 2 s fxed at ts equlbrum poston, R e =2.072a 0. In ths case, the order of onzaton probablty s FIG. 12. The tme-dependent populaton of electrons n dfferent spn orbtals of N 2 n W/cm 2, 1064 nm, sn 2 pulse laser feld wth 20 optcal cycles n pulse duraton. found to be 1 u 2 g 2 u 3 g. On the other hand, for the W/cm 2 and 1064 nm pulses not shown, the order of onzaton probablty s 2 g 1 u 2 u 3 g. Thus wthn the electrons, the lower the electron orbtal bndng energy onzaton potental s, the more wll be the electron onzaton probablty. However, although the onzaton potental of 1 u electrons s lower than that of 2 u electrons, the onzaton probablty of 1 u electrons turns out to be less than that of 2 u electrons n all the cases. Ths can be attrbuted to the fact that 2 u orbtal s along the electrc feld drecton Ê, whle that of 1 u s perpendcular to Ê. We thus see two dfferent effects that can contrbute to the onzaton: the onzaton potental electron bndng energy effect and the orbtal orentaton effect. The onzaton potental effect makes the electrons wth lower onzaton potentals easer to onze. The orentaton effect makes the onzaton easer for those electrons whose orbtal orentatons are parallel to the electrc feld. These two effects are clearly competng. We now dscuss brefly the results of the all-electron HHG calculatons of N The relatve contrbuton of ndvdual spn orbtal to HHG, d, depends on the harmonc frequency range but n general t follows roughly the same trend as the order of tme-dependent nduced dpole moment. 87 The total HHG power spectrum s obtaned by the sum of ndvdual spn-orbtal HHG power spectrum d plus the nterference terms. For the case of N 2, we observe nterestng constructve and destructve nterferences between the two hghest occuped bond 3 g and ant-bondng 2 u orbtals: It s the nterference between these two largest nduced dpoles d 3g and d 2u that contrbutes domnantly to the overall HHG power spectrum of N Thus for manyelectron molecular systems such as N 2, the conventonal sngle-actve-electron SAE model s not vald, snce there s no sngle electron molecular orbtal whch domnates the total HHG process. V. GENERALIZED FLOQUET FORMULATION OF TIME- DEPENDENT DENSITY-FUNCTIONAL THEORY IN PERIODIC OR QUASIPERIODIC FIELDS In the last few sectons, the tme-dependent equatons n the self-nteracton-free TDDFT formulatons are solved nu-