Kinetic-Energy Density-Functional Theory on a Lattice

Transcription

1 h an open acce artcle publhed under an ACS AuthorChoce Lcene, whch permt copyng and redtrbuton of the artcle or any adaptaton for non-commercal purpoe. Artcle Cte h: J. Chem. heory Comput. 08, 4, pub.ac.org/jcc Knetc-Energy Denty-Functonal heory on a Lattce Ir heophlou,*, Floran Buchholz, F. G. Ech, chael Ruggenthaler, and Angel Rubo, ax Planck Inttute for the Structure and Dynamc of atter and Center for Free Electron Laer Scence, Hamburg 76, Germany Center for Computatonal Quantum Phyc (CCQ), Flatron Inttute, New York, New York 000, Unted State Downloaded va on September 9, 08 at 0:06:39 (UC). See for opton on how to legtmately hare publhed artcle. ABSRAC: We preent a knetc-energy denty-functonal theory and the correpondng knetc-energy Kohn Sham () cheme on a lattce and how that, by ncludng more obervable explctly n a denty-functonal approach, already mple approxmaton tratege lead to very accurate reult. Here, we promote the knetc-energy denty to a fundamental varable alongde the denty and how for pecfc cae (analytcally and numercally) that there a one-to-one correpondence between the external par of on-te potental and te-dependent hoppng and the nternal par of denty and knetc-energy denty. On the ba of th mappng, we etablh two unknown effectve feld, the mean-feld exchange-correlaton potental and the mean-feld exchangecorrelaton hoppng, whch force the ytem to generate the ame knetc-energy denty and denty a the fully nteractng one. We how, by a decompoton baed on the equaton of moton for the denty and the knetc-energy denty, that we can contruct mple orbtal-dependent functonal that outperform the correpondng exact-exchange Kohn Sham () approxmaton of tandard denty-functonal theory. We do o by conderng the exact and ytem and comparng the unknown correlaton contrbuton a well a by comparng elf-content calculaton baed on the mean-feld exchange (for the effectve potental) and a unform (for the effectve hoppng) approxmaton for the and the exactexchange approxmaton for the ytem, repectvely.. INRODUCION Denty-functonal theory (DF) ha become over the pat decade a tandard approach to the quantum many-body problem. It ucce come from the fact that t combne low computatonal cot wth a reaonable accuracy, whch help to undertand and predct expermental reult for ytem not acceble wth wave functon-baed method. DF avod the exponental numercal cot of wave functon-baed method by reformulatng quantum mechanc n term of the denty. he major drawback of DF that the exact energy expreon of the quantum ytem n term of the denty not avalable, and n practce approxmaton need to be employed. Already before the rgorou formulaton of DF, a heurtc method baed on the denty ntead of the wave functon exted, whch wa called the homa Ferm theory.,3 Whle th theory proved to be very mportant for the dervaton of fundamental reult, for example, the tablty of quantum matter, 4 n practce t not very accurate (only n the lmt of atom wth arbtrarly hgh atomc number or for homogeneou ytem) and doe not provde bac properte uch a the hell tructure of atom or the bndng of molecule. A t wa quckly realzed, t the approxmaton to the knetc-energy expreon that prevent homa Ferm denty-functonal approxmaton from leadng to accurate reult. What ha made DF popular for determnng properte of complex many-body ytem the Kohn Sham () contructon, 5 where ntead of modelng the knetc energy drectly n term of the denty an auxlary nonnteractng quantum ytem ued that ha the ame denty. he knetc energy of th computatonally cheap auxlary ytem then corrected by o-called Hartreeexchange-correlaton (Hxc) contrbuton that ncorporate the mng nteracton and knetc-energy contrbuton. Already mple approxmaton to th unknown expreon gve reaonably accurate anwer. However, t hard to ytematcally ncreae the accuracy of approxmaton whle tll epng the favorable numercal cot. 6 oreover, t ha been hown recently that numerou functonal, although accurate when t come to total energe, fal to reproduce the true denty. 7 he dffculty n functonal contructon can be attrbuted to the fact that t not eay to fnd the approprate expreon of Hxc contrbuton n term of the auxlary wave functon or the denty. here are everal other approache for dealng wth the quantum many-body problem that alo avod the many-body wave functon, whle the bac varable ued ma t eaer to model the dered phycal quantte. Green functon technque can be ytematcally mproved n accuracy by ncludng hgher-order Feynmann dagram, but are computa- Receved: arch 3, 08 Publhed: July 3, Amercan Chemcal Socety 407 DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,

2 Journal of Chemcal heory and Computaton tonally much more expenve. 8,9 Reduced denty-matrx (RD) functonal theore 0, provde a comprome between accuracy and computatonal cot. In one-body RD (RD) functonal theory, the knetc energy an explct functonal of the RD, thu only part of the nteracton energy need to be approxmated, whle n the two-body cae 0 even the nteracton gven by an explct functonal. Although the explct ue of wave functon can be avoded n thee cae, t tll neceary for the RD to be repreentable by a wave functon. However, the o-called N-repreentablty condton that guarantee an underlyng wave functon aocated wth an RD are anythng but trval. 5 oreover, t not poble to aocate to every RD an auxlary ytem of nonnteractng partcle that would allow one to replace the N- repreentablty condton by a numercally mpler auxlary wave functon, l n the DF cae. he Bogolubov Born Green Krkwood Yvon herarchy, where the tme propagaton of an RD of certan order related to the RD of the next order, uffer from mlar N-repreentablty ue. 6 here are now everal poble way to remedy the abovementoned defcence. For RD theory, t helpful to conder the many-body problem at fnte temperature and ndefnte number of partcle. 7 9 In th cae, the repreentablty condton n term of an enemble of wave functon are known and eay to mplement, and one can even fnd a nonnteractng auxlary ytem that generate the ame RD. Another poblty to contruct approxmate natural orbtal, whch are egenfuncton of ngle-partcle Hamltonan wth a local effectve-potental. 0, On the DF de, bede changng the auxlary ytem for the contructon, 4 a poble way out to nclude the knetc-energy denty a a bac functonal varable along wth the denty, mplfyng the modelng of the exchange-correlaton potental becaue they wll not nclude any more knetc-energy contrbuton. h however mple that an addtonal auxlary potental, whch couple to the knetc-energy denty, ha to be ntroduced. A mlar approach ha recently appeared n a dfferent context, that, n thermal DF, 5,6 where the addtonal auxlary potental correpond to a proxy for local temperature varaton and couple to the entre energy denty, ncludng knetc and nteracton contrbuton. he concept of local temperature wa alo ntroduced n the local thermodynamc anatz of DF. 7 9 Furthermore, t mportant to note that the knetc-energy denty already ued extenvely n DF, for ntance, a an ntegral part of the o-called meta-gga. When treated wthn the generalzed framework, 30 meta-gga lead to a local potental couplng to the knetc-energy denty, whch can be nterpreted a a poton-dependent ma. 3 In th Artcle, we nvetgate the poblty to nclude the knetc-energy denty a a bac functonal varable n DF alongde the denty. he dea that by dong o one can ncreae the accuracy of denty-functonal approxmaton. We nvetgate th by contructng the exact denty functonal of tandard DF and comparng them to the combned knetcenergy denty and denty functonal of th extended approach we call knetc-energy denty-functonal theory (DF). In th way, we want to ae poble advantage of uch an approach when conderng trongly correlated ytem. he o-called knetc contrbuton 3 to the exchange correlaton potental mportant for the decrpton of uch ytem. It ha been hown that tandard DF functonal fal to decrbe the effect of th knetc contrbuton uch a the band narrowng due to nteracton. 33 By ncludng the knetcenergy denty a a fundamental varable, th contrbuton tan nto account explctly. Further, we want to conder the qualty of poble approxmaton cheme to DF baed on a knetc-energy () contructon and tet them n practce. A clear from the extent of the propoed program, th not poble for real ytem. Smlar to nvetgaton of the exact functonal n DF and other extenon of DF, 37 we retrct our tudy to a fnte lattce approxmaton for the Hamltonan, where the partcle are only n pecfc tate/ poton. We therefore conder lattce DF. In th way, we not only avod the prohbtvely expenve calculaton of reference data for realtc nteractng many-body ytem but alo avod mathematcal ue connected to the contnuum cae, l the nonextence of ground tate and nondfferentablty of the nvolved functonal 38,39 or havng to deal wth the knetc-energy operator, whch unbounded. 40 All of the operator that appear on the lattce are Hermtan matrce, whch yeld lowet energy egentate, and exact oluton can be ealy calculated contrary to the contnuum where one alway ha to reort to ba et approxmaton. We alo hghlght how mple approxmaton carry over from our model ytem to more complex lattce ytem and even to the full contnuum lmt. he reult hnt at the poblty to treat weakly and trongly correlated ytem wth the ame mple approxmaton to DF. h Artcle tructured a follow: In ecton we ntroduce our lattce model, defne the denty and knetcenergy denty on the lattce, and hghlght for a mple twote cae that the knetc-energy denty a natural quantty to be reproduced by an extended contructon. We then ntroduce the reultng contructon aumng the extence of the underlyng map between dente and feld. In ecton 3 we dcu thee mappng and how how by allowng a patally dependent ma/hoppng a large gauge freedom ntroduced. Stll we can provde a bjectve mappng between dente and feld for pecfc cae. In ecton 4 we then how how we numercally contruct the mappng beyond thee pecfc cae and hence fnd that DF on a lattce can be defned alo for more general tuaton. In ecton 5 we then ue the contructed mappng to determne the exact correlaton expreon for the and the contructon, repectvely. In ecton 6 we then compare the reult of elfcontent calculaton for mlar approxmaton for the and the ytem, repectvely. Fnally, we conclude n ecton 7.. FORULAION OF HE LAICE PROBLE In the followng, we conder quantum ytem contng of N Fermon (electron) on a one-dmenonal lattce of dcrete te. We aume that thee partcle can move from te to te only va nearet-neghbor hoppng (correpondng to a econd-order fnte-dfferencng approxmaton to the Laplacan) and employ zero boundary condton for defntene (the extenon to perodc boundary condton traghtforward). h lead to a Hamltonan of the followng type: σ σ ( h.c.) H = t c c + + vn + U n n =, σ=, + = = Artcle () 4073 DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,

3 Journal of Chemcal heory and Computaton he nonlocal frt term correpond to the knetc energy. Wthout lo of generalty, we can aume that the hoppng ampltude obey t > 0. Let u pont out that uually the hoppng ampltude te-ndependent. We employ th more general form (correpondng to a te-dependent ma) to etablh the neceary mappng (ee eq 0). However, when we numercally conder nteractng ytem, we alway employ a te-ndependent hoppng, whch correpond to the tandard Hubbard Hamltonan. he econd term correpond to a local calar electrotatc potental v actng on the charged partcle at te. U 0 the on-te Hubbard nteracton between the Fermon, whch remncent of the Coulomb nteracton. Further, the Fermonc creaton and annhlaton operator obey the antcommutaton relaton {c σ î,c σ ĵ }=δ j δ σσ, where σ correpond to the pn degree of freedom of the partcle, n σ î = c σ î c σ î the pn-denty operator, and n î = n î + n î the denty operator that couple to the electrotatc potental. Becaue we fx the number of partcle, the potental v phycally equvalent to a potental that dffer by only a global contant. In the followng, th arbtrary contant fxed by requrng v = 0 = () Now, f Ψ the ground-tate wave functon of Hamltonan, we can aocate to every pont n pace a ground-tate denty n = Ψ n î Ψ. From the lattce-veron of DF, 4 we know that for every fxed et of parameter (t,u), there a bjectve mappng between the et of all poble potental (n the above gauge) to all poble dente for a fxed number of partcle. o eae notaton, we ntroduce a vector for the denty n (n,...,n ) and accordngly for the potental v (v,...,v ), whch allow u to wrte the underlyng mappng a : n v. Accordngly, for the potental of an nteractng ytem (U > 0) a a functonal of the denty, we wrte v [n]. We further note that becaue the total number of partcle fxed to N, the denty contraned by = n = N. h mean that ntead of the denty at every pont one can equvalently ue the denty dfference between te Δn = Ψ n î n î+ Ψ to etablh the above mappng at a fxed number of partcle. Smlarly, knowng the local potental v at every te together wth the gauge condton equvalent to knowng Δv = v v +. In certan tuaton, for example, for fgure, t more convenent to ue the denty and potental dfference ntead of the denty and potental. Clearly a mlar mappng between denty and potental alo hold for a nonnteractng Hamltonan, that, U = 0. Becaue t bjectve, we can nvert the mappng and fnd a potental v (where we follow the uual conventon and denote the potental of a nonnteractng ytem wth an ) for a gven denty n. he nonnteractng mappng allow one to defne v [n], whch n turn lead to [ ] = σ σ + + H n t( c c h.c.) v [ n ] n =, σ=, + = he nonnteractng Hamltonan reproduce the precrbed denty n a t ground tate by contructon. h not yet the contructon, becaue we need to know the target denty n advance. Only upon connectng the nteractng wth the nonnteractng ytem by ntroducng the Hxc potental: (3) Hxc v [ n] = v [ n] v[ n] whch can alo be defned a a dervatve of the correpondng Hxc energy functonal wth repect to n, we fnd the nonlnear equaton for a gven and fxed external potental v of the nteractng ytem: = σ σ Hxc H t( c c h.c.) ( v v [ n ] ) n =, σ=, + = (4) h problem ha a the unque oluton the nonnteractng wave functon that generate the denty of the nteractng problem wthout knowng t n advance. 4 It mperatve at th pont to undertand the (often overlood) dfference between H [n] together wth the denty functonal v [n] and H together wth the -potental functonal v [v;n] =v + v Hxc [n]. Only the latter provde an teratve cheme to predct the denty of an nteractng reference ytem. Alo, only at the unque fxed pont of the teraton procedure, where v = v [n], do both Hamltonan gve re to the ame nonnteractng wave functon. When we later preent reult for the exact contructon n ecton 5, we refer alway to the reult at the unque fxed pont of the contructon. In practce, however, we do not have the exact v [v;n] avalable, and hence we need to deve approxmaton to the unknown Hxc functonal. he mplet uch approxmaton would be a mean-feld anatz of the form v Hxc [n] Un. o comply wth the choen gauge of eq, we could ue v Hxc [n] U(n N/ ). A we wll ee n ecton 5, the major problem n thee approxmaton that the knetc-energy denty of the and the nteractng ytem become dramatcally dfferent wth an ncreang U. Here, the knetc-energy denty at te defned nonlocally (becaue t nvolve the hoppng) wth the help of the frt off-dagonal of the (pn-ummed) RD n te ba repreentaton: = t ( + ) (5), + +, (6) where,+ gven by σ σ σ σ, +, + j, j, j, j σ = Ψ Ψ wth = and = c c By analogy to the contnuum cae, one can alo defne the charge current J a J = t( ) Artcle (7), + +, (8) Wth no external magnetc feld preent, that, no complex phae of the hoppng ampltude, the ground-tate wave functon are real valued, whch mple,+ = +,, leadng to zero current. We note that the current obey the lattce veron of the contnuty equaton: n = J (9) n a tme-dependent tuaton, where J = J J the backward dervatve of J. Equaton 9 an equaton of moton (EO) (ee alo Appendx B for further EO) that phycal wave functon need to adhere to. It mportant to note that t not only the varatonal (mnmum-energy) prncple that ground tate have to fulfll, but there are many more exact relaton. Whle for the cae of the ground tate the EO of eq 9 trval becaue both de are ndvdually zero, there are 4074 DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,

4 Journal of Chemcal heory and Computaton many other nontrval exact relaton that can be baed on EO and that provde u wth exact relaton between the dente, the feld, and other phycal quantte. For ntance, the econd-tme dervatve of the denty provde u wth the local force balance of the equlbrum quantum ytem, 43 whch we wll ue n ecton 4. Alo, whle the mot common way to fnd approxmaton for the Hxc potental by obtanng approxmate Hxc energy expreon and then takng a functonal dervatve, the EO provde an alternatve way to contruct approxmate Hxc potental wthout the need to perform functonal varaton. 44,45 In ecton 5 we wll how how one can obtan uch approxmaton, for ntance, the exact-exchange approxmaton of tandard DF. Clearly, f we could enforce that an auxlary nonnteractng ytem ha the ame RD a the nteractng one, then alo the knetc-energy dente of the two ytem would concde. h ugget that one can etablh a mappng between the nteractng RD and a nonlocal potental, that, a v,j that connect any two te of the lattce and thu couple drectly to the full RD. However, n general th not poble a ha been realzed early on n RD functonal theory. 46 A concrete example the two-te homogeneou Hubbard problem at half fllng formng a nglet. In th cae, we have =, and v = 0. So we have a homogeneou denty n =, and we can analytcally determne all egenfuncton of the nteractng and nonnteractng ytem. Further, n the cae of only two te, the full pn-ummed RD a matrx, where the dagonal are merely, = n = and the offdagonal are gven explctly by = 4t =,, (4 t) + U. Becaue the denty fxe the potental of the nteractng and ytem to be exactly zero, our only freedom to adopt the nonlocal potental, whch equvalent to jut adoptng the hoppng of the ytem (n th cae, the nonlocal potental v, t). Yet becaue the off-dagonal for the ytem are, =, rrepectve of the hoppng ampltude, no nonlocal potental ext that reproduce the nteractng RD. h alo true n more general lattce tuaton a ha been hown n, for example, ref 47. For the RD, two oluton to th problem are known. One to nclude temperature and pobly an ndefnte number of partcle, whch ntroduce off-dagonal that depend on the temperature and the hoppng, that, the nonlocal potental. 9 We note that for the homogeneou two-te cae, th can tll be olved analytcally and verfed explctly. he other poblty to ma the ytem degenerate uch that we can reproduce any denty matrx. 9 Here, we apply a dfferent trategy. Whle we cannot force the denty matrce to concde, t poble to requre the knetc-energy dente to be the ame. he crucal dfference that we nclude the couplng n the Hamltonan n the defnton of the quantty to be reproduced by the ytem. For example, n the two-te cae, we merely need to ue an nteracton-dependent hoppng t = 4t (4 t) + U. hu, the auxlary nonnteractng ytem reproduce now the par (n,) of the nteractng ytem. Before we move on, let u note that mlarly to the contnuum cae, one could ue RD functonal theory at zero temperature alo on the lattce f one avod the ue of a nonnteractng auxlary ytem and merely ue functonal baed drectly on the nteractng RD. 48,49 Note that N-repreentablty condton would tll need to be enforced n uch a cheme. Let u now aume that mlar to DF we can etablh a bjectve mappng: : ( v, t) ( n, ) (0) whch would allow u to defne hoppng parameter and potental that generate a gven knetc-energy denty and denty, that, (t [n,], v [n,]). Specfcally we can then conder a nonnteractng auxlary problem that generate a precrbed par (n,): H [ n, ] = t [ n, ] ( + h.c.) + v [ n, ] n =, + = () by t ground tate. Whether we can contruct uch an auxlary ytem that reproduce the denty and knetc-energy denty of an nteractng ytem omethng we do not know a pror. In th Artcle, we provde numercal evdence a well a proof for pecfc tuaton that ugget that uch a contructon poble (ee ecton 3). If we ntroduce then the correpondng mappng dfference mlar to eq 4 and denote them by mean-feld exchange-correlaton (xc): xc v [ n, ] = v [ n, ] v[ n, ] xc t [ n, ] = t [ n, ] t[ n, ] we fnd the correpondng ytem: xc H = ( t + t [ n, ] )( + h.c.) =, σ=, + ( v + v [ n, ] ) n =, + xc uch that = Ψ t Ψ + c.c. and, +, + = t [ n, ] Φ Φ + c.c. n = Ψ n Ψ = Φ n Φ Artcle () (3) (4) (5) (6) where Φ the correpondng ground tate. h contructon gve re to the hoppng t [t;n,] and the potental v [v;n,]. Smlarly to tandard DF, t mportant to realze the dfference between H [n,] together wth (t [n,],v [n,]) and the Hamltonan H together wth the functonal (t [t;n,],v [v;n,]). Only the latter provde an teratve cheme to predct the phycal par (n,) of the nteractng reference ytem. At the unque fxed pont of the teraton procedure, where v = v [n,] and t = t [n,], both Hamltonan concde and gve re to the ame nonnteractng wave functon. When we n the followng preent reult for the exact contructon, we refer alway to the reult at the unque fxed pont of the contructon. h alo allow u n the followng to only ue t and v to hghlght the dfference between the uual and the contructon. o ma the cheme practcal, we now need two approxmaton: one for the xc potental and one for the xc hoppng. Poble route on how to contruct approxmaton and how th could help to more accurately capture trongly correlated ytem we conder n ecton DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,

5 Journal of Chemcal heory and Computaton At th pont, we want to ma a frt connecton to the contnuum by conderng the approprate choce of the knetc-energy denty for that cae. here are dfferent poble defnton for a local knetc-energy denty, whch wll gve re to the ame total knetc energy. 50 For ntance, we can chooe the gauge-ndependent defnton 5 ( r) = d r..d r ( ) Ψ ( r, r,.., r ) m( r) N r N, where Ψ correpond to the nteractng wave functon, uch that the knetc-energy denty potve at every pont n pace. Here, we have defned a patally dependent ma m(r) > 0 that ta the role of the te-dependent hoppng n the lattce cae. For the nonnteractng ytem, the correpondng knetc-energy denty (provded we aume a Slater determnant) read N () r = ϕ() r m ( r) =, where m (r) > 0 the patally dependent nonnteractng ma and ϕ the ngle partcle orbtal. he ngle-partcle knetc-energy operator then Ä É become accordngly, where m(r) hould be ÇÅ m() r ÖÑ ubttuted wth m (r) n the nonnteractng cae. 3. GENERALIZED APPINGS FRO DENSIIES O POENIALS Smlarly to fxng the contant of the local potental, one need to fx the gauge of the hoppng parameter t () (where the upercrpt n parenthee ued to denote that we refer both to nteractng and to nonnteractng ytem). One of the frt thng to note that by lettng t () change from te to te, we encounter a large equvalence cla for the tedependent hoppng parameter. Indeed, we can arbtrarly change the gn of the hoppng from t () t () wthout changng the denty and the knetc-energy denty. However, the wave functon and alo, for example, the RD change. For ntance, for the nonnteractng ngle partcle Hamltonan, we ee that changng the gn locally, ay at te, wll tranform the ngle-partcle wave functon at th te ϕ to ϕ (ee Fgure for an example and Appendx D for further detal). h leave the denty unchanged, a t jut a um of the quared abolute value of the ngle-partcle wave Fgure. Doubly occuped orbtal ϕ that correpond to a twoelectron nglet-tate of a ngle partcle Hamltonan wth all hoppng parameter t = t potve and the correpondng one ϕ t t wth alternatng hoppng parameter ±t from te 7 to 9. Every tme we alternate t to t at te, the orbtal ϕ change gn from that te on. Becaue we replace t to t from te to te, the orbtal wll recover t orgnal gn after two te. A one can readly ee, the denty tay the ame n both cae, a a conequence of the gn of t beng only a gauge choce. Artcle functon. Alo, the knetc-energy dente tay the ame, becaue the RD wtche gn at the ame place a the hoppng ampltude. A t follow from the dcuon above, the gn of t () jut a gauge choce, and we need to fx the gauge to etablh the ought-after mappng. In the followng, we chooe t () >0. A further complcaton that one encounter n etablhng the neceary mappng that the uual Hohenberg Kohn approach doe not work n our cae. he reaon that the control feld t now become explctly part of the control object. A mlar problem encountered n current-dentyfunctonal theory, when tryng to etablh a mappng n term of the gauge-ndependent phycal charge current Whle n the tme-dependent cae havng the feld a part of the control objectve actually an advantage and a general proof ha been etablhed, 55 thee complcaton unfortunately prohbt a mple general proof of the extence of the mappng (n,) (v,t) for the tme-ndependent cae. However, for pecfc tuaton, we are able to how that the dcued mappng poble. he mot mportant one n our context the cae of the two-te Hubbard model (ee Appendx A for detal). In th cae, we only have a ngle potental dfference Δv and denty dfference Δn. So we can mply recale the auxlary Hamltonan and thu prove the extence of the mappng n the nonnteractng cae by employng the Hohenberg Kohn reult. A further mple cae two nonnteractng partcle, formng a nglet, n a general -te lattce. Here, the denty fxe the ngle-partcle orbtal (doubly occuped) up to a gn, and thu for a gven only a unque te-dependent hoppng t poble. Fnally, n the homogeneou cae, where the local potental v = 0 and perodc boundary condton are employed, the denty N n = and the knetc-energy denty of the nteractng ytem wll be contant at every te, =. he matrx element,+ wll alo be contant from te to te,,+ =.In th cae, the mappng nvertble and a unque (up to a gn choce) t = aocated from te to te. Note that n th cae the ytem and the ytem yeld the ame t wave functon and = t. h lat example, although t only how the nvertblty of the mappng (v,t) (n,) at the pecfc pont t = t > 0 and v = 0, ha very mportant conequence. It allow u n a mple yet exact way to connect the auxlary ytem to the nteractng ytem. We wll ue th later to contruct a frt approxmaton to t xc. o how that the DF mappng can alo be defned for other, more general cae, we contruct n the followng the mappng numercally. Afterward, we ma ue of the contructed mappng to nvetgate the properte of the xc potental and the bac functonal, whch for the contnuum cae would be numercally prohbtvely expenve. 4. INVERSION OF (n,) Becaue, a dcued above, t not traghtforward to how that the mappng 0 : n general, we nvetgate th queton numercally. herefore, we contruct et of dente and knetc-energy dente (n,) by olvng the nteractng problem pecfed by the Hamltonan gven n eq (wth a te-ndependent hoppng, whch correpond to the uual Hubbard Hamltonan), and for every et we determne the potental (v,t) of the nonnteractng Hamltonan pecfed n eq 4, whch yeld the target dente (n,). o determne 4076 DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,

6 Journal of Chemcal heory and Computaton thee potental, we et up an nveron cheme by ung the EO for the denty and the knetc-energy denty, repectvely. hee provde not only phycal relaton that connect the quantte (v,t) wth (n,), but they are alo utable to defne correlaton potental, a we wll explan n the followng. Note that n prncple the nveron can be done wth other technque, whch are ued to fnd the exact local potental for a gven nteractng target denty However, t not traghtforward how to tranfer thee technque to the current tuaton. For ntance, n ref 57, an teraton cheme ntroduced that adopt the potental baed on the ntuton that where the denty too low the potental made more attractve and where the denty too hgh t made le attractve. It not o clear how to tranfer th ntutve procedure to the knetc-energy denty, whch nonlocal, and the control feld part of the obervable telf. In the contnuum, one could perform an nveron and defne the correpondng auxlary potental agan by EO. 4,6 Another poblty would be to explot technque where the knetc-energy denty of the and the nteractng ytem are ued to model the exchange-correlaton potental. 60 Becaue the frt-order EO for the denty, that, the contnuty eq 9, trvally atfed a the current jut zero n the ground tate, we conder the econd tme dervatve of the denty n. Becaue the frt tme dervatve of the knetcenergy denty vanhe for ground-tate wave functon, we ue agan the econd-order EO. A example, we gve here the EO for n and for two te n the nonnteractng cae that we ue n our numercal nveron cheme: n = ( t ) Δn Δv = Δv (( t ) Δn Δ v ) = Δv n (7) (8) In Appendx B, the general expreon for any number of te can be found. Here, we have dropped the te ndex becaue everythng correpond to te, Δn =(n n ) the denty dfference between the two te, Δv =(v v ) the local potental dfference, and = t,. A one can readly ee for the two-te cae, there no addtonal nformaton n the equaton for, a once n 0 alo 0. Neverthele, once we go to more te, wll alo gve u new equaton. For a detaled dcuon of th ue, ee Appendx B. he nveron cheme we employ an teratve procedure baed on the above ntroduced EO (ee eq 44 and 48 n Appendx B for the general expreon), whch provde u wth relaton between (Δv,t ) and the target quantte (n,). We obtan the target quantte (n,) byfndng the ground tate of the correpondng nteractng Hamltonan of eq, wth a poton-ndepent hoppng t = t. We then chooe a an ntal gue for the auxlary ytem the value of the nteractng ytem v,0 = v and t,0 = t. (a) We olve the auxlary nonnteractng Schro dnger eq wth the value v,0 and t,0 : j k = y,0,0,0,0 t ( + h.c.) + v n Φ = ε Φ, + z = { (9) (b) We next calculate the denty and knetc-energy denty that correpond to the tate Φ,0, that, n 0 = Φ,0 n î Φ,0 and 0 = t,0 Φ,0 î,+ Φ,0 a well a the matrx element 0,j that enter the EO 44 and 48. (c) In a lat tep, we then calculate the varable of the next teraton v, and t,. he EO for n =0ofeq 44 provde u wth analytc expreon of v, n term of the target dente, the hoppng ampltude t,0, and the reduced denty matrx element 0,j of the prevou teraton. For calculatng the t,, we ue a numercal olver on all of the avalable EO for n = 0(eq 44) and = 0(eq 48), wth the target knetc-energy dente, but updated dente n 0 0 and,j from the lat teraton and the renewed local potental v,. We repeat tep (a) (c) untl convergence of the calculated feld. A an example n the two-te cae, one can update n every teraton the local potental:, Δ v =, ( t ) Δn (0) and the hoppng parameter: t, Δv = j k Δn, / y z { Artcle () where Δn the target denty dfference between the two te and the target knetc-energy denty. We want to pont out that the procedure to update v, and t, not the only one poble. For example, one could have ued ntead of the EO that we get for n = 0 the one for J = 0. Further note that there are alway ndependent equaton from n = 0 becaue of partcle number conervaton, thu a many a the ndependent v that we have (although t not clear that we need all of them a we do not have a lnear ytem of equaton). he number of EO that we get for the knetc-energy denty, a we explan n Appendx B. he nteractng ground tate wa obtaned ung the nglete DRG 6 routne, mplemented n the Syen toolkt. 63 We uccefully performed nveron for ytem of up to four te wth dfferent total number of electron for dfferent on-te nteracton trength U and local potental v. Some repreentatve reult for half-fllng are hown n the next ecton, where we ue the contructed mappng to conder the exact ytem. Note that we alo uccefully performed nveron beyond half-fllng. We alo performed ucceful nveron for the ame ytem for the nteractng problem; that, we choe random value (n,) and reproduced them wth a nonzero Hubbard nteracton. h ma the equaton nvolved lghtly more complex (and we refraned from howng them here explctly), but the nveron procedure tay the ame. he fact that we could ndeed contruct a auxlary ytem for thee cae a well a perform nveron for the nteractng problem provde u wth ndcaton for the extence of a ytem for an arbtrary number of electron/te. 5. COPARING HE EXAC AND CONSRUCION Next, we ae the practcal mplcaton of ung the knetcenergy denty a bac functonal varable along wth the denty. Frt, we ue the contructon of the exact ytem and the correpondng ytem to compare the Hxc energy E Hxc of the ytem wth the correpondng quantty E xc of the ytem. h gve u a frt ndcaton of whether a 4077 DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,

7 Journal of Chemcal heory and Computaton approach mght help to capture alo trong correlaton effect more ealy. For the ytem, the Hxc energy Hxc E = E vn g = = () where = t Φ î,+ Φ the knetc-energy denty of the ytem and Φ t ground-tate wave functon. By E g, we denote the total ground-tate energy of the nteractng ytem and v t external potental. he correpondng energy contrbuton of the ytem read xc E = E vn g = = (3) where = t Φ î,+ Φ. Becaue the knetc energy of the ytem dentcal to the nteractng one by contructon, the E xc E nt = U = Ψ n î n î Ψ equal only to the nteracton energy n th cae. he correpondng term of the ytem nclude knetc-energy contrbuton a well. In Fgure, we plot E xc and E Hxc for a Hubbard dmer at Fgure. Hxc energy E Hxc (dahed lne) and the correpondng energy term of the ytem E xc (contnuou lne) for a Hubbard dmer at half-fllng wth local potental Δv/t = a a functon of U/t. We ee that for U > 0 t hold that E xc < E Hxc. half-fllng wth local potental Δv/t = a a functon of the nteracton trength U/t. Note that the data from the numercal nveron are ued. hu, both energy quantte are exact, and there no approxmaton nvolved. In Fgure 3, we how the correpondng plot for a four-te Hubbard ytem at half-fllng wth Δv /t = Δv 3 /t = 0.65 and Δv /t = hee two ytem for two-te and fourte wll erve a our tet ytem, and n the followng we wll refer to them a the two-te cae and four-te cae, repectvely. A one can readly ee n Fgure and 3, for every nteracton trength U > 0 t hold that E xc < E Hxc.Inthe trong correlaton lmt, the knetc-energy of the ytem far from the nteractng one. Havng n mnd the followng relaton: xc Hxc E E = 0 (4) t become apparent why E xc and E Hxc are o dfferent for trong nteracton. A a conequence, the exchangecorrelaton potental derved from E Hxc wll need to ta nto account th dfference n the trong nteracton regme. In the ytem, on the other hand, one need to ntroduce a econd feld t xc, whch reponble for reproducng the Fgure 3. Hxc energy E Hxc (dahed lne) and E xc (contnuou lne) for a four-te Hubbard model at half-fllng wth local potental Δv /t = Δv 3 /t = 0.65 and Δv /t = a a functon of U/t. We agan fnd that for U > 0 t hold that E xc < E Hxc. knetc-energy denty along wth the potental v xc that enure the denty reproduced. Furthermore, due to the fact that E xc doe not contan knetc contrbuton, t offer a mple calng relaton n contrat to E Hxc. 64 Whle the energy functonal are an nteretng frt ndcaton that the approach can be ueful to treat alo trongly correlated ytem, the real quantte of nteret are the effectve feld that the and contructon employ, epecally thoe part of the Hxc potental and of the xc hoppng and potental that are not acceble by mple approxmaton tratege. hoe part, whch one uually aume to be mall n practce, we wll denote a correlaton term. Let u n the followng, baed on the EO we ued to derve the teraton cheme, defne part of the effectve feld that we can expre explctly n term of the and wave functon. Smlar contructon baed on the EO of the denty have been employed n DF and DDF. 44,45 For mplcty, we preent the expreon only for the two-te cae. he expreon for four-te are gven n Appendx C. he Hxc potental defned a Δv Hxc [n] =Δv [n] Δv[n] (eq 4), where n the target denty of the nteractng ytem, Δv [n] the local potental dfference of the ytem, and Δv[n] jut the external potental of the nteractng ytem. he EO for the nonnteractng/nteractng denty, eq 7/ 53, provde expreon for the local potental Δv [n] and Δv[n] of the nonnteractng/nteractng ytem. hu, the Hxc potental n the two-te cae read Hxc t Δn t Δn Δ v [ n] = Ut Ψ ( n n ) Ψ, We can decompoe Δv Hxc Δv Hx [n,φ ]: (5) n a Hartree-exchange part Hx Ut Δ v [ n, Φ ] = Φ ( n n ) Φ, Artcle (6) whch correpond to the uual Hartree plu exchange approxmaton n tandard DF, and a remanng correlaton part: 4078 DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,

8 Journal of Chemcal heory and Computaton t Δn t Δn Δ vc [ n, Φ] = Ut Ψ ( n n ) Ψ, Ut + Φ ( n n ) Φ, (7) Here, we nclude the wave functon n the functonal dependence to hghlght that t an orbtal functonal; that, t depend on the wave functon. We note, however, that n the exact cae the wave functon unquely determned by the denty. he above decompoton mlar to that of the contnuum cae ntroduced n ref 44 and later ued n, for example, ref 45 and 65. Ineq, we have defned the xc potental v xc for the ytem, whch n the two-te cae (by ung the ame EO a before) read xc t Δn t Δn Δ v [ n, ] = Ut Ψ ( n n ) Ψ, (8) We ee that the frt two term are completely determned by the ytem, contrary to Δv Hxc of the ytem, where the econd term cannot be gven n term of Δn or Φ explctly. he frt term, however, depend explctly on the hoppng n the ytem, whch ha to be approxmated n practce. Keepng th n mnd, one can dentfy a mean-feld exchange potental, mlarly to the Hartree-exchange potental of the ytem: Ut Δ vc [ n,, Φ ] = Ψ ( n n ) Ψ, Ut + Φ ( n n ) Φ, Artcle (30) We tre that the defnton of the correlaton contrbuton to the local potental 30 only contan part of what uually referred to a correlaton n the context of DF. he ocalled knetc correlaton tan care of n the hoppng ampltude t and va defnton 9 n v x.infgure 4 we plot Fgure 4. Correlaton potental of the Δv c (contnuou curve) and ytem Δv c (dahed curve) for the two-te cae a a functon of the nteracton trength U/t. Apart from a mall regon at vanhng nteracton trength U, Δv c < Δv c. for the two-te cae the correlaton and potental, whch are gven by eq 7 and 30, repectvely. In Fgure 5 we x t Δn t Δn Δ v [ n,, Φ ] = Ut Φ ( n n ) Φ, (9) whch depend explctly on the denty, the knetc-energy denty, and the ground tate of the ytem. Let u at th pont remark that f there no approxmaton for the hoppng parameter nvolved, that, when t = t, the expreon of v x n eq 9 dentcal to the expreon for v Hx n eq 6. In the mean-feld approxmaton,,+,+, t follow mmedately that t = t, becaue we requre the knetc-energy dente to be the ame, that, t,+ = t,+. We note that for the exact cae we conder here, that, at the oluton pont of the exact nonlnear equaton (ee alo dcuon about the contructon below eq 6), t can be explctly gven n term of t and the exact Φ. In practce, however, we do not know t [t,n,] =t + t xc [n,] a pror, and we need to nclude further an extra approxmaton for t xc [n,]. Whch approxmaton are poble (and how accurate they are) wll be dcued next, and n the followng ecton we wll ee how the practcal form of v x [n,,φ ], that, ncludng an approxmate t xc [n,], perform. he remanng local potental correlaton term contan now only contrbuton from the dfference n nteracton: Fgure 5. Correlaton potental of the Δv c, (dahed) and ytem Δv c, (dotted-dahed) for the four-te cae a a functon of the nteracton trength U/t. Agan, we fnd that, apart from a mall regon at vanhng nteracton trength U, Δv c, < Δv c,. plot the correlaton potental for the four-te cae, whch are gven for the ytem by eq and for the by eq A one can readly ee for the two te, the correlaton potental maller n abolute value for the ytem than n the one for all nteracton trength teted, apart from a mall regon at vanhng nteracton. h follow from the fact that the knetc contrbuton are ncluded n the mean-feld exchange potental Δv x n the cae. For the four te we ee the ame trend. However, n the contructon, we have a econd effectve feld, whch we o far dd not ta nto account n our comparon. One need to fnd an analogou decompoton nto a term that correpond to the v Hx n the cae, whch can be approxmated wth a relatvely mple 4079 DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,

9 Journal of Chemcal heory and Computaton functonal, and a term that requre more advanced approxmaton, n correpondence to v c. Of coure, f the latter part large a compared to the former, we dd not gan anythng by ntroducng the addtonal feld t. Startng from the defnton of t xc [n,] =t [n,] t [n,], and ung the fact that the knetc-energy denty ha to be the ame n the nteractng and ytem, we get that y xc [ ] = j [ ] t n, n, [ n, ] z k, +, + { (3) whch by ubttutng = t [n,],+ [n,] and reorderng the varou term yeld the form: t δ [ ] = [ n, ] + t [ n, n ] xc,, [ n, ] (3), + where we have defned δ,+,+,+. Up to here, there no approxmaton nvolved. A the term δ,+ nvolve the oluton of an nteractng and nonnteractng problem, an approxmaton baed on a reference oluton ugget telf. he mplet uch reference oluton would be to ue the homogeneou cae of the nteractng and the ytem, repectvely, mlar to the local-denty approxmaton n tandard DF. Becaue n the homogeneou cae wth perodc boundary condton, a dcued n ecton 3, the and the denty matrce are the ame, we can drectly ue wellknown reult uch a the Bethe-anatz oluton at half fllng. In th way, t become alo traghtforward to extend the ntroduced approxmaton to the contnuum cae, where we can ue reference calculaton for nteractng homogeneou contnuum ytem. Let u alo do the ame unform approxmaton for the zero boundary condton cae that we dcu here, although the unform and denty matrce wll not be the ame apart from the two-te cae. Here, we aume that we have reference data for the homogeneou problem for dfferent local hoppng parameter t > 0, local fllng 0 < n <, and for the local nteracton U > 0. Further we gnore the dependence of the t n the numerator on the nternal par (n,) and ue an explct dependence on Φ n the denomnator. In the followng, we wll mplfy the explct part even further and wll jut ta the homogeneou oluton at half fllng, that, calculate δ,+ for two-te and four-te cae wth dfferent U. he update formula for t xc, unf whch we wll denote a t to tre that the approxmaton come from ung unform reference data for the RD, read: unf tδ unf, + t [Φ ] = (33), + Let u pont out here that when 33 ued n a elf-content loop, the t unf updated ung the,+ of the prevou teraton a the enumerator fxed and tan from the reference calculaton. In the two-te cae uch an anatz eem approprate, becaue depte the zero-boundary condton the and the ytem are the ame by contructon. For the four-te cae, however, the zero-boundary condton ma the and denty matrce dfferent. Hence, the fourte cae a very challengng tet for the accuracy of uch a mple approxmaton. In accordance to the above ntroduced approxmaton, we wll then defne the remander of the hoppng feld a t c, [n,,φ ]=t xc [n,] t unf [Φ ], where we emphaze that trctly peakng t c, nclude correlaton Artcle beyond the correlaton preent n a unform ytem. In Fgure 6 we plot the (beyond unform) correlaton hoppng feld t c /t Fgure 6. Correlaton part of the hoppng t c, n unt of t, for the twote cae a a functon of nteracton trength U/t. For trong nteracton trength, the ytem reemble a homogeneou one o that the unform type of approxmaton we employed become very good. a a functon of the nteracton trength U/t for the two-te cae. We ee that the value of t c mall a compared to the choen t for all nteracton trength and epecally for weak and trong nteracton. For trong nteracton, the ytem reemble a homogeneou one a the nteracton trength become more promnent n comparon to the local potental dfference, and thu t c become maller n th regme. From th we can nfer that for the cae of a general ytem wth perodc boundary condton, the homogeneou anatz wll capture not only the weak but alo the trong-nteracton lmt accurately. In Fgure 7 we turn to the more challengng cae of four te wth zero boundary condton and plot the three dfferent t c, /t Fgure 7. Correlaton part of the hoppng t c, n unt of t, for the fourte cae a a functon of nteracton trength U/t. component a a functon of the nteracton trength. A one can readly ee, all three t c, are mall for every nteracton trength U. However, only two of them eem to converge to a value that cloe to zero for trong nteracton, at leat for the parameter range we nvetgated. (We want to pont out that for larger value of U we encountered ome convergence ue n the four-te cae. he reaon that whle the denty matrce are not homogeneou, the denty, whch caue ome problem n the teraton cheme, where we dvde by the denty dfference between two neghborng te n each teraton. h problem, however, can be potentally overcome by ung dfferent update equaton.) We remnd the reader 4080 DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,

10 Journal of Chemcal heory and Computaton Artcle that th dfference n accuracy between the two-te and the four-te cae not urprng. In the two-te cae, the unform,+, whch ued to contruct the approxmaton for t unf, concde wth the correpondng unform,+ of the ytem. For four te th no longer the cae. Stll, comparng the numercal value of the correlaton hoppng t c, to thoe of the correlaton potental v c,, there an order of magntude dfference. h gve ome hope that crude unf approxmaton l the t can tll lead to accurate predcton. Let u tet th n the followng ecton. 6. COPARING A SELF-CONSISEN AND CALCULAION Whle the above conderaton about the exact correlaton energe, potental, and hoppng parameter are crucal to undertand what the dfferent approxmaton to the unknown exchange-correlaton term are able to capture, t not ther performance at the exact oluton that matter n practce. A elf-content calculaton wth the approxmate functonal not at all clear that wll converge to a enble oluton or even converge at all. For ntance, even for the prme example of a nonlnear problem n quantum chemtry, that, the groundtate Hartree Fock equaton, the convergence to a unque oluton ha not been hown except for hghly unuual cae. 66 o fnally tet whether the propoed DF and t contructon can be ued n practce to predct the properte of correlated many-electron ytem, we perform elfcontent calculaton for our two-te and four-te Hubbard model. We ue the mean-feld exchange approxmaton of eq 9 for two te and of eq 69 7 for four te together wth the unform approxmaton for the hoppng term of eq 33. h lead to j k = unf ( t + t [Φ ])( + h.c.), + y x + ( v + v [ n,, Φ ]) n Φ = ε Φ z = { (34) where we update the nvolved effectve feld n every teraton untl convergence acheved. We then compare the dente and knetc-energy dente that we get wth the one wthn the exact-exchange approxmaton (thu t = t and v Hx gven by eq 6 for two te and eq 60 6 for four te). We do o a n th way we have for both the and the contructon the ame level of approxmaton a v x wll reduce to v Hx for t = t. h allow u to judge whether ncludng the knetc-energy denty n the modelng of many-partcle ytem ha any advantage over the uual denty-only approach. We frt quantfy the denty dfference between the calculated quantte and the exact one ung the followng meaure: δn / = = n / n, where n the nteractng denty at te whle n / the correpondng denty of the / ytem. Indeed, for both the two-te cae (ee Fgure 8) a well a the more challengng four-te cae (ee Fgure 9), the elfcontent approxmaton perform better than the correpondng elf-content exact-exchange approxmaton. Becaue the man dfference le n the error correcton to the local knetc-energy denty, we next alo compare a meaure for the dfference n local knetc-energy denty: Fgure 8. Denty dfference δn / between the elf-content calculaton n the ytem and the exact one (contnuou, blue lne), a well a for the elf-content oluton n the ytem and the exact one (dotted, red lne), for the two-te cae a a functon of nteracton trength U/t. Fgure 9. Denty dfference δn / between the elf-content calculaton n the ytem and the exact one (contnuou, blue lne), a well a for the elf-content oluton n the ytem and the exact one (dotted, red lne), for the four-te cae a a functon of nteracton trength U/t. δ / = = /, where / the knetc-energy denty between te and +, whle the correpondng nteractng one. Not urprngly, n both cae (ee Fgure 0 and ), the approxmate knetc-energy denty of the ytem much cloer to the actual one than the bare energy denty. We ee that for large nteracton trength the error bacally zero for the two-te cae becaue n th lmt the nteracton much larger than the aymmetry nduced by the local potental. In the four-te cae, our approxmaton for the knetc-energy denty not a accurate, although t tll better than the correpondng one. he reaon for th drop n accuracy, a dcued around eq 33, the aumpton that unf,,+ = unf,,+, whch volated for the four-te cae. Neverthele, for large ytem (where the boundare wll not be gnfcant) or for ytem wth perodc boundary condton, th ue wll eentally vanh becaue then the unform reference ytem obey unf,,+ = unf,,+ and the trongnteracton lmt captured hghly accurately. Conequently, t can be expected that ncludng the knetc-energy denty can help to treat multpartcle ytem accurately from the weak to trong nteracton regme. 408 DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,

11 Journal of Chemcal heory and Computaton Fgure 0. Knetc-energy denty dfference δ / between the elfcontent calculaton n the ytem and the exact one (contnuou lne), a well a for the elf-content oluton n the ytem and the exact one (dotted lne), for the two-te cae a a functon of nteracton trength U/t. Fgure. Knetc-energy denty dfference δ / between the elfcontent calculaton n the ytem and the exact one (contnuou lne), a well a for the elf-content oluton n the ytem and the exact one (dotted lne), for the four-te cae a a functon of nteracton trength U/t. 7. CONCLUSION AND OULOOK In th work, we have ntroduced a knetc-energy dentyfunctonal theory (DF) and the reultng knetc-energy Kohn Sham () cheme on a lattce. he dea wa that by lftng the knetc-energy denty to a fundamental varable along wth the denty n, the reultng effectve theory become eaer to approxmate becaue more part are known explctly. Becaue the new external feld, a te-dependent hoppng t, part of the knetc-energy denty, the uual Hohenberg Kohntype proof trategy to etablh the neceary one-to-one correpondence between (v,t) and (n,), where v the uual on-te potental, doe not work. However, bede gvng proof for pecfc cae and dcung the gauge freedom of the approach, we provded an ndcaton that the neceary bjectvty hold by numercally contructng the nvere map from a gven par (n,) to(v,t) for two- to four-te Hubbard model. We dd o by ntroducng an teratve cheme baed on the equaton of moton (EO) of the denty and the knetc-energy denty. We then ntroduced a decompoton of the two unknown effectve feld of the cheme, the mean-feld-exchange-correlaton potental v xc [n,] and the mean-feld exchange-correlaton hoppng t xc [n,], nto explctly known mean-feld exchange (for the effectve potental) and unform (for the effectve hoppng) a well a unknown correlaton part. By comparng the unknown part of the tandard Kohn Sham () approach to the approach, we aw that ncludng the knetc-energy denty n the fundamental varable reduced the unknown part conderably. Fnally, we teted the approach n practce by olvng the reultng nonlnear equaton wth the ntroduced approxmaton. We found that the mean-feld exchange and unform outperform the correpondng exact-exchange from weak to trong nteracton and hence hold prome to become an alternatve approach to treat manypartcle ytem effcently and accurately. Whle the preented approach wa thoroughly nvetgated only for mple few-te problem, t extenon to many de, arbtrary dmenon, and even the contnuum traghtforward. Followng ref 50 n the contnuum, we can chooe a gauge-ndependent and trctly potve defnton of the knetc-energy denty wth a patally dependent ma term. he man reaon why the cheme can be more accurate than the uual cheme alo n the contnuum that we can model explctly the knetc-energy denty n th cae. Becaue the mple knetc-energy denty approxmaton we ntroduced proved to be already qute reaonable, the extenon to the contnuum eem epecally promng. For homogeneou ytem, many reference calculaton ext that can be ued to derve a unveral local knetc-energy denty approxmaton that reemble the unform approxmaton ntroduced n th work. APPENDIX A: WO-SIES PROOF OF (Δv,t ) (Δn,) In th Appendx, we provde a proof of the bjectvene of the mappng between denty and knetc-energy denty and the correpondng feld n the cae of a nonnteractng ytem of up to N 3 electron on a two-te lattce. For th cae, the nonnteractng Hamltonan read: = + + Δ v H t ( h.c.) n n, ( ) (35) where Δv and t > 0. he mappng that we wh to how that bjectve the followng: : ( t, Δv ) (, Δn) Artcle (36) wth > 0 and Δn [ N,N]. Here, Δv v v and Δn n n, where the lower ndexe refer to dfferent te pont. Note that t 0 a certan gauge choce a {(t,δv ), ( t,δv )} (,Δn). oreover, due to partcle hole ymmetry, t hold alo that {( t, Δv ),(t, Δv )} (,Δn). hu, from now on, when we refer to dfferent potental and hoppng parameter, we wll mean that they dffer by more than a gn change. For the local potental, the gauge choce v + v = 0 a throughout the document. Proof: We are gong to prove eq 36 through dfferent cae. Cae : wo Hamltonan H, H have the ame hoppng parameter t = t but dfferent local potental Δv Δv. From the Hohenberg Kohn theorem, we then have that the correpondng wave functon are dfferent, that, Φ Φ, and the ame hold alo for the dente Δn Δn and conequently (,Δn) (,Δn ). Cae : wo Hamltonan H, H have dfferent hoppng parameter t t but the ame local potental Δv Δv. Aume that (,Δn) =(,Δn ). We then have two wave functon Φ and Φ that are ground tate of the correpondng Hamltonan: 408 DOI: 0.0/ac.jctc.8b009 J. Chem. heory Comput. 08, 4,