HYBRID EXCHANGE CORRELATION FUNCTIONALS AND POTENTIALS: CONCEPT ELABORATION INTRODUCTION. A. V. Arbuznikov UDC

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1 Jounal of Stuctual Chemsty, Vol. 48, Supplement, pp. S1-S31, 007 Ognal Russan Tet Copyght 007 by A. V. Abuznkov HYBRID EXCHANGE CORRELATION FUNCTIONALS AND POTENTIALS: CONCEPT ELABORATION A. V. Abuznkov UDC Ths pape deals wth hybd functonals that contan eact ehange enegy and ae the most popula and effectve functonals n moden densty functonal theoy. Emphass s lad on genealzaton of the noton of a hybd functonal, whch ases fom the ntoducton of the spatal dependence of the eact ehange admtue (local hybd functonals). Poblems nheent n hybd functonals ae consdeed along wth poblems nheent n a wde class of so-called obtal-dependent functonals. In patcula, the technque fo constuctng the local and multplcatve potentals, ncludng the optmzed effectve potental method, s consdeed n detal. The theoetcal appoaches unde study ae llustated by calculatons of atomzaton molecula eneges and magnetc esonance paametes. Keywods: densty functonal theoy, hybd functonal, local hybd functonal, localzed local hybd potental, optmzed effectve potental method, atomzaton enegy, chemcal sceenng constant, g tenso. INTRODUCTION Ove the past decade o two, densty functonal theoy (DFT) has become one of the most popula methods of electonc stuctue calculatons of atoms, molecules, clustes, solds, etc. [1, ]. The gowng populaty of DFT s pmaly detemned by the combnaton of hgh accuacy, occasonally eeedng the accuacy of stct many-patcle methods, and athe modeate equements to computatonal esouces, whch today allow calculatons fo lage systems of nteest n nanotechnology, namely, bologcally actve molecules contanng hundeds of atoms, etc. Moden DFT s based on two Hohenbeg Kohn s theoems [3]. The fst theoem sets one-to-one coespondence between the electon densty of a many-electon system n the gound state and the etenal potental,.e., the electostatc potental of nucle, whch ae mmoble wthn the famewok of the adabatc appomaton, and the etenal statc electc feld. The second Hohenbeg Kohn theoem s essentally the vaatonal pncple fomulated n tems of electon densty as a basc quantty. In contast to the many-electon wave functon ( 1,,, N ; 1,,, N ) (whee = (, y, z ) ae the spatal coodnates of an electon wth the nde ; s ts spn vaable; and N s the numbe of electons n the system), ncludng a lot of edundant nfomaton, the electon densty 1 N 1 N 1 N 1 N 1 N d d Nd 1 d N ( ) (,,, ;,,, ) (,,, ;,,, ) (1) s pmaly attactve because t s easly pecevable (coesponds to the ntutve concepts of the electonc stuctue of atoms, molecules, etc.) and elatvely smple fom mathematcal vewpont (detemned by a tplet of spatal vaables). Fom the vewpont of pactcal calculatons, howeve, the stuaton s much moe poblematc. Regetfully, an Wüzbug Unvesty, Gemany; abouznkov@mal.un-wuezbug.de. Tanslated fom Zhunal Stuktuno Khm, Vol. 48, Supplement, pp. S5-S38, 007. Ognal atcle submtted Mach 0, /07/48 Supplement Spnge Scence+Busness Meda, Inc. S1

2 equaton smla to the Schödnge equaton wth electon densty nstead of the many-electon wave functon has not yet been fomulated. The majo dffculty les n mpossblty of epessng the knetc enegy of the system wth an acceptable accuacy n tems of densty alone (the fundamental Thomas Fem model [4, 5] faled to povde an accuacy needed fo eal chemcal applcatons). The Kohn Sham fomalsm poved to be a soluton to ths poblem [6]. In ths fomalsm, an magnay many-electon system s ntoduced whch has the same electon densty dstbuton as the eal system n queston, but dffes fom t n the absence of nteelecton nteacton. Ths fcttous nonnteactng system s descbed by a sngledetemnantal wave functon composed of the (KohnSham) one-electon obtals () Nocc (N occ s the numbe of occuped spatal obtals among whch thee may be ecuent doubly occuped obtals) wth whch the total electon enegy of a eal system can be ecoded as Nocc 1 ( ) 1 ( ) ( ) K 1 K RK Nocc Ts( ) V [ ] [ ] 1 ne J E[ ], E () () d Z d dd E [] whee Z K s the chage on an (mmoble) nucleus wth the nde K, and R K ae the spatal coodnates of ths nucleus; the electon densty s defned as Nocc =1 1 () (). (3) In the ght pat of (), the fst tem, T s, descbes the knetc enegy of a nonnteactng system; the second tem, V ne, s the attacton of electons to nucle; and J s the classc contbuton of electon epulson to the enegy. Fnally, E, whch s called an ehange coelaton functonal, s the esdue of the electon enegy of the eal system, namely, the nonclassc contbuton of nteelecton nteacton to the potental enegy and the dffeence between the knetc eneges of the eal and magnay systems: E ( Vee J) ( T Ts ). (4) DFT s an essentally stct theoy povded that an eact ehange coelaton functonal s known. Regetfully, an deal functonal of ths knd s naccessble; constucton of nceasngly eact appomatons to E emans the cental poblem of moden densty functonal theoy. The ehange coelaton functonal s vtually the coe of DFT because t eventually affects the pedctve stength and effcency of the theoy. Ths wok does not clam to gve a full descpton of the evoluton of the ehange coelaton functonals. Instead we attempt to tace the ogn of so-called hybd functonals and pospects fo the development because these ae now the most popula and effectve functonals. We also ty to undestand easons fo the success and to demonstate seveal ptfalls n the ealzaton, whch often escape the attenton of eseaches. Ths wok deals wth poblems nheent n hybd functonals and obtal-dependent functonals, whch have ecently become elable tools of appled DFT; n patcula, ths wok consdes n detal the technque fo constuctng the local and multplcatve potentals, ncludng the optmzed effectve potental method and ts appomate vesons. To avod cumbesome equatons we gve all elatons fo the case of eal obtals, esponse functons, etc. (whch can be eadly genealzed to comple objects). Hee we consde only fnte systems (atoms, molecules, clustes, etc.). LOCAL AND SEMILOCAL DENSITY FUNCTIONALS Befoe descbng the densty functonals t s necessay to ntoduce seveal addtonal defntons. The spn densty functonal theoy (SDFT) s a natual genealzaton of DFT fo descbng open-shell systems o any systems n an etenal magnetc feld. In ths theoy, the ehange-coelaton enegy s defned by two vaables, namely, by electon denstes wth spns ( ) and ( ): E E [ ; ], (5) () S

3 whee* N,occ =1 ( ) ( ),, ; (6) ( ) ( ) ( ). (10) The ntoducton of dmensonless spn polazaton s an altenatve method fo ncludng spn effects n calculatons: ( ) ( ) ( ). (8) () It can easly be seen that the pa of vaables ( ; ) s qute equvalent to the pa (; ). Natually, n closed-shell systems, = = / (o = 0) at each pont of space. Usng the vaatonal pncple fo enegy functonal () and mposng an addtonal condton of othonomal obtals yelds Kohn Sham one-electon equatons (smla n fom to Hatee Fock equatons [7]): whee s the Kohn Sham potental, and 1 v s () () (), (9) ZK ( ) s() d () R K K () E (11) ( ) s the ehange coelaton potental defned as a functonal devatve [8] of the ehange coelaton functonal wth espect to densty**. In the local densty appomaton (LDA) [9-13], whch has ecently been wdely used fo a long tme, the ehange coelaton functonal was ecoded as an ntegal of a cetan functon of spatal vaables. The latte, n tun, s only detemned by electon densty (and spn polazaton) at a gven pont: The ntegand functons LDA and LDA LDA LDA E [ ( ); ( )] d e [ ( ); ( )] ( ) d. (1) LDA e ae two altenatve defntons of the ehange coelaton enegy densty ( e LDA s often called, moe eplctly, enegy densty pe electon); hee we use the fst defnton ( LDA ). LDA natually appeaed n electon gas theoy (see, e.g., [14]) and poved a vey effectve method fo descbng solds as thee-dmensonal peodc systems. On the othe hand, LDA s a athe ough appomaton fo descbng fnte objects (atoms and molecules), whose electon densty dstbuton has almost nothng n common wth that n a homogeneous electon gas. Befoe statng to dscuss moe eact methods, t should be noted that t s common pactce to dvde all ehange coelaton functonals nto goups of ehange and coelaton functonals: c, (10) E E E (13) and model each contbuton sepaately. Stctly speakng, ths dvson s athe conventonal and even patly equvocal [15] and s dctated manly by pactcal consdeatons of convenence (fo dscusson, see the net secton). Nevetheless, based *The densty can also be obtaned dectly fom (1) f ntegaton (summaton) ove the spn vaable 1 s not caed out. **Fo open-shell systems descbed wthn the spn-unestcted Kohn Sham method, two coupled-petubed equatons appea nstead of one equaton (9): one fo spn, and the othe, fo spn, whch accodngly nclude dffeent spn-dependent ehange coelaton potentals,,. S3

4 3 on the known epanson n a sees of the ecpocal powes of the Setz adus s 4 1/3 fo the homogeneous electon gas enegy (see Eq. (3.37) fom [14]) n LDA one can natually detemne the local ehange enegy (often called Slate S, ehange [10] and desgnated as E although t was suggested by Dac [9]), whch s attactve n havng a vey smple epesson: LDA S LDA LDA E E E,, [ ( )] d, (14),, whee LDA 4 / 3 3 3, C ( ), C (15) 4 In contast to E, the coelaton enegy E c of a homogeneous electon gas [11-14] cannot be epoduced n analytcal fom. The most popula paametzatons of E c, VWN* [1] and PW91 [13], ae constucted by analytcally ntepolatng the esults of Monte Calo stochastc smulaton of electon coelaton [16]. Snce the md-1980s, DFT became even moe popula as a quantum chemcal-method because of the appeaance of the genealzed gadent appomaton (GGA) [17-0], whch substantally nceased the accuacy of descpton. In the GGA, the ehange coelaton functonal ncludes not only electon densty, but also nfomaton about ts nhomogenety n the fom of the absolute value of the densty gadent: GGA 1/3 GGA d (16) E [ ( ), ( ) ; ( ), ( ) ]. Moe ecently, a numbe of new GGA functonals wee ceated and allowed hghe accuacy of calculatons of defnte popetes of molecula systems** [1-5]. Wthn the famewok of the GGA, the most popula functonals ae Becke s ehange functonal (B88) [0], Handy Cohen s optmzed ehange [6]; Lee Yang Pa s (LYP) [1, ] and Pedew Wang s coelaton functonals (PW91) [13, 3, 4], and Pedew Buke Enzehof s (PBE) ehange coelaton functonal [5]. Futhe development of the ehange coelaton functonals followed two man tends: ncluson of the new nhomogenety paametes of electon densty o addton of the eact (Hatee Fock) ehange enegy. The fst decton was called meta-gga (o MGGA) [7-35] and used, along wth the densty gadent, the Laplacan of densty and/o the postvely defned local densty of knetc enegy *** N,occ 1 ( ) ( ),,. (17) =1 Thus, the meta-gga functonal can be ecoded n geneal fom as MGGA MGGA E [ ( ), ( ), ( ), ( ); ( ), ( ), ( ), ( )] d. (18) *As a matte of fact, thee ae two vesons of the VWN paametzaton: the standad VWN-5 paametzaton, ecommended fo use n the ognal wok, and VWN-3, whch s used moe aely. **The accuacy of calculaton of vaous popetes (atomzaton eneges of molecules n the equlbum state, baes of eactons, geometcal paametes, vbatonal fequences, optcal popetes, magnetc esonance paametes, etc.) geneally dffes substantally between functonals. Theefoe, pefeence should be gven to the functonal that yelds the hghest accuacy of calculaton fo the popety. ***The phase postvely defned s needed n vew of the fact that any enegy densty n geneal and the knetc enegy densty n patcula ae bascally ambguous. Fo eample, dentcal esults wll evdently be povded by ntegaton N,occ N,occ (1 ) ove the ente space of the epesson * ( ) ( ) ( ) fo any value of the paamete (see =1 =1 below). S4

5 Fom pactcal vewpont, today the Tao Pedew Staoveov Scusea (TPSS) functonal s the most effectve functonal fom ths class [35]. The second tend, namely, addton of eact ehange led to the ceaton of especally effectve hybd functonals, whch ae consdeed n the net secton of ths pape. The LDA, GGA, and meta-gga functonals ae geneally classfed as semlocal ; the densty of ehange coelaton enegy n them s gven n tems of the functon that depends, at each pont of space, on the value of electon densty and ts devatves (and also, possbly, on (17)) obtaned elusvely fo that pont (Eq. (18)). In othe wods, n contast to the essentally nonlocal eact ehange (see below), the semlocal contans no ntegaton ove the othe ndependent set of spatal coodnates. Fnally, t s woth mentonng that dect modelng of ehange coelaton potentals avodng the stage of enegy functonals s possble [36-40]. Model potentals of ths knd ae constucted based on the pncple of the coect behavo both n the nfntely emote (asymptotc) egon and n the (sub)valent and coe egons. Ths makes t possble to epoduce, wth a hgh degee of accuacy, esponsve molecula popetes such as the vetcal etaton enegy, dpole polazablty and hypepolazablty, vetcal onzaton potental, and electon affnty. The majo dsadvantages of ths appoach ae lmted genealty and applcablty; snce the enegy functonal s unknown, the total enegy and hence the themochemcal popetes of molecula systems cannot be obtaned wth these potentals. TRADITIONAL ( GLOBAL ) HYBRID FUNCTIONALS It s wdely known that the ehange enegy of electons wth the same spn (o so-called Fem coelaton, whch s a dect consequence of the Paul pncple) s the pedomnant contbuton to the total ehange coelaton enegy (85-95% of the latte). Theefoe, t would be easonable to epect good esults f the ough semlocal ehange functonal s eplaced wth an eact (Hatee Fock) equaton fo ehange enegy, occ eact 1 ( ) j ( ) ( ) j ( ) E d d (19),, j and futhe modeled wth the densty functonals elatve to the small esdue,.e., Coulomb coelaton enegy E c (Eq. (13))*. It should be emphaszed that eact ehange (19) has a numbe of advantages ove any semlocal ehange functonal; the majo advantage s total elmnaton of the absud, nonphyscal self-nteacton. Ths effect can eadly be undestood fom (): fo one-electon systems (fo eample, fo the hydogen atom o H molecula on) the classcal electon epulson enegy J() should be completely compensated by the ehange coelaton enegy E (n othe wods, the delocalzed poston of an electon cannot be ntepeted n the statc sense because the electon cannot epeence electostatc epulson fom tself lyng at a dffeent pont of space at a dffeent moment of tme!). As a matte of fact, n any of the semlocal ehange functonals mentoned n the pevous secton, self-nteacton s compensated only patally**. The pesence of self-nteacton s also detmental to descpton of many-electon systems, and especally to descpton of nonthemochemcal popetes that ae senstve to the subtle featues of electon densty dstbuton [41, 4]. Elmnaton of self-nteacton by eplctly subtactng t [43] s qute effectve [44], but has one mpotant dawback, namely, the scheme of [43] s not nvaant unde the untay tansfomatons of the occuped one-electon obtals. The ablty of eact ehange (19) to fully compensate self-nteacton follows fom ts defnton. Ths popety can also dectly povde a egula asymptotc behavo of the coespondng ehange coelaton potentals, whch s especally *Hee and below, summaton ove occuped obtals s desgnated fo bevty as occ (occuped). **Self-nteacton can be elmnated fom the coelaton functonals ( self-coelaton ) much moe easly, whch was acheved n seveal functonals [33-35] usng the knetc enegy densty (Eq. (17)). S5

6 mpotant fo coect descpton of a numbe of popetes calculated wthn the famewok of lnea esponse theoy and/o second ode petubaton theoy (polazablty, magnetc esonance paametes, etc.). Regetfully, wth all postve featues of eact ehange, attempts to constuct an ehange coelaton functonal n the fom of eact c, E E E (0) poved napplcable to descpton of chemcal bonds n molecules [45] and gave acceptable esults only fo one-cente systems (atoms, ons) [46-48]. Ths stuaton can be eplaned n the followng way. In DFT, the (Coulomb) coelaton enegy, modeled wth the coelaton functonals E c, s geneally mpled to be only ts dynamc (shot-ange) component whch s esponsble fo lowe pobablty of stuatons n whch electons wth opposte spns appoach one anothe to shot dstances*. Anothe type of coelaton s nondynamc (long-ange) coelaton, whch s especally ponounced n descpton of homolytc dssocaton of molecules. Let us consde the smple two-electon case, namely, the dssocated H molecule. Thee ae two ntepetatons of the nondynamc coelaton: one petans to eal space and the othe, to Hlbet space. In the nd fst ntepetaton, the enegy gan E c ases fom futhe decease n Coulomb epulson of electons due to the hghe pobablty of occuence of one electon nea, say, the ght nucleus (f the dssocated molecule s consdeed to be oentated hozontally ) povded that anothe electon les nea the left nucleus**. In Hlbet space of many-electon wave functons, the one-confguaton contbuton of the eted confguaton of complete dssocaton, the 1 g and g d E c, (1 ) descpton of an etended H molecule s evdently nadequate; the 1 u to the eact wave functon nceases wth the ntenuclea dstance (n the lmt 1 g confguatons become degeneate, and the contbutons become dentcal). Thus, the deceased enegy E nd c can be ntepeted as admng of the low-lyng eted confguatons. Ths effect s obvously epected not only fo dssocated molecules, but also fo many tanston metal compounds, whee the effects of quasdegeneaton ae especally ponounced. By defnton, eact ehange (19) ncludes only the Fem electon coelaton wth dentcal spns, and hence (0) has no nondynamc Coulomb coelaton. It s mpotant to emphasze that eact ehange s essentally nonlocal. Ths becomes qute evdent f we ewte (19) to obtan an equaton that esembles (1), (14), (16), and (18): (1) eact eact eact,,,, E E ( ) d, whee occ eact 1 ( ) j ( ) ( ) ( ) j ( ) d (), j s the eact ehange enegy densty. A compason of Eq. () wth Eqs. (14), (16), and (18) shows that these equatons ae qute dffeent. In contast to the ehange enegy densty n LDA, GGA, o meta-gga, the eact ehange enegy densty at each pont of space conveys nfomaton (ntegated ove the second ndependent set of spatal vaables ()) about the behavo of the one-electon obtals ove ente eal space. eact DFT As opposed to the case of eact ehange E, the semlocal ehange functonals (denoted fo bevty as E, whee DFT = LDA, GGA, MGGA) can effectvely take nto account nondynamc coelaton. Theefoe, (13) can be conventonally ewtten as [6, 49]: *The Coulomb coelaton of electons wth dentcal spns makes a much lowe contbuton, but s also ncluded n seveal models (e.g., [7]). ** Left-ght coelaton. S6

7 DFT DFT eact nd DFT c c c ; E E E E E E (3) n othe wods, t wll be moe appopate to ntepet the ehange densty functonals as the functonals that descbe electon ehange togethe wth nondynamc coelaton: E DFT eact nd E E c. (4) The popety of the ehange functonals epessed by (4) follows eactly fom the local natue of these functonals (fo detals of ths dscusson, see [49]). Also, t should be noted that pattonng (4) s of methodologcal athe than pactcal mpotance because t s dffcult to sepaate ehange fom nondynamc coelaton eplctly. Summazng the afoesad and compang (0) and (3), one can easly see that the fome s fee fom any nonphyscal self-nteacton and contans no nondynamc coelatons, whle the latte contans both. Consequently, a coect choce of a combnaton of the ght pats of (0) and (3) could be a compomse to ensue balance between the elmnaton of self-nteacton and the ncluson of nondynamc coelaton. The smplest choce s a lnea combnaton, natually leadng to the noton of a hybd ehange coelaton functonal hyb eact DFT E a0e (1 a0) E Ec, 0 a0 1. (5) DFT LDA In ths fom (wth E E and a 0 = 0.5) the hybd functonal was ntoduced by Becke n 1993 [50]. He used a dffeent theoetcal atonale fo the hybd functonal, whch was based on the adabatc bndng fomalsm of KohnSham's fcttous system (n whch nteelecton nteacton was absent) wth a eal system [51]. The hybd functonal based on Becke's thee-paamete scheme (B3) became one of the most popula functonals [5]: hyb eact LDA B88 LDA GGA E a0e (1 a0 ) E ae Ec acec, (6) a0 0., a 0.7, ac 0.81, B88 GGA whee E s the gadent coecton to the ehange enegy taken fom the B88 functonal [0], and E c s the gadent coecton to the coelaton enegy*. The numecal paametes a 0, a, and a c wee obtaned by fttng the themochemcal data obtaned by usng functonal (6) to the coespondng epemental data. In [5] fo GGA E c we used PW91 E c [3, 4]. Late t appeaed [53] that a combnaton of the B3 scheme wth the LYP coelaton functonal [1, ] led to slghtly moe eact esults **. The successful use of B3LYP ntated the development of new hybd functonals. Of hybds that appeaed ecently one can menton the Becke-97 (B97) [54] and HampechtCohenTozeHandy (HCT) functonals [55]. The majo dsadvantage of these functonals s that they ae ovecowded wth the empcal (fttng) paametes whle the accuacy of the descpton of molecula systems s modeate (e.g., B97 ncludes 10 paametes, and HCTC ncludes 15). On the othe hand, thee wee qute opposte attempts such as the PBE0PBE functonal [56]. Ths functonal has no fttng paametes, and a functonal of the fom of (5) s used fo whch the choce of the value of a 0 was based on cetan geneal theoetcal analyss [57]. Nevetheless, the B3LYP thee-paamete functonal emaned the most popula ehange *In GGA, the ehange (coelaton) functonal s geneally epesented as the sum of two tems: GGA LDA GGA E E E. LYP E c LYP **Stctly speakng, fo the LYP coelaton functonal, thee s no eplct gadent coecton ; theefoe, one LYP LDA can use the followng atfcal method. The coelaton enegy s fomally ecoded as Ec Ec E c. Afte that, n LYP LYP LDA LDA VWN (6) we pefom a substtuton Ec ac Ec Ec, whee fo E c one can use E c [1]. The uncontollable ac altenatve vesons n usng the two dffeent vesons of the VWN paametzaton (see footnote * on p. S4) n dffeent quantum-chemcal pogams ae often a souce of confuson and epoducble esults obtaned by usng the B3LYP functonal. S7

8 coelaton functonal eve used n DFT fo solvng physcochemcal poblems*; ths s a good compomse between the small numbe of empcal fttng paametes and the wde spectum of popetes t descbes (wth hgh accuacy). The hybd functonals owe much of the success not only to the unque ablty to pedct themochemcal esults and molecula stuctue, but even, occasonally, to the unque chance they gve us n ou effots to obtan acceptable accuacy n calculatons of moe delcate popetes that ae senstve to the subtle detals of electon densty dstbuton**. These ae, e.g., the paametes of nuclea magnetc esonance (NMR) and electonc paamagnetc esonance (EPR) of tanston metal complees. As eamples we can cte the on ( 57 Fe) and uthenum ( 103 Ru) chemcal shfts n the coespondng comple compounds [58, 59], as well as the electonc g tensos of metal complees fom the fst tanston ow [60]. As mentoned above, the hybd functonals ae assocated wth an undesable effect of hypepaametzaton. It appeaed that thee was a adcally dffeent way, leadng to genealzaton of the noton hybd functonal analyzed n ths wok. We emphasze that the tadtonal hybd functonals (5) and (6) ae called below global hybd functonals (global hybds fo bevty); unde globalty we undestand the constancy of the paamete a 0, whch detemnes the value of eact eact ehange E (19). LOCAL HYBRID FUNCTIONALS In hs poneeng wok on hybd functonals [50], Becke stated that the same enegy facton of eact ehange ove the ente eal space (a 0 = 0.5 and 0. n (5) and (6), espectvely) could only be egaded as a fst appomaton. In quantum-chemcal calculatons, ths fst appomaton became wdespead pactce. Recently, t was suggested that the constancy of the eact ehange admtue should be ejected, whch led to the appeaance of a new noton the local hybd functonal [61]. In ths concept, the ehange coelaton functonal s ecoded as loc=hyb eact DFT DFT,, c, E g ( ) ( ) [1 g ( )] ( ) d E, (7) eact whee, s the eact ehange enegy densty defned n (). Thus, the eact ehange admtue becomes spatally dependent. The functon g, whch contols the value of ths admtue, s called the local mng functon (LMF). Evdently, the LMF should satsfy the condton 0 ( ) 1. (8) g In the pevous secton, balance between the elmnaton of self-nteacton and ncluson of nondynamc coelaton was gven as one of the easons fo success of the (global) hybds. Obvously, local hybd (7) can n pncple povde subtle adjustment of ths balance povded that the choce of LMF s coect and physcally detemned***. Fo LMF, the authos of [61] offeed the ato of Wezsäcke knetc enegy densty W to the local knetc enegy densty : W, ( ) g( ) ( ), (9) ( ) whee W, 1 ( ) ( ), 8 ( ) (30) *Ctaton of B3LYP amounts to dozens of thousands. **The total enegy of the system s a too hghly ntegated chaactestc n ths sense; one can easly magne a stuaton n whch the local vaatons of densty, balanced n a cetan way, do not cause any ponounced changes n enegy. ***The patcula case of the LMF s the constant (g() = const = a 0 ), whch coesponds to the local hybd (7) degeneated nto the global one (5), (6). S8

9 and () s defned by (17). LMF (9) satsfes condton (8); accodng to the defntons of W, and, t s nonnegatve and vanshes at those ponts at whch electon densty s locally homogeneous: = 0. Then W, neve eeeds and becomes equal to t n those egons of space n whch the contbuton to electon densty s due only to one of the occuped obtals,.e., whee the contbuton fom the othe obtals s vanshngly small: ( ) k ( ), 1 k N,occ. (31) The latte condton s satsfed fo the asymptotc egon of any molecula system (n ths case, k = N,occ ;.e., the densty s detemned by only the hghest occuped molecula obtal (HOMO)) and, dentcally, fo any system contanng no moe than one electon wth spn and one electon wth spn. Systems of ths knd nclude not only the hydogen atom o the H molecula on (fo whch the gven stuaton s desable because t coesponds to 100 % eact ehange and hence to complete elmnaton of self-nteacton), but also (egetfully) any two-electon closed-shell system (e.g., the neutal hydogen molecule H, helum atom, etc.). Thus, n these two-electon systems, a local hybd wth LMF (9) always degeneates nto a global hybd wth a 0 = 1. It was shown [61] that the local hybd functonals wth LMF (9) allow one to obtan qute acceptable esults. Good esults wee obtaned fo the enegy of thee-electon bonds n He, Ne, A, (HO), and othe symmetcal adcal catons and fo the bae heght n hydogen atom tansfe eactons, that s, fo those popetes that cannot be eadly epoduced wthn the famewok of DFT wth the tadtonal semlocal functonals o global hybds. At the same tme, the fundamental popetes used n computatonal themochemsty, namely, the atomzaton enegy of molecules could not be epoduced satsfactoly; fo the standad G-1 set of 55 molecules and adcals [6, 63], the mean absolute devaton fom the epemental values was 13-0 kcal/mol (whch depended on the DFT and DFT c used), but these values eeeded the values obtaned, e.g., wth BLYP [0-] and B3LYP [1,, 5] functonals seveal fold. Recently, we showed [64] that bette ageement of atomzaton enegy wth epement was obtaned by smply scalng LMF (9), g () t (), 1. (3) The best esults wee obtaned wth = 0.48 when local ehange was combned wth the local coelaton DFT VWN [9, 10], c c [1]). Wth LMF (3) one could also obtan hgh accuacy fo the bae heghts of eactons [65]. Theefoe, ths functon may be consdeed a beakthough n the development of local hybd functonals (see the Bef summay of the esults secton). The only seous dsadvantage of LMF (3) s the loss of the egula asymptotc behavo of the potentals coespondng to the local hybd functonals of ths knd. In the asymptotc egon, the eact ehange admtue eached 0.48, but not 1, and elmnaton of self-nteacton, theefoe, was ncomplete (ths leads to wose esults n descpton of the nonthemochemcal popetes calculated wth second ode petubaton theoy o lnea esponse theoy). To elmnate ths dsadvantage we offeed a new type of local mng functons, whch dffeed adcally fom (9) and (3) [66]. Instead of t (9) we suggested usng the dmensonless gadent of electon densty as a basc vaable ( ) ( ) ( ) s ( ). (33) 1/3 4/3 4/3 kf ( ) ( ) (3 ) ( ) ( ) Based on (33) and eponentally deceasng densty of fnte systems n the asymptotc egon, s changes fom zeo to nfnty; theefoe, sutable contnuous, monotonous mappng of ay [0;) to a segment [0;1] could be a canddate fo LMF. We consdeed the followng functonal foms as LMFs: s m /( + s m ), [s/( + s)] m, [1 ep( s)] m, [ef(s)] m, [th(s)] m, [(/)ac tan(s)] m, (m = 1, ), etc. (s = s, s ) wth a postve vaable paamete. All these povde compaable levels of themochemcal accuacy when local ehange s combned wth coelaton of LYP [1, ] n functonal (7). The best esults wee obtaned fo the followng choce of LMF [66]: ( DFT LDA g ( s ) [ s /( s )]. (34) S9

10 Fg. 1. Local mng functons (LMF) of two types fo the cabon monosulfde molecule (the t and s vaables ae detemned by the condtons of (9) and (33), espectvely). Calculaton wth the LDA functonal (S-VWN) n the cc-pvqz bass. The LMF values coespond to the as that connects the nucle. LMF (34) pemts the same level of accuacy as LMF (3) n obtanng atomzaton eneges, and s advantageous n havng a egula asymptotc behavo of local hybds (7) constucted fom t. As an llustaton, Fg. 1 shows LMFs of both types fo the cabon monosulfde molecule (the calculated values of LMF le on the as though the cabon and sulfu nucle). It can eadly be seen that both (t- and s-dependent) LMFs eflect the shell stuctue of atoms n the molecule, but they do t n qualtatvely dffeent ways; t-lmf has a mamum on the nucle, whle s-lmf has a mnmum (t s not clea as yet whch s pefeable). Ths dffeence n the asymptotc behavo of the two local mng functons s evdent fom Fg. 1. It should be mentoned that the conceptually local hybd functonal (7) s close to the nondynamc coelaton model ecently suggested by Becke [67, 68]. Fo closed-shell systems, the ehange coelaton enegy functonal n the latte fomally concdes wth (7) (LMF s much moe comple n fom than (9), (3), o (33)). Dscussng the local hybd functonals, t should be mentoned that the defnton of the ehange enegy densty s ambguous, whch concens both eact, and DFT, (see footnote *** on p. S4). To any enegy densty (()d = E) one can add a cetan calbatng functon, whose ntegal ove the ente space s zeo; then the functon can evdently be egaded as the densty of the same enegy (o one can tansfom the ntegal usng thee-dmensonal ntegaton by pats and obtan anothe densty equaton, see below). A method fo constuctng an unambguous ehange coelaton enegy densty was offeed n [69], but ths method s of conceptual athe than pactcal value. The ambguty, howeve, does not seem to be a seous obstacle to wde use of the local hybd functonals (especally n those cases when S10 DFT, s a smooth LDA functon, (15)) and can be efomulated as the poblem of selectng an appopate LMF; but ths eques stct defnton of the ehange enegy densty, whch should be followed n all constuctons. The dscusson above mplctly mpled self-consstent solutons of (9) and (10) wth the ehange coelaton potental defned by (11). As wll be shown below, devaton of potental (11) s often a seous (techncal and conceptual) poblem; theefoe, non-self-consstent calculaton of the total eneges s common pactce n pelmnay evaluaton of the qualty of the new ehange coelaton functonals of dffeent types (16), (18), (5), (7). Densty s substtuted nto Eq. (), whee E s the functonal of nteest to us, and, f necessay, we also substtute nto ths equaton the ndvdual obtals obtaned n self-consstent calculaton (9) wth a potental coespondng to any othe functonal that s smple fom the vewpont of calculatng the functonal devatve such as LDA (14), (15). Non-self-consstent calculatons of ths knd,

11 whch may be called post-lda n ths case, cetanly gve lowe accuacy. Howeve, the pecson s hgh enough to obtan the geneal dea about the themochemcal advantages of the new functonal. Ths method s applcable only to the total enegy calculatons; evaluaton of any othe popety eques knowledge of the coespondng potental because the latte emans the only method of conveyng the nfomaton about ths functonal, fo eample, n the fom of one-electon obtals and the eneges, whch ae solutons of (9) late employed n popety calculatons usng petubaton theoy. At the same tme, devaton of potental (11) s a nontval poblem evey tme when the functonal nvolves not only densty and ts devatves, but also obtal-dependent contbutons such as the local densty of knetc enegy (17) o eact ehange enegy (19) (and/o the densty of the latte ()). HYBRID FUNCTIONALS WITH DIFFERENT METHODS OF INCLUDING THE SHORT- AND LONG-RANGE COMPONENTS OF INTERELECTRON INTERACTION In summay of ths evew of methods fo genealzng the noton of hybd functonal, we befly consde an nteestng and ognal method fo constuctng ehange functonals (the coelaton component s also consdeed ndependently, see above), whch s now based on the dea of decomposng the epulson enegy of a pa of electons (lyng at ponts and ) nto the long-ange (l) and shot-ange (s) components [70-75] (ange-sepaated hybd functonals): 1 s. E E E (35) The statng pont n ths method s mathematcally vey smple; the quantty that s nvesely popotonal to the nteelecton dstance s epesented as the sum of two complementay contbutons: 1 ef ( ) 1ef ( ), whee ef s the Gaussan functon of eo tem n the ght pat of (36) s elated to 0 (36) ef( ) ( / ) ep( t ) dt, and µ s the numecal fttng paamete. The fst l E fom (35), and the second, wth s. E The Hatee Fock (eact ehange) l method can be used fo E, and the local densty appomaton, fo s E *. Thus, the long-ange contbuton to the ehange enegy becomes (cf. (1) and ()): occ l 1 ( ) j ( )ef( ) ( ) j ( ) E d d. (37),, j A compason of Eq. (37) wth Eqs. (19), (1), and () fo eact ehange shows that fo the fome s educed to the latte. In evey patcula case, eplct equatons fo the shot-ange component of ehange enegy depend on the type of the appomaton used wthn the famewok of DFT (LDA, GGA, etc.). These ae geneally athe cumbesome equatons, whch ae not specfed hee. The equed nfomaton can be found n efeences cted above n ths secton. Those efeences (and also [76]) also gve the esults of detaled evaluaton of the ablty of these functonals to epoduce dffeent popetes of dffeent physcochemcal objects. *It should be emphaszed that ths method s evdently beyond the scope of the hybd functonal concept; n ts l ognal veson [70], the long-ange component E was descbed n tems of the confguaton nteacton (CI) method. S11

12 EXCHANGE CORRELATION POTENTIALS OF THE ORBITAL-DEPENDENT FUNCTIONALS Ths secton dscusses poblems of calculatng the functonal devatves wth espect to densty (potentals) (11) of obtal-dependent functonals (fo the sake of smplfcaton, the spn nde s omtted evey tme when t s nsgnfcant n the cuent contet). Recall the defnton of the functonal devatve fo a cetan ntegated functonal n d E[ ] [ ( ), ( ), ( ),, ( )], (38) whee () s a cetan contnuous functon of spatal vaables. An nfntely small vaaton of the latte, (), leads to vaaton of the functonal E E[ ] E[ ]. (39) If E s epesentable as E v( ) ( ) d, (40) then v() s called the functonal devatve of the E functonal wth espect to the functon: E v( ). (41) ( ) Fo a composton of functonals, fo eample, fo E[f[]] (E s a functonal of f, whose value at each pont s, n tun, a functonal of ) the followng equaton s vald: E E f( ) d, (4) ( ) f ( ) ( ) whch esembles the ule of comple functon dffeentaton and can eadly be genealzed to comple functonals of any degee of embeddedness. Fo functonal (38), the followng s vald*: E n n ( ) ( ) ( ) ( 1) ( ). n ( ) (43) Evdently, fo enegy functonals LDA (14), (15) and GGA (16), as well as meta-gga (18), devaton does not pesent any methodologcal poblems only f the latte functonals do not contan a dependence on the knetc enegy densty (17). In othe wods, the ehange coelaton potental s accessble n eplct fom n all cases when the coespondng functonal ncludes a dependence only on electon densty and ts devatves (of any ode). Fo bevty these functonals ae called pue densty functonals and desgnated by E ; hee, denotes densty wth ts devatves. Complcatons ase fo dependent meta-gga and hybd (both global (5), (6) and local (7)) functonals. Ths s assocated wth the pesence of eact contbutons fom the denstes of knetc enegy (17) and eact ehange enegy E (19) (eact ehange enegy eact densty ()) espectvely, o wth the pesence of both, as n the case of local hybds wth t-dependent LMF (9), (3)). The poblem les n the fact that nethe no eact E (o eact ae eplct electon densty functonals** (fom (17), ) *In ths scala equaton, we use an abbevated fom of the vecto ecod; e.g., the tem (/) should be ntepeted as ( / )/ ( / y)/ y( / z)/ z, whee u / u, u =, y, z. **Accodng to the fst Hohenbeg Kohn theoem [3], any quantty can be consdeed a densty functonal, ncludng a cetan Kohn Sham obtal at an abtay pont of space. Regetfully, ths theoem does not contan any ndcatons on the constucton of an eplct (analytcal) fom of ths functonal. S1

13 ( ) N o the (19), and () t can eadly be seen that the dependence of these functons on the KohnSham obtals occ devatves can neve be educed to the dependence on densty o ts devatves). In the lteatue, the -dependent and hybd functonals ae called obtal-dependent functonals (hee we use fo them the geneal notaton E ). Incdentally, fom fomal vewpont, even equatons fo the total electon enegy n tems of many-patcle petubaton theoy (e.g., second ode Mölle Plesset theoy MP [7]) can be assgned to obtal-dependent functonals; n ths case, howeve, the enegy functonal ncludes not only occuped, but also vacant obtals (so-called ab nto densty functonals) [77-80]. Evdently, fom any obtal-dependent functonal one can always obtan ts functonal devatve wth espect to the KohnSham obtals ob (FDO) E /. ob 1 Let us fst consde the FDO of pue densty functonals. Takng nto account the eplct densty equaton n tems of obtals () and the fact that v E / s defned eplctly n tems of (43), one can ecod E E ( ) v ( ) ( ). () () v In (44) we emphasze that the potental s multplcatve (the acton of the potental on the obtal s educed to multplcaton of the latte on t); consequently, t can be ewtten n the followng way: l E v ( ). (45) ( ) ( ) It s nteestng to note that (45) s vald fo any occuped obtal, whch eflects the known fact that n the Kohn Sham fomalsm, the electons belongng to dffeent obtals epeence the acton of the same potental (ths seemngly tval popety s not obseved, e.g., n the Hatee Fock method, see below). Thus, n the case of pue densty functonals, t makes no dffeence whethe the potental s evaluated wth (43) o ecalculated fom FDO accodng to (45). The stuaton s qute dffeent fo obtal-dependent functonals. Let us consde the functonals that nclude the dependence on the local densty of knetc enegy (17) ( -dependent functonals; fo these, we use notaton shown n [81],,,, E n n () () (1) () () n (),, d ˆ, () () () () [v ](). d E, ). The fst tem on the ght sde of (46) s the famla multplcatve contbuton to FDO (ght sde of (43)); the othe tems (whose ogn s assocated elusvely wth the dependence on ) contan the dffeental opeatos. Thus, FDO (46) s the esult of the acton of a cetan nonmultplcatve, but stll local opeato (o semlocal because of the absence of ntegaton ove anothe ndependent set of spatal vaables, see above) on the obtal. As a esult, etacton of the multplcatve potental wth (45) becomes mpossble. Howeve, the mechancal substtuton v ob (44) As (46) l E (47) mples a stuaton n whch the electons that occupy dffeent obtals tavel at dffeent etenal potentals, whch s obvously a depatue fom the Kohn Sham fomalsm (we emphasze that Eqs. (9)-(11) cetanly mply that the Kohn Sham potental v () s local and multplcatve). S13

14 Fnally, let us consde hybd functonals (begnnng wth global functonals, e.g., (5)) wth a pue densty DFT functonal as a contbuton (1 a0) E Ec). Evdently, an obtal-dependent contbuton, namely, eact ehange (19) wll be a souce of poblems n constuctng the coespondng potental. Theefoe, an eplct equaton fo FDO s as follows: whee hyb E eact DFT hyb a0[v ˆ ]() [(1 a0)v () v()] c () [v ˆ ](), () occ eact ( ) ( ) [v ˆ ]( ) j( ) d. j eact Equaton (49) s a defnton of the eact ehange opeato ˆv, whch s nonmultplcatve and nonlocal (ntegated). hyb Accodngly, the ente ˆv opeato n (48) s nonlocal and nonmultplcatve. As n the case of the -dependent opeatos, hyb etacton of the Kohn Sham potental fom E / s mpossble. It s nteestng to note that f n (5) and (48) we set that a 0 = 1 and E c = 0 (accodngly, v c () 0), we obtan eactly the HateeFock method, n whch the vaatonal poblem s fomulated n tems of obtals, but not electon densty. Accodngly, the Hatee Fock equatons M 1 ZK ( ) eact d () [v ˆ ]() () K 1 K R dffe fom Kohn Sham equatons (9)-(11) pmaly n havng the nonmultplcatve and nonlocal eact ehange opeato eact ˆv. It should be emphaszed that n most moden standad quantum-chemcal pogams (e.g., GAUSSIAN-03 [8]) the hybd potentals ae ealzed as the ˆv hyb nonlocal opeatos. A smla substtuton, (47), s almost nsgnfcant to the calculated total eneges, whle fo seveal nonthemochemcal popetes, thee ae consdeable complcatons, qualtatvely changng the ultmate soluton. Fo eample, n calculatons of magnetc esonance paametes such as the nuclea chemcal sceenng constants [83, 84] o electonc g tensos [85], the etenal magnetc feld consdeed as mno petubaton can eact ˆv gve se to the coectons to the opeato, whch ae of the same ode as the coectons to obtals. In othe wods, n contast to the local potental (11) and all othe contbutons to the Kohn Sham potental (10), the eact ehange opeato acques lnea esponse when an etenal magnetc feld s appled (stctly speakng, n magnetc felds, a moe geneal theoy ncludng electon cuent densty n addton to should be used nstead of conventonal DFT [86, 87]). As a esult, nstead of the dect one-step applcaton of petubaton theoy we have to stat an teatve pocess of solvng coupledpetubed equatons (see [83, 84] fo detals of the dscusson). Ths s undesable not only because of geate computaton tmes, but also because of the loss of dect coelaton between the qualty of the self-consstent soluton (n the fom of obtals and obtal eneges), on the one hand, and the accuacy of the calculated magnetc esonance paametes, on the othe, n vew of the fact that the outcome of any teatve pocess s unpedctable. Let us now consde pactcal chances to avod the nonlocal and/o nonmultplcatve contbutons to the ehange coelaton potental,.e., to the poblem of functonal dffeentaton of the obtal-dependent functonal wth espect to ob densty. Snce the E / FDO s accessble and takng nto account equaton (4), one can fomally ecod the functonal devatve E ob / as follows: ob ob * j (48) (49) (50) ( ) v ( ) d. () ( ) () occ ob E E (51) S14

15 Fo evaluatng /, n tun, fo the bndng element one geneally uses the Kohn Sham one-electon potental v s (10) [77, 88]: ( ) ( ) v s ( ) d. ( ) v ( ) ( ) s In (5), the functonal devatve /v s can be evaluated n tems of petubaton theoy statng fom Eq. (9) (.e., consdeng v s as a small petubaton of the v s potental, and, as the coespondng fst ode coecton to the I oneelecton functon): ( j ) v s ( ) ( ) d j( ) ( j ) ( ) j( ) v s ( ) ( ) d. (53) j j j j By the defnton of the functonal devatve (40), (41), () ( ) ( ) j j ( ) G(, ) ( ), v( ) s j j (5) (54) whee G s the statc one-electon Geen functon coespondng to the obtal wth the nde. To fnd the last component v s / equed, one uses the statc lnea esponse functon, occ () (, ) () G(, ) ( ). v ( ) s (55) The physcal sense of the functon s eflected n ts name because t descbes the eacton of the electon densty of the system (n fst ode petubaton theoy) to a mno change n the etenal potental*. Accodngly, the desed functonal devatve v s / can be fomally ecoded as an nveted lnea esponse functon v() s 1 (, ). (56) ( ) Equaton (56) mples that the followng elaton s satsfed: 1 1 (, ) (, ) d (, ) (, ) d ( ). (57) Inveson of the lnea esponse functon s a techncally comple poblem, whch can be solved by pefomng epanson n a fnte aulay bass (whch dffes fom the bass of atomc obtals) [89], but s numecally unstable n calculatons fo molecules. Moeove, constucton of the lnea esponse functon tself (55), whch, n tun, eques the constucton of the Geen functon G (see Eq. (54)), s a athe epensve pocedue, especally because all of the above opeatons should be pefomed at each teaton of the self-consstency pocess. Recall that the ntal poblem was to constuct the local and multplcatve potental v ob (51) to substtute the latte nto (9) and (10). The whole pocedue can only be fulflled ob ob numecally (v cannot be obtaned n analytcal fom). In pactce, howeve, v s constucted by an altenatve technque (whch also mples only numecal soluton) called the optmzed effectve potental (OEP) method and dscussed below. OPTIMIZED EFFECTIVE POTENTIAL METHOD (OEP) In the OEP method [90, 91], the vaatonal poblem s fomulated dffeently; nstead of seekng the statonay ponts of the total electon enegy wth espect to vaaton of densty (E/ = 0), the Kohn Sham potental (10) s vaed fo the same pupose; that s, the poblem s educed to seekng an optmzed effectve potental OEP v s such that *Equaton (55) can eadly be obtaned fom (54) f densty equaton (3) s taken nto account. S15

16 v E OEP s 0. (58) These vaatonal poblems ae equvalent because theoetcally they lead to dentcal esults: as shown ecently [9], n the famewok of DFT fo the basc vaable one can use not only electon densty, but also othe objects, n patcula, the KohnSham potental (10). The functonal devatve E/v s can be convenently epesented as occ E E ( ) d, v ( ) v ( ) OEP s OEP s (59) OEP whee / v s s detemned n tems of (51); takng nto account (), (9), and (10), E/ can be ecoded as ob ob E ZK ( ) E ob E d ( ) [ v ( )] ( ), () K K R () () (60) ob whee v s the desed local and multplcatve ehange coelaton functonal. Substtutng (60) and (54) nto (58), (59) and takng nto account (55) and the fact that the Geen functon G (by constucton) s a lnea combnaton of obtals othogonal to, we obtan (afte dentcal tansfomatons) an ntegal equaton of the OEP method, occ ob ob E (, )v ( )d ( ) G( )d. ( ) In the case of pue densty functonals, by vtue of (44), Eq. (61) becomes an dentty. The numecal method fo solvng (61) s not effcent and not stable fom computatonal vewpont; t s sutable only fo calculatons of atoms [91]. Moeove, fom pactcal vewpont, the equaton tself has no advantages ove the eplct equaton fo v ob (51). Nevetheless, the natue of ntegal equaton (61) admts the ntoducton of qute effectve ob appomatons, leadng to a consdeable smplfcaton of the constucton pocedue of the v potental. We wll dscuss n detal one of the appomatons actvely used n ou studes, namely, the common enegy denomnato appomaton (CEDA) [93] (whch s deologcally fully equvalent to the localzed HateeFock method (LHF)* [94]). The CEDA fomalsm s also vey convenent fo dscussng a athe popula but oughe appomaton used pevously, namely the KegeLIafat (KLI) appomaton [95]. The majo dea of the CEDA [93] s appomaton fo the one-electon Geen functon, accodng to whch the dffeences between the obtal eneges coespondng to the occuped vacant obtal pas n the denomnato ae eplaced wth a cetan common mean value: ( ) ( ) occ ( ) ( ) vac j j j j 1 G(, ) j() j( ). (6) j j j j j The net step s etenson of the common enegy denomnato appomaton to the occuped occuped obtal pas; t leads to the KLI appomaton [95], n whch due to the completeness of the space of the (othonomalzed) molecula obtals, the Geen functon 1, (63a) o (61) ( ) ( ) ( ), (63b) *Ths name s not qute sutable because ths pocedue s applcable to any obtal-dependent functnonals, and not only to eact (Hatee Fock) ehange. S16

17 becomes vey smple, 1 1 G(, ) j( ) j( ) [ ( ) ( ) ( )]. (64) j Wthout detalng the wokng equatons of the KLI appomaton, we note that whle beng smple and attactve fom the vewpont of computatonal esouces, ths s a athe ough appoach (essentally admttng aveagng of values wth opposte sgns!). It has the same dsadvantage as the method of dect subtacton of self-nteacton [43], namely, t s not nvaant unde the untay tansfomatons of the occuped one-electon obtals. Let us now etun to the CEDA LHF fomalsm. We omt the cumbesome ntemedate constuctons and tansfomatons and gve only the fnal wokng equatons. The desed local and multplcatve ehange coelaton potental s constucted as the sum of two tems [94, 93]: whee and av v s the obtal-aveaged FDO (46) o (48), av co v ( )=v ( ) v ( ), (65) occ ob ()[v ˆ ]() av ob, hyb ˆ ˆ ˆ v ( )=, v v, v, () v co s the coecton potental also called the esponse potental [96], occ ob ' ˆ ( ) j( ) j v v ] co, j v ( )= (67) () (the pme at the summaton symbol mples that n the sum of (67) we omtted the dagonal tem that coesponds to the hghest occuped molecula obtal;.e., = j = N occ ). The aveage potental v av s calculated fom (66) dectly, whle the co coecton tem v s calculated by usng teatons, whch geneally quckly convege [94]. Note that the stuctue of the soluton n the KLI appomaton s smla to that of (65)-(67); the only dffeence s that n coecton potental (67) we neglect all of the off-dagonal tems (.e., the tems coespondng to the condton j). In the case of 100% eact ehange and neglect of dynamc coelaton (vˆ vˆ (49)) the aveage potental (66) S has a specal name: Slate potental v [97]. It can eadly be seen that the latte s elated by a smple equaton to the eact ehange enegy densty (): eact 1 S ()= ()v(). (68) Recently, the dect optmzaton method has become popula. Ths method uses the OEP epesentaton n the fom of the sum of a cetan fed potental and a coecton, whch can be epanded n an aulay bass that s geneally dffeent fom the atomc obtal bass [98]. Howeve, today the senstvty of ths method to the qualty of an aulay bass, n patcula, to ts completeness, s not completely studed (see, e.g., ecent publcatons [99, 100]). Usng the OEP method was descbed n the lteatue fo tansfomng the nonlocal eact ehange opeato (49) pe se* nto the local and multplcatve potental as appled to fnte systems n both the KLI [48, 78, ] and CEDA- LHF [15, 94, 105] appomatons. Fo tansfomng the eact ehange opeato wthn the nonlocal hybd potental (48), usng the OEP method was descbed n the fom of the CEDA-LHF appomaton [ ] and n the fom of a decomposton nto the bass functons [11, 113]. ob eact (66) *In cases when 100% eact ehange (19) s used as ehange enegy. S17

18 Fg.. Localzaton effect of the authentcally nonlocal hybd potental (opeato) fo the ntogen molecule: the total enegy of the molecule and the occuped one-electon obtal eneges vay nsgnfcantly, whle the vtual obtal eneges decease substantally. Self-consstent calculaton wth the B88-EXX-LYP hybd functonal (a 0 = 0.5, ths s often called the Becke-Half-and-Half-LYP o, n abbevated fom, BHandHLYP n the lteatue) n the IGLO-IV bass. Let us consde detals of the localzaton effect of the nonlocal hybd opeato, takng the ntogen molecule as an llustaton. As can be seen n Fg., the dffeence n the natue of the potentals (opeatos) s almost nsgnfcant to the total enegy of the molecule and to the eneges of the occuped one-electon obtals. At the same tme, the vtual obtal eneges decease consdeably, whch esults n the appeaance of geate numbes of bound one-electon states (descpton of the vtual subspace s much moe adequate n the case of localzed potentals; see [94] fo detaled dscusson). Ths effect s vey mpotant fo calculatng the esponse popetes n second ode petubaton theoy (fo eample, nuclea chemcal sceenng constants) snce the one-electon enegy dffeences (and the mat elements between the occuped and vtual obtals) change sgnfcantly. The constucton of the local and multplcatve -dependent potental was fst descbed n ou wok [114], whch epoted the tansfomaton of FDO (46) by the OEP method (CEDA-LHF appomaton). Late we succeeded n obtanng the Kohn Sham potentals fo the moe geneal case of obtal-dependent functonals (see the net secton). To summaze, the Zhao Moson Pa (ZMP) appoach [115] s an altenatve to the OEP method fo the constucton of the local and multplcatve potentals. Accodng to ths appoach, the KohnSham potental can be constucted dung teatons, statng fom any electon densty specfed as an epanson nto the functons of a cetan bass. In ths case, the ogn of densty s unmpotant because t can be obtaned n both self-consstent calculaton wth a nonlocal hybd potental and beyond the DFT o Hatee Fock method, that s, fo eample, usng any hghly coelated post-hatee Fock method [7] (of the many-patcle petubaton theoy, multconfguaton method of self-consstent feld, confguaton nteacton, coupled cluste method, etc.). The ZMP technque was used [108, ] eactly fo constuctng the Kohn Sham potentals fom the denstes obtaned n a self-consstent calculaton wth nonlocal hybd potentals. Whle the ZMP method s unvesal, t s athe labo-consumng and s hadly applcable to lage systems; moeove, ts numecal stablty and the elablty of esults obtaned wth t ae dubous. The same popetes (chemcal sceenng constants) obtaned wth the local and multplcatve potentals obtaned fom the same hybd FDO (48) n dffeent ways (wthn the famewok of S18