Conserving GW scheme for nonequilibrium quantum transport in molecular contacts

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1 Downloaded fom obit.dtu.dk on: Sep 3, 28 Conseving scheme fo nonequilibium quantum tanspot in molecula contacts Thygesen, Kistian Somme; Rubio, Angel Published in: Physical Review B Condensed Matte Link to aticle, DOI:.3/PhysRevB Publication date: 28 Document Vesion Publishe's PDF, also known as Vesion of ecod Link back to DTU Obit Citation (APA): Thygesen, K. S., & Rubio, A. (28). Conseving scheme fo nonequilibium quantum tanspot in molecula contacts. Physical Review B Condensed Matte, 77(), DOI:.3/PhysRevB Geneal ights Copyight and moal ights fo the publications made accessible in the public potal ae etained by the authos and/o othe copyight ownes and it is a condition of accessing publications that uses ecognise and abide by the legal equiements associated with these ights. Uses may download and pint one copy of any publication fom the public potal fo the pupose of pivate study o eseach. You may not futhe distibute the mateial o use it fo any pofit-making activity o commecial gain You may feely distibute the URL identifying the publication in the public potal If you believe that this document beaches copyight please contact us poviding details, and we will emove access to the wok immediately and investigate you claim.

2 Conseving scheme fo nonequilibium quantum tanspot in molecula contacts Kistian S. Thygesen and Angel Rubio 2 Cente fo Atomic-scale Mateials Design (CAMD), Depatment of Physics, Technical Univesity of Denmak, DK-28 Kgs. Lyngby, Denmak 2 Euopean Theoetical Spectoscopy Facility (ETSF), Depatamento de Física de Mateiales, Edificio Kota, Univesidad del País Vasco, Cento Mixto CSIC-UPV, and Donostia Intenational Physics Cente (DIPC), Avenida de Tolosa 7, E-28 Donostia-San Sebastián, Spain Received Septembe 27; evised manuscipt eceived 4 Januay 28; published 8 Mach 28 We give a detailed pesentation of ou ecent scheme to include coelation effects in molecula tanspot calculations using the nonequilibium Keldysh fomalism. The scheme is geneal and can be used with any quasipaticle self-enegy, but fo pactical easons, we mainly specialize to the so-called self-enegy, widely used to descibe the quasipaticle band stuctues and spectoscopic popeties of extended and lowdimensional systems. We estict the self-enegy to a finite, cental egion containing the molecule, and we descibe the leads by density functional theoy DFT. A minimal basis of maximally localized Wannie functions is applied both in the cental egion and the leads. The impotance of using a conseving, i.e., fully self-consistent, self-enegy is demonstated both analytically and numeically. We intoduce an effective spin-dependent inteaction which automatically educes self-inteaction eos to all odes in the inteaction. The scheme is applied to the Andeson model in and out of equilibium. In equilibium at zeo tempeatue, we find that descibes the Kondo esonance faily well fo intemediate inteaction stengths. Out of equilibium, we demonstate that the one-shot G W appoximation can poduce sevee eos, in paticula, at high bias. Finally, we conside a benzene molecule between featueless leads. It is found that the molecule s highest occupied molecula obital lowest unoccupied molecula obital gap as calculated in is significantly educed as the coupling to the leads is inceased, eflecting the moe efficient sceening in the stongly coupled junction. Fo the I-V chaacteistics of the junction, we find that Hatee Fock HF and G W G HF yield esults close to than does DFT and G W G DFT. This is explained in tems of selfinteaction effects and lifetime eduction due to electon-electon inteactions. DOI:.3/PhysRevB PACS numbe s : b, 72.. d, 7.. w I. INTRODUCTION Since the fist measuements of electon tanspot though single molecules wee epoted in the late 99s, 3 the theoetical inteest fo quantum tanspot in nanoscale systems has been apidly gowing. An impotant diving foce behind the scientific developments is the potential use of molecula devices in electonics and senso applications. On the othe hand, it is clea that a successful intoduction of these technologies is heavily dependent on the availability of theoetical and numeical tools fo the accuate desciption of such molecula devices. So fa, the combination of density functional theoy DFT and nonequilibium Geen s functions NEGF has been the most popula method fo modeling nanoscale conductivity. 4 7 Fo stongly coupled systems such as metallic point contacts, monatomic chains, and contacts with small chemisobed molecules, this combination has been emakably successful, 8 but in the opposite limit of weakly coupled systems whee the conductance is much smalle than the conductance quantum, G =2e 2 /h, the NEGF-DFT method has been found to oveestimate the conductance elative to expeiments. 3 Pat of this discepancy might esult fom the use of inappopiate exchange-coelation xc functionals. 4 Howeve, it is impotant to emembe that the application of gound state DFT to nonequilibium tanspot cannot be igoously justified even with the exact xc functional. In paticula, a beakdown of the effective singlepaticle DFT desciption is expected when coelation effects ae impotant o when the system is diven out of equilibium. Ove the yeas, seveal diffeent schemes have been poposed as altenatives to NEGF-DFT. Histoically, the fist DFT based tanspot methods used an equivalent fomulation in tems of scatteing states athe than Geen s functions. 5 7 A moe ecent appoach still within DFT solves a maste equation fo the density matix of an electon system exposed to a constant electic field and coupled to a damping heat bath of auxiliay phonons. 8 A few attempts have been made to calculate the cuent in the pesence of electonic coelations. In one appoach, the density matix is obtained fom a many-body wave function and the nonequilibium bounday conditions ae invoked by fixing the occupation numbes of left- and ight-going states. 9 Exact diagonalization within the molecula subspace has been combined with ate equations to calculate tunneling cuents to fist ode in the lead-molecule coupling stength. 2 The linea esponse conductance of jellium quantum point contacts has been addessed on the basis of the Kubo fomula. 2,22 Although this method is esticted to the low bias egime, it has the advantage ove the NEGF method that inteactions outside the device egion can be natually included. The time-dependent vesion of density functional theoy has also been used as famewok fo quantum tanspot This scheme is paticulaly useful fo simulating tansients and high fequency ac esponses. Within the NEGF fomalism, the many-body appoxi- 98-2/28/77 / The Ameican Physical Society

3 KRISTIAN S. THYGESEN AND ANGEL RUBIO mation has been used to addess coelated tanspot both unde equilibium 26 and nonequilibium 27 conditions. Within the famewok of many-body petubation theoy, electonic coelations ae descibed by a self-enegy which in pactice must be obtained accoding to some appoximate scheme, e.g., by summing a esticted set of Feynman diagams. The impotant question then aises whethe the quantities calculated fom the esulting Geen s function will obey the simple consevation laws. In the context of quantum tanspot, the continuity equation, which ensues chage consevation, is obviously of special inteest. An elegant way of invoking the consevation laws is to wite the self-enegy as the functional deivative of a so-called functional, i.e., G = G / G. Since the self-enegy in this way becomes dependent on the Geen s function GF, it must be detemined self-consistently in conjunction with the Dyson equation. 28 Due to the lage computational demands connected with the self-consistent solution of the Dyson equation, pactical band stuctue calculations usually evaluate the selfenegy at some appoximate noninteacting G. This nonself-consistent scheme does not constitute a conseving appoximation. While this might not be impotant fo the calculated spectum, self-consistency has been demonstated to be fundamental fo out-of-equilibium tanspot. 27 In addition to its conseving natue, anothe nice featue of the self-consistent appoach is that it leads to a unique GF and, thus, emoves the G dependence inheent in the non-selfconsistent appoach. A eliable desciption of electon tanspot though a molecula junction equies, fist of all, a eliable desciption of the intenal electonic stuctue of the molecule itself, i.e., its electon addition and emoval enegies. The appoximation has been widely and successfully used to calculate such quasipaticle excitations in both semiconductos, insulatos, and molecules, and on this basis, it seems natual to extend its use to tanspot calculations. Thee ae two main obstacles elated to the extension of the method to chage tanspot. Fist, the conventional application of the method has been on gound state poblems, wheeas tanspot is an inheent nonequilibium poblem. Second, it is not obvious how to teat electonelecton inteactions in the leads within the NEGF fomalism. In Ref. 27, we poposed to ovecome these poblems by extending the self-enegy to the Keldysh contou and by esticting it to a finite cental egion whee coelation effects ae expected to be most impotant. In the pesent pape, we povide an extended pesentation of these ideas. When a molecule is bought into contact with electodes, a numbe of physical mechanisms will affect its electonic stuctue. Some of these mechanisms ae single paticle in natue and ae aleady well descibed at the DFT Kohn Sham level, but thee ae also impotant many-body effects which equie a dynamical teatment of the electonic inteactions. One example is the enomalization of the highest occupied molecula obital lowest unoccupied molecula obital HOMO-LUMO gap induced by the image chages fomed in the electodes when an electon is added to o emoved fom the molecule. 29,34 Anothe example is the Kondo effect which esults fom coelations between a localized spin on the molecule and delocalized electons in the electodes. 35,36 Thid, as we will show hee, the coupling to noninteacting electodes enhances the sceening on the molecule leading to chaacteistic eduction of the HOMO- LUMO gap as function of the electode-molecule coupling stength. In this pape, we focus on impoving the desciption of quantum tanspot in molecula junctions by impoving the desciption of the intenal electonic stuctue of the molecule while peseving a nonpetubative teatment of the coupling to leads. We do this within the NEGF fomalism by using a self-consistent self-enegy to include xc effects within the molecula subspace which, in tun, is coupled to noninteacting leads. The ationale behind this division is that the tanspot popeties, to a lage extent, ae detemined by the naowest pat of the conducto, i.e., the molecule, while the leads mainly seve as paticle esevois. Stictly speaking, this is coect only when a sufficiently lage pat of the leads is included in the egion. If the cental egion is too small, spuious backscatteing at the inteface between the and the mean-field egions might affect the calculated conductance. Futhemoe, the dynamical fomation of image chages in the electodes equies that pat of the electodes ae included in the egion. In the pesent wok, howeve, we do not attempt to addess this latte effect. The pape is oganized as follows. In Sec. II, we intoduce the model used to descibe the tanspot poblem and eview the basic elements of the Keldysh Geen s function fomalism. In Sec. III, we intoduce an effective inteaction, discuss the poblem of self-inteaction coection in diagammatic expansions, and deive the nonequilibium equations fo an inteacting egion coupled to noninteacting leads. In Sec. IV, we intoduce the cuent fomula and show that chage consevation is fulfilled within the NEGF fomalism fo deivable self-enegies also when incomplete basis sets ae used. The pactical implementation of the tanspot scheme using a Wannie function basis obtained fom DFT is descibed in Sec. V. In Secs. VI and VII, we pesent the esults fo the nonequilibium tanspot popeties of the Andeson impuity model and the benzene molecule between jellium leads, espectively. In Sec. VIII, we pesent ou conclusions. II. GENERAL FORMALISM In this section, we eview the elements of the Keldysh Geen s function fomalism necessay to deal with the nonequilibium tanspot poblem. To limit the technical details, we specialize to the case of othogonal basis sets and efe to Ref. 37 fo a genealization to the nonothogonal case. A. Model We conside a quantum conducto consisting of a cental egion C connected to left L and ight R leads Fig.. Fo times t t, the thee egions ae decoupled fom each othe, each being in themal equilibium with a common tempeatue T and chemical potentials L, C, and R, espectively. At t=t, the coupling between the thee sub

4 CONSERVING SCHEME FOR NONEQUILIBRIUM Enegy µ L µ R µ C Left lead (L) systems is switched on and a cuent stats to flow as the electode with highe chemical potential dischages though the cental egion into the lead with lowe chemical potential. Ou aim is to calculate the steady state cuent which aise afte the tansient has died out. We denote by i an othonomal set of single-paticle obitals and by H the Hilbet space spanned by i. The obitals i ae assumed to be localized such that H can be decomposed into a sum of othogonal subspaces coesponding to the division of the system into leads and cental egion, i.e., H=H L +H C +H R. We will use the notation i to indicate that i H fo some L,C,R. The noninteacting pat of the Hamiltonian of the connected system is witten ĥ = i,j Cental egion (C) µ L µ C µ R L,C,R h ij c i c j, = whee i, j un ove all basis states of the system. Fo, L,C,R, the opeato ĥ is obtained by esticting i to egion, and j to egion in Eq.. Occasionally, we shall wite ĥ instead of ĥ. We assume that thee is no diect coupling between the two leads, i.e., ĥ LR =ĥ RL = this condition can always be fulfilled by inceasing the size of the cental egion since the basis functions ae localized. We intoduce a special notation fo the diagonal of ĥ, ĥ = ĥ LL + ĥ CC + ĥ RR. Right lead (R) FIG.. Befoe the coupling between the thee egions is established, the thee subsystems ae in equilibium with chemical potentials L, C, and R, espectively. It is instuctive to note that ĥ does not descibe the thee egions in isolation fom each othe, but athe the contacted system without inte-egion hopping. We allow fo inteactions between electons inside the cental egion. The most geneal fom of such a two-body inteaction is 2 Vˆ = ijkl C V ij,kl c i c j c l c k. 3 The full Hamiltonian descibing the system at time t can then be witten. = ĥ + Vˆ fo t t Ĥ t = Ĥ 4 Ĥ = ĥ + Vˆ fo t t Notice that we use small lettes fo noninteacting quantities and the subscipt fo uncoupled quantities. The specific fom of the matix elements h ij and V ij,kl defining the Hamiltonian is consideed in Sec. V. Having defined the Hamiltonian, we now conside the initial state of the system, i.e., the state at times t t. Fo such times, the thee subsystems ae each in themal equilibium and, thus, chaacteized by thei equilibium density matices. Fo the left lead, we have with ϱˆ L = Z L exp ĥ L L Nˆ L Z L =T exp ĥ L L Nˆ L. Hee, is the invese tempeatue and Nˆ L=,i L c i c i is the numbe opeato of lead L. ϱˆ R and Z R ae obtained by eplacing L by R. Fo ϱˆ C and Z C, we must add Vˆ to account fo coelations in the initial state of the cental egion. The initial state of the whole system is then given by ϱˆ = ϱˆ Lϱˆ Cϱˆ R. If Vˆ is not included in ϱˆ C, we obtain the uncoelated noninteacting initial state ϱˆ ni. We note that the ode of the density matices in Eq. 7 plays no ole since they all commute due to the othogonality of the system i. Because Ĥ ĥ descibes the contacted system without inte-egion hopping, ϱˆ ϱˆ ni does not descibe the thee egions in physical isolation. In othe wods, the thee egions ae only decoupled at the dynamic level fo times t t. B. Contou-odeed Geen s function In this section, we intoduce the contou-odeed GF, which is the cental object fo the many-body petubation theoy in nonequilibium systems. Fo moe detailed accounts of the NEGF theoy, we efe to Refs. 38 and 39. The contou-odeed GF elevant fo the model intoduced in the pevious section is defined by G i,j, = i T ϱˆ T c H,i c H,j. 8 Hee, and ae points on the Keldysh contou, C, which uns along the eal-time axis fom t to and back to t, and T is the time-odeing opeato on the contou. The ceation and annihilation opeatos ae taken in the Heisenbeg pictue with espect to the full Hamiltonian in Eq. 4. Wedo

5 KRISTIAN S. THYGESEN AND ANGEL RUBIO not conside spin-flip pocesses and, thus, suppess the spin indices in the following. In ode to obtain an expansion of G ij, in powes of Vˆ, we switch to the inteaction pictue whee we have G ij, = i T ϱˆ T e i d Vˆ C h c h,i c h,j. 9 By extending C into the complex plane by a vetical banch unning fom t to t i, we can eplace ϱˆ by the uncoelated ϱˆ ni. 39 Neglecting the vetical banch then coesponds to neglecting coelations in the cental egion s initial state. While it must be expected that the pesence of initial coelations will influence the tansient behavio of the cuent, it seems plausible that they will be washed out ove time such that the steady state cuent will not depend on ϱˆ C. Futhemoe, in the special case of equilibium L = C = R and zeo tempeatue, the Gellman Low theoem ensues that the coelations ae coectly intoduced when stating fom the uncoelated initial state at t =. 4 In pactice, the neglect of initial coelations is a majo simplification which allows us to wok entiely on the eal axis, avoiding any efeence to the imaginay time. Fo these easons, we shall adopt this appoximation and neglect initial coelations in the est of this pape. Equation 9 with ϱˆ eplaced by ϱˆ ni constitute the stating point fo a systematic seies expansion of G ij in powes of Vˆ and the fee popagato, g ij, = i T ϱˆ ni T c h,i c h,j, Notice the slight abuse of notation: is not the submatix of. In fact, L and R ae both matices in the centalegion indices. Combining Eqs. and 2, we can wite G C, = g,c, + C d d 2 g,c, tot, 2 G C 2,, 4 which expesses G C in tems of the equilibium popagato of the noninteacting, uncoupled system, g, and the total self-enegy tot = + L + R. C. Real-time Geen s functions 5 In ode to evaluate expectation values of single-paticle obsevables, we need the eal-time coelation functions. We wok with two coelation functions, also called the lesse and geate GFs and defined as G ij t,t = i T ϱˆ ni c H,j t c H,i t, 6 G ij t,t = i T ϱˆ ni c H,i t c H,j t. 7 Two othe impotant eal-time GFs ae the etaded and advanced GFs, defined by which descibes the noninteacting electons in the coupled system. The diagammatic expansion leads to the identification of a self-enegy,, which elates the inteacting GF to the noninteacting one though Dyson s equation G ij t,t = t t G ij t,t G ij t,t, G a ij t,t = t t G ij t,t G ij t,t. 8 9 G, = g, + C d d 2 g,, 2 G 2, matix multiplication is implied. As we will see in Sec. IV A, only the Geen s function of the cental egion is needed fo the calculation of the cuent, and we can, theefoe, focus on the cental-egion submatix of G. Due to the stuctue of Vˆ, the self-enegy matix ij will be nonzeo only when both i, j C, and fo this eason, C subscipts can be added to all matices in Eq.. Having obseved this, we will, nevetheless, wite instead of C fo notational simplicity. The fee popagato g C,, which is still a nonequilibium GF, satisfies the following Dyson equation: g C, = g,c, + C d d 2 g,c, L, 2 + R, 2 g C 2,, 2 whee g is the equilibium GF defined by ϱˆ ni and ĥ. The coupling self-enegy due to lead =L,R is given by, = h C g,, h C. 3 The fou GFs ae elated via G G = G G a. 2 The lesse and geate GFs ae just special cases of the contou-odeed GF. Fo example, G t,t =G, when =t is on the uppe banch of C and =t is on the lowe banch. This can be used to deive a set of ules, sometimes efeed to as the Langeth ules, fo conveting expessions involving contou-odeed quantities into equivalent expessions involving eal-time quantities. We shall not list the convesion ules hee, but efe to Ref. 39 no initial coelations o Ref. 38 including initial coelations. The usual pocedue in nonequilibium is then to deive the elevant equations on the contou using the standad diagammatic techniques and subsequently convet these equations to eal time by means of the Langeth ules. An example of this pocedue is given in Sec. III B, whee the nonequilibium equations ae deived.. Equilibium In equilibium, the eal-time GFs depend only on the time diffeence t t. Fouie tansfoming with espect to this time diffeence then bings out the spectal popeties of the system. In paticula, the spectal function

6 CONSERVING SCHEME FOR NONEQUILIBRIUM A = i G G a = i G G 2 shows peaks at the quasipaticle QP enegies of the system. In equilibium, we futhemoe have the fluctuationdissipation theoem, G = if A, 22 G = i f A, 23 elating the coelation functions to the spectal function and the Femi Diac distibution function, f. The fluctuationdissipation theoem follows fom the Lehman epesentation which no longe holds out of equilibium, and as a consequence, one has to wok explicitly with the coelation functions in nonequilibium situations. 2. Nonequilibium steady state We shall wok unde the assumption that in steady state, all the eal-time GFs depend only on the time diffeence t t. Taking the limit t, this will allow us to use the Fouie tansfom to tun convolutions in eal time into poducts in fequency space. Applying the Langeth convesion ules to the Dyson equation 4 and Fouie tansfoming with espect to t t then leads to the following expession fo the etaded GF of the cental egion: G C = g,c + g,c tot G C. This equation can be inveted to yield the closed fom 24 G C = + i I C h C L R. 25 The equation fo G a is obtained by eplacing by a and by o, altenatively, fom G a = G. Fo the lesse coelation function, the convesion ules lead to the expession whee G / C = G C / tot G a C + /, 26 / = I C + G C tot g /,C I C + a tot G a C. 27 The dependence has been suppessed fo notational simplicity. Using /a tot = g /a,c G /a C togethe with the equilibium elations g,c = f C g,c g a,c and g,c = f C g,c g a,c,wefind =2i f C G C G a C, 28 =2i f C G C G a C. 29 If the poduct G G a is independent of, we can conclude that in the elevant limit of small. Howeve, as explained below, this is not always the case. 3. Bound states and the tem We fist focus on noninteacting electons. In this case, the nonequilibium coelation functions g / must be evaluated fom Eq. 26 with tot = L + R. Fo enegies outside the bandwidth of the leads, we have a = such that no boadening of the noninteacting levels is intoduced by the coupling to the leads. At such enegies we have g C g C a =2i g C g C a, and we conclude fom Eqs. 28 and 29 that / becomes popotional to the spectal function A=g C g C a. Since A does not necessaily vanish outside the bandwidth of the leads it has delta peaks at the position of bound states, it follows that / should be included in the calculation of g / to popely account fo the bound states. It is inteesting to notice that C, which defines the initial state of the cental egion, dops out of the equations fo g if and only if thee ae no bound states. When inteactions ae pesent in the cental egion, coelation effects will educe the lifetime of any single-paticle state in C. Mathematically, this is expessed by the fact that a will be nonzeo fo all physically elevant enegies. Consequently, the poduct G G a will appoach a finite value as, leading to a vanishing /. In conclusion, the tems of Eqs. 28 and 29 always vanish when inteactions ae pesent in C, while fo the noninteacting electons, they vanish eveywhee except fo coesponding to bound states. We mention that it has ecently been shown in the time-dependent NEGF famewok that the pesence of bound states can affect the long time behavio of the cuent in the noninteacting case. 4 III. EQUATIONS In this section, we deive and discuss the nonequilibium and second-ode Bon 2B appoximations. Howeve, befoe addessing the expessions fo the self-enegies, we intoduce an effective inteaction which leads to a paticulaly simple fom of the equations and, at the same time, povides a means fo educing self-inteaction eos in highe-ode diagammatic expansions. A. Effective inteaction The diect use of the full inteaction Eq. 3 esults in a fou-index polaization function. The numeical epesentation and stoage of this fequency-dependent fou-index function ae vey demanding, and fo this eason, we conside the effective inteaction defined by Vˆ eff = Ṽ i,j c i c j c j c i, 3 ij, whee Ṽ i,j = V ij,ij V ij,ji. 3 This expession follows by esticting the sum in the full inteaction Eq. 3 to tems of the fom V ij,ij c i c j c j c i and V ij,ji c i c j c j c i. The effective inteaction is local in obital space, i.e., it is a two-point function instead of a fou-point function and, thus, esembles the eal-space epesentation. Note, howeve, that in contast to the eal-space epesentation Ṽ i,j is spin dependent. In paticula, the self-inteactions Ṽ i,i ae zeo by constuction and, consequently, self-inteaction in the obital basis is avoided to all odes in a petubation expan

7 KRISTIAN S. THYGESEN AND ANGEL RUBIO sion in powes of Ṽ. Since the off-diagonal elements i j of the exchange integals V ij,ji ae small, one expects that the main effect of the second tem in Eq. 3 is to cancel the self-inteaction in the fist tem. It is not staightfowad to anticipate the quality of a calculation based on the effective inteaction 3 as compaed to the full inteaction 3. Clealy, if we include all Feynman diagams in, we obtain the exact esult when the full inteaction 3 is used, while the use of the effective inteaction 3 would yield an appoximate esult. The quality of this appoximate esult would then depend on the basis set, becoming bette the moe localized the basis functions and is equal to the exact esult in the limit of completely localized delta functions, whee only the diect Coulomb integals V ij,ij will be nonzeo. Howeve, when only a subset of all diagams ae included in, the situation is diffeent: In the appoximation, only one diagam pe ode in Vˆ is included, and thus cancellation of self-inteaction does not occu when the full inteaction is used. On the othe hand, the effective inteaction 3 is self-inteaction-fee in the obital basis by constuction. The situation can be undestood by consideing the lowest-ode case. Thee ae only two fist-ode diagams the Hatee and exchange diagams and each cancel the self-inteaction in the othe. Moe geneally, the pesence of self-inteaction in an incomplete petubation expansion can be seen as a violation of identities of the fom c k c i c i c j = when not all Wick contactions ae evaluated. Such expectation values will coectly vanish when the effective inteaction is used because the pefacto of the c i c i opeato, Ṽ i,i, is zeo. The pesence of selfinteaction eos in non-self-consistent calculations was ecently studied fo a hydogen atom. 42 In Appendix B, we compae the pefomance of the effective inteaction with exact esults fo the Hatee and exchange self-enegies of a benzene molecule. These fist-ode esults indicate that the accuacy of calculations based on the effective inteaction 3 should be compaable to calculations based on the full inteaction 3. We stess, howeve, that in pactice only the coelation pat of the self-enegy second- and highe-ode tems is evaluated using Vˆ eff, while the Hatee and exchange self-enegies ae teated sepaately at a highe level of accuacy see Sec. VC. B. Nonequilibium self-enegy It is useful to split the full inteaction self-enegy into its Hatee and exchange-coelation pats, = h, + xc,. 32 The Hatee tem is local in time and can be witten h, = h C,, whee C is a delta function on the Keldysh contou. Within the appoximation, the exchange-coelation tem is witten as a poduct of the Geen s function G and the sceened inteaction W, calculated in the andom-phase appoximation RPA. With the effective inteaction 3, the sceened inteaction and the Φ Σ Φ 2B Σ 2B = = = = _ 4 2 polaization ae educed fom fou- to two-index functions. Fo notational simplicity, we absob the spin index into the obital index, i.e., i i but we do not neglect it. The equations on the contou then ead,ij, = ig ij, + W ij,, W ij, = Ṽ ij C, + kl _ d Ṽ ik P kl, W lj,, C P ij, = ig ij, G ji, It is impotant to notice that in contast to the conventional eal-space fomulation of the method, the spin dependence cannot be neglected when the effective inteaction is used. The eason fo this is that Ṽ is spin dependent and, consequently, the spin off-diagonal elements of W will influence the spin-diagonal elements of G,, and P. A diagammatic epesentation of the appoximation is shown in Fig. 2. As they stand, Eqs involve quantities of the whole system leads and cental egion. Howeve, since Ṽ ij is nonzeo only when i, j C, it follows fom Eq. 34 that W and, hence, also have this stuctue. Consequently, the subscipt C can be diectly attached to each quantity in Eqs ; howeve, fo the sake of geneality and notational simplicity, we shall not do so at this point. It is, howeve, impotant to ealize that the GF appeaing in the equations includes the self-enegy due to the leads FIG. 2. The and second Bon self-enegies, and 2B, can be obtained as functional deivatives of thei espective functionals, G and 2B G. Staight lines epesent the full Geen s function G, i.e., the Geen s function in the pesence of coupling to the leads and inteactions. Wiggly lines epesent the inteactions

8 CONSERVING SCHEME FOR NONEQUILIBRIUM Using the Langeth convesion ules, 39 the etaded and lesse self-enegies become on the time axis,ij t = ig ij t W ij t + ig ij t W ij t, 36 /,ij t = ig / ij t W / ij t, 37 whee we have used the vaiable t instead of the time diffeence t t. Fo the sceened inteaction, we obtain in fequency space W = Ṽ I P Ṽ, W / = W P / W a, whee all quantities ae matices in the indices i, and matix multiplication is implied. Notice that the spin offdiagonal pat of Ṽ will affect the spin-diagonal pat of W though the matix invesion. Finally, the eal-time components of the ieducible polaization become P ij t = ig ij t G ji t ig ij t G a ji t, P ij / t = ig ij / t G ji / t. 4 4 Fom thei definitions, it is clea that both the polaization and the sceened inteaction obey the elations P a ij = P ji and W a ij =W ji, while fo the self-enegy and a GFs, we have = and G a =G.Inaddition, all quantities fulfill the geneal identity X X =X X a. We mention that equations simila to those deived above without the exta complication of coupling to extenal leads have peviously been used to calculate bulk band stuctues of excited GaAs. 43 In deiving Eqs. 38 and 39, we have made use of the convesion ules / C t,t = and /a C t,t = t t. With these definitions, the applicability of Langeth ules can be extended to functions containing delta functions on the contou. Notice, howeve, that with these definitions, elation 8 does not hold fo the delta function. The eason why the delta function equies a sepaate teatment is that the Langeth ules ae deived unde the assumption that all functions on the contou ae well behaved, e.g., do not contain delta functions. We stess that no spin symmety has been assumed in the above equations. Indeed, by eintoducing the spin index, i.e., i i and j j, it is clea that spin-polaized calculations can be pefomed by teating G and G independently. Within the appoximation, the full inteaction selfenegy is given by, = h, +,, 42 whee the self-enegy can be futhe split into an exchange pat and a coelation pat,, = x C, + co,. Due to the static natue of h and x, we have 43 / h = / x =. 44 The etaded components of the Hatee and exchange selfenegies become constant in fequency space, and we have note that fo h and x we do not use the effective inteaction 3 h,ij x,ij = i G kl t = V ik,jl, kl = i G kl t = V ik,lj. kl Due to Eq. 44, it is clea that Eq. 37 yields the lesse and/o geate components of co. Since co, does not contain delta functions, its etaded component can be obtained fom the elation co t = t t t. 47 The sepaate calculation of x and co fom Eqs. 46 and 47, as opposed to calculating thei sum diectly fom Eq. 36, has two advantages: i It allows us to teat x, which is the dominant contibution to, at a highe level of accuacy than co see Appendix A. ii We avoid numeical opeations involving G and W in the time domain see Appendix E. C. Nonequilibium second Bon appoximation When sceening and/o stong coelation effects ae less impotant, as, e.g., in the case of small molecules, the higheode tems of the appoximation ae small and it is moe impotant to include all second-ode diagams. 33 The full second-ode appoximation, often efeed to as the 2B, is shown diagammatically in Fig. 2. As we will use the 2B fo compaison with the esults, we state the elevant expessions hee fo completeness. The nonequilibium 2B has ecently been applied to study atoms in lase fields. 44 On the contou, the 2B self-enegy eads with the effective inteaction 3 2B,ij, = G ij, G kl, G lk, Ṽ ik Ṽ jl kl G ik, G kl, G lj, Ṽ il Ṽ jk. kl 48 Notice that the fist tem in 2B is simply the second-ode tem of the self-enegy. Fom Eq. 48, it is easy to obtain the lesse and/o geate self-enegies, / t = 2B,ij kl G / ij t G / kl t G / lk t Ṽ ik Ṽ jl G / ik t G / kl t G / lj t Ṽ il Ṽ jk, kl whee t has been used instead of the time diffeence t t. Since these second-ode contibutions do not contain delta functions of the time vaiable, we can obtain the etaded self-enegy diectly fom the Kames Konig elation

9 KRISTIAN S. THYGESEN AND ANGEL RUBIO 2B t = t 2B t 2B t see Appendix E. 49 IV. CURRENT FORMULA AND CHARGE CONSERVATION In this section, we addess the question of chage consevation in the model intoduced in Sec. II A. In paticula, we ask unde which conditions the cuent calculated at the left and ight sides of the cental egion ae equal, and we show in Sec. IV D that this is fulfilled wheneve the self-enegy used to descibe the inteactions is deivable, independent of the applied basis set. A. Cuent fomula As shown by Mei and Wingeen, 45 the paticle cuent fom lead into the cental egion can be expessed as I = d 2 T G C G C, 5 whee matix multiplication is undestood. By witing I = I L I R /2, one obtains a cuent expession symmetic in the L, R indices, I = 4 i T L R G C + f L L f R R G C G a C d, 5 whee we have suppessed the dependence and intoduced the coupling stength of lead, =i a. We note in passing that fo noninteacting electons, the integal has weight only inside the bias window, wheeas this is no longe tue when inteactions ae pesent. B. Chage consevation Due to chage consevation, we expect that in steady state I L = I R =I, i.e., the cuent flowing fom the left lead to the molecule is the negative of the cuent flowing fom the ight lead to the molecule. We deive a condition fo this specific fom of paticle consevation. Fom Eq. 5, the diffeence between the cuents at the left and ight intefaces, I=I L +I R, is given by I = d 2 T L + R G C L + R G C. 52 To obtain a condition fo I= in tems of, we stat by poving the geneal identity d 2 T tot G C tot G C =. 53 To pove this, we inset G / =G C / tot G a C + / fom Eq. 26 in the left hand side of Eq. 53. This esults in two tems involving G / tot G a and two tems involving /. The fist two tems contibute by d 2 T totg tot G a tot G tot G a. 54 Inseting tot = tot + G a G see Ref. 46 in this expession and using the cyclic invaiance of the tace, it is staightfowad to show that Eq. 54 vanishes. The two tems involving / contibute to the left hand side of Eq. 53 by d 2 T tot tot. 55 As discussed in Sec. II C 3, and ae always zeo when inteactions ae pesent. In the case of noninteacting electons, we have / tot = / L + / R, which vanishes outside the bandwidth of the leads. On the othe hand, / is only nonzeo at enegies coesponding to bound states, i.e., states lying outside the bands, and thus we conclude that the tem 55 is always zeo. Fom Eqs. 52 and 53, it then follows that I = d 2 T G C G C. 56 We notice that without any inteactions, paticle consevation in the sense I= is tivially fulfilled since =. When inteactions ae pesent, paticle consevation depends on the specific appoximation used fo the inteaction self-enegy. C. Conseving appoximations A self-enegy is called conseving, o deivable, if it can be witten as a functional deivative of a so-called functional, G = G / G. 28 Since a -deivable selfenegy depends on G, the Dyson equation must be solved self-consistently. The esulting Geen s function automatically fulfills all impotant consevation laws including the continuity equation, which is of majo elevance in the context of quantum tanspot. The exact G can be obtained by summing ove all skeleton diagams, i.e., closed diagams with no self-enegy insetions, constucted using the full G as popagato. Pactical appoximations ae then obtained by including only a subset of skeleton diagams. Two examples of such appoximations ae povided by the and second Bon functional and associated self-enegies, which ae illustated in Fig. 2. Solving the Dyson equation self-consistently with one of these self-enegies, thus, defines a conseving appoximation in the sense of Baym. The validity of the consevation laws fo -deivable selfenegies follows fom the invaiance of unde cetain tansfomations of the Geen s function. Fo example, it follows fom the closed diagammatic stuctue of that the tansfomation 28 G, e i G, e i, 57 whee is any scala function, leaves G unchanged. Using the compact notation, =, the change in when the GF is changed by G can be witten as = dd2,2 G 2, + =, whee we have used = G / G. To fist ode in, we then have

10 CONSERVING SCHEME FOR NONEQUILIBRIUM = i dd2,2 2 G 2, + = i dd2,2 G 2, + G,2 + 2,. Since this holds fo all by a scaling agument, we conclude that d2,2 G 2, + G,2 + 2, =. 58 It can be shown that this condition ensues the validity of the continuity equation on the contou at any point in space. 28 D. Chage consevation fom -deivable self-enegies We show that I of Eq. 56 always vanishes when the self-enegy is deivable, i.e., the geneal concept of a conseving appoximation caies ove to the discete famewok of ou tanspot model. We stat by noting that Eq. 58 holds fo any pai G,2, G,2 povided is of the -deivable fom. In paticula, Eq. 58 does not assume that the pai G, G fulfills a Dyson equation. Theefoe, by taking any othonomal, but not necessaily complete set, i, and witing G,2 = ij i G ij, 2 j * 2, we get fom Eq. 58 afte integating ove, j d ij, G ji, C + G ij, ji, =, 59 which in matix notation takes the fom G ij, e i i G ij, e i j, 6 whee is now a discete vecto. By adapting the aguments following Eq. 57 to the discete case, we aive at Eq. 58 with the eplacements i and 2 j and with the integal eplaced by a discete sum ove j. Summing also ove i leads diectly to Eq. 6, which is the desied esult. To summaize, we have shown that paticle consevation in the sense I L = I R is obeyed wheneve a -deivable selfenegy is used and eithe i all Coulomb matix elements V ij,kl o ii the tuncated two-point inteaction of Eq. 3 is used to evaluate. V. IMPLEMENTATION In this section, we descibe the pactical implementation of the Wannie- tanspot scheme. Afte a bief sketch of the basic idea of the method, we outline the calculation of the noninteacting Hamiltonian matix elements and Coulomb integals in tems of Wannie obitals. The explicit expession fo the Geen s function is given in Sec. V D, and in Sec. V F, we descibe ou implementation of the Pulay mixing scheme fo pefoming self-consistent Geen s function calculations. We end the section with a discussion of the pesent limitations and futue impovements of the method. A. Inteactions in the cental egion Most fist-pinciples calculations addessing tanspot in molecula contacts ae based on the assumption that the chage caies electons can be consideed as independent paticles govened by an effective single-paticle Hamiltonian. A popula choice fo the effective Hamiltonian is the Kohn Sham Hamiltonian of DFT, C d T, G, + G,, =. ĥ s = v ext + v h + v xc, 62 6 Hee, ij is exactly the self-enegy matix obtained when the diagams ae evaluated using G ij and the V ij,kl fom Eq. 3. The left hand side of Eq. 6, which is always zeo fo a -deivable, can be witten as T A t,t when A is given by Eq. C, with B= and C=G. It then follows fom the geneal esult C2 and the condition 56 that cuent consevation in the sense I L = I R is always obeyed when is deivable. The above deivation of Eq. 6 elied on all the Coulomb matix elements, V ijkl, that ae included in the evaluation of. Thus, the poof does not cay though if a geneal tuncation scheme fo the inteaction matix is used. Howeve, in the special case of a tuncated inteaction of the fom 3, i.e., when the inteaction is a two-point function, Eq. 6 emains valid. To show this, it is moe appopiate to wok entiely in the matix epesentation and, thus, define G ij, as the sum of a set of skeleton diagams evaluated diectly in tems of G ij and Ṽ ij. With the same agument as used in Eq. 57, it follows that is invaiant unde the tansfomation whee v ext is the extenal potential fom the ions, v h is the classical Hatee field, and v xc is the exchangecoelation xc potential which to some degee includes e-e inteaction effect beyond the Hatee level. In the pesent method, we ely on the Kohn Sham KS Hamiltonian to descibe the metallic electodes as well as the coupling into the cental egion, but we eplace the local xc potential by a many-body self-enegy inside the cental egion whee coelation effects ae expected to be most impotant. Clealy, this division does not teat all pats of the system on the same footing, and one might be concened that electons can scatte off the atificial inteface defined by the tansition egion between the mean-field and many-body desciption and, thus, intoduce an atificial contact esistance. Such unphysical scatteing is cetainly expected to affect the calculated popeties if the tansition egion is vey close to the constiction of the contact. On the othe hand, the cental egion can, at least in pinciple, be chosen so lage that the tansition egion occus deep in the electodes fa away fom the constiction. In this case, the lage numbe of available conductance channels in the electodes should ensue that the calculated popeties ae not dominated by

11 KRISTIAN S. THYGESEN AND ANGEL RUBIO inteface effects and the noninteacting pat of the electodes will mainly seve as paticle esevois whose pecise stuctue is unimpotant. Thus, the assumption of inteactions in the cental egion seems justified in pinciple although it might be difficult to fully avoid atificial backscatteing in pactice. B. Wannie Hamiltonian and Coulomb integals In ode to make the evaluation and stoing of the self-enegy feasible, we use a minimal basis set consisting of maximally localized, patially occupied Wannie functions 47 obtained fom the plane-wave pseudopotential code DACAPO. 48 Below we outline how the Hamiltonian is evaluated in the Wannie function WF basis, and we efe to Ref. 49 fo moe details. The WFs used to descibe the leads ae obtained fom a bulk calculation o supecell calculation if the leads have finite coss section. We define the extended cental egion C2 as the molecule itself plus a potion of the leads. C2 should be so lage that it compises all petubations in the KS potential aising fom the pesence of the molecula contact such that a smooth tansition fom C2 into the bulk is ensued. The WFs inside C2 ae obtained fom a DFT calculation with peiodic bounday conditions imposed on the supecell containing C2. The esulting WFs will inheit the peiodicity of the eigenstates; howeve, due to thei localized natue, they can be unambiguously extended into the lead egions. Thanks to the lage size of C2, hybidization effects between the molecule and the metal leads will automatically be incopoated into the WFs. With the combined set of WFs lead+c2, we can then epesent any KS state of the contacted system up to a few electon volts above the Femi enegy. 47 In pactice, the equiement of complete sceening means that 3 4 atomic layes of the lead mateial must be included in C2 on both sides of the molecule. While this size of systems can be easily handled within DFT, it may well exceed what is computationally feasible fo a many-body teatment such as the method even with the minimal WF basis. Fo this eason, we shall allow the cental egion C to consist of a pope subset of the WFs in C2, subject to the equiement that thee is no diect coupling acoss it, i.e., i ĥ s j = fo i L and j R, whee the left ight lead by definition is all WFs to the left ight of C. With this definition of C, the KS potential outside C is not necessaily peiodic this is, howeve, always the case outside C2, and consequently, the calculation of the coupling self-enegies becomes somewhat moe involved as compaed to the usual situation of peiodic leads see discussion in Appendix D. We stess that the tansmission function fo the noninteacting KS poblem is exactly the same whethe C o C2 is used as the cental egion as long as thee is no diect coupling acoss egion C. Having constucted the WFs, we calculate the matix elements of the effective KS Hamiltonian of the contacted, unbiased system, i ĥ s j. To coect fo double counting when the self-enegy is added, we also need the matix elements, i v xc j, fo WFs belonging to the cental egion. The matix elements defining the inteaction Vˆ in Eq. 3 ae calculated as the unsceened Coulomb integals V ij,kl = dd i * j * k l 63 fo WFs belonging to the cental egion. The Coulomb integals ae evaluated in Fouie space using neutalizing Gaussian chage distibutions to avoid contibutions fom the peiodic images see Ref. 5. C. Hatee and exchange As aleady mentioned, it is not feasible to include all the inteaction matix elements when evaluating the fequencydependent pat of the many-body self-enegy, co, which is theefoe calculated using the effective inteaction of Eq. 3. Howeve, the exchange tem, which can be unambiguously sepaated fom the self-enegy, is evaluated fom Eq. 46 using all Coulomb elements of the foms V ij,ij, V ij,ji, V ii,jj, V ii,ij. As shown in Appendix A, this poduces esults within 5% of the exact values. The KS Hamiltonian aleady includes the Hatee potential of the DFT gound state. In a self-consistent, finite-bias calculation, the elevant Hatee potential will deviate fom the DFT Hatee potential due to the finite bias and the fact that the xc potential is eplaced by the self-enegy. This coection, which is much smalle than the full Hatee potential, is teated in the same way as the exchange tem, i.e., calculated fom Eq. 46 with all Coulomb elements of the fom V ij,ij, V ij,ji, V ii,jj, V ii,ij. As fo the exchange tems, this yields esults within 5% of the exact values see Appendix A. D. Expession fo G To simplify the notation, in the following we omit the subscipt C as all quantities will be matices in the cental egion. The etaded GF of the cental egion is obtained fom G = + i I h s v xc L R h G h g eq s G. 64 Seveal comments ae in ode. Fist, we notice that all quantities except fo v xc, h s, and h g eq s ae bias dependent; howeve, to keep the notation as simple as possible, we omit any efeence to this dependence. The tems L and R account fo the coupling to the leads. By subtacting v xc fom h s, we ensue that exchange-coelation effects ae not counted twice when we add the self-enegy,. The tem v h = h G h g eq s is the change in Hatee potential elative to the equilibium DFT value. This change is due to the applied bias and the eplacement of v xc by even in equilibium, the Hatee field will change duing the self-consistency cycle. The Hatee potential in C oiginating fom the electon density in the electodes, which entes 5333-

12 CONSERVING SCHEME FOR NONEQUILIBRIUM G though h s, is assumed to stay constant when the system is diven out of equilibium, i.e., the out-of-equilibium chage distibution in the leads is assumed to equal the equilibium one. Finally, in ode to make contact with the geneal fomalism of Sec. II, and in paticula Eq. 25, we note that the matix elements h ij defining the effective single-paticle Hamiltonian in Eq. ae elated to the quantities intoduced above via i ĥ s vˆ xc j h h ij = g eq s ij fo i, j both in C i ĥ s j + L R F ij fo i, j both in L R i ĥ s j othewise. E. Fequency dependence To epesent the tempoal dependence of the Geen s functions and self-enegies, we use an equidistant fequency gid with N g gid points and gid spacing. Thus, the GFs and the self-enegies ae epesented by N w N w N g matices. At each of the discete fequencies i =n i, n i =,...,N g, we have an N w N w matix epesentation of G i in the WF basis. The gid spacing should be small enough that all featues in the fequency dependence of the GFs and self-enegies can be esolved. At the same time, the fequency gid should be lage enough contain enough points to popely descibe the asymptotic behavio the tail of the GFs. Although the tail is ielevant fo the cuent in Eq. 5, it contibutes to the self-enegy, G. In pactice, N g and should be inceased and deceased, espectively, until the esults do not change. To avoid time consuming convolutions on the fequency gid, we use the fast Fouie tansfom FFT to switch between fequency and time domains. An impotant but technical issue concening the evaluation of etaded functions is discussed in Appendix E. F. Self-consistency Since depends on G, and G depends on, the Dyson equations 26 and 64 must be solved self-consistently in conjunction with the equations fo the, Hatee, and exchange self-enegies. In pactice, this self-consistent poblem is solved by iteation. Clealy, the iteative appoach elies on the assumption that the poblem has a unique solution and that the iteative pocess conveges to this solution. Fo all applications we have studied so fa, this has been the case. In ode to stabilize the iteative pocedue, we use the Pulay scheme 5 to mix the GFs of the pevious N iteations, vey simila to what is done fo the electon density in many DFT codes. Moe specifically, the input GF at iteation n is obtained accoding to n G in X,n = j=n N n c j n G in X,j + j=n N c n j G X,j out, X. 65 To detemine the optimal values fo the expansion coefficients, c n, we fist define an inne poduct in the space of etaded GFs Bulk G,i,G,j = n (C) (C2) Bulk FIG. 3. The extended cental egion C2 is chosen so lage that it compises all petubations in the effective DFT potential aising fom the molecula contact. The cental egion C can be a pope subegion of C2, but it must be so lage that thee is no diect coupling acoss it. We solve fo the self-consistent Kohn Sham potential within C2, but we eplace the static xc potential by the self-enegy inside C. Im G,i nn * Im G,j nn d. 66 Equivalent inne poducts can be obtained, e.g., by using the eal pat of the GF instead of the imaginay pat o the lesse component instead of the etaded pat. The Pulay esidue matix detemining the coefficients c n is then given by A n ij = G,i in G,i out,g,j in G,j out, 67 whee i, j=n N,...,n. We typically use a mixing facto aound.4. Duing the mixing pocedue, one must keep tack of both the etaded and lesse GFs since one does not follow diectly fom the othe. Howeve, it is impotant that the same coefficients, c n, ae used fo mixing the two components. If sepaate coefficients ae used fo G and G, the fundamental elation 2 is not guaanteed duing the selfconsistent cycle. As noted above, we define the esidue exclusively fom the etaded GF. In pactice, we always find that once the etaded GF has conveged, the lesse GF has conveged too, and this justifies the use of common expansion coefficients fo the two GF components. G. Oveview We give an oveview of the vaious steps involved in pefoming a self-consistent nonequilibium tanspot calculation as follows: Pefom DFT calculations fo the electodes and the extended cental egion egion C2 in Fig Constuct the Wannie functions and obtain the matix epesentation of the KS Hamiltonian fo the contacted system in equilibium. Evaluate the matix elements fo v xc and elevant Coulomb integals fo Wannie functions belonging to the cental egion C. 3 Fix the bias voltage and calculate the coupling selfenegies Eq. 3 as descibed in Appendix D these stay unchanged duing self-consistency. 4 Evaluate the initial noninteacting Geen s functions, G C and G C, e.g., fom the KS Hamiltonian. 5 Fom G C and G C, constuct the desied inteaction self-enegies h, x,,o 2B. 6 Test fo self-consistency. In the negative, obtain a new set of output Geen s functions fom Eqs. 64 and 26, and 5333-

13 KRISTIAN S. THYGESEN AND ANGEL RUBIO mix with the pevious GFs as descibed in Sec. V F. H. Limitations and futue impovements The main appoximation of the pesent implementation is the use of a fixed, minimal basis set. We have used WFs obtained fom the DFT-PBE obitals whee PBE denotes Pedew Buke Enzehof ; howeve, one could also use Hatee Fock o some othe mean-field obitals. Out of equilibium, the WFs will be distoted due to the change in electostatic potential; howeve, this effect is not included. Although the manifold spanned by the WFs, i.e., the KS eigenstates up to a few electon volts above the Femi level, ae expected to epesent the quasipaticle wave functions of the same enegy ange quite well, an accuate epesentation of the sceened inteaction might equie inclusion of high-enegy eigenstates. With the pesent implementation of the scheme, it is not feasible to include moe than a few electode atoms in addition to the molecule itself in the egion egion C in Fig. 3. The use of a small C egion might affect the desciption of image chage fomations in the electode, and it might intoduce atificial backscatteing at the DFT- inteface. The use of lage and moe accuate basis sets as well as the inclusion of moe electode atoms in the egion ae not fundamental but pactical limitations of the method, which, in pinciple, could be emoved by invoking efficient simplifications and/o appoximations into the pesent fomalism. VI. ANDERSON MODEL Since its intoduction in 96, the Andeson impuity model 52 has become a standad tool to investigate stong coelation phenomena such as local moments fomation, Kondo effects, and Coulomb blockade. The Andeson model descibes a localized electonic level of enegy c and coelation enegy U coupled to a continuum of states. Thus, the cental-egion pat of the Hamiltonian eads Ĥ C = c c c + Un n. 68 In equilibium, accuate esults fo the themodynamic popeties of the Andeson model have been obtained fom the Bethe ansatz, 53,54 quantum Monte Calo simulations, 55,56 and numeical enomalization goup theoy. 36,57 Out of equilibium, the low-tempeatue popeties of the Andeson model have been much less studied. The ealiest wok addessed the poblem by applying second-ode petubation theoy in the inteaction stength U. 58,59 Despite the simplicity of this appoach, it povides a supisingly good desciption of the equilibium spectal function. Thee ae, howeve, seveal fundamental poblems elated to the nonself-consistent low-ode petubative appoach: i the esult depends on the stating point aound which the petubation is applied, ii it inevitably violates the consevation laws, and iii it applies only in the small-u limit. Methods elying on the slave-boson technique 6 have been developed to exploe the stong coelation egime of the model. The noncossing appoximation is believed to wok well in the infinite-u limit and fo sufficiently small tunneling stength,, but it fails to epoduce the coect Femi liquid behavio at low tempeatues. 6,62 Moe ecently, a finite-u slaveboson mean-field appoach 63 has been poposed. Finally, we mention that a numbe of moe advanced schemes have been used to addess nonequilibium Kondo-like phenomena focusing on the low-enegy popeties of the Andeson model in the limit whee U is much lage than the hybidization enegy, While the Andeson model is nomally used to descibe stongly coelated systems, the main application of the appoximation has been on weakly inteacting quasipaticles in closed shell systems such as molecules, insulatos, and semiconductos. In view of this, one could ague that the method is inappopiate fo the Andeson model. Nevetheless, we find this application athe instuctive as it illustates some geneal featues of the appoximation including the ole of self-consistency both in elation to chage consevation and the line shape of spectal functions. Moeove, as many impotant tanspot phenomena, such as Kondo effects and Coulomb blockade, ae well descibed by the Andeson model, it should always be of inteest to benchmak a tanspot scheme against this model. In a vey ecent study, 67 the appoximation was applied to the Andeson model in equilibium fo inteaction stengths U/ up to 8.4/.65 3 and vaious tempeatues. Fo the lagest inteaction stength, it was found that pefes to beak the spin symmety, leading to diectly eoneous esults in the Kondo egime. Fo intemediate inteaction stengths U/ =4.2/ whee does not beak the spin symmety, it was concluded that does not descibe the T dependence of the Kondo effect well. Nevetheless, we show hee that at T=, the width of the Kondo-like esonance follows the analytical esult fo T K quite well fo intemediate inteaction stengths. Hee, as in ou pevious pape, 27 we focus on the zeo tempeatue, nonequilibium situation. We conside inteaction stengths of U/ up to 8 we keep U=4 fixed and vay. Fo these inteaction stengths, we always find a stable nonmagnetic solution, i.e., G =G. In contast, the HF solution can develop a magnetic moment fo U/ depending on bias voltage and c. We adopt the wide-band appoximation whee the coupling to the continuum is modeled by constant imaginay self-enegies L + R = i. Without loss of geneality, we set E F =. In all calculations, the fequency gid extends fom 5 to 5 with the gid spacing anging fom. to.5. A. Equilibium spectal function In Fig. 4, we show the c dependence of the equilibium spectal function, A = ImG, fo U=4 and =.65. The HF solutions ae Loentzians centeed at HF = c +U nˆ with a full width at half maximum FWHM given by 2. As can be seen, the position of the HF peaks do not vay linealy with c. Instead, thee is a chaging esistance fo the peak to move though the Femi level due to the cost in Hatee enegy associated with the filling of the level. This effectively pins the level to E F

14 CONSERVING SCHEME FOR NONEQUILIBRIUM.5 ε c =-4.4 ε c =-3.6 ε c =-2.8 HF.5 U/Γ=2. U/Γ=4. U/Γ=8. HF ε c =-2. A(ω)Γ.5 2B A(ω)Γ.5 2B ω FIG. 4. Colo online Spectal function of the cental site fo =.65, U=4., and diffeent values of c. The inset in the lowe panel is a zoom of the spectal peak aound = ω FIG. 6. Colo online Spectal function fo U=4., c = U/2, and thee diffeent values of =2.,., and.5 coesponding to stong, intemediate, and weak coupling to the leads. Moving fom HF to the second Bon appoximation, the Loentzian shape of the spectal peak is distoted due to the dependence of the 2B self-enegy. We can obseve a geneal shift of spectal weight towad the chemical potential as well as a naowing of the esonance as it comes close to E F. The edistibution of the spectal weight towad the chemical potential becomes even moe ponounced in the appoximation. Fo U c the so-called Kondo egime, a shap peak develops at E F. Fo U/ sufficiently lage, the Kondo effect should eveal itself as a peak in the spectal function with a FWHM given appoximately by the Kondo tempeatue 68 T K.5 2 U /2 exp c c + U /2 U. 69 In Fig. 5, we compae the above expession fo T K with the FWHM. (Γ=.5) (Γ=.65) Analytic (Γ=.5) Analytic (Γ=.65) πε c (ε c +U)/(2UΓ) FIG. 5. Colo online FWHM of the Kondo esonance as calculated in the appoximation and fom the analytical esult Eq. 69. The inteaction stength is U=4 and c is vaied in the Kondo egime. FWHM of the Kondo peak. The exponential scaling of T K is supisingly well epoduced. Deviations fom the exponential scaling natually occu fo smalle values of U/ not shown, whee the Kondo effect does not occu and 69 does not apply. In accodance with ecent wok, 67 we wee not able to obtain nonmagnetic solutions in the stong inteaction egime U/ 8. In Fig. 6, we show the dependence of the spectal function on the atio U/ fo the cental level at the symmetic position c = U/2= 2. Fo U/ =2, thee is no significant diffeence between the thee desciptions. This is to be expected since the coelation plays a mino ole compaed to the hybidization effects. In the weakly coupled limit, howeve, coelations become significant and, as a consequence, the 2B and esults change makedly fom the Loentzian shape and show a Kondo-like peak at the metal Femi level. The 2B appoximation significantly oveestimates the width of the Kondo peak, indicating, as expected, that the higheode RPA tems enhance the stong coelation featues. Fo lage U/, it is known 36,57 that the spectal function, in addition to the Kondo peak, should develop peaks at the atomic levels c and c +U. We find that the self-consistent 2B and appoximations always fail to captue these sidebands and instead distibute the spectal weight as a boad slowly decaying tail. These findings agee well with pevious esults obtained with the fluctuation-exchange appoximation 69 and with studies of the homogeneous electon gas, which showed that self-consistency in the self-enegy washed out the satellite stuctue in the spectum. 7 B. Nonequilibium tanspot We now move to the nonequilibium case and intoduce a diffeence in the chemical potentials of the two leads. In Fig. 7, we show the zeo-tempeatue diffeential conductance unde a symmetic bias, L/R = V/2, as a function of c fo U=4 and =.65. The di/dv at bias voltage V has been

15 KRISTIAN S. THYGESEN AND ANGEL RUBIO Diffeential conductance (2e 2 /h) HF 2B V= V=.8 V=.4 V=.8 V= ε c FIG. 7. Colo online Diffeential conductance, di/dv, asa function of the cental site enegy, c, fo diffeent applied biases, U=4 and =.65. calculated as a finite diffeence between the cuents obtained fom Eq. 5 fo bias voltages V and V+ V, espectively. The 2B esult falls in between the HF and esults, and fo this eason, we will focus on the latte two in the following discussion. Fo V=, thee is only little diffeence between the thee esults, which all show a boad conductance peak eaching the unitay limit at the symmetic point c = U/2. The physical oigin of the conductance tace is, howeve, vey diffeent: While the HF esult is poduced by coheent tanspot though a boad spectal peak moving igidly though the Femi level, the esult is due to tanspot though a naow Kondo peak which is always on esonance fo c in the Kondo egime. In all cases, the width of the di/dv cuve is appoximately U. In the case, this is because the Kondo peak develops only when the cental level is half occupied, i.e., U c. In HF, on the othe hand, the di/dv peak acquies a width on the ode of U due to the chage pinning effect discussed in Sec. VI A. The diffeence in the mechanisms leading to the HF and esults is bought out clealy as V is inceased: fo V, the bias has little effect on the HF conductance, while the conductance dops damatically aleady at biases compaable to T K due to the suppession of the Kondo esonance at finite bias. The suppession of the Kondo esonance is due to quasipaticle QP scatteing. While QP scatteing does not affect the lifetime of QPs at E F in equilibium, it does so at finite bias, whee Im E F becomes nonzeo. We mention that we do not obseve a splitting of the Kondo esonance at finite V. 62 The peaks appeaing in the di/dv at the lagest bias V =4 occu when the cental level is aligned with eithe the lowe o uppe edge of the bias window. It is woth noticing that the height of these peaks ae smalle than the value of G expected fom on-esonant tanspot though a single level. The eason fo this is twofold: i The bias window only hits the esonance with one edge eithe uppe o lowe edge, and consequently, only half the spectal weight entes the bias window when the voltage is inceased by V as compaed to the low-bias situation. ii The self-consistent chaging esistance discussed in Sec. VI A pins the level to the edge of the bias window, making the esonance follow the bias. C. G W appoximation Non-self-consistent, o one-shot, calculations can be pefomed by evaluating the sceened inteaction and self-enegy fom some tial noninteacting Geen s function G. The esulting G W appoximation, with G obtained fom a local density appoximation LDA and/o genealized gadient appoximation GGA calculation, has been found to yield vey satisfactoy esults fo the band gaps of insulatos and semiconductos. 3,32 Fo this eason, and due to its significantly lowe computational cost, this G W appoach has geneally been pefeed ove the self-consistent. One athe unsatisfactoy featue of the petubative G W method is its G dependence. Howeve, as will be demonstated below, a just as citical poblem in nonequilibium situations is its nonconseving natue. Befoe we apply the G W appoximation to the Andeson model, we need to addess a cetain issue which unfotunately has led to an eo in ou pevious pape. 27 All conclusions fom that pape ae, howeve, unaffected by the mistake.. Instability of the nonmagnetic gound state Conside a system which admits a spin-polaized gound state at the Hatee level notice that Hatee and HF is equivalent fo the Andeson model when the effective inteaction of Eq. 3 is used, and let G denote the GF obtained fom spin-unpolaized Hatee calculation. It tuns out that the analytical popeties of the sceened inteaction, W G, evaluated fom G will be wong. In paticula, W G will not be etaded as it should be. The eason is that the RPA esponse function is ill defined aound the nonmagnetic, and thus unstable, G. The poblem has been peviously mentioned by White 69 and was bought to the authos attention by Spatau. Fo cetain paamete values, the HF gound state of the Andeson model develops a finite magnetic moment. As a consequence, the analytic popeties of W as calculated fom Eq. 38 with the unpolaized G HF become wong. In ou pevious pape, 27 this poblem was not ecognized because we, fo numeical efficiency, applied the Kames Konig elation 47 to obtain fom, instead of using Eq. 36. Thus, by constuction, ou was etaded. Specifically, this implies that the G W spectal function plotted in Fig. of Ref. 27 as well as the di/dv cuves in the middle panel of Fig. 2 fo c in the inteval 3.6 to.4 ae incoect. In fact, thee exists no nonmagnetic G W G HF solution in these cases. We stess, howeve, that all conclusions fom ou pape ae unaffected by this mistake. In paticula, we show below that fo paamete values leading to a stable nonmagnetic HF gound state, the G W appoxima

16 CONSERVING SCHEME FOR NONEQUILIBRIUM di/dv (2e 2 /h) I /I HF G W V FIG. 8. Colo online Diffeential conductance as a function of applied bias fo U=4, =.65, and c = 4. Fo these paametes, the nonmagnetic HF solution is stable fo bias voltages smalle than.6. The G W appoximation yields diffeent cuents at the left and ight intefaces I and yields negative diffeential conductance at finite bias. tion still violates chage consevation and gives unphysical esults such as negative diffeential conductance. Moeove, we aive at the same conclusions fo G W self-enegies constucted fom the spin-polaized HF Geen s function, in which case the instability poblem does not occu at all. 2. Results of the G W appoximation In Fig. 8, we show the calculated di/dv fo the Andeson model with =.65 and c = 4 fo the HF,, and G W G HF appoximations. Fo these paametes, the nonmagnetic HF solution is stable fo bias voltages smalle than.6 such that the G W appoximation based on a nonmagnetic G HF is indeed meaningful in this paamete ange. The G W conductance has been obtained as a finite diffeence between the cuents obtained fom Geen s functions with self-enegies G HF V and G HF V+ V, espectively, whee G HF V is the HF Geen s function evaluated self-consistently unde a bias voltage V. Fom Fig. 8 we conclude that the G W appoximation leads to unphysical esults in the fom of stong negative diffeential conductance. Moeove, as shown in the lowe panel of the figue, the G W appoach gives diffeent values fo I L and I R. We note in passing that this symmety beak comes fom the diffeent chemical potentials of the left and ight leads. Finally, we mention that the inceasing behavio of I/I as a function of bias voltage seems to be a geneal effect. As aleady mentioned, the HF solution beaks the spin symmety fo cetain paamete values. Meaningful G W esults can still be obtained in this case povided the selfenegy is constucted fom the spin-polaized HF Geen s function. Figues 9 and compae the esult of such calculations with self-consistent fo two diffeent values of the bias voltage. Fom the figues, we daw the following conclusions: i The G W and cuents agee when the level is almost empty o filled. ii The cuent calculated in Occupation Cuent I HF (up) HF (down) G W G W ε c 2 4 FIG. 9. Colo online Uppe panel: Occupation of the cental site as function of c fo U=4, =.65, and bias V=.8. Notice that the HF solution beaks the spin symmety fo some c values. Middle panel: Cuent calculated in self-consistent and G W G HF,,G HF,. Lowe panel: Violation of the continuity equation measued as the diffeence between the cuents in the left and ight leads. G W show unphysical behavio in and close to the magnetic egime. iii The violation of chage consevation in G W is moe sevee when the cuent is lage. Occupation Cuent I HF (up) HF (down) G W G W ε c FIG.. Colo online Same as Fig. 9, but fo bias voltage V =

17 KRISTIAN S. THYGESEN AND ANGEL RUBIO VII. BENZENE JUNCTION In this section, we apply the Wannie- method to a moe ealistic nanojunction, namely, a benzene molecule coupled to featueless leads. In contast to the Andeson model consideed in the peceding section, the benzene junction epesents a closed-shell system with the Femi level lying within the HOMO-LUMO gap, leading to athe low tansmission fo all but the stongest molecule-lead coupling stengths. The use of featueless wide-band electodes is convenient as it allows us to isolate the effects of the electonelecton inteactions. The use of moe ealistic contacts with enegy dependent spectal featues would lead to an additional enomalization of the molecula levels, making a clea sepaation between xc and contact effects moe difficult. We stess, howeve, that the contacts only ente the theoy though the coupling self-enegies, which can be calculated once and fo all as in the standad NEGF-DFT appoach. Thus, the use of moe ealistic contact self-enegies is staightfowad. To descibe the benzene molecule, we fist pefom a DFT calculation fo the isolated molecule see Ref. 74. The KS eigenstates ae then tansfomed into maximally localized WFs, and the KS Hamiltonian and Coulomb integals ae evaluated in the WF basis. Fo the inteactions, we use the tuncation scheme Vˆ 2 defined in Appendix A to evaluate Hatee and exchange self-enegies. As shown in Table I, this leads to esults within 5% of the exact values. We use the effective inteaction Eq. 3 fo the coelation pat of the self-enegy. In all calculations, we have applied a fequency gid extending fom to ev and gid spacings in the ange.2 to.2, depending on the value of. In Sec. VII A, we show that the expeimental ionization potential of the isolated benzene molecule is vey well epoduced with ou scheme. In Sec. VII B, we investigate the ole of the coupling stength on the spectum of the benzene junction. Finally, in Sec. VII C, we calculate the nonequilibium conductance of the junction and we compae vaious appoximations fo the xc self-enegy. b) Fock, and iii fully self-consistent. We stess that ou calculations include the full dynamical dependence of the self-enegy as well as all off-diagonal elements. Thus, no analytic extension is pefomed, and we do not lineaize the self-enegy aound the DFT eigenvalues to obtain an appoximate quasipaticle equation as is done in standad calculations. The spectal peaks seen in Fig. 2 occuing above below the Femi level coespond to electon addition emoval enegies. In paticula, the HOMO level should coina) µ L FIG.. Colo online a Illustation of a benzene molecule coupled to featueless electodes with diffeent chemical potentials. b Isosufaces fo the 8 patially occupied Wannie functions used as basis functions in the calculations. The WFs ae linea combinations of Kohn Sham eigenstates obtained fom a DFT-PBE planewave calculation. µ R PBE A. Spectum of isolated benzene Within ou geneal tanspot fomalism, we model the situation of a fee molecule by using a vey weak coupling to the wide-band leads see Fig. a. The contacts meely act as paticle esevois fixing the numbe of electons on the molecule and poviding an insignificant boadening =.5 ev of the discete enegy levels. We fix the Femi levels of the electodes to E F = 3 ev, which is appoximately halfway between the HOMO and LUMO levels the pecise position of E F within the gap is unimpotant fo the esults pesented in this section. In Fig. 2, we show the total density of states DOS, DOS (ab. units) µ L = µ R Enegy (ev) HF N w D = n= Im G nn, 7 whee the sum uns ove all WFs on the molecule. We use thee diffeent appoximations: i DFT-PBE, ii Hatee FIG. 2. Density of states fo a benzene molecule weakly coupled to featueless leads =.5. The common Femi levels of the leads is indicated. Notice the chaacteistic opening of the band gap when going fom DFT-PBE to HF and the subsequent slight eduction when coelations ae included at the level