Van Leeuwen, Robert

English

We give an overview of the underlying concepts of time-dependent current-density functional theory (TDCDFT). We show how the basic equations of TDCDFT can be elegantly derived using the time contour method of nonequilibrium Green function theory. We further demonstrate how the formalism can be used to derive explicit equations for the exchange-correlation vector potentials and integral kernels for the Kohn-Sham equations and their linearized form.</p

ARTS repository - University of Groningen

   University of GroningenNonequilibrium green functions in time-dependent current-density-functional theoryVan Leeuwen, RobertPublished in:Proceedings of the Conference Progress in Nonequilibrium Green's Functions IIIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2003Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Van Leeuwen, R. (2003). Nonequilibrium green functions in time-dependent current-density-functionaltheory. In M. Bonitz, & D. Semkat (Eds.), Proceedings of the Conference Progress in NonequilibriumGreen's Functions II (pp. 427-435). World Scientific Publishing.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 06-06-2022NONEQUILIBRIUM GREEN FUNCTIONS INTIME-DEPENDENT CURRENT-DENSITY-FUNCTIONALTHEORY∗ROBERT VAN LEEUWENTheoretical Chemistry, Materials Science Centre, Rijksuniversiteit Groningen9747 AG Nijenborgh 4,Groningen, The NetherlandsE-mail: R.van.Leeuwen@chem.rug.nlWe give an overview of the underlying concepts of time-dependent current-density-functional theory (TDCDFT). We show how the basic equations of TDCDFT canbe elegantly derived using the time contour method of nonequilibrium Green func-tion theory. We further demonstrate how the formalism can be used to deriveexplicit equations for the exchange-correlation vector potentials and integral ker-nels for the Kohn-Sham equations and their linearized form.1. IntroductionTime-dependent density-functional theory (TDDFT) is a method for cal-culating properties of many-electron systems in time-dependent externalfields (for reviews see 1,2). In TDDFT one introduces a noninteracting sys-tem with the same density n(rt) as the true system. This system is calledthe Kohn-Sham system and its external potential, which is a functional ofthe density, is denoted as vs(rt). Once an approximate functional is giventhe Kohn-Sham equations can be solved self-consistently as has been donevery successfully. Nevertheless, the theory has the limitation that it cannot deal with transverse fields. This limitation is lifted in time-dependentcurrent-density-functional theory (TDCDFT), 3 whose basic variable is thecurrent-density in addition to the density and in which an effective vectorpotential As appears as well as a scalar potential. This theory has shownto be very successful in describing macroscopic polarization in solids 4,plasmon line widths 5 and polarizabilities of polymers 6. In this work we∗based on a poster presented at the conference “progress in nonequilibrium greens func-tions, dresden, germany, 19.-22. august 2002”12will describe how systematic approximations for As can be obtained fromperturbation theory on the basis of Keldysh Green functions.2. The field-matter HamiltonianThe full Hamiltonian in the Heisenberg picture for a quantized electromag-netic field and nonrelativistic electrons in the presence of a classical externalelectromagnetic field (φext,Aext) is given by 8,7Ĥ =18π∫d3r((∇φ̂+ ∂tÂ)2 + c2(∇× Â)2)+∑σ∫d3r(12ψ̂†σ(−i∇− e(Â + Aext))2ψ̂σ + eψ̂†σψ̂σφext)(1)where σ is a spin index, e is the electron charge and we use the Coulombgauge ∇ · Â = ∇ ·Aext = 0. We formally obtain the operator equationsφ̂(rt) = e∫d3r′n̂(r′t)|r− r′|(2)Â(rt) =ec2∫d3r′ĵT (r′, t− |r− r′|/c)|r− r′|(3)where n̂ is the number density operator and jT is the transverse part ofthe particle current operator ĵ(rt) = ĵp(rt) − en̂(rt)Âtot(rt) where we de-fined the total vector potential by Âtot = Â + Aext and the density andparamagnetic current (Heisenberg) operators byn̂(rt) =∑σψ̂†σ(rt)ψ̂σ(rt) (4)ĵp(rt) =∑σ12i(ψ̂†σ(rt)∇ψ̂σ(rt)− [∇ψ̂†σ(rt)]ψ̂σ(rt)) (5)If we now insert the explicit form of operator φ̂ into the full Hamiltonianand replace Â in the mean field approximation with its expectation value〈Â〉 and treat this field classically, we obtain (using Gauss’ theorem totransform (∇φ̂)2 to φ̂∇2φ̂ ) the following Hamiltonian in the Schrödingerpicture for the electronsĤ =∑σ12∫d3rψ̂†σ(r)(−i∇− e〈Âtot(rt)〉)2ψ̂σ(r) + en̂φext(rt)+∑σ,σ′e22∫d3rd3r′ψ̂†σ(r)ψ̂†σ′(r′)ψ̂σ′(r′)ψ̂σ(r)|r− r′|(6)This is the Hamiltonian that will be used as the basis of our theory. Inthe following we put e = −1 and we will neglect the induced field 〈Â〉 such3that 〈Âtot〉 = Âext and call this field A(rt). Note that this is allowed infinite systems, but not in infinite solids where there will be a macroscopiccontribution to the induced field 4. We will also choose a gauge in which φextis the potential of the nuclear framework of a solid or molecule. All otherpotentials (externally applied or effective) will be described by a vectorpotential A which in general will also contain longitudinal components.3. The fundamental equations of TDCDFTWe start out by defining the Schwinger-Keldysh contour 9,10,11,12 . This isa parametrization of real time t(τ) by an underlying pseudotime parametersuch that if τ runs from a certain initial τi to a final τf then the real timeruns from t0 to t1 and back to t0 (in practice we will often choose t0 = −∞and t1 = +∞). We then define a generating functional, resembling thepartition function of statistical mechanics:F̃ [A] = i ln〈Ψ0|V (τf , τi)|Ψ0〉 (7)where V is the τ - or contour ordered evolution operator of the systemV (τ2, τ1) = TC exp [−i∫ τ2τ1dτt′(τ)Ĥ(τ)] (8)with t′(τ) = dt/dτ and where TC denotes ordering in τ , Ĥ is the Hamilto-nian of Eq.(6) and the state |Ψ0〉 is the initial state. A point on the contourwill be denoted as 1̄ = r1τ1, whereas a point in physical time is denotedas 1 = r1t1. Here we extend the original definition of the Hamiltonian andallow A(1̄) to take different values of the forward and backward parts ofthe contour. Functional derivatives will eventually be evaluated at physicalvector potentials with identical values on both sides of the contour. Toevaluate the functional derivative we useδV (τ2, τ1)/δA(3̄) = −iV (τ2, τ )̂j(3̄)V (τ, τ1) (9)where τ1 < τ < τ2. These equations follow directly from the Schrödingerequation. Using these equations we findδF̃ /δA(1̄) = 〈̂jH(1̄)〉 = j(1̄) (10)where we defined the Heisenberg representation of an operator as usual byÔH(τ) = V (τi, τ)ÔV (τ, τi) and the expectation value by〈ÔH(τ)〉 =〈Ψ0|TC [V (τf , τi)ÔH(τ)]|Ψ0〉〈Ψ0|V (τf , τi)|Ψ0〉(11)4Therefore the derivative of F̃ at the physical potential A is the gaugeinvariant current j(rt) of the system in the external field A(rt). We cannow calculate higher order response functions by repeated differentiation.To not overcrowd notation we drop the subindex H from the operators.The current-current response function χµν is then given byχµν(1̄, 2̄) =δ2F̃δAµ(1̄)δAν(2̄)= δµνn(1̄)δC(t1 − t2)− i〈TC [∆ĵp,µ(1̄)∆ĵp,ν(2̄)]〉 (12)where δC(t1 − t2) = δ(τ1 − τ2)/t′(τ1) is the contour delta function 11 andwhere the current-density fluctuation operator ∆ĵp,µ(1̄) = ĵp,µ(1̄)−〈̂jp,µ(1̄)〉enters due to the derivatives of the denominator in Eq.(10). The responsefunction is a symmetric function of its arguments as it should (being asecond order functional derivative) and can be regarded as a integral kernelin pseudotime. It will however become a retarded function acting in physicaltime as can be seen from the current response δj(rt) due to a physicalpotential variation δA(rt):δjµ(1) =∑ν∫Cd2̄χµν(1̄, 2̄)δAν(2) =∑ν∫d2χR,µν(1, 2)δAν(2) (13)Here the function χR,µν is the usual retarded response functionχR,µν(1, 2) = n0(r1)δµνδ(t1 − t2)δ(r1 − r2)− iθ(t1 − t2)〈Ψ0|[̂jp,µ(1), ĵp,ν(2)]|Ψ0〉 (14)as it usually appears in response theory. The outlined procedure appliesto all higher order derivatives as well, i.e. all higher order response func-tions are symmetric functions in pseudotime and become causal or retardedfunctions in physical time. This feature has been used to resolve a recentparadox in TDDFT 13.We now want to use the current j(rτ) as our basic variable and we performa Legendre transform and defineF [j] = −F̃ [A] +∫Cd1̄j(1̄) ·A(1̄) (15)and hence δF/δj(1̄) = A(1̄). The Legendre transformation assumes thatthere is a one-to-one relation between j(1̄) and A(1̄) such that Eq. (10) is in-vertible (up to gauge). One can prove, by a direct generalization of the proofgiven for the density-density response function in ref.2, that the Keldyshcurrent-current reponse function is invertible for switch-on processes. We5now define an action functional for a noninteracting system with the Hamil-tonian Ĥs obtained from Eq.(6) by removal of the Coulomb interaction andreplacement of A by an effective vector potential As (subindex s refers tosingle-particle, as the system can now be descibed by single-particle equa-tions).Ĥs =∑σ12∫d3rψ̂†σ(r)(−i∇− eÂs(rt))2ψ̂σ(r) + en̂φext(rt) (16)We then defineF̃s[As] = i ln〈Φ0|Vs(τf , τi)|Φ0〉 (17)The evolution operator Vs(τf , τi) is defined similarly as in Eq.(8) with Ĥreplaced by Ĥs. The initial wave function Φ0 at t = t0 is a noninteractingstate and will often be a Slater determinant. We can now do again aLegendre transform and defineFs[j] = −F̃s[As] +∫Cd1̄j(1̄) ·As(1̄) (18)The exchange-correlation part Fxc of the action functional is then definedbyF [j] = Fs[j]− Fxc[j]−12∫Cd1̄∫Cd2̄v(1̄, 2̄)n(1̄)n(2̄) (19)where v(1̄, 2̄) = δC(t1 − t2)/|r1 − r2|. The Keldysh density n is a func-tional of the initial state and the current j through the continuity equation.Functional differentiation of Eq. (19) with respect to j(1̄) yieldsA(1̄) = As(1̄)−Axc(1̄)−AH(1̄) (20)∂tAH(1̄) = −∇∫d3r′n(1̄)|r− r′|(21)Axc(1̄) = δFxc/δj(1̄) (22)where AH and Axc are the Hartree and exchange-correlation vector poten-tial. By construction the vector potential As of the noninteracting systemyields the same density as the vector potential A in the fully interactingsystem. The noninteracting system is thus to be identified with the time-dependent Kohn-Sham system. Let us now see how the current-currentresponse function can be obtained from this formalism. We can writeδjµ(1̄)δAν(2̄)=∑λ∫Cd3δjµ(1̄)δAs,λ(3̄)δAs,λ(3̄)δAν(2̄)=δjµ(1̄)δAs,ν(2̄)+∑λκ∫Cd3d4δjµ(1̄)δAs,λ(3̄)δ(AH,λ(3̄) +Axc,λ(3̄))δjκ(4̄)δjκ(4̄)δAν(2̄)(23)6Hence we obtainχµν(1̄, 2̄) = χs,µν(1̄, 2̄)+∑λκ∫Cd3d4χs,µλ(1̄, 3̄)(fH,λκ(3̄, 4̄) + fxc,λκ(3̄, 4̄))χκν(4̄, 2̄) (24)where χs,µν is the current-current response function of the Kohn-Shamsystem and where fH,µν and fxc,µν are the functional derivatives of AHand Axc with respect to j. We have therefore found a relation betweenthe full and the Kohn-Sham current-current response function. An explicitform of fH,µν can be obtained from Eq.(21). Eq.(24) is the basic equationof time-dependent current response theory. When transformed to physicaltime this equations retains the same form with all response functions andkernels replaced by their retarded equivalent.4. Perturbation theoryLet us now see how we can obtain Axc and fxc,µν from perturbation theory.The potential Axc is determined by the requirement that the Kohn-Shamand real system have the same current, i.e.〈̂jp(r1)〉+ 〈n̂(1̄)〉A(1̄) = 〈̂jp(r1)〉0 + 〈n̂(1̄)〉0As(1̄) (25)where the subindex 0 denotes the expectation value with respect to theKohn-Sham state |Φ0〉. Note that by construction 〈n̂〉0 = 〈n̂〉 = n. Toconnect the fully interacting initial state |Ψ0〉 to the Kohn-Sham state |Φ0〉we use the following adiabatic switch-onĤ(τ) = Ĥs(τ) + fε(t)H1(τ) (26)where fε = eε(t−t0) for t < t0 and fε = 1 for t ≥ t0. We further definedĤ1(τ) = Ĥ(τ)− Ĥs(τ). Note that in the interval [−∞, t0] the Hamiltonianis the same on both sides of the contour. Using the Gell-Mann-Low theoremwe then have|Ψ0〉 = limε→0A−1V (τi,−∞)|Φ0〉 = limε→0A−1V (τf ,+∞)|Φ0〉 (27)with the phase factor A = 〈Φ0|V (τi,−∞)|Φ0〉 = 〈Φ0|V (τf ,+∞)|Φ0〉. Wethen have (see Eq.(11)〈jp(1̄)〉 = limε→0〈Φ0|V (+∞, τ)jp(r1)V (τ,−∞)|Φ0〉〈Φ0|V (+∞,−∞)|Φ0〉(28)7where we used that |A| = 1. Expanding the evolution operator V in powersof Ĥ1 we find thatn(1̄)(As(1̄)−A(1̄)) = limε→0∞∑n=1(−i)nn!∫Cd2̄ . . .∫Cd(n+ 1)× fε(2) . . . fε(n+ 1)〈TC∆jp(1̄)∆Ĥ1(2̄) . . .∆Ĥ1(n+ 1)〉0 (29)where the fluctuation operators ∆Ô = Ô−〈Ô〉0 appear from the expansionof the denominator in Eq.(28). The right hand side can now by Wick’stheorem be expanded in Keldysh Green functions defined byG(1̄, 2̄) = 〈Φ0|TC ψ̂(1̄)ψ̂†(2̄)|Φ0〉= θ(τ1 − τ2)G>(1, 2) + θ(τ2 − τ1)G<(1, 2) (30)Eq.(29) is not directly usable, since we use a reference Hamiltonian that isdetermined self-consistently. This can however be resolved by an expansionof As in powers of the Coulomb interaction. If we denote the n-th orderterm by A(n)s we have (the Hartree part is explicitly a first order term) tofirst order:n(1̄)(AH + A(1)xc ) = −i limε→0∫Cd2̄fε(2̄)〈TC∆jp(1̄)∆Ŵ (2̄)〉0+ i limε→0∫d2̄fε(2̄)〈TC∆jp(1̄)∆jp(2̄)〉0 · (AH(2̄) + A(1)xc (2̄))(31)This can be written aslimε→0∑ν∫Cd2̄fε(2̄)χs,µν(1̄, 2̄)(AH,ν(2̄) +A(1)xc,ν(2̄)) =−i limε→0∫Cd2̄fε(2̄)〈TC∆ĵp,µ(1̄)∆Ŵ (2̄)〉0 (32)If we evaluate this at physical potentials, then we can replace ∆Ô by Ô(the average value is the same on both sides of the contour and integratesto zero). If we then use Wick’s theorem and the explicit form of AH weobtainlimε→0∑ν∫Cd2̄fε(2̄)χs,µν(1̄, 2̄)A(1)xc,ν(2̄) =limε→02∫Cd2̄∫Cd3̄fε(2)v(3̄, 2̄)G(3̄, 2̄)G(2̄, 1̄)jop,µ(r1)G(1̄, 3̄) (33)where we defined the operator jop,µ = 1/2i(−→∂ µ−←−∂ µ) and where the Greenfunctions are for the spin unpolarized case. The first order change A(1)xc is8only due to the exchange diagram. We therefore denote A(1)xc = Ax. Let usnow write these equations in physical time:limε→0∑ν∫ +∞−∞dt2∫d3r2fε(t2)χR,s,µν(1, 2)Ax,ν(2) == limε→02∫d3r2∫d3r3∫ t1−∞dtfε(t)G<(r2t, r3t)|r2 − r3|×[G<(r2t, r1t)jop,µ(r1)G>(r1t, r3t)− (>↔<)](34)These equations are the extension of the time-dependent optimized poten-tial method14 (TDOPM) to current-density-functional theory. Note thatfor times t ≤ t0 the time integrations can be done analytically as in thisinterval the time-dependence of the orbitals is known. After taking thelimit ε → 0 we then obtain the static x-only OPM equations (which canbe gauged to scalar form as there are no transverse external fields in theground state). By collecting terms higher order in the two-particle inter-action we can obtain equations for the higher order terms A(n)xc . On theother hand, we can by functional differentiation of Eq.(33) also obtain anintegral equation for the kernel f (1)xc,µν . It would go to far to represent thedetails in the short presentation here. The function has recently been ana-lyzed in detail for the electron gas 15 with the aim of providing an analyticform that can be used in the local density approximation for response cal-culations on inhomogeneous systems. We finally remark that the TDOPMequations can also be obtained from the so-called Keldysh Sham-Schlüterequation 16 in which case one can derive conserving approximations bymeans of Φ-derivable 17 and Ψ-derivable functionals 18.References1. E.K.U.Gross, J.F.Dobson and M.Petersilka, in Topics in Current Chemistry.(ed.) R.F.Nalewajski, Springer, (1996)2. R.van Leeuwen, Int.J.Mod.Phys. B15, 1969 (2001).3. G.Vignale and W.Kohn, Phys.Rev.Lett. 77, 2037 (1996)4. F.Kootstra, P.L.de Boeij and J.G.Snijders, Phys.Rev. B62, 7071 (2000)5. C.A.Ullrich and G.Vignale, Phys.Rev.B65, 245102 (2002)6. M.van Faassen, P.L.de Boeij, R.van Leeuwen, J.A.Berger and J.G.Snijders,Phys.Rev.Lett. 88, 186401 (2002)7. L.I.Schiff, Quantum Mechanics, McGraw-Hill, (1968)8. J.D.Bjorken and S.D.Drell, Relativistic Quantum Fields, McGraw-Hill, (1965)9. J.Schwinger, J.Math.Phys. 2, 407 (1961)10. L.V.Keldysh, Sov.Phys.JETP 20, 1018 (1965)11. P.Danielewicz. Ann.Phys. 152, 239, (1984)12. J.Rammer and H.Smith. Rev. Mod. Phys., 58, 323 (1986)913. R.van Leeuwen, Phys. Rev. Lett. 80, 1280 (1998).14. C. A. Ullrich, U. J. Gossmann, and E. K. U. Gross Phys. Rev. Lett. 74, 872(1995).15. R.van Leeuwen, S.Kurth and U.von Barth, to be published16. R.van Leeuwen, Phys.Rev.Lett. 76, 3610 (1996).17. G.Baym Phys.Rev. 127, 1391 (1962)18. C.-O. Almbladh, U.von Barth and R.van Leeuwen, Int.J.Mod.Phys.B13, 535(1999)

Nonequilibrium green functions in time-dependent current-density-functional theory

https://pure.rug.nl/ws/files/10388602/2003ProgNoneqGrFvLeeuwen.pdf

Nonequilibrium green functions in time-dependent current-density-functional theory

Abstract

Similar works

Full text

Available Versions

ARTS repository - University of Groningen