Quasi-low-dimensional electron gas with one populated band as a testing ground for time-dependent density-functional theory
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Quasi-low-dimensional electron gas with one populated band as a testing ground fortime-dependent density-functional theory
Vladimir U. Nazarov
Research Center for Applied Sciences, Academia Sinica, Taipei 11529, Taiwan
We find the analytical solution to the time-dependent density-functional theory (TDDFT) problemfor the quasi-low-dimensional (2D and 1D) electron gas (QLDEG) with only one band filled in thedirection perpendicular to the system extent. The theory is developed at the level of TD exactexchange and yields the exchange potential as an explicit nonlocal operator of the spin-density.The dressed interband (image states) excitation spectra of the Q2DEG are calculated, while thecomparison with the Kohn-Sham (KS) transitions provides the insight into the qualitative andquantitative role of the many-body interactions. Important cancellations between the Hartree f H and the exchange f x kernels are found in the low-density limit, shedding light on the interrelationsbetween the KS and many-body excitations. PACS numbers: 73.21.-b, 73.21.Fg, 73.21.Hb
Density-functional theory (DFT) [1] and its time-dependent counterpart TDDFT [2] are presently, byfar, the most popular methods to conceive the ground-state and excitation, respectively, properties of atomic,molecular, and condensed matter systems. Both DFTand TDDFT require the knowledge of the exchange-correlation (xc) potentials, v xc ( r ) and v xc ( r , t ), respec-tively. Although the exact xc potentials exist in princi-ple, they are never known for non-trivial systems, makingus resort to approximations.The potentials now overwhelmingly used in applica-tions are local functions of the electron density or alsoof its spatial derivatives, the local-density approxima-tion (LDA) [1] and the generalized-gradient approxima-tion (GGA) [3], respectively. While very simple and ef-ficient in implementations, these approximations sufferfrom well known deficiencies. The one of our concern herewill be the inherent dimensionality dependence of bothLDA and GGA, i.e., their having distinct 3D, 2D, and 1Dversions, which makes them unreliable and even poorlysubstantiated in the case of the systems of intermediatedimensionality, such as quasi-low-dimensional materials.A truly first-principles xc functional, being one and thesame for all systems, must work equally well for differentdimensionalities, including the intermediate ones. Theexact exchange (EXX) [or optimized-effective potential(OEP)] [4, 5] stands out in DFT as a first-principles po-tential not, in particular, bound to any specific dimen-sionality. This potential obeys a number of importantrequirements of the exact theory, such as the correctasymptotic behavior − e /r for finite systems, the sup-port of image states at surfaces and in low-dimensions[6–8], it produces [9, 10] the derivative discontinuity inthe energy dependence on the fractional electrons num-ber [11], and it is free from self-interaction. The time-dependent version of the EXX theory has been developed[12, 13] and found to support the excitonic effect in semi-conductors [14]. For all the advantages, an unfortunatedrawback of the EXX theory is the extreme complexityof its implementation. It is the orbital-dependent for- malism which involves the solution of the notoriously te-dious OEP integral equation [4, 5]. This has preventedEXX from becoming widely used in applications, andeven qualitative insights are often obscured by heavy nu-merical difficulties.It, therefore, came recently as a surprise that forthe quasi-low-dimensional electron gas (QLDEG) withonly one band populated in the transverse direction, theground-state EXX problem has a simple explicit solu-tion in terms of the (spin-) density [8]. A natural ques- ( x,y ) zv ext ( z, t ) FIG. 1. (color online) Schematics of the Q2DEG under theaction of a time-dependent external potential. tion arises whether the same route can be taken to buildthe analytical EXX theory of many-body excitations inQLDEG. In this Letter we give to this a positive answerby finding an explicit solution to the TD exchange po-tential in terms of the TD spin-density for QLDEG withone band populated. For the solution to be expressiblethrough the density, the applied perturbation must notchange the symmetry of the QLDEG, as is discussed be-low.We start from the ground-state of a Q2DEG (for 1Dcase, see below), uniform in the xy -plane and confined inthe z direction by a potential v ext ( z ). The in-plane andthe perpendicular variables separate in this case. Wefurther assume that only the states µ ↑ ( z ) and µ ↓ ( z ), onefor each spin orientation, are occupied in the z -direction[8], leading to the electrons’ wave-functions of the form ψ σ k k ( r ) = 1 √ Ω e i k k · r k µ σ ( z ) , (1)where Ω is the normalization area To this system we ap-ply a TD potential, which is assumed to depend on the z coordinate only (see Fig. 1) and, by this, it preservesthe system’s lateral uniformity during the time-evolution.We will see that a wealth of many-body phenomena arepreserved within these constraints, while the gain is thesystem admitting an analytical solution.The main result of this Letter is that, with the abovesetup, the TDEXX potential is v σx ( z, t ) = − n σ D Z F ( k σF | z − z ′ | ) | z − z ′ | n σ ( z ′ , t ) dz ′ , (2)where n σ ( z, t ) is the spin-density, F ( x ) = 1 + L (2 x ) − I (2 x ) x ,L and I are the first-order modified Struve and Besselfunctions [15, 16], respectively, n σ D = R ∞−∞ n σ ( z, t ) dz isthe 2D spin-density, which does not change during thetime-evolution, and k σF = p πn σ D is the corresponding2D Fermi radius. We derive Eq. (2) in Appendix A withthe use of the adiabatic-connection method [13, 17]. Inthe linear-response regime, Eq. (2) gives immediately forthe exchange kernel f σσ ′ x ( z, z ′ , ω ) = δv σx ( z, ω ) δn σ ′ ( z ′ , ω ) = − n σ D F ( k σF | z − z ′ | ) | z − z ′ | δ σσ ′ . (3)Notably, f x of Eq. (3) is frequency-independent. We,however, emphasize that Eqs. (2) and (3) are by no meansan adiabatic approximation: Our detailed derivation inAppendix A shows that they hold exactly within the fullydynamic TDEXX for QLDEG with one band filled, pro-vided that the exciting field is applied perpendicularly tothe layer.We use the kernel of Eq. (3) with the basic linear-response TDDFT equality [18, 19] (cid:0) χ − (cid:1) σσ ′ ( z, z ′ , ω ) = (cid:0) χ − s (cid:1) σσ ′ ( z, z ′ , ω ) − f H ( z, z ′ ) − f σσ ′ x ( z, z ′ , ω ) , (4)where χ and χ s are the interacting-electrons and Kohn-Sham (KS) spin-density-response functions, respectively,the latter given in our case by χ σσ ′ s ( z, z ′ , ω ) = n σ D µ σ ( z ) µ σ ( z ′ ) ∞ X n =1 (cid:18) ω + λ σ − λ σn + i + − ω − λ σ + λ σn + i + (cid:19) µ σn ( z ) µ σn ( z ′ ) δ σσ ′ , (5) − + − + ··· . . . . . − + − + ··· . . . . − + − + ··· . . . − + − + ··· E x c i t a t i o n e n e r g y ( e V ) TDEXXRPAKS r s = 2 r s = 5 Excited state number r s = 10 r s = 50 FIG. 2. Excitation energies of a spin-neutral Q2DEG withone transverse band filled. Circles, squares, and triangles areTDEXX, RPA, and KS excitation energies, respectively. Plusand minus signs mark even and odd excitations, respectively.Dashed lines connect the energies of the even and odd self-oscillations, separately. − + − + ··· . . . . . . − + − + ··· . . . . − + − + ··· . . . . − + − + ··· E x c i t a t i o n e n e r g y ( e V ) TDEXXRPAKS r s = 2 r s = 5 Excited state number r s = 10 r s = 50 FIG. 3. The same as Fig. 2, but for the fully spin-polarizedQ2DEG. where λ σn and µ σn ( z ) are the eigenenergies and the eigen-functions of the perpendicular motion, respectively. Aremarkable property of the KS response function of oursystem is that it is immediately invertible to (see Ap-pendix B)( χ − s ) σσ ′ ( z, z ′ , ω ) = δ σσ ′ n σ µ σ ( z ) µ σ ( z ′ ) × (cid:2) ω X σ ( z, z ′ ) − X σ ( z, z ′ ) (cid:3) , (6)with X σ ( z, z ′ ) = ∞ X n =1 µ σn ( z ) µ σn ( z ′ ) λ σn − λ σ = (cid:16) ˆ h σs − λ σ (cid:17) − [ δ ( z − z ′ ) − µ σ ( z ) µ σ ( z ′ )] , (7) X σ ( z, z ′ ) = ∞ X n =1 ( λ σn − λ σ ) µ σn ( z ) µ σn ( z ′ )= (cid:16) ˆ h σs − λ σ (cid:17) δ ( z − z ′ ) , (8)where ˆ h σs is the static KS Hamiltonian [20]. The Hartreepart of the kernel is f σσ ′ H ( z, z ′ ) = − π | z − z ′ | . (9)The many-body excitation energies ω are found from theequation X σ ′ Z (cid:0) χ − (cid:1) σσ ′ ( z, z ′ , ω ) δn σ ′ ( z ′ , ω ) dz ′ = 0 , (10)where δn σ ( z ) is the self-oscillation of the spin-densityWith the use of Eqs. (4) and (6)-(8), Eq. (10) can berewritten as the following eigenvalue problem (cid:16) ˆ h σs − λ σ (cid:17) h(cid:16) ˆ h σs − λ σ (cid:17) y σ ( z ) + 2 n σ D × Z µ σ ( z ) X σ ′ f σσ ′ H ( z, z ′ ) µ σ ′ ( z ′ ) y σ ′ ( z ′ ) dz ′ + 2 n σ D × Z µ σ ( z ) f σσx ( z, z ′ ) µ σ ( z ′ ) y σ ( z ′ ) dz ′ (cid:21) = ω y σ ( z ) , (11)where y σ ( z ) = δn σ ( z ) /µ σ ( z ).We have found the eigenvalues and eigenfunctions ofEq. (11) numerically on a z -axis grid for a number of theEG densities. The confining potential v ext ( z ) was chosenthat of the 2D positive charge background. The static KSproblem was solved self-consistently with the use of theEXX potential, which is that of Eq. (2) with the ground-state density in place of the TD one [8]. Results for theeigenenergies of the excited states are presented in Figs. 2and 3, for spin-neutral and fully spin-polarized Q2DEG,respectively, where TDEXX is compared to the random-phase approximation (RPA) [setting f x = 0 in Eq. (11)]and with the KS transitions [setting f x = f H = 0 inEq. (11)]. Obviously, the first excited state is influencedstrongly by the many-body interactions, resulting in theboth TDEXX and RPA being very different from thesingle-particle KS transition. This effect, however, weak-ens for higher excited states. Secondly, the difference be-tween the TDEXX and RPA increases with the growth of r s (decrease of the density), the former moving to the KSvalues, which is more pronounced for the spin-polarizedthan for the spin-neutral EG. This has an elegant expla-nation: Expanding Eq. (3) in powers of k σF , we can writeat small k σF f σσ ′ x ( z, z ′ , ω ) ≈ (cid:20) − k σF + 2 π | z − z ′ | (cid:21) δ σσ ′ . (12)Noting that the first term in Eq. (12) is a constant and,consequently, it does not play a role in f x , and compar-ing with Eq. (9), we conclude that, for a dilute EG, the − − − r s = 5 δ n ( z )( a r b . u . ) z (a.u.) δn ( z ) 1234 FIG. 4. (color online) Self-oscillations of the density of thespin-neutral EG of r s = 5, corresponding to the transitionsto the first four excited states. exchange part of the kernel by a half and fully cancels theHartree part, for the spin-neutral and fully spin-polarizedEG, respectively . In the fully spin-polarized case, at lowdensities, this brings the many-body excitation energiesback to the KS values, as can be observed in Fig. 3.In contrast to the KS transitions, TDEXX and RPAexcitation energies split into the even and odd series,the values changing smoothly within each, while jump-ing across the series. In Figs. 2 and 3, the points withineach series are connected with dashed lines serving aseye-guides. In Fig. 4, we plot the even and odd self-oscillations δn ( z ) themselves.The quasi-1D electron gas admits the same treatmentas the Q2DEG above, leading to the following results (cf.the static case [8]). For the exchange kernel we have f σσ ′ x ( ρ , ρ ′ , ω ) = − n σ D F ( k σF | ρ − ρ ′ | ) | ρ − ρ ′ | δ σσ ′ , (13)where ρ = ( x, y ), the wire is stretched along the z -axis, F ( x ) = 12 π G , , (cid:20) x (cid:12)(cid:12)(cid:12)(cid:12) , , , − , (cid:21) , (14)and G m,np,q (cid:20) x (cid:12)(cid:12)(cid:12)(cid:12) a , ..., a p b , ..., b q (cid:21) is the Meijer G-function [15,16]. The Hartree kernel is f σσ ′ H ( ρ , ρ ′ ) = − k σF | ρ − ρ ′ | ) . (15)As shown earlier [8], the assumption of the QLDEGhaving one spin-state occupied in the perpendicular di-rection is not very restrictive: This is a regime actuallyrealizing at r s > .
46 and r s > .
72, for the Q2D andQ1D cases, respectively, provided the confining potentialis that of the positive 2D (1D) uniform background. Thesecond feature of our setup, that of the perturbation fieldbeing applied perpendicularly to the layer, is important:By this we do not study the excitation spectra of the2D (1D) EG proper, which problem has been extensivelyaddressed in the literature before [19, 21], but we areconcerned with the interband excitations, which are theexcitations to the image states of QLDEG. The latterexcitations we handle as dressed, i.e., accounting for themany-body dynamic interactions, doing this at the levelof TDEXX. With the understanding of the above, ourtheory is exact.The localized Hartree-Fock potential (LHF) [22] has re-cently attracted new attention as a single-particle poten-tial providing the best possible fulfillment of the many-body TD Schr¨odinger equation by a Slater determinantwave-function [23, 24] and, in the spirit of the “direct-energy” potentials [25], yielding the energy as a sum ofKS eigenvalues. It has been recently shown [8] that forQLDEG with one populated band in its ground-state,LHF potential coincides exactly up to a constant withthe EXX one. The same is true in the TD case, the proofof which is a repetition of that given in Sec. V of Ref. [8]for the static case, with all the functions acquiring anadditional time-argument t .In conclusions, we have identified the quasi-low-dimensional electron gas with one occupied band as aunique system admitting analytical or semi-analytical so-lution of the many-body excitation problem by means of the time-dependent density-functional theory at the levelof the time-dependent exact-exchange. The fundamen-tal quantities of TDDFT, such as the time-dependentexchange potential and the exchange kernel, have beenconstructed as an explicit nonlocal operator of the spin-density and purely analytically, respectively. We have ap-plied our theory to obtain the interband excitation spec-tra (excitation to image states) of Q2DEG. The low-lyingexcited states are shown to be strongly affected by themany-body interactions for the EG of higher densities. Inthe low-density regime, we have shown that the exchangekernel cancels the Hartree one by a half and entirely,in the case of the spin-neutral and fully spin-polarizedEG, respectively. This demonstrates how qualitativelywrong and inconsistent the often used random-phase ap-proximation (i.e., the account of the Hartree part of thekernel only) may be. For the dilute fully spin-polarizedQLDEG this leads to an important conclusion that theKohn-Sham excitation energies can be, at the same time,the true excitation energies of a many-body system.We, finally, argue that QLDEG with one populatedband has a promise to be extendable to yield analyticalor semi-analytical results with inclusion of correlations,further enriching our understanding of DFT and TDDFTin mesoscopic physics. [1] W. Kohn and L. J. Sham, “Self-consistent equations in-cluding exchange and correlation effects,” Phys. Rev. , A1133–A1138 (1965).[2] E. Runge and E. K. U. Gross, “Density-functional theoryfor time-dependent systems,” Phys. Rev. Lett. , 997(1984).[3] J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalizedgradient approximation made simple,” Phys. Rev. Lett. , 3865–3868 (1996).[4] R. T. Sharp and G. K. Horton, “A variational approachto the unipotential many-electron problem,” Phys. Rev. , 317–317 (1953).[5] J. D. Talman and W. F. Shadwick, “Optimized effectiveatomic central potential,” Phys. Rev. A , 36–40 (1976).[6] C. M. Horowitz, C. R. Proetto, and S. Rigamonti,“Kohn-Sham exchange potential for a metallic surface,”Phys. Rev. Lett. , 026802 (2006).[7] E. Engel, “Exact exchange plane-wave-pseudopotentialcalculations for slabs,” The Journal of Chemical Physics , 18A505 (2014).[8] V. U. Nazarov, “Exact exact-exchange potential of two-and one-dimensional electron gases beyond the asymp-totic limit,” Phys. Rev. B , 195432 (2016).[9] T. Grabo, T. Kreibich, and E. K. U. Gross, “Optimizedeffective potential for atoms and molecules,” MolecularEngineering , 27–50 (1997).[10] P. Mori-S´anchez, A. J. Cohen, and W. Yang, “Many-electron self-interaction error in approximate densityfunctionals,” The Journal of Chemical Physics ,201102 (2006).[11] J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz,“Density-functional theory for fractional particle num- ber: Derivative discontinuities of the energy,” Phys. Rev.Lett. , 1691–1694 (1982).[12] C. A. Ullrich, U. J. Gossmann, and E. K. U. Gross,“Time-dependent optimized effective potential,” Phys.Rev. Lett. , 872–875 (1995).[13] A. G¨orling, “Time-dependent Kohn-Sham formalism,”Phys. Rev. A , 2630–2639 (1997).[14] Y.-H. Kim and A. G¨orling, “Exact Kohn-Sham exchangekernel for insulators and its long-wavelength behavior,”Phys. Rev. B , 035114 (2002).[15] A. P. Prudnikov, O. I. Marichev, and Yu. A. Brychkov, Integrals and Series , Vol. 3:
More Special Functions (Gor-don and Breach, Newark, NJ, 1990).[16] Wolfram Research,
Mathematica , version 11.0 ed. (Cham-paign, Illinios, 2016).[17] A. G¨orling and M. Levy, “Exact Kohn-Sham schemebased on perturbation theory,” Phys. Rev. A , 196–204 (1994).[18] E. K. U. Gross and W. Kohn, “Local density-functionaltheory of frequency-dependent linear response,” Phys.Rev. Lett. , 2850–2852 (1985).[19] G. F. Giuliani and G. Vignale, Quantum Theory ofthe Electron Liquid (Cambridge University Press, Cam-bridge, 2005).[20] The operator ˆ h σs − λ σ is invertible on the subspace offunctions orthogonal to µ σ ( z ), to which the function inthe brackets on the right-hand side of Eq. (7) belongs.[21] F. Stern, “Polarizability of a two-dimensional electrongas,” Phys. Rev. Lett. , 546–548 (1967).[22] F. Della Sala and A. G¨orling, “Efficient localized Hartree-Fock methods as effective exact-exchange Kohn-Shammethods for molecules,” The Journal of Chemical Physics , 5718–5732 (2001).[23] V. U. Nazarov, “Time-dependent effective potential andexchange kernel of homogeneous electron gas,” Phys.Rev. B , 165125 (2013).[24] V. U. Nazarov and G. Vignale, “Derivative discontinuitywith localized Hartree-Fock potential,” The Journal of Chemical Physics , 064111 (2015).[25] M. Levy and F. Zahariev, “Ground-state energy as a sim-ple sum of orbital energies in Kohn-Sham theory: A shiftin perspective through a shift in potential,” Phys. Rev.Lett. , 113002 (2014). Appendix A: Time-dependent exchange potential [Proof of Eq. (2)]
We follow the adiabatic connection method [13, 17]. The adiabatic connection Hamiltonian is (for brevity, in thefollowing we omit the spin index)ˆ H λ ( t ) = X i (cid:20) −
12 ∆ i + v ext ( r i , t ) + ˜ v λ ( r i , t ) (cid:21) + X i 12 ∆ i + v ext ( r i , t ) + ˜ v ( r i , t ) (cid:21) , (A7)ˆ H ( t ) = X i ˜ v ( r i , t ) + X i 0[ ˆ H , ˆ ρ ] + [ ˆ H , ˆ ρ ] = 0 , (A14)and which, solved with respect to ˆ ρ , produces h α | ˆ ρ | β i = δ β − δ α E β − E α h α | ˆ H | β i , (A15)where E α are the eigenvalues of ˆ H . Together Eqs. (A13) and (A15) give h α ( t ) | ˆ ρ ( t ) | β ( t ) i = ( δ β − δ α ) " h α | ˆ H | β i E β − E α − i Z t h α ( t ′ ) | ˆ H ( t ′ ) | β ( t ′ ) i dt ′ . (A16)Further, we calculate the TD density to the 1st order in λn ( r , t ) = Sp ( ˆ ρ ( t ) X i δ ( r − r i ) ) (A17)or written through the matrix elements and with account of the identity of electrons n ( r , t ) = N X αβ h α ( t ) | ˆ ρ ( t ) | β ( t ) i h β ( t ) | δ ( r − r ) | α ( t ) i . (A18)The following facts will play a critical role below:1. The density operator is a single-particle operator and, therefore, only the determinants | α ( t ) i and | β ( t ) i whichdiffer by one orbital at most contribute to Eq. (A18);2. Because of Eq. (A16), only the elements h t ) | ˆ ρ ( t ) | α ( t ) i and h α ( t ) | ˆ ρ ( t ) | t ) i , α = 0, are non-zero;3. Due to the symmetry of our system and the external potential varying in the z direction only, the density n ( r , t )can be a function of the z coordinate only. We can, therefore, average Eq. (A18) in ( x, y ) over the normalizationarea Ω without changing this equation.Then, with account of the facts 2 and 3, n ( z, t ) = 2 N Re X α =0 h α ( t ) | ˆ ρ ( t ) | t ) i h t ) | δ ( z − z ) | α ( t ) i , (A19)Since during the time-evolution of the KS system the orbitals remain of the form φ i ( r , t ) = µ n i ( z, t ) e i k i k · r i k Ω / , (A20)and with account of the facts 1 and 2, we conclude that only the matrix elements h α ns ( t ) | ˆ ρ ( t ) | t ) i = h t ) | ˆ ρ ( t ) | α ns ( t ) i ∗ , where | α ns ( t ) i = 1( N !Ω N ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ ( z , t ) e i k · r k . . . µ ( z N , t ) e i k · r N k ... . . . ... µ ( z , t ) e i k s − · r k . . . µ ( z N , t ) e i k s − · r N k µ n ( z , t ) e i k s · r k . . . µ n ( z N , t ) e i k s · r N k µ ( z , t ) e i k s +1 · r k . . . µ ( z N , t ) e i k s +1 · r N k ... . . . ... µ ( z , t ) e i k N · r k . . . µ ( z N , t ) e i k N · r N k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A21) n = 1 , , ... , s = 1 ...N , and | t ) i = 1( N !Ω N ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ ( z , t ) e i k · r k . . . µ ( z N , t ) e i k · r N k ... . . . ... µ ( z , t ) e i k N · r k . . . µ ( z N , t ) e i k N · r N k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A22)contribute to Eq. (A19).We can write by Eq. (A8) h α ns ( t ) | ˆ H ( t ) | t ) i = h α ns ( t ) | N ˜ v ( r , t ) + N ( N − | r − r | | t ) i , (A23)and we evaluate straightforwardly h α ns ( t ) | ˜ v ( z , t ) | t ) i = 1 N Z µ ∗ n ( z , t )˜ v ( z , t ) µ ( z , t ) dz , (A24) h α ns ( t ) | | r − r | | t ) i = 2Ω N ( N − Z µ ∗ n ( z , t ) µ ( z , t ) | µ ( z , t ) | | r − r | h N − Ω e i k s · ( r k − r k ) ρ ∗ ( r k − r k ) i d r d r , (A25) h α ns ( t ) | δ ( z − z ) | t ) i = 1 N µ ∗ n ( z, t ) µ ( z, t ) , (A26)where ρ ( r k ) = 1Ω X | k |≤ k F e i k · r k . (A27)Then, by Eqs. (A23)-(A25), and with an integration variable substitution, h α ns ( t ) | ˆ H ( t ) | t ) i = Z µ ∗ n α ( z , t ) µ ( z , t ) ˜ v ( z , t ) + 1Ω Z | µ ( z , t ) | q ( z − z ) + r k (cid:2) N − Ω e i k sα · r k ρ ∗ ( r k ) (cid:3) d r k dz dz . (A28)By virtue of Eqs. (A16) and (A28) we can write h α ns ( t ) | ˆ ρ ( t ) | t ) i = − i t Z dt ′ Z µ ∗ n ( z , t ′ ) µ ( z , t ′ ) ˜ v ( z , t ′ ) + 1Ω Z | µ ( z , t ′ ) | q ( z − z ) + r k (cid:2) N − Ω e i k s · r k ρ ∗ ( r k ) (cid:3) d r k dz dz + 1 ǫ − ǫ n α Z µ ∗ n ( z ) µ ( z ) ˜ v ( z ) + 1Ω Z | µ ( z ) | q ( z − z ) + r k (cid:2) N − Ω e i k s · r k ρ ∗ ( r k ) (cid:3) d r k dz dz . (A29) n ( z ) = 2 Re ∞ X n =1 N X s =1 h α ns ( t ) | ˆ ρ ( t ) | t ) i µ n ( z, t ) µ ∗ ( z, t ) . (A30)In the spirit of the adiabatic connection method, with the change of λ from 0 to 1, the TD density should remainunchanged. Since, by Eqs. (A27) and (A29), N X s =1 h α ns ( t ) | ˆ ρ ( t ) | t ) i = − i t Z dt ′ Z µ ∗ n ( z , t ′ ) µ ( z , t ′ ) N ˜ v ( z , t ′ ) + 1Ω Z | µ ( z , t ′ ) | q ( z − z ) + r k (cid:2) N − Ω | ρ ( r k ) | ) (cid:3) d r k dz dz + 1 ǫ − ǫ n α Z µ ∗ n ( z ) µ ( z ) N ˜ v ( z ) + 1Ω Z | µ ( z ) | q ( z − z ) + r k (cid:2) N − Ω | ρ ( r k ) | (cid:3) d r k dz dz , (A31)we see that n ( z, t ) = 0 if˜ v ( z , t ) = − N Z | µ ( z , t ) | q ( z − z ) + r k (cid:2) N − Ω | ρ ( r k ) | ) (cid:3) d r k dz . (A32)According to Eq. (A1), ˜ v ( r , t ) = v H ( r , t ) + v xc ( r , t ) and ˜ v ( r , t ) = 0, where v H ( r , t ) and v xc ( r , t ) are Hartree andthe exchange-correlations potentials, respectively. To the first order in λ (A9) this gives˜ v ( r , t ) = − v H ( r , t ) − v x ( r , t ) , (A33)where in the notation we have taken into account that to the first order we have, by definition, exchange only [13, 17].On the other hand, it is easy to see that for our system v H ( r , t ) = N Ω Z | µ ( z ′ , t ) | q ( z − z ′ ) + r ′ k d r ′k dz ′ , (A34)leading us to v x ( z, t ) = − Ω N Z | µ ( z ′ , t ) | q ( z − z ′ ) + r ′ k | ρ ( r ′k ) | d r ′k dz ′ , (A35)The proof of Eq. (2) by the explicit integration over r ′k in Eq. (A35) with the account of Eq. (A27). Appendix B: Kohn-Sham spin-density-response function and its inverse [Proof of Eqs. (6)-(8)] We construct the operator( χ − s ) σσ ′ ( z, z ′ , ω ) = 1 n σ D ∞ X n =1 (cid:18) ω + λ σ − λ σn + i + − ω − λ σ + λ σn + i + (cid:19) − µ σn ( z ) µ σn ( z ′ ) µ σ ( z ) µ σ ( z ′ ) δ σσ ′ (B1)and directly check that for an arbitrary function g ( z ) such that Z g ( z ) dz = 0 , (B2)the equality holds Z χ σs ( z, z ′′ , ω )( χ σs ) − ( z ′′ , z ′ , ω ) g ( z ′ ) dz ′′ dz ′ = g ( z ) , (B3)where χ s is given by Eq. (5). In arriving at Eq. (B3) we have used the completeness relation ∞ X n =0 µ σn ( z ) µ σn ( z ′ ) = δ ( z − z ′ ) . (B4)On the other hand, the operator ( χ σs ) − χ σs is defined on any function h ( z ) of the Hilbert space, and Z ( χ σs ) − ( z, z ′′ , ω ) χ σs ( z ′′ , z ′ , ω ) h ( z ′ ) dz ′′ dz ′ = h ( z ) − Z [ µ σ ( z ′ )] h ( z ′ ) dz ′ , (B5)where the second term on the right-hand side is a constant. Equations (B3) and (B5) prove that χ s of Eq. (5) and χ − ss