Time-dependent potential through an Ansatz for the Kohn-Sham orbitals
aa r X i v : . [ phy s i c s . c o m p - ph ] N ov Time-dependent potential through an Ansatz forthe Kohn-Sham orbitals
R. J. Magyar ∗ Center for Computing Research & the Center for Integrated Nanotechnologies, SandiaNational Laboratories, Albuquerque NM 87185
E-mail: [email protected]
Abstract
Given the time-evolution of an electron charge density, the local potential in Kohn-Sham time-dependent density functional theory (KS-TDDFT) can be modeled as asum of instantaneous and dynamic contributions by assuming a certain form of thetime-dependent Kohn-Sham (TD-KS) orbitals. The instantaneous part is obtainednumerically using methods from ground-state density functional theory (DFT) andthe dynamic part is expressed in terms of a velocity potential that depends on theelectron current density. The suggested form of the TD-KS orbitals satisfies severalknown constraints (orthonormality, N-representability, J-representability), and the do-main of validity is shown to depend on the evolution of the instantaneous quanti-ties. Through this decomposition, we can relate time-dependent and ground -stateV-representability. The resulting potentials are shown numerically to approximate theexact time-dependent Kohn-Sham potentials for a set of 3 non-singlet two-particle sys-tems (a Kohn-mode, a Coulomb explosion, and a double quantum well) where theexact solutions and reference densities are known or obtained through configurationinteraction (CI) approaches. ntroduction Time-dependent density functional theory (TDDFT) within the Kohn-Sham representation enables the efficient computation of the dynamics of interacting electrons thereby providinginformation for decision support in developing technologies such as photovoltaics, fuel cells,and qubits. However, the accuracy of predictions is limited by the quality of approximationsto the unknown time-dependent Kohn-Sham potential. It is very rare that the exact potentialis known but when available, comparisons between approximations and the exact can provideinvaluable insight advancing the development of more reliable approximate potentials. Inthis manuscript, we assume a form for the Kohn-Sham (KS) orbitals that is guaranteedto provide 1. a given time-dependent density from, for example, highly accurate quantummechanics solutions, 2. the exact time-dependent current density, and 3. orbitals thatare orthogonal to each other at all moments. When the orbitals do have this form, theTD-KS potential can be decomposed into instantaneous and dynamic components that canbe extracted independently using known techniques. The impact of these results is that abody of high precision quantum mechanical results can be made to provide exact TD-KSpotentials, and the results can be compared to existing models and used to improve them.This decomposition has implications for the structure of the theory by providing constraintson the resulting TD-KS potential.It is difficult to extract the TD-KS potential from high quality numeric results evenapproximately. For singlet states of two electrons, the inversion can be done analytically.Ullrich and others have examined the spin singlet state of 2 electrons in various externalpotentials noting the decomposition into adiabatically exact and non-adiabatic pieces. K¨ummel and coworkers have studied exact models including 1D screened Coulomb sys-tems to extract exact TDDFT results for spin-singlet, single KS orbital calculations.
Itis also possible to invert exact solutions to study excited-states in the linear response for-mulation.
We choose to explore the case when multiple Kohn-Sham orbitals of differentcharacters are populated. 2he most straightforward approach to solve the inverse problem is to modify a trialpotential self-consistently at each time step so that an initial state propagates to a givendensity at the next time step; however, this approach is numerically demanding as thevariations in the density are often quite small and smaller than significant variations in theKS orbitals.
Numerically, this problem can be reformulated in the language of optimalcontrol theory allowing inversion for a general number of particles but with a substantialcomputational overhead.
These schemes share a high sensitivity to small variations ofthe exact density especially in the low density / large external potential regions. We takeanother approach. By restricting ourselves to a specific form of the orbitals, we find that wecan describe the potential in terms of quantities that can be calculated more easily.Our work differs from earlier attempts to find the TD-KS potential in that we postulatean underlying relationship between time-dependent (TD) orbitals and instantaneous densitythrough a Ansatz form of the orbitals. This structure provides a direct route to the TDpotential that is free of the two implementation problems suggested by Jensen. Theseproblems are the step by step integration and the division by infinitesimally small densitycontributions. The former problem is mitigated by the inclusion of the velocity potential.The later problem is avoided only in part since the instantaneously exact solutions can besolved in a way that avoids the inversion entirely; however, the problem does persist in thelow density regime. While it quite possible this Ansatz approach will not be valid in manycases of electron dynamics, the approach could provide potentials for analysis purposes aswell as initial starting guesses for iterative processes.The introduction of hydrodynamic variables such as the local velocity field stems backto the early days of many-body theory and the relationship between effective potentials andthe derivatives of the local velocity potential have also been suggested.
Here we showseveral important corollaries of this earlier work. First, that the decomposition of the TD-KSpotential breaks into instantaneous and dynamic pieces with the velocity field completelydefining the later. Second, that for many models problems this decomposition is a way to3xtract a TD-KS potential for more than two particles or for a two-particle non-singlet state.In the following, we restrict ourselves to the Schr¨odinger equation for spin-less electronsin 1D for clarity and numerical convenience, and we use atomic units (¯ h = m e = e = 1),but the results can be generalized to the usual 3D case. Two Time-Dependent KS-like Systems
We will describe the time-dependent electron density in two ways. First, through the TD-KSformalism and then through what we define as the instantaneously exact KS formalism . TheTD-KS equations for a set of orbitals are − d dx + v S ( x, t ) ! φ i ( x, t ) = i ∂∂t φ i ( x, t ) , N X i φ ∗ i ( x, t ) φ i ( x, t ) = n ( x, t ) . (1)The KS orbitals, φ i ( x, t ), are orthogonal to each other, and n ( x, t ) is the time-dependentelectron density. i runs from 1 to N , the number of electrons. The TD-KS potential, v S ( x, t ), is assumed to exist and to be real. In practice, the potential has contributionsfrom the external potential, the Hartree term, and the exchange-correlation potential. Inthis work, we intend to find an accurate representation for this potential as a whole. Whilethe Hartree term can be reliably constructed from an accurate density, the separation ofthe external potential and exchange-correlation potential from a reference density representsa further challenge. If we generate our exact density from the solution of a many-bodySchr¨odinger equation, then we also know the external potential that is associated with themany-body density. In this case, we can also find the exchange-correlation contribution tothe potential.We define the instantaneously exact equation as − d dx + v inst S ( x, t ) ! ψ insti ( x, t ) = ǫ i ( t ) ψ inst ( x, t ) , N X i ψ insti ( x, t ) = n ( x, t ) (2)4he instantaneous orbitals, ψ insti ( x, t ), are orthogonal to each other and real (in 1D) if theyexist. The ǫ i ( t ) are the eigenvalues of the instantaneous Kohn-Sham Hamiltonian at time t .The instantaneously exact potential is the potential that provides a stationary solution for theinstantaneous density and can be obtained using methods similar to those suggested in Ref. Two important assumptions here are that the density is stationary v-representable and TDv-representable. The designation stationary is important here since the instantaneously exactpotential need not be the ground-state KS potential for the given density.
It must however bean effective potential that produces a set of orbitals that reconstruct the given density, and thepotential is continuously deformable from an earlier time’s instantaneously exact potential .It is possible that while tracking a time-dependent density, several potentials might providea solution to the instantaneously exact equation apparently violating a KS-like assumption.However, the potential must be continuously deformable from the potential at the previousinstance. So if the system starts in a unique density functional ground-state, we will havea unique definition of the potential. We point out that it has been shown that a KS-likevariational minimum can not always be found to be unique for excited state densities.
It is unclear how this applies to the instantaneously exact system as the assumption ofbeing continuously deformable from an earlier time’s potential may be sufficient to uniquelydefine this potential, and it is not required that the orbitals be occupied according to thelowest instantaneous eigenvalues as would be expected for the ground state problem. In thispaper we show some examples where this potential exists, but in general, the existence ofthe instantaneously exact potential is a subject of needed inquiry. Considerable informationabout the system may also be extracted from the instantaneously exact eigenvalues, ǫ i ( t ). Ifthe state is stationary ground-state, these are the usual Kohn-Sham eigenvalues, but for ageneral density these provide additional information about the dynamics of the system. Incases, where the eigenvalues cross in time, we might expect the dynamics to change character,where in a wave-function based theory, the single determinantal picture might be replacedby a multi-reference one. 5 ey Ansatz : The TD-KS orbitals can be directly expressed in terms of the instanta-neously exact orbitals and can be written as φ i ( x, t ) = exp (cid:18) iS ( x, t ) − i Z tt dτ ǫ i ( τ ) (cid:19) ψ insti ( x, t ) (3)where the modulus squared sum of these orbitals reproduces the density at a given snapshotin time and remains orthonormal, while the paramagnetic current, J ( x, t ) = − i N X i =1 φ ∗ i ( x, t ) ddx φ i ( x, t ) − ( ddx φ i ( x, t )) ∗ φ i ( x, t ) ! , (4)satisfies the continuity equation. Note that the inclusion of the position dependent phasein the wave-function can be compared to early work by Bohm but the interpretationhere is quite different. The position-dependent phase can be related to the current, Eq.4. Conservation of electron charge requires that ∂∂t n ( x, t ) + ddx J ( x, t ) = 0 where n ( x, t ) isthe time-dependent density and the local velocity field is V ( x, t ) = J ( x, t ) /n ( x, t ). We willshow the velocity potential, S , in our Ansatz Eq. 3 can be related to the current through V ( x, t ) = ddx S ( x, t ) = J ( x, t ) /n ( x, t ). Note that the time derivative of the velocity field ˙ V = 0in the static limit. For example if globally ˙ V vanishes, then we have either a stationary stateat rest or a steady translational motion of a stationary distribution.To find the potential, we invert Eq. 1, v S ( x, t ) = 12 d dx φ i ( x, t ) /φ i ( x, t ) + i ∂∂t φ i ( x, t ) /φ i ( x, t ) , (5)and insert Eq. 3 for a TD-KS orbital, φ i ( x ). The TD-KS potential is then v S ( x, t ) = 12 d dx ψ insti ( x, t ) /ψ insti ( x, t ) − ddx S ( x, t ) ! − ∂∂t S ( x, t ) + ǫ i ( t )+ i d dx S ( x, t ) + 2 ddx S ( x, t ) ddx ψ insti ( x, t ) /ψ insti ( x, t ) + 2 ∂∂t (cid:16) ln ψ insti ( x, t ) (cid:17)! (6)6y inverting the instantaneously exact KS-equation Eq. 2, we have d dx ψ insti ( x, t ) /ψ insti ( x, t ) = 2( v inst S ( x, t ) − ǫ i ( t )) . (7)For Eq. 6 to describe a physically meaningful TD-KS potential, the imaginary part of theright-hand side must vanish. Using the total derivative operator DDt = ∂∂t + ddx S ddx , we expressthe constraint as DDt ψ insti ( x, t ) = − d dx S ( x, t ) ψ insti ( x, t ) . (8)This condition can be tested at each time step since all the required quantities are availablefrom the instantaneous calculation if the instantaneous solution exists. Note that the resultis consistent with the continuity equation for the charge density. Using the total derivativenotation, DDt n ( x, t ) = − n ( x, t ) ddx v ( x, t ) ! = − n ( x, t ) d dx S ( x, t ) . (9) n can be written n ( x, t ) = P Ni ψ insti ( x, t ) . So we find a dynamic equation for the instanta-neously exact orbitals, N X i ψ insti ( x, t ) DDt + 12 d dx S ( x, t ) ! ψ insti ( x, t ) = 0 (10)Plug Eq. 7 into Eq. 6 and take the real part. The TD-KS potential is thus v S ( x, t ) = v instS [ n ]( x ) | n = n ( x,t ) + v dynS ( S, ˙ S )( x, t ) (11)The dynamic contribution to the potential is a function of the velocity potential, v dyn S ( x, t ) = − ∂∂t S ( x, t ) − ddx S ( x, t ) ! . (12)7his result is general and does not require the solution to be a single electron or a 2-electronspin-singlet state. We will demonstrate that for three non-trivial examples, this Ansatz holdswithin our numerical fidelity. Exact Results
The exact time-dependent densities are obtained by theory or by diagonalization of thefew-body Hamiltonian within a local orbital many-particle basis. The initial state can beobtained to arbitrary accuracy by augmenting the basis set until convergence. Our singleparticle basis set is the the set of solutions to the 1D non-interacting harmonic oscillator,and the 2-body basis is constructed as the set of antisymmetric products of one-body ba-sis functions. In the CI solutions, the entire product basis set up to a given upper limitin energy in the single particle basis is considered. The dimension of the matrix being di-agonalized is ( M − M ) where M is the number of basis functions. The bottleneck forthe CI calculation are the calculation and storage of the 4-point integrals which scales as M . The use of symmetries and fast identification of vanishing terms can reduce this costsubstantially; nevertheless, the entire calculation is often limited to single particle basesof about 100 terms. All matrix elements are evaluated analytically. The 4-point integralscan be reduces to products of special functions using elementary rules and the identity R ∞−∞ du exp( − / u ) / √ ǫ + u = exp( − ǫ / K ( ǫ / S ( x, t ) = Z x −∞ dx ′ n ( x ′ , t ) "Z x ′ −∞ dx ′′ ∂∂t n ( x ′′ , t ) (13)assuming the charge density and current vanish sufficiently fast with x that the spatialderivative of S vanishes far from the region of interest. Solving Eq. 13 is not ideal sinceone over the density is expected to diverge in regions far removed from significant density.However, when analytic results are known, or when asymptotic limits for S are available,this is an efficient way to obtain S .We investigate 3 systems. First, we consider the collective motion of a density distributionin a harmonic well. This system yields explicit analytic results and has been studied in thecontext of non-adiabatic functionals. Since exact results are known here, the model offersa trial case for the methods proposed. The second system models a
Coulomb explosion.Two initially harmonically confined electrons are released when the confinement potentialis suddenly removed. The system admits an exact solution in the non-interacting limit andis important for radiation damage in solids. The third system is an x well with two localminima chosen to locally mimic the harmonic well of system 1. The ground state localizescharge on discrete sites, but a linear perturbation can transfer charge from one site to theother. Thus, this model allows us to examine an exact solution in a case when the derivativediscontinuity should matter. Additionally, we would transition from a system that goes fromdouble reference to single reference directly probing some of the most challenging aspects ofTDDFT.We chose to model the Coulomb interaction in 1D using the soft form v ( x ) = λ/ √ x + ǫ with λ = 1 and ǫ = 0 .
01. These parameters differ from those chosen in other work. The advantages of this interaction potential choice are the numerical efficiency by whichintegrals can be solved and the long ranged nature of the interaction; however, the potentialviolates expected scaling laws for a Coulomb interaction. The small ǫ is deliberately chosen9o enhance the short range effects of the interaction that are likely more accurately describedusing our local basis set. Example 1: Hooke’s Atom
The one-dimensional analog of Hooke’s atom has two electrons occupying a harmonic well-potential and interacting through the screened Coulomb interaction. The related Hooke’satom model in 3D has been used extensively to study exchange-correlation in TDDFT andthe model is a natural starting place for study since it admits exact solutions. Dobsonused the Harmonic potential theorem to study the adiabatic local density approximation providing results exploited heavily here to validate our calculations and to illustrate thegenerality of the theorem. A linear term simulating a constant electric field is appliedinstantaneously at t = 0. v ext. ( x, t ) = ω x , t < ω x + κx, t ≥ X ( t ) = κω (1 − cos( ωt )). The exact density of a many-fermion system time evolves according to n ( x, t ) = n ( x − X ( t )) where n is the initial densityprofile either interacting or not. The velocity potential is S ( x, t ) = κω x sin( ωt ). Note thatsince d dx S ( x, t ) = 0 this is incompressible flow of the electron liquid. The instantaneouslyexact KS potential, v instks ( x, t ) = ω x − X ( t )) + v HXC ( x − X ( t )) (15)= ω x − ω xX ( t ) + v HXC ( x − X ( t )) + ω X ( t ) , (16)is just the translated potential since in a Kohn-model oscillation only the center of chargechanges position but the distribution retains the same shape. Note that the final term is10ust a time dependent constant shift of the potential and does not affect the dynamics of thedensity. The exact TD-KS potential is v ks ( x, t ) = ω x + v HXC ( x − X ( t )) − κx (17)Plugging S into Eq. 12, the dynamic potential is v dynks ( x, t ) = ω xX ( t ) − κx (18)which can be added to the instantaneous potential to provide the exact KS potential. Figure1 shows the time-evolution in a harmonic ( ω = 1) potential after the the instantaneous ap-plication of an electric field (linear potential κ = 1 / t = π ) and opposite turningpoints (top right, short-dashed)( t = π ). The bottom quadrants present elements of the ef-fective time-dependent potential. The time-dependent KS potential ( v S ), the instantaneouspotential ( v inst ), the dynamic potential ( v dyn ) at initial and maximum velocity times and forthe other turning point. In this case, the instantaneous potential follows along with the den-sity effectively maintaining the same shape. The dynamic potential acts as a linear drivingpotential enforcing simple Harmonic motion. At the maximum velocity point, there is inter-estingly no dynamic potential in this case; instead, the density’s inertia drives the continuedmotion through the centroid. These images are consistent with Dobson’s earlier work, butdemonstrate that this decomposition as implemented provides numerically expected resultsin a limit where they are available. 11 xample 2: A Dispersing Wave-packet The second case to consider is the released and dispersing wave-packet. In the interactingcase, this can be thought of as a Coulomb explosion, v ext. ( x, t ) = ω x , t < , t > n ( x, t ) = 1 / √ ω t n ( x/ √ ω t ) (20)and S ( x, t ) = ω x t ω t ) . The time-evolved density is identical to the ground-state densitywith a different constant Ω( t ) = ω/ (1 + ω t ). Our numerical studies suggest that theinteracting density obeys a similar scaling to high accuracy. The figure 2 shows the time-evolution after the instantaneous annihilation of a harmonic ( ω = 1) confinement potential.The plots on the left are for λ = 0 the non-interacting and analytically solvable system,and on the right, the interacting-system λ = 1. The densities are shown on the top. Thedashed line is the initial density, the short-dashed blue line is the exact result, and themedium length, green-dashed line is the result of our TDDFT inversion. On the bottomare shown the corresponding potentials v S , v inst , v dyn (dotted-blue, turquoise-dashed, short-green-dashed), the long black dashed in the original v S . As in the previous case, we find forthe non-interacting case that the dynamic potential exactly cancels the instantaneously exactone correctly predicting the total potential to vanish. This is also calculated numerically forthe interacting case. For the non-interacting case, v inst S = ω x ω t and v dyn S = − ω x ω t .A Harmonic density expands, to a good approximation, according to a scaling formula evenwhen interactions are present. If this decomposition holds more generally for non-harmonicexternal potentials and in 3D, a new universal constraint on the TD-KS wave-functionsmust hold, the spatially-varying phase on the KS orbitals must be the same for all occupied12rbitals. Plot 2 represents a Coulomb explosion, in this case, because we use a screened-Coulomb interaction, this is more properly labeled an interaction explosion.
Example 3: The Double Well
To illustrate utility, we apply this to 1D problem that does not admit explicit solutions evenin the non-interacting case. The problem is the 1D double well with an linear potentialdesigned to shuffle one electron from the left well to the right well sharing the right site.This model represents a charge-transfer event, v ext. ( x, t ) = ω x x − ω x , t < ω x x − ω x + κx, t > ± x being the position of the 2 wells and ω being the approximate angular frequencyof the lowest state in each well. Initially, each electron sits predominantly in a separatequantum well. The perturbing potential is designed to transform the initial double well intoa single doubly occupied well. We have simulated the dynamics of 2 cases. The first is a setof non-interacting electrons. The second is a pair of interacting Fermi electrons. We presentresults at an instant of time when a significant fraction of the charge has transferred to thecombined-well.Figure 3 shows a charge transfer excitation: Time-evolution of two electrons in a 1Ddouble-well potential ( x = √ ω = 1 ) after being perturbed by an electric field. The leftside shows the density and potential for non-interacting electrons ( λ = 0). The right sideshows the density and potential for interacting ones ( λ = 1). The densities and potentialsare shown for the initial conditions (long-dashed). For the time-evolved densities, the topplots present the exact time-evolved results (short-blue dashed), and the results of the TDinversion scheme noted in the text (medium green dashed). In the lower plots, we have theinstantaneous contribution (alternating blue dashes), the dynamic contribution (medium13reen dashes), and the total potential (short dark blue dashes).This process moves fastest for the non-interacting case; for in the interacting case,Coulomb interaction discourages the free flow of electrons to the occupied site. In bothcases, the proposed decomposition provides a TD-KS potentials that accurately describesthe dynamics of the electrons. It is important to note the step in the dynamic potential .This is shown in Fig. 3 bottom plots. The green-medium dashed line exhibits a dynamicallychanging step around x = 0 ensuring that 2 wells have a correct chemical potential differ-ence. The results manifests in the dynamic contribution to the TD-KS potential. We showboth the exact time-evolved densities and the results of our Ansatz for the TD-KS orbitals.While the suggested approach proves accurate results for this highly non-trivial example,the density does drift from the exact one. This is in part due to the difficulty in obtainingthe instantaneously exact potential in the asymptotic regimes through Eq. 2. The samecan be said for the numeric calculation of the velocity potential. Ultimately, the numericchallenges limit the general applicability. However, the underlying structure of the TD-KSorbitals when true provides an strong insight into the requirements that need to be buildinto TD-KS potentials. Conclusions
In this paper for three 1D systems, the exact TD Kohn-Sham potential can be decomposedinto instantaneous and dynamic contributions with variational theory providing the formerand fluid-dynamic-like equations providing the latter. This decomposition, when true, pro-vides a route to new functionals by combining state-of-the-art ground-state functionals withapproximations from fluid dynamics.Additionally, the TD-KS orbitals can be related to the instantaneously exact orbitals ateach time step. This reduces the problem of TD V-representability to a form of ground-stateV-representability. There is no paradox with initial-state dependence as the instantaneously xact potential is assumed to be continuously deformable in time from the initial state, butthe history dependence is contained somewhat paradoxically in the ground-state-like term.It is expected that the consequences of this connection are that the instantaneous orbitalsare not necessarily filled in the order of increasing instantaneous eigenvalues.One of the potential drawbacks for using this Ansatz and decomposition to determinetime-dependent potentials is that the instantaneous inversion requires the repeated solutionof an eigenvalue problem scaling poorly with system system. This problem is somewhatmitigated by the assumption that the instantaneous potential various continuously in timethereby providing a convenient starting place for the instantaneous solutions. However, theability to calculate the instantaneous solutions will eventually limit the applicability of thismethod to systems below some critical size. A related objection is that the solution of Eq.13 involves a double integral and inverses of exponentially small densities. This is not a severproblem, and Eq. 13 can be solved using numerical methods for partial differential equations.In greater than 1D, the use of Eq. 13 to find the velocity potential is problematic. Insteaddirect solution of Eq. 13 is required to provide the velocity field. In this case the dynamicpotential will capture the singularities. However, instantaneously exact V-representabilitywill probably not hold for arbitrary time-evolving electron densities.In this paper, we have shown that an explicit representation of the time-dependent Kohn-Sham orbitals in terms of a position-dependent phase term and instantaneously exact or-bitals can provide an alternate scheme to construct the TDDFT potential from a targettime-dependent density. This approach avoids many of the instabilities that plague moredirect inversion schemes. However, there exists some limitation to this approach as the ex-istence of the instantaneously exact potential is not always guaranteed, and even then, theinstantaneous solution might still be numerically challenging. Additionally, the scheme doesnot distinguish between external and exchange-correlation contributions to the potential. Ifthe driving potential for the exact density is known then the exchange-correlation potentialmay be extracted independently. We plan to generalize this procedure to 3D and to explore15ore realistic Coulomb interacting potentials.We would like to thank A.D. Baczewski, N. Maitra, and D. Jensen for insightful discus-sions. This work was funded through the Sandia National Laboratories LDRD office underproject 151362 in the Science of Extreme Environments research area. Sandia NationalLaboratories is a multi-program laboratory managed and operated by Sandia Corporation,a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department ofEnergy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. References (1) Runge, E.; and Gross, E.K.U.
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8, 247.18 .00.10.20.30.40.50.60.7 -3 -2 -1 0 1 2 3 ρ [ boh r - ] x [Bohr]t=0t= π /2 0.00.10.20.30.40.50.60.7 -3 -2 -1 0 1 2 3 ρ [ boh r - ] x [Bohr]t=0t= π -2.0-1.00.01.02.03.04.05.0 -3 -2 -1 0 1 2 3 v [ H a r t . ] x [Bohr]Vs t=0Vadia t= π /2Vdyn t= π /2Vs t= π /2 -2.0-1.00.01.02.03.04.05.0 -3 -2 -1 0 1 2 3 v [ H a r t . ] x [Bohr]Vs t=0Vadia t= π Vdyn t= π Vs t= π Figure 1: Harmonic Potential Motion: Time-evolution in a harmonic ( ω = 1) potentialafter the the instantaneous application of an electric field (linear potential κ = 1 / t = π ). Top right: density at initial timeand at the other turning point ( t = π ). Bottom left: the time-dependent KS potential ( v S ),the instantaneous potential ( v inst ), the dynamic potential ( v dyn ) at initial and maximumvelocity times. Bottom right: same but for the other turning point.19 .00.10.20.30.40.50.60.7 -3 -2 -1 0 1 2 3 ρ [ boh r - ] x [Bohr] λ = 0 t=0t= π /4scaled 0.00.10.20.30.40.50.60.7 -3 -2 -1 0 1 2 3 ρ [ boh r - ] x [Bohr] λ = 1 t=0t= π /4scaled-5.0-4.0-3.0-2.0-1.00.01.02.03.04.05.0 -3 -2 -1 0 1 2 3 v [ H a r t . ] x [Bohr] λ = 0 Vs t=0Vadia t= π /4Vdyn t= π /4Vs t= π /4 -5.0-4.0-3.0-2.0-1.00.01.02.03.04.05.0 -3 -2 -1 0 1 2 3 v [ H a r t . ] x [Bohr] λ = 1 Vs t=0Vadia t= π /4Vdyn t= π /4Vs t= π /4 Figure 2: The
Coulomb
Explosion: Time-evolution after the instantaneous annihilation ofa harmonic ( ω = 1) confinement potential. The plots on the left are for λ = 0 the non-interacting and analytically solvable system, and on the right, the interacting-system λ = 1.The densities are shown on the top. The dashed line is the initial density, the short dashedblue line is the exact result, and the medium green-dashed line is the result of our TDDFTinversion. On the bottom are shown the corresponding potentials v S , v inst , v dyn (dotted-blue,turquoise-dashed, short-green-dashed), the long black dashed in the original v S .20 .00.10.20.30.40.50.60.7 -3 -2 -1 0 1 2 3 ρ [ boh r - ] x [Bohr] λ = 0 t=0t=2 π TDDFT 0.00.10.20.30.40.50.60.7 -3 -2 -1 0 1 2 3 ρ [ boh r - ] x [Bohr] λ = 1 t=0t=2 π TDDFT-2.0-1.00.01.02.0 -3 -2 -1 0 1 2 3 v [ H a r t . ] x [Bohr] λ = 0 Vs t=0Vadia t=2 π Vdyn t=2 π Vs t=2 π -2.0-1.00.01.02.0 -3 -2 -1 0 1 2 3 v [ H a r t . ] x [Bohr] λ = 1 Vs t=0Vadia t=2 π Vdyn t=2 π Vs t=2 π Figure 3: Charge transfer excitation: Time-evolution of two electrons in a 1D double-wellpotential ( x = √ ω = 1 ) after being perturbed by an electric field. The left sideshows the density and potential for non-interacting electrons ( λ = 0). The right side showsthe density and potential for interacting ones ( λλ