Johanna I. Fuks* and Neepa T. Maitra* · 2020. 10. 2. · approximation is, what features of the exact potential are missing, etc. In a few cases, the AE potential was used to self-consistently

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Cite this:DOI: 10.1039/c4cp00118d

Challenging adiabatic time-dependent densityfunctional theory with a Hubbard dimer: the caseof time-resolved long-range charge transfer

Johanna I. Fuks* and Neepa T. Maitra*

We explore an asymmetric two-fermion Hubbard dimer to test the accuracy of the adiabatic approximation of

time-dependent density functional theory in modelling time-resolved charge transfer. We show that the

model shares essential features of a ground state long-range molecule in real-space, and by applying a

resonant field we show that the model also reproduces essential traits of the CT dynamics. The simplicity of

the model allows us to propagate with an ‘‘adiabatically-exact’’ approximation, i.e. one that uses the exact

ground-state exchange–correlation functional, and compare with the exact propagation. This allows us to

study the impact of the time-dependent charge-transfer step feature in the exact correlation potential of real

molecules on the resulting dynamics. Tuning the parameters of the dimer allows a study both of charge-

transfer between open-shell fragments and between closed-shell fragments. We find that the adiabatically-

exact functional is unable to properly transfer charge, even in situations where the adiabatically-exact

resonance frequency is remarkably close to the exact resonance, and we analyze why.

I. Introduction

The transfer of an electron across a molecule is an essential processin biology, chemistry, and physics, which needs to be accuratelydescribed in order to computationally model phenomena in manytopical applications, e.g. photovoltaics, vision, photosynthesis, mole-cular electronics, and the control of coupled electron–ion dynamicsby strong lasers (e.g. ref. 1–7). For most of these applications, a time-resolved picture of the charge transfer (CT) is extremely useful, andoften necessary, as has been stressed in recent work, and thecorrelation between electrons as well as between electronsand ions play a crucial role.8 The systems are large enough thattime-dependent density functional theory (TDDFT) is the onlycalculationally feasible approach.9–11 It is well-known thatthe standard functional approximations considerably under-estimate CT excitations, and there has been intense develop-ment of improved functionals for this; in particular the optimallytuned hybrids present a useful non-empirical approach.12

However the transfer of one electron from one region of spaceto another is clearly a non-perturbative process and calls forcalculations that go beyond linear response and excitations.The success of TDDFT to date rests on its performance in thelinear regime, however the theory applies also to dynamics farfrom equilibrium. The performance of functionals for CT in this

regime paints a more hazy picture: there have been calculationsin good agreement with experiment (e.g. ref. 6) but failures havebeen reported too (e.g. ref. 13). It would be fair to say that it is notalways clear to what accuracy the TDDFT results can be trusted.Part of the problem is that there are very few alternate practicalcomputational methods for correlated electronic dynamics totest against. Calculations on simplified model systems that canbe solved exactly, e.g. two-electron systems in one-dimension,have highlighted prominent features that the approximate func-tionals lack, not just for CT dynamics,14 but also more generallyin the non-linear regime.15–18 The errors that result from the lackof these features appear to be sometimes very significant, andother times less so.

TDDFT in practise is almost always synonymous with adiabaticTDDFT, certainly in the non-linear regime. That is, the Kohn–Sham (KS) system is propagated using an adiabatic exchange–correlation potential, where the evolving density at time t is inputinto a ground-state (gs) functional: nAXC[n; C0, F0](r, t): =ngsXC[n(t)](r). There are two distinct sources of error in such anapproximation: one is from the choice of the gs functionalapproximation, while the other is the adiabatic approximationitself. To separate these the adiabatically-exact (AE) approximation18

is defined: the instantaneous density is input into the exact gsfunctional, nAEXC[n; C0, F0](r, t) = n

AEXC[n](r, t) := n

exact gsXC [n(t)](r).

This approximation neglects memory-dependence that the exactfunctional is known to possess (dependence on the density’shistory and true and KS initial states C0 and F0) but is fullynon-local in space, and, if the true and KS states at time t were

Department of Physics and Astronomy, Hunter College and the City University of

New York, 695 Park Avenue, New York, New York 10065, USA.

E-mail: [email protected], [email protected]

Received 10th January 2014,Accepted 4th March 2014

DOI: 10.1039/c4cp00118d

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actually gs’s of some potential, it would be instantaneouslyexact at time t.

Since the exact gs exchange–correlation functional isnot known, even for one-dimensional two electron systems,nAEXC[n](r, t) must be found via a numerical scheme, of an inverseproblem type. A handful of papers14,17,18 have found nAEXC[n](r, t)using an iterative scheme for some model systems: the exactdensity n(t), found by solving the interacting Schrödingerequation, provides the input to an iterative procedure thatfinds at each t of interest the interacting and non-interactinggs’s of density n(t), along with the potentials in which they arethe gs. Then, nAEXC[n(t)](r) = n

exact gsext [n(t)](r) � nexact gsS [n(t)](r) �

nH[n(t)](r) where nH[n](r) is the electrostatic Hartree potential.In most cases studied so far the AE potential nAEXC[n](r, t) hasbeen evaluated on the exact density n(t), and compared with theexact (memory-dependent) exchange–correlation potentialnXC[n,C0, F0](r, t) at that time, to analyze how good the AEapproximation is, what features of the exact potential aremissing, etc. In a few cases, the AE potential was used to self-consistently propagate the KS orbitals, using at each time-step,the AE potential evaluated on the self-consistent instantaneousdensity. Such a propagation provides a more useful assessmentof the accuracy of the AE, as it measures directly the impact ofthe AE on the resulting dynamics. For example, it is possiblethat some features that might make the AE potential looksignificantly different from the exact may in fact have a limitedeffect on the propagation. However, self-consistent AE propaga-tion clearly requires much more numerical effort, as manyiterations need to be performed at every time-step to find thepotential to propagate in, and it has only been done in a fewexamples18–20 in one-dimensional model systems.

In particular, for CT dynamics it is particularly challengingto converge the iterative density-inversion scheme due to thevery low density region between the atoms. Yet, such a calcula-tion is of great interest for CT dynamics: not only because of itssignificance in the phenomena mentioned earlier, but alsobecause it is known that the exact functional develops featuresthat the usual approximations lack. Ref. 14 showed that for atwo-electron model molecule composed of closed-shell atoms(meaning that in the dissociation limit, each of the two atomshave spin-paired electrons) and driven at the CT resonance, astep associated with the CT process gradually builds up overtime in the exact correlation potential. A dynamical oscillatorystep is superimposed on this (see ref. 17 and 21), and is ageneric feature of non-linear dynamics, not only in CTdynamics, which has a non-adiabatic density-dependence.The AE approximation fails to capture the dynamical stepbut, when evaluated on the exact density, does yield a CT stepalthough of a smaller size than the exact. Such steps requirefunctionals with a spatially non-local dependence on thedensity. The available approximations do not yield any stepstructure whatsoever: the dismal failure of ALDA, ASIC-LDA,and AEXX, none of which contain any step in the correlationpotential, to transfer any charge was shown (Fig. 3 of ref. 14)and attributed to this lack of step structure. We expect someblame must go to the adiabatic approximation itself, but a

question arises: is the partial step of the AE approximation enoughto give a reasonable description of the CT dynamics? If yes,this would greatly simplify the on-going search for accuratefunctionals for non-perturbative CT: it would mean that onedoes need to build in spatial non-local density-dependence intothe correlation functional approximation, but that one couldget away with a time-local, i.e. adiabatic approximation. Toanswer the question, we would need to propagate with the AEself-consistently, but as discussed above, this procedure isnumerically very challenging for CT dynamics. Recently22

we have shown that the answer is no, by studying CT dynamicsin a two-fermion asymmetric Hubbard dimer, which sharesthe essential features of CT dynamics with real-space mole-cules. Due to the small Hilbert space of the dimer the exactgs functional can be found via a constrained search,28 para-metrized into an analytic form, and then used in nAEXC(t) to self-consistently propagate the system. No iterative scheme isneeded because the exact functional form of the gs Hartree-exchange–correlation (HXC) potential is known. This enabledus to assess errors in the adiabatic approximation for CTdynamics independently of those resulting from errors in thegs approximation used. In this paper we give more details onthe dimer model, and the procedure followed. Like in ref. 22,we study both the cases of resonant CT between closed-shellsand between open-shells, by tuning the potential-differencebetween the two sites. However, unlike ref. 22, we choose thisasymmetry such that the CT state of the first case has a verysimilar density to the gs of the second, and vice versa. Althoughthis choice leads to the exact density-dynamics in one casebeing a time-reversed version of the dynamics in the other case,we find the AE dynamics does not have this property. The AEapproximation in the closed-shell case is better for longer thanfor the open-shell case, where it fails almost immediately; yet ineither case, it fails to properly transfer the charge. The hopethat the step seen in the AE approximation evaluated on theexact density, albeit smaller than the exact, is enough to do areasonable job for CT processes is dashed. A further result is anexpression for the interacting frequencies of the system interms of the KS ones and the HXC kernel. Using this, wecompare the exact resonant frequency of the interacting systemwith that predicted by the AE approximation.

In Section II we introduce the model, its ground-state energyfunctionals and potentials, and the exact time-dependent KSpotential. In Section III, we present the parameters used tostudy CT between closed-shells, give details of the eigenstatesof the interacting and KS systems, and propagate the systemwith a resonant field to induce Rabi oscillations between theground and CT excited state. We compare the exact propagationwith that resulting from the AE propagation and discussfeatures of the potentials. Section IV contains the analogousanalysis for the case of CT between open-shells. In Section V wederive a formula for the interacting frequencies of the system interms of the KS ones and the HXC kernel. This is used to findthe AE resonant frequency, compare with the exact in each case,and discuss the connection to the time-resolved dynamics. Wefinish with conclusions and outlook.

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II. The model

The Hamiltonian of the two-site interacting Hubbard modelwith on-site repulsion U and hopping parameter T22–30 reads:

Ĥ ¼ � TXs

ĉyLsĉRs þ ĉ

yRsĉLs

� �þU n̂L"n̂L# þ n̂R"n̂R#

� �

þ DnðtÞ2

n̂L � n̂Rð Þ;(1)

where ĉyLðRÞs and ĉL(R)s are creation and annihilation operators

for a spin-s electron on the left(right) site L(R), respectively, and

n̂LðRÞ ¼P

s¼";#ĉyLðRÞsĉLðRÞs are the site-occupancy operators.

The occupation difference hn̂L � n̂Ri = Dn represents thedipole in this model, d = Dn, and is the main variable;28 thetotal number of fermions is fixed at N = 2. A static potentialdifference, Dn0 ¼

Ps

v0Ls � v0Rs� �

, renders the Hubbard dimer

asymmetric. The total external potential Dn(t) is given by Dn(t) =Dn0 + 2e(t), where the last term represents an electric fieldthat we will tune to induce CT between the sites. An infinitelylong-range molecule is modelled by T/U - 0: in our calcula-tions, we fix the interaction strength to be unity, U = 1, andmake the hopping parameter T small, corresponding to a largeseparation between the sites (equivalent to the strongly corre-lated limit U/T - N). We use �h = e = 1 throughout, and allenergies are given in units of U.

The singlet sector of the two-electron vector space is three-dimensional (depicted in Fig. 1),

C1j i ¼1ffiffiffi2p ĉy1"ĉ

y2# � ĉ

y1#ĉy2"

� �0j i ¼ 1ffiffiffi

2p "; #j i � #; "j i½ � (2)

C2j i ¼ ĉy1"ĉy1# 0j i ¼ "#; 0j i (3)

C3j i ¼ ĉy2"ĉy2# 0j i ¼ 0; "#j i (4)

For fixed T/U a constrained search over all gs wavefunctions|Ci = a1|C1i + a2|C2i + a3|C3i that yield a given Dn32,33 canbe straightforwardly performed due to the small size of the

Hilbert space. This results in the Hohenberg–Kohn (HK) energyfunctional:31–33

FHK½Dn� ¼ minC!Dn

CDnh jT̂ þ Û CDnj i ¼ EHXC½Dn� þ Ts½Dn�; (5)

where T̂ and Û are the first two terms in eqn (1), Ts[Dn] =minF-DnhFDn|T̂|FDni is the non-interacting kinetic energy, andF denotes a single Slater determinant. EHXC[Dn] is the HXCenergy functional, which must in practise be approximated forreal systems, but here for the Hubbard model we can computethe numerically exact functional explicitly. The HK functionalFHK[Dn] completely determines the gs energy Egs,

Egs ¼ minDn

FHK½Dn� þDn0

2Dn

� �: (6)

The gs occupation difference Dngs of all possible asymmetric (and

symmetric) Hubbard dimers is determined by@FHK@Dn

Dngs

¼ �Dn0

2.

The minimization eqn (5) was carried out in Mathematica;the resulting discrete function F jHK(Dnj) was fitted and derived

using splines to obtain the exact gs HXC potential DngsHXC½Dn� ¼

2@ FHK½Dn� � Ts½Dn�ð Þ

@Dn(see Fig. 2). (A factor of 2 results from

expressing the energy functional in terms of the variable

Dn = nL � nR, namely Dn gsHXC½Dn� ¼ nLHXC½Dn� � nRHXC½Dn� ¼

dEHXC½Dn�dðDnÞ

dDndnL�dEHXC½Dn�

dðDnÞdDndnR

).

In Fig. 2 the different components of the energy as afunction of the occupation difference Dn for different T andfixed Hubbard strength U = 1 are plotted. Note that the Hartree-exchange (HX) part of the energy functional is independent ofT, EHX[Dn] = (N

2 + Dn2)U/8 (with N being the number ofparticles).27 In the limit where the two electrons are sitting onthe same site the HXC energy is entirely due to HX and equal to

Fig. 1 The three states pictured form a complete basis of the singletsector of the Hubbard dimer.

Fig. 2 Correlation energy functional EC[Dn] for T = 1 (red dotted) T = 0.1(green dashed) and T = 0.05 (black solid). Insets: EHXC[Dn] (left) and non-interacting kinetic energy functional TS[Dn] for the same parameters.Energies are in units of U.

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the on-site repulsion U, EHXC[Dn = �2] = EHX = U (see the leftinset in Fig. 2). The hopping parameter T plays a role as soon asthe electronic density delocalizes (see the right inset in Fig. 2).

For small occupation differences Dn - 0 (one electron oneach site), in the infinite separation limit T/U - 0, the correla-tion energy EC[Dn] develops a discontinuity in its derivative (seeFig. 2). This discontinuity manifests in a step-like function in

the correlation potential difference DngsC ½Dn� ¼ 2@DEC@Dn

(see the

left inset Fig. 3). This feature is related to the derivativediscontinuity of the isolated 1-electron site for the followingreason. The variable Dn plays the role of the density-variable, aswell as directly giving the particle number on each site, nL,R =1 � Dn/2. So, in the isolated-site limit T/U - 0, a variation dnnear Dn = 0 can be thought of as adding(subtracting) a fractionof charge dn to the one-fermion site on the left(right):

2dEC½Dn�dðDnÞ

Dn¼0þ

�2dEC½Dn�dðDnÞ

Dn¼0�

¼DngsC Dn¼ 0þ½ ��DngsC Dn¼ 0

�½ �

� 2D1�siteC ðN¼ 1Þ (7)

The difference in the correlation potential as one crossesDn = 0, therefore, coincides with the derivative discontinuity atN = 1 of one site; the value of D1-siteC (N = 1) = U. The discontinuityonly shows up in the infinite separation limit; if instead thetwo sites lie closer to each other they cannot be consideredas two separated one-electron systems and thus moving afraction of electron back or forth represents a smooth changein the energy.

In Section III we will study CT dynamics between two closed-shell fragments (cs–cs) by applying a relatively large staticpotential difference such that Dngs E 2. One electron will betransferred to the other site by turning on a field resonant withthe CT excitation frequency. Looking at Fig. 2 and 3 thiscorresponds to scanning the densities starting at the outerright region and finishing at the central region once the CT

state (consisting of two now open-shell sites) is reached. TheAE propagation is performed using the exact gs HXC potentialDngsHXC[Dn] shown in Fig. 3, i.e. assuming that at every time tthe density Dn(t) is the gs density of some potential Dn.In Section IV we study instead CT between two open-shellfragments (os–os), starting with a gs consisting of two open-shell sites each with approximately one electron (Dngs E 0) thatevolves to a CT state with Dn E 2; thus scanning the densitiesin a ‘‘time-reversed’’ way compared to Section III, moving fromthe central region in Fig. 2 and 3 to the outer region. We havechosen the Dn0’s such that the CT density of the cs–cs system isclose to the gs density of the os–os system, Dncs–csCT E Dn

os–osgs and

vice versa, Dnos–osCT E Dncs–csgs .

A. Time-dependent Kohn–Sham potential

The KS Hamiltonian has the form of eqn (1) but with U = 0 andDn(t) replaced by the KS potential difference,

Dns[Dn,F(t0)](t) = nHXC[Dn,C(t0),F(t0)](t) + Dn(t), (8)

defined such that the interacting Dn(t) is reproduced. The exacttime-dependent KS potential can be found by inversion of thetime-dependent KS eqn (10) assuming a doubly-occupied sing-let state. This yields26,29

Dns½Dn;Fðt ¼ 0Þ� ¼ �€Dnþ ð2TÞ2Dnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð2TÞ2 4� ðDnÞ2� �

� ð _DnÞ2r

0BB@

1CCA (9)

when the KS initial state is the KS gs. Dn(t) is time-dependentnon-interacting v-representable as long as the denominator ineqn (9) does not vanish,

_Dn o 2T

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4� ðDnÞ2

q: (10)

Condition (10) fixes an upper bound to the absolute value of

the link-current j _Dnj, which can be identified with the sum ofcurrents flowing along links attached to the site. On a latticethe maximum link-current depends on T (see ref. 29 andreferences therein).

III. Closed-shell to closed-shell CT

To model CT between two closed-shell fragments we choose thestatic external potential difference in the Hubbard dimer to beDn0 = �1.5 U, which results in a gs with almost two fermionssitting on the left site Dngs = 1.9620 (see Fig. 4). The vector spaceis built from the three singlet states introduced in Fig. 1 andeqn (4), the eigenstates are presented in detail in Table 1. Theinteracting states in this basis were found by first computingthe matrix elements of the static Hamiltonian eqn (1) (E(t) = 0)in basis eqn (4) and then diagonalizing the matrix. The calculationwas performed in a self-developed code written in the secondquantization formalism. The interacting gs is predominantly |C2i,while the first excited state is mainly |C1i, i.e. a CT excitationwith about one electron on each site. There is a second CTexcited state Ccs–cse2 , dominated by |C3i. The KS states F are

Fig. 3 Ground-state HXC potential functional DngsHXC

[Dn] for T = 1 (reddotted) T = 0.1 (green dashed) and T = 0.05 (black solid). Left inset:Ground-state correlation potential functional Dngs

C[Dn]. Right inset:

Df gsHXC½Dn� ¼d2EHXC½Dn�

dðDnÞ2.

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obtained from diagonalization of the exact gs KS Hamiltonian,which corresponds to taking U - 0 and Dn = Dn0 + DnHXC[Dn] in

eqn (1), with DnHXC½Dn� ¼ 2@EHXC½Dn�

@Dnand EHXC[Dn] found by

constrained search as discussed in Section II. The non-interacting two-electron KS gs is also predominantly |C2i, whilethe first KS excited state, predominantly |C1i, corresponds toa single excitation to a CT state. The second KS excited stateFcs–cse2 is dominated by |C3i and is actually a double excitation.Comparing the KS states with the interacting ones for the cs–cscase we observe that they are very similar.

Table 2 contains the energies, site-occupation differences

Dn ¼ Ccs�csjD̂njCcs�csD E

of the three interacting and KS states

enumerated in Table 1, and the transition matrix elements from

the gs to the CT excited states, dgs!e1ð2Þ ¼ Ccs�cse1ð2Þ jD̂njCcs�csgsD E

for

both interacting and Kohn–Sham systems. By construction theexact gs HXC functional EHXC reproduces the exact gs energy Egsand gs density Dngs. All other KS variables shown such asinteracting excitation frequencies os and transition matrixelements dsgs-e1(2) have limited physical meaning. For the caseof the cs–cs CT we are studying in this section, however,they turn out to be good approximations to the exact quantities,

ose(2) = ese(2) � eSgs E o, and dgs!e1ð2Þ ¼ FgsjD̂njFeð2Þ

D E� dgs!e1ð2Þ.

In contrast to the interacting system the non-interacting Kohn–Sham system has equidistant excitations ese2 � ese1 = ese1 � esgs.That is, the second KS excitation is a pure double-excitation outof the doubly-occupied gs KS orbital; consequently, its dipoletransition matrix element is exactly zero. In the interactingsystem, the second excitation has a very small but non-zerotransition matrix element (see Table 2). We will choose a fieldresonant with the first excitation, weak enough that only theground and first excited interacting states get occupied duringthe dynamics.

We induce the CT dynamics by turning on a field resonantwith the lowest excitation, e(t) = 0.09 sin(ot), with o = ogs-e1 =0.5177 U. All propagations were performed using a Crank–Nicolson scheme, time-step is 0.01/U. The TDDFT calculationshave been performed using a predictor–corrector scheme. Weevolve the interacting gs in the Hamiltonian of eqn (1) to obtainthe exact dipole shown in Fig. 4 for a little over half a Rabiperiod; the CT excited state is reached at around t = 128/U. Thephysics is similar to the real-space CT dynamics in the long-range one-dimensional molecule shown in Fig. 4 of ref. 14 (seealso Fig. 3 in ref. 22) and also in the three-dimensional LiCNmolecule in Fig. 3 and 4 in ref. 13. (Note that Fig. 3 of ref. 22,which was for the Hubbard dimer with a larger asymmetry,happens to have a closer match with the real-space case. The

Fig. 4 Model of CT between two closed-shell fragments at large separa-tion: a large static potential difference Dn0 = �1.5 U is chosen such that inthe gs almost two electrons are sitting on the left site (Dng E 2) and T/U =0.05. In the top panel the results of the propagation are shown: exactdipole Dn(t) (black solid), self-consistent AE dipole DnAEsc (t) (red dashed) andself-consistent adiabatic EXX dipole DnAEXXsc (t) (pink dotted). Time is given inunits of 1/U, the CT state is reached at TR/2 E 128/U.

Table 1 Interacting and Kohn–Sham states for the asymmetric dimer withthe cs–cs ground state

Dn0 = �1.5 U |C1i |C2i |C3i

|Ccs–csgs i �0.13781 0.99045 0.00323|Ccs–cse1 i 0.990054 �0.13785 0.02809|Ccs–cse2 i �0.02827 0.000666 0.9996

|Fcs–csgs i 0.13724 0.99049 0.0095073|Fcs–cse1 i 0.980985 �0.13724 0.13724|Fcs–cse2 i �0.137236 0.0095072 0.990492

Table 2 Eigenstates, energies, and transition matrix elements for thedimer with Dn0 = �1.5 U, T = 0.05, U = 1. Energies are given in units ofU. Note that e1 and e2 are CT excitations for both the interacting and KSsystems

Dn = �1.5 U

Interacting Kohn–Sham

Dngs 1.9620 1.9620Egs �0.5098 �0.5098egs �0.5152Dne1 0.0364 0.0000ee1 0.0078 0.0000ogs-e1 0.5177 0.5152dgs-e1 �0.2733 �0.2745Dne2 �1.9980 �1.9620ee2 2.5020 0.5152ogs-e2 3.0118 1.0304oe1-e2 2.4942 0.5152dgs-e2 �0.0052 0.0000de1-e2 �0.0563 �0.2745

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resonant frequency was larger, so the dipole oscillates faster inthe half-Rabi period and also the transition matrix elementdgs-e1 was smaller, making the amplitude of the fast oscilla-tions smaller.) Fig. 4 shows also the dipole under propagationwith the adiabatic exact-exchange (AEXX) approximation,

DvAEXXHXC ¼ DvHX ¼ 2@EHX Dn½ �@Dn

¼ U2DnðtÞ.27 DnAEXXsc does not

show any charge-transfer, resembling the real-space AEXX caseof ref. 14. Other adiabatic approximations were also shown tofail in a similar way.14 However in ref. 14 and 13 it was notpossible to determine whether the culprit was the adiabaticapproximation itself or the chosen gs approximation. For theHubbard dimer, with its vastly reduced Hilbert space, andthe exact HXC potential found by the constrained search(Section II) we are able to propagate the KS system with theAE functional; at each time-step inserting the instantaneousdensity DnAEsc into the exact gs HXC potential Dn

gsHXC[Dn

AEsc ]

(Fig. 3). The result is DnAEsc in Fig. 4: DnAEsc closely follows the exact

density for a longer time than the AEXX does, but ultimately failsto transfer the charge. As was concluded in ref. 22, it is essentialto have a memory-dependent functional in order to correctlydescribe a full charge transfer.

Ref. 22 plotted the exact and AE potentials for the case ofcs–cs CT studied there, which illuminated some of the aspectsof the dynamics and strengthened the comparison with thereal-space molecular case. Although the case studied in ref. 22was for a more asymmetric dimer (Dn = �2), resulting in ahigher resonant field frequency and more oscillations over theRabi period, the essential observations carry over to the presentcase, and the potentials follow similar features to those shownin ref. 22. In particular, (i) the exact correlation potential dropsfrom its gs value to that of �Dn0 after half a Rabi cycle, suchthat the total KS potential DnS = Dn

0 + DnHXC goes to zero,equalizing the levels on each site. This exactly mirrors the real-space case, where a spatial step in the correlation potential inthe intermolecular bonding region develops such that at thehalf-Rabi cycle, the two atomic levels are aligned with eachother, i.e. the step has a size equal to the difference in theionization potentials of the (N � 1)-electron ions. (ii) The AEcorrelation potential evaluated on the exact density, DnAE[Dn],tracks the ground-state correlation potential shown in Fig. 3,moving from the right inwards to the central region, gentlyoscillating around it, in synch with the density. As Dn(t) - 0and the CT state is reached it tracks the approaching disconti-nuity, which in the limit of T/U - 0 is equal to the one-site one-fermion derivative discontinuity,22 in complete analogy withthe infinite-separation limit of the real-space molecular case.14

The donor potential is shifted upwards relative to the acceptorby an amount equal to the derivative discontinuity of the donor,and in both the real-space and Hubbard cases, this under-estimates the shift provided by the exact correlation potential.(iii) The self-consistent AE correlation potential, n AEC [Dnsc](t),deviates from the true potential quite early on. As a conse-quence of this, the two sites remain far from being ‘‘aligned’’,forbidding the possibility that a stable CT state with one electronon each, can be approached in the self-consistent AE propagation.

We now turn to one aspect of the exact correlation potentialthat was discussed only briefly in ref. 22. It was found thatthe exact correlation potential after a very short time developslarge oscillations which appear to be related to maintainingnon-interacting v-representability. Since the system begins withDn = 1.960, close to 2, and T is small, the right-hand-side ofeqn (10) starts out quite small. The left-hand-side starts fromzero and increases but it does not take a very large link-currentfor the two sides of eqn (10) to approach each other, leading tothe denominator of DnS to approach zero and hence becomingclose to violating the non-interacting v-representability condi-tion. Fig. 5 shows the early-time behavior of the KS potential:DnS first swings sharply to �DnS (near time of 19/U) when thedenominator gets very close to zero, causing the acceleration €Dnto change the direction (smoothly) and a consequent decreasein the current. This moves the denominator away from zero,escaping the violation of v-representability. The system oscil-lates due to the field, and again the denominator becomes verysmall at around time of 22/U, when again the DnS changes thedirection, avoiding again the crash into non-v-representability.As time evolves the density transfers, Dn moves further from 2,and so larger currents are possible without danger of thev-representability condition being violated. The potential oscilla-tions then become more gentle, as shown in the figure.

IV. Open-shell to open-shell CT

To study CT between two open-shell fragments we choose thestatic external potential difference to be Dn0 = �0.5 U, whichresults in a gs with about one electron on each site, Dn = 0.0329(see Fig. 6). The eigenstates for the os–os ground-state caseare presented in detail in Table 3. The value of Dn0 has beenchosen such that DnDn =�0.5gs E Dn

Dn =�1.5CT of the cs–cs case in the

previous section, and the CT excitation in the present os–oscase has DnDn=�0.5CT EDn

Dn=�1.5gs of the os–os case (compare Tables 2

and 4). The gs of this problem is dominated by |C1i, while the CT

Fig. 5 Dn0 = �1.5 U cs–cs CT: KS potential DnS (black solid), denominatorof DnS (red dashed), link-current

:Dn (blue dotted), acceleration €Dn (green

dashed-dotted) and dipole Dn (pink dotted).

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excited state is mainly |C2i, with almost two electrons on the leftsite. Again there is a second CT state, with much smaller gs dipoletransition matrix element dg-CT2 { dg-CT (see Table 4), and closeto both electrons on the right (|C3i). The ground and excited KSstates have a very different form: instead of the predominantlyHeitler–London-like nature of the interacting gs, the ground-KSstate is a single Slater determinant. This is quite analogous to thereal-space molecular case where the KS state is a doubly-occupiedbonding orbital, a single Slater determinant, while the interactionis of Heitler–London form in the infinite separation limit, requiringminimally two determinants to describe. The KS excitations are

also similar to those of the real-space case: the first KS excita-tion is not a CT state, but rather a single-excitation to theantibonding state, and second KS excitation is a double-excitation to the antibonding state. As a consequence the KSexcitation energies become very small as T/U - 0 in contrastwith the true energies, just as in the infinite-separation limit ofthe real-space case, and it can also be understood from theabove that the transition matrix elements are large in the KScase while small in the true case (Table 4).

We now turn to the dynamics, taking e(t) = 0.09 sin(ot),resonant with the lowest CT excitation resonance, o = ogs-e1 =0.5228 U, and compare the exact and AE dipoles, as in theprevious section. Due to the present choice of parameters, theexact dipole looks almost exactly like a mirror image of the cs–cscase in the previous section, i.e. the dipole dynamics in the os–oscase resembles that of the cs–cs case starting at the CT state.However the AE dipole does not at all. The AE dipole for the os–oscase fails badly even after a very short time, as shown in Fig. 6; forall times one electron more or less remains on each site duringthe AE propagation, while in the exact propagation, (almost) oneelectron transfers from the left to the right site.

The exact and AE potentials are similar to the os–os case studiedin ref. 22 for a slightly smaller asymmetry (Dn0 = �0.4 in ref. 22).The essential features are as follows. The exact correlation potentialhas the same property as in the real-space case: it starts with a valueto exactly cancel the asymmetry in the external potential, such thatthe KS potential sees the two sites aligned. In the real-space case,the HXC gs potential of a long-ranged heteroatomic diatomicmolecule has a step in the bonding region that aligns the highestoccupied orbital energies on each atom.34–36 In both real-space andHubbard dimer cases, this is a ground-state correlation effect.Then, as the charge transfers, the relative shift in the correlationpotential between the sites oscillates on the optical scale whiledropping to the value predicted by subtracting the external

potential from eqn (9), putting _Dn ¼ €Dn ¼ 0, when the CT excited

Fig. 6 Model for CT between two open-shell fragments at large separa-tion: a small static potential difference Dn0 = �0.5 U is chosen such thatthe gs is close to homogenous (Dng E 0), and again T/U = 0.05. In the toppanel the results of the propagation are shown: exact dipole Dn(t) (blacksolid), self-consistent AE dipole DnAEsc (t) (red dashed). Time is given in unitsof 1/U; the CT state is reached at TR/2 E 129/U. (The AEXX dipole is notincluded, because of difficulties in converging to a stable os–os gs forthese parameters).

Table 3 Interacting and Kohn–Sham states for the asymmetric dimerwith the os–os ground state

Dn0 = �0.5 U |C1i |C2i |C3i

|Cos–osgs i 0.989568 0.136387 0.04625|Cos–ose1 i �0.13608 0.99065 �0.0097168|Cos–ose2 i �0.0047131 0.0033222 0.99888

|Fos–osgs i 0.707011 0.50823 0.49177|Fos–ose1 i �0.016462 0.707011 �0.707011|Fos–ose2 i �0.707011 0.49177 0.50823

Table 4 Eigenstates, energies, and transition matrix elements for thedimer with Dn0 = �0.5 U, T = 0.05, U = 1. Energies are given in units ofU. Note that e1 and e2 are CT excitations for the interacting system

Dn = �0.5 U

Interacting Kohn–Sham

Dngs 0.0329 0.0329Egs �0.0131 �0.0131egs �0.1000

Dne1 1.9626 0.000ee1 0.5097 0.000ogs-e1 0.5228 0.1000dgs-e1 0.2711 1.4140

Dne2 �1.9955 �0.0329ee2 1.5033 0.1000

ogs-e2 1.5164 0.2000oe1-e2 0.9936 0.1000dgs-e2 �0.0915 0.0000de1-e2 0.0260 1.4140

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state is reached. As for Dn AEC [Dn](t), it tracks DngsC [Dn(t)] of Fig. 3

moving from near the center out to the right; with gentleoscillations reflecting the oscillations in Dn(t). Again we notethat its value at the CT excited state is the correlation potential ofa gs of density Dn = 1.9626 as opposed to the exact correlationpotential which is that for an excited-state of the same density.On the other hand, the self-consistent AE potential, although itstarts correctly (both KS and interacting initial states beingground-states) and captures the relative shift between the sites,quite quickly deviates from the exact. This is because the energyof the lowest excitation of the KS system is very close to the gs(see Table 4); the system becomes increasingly degenerate asT/U - 0. This is quite in contrast to the true interacting systemwhich has Heitler–London form in the gs and a finite gapogs-e1. To open the vanishing KS gap o

sgs-e1 strong non-

adiabaticity is required in the linear response kernel;37,38 thereason is that the double excitation is nearly degenerate with thesingle excitation and thus critical to incorporate. Given that atshort times the dynamics is close to the linear response regime,this might explain why the adiabatic propagation of the os–ossystem fails so early. Given the analogous structure of the statesfor a real-space molecule composed of open-shell fragments, weexpect that also in real space a self-consistent AE calculation willlead to a very poor dipole.

V. Linear response formula

The results above show the failure of AE TDDFT to yieldaccurate CT dynamics in both the case when the CT is betweenclosed-shell sites and when it is between open-shell sites. In theformer case we noted that the KS excitation frequencies wereclose to the exact, while in the latter case they were significantlydifferent. We now ask what the AE TDDFT frequencies are ineach case, i.e. when an AE kernel is used in linear response, tocheck whether there is an indication of the bad CT dynamics ofthe AE in its predicted excitation energies.

First we derive a general expression for the TDDFT excitationenergies of the Hubbard dimer, based on the dipole–dipoleresponse function:

wD̂n;D̂nðoÞ ¼dDn

dðDn=2Þ

Dngs

; (11)

where the factor of 1/2 comes from the fact that the potential

difference couples to the dipole operator D̂n ¼ n̂L � n̂R with afactor of 1/2 in Hamiltonian in eqn (1). From the relation DnS =Dn + DnHXC, we then find a Dyson-like equation relatingwD̂n;D̂nðoÞ to the KS linear response function and the kernel:

wD̂n;D̂n�1ðoÞ ¼ w

s;D̂n;D̂n�1ðoÞ � DfHXCðoÞ; (12)

where DfHXC[Dn] = d(DnHXC[Dn]/2)/d(Dn). In the KS linearresponse function,

ws;D̂n;D̂nðoÞ ¼

Xj

Fgs

D̂n Fej � Fej D̂n Fgs �

o� ogs!ej þ iZþ c:c: (13)

there is only one term in the sum, since only one excitationcontributes, that due to the KS single-excitation e1, as thedouble-excitation e2 yields a zero numerator. At a true excitation,wD̂n;D̂nðoÞ has a pole in o, and wD̂n;D̂n

�1ðoÞ vanishes. So puttingthe right-hand-side of eqn (12) to zero, we obtain the excitationfrequencies of the interacting system from:

o2 = oS2 + 2oS|d

sgs-e1|

2DfHXC[Dn](o), (14)

where oS is the KS eigenvalue difference oS = osgs-e1. Eqn (14)

has the same form as the ‘‘small matrix approximation’’ of thereal-space TDDFT linear response eqn (10) except for a factorof 2, again due to the use of Dn as a main variable. But animportant difference is that eqn (14) is exact, since there is onlyone KS single-excitation in the Hubbard dimer. The correctionto the bare KS eigenvalue difference (second term in eqn (14)) ismost significant for os–os ground states, because, as discussedat the end of Section IV, the exact resonant frequency ofthe interacting os–os system is finite, while the resonantfrequency of the KS system is very small (bonding–antibondingtransition). As a consequence the exact, frequency-dependentkernel DfHXC[Dn](o) must be very large in the os–os case. On theother hand, if we consider the exact gs HXC kernel (shown ininset of Fig. 3), which yields the TDDFT frequency in the AEapproximation, it also becomes large around Dn = 0 (see theinset of Fig. 3). That is,

Df AEHXC ¼d DngsHXC=2� �

dDn

Dngs

¼ d2EHXC

dDn2

Dngs

(15)

has a sharp peaked structure at Dn = 0. In the limit that T/U -0, it becomes proportional to a d-function. This divergence ofthe static kernel is consistent with what is found for real os–osmolecules at large separation, ref. 39 and 38.

Using eqn (15) in eqn (14) gives the AE resonant frequencyoAE. For the Dn0 = �0.5 U os–os CT of Section IV we find theAE resonance oAEos–os = 1.1681 U overestimates the physicalresonance oos–os = 0.5228 U significantly. There is a largenon-adiabatic correction to the static kernel in this case. Onthe other hand, for the cs–cs CT of Section III, the bare KSeigenvalue difference is already a good approximation to thetrue resonance (see Table 2), and the correction due to Df AEHXCbrings the AE resonance even closer, oAEcs–cs = 0.5187 U, only0.001 U away from the true exact resonance. This is consistentwith our findings that for short times, the AE cs–cs dipolefollowed the exact one closely, while the AE os–os one did not(at short enough times the system responds in a linear way).Similarly in ref. 13 it was shown that despite relatively good LRspectra (Fig. 5), the time-resolved CT within LiCN moleculeswas not predicted by any of the approximate adiabatic func-tionals tested. The linear response performance is proved to beirrelevant for the highly non-linear process of the transfer ofone electron across a molecule.

These findings are analogous to the real-space case: hereCT excitation energies of a long-range molecule composed ofclosed-shell fragments can be well-captured by an adiabaticapproximation (e.g. ref. 40), but the non-linear process of fully

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time-resolved CT, requires a non-adiabatic approximation.When the molecule consists of two open-shell fragments, non-adiabaticity is essential even in the linear response regime.38

VI. Conclusions and outlook

The Hubbard dimer with small T/U parameters is useful forstudying real-time CT dynamics in a long-range molecule.Due to its small Hilbert space much can be done numericallyexactly or even analytically, so enabling a thorough study ofthe performance of the adiabatic approximation in TDDFT,which cannot be easily studied in real-space. In particular, weexamined here the performance of the AE propagation todescribe time-resolved CT dynamics. Although previous workon real-space molecules has shown that the usual adiabaticapproximations perform poorly,13 whether this is largely due tothe choice of gs functional or to the adiabatic approximationitself was not known. Ref. 14 showed that the AE approximationwhen evaluated on the exact density yields a step structureknown to be important in CT dynamics. This AE step hasexactly the right size in the case of CT between two open-shellatoms, where the step appears in the initial potential, but thewrong step-size for CT between two closed-shell atoms whenthe step appears in the final CT excited state. By propagating theHubbard dimer self-consistently with the AE approximation, anumerically very challenging task in real-space, we were able toshow that the AE approximation qualitatively fails to describetime-resolved CT dynamics. In the case of CT between open-shellfragments AE fails very early, and actually does not transfer anycharge. In the case of CT between closed-shells the collapse ofthe adiabatic approximation shows up later in the dynamics: theAE dipole follows the exact one for a significant part of the Rabicycle, but it drops back to its initial value way before the physicalsystem has reached the CT state. One may think that the failureis due to the AE resonant frequency being detuned from theexact one, but for the cs–cs case the AE resonance is actually veryclose to the exact resonance! Clearly memory effects are essentialto describe time-resolved CT.

In both the cs–cs CT and the os–os CT, the form of theinteracting state undergoes a fundamental change: in the cs–cscase, from approximately a single-Slater determinant initially toa double-Slater determinant of the Heitler–London type in theCT state, while the reverse occurs for the os–os case. The KSstate however remains a single Slater determinant throughout(a doubly-occupied orbital singlet state). In a sense, this is theunderlying reason for the development (or loss) of the stepstructure in the exact potential in real-space, reflected in theHubbard model by the realignment of the two sites, signifyingstrong correlation. An AE approximation does capture thisstrong correlation effect perfectly when it occurs in the gs,but our work here shows it cannot propagate well. In the cs–cscase, the AE potential was ultimately unable to develop the shiftneeded for the CT state. In the os–os case it begins with thecorrect shift but the near-degeneracy in the KS system meantthat even as soon as we begin to evolve away from the gs, the AE

approximation fails. The main features of the exact time-dependent HXC potential and the exact gs potential are analo-gous to the real-space case, in particular the relative shiftsbetween the donor and the acceptor and the relation withthe derivative discontinuity. This shift appears to be an inter-molecular step in the real-space case, but we show here thatan ‘adiabatic step’ is not enough to model the dynamics: theresults here suggest that its nonlocal dependence on both spaceand time must be modelled to yield accurate CT dynamicsin molecules.

Of course there are many aspects of a real CT within amolecule that cannot be modeled by a two-site lattice, never-theless we stress here that even for such a simple modelrelevant physics of the electronic process is missed if anadiabatic approximation is used. The impact of the step struc-ture is likely to be dampened by the effect of many electrons,three-dimensions, coupling to ionic motion, etc., but there is noreason to believe that the shortcomings of the adiabaticapproximation to describe time-resolved long-range CT startingin the ground-state will completely disappear when more com-plexity is added to the model.

We gratefully acknowledge financial support from theNational Science Foundation CHE-1152784 (NTM) and USDepartment of Energy Office of Basic Energy Sciences, Divisionof Chemical Sciences, Geosciences and Biosciences underAward DE-SC0008623 (JIF).

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Johanna I. Fuks* and Neepa T. Maitra* · 2020. 10. 2. · approximation is, what features of the exact potential are missing, etc. In a few cases, the AE potential was used to self-consistently