Chemical Physics 391 (2011) 157–163

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Chemical Physics

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Time-dependent transition density matrix

Yonghui Li, C.A. Ullrich ⇑Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA

a r t i c l e i n f o a b s t r a c t

Article history:Available online 21 February 2011

Keywords:Time-dependent density-functional theoryExcitonsTransition density matrix

0301-0104/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.chemphys.2011.02.001

⇑ Corresponding author.E-mail address: ullrichc@missouri.edu (C.A. Ullrich

The transition density matrix (TDM) is a useful tool for analyzing and interpreting electronic excitationprocesses in molecular systems. For any transition between two eigenstates of a many-body system,the TDM provides a characteristic spatial map which indicates the distribution of the associatedelectron–hole pairs and allows one to identify their delocalization and coherence lengths. This is partic-ularly useful for characterizing charge-transfer excitations in large molecular chains or light-harvestingmolecules. We here extend these concepts into the real-time domain and define the time-dependentTDM and discuss it in the context of TDDFT. An approximation is proposed in terms of the Kohn–ShamSlater determinants. This provides a new tool for the real-time visualization of electronic excitation pro-cesses such as exciton formation, diffusion, recombination, or charge separation. We illustrate the time-dependent TDM for simple one-dimensional lattice systems with two spinless electrons which are eithernoninteracting of fully interacting.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

A central objective of electronic structure theory is to describethe ground-state, or equilibrium, properties of matter. This in-volves the calculation of integrated quantities such as total ener-gies, dipole moments, magnetic moments, or polarizations. But togain a more comprehensive picture of the properties of atoms,molecules or solids, spatially local observables are also needed,e.g., to provide information on the nature of chemical bonds, thelocation of lone pairs, or to identify functional groups in molecules.For this purpose, local quantities such as the charge or spin density,stresses in the electron liquid [1], or the electron localization func-tion [2] have proved to be very useful.

Nonequilibrium processes pose additional challenges not pres-ent in the static case. The electronic response to time-dependentperturbations involves transitions to excited states; these areaccompanied by the flow of currents, dynamic polarization, thebreaking and formation of bonds, or ionization. Again, integratedglobal observables are of great interest, such as total particle num-bers or induced dipole moments and the resulting absorption oremission spectra. But even more than in the static case, interpret-ing the dynamical behavior of the system acted upon by a time-dependent perturbation requires detailed spatial visualizationtools. For instance, the time-dependent density and current den-sity, n(r, t) and j(r, t), contain information about charge rearrange-ment and current flow in a system. The time-dependent electron

ll rights reserved.

).

localization function [3] is a powerful visual tool to observe bondbreaking and formation in real time.

However, there are important questions that cannot beanswered based on these local observables alone. One of the keyaspects of electron dynamics is the creation of electron–hole pairsor excitons, their motion, and their eventual recombination or sep-aration. Many important processes in nature and technology arecharacterized by exciton dynamics; excitons play a role in organicand inorganic materials, in bulk and in the nanoscale, in polymers,quantum dots, nanotubes, or biological materials such as light-har-vesting molecules [4–9].

The challenge of calculating excitonic effects in the optical spec-tra of materials has been successfully addressed using many-bodytechniques as well as time-dependent density-functional theory(TDDFT) [10,11]. The central object in linear-response TDDFT, theexchange–correlation (xc) kernel fxc(r,r0,x), must be sufficientlylong-ranged in order to produce the electron–hole interaction thatleads to exciton formation. Much progress has been made in recentyears in developing approximations for fxc that are suitable for cap-turing excitonic effects [4,12–16].

In this paper we address a different but equally important ques-tion: assuming we have a computational approach (TDDFT orother) which correctly describes excitonic effects, how can we visu-alize the dynamics of electron–hole pairs? The transition densitymatrix (TDM) [17] has proven to be a useful tool for this purpose[18–21]. For a specific electronic excitation process connectingthe ground state W0 with an excited state Wn, the TDM can be rep-resented as a spatial map that indicates where electron–hole pairsare formed and over what spatial ranges they extend. Quantitativemeasures of excitons are their delocalization length and their

158 Y. Li, C.A. Ullrich / Chemical Physics 391 (2011) 157–163

coherence length, which can be read off the TDM in a straightfor-ward manner; this would not be possible from a map of the densityfluctuations because of a lack of nonlocality and phase information.

The goal in this paper is to introduce a time-dependent exten-sion of the TDM. As we will show, the time-dependent TDM ismore general than the stationary TDM and reduces to the latterin the appropriate limit. A rigorous definition of the time-depen-dent TDM is given in terms of many-body wave functions; withinTDDFT, it can be approximately calculated from the time-depen-dent Kohn–Sham Slater determinant. The time-dependent TDM al-lows one to visualize exciton dynamics in real time: the formationof electron–hole pairs following a short-pulse excitation, the mo-tion of electron–hole pairs, and their recombination or charge sep-aration. We will illustrate this with simple lattice examples.

This paper is organized as follows. In Section 2 we define thestationary and time-dependent TDM and explain their physicalmeaning. Section 3 makes a connection with TDDFT and givesapproximate expressions of the TDM in terms of Kohn–Sham orbi-tals. Section 4 presents numerical examples for one-dimensionallattice systems. We conclude in Section 5. Hartree atomic units(m = e = ⁄ = 1) are used throughout.

2. Definition of the TDM

2.1. Stationary TDM

Consider an interacting N-electron system such as an atom or amolecule, with many-body Hamiltonian bH0 and eigenstates Wn,wherebH0Wn ¼ EnWn; n ¼ 0;1;2; . . . ð1Þ

Let W0 be the electronic ground state of the system; all other statesWn with n > 0 are the excited states.

The TDM associated with an electronic transition between theground state and the nth excited state can be defined as follows[22]:

Cnðr; r0Þ ¼ hWnjqðr; r0ÞjW0i; ð2Þ

where qðr; r0Þ ¼ wyðr0ÞwðrÞ is the reduced one-particle density ma-trix operator, written here in second quantization in terms of theusual fermionic field operators.

2.2. Time-dependent TDM

Let us now consider a situation in which the system is in itsground state for all times t < t0, and a time-dependent perturbationis switched on at the initial time t0. The Hamiltonian of the systemthen becomes time-dependent:bHðtÞ ¼ bH0 þ kbH1ðtÞhðt � t0Þ; ð3Þ

where k is a parameter which can be chosen to be small. The systemobeys the time-dependent many-body Schrödinger equation

i@

@tWðtÞ ¼ bHðtÞWðtÞ; ð4Þ

with W(t0) = W0. We define the time-dependent TDM as follows:

Cðr; r0; tÞ ¼ hWðtÞjqðr; r0ÞjW0e�iE0ti: ð5Þ

In the absence of any perturbation (bH1 ¼ 0) we trivially haveC(r,r0, t) = C0(r,r0), the ground-state density matrix. This means thatno transition is taking place at all. For finite perturbations we ex-pand the many-body wave function at time t > t0 in terms of theeigenstates of the unperturbed Hamiltonian bH0:

WðtÞ ¼X

k

ckðtÞWke�iEkt ; ð6Þ

with ck(t0) = dk0. With this, the time-dependent TDM can be writtenas

Cðr; r0; tÞ ¼X

k

c�kðtÞCkðr; r0ÞeiðEk�E0Þt; ð7Þ

i.e., it is a superposition of stationary TDMs, each one multipliedwith their respective time-dependent phase factors and scaled withthe complex conjugates of the expansion coefficients ck(t). The diag-onal of the TDM, C(r,r, t), contains the time-dependent density fluc-tuations. Clearly, we haveZ

d3r Cðr; r; tÞ � C0ðr; rÞ½ � ¼ 0; ð8Þ

due to particle conservation, where the ground-state density is gi-ven by C0(r,r) = n0(r).

To see how the time-dependent TDM reduces to the stationaryTDM in the appropriate limit, we use first-order time-dependentperturbation theory. The wave function is expanded as

WðtÞ ¼ W0e�iE0t þ kX

k

cð1Þk ðtÞWke�iEkt þOðk2Þ; ð9Þ

where

cð1Þk ðtÞ ¼ �iZ t

t0

dt0hWkjbH1ðt0ÞjW0ieiðEk�E0Þt: ð10Þ

With this, the time-dependent TDM becomes

Cðr; r0; tÞ ¼ C0ðr; r0Þ þ kX

k

cð1Þ�k ðtÞCkðr; r0ÞeiðEk�E0Þt þOðk2Þ: ð11Þ

Now choose the special case of a perturbation of the formbH1ðtÞ ¼ v1ðrÞf ðtÞ sin½ðEn � E0Þt�, where f(t) is some envelope func-tion (e.g., a Gaussian pulse), and the matrix element hWnjbH1ðtÞjW0iis nonvanishing. The first-order term in Eq. (11) then reduces toCn(r,r0) after Fourier transformation, in the long-pulse limit wherethe perturbation becomes monochromatic.

Let us emphasize that even though we have made use of lowest-order perturbation theory in the argument above, the time-depen-dent TDM is in general not restricted to this limit and can be de-fined, in principle, for perturbations of arbitrary strength.

2.3. Physical meaning

Let us now discuss the physical meaning of the time-dependentTDM. A well-known related quantity is the state-to-state transitionamplitude [23]

Si;f ðtÞ ¼ hWðtÞjWf i; ð12Þ

where it is assumed that the system starts from an initial eigenstateWi. The transition amplitude is the overlap of the time-dependentwave function with some final eigenstate Wf of the (unperturbed)many-body system. One usually considers the limit t ?1, assum-ing that the perturbation only lasts for a finite time; this thendefines the so-called S-matrix. The state-to-state transition proba-bility associated with the particular process under study is obtainedas

Pi;f ðtÞ ¼ hWðtÞjWf i�� ��2: ð13Þ

It expresses the probability that a system which starts out from agiven initial eigenstate can be found in a given eigenstate at time t.

Let us now consider the special case where the system is ini-tially in the ground state and where the final state can be charac-terized as a pure single-particle excitation, jWf i ¼ cyj ckjW0i.Carrying out a unitary transformation from the creation/annihila-tion operators cyj and ck (which are defined with respect to some

Y. Li, C.A. Ullrich / Chemical Physics 391 (2011) 157–163 159

single-particle basis {u}) to the field operators, we can rewrite thetransition amplitude as

Si;f ðtÞ ¼Z

d3rZ

d3r0u�kðrÞujðr0ÞhWðtÞjwyðr0ÞwðrÞjW0i: ð14Þ

The transition amplitude thus emerges as a continuous sum of pro-cesses where a particle is destroyed at position r and recreated atposition r0, with amplitudes

Sðr; r0; tÞ ¼ hWðtÞjwyðr0ÞwðrÞjW0i: ð15Þ

In addition, each such process is weighted in Eq. (14) by the single-particle orbitals which are involved in the specific transition understudy. This establishes the desired connection with definition (5) ofthe time-dependent TDM:

Cðr; r0; tÞ ¼ Sðr; r0; tÞe�iE0t : ð16Þ

In light of this discussion, the TDM has the physical meaning of aspatial map which shows how electrons move between positionsr and r0 during a dynamical process.

The time-dependent TDM C(r,r0, t) is a complex quantity; in or-der to visualize it graphically, it is convenient to take its modulussquare and subtract from it the square of the ground-state densitymatrix. In other words, we consider the following object, which weshall call the TDM difference modulus:

DPðr; r0; tÞ ¼ jCðr; r0; tÞj2 � jC0ðr; r0Þj2: ð17Þ

A very simple yet instructive illustration is provided by a single par-ticle in a one-dimensional box of length 1. Let the time-dependentwave function be

Wðx; tÞ ¼ W0ðxÞe�iE0t þ kW1ðx; tÞe�iE1t ; ð18Þ

where W0 and W1 are the ground and first excited state of the par-ticle in the box. To first order in k we find

DPðx; x0; tÞ ¼ 8k sinðpx0Þ sinð2px0Þ sin2ðpxÞ cos3p2

2t

� �: ð19Þ

A snapshot of his function is plotted in Fig. 1. The TDM differencemodulus is a two-dimensional map, consisting of a broad maximumand a broad minimum, which indicates the transfer of electronicprobability from one side of the box to the other. This, in and by it-

Fig. 1. Time-dependent TDM for a single electron in a box, see Eq. (19), indicatingthe characteristic length scales of a particle-hole pair.

self, would be a rather unremarkable piece of information, andcould have simply been obtained from the density fluctuations.However, as seen in Fig. 1, the two-dimensional map associatedwith the TDM contains much more information on the characterand behavior of the electron–hole pairs involved in the dynamics.

To see this, we introduce two length scales: the coherencelength and the delocalization length [19]. The coherence lengthindicates the typical separation (or the ‘‘size’’) of the electron–holepair which is involved in the transition. It can be defined as the dis-tance between the respective ‘‘center of mass’’ of the electron andthe hole distribution. The delocalization length indicates the typi-cal range of positions where the electron–hole pair can be found.Due to symmetry, the electron and the hole are equally delocal-ized. The horizontal axis of Fig. 1 tells us where the electron inthe electron–hole pair is likely to come from, and the vertical axistells us where it is likely to go during the transition. As we will seebelow, these two length scales are very useful to characterize andinterpret the electron dynamics.

3. The TDM in TDDFT

3.1. TDDFT formalism

TDDFT is a formally exact approach to the time-dependentmany-body problem [10,11]. According to it, all physical observ-ables can, in principle, be expressed as functionals of the time-dependent density n(r, t) (assuming the system starts from theground state; otherwise there is an additional dependence on theinitial state). In practice, one works with the time-dependentKohn–Sham (TDKS) formalism:

i@

@tujðr; tÞ ¼ �r

2

2þ vðr; tÞ þ vHðr; tÞ þ vxc½n�ðr; tÞ

!ujðr; tÞ;

ð20Þ

where vH is the time-dependent Hartree potential and vxc[n] is thetime-dependent xc potential, which is a functional of the densityand has to be approximated in practice. The TDKS orbitals repro-duce, in principle, the exact time-dependent density:

nðr; tÞ ¼XN

j¼1

jujðr; tÞj2: ð21Þ

The TDKS equation has to be propagated self-consistently in time.Formally, it is an initial value problem: uj(r, t0) = uj(r), where theinitial orbitals follow from the static Kohn–Sham equation:

�r2

2þ v0ðrÞ þ vHðrÞ þ vxc½n0�ðrÞ

!ujðr; tÞ ¼ ejujðrÞ; ð22Þ

where vxc[n0](r) is the static xc potential and a functional of theground-state density n0(r).

One thus proceeds in two steps: first, determine the Kohn–Sham ground state by solving Eq. (22), and then carry out the timepropagation of Eq. (20) under the influence of the externalpotential

vðr; tÞ ¼ v0ðrÞ þ v1ðr; tÞhðt � t0Þ: ð23Þ

All physical observables of interest should be calculated from thetime-dependent density (21); the TDKS orbitals uj(r, t), and the Sla-ter determinant that can be constructed from them, are merely aux-iliary quantities and have no rigorous physical meaning. However,this caveat is often ignored in practice: there are many situationswhere observables of interest are only an implicit density function-als but can be easily expressed in terms of the orbitals. As we shallsee, the TDM is such an example.

160 Y. Li, C.A. Ullrich / Chemical Physics 391 (2011) 157–163

3.2. Linear-response TDDFT and stationary TDM

The most common application of TDDFT is the calculation ofexcitation energies [24]. Formally, the linear density responsecan be rigorously calculated with TDDFT [25], and this can thenbe used to extract excitation energies in principle exactly [26]. Inpractice, most calculations of excitation energies are done usingthe Casida formalism [27]:

A KK� A�

� �XY

� �¼ Xn

�1 00 1

� �XY

� �; ð24Þ

where the matrices A and K are defined as follows:

Aia;i0a0 ¼ dii0daa0 ðea � eiÞ þ Kia;i0a0 ð25Þ

and

Kia;i0a0 ¼Z

d3rZ

d3r0u�i ðrÞuaðrÞ

� 1jr� r0j þ fxcðr; r0;xÞ� �

ui0 ðr0Þu�a0 ðr0Þ: ð26Þ

We use the standard convention that i, i0, . . . and a,a0, . . . are indicesof occupied and unoccupied orbitals, respectively. fxc(r,r0,x) is theso-called xc kernel. It is a frequency-dependent quantity, but inpractice it is often approximated using frequency-independentexpressions, which is known as the adiabatic approximation.

Eq. (24) represents an anti-Hermitian eigenvalue problem,where the eigenvalues Xn are in principle the exact excitationenergies. From the eigenvectors one obtains the density fluctua-tions associated with a given excitation Xn:

n1ðr;XnÞ ¼X

ia

uiðrÞu�aðrÞXiaðXnÞ�

þu�i ðrÞuaðrÞYiaðXnÞ�; ð27Þ

which are also known as transition densities. They are identicalwith the diagonal of the transition density matrix:

n1ðr;XnÞ ¼ Cnðr; rÞ: ð28Þ

We can now define the Kohn–Sham TDM as follows [22]:

CKSn ðr; r0Þ ¼

Xia

uiðr0Þu�aðrÞXiaðXnÞ þu�i ðrÞuaðr0ÞYiaðXnÞ� �

: ð29Þ

Unfortunately, in general we have CKSn ðr; r0Þ – Cnðr; r0Þ for r – r

0,

even if the exact Kohn–Sham orbitals and the exact excitation ener-gies and eigenvectors Xia and Yia were used in Eq. (29). This is be-cause the Kohn–Sham Slater determinant, and the reduceddensity matrix following from it, are not meant to reproduce the ex-act many-body wave function, and can therefore in general not beexpected to even be good approximations.

In spite of this, the Kohn–Sham TDM, CKSn ðr; r0Þ, has been suc-

cessfully used to describe excitation processes in a wide varietyof systems [18–20], giving valuable information on charge transferand exciton dynamics.

3.3. Time-dependent TDM

The time-dependent TDM (5) is an implicit functional of thetime-dependent density. The simplest approximation is to replacethe full many-body wave functions W0 and W(t) by the respectiveKohn–Sham Slater determinants U0 and U(t). This yields

CKSðr; r0; tÞ ¼ hUðtÞjqðr; r0ÞjU0e�iEKS0 ti; ð30Þ

where EKS0 ¼

Poccj ej. We can carry out an expansion of U(t) in terms

of the complete set of Kohn–Sham Slater determinants:

UðtÞ ¼X

k

bkðtÞUke�iEKSk t ; ð31Þ

with bk(t0) = dk0. In practice one works with the time-dependentsingle-particle Kohn–Sham orbitals rather than with the Slaterdeterminant U(t). A connection can be made by expanding

ujðr; tÞ ¼X

l

cjlðtÞulðrÞe�iel t: ð32Þ

The index k in Eq. (31) thus becomes a compound index, k= {k1,k2, . . . ,kN}, indicating the configuration of the Uk, and weobtain

bkðtÞ ¼ det cikiðtÞ

ð33Þ

and

EKSk ¼

XN

i¼1

eki: ð34Þ

With this, the TDKS TDM can be written as

CKSðr; r0; tÞ ¼X

k

b�kðtÞhUkjqðr; r0ÞjU0ieiðEKSk �EKS

0 Þt: ð35Þ

The matrix elements hUkjqðr; r0ÞjU0i, however, are only nonvanish-ing for singly-excited Kohn–Sham Slater determinants Uk. Thus,we obtain

CKSðr; r0; tÞ ¼ b�0ðtÞX

i

uiðrÞu�i ðr0Þ

þX

ia

fiab�iaðtÞuiðrÞu�aðr0Þeiðea�eiÞt; ð36Þ

where, as before, i and a are indices running over occupied andunoccupied orbitals, and fia is +1 or �1, depending on the orbitalconfiguration of Uk.

The TDKS TDM CKS(r,r0, t) reduces to the stationary Kohn–ShamTDMs CKS

n ðr; r0Þ using a similar procedure as for the exact case, i.e.,applying a time-dependent perturbation whose driving frequencycorresponds to an excitation energy of the many-body system(which can be obtained in principle exactly using linear-responseTDDFT, see previous section), and carrying out a Fourier transformof the first-order change of CKS(r,r0, t).

Like in the exact case, the quantity we will actually plot is theTDM difference modulus:

DPKSðr; r0; tÞ ¼ jCKSðr; r0; tÞj2 � jCKS0 ðr; r0Þj

2: ð37Þ

4. One-dimensional lattice examples

In this section we will look at some numerical examples on one-dimensional lattices containing two spinless electrons to illustratethe performance of the time-dependent TDM and the kind of infor-mation about real-time electron dynamics one can extract from it.The focus will not be on TDDFT (we will treat the electrons eitheras completely noninteracting, or as fully interacting), but on under-standing what the TDM can tell us.

4.1. Triple-well systems

Fig. 2 shows two triple-well systems of unequal well widths anddifferent depths, together with the first three single-particle eigen-functions. The two systems are given on a linear 30-point lattice.To obtain the eigenfunctions, the noninteracting single-particleSchrödinger equation was discretized using a standard three-pointfinite-difference scheme [28].

Two situations are considered: the electrons are either com-pletely noninteracting, or they interact via a soft-core Coulombinteraction of the type wðxÞ ¼ 1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ 1p

. In both cases the initialground state is obtained in the presence of a weak linear potentialv(x) = 0.01x, which is suddenly switched off at the initial time t0.

a

b

Fig. 2. Lowest three single-particle wave functions (empty circles, squares, and fullcircles) on a 30-point lattice with a triple-well potential of depth (a) 5 a.u. and (b)2 a.u.

Fig. 3. Snapshots of the time-dependent TDM for two spinless electrons on a 30-point lattice (left: noninteracting, right: interacting) in the deeper triple wellpotential of Fig. 2 (a). The faint lines indicate the positions of the wells.

Y. Li, C.A. Ullrich / Chemical Physics 391 (2011) 157–163 161

The nonequilibrium initial state on the lattice is then propagated intime without any further time-dependent perturbation.

In the noninteracting case, the time propagation of the single-particle orbitals on the grid is carried out using a standardCrank–Nicholson algorithm. In the interacting case we expandthe two-electron wave function in a complete basis of two-electronSlater determinants on the 30-point grid (i.e., ground state plus 56single and 378 double excitations), and then carry out static andtime-dependent configuration-interaction (CI) calculations.

Figs. 3 and 4 show snapshots of the time-dependent TDM forthe noninteracting and the interacting case. The snapshots are ta-ken at times 0.72 a.u., 1.68 a.u., 3.44 a.u., and 4 a.u. For comparison,a characteristic time scale can be defined as T = 2p/(e2 � e1), featur-ing the difference of the lowest two single-particle eigenvalues ofthe system. For the wells of depth 5 a.u. and 2 a.u. we haveT = 13.8 a.u. and T = 24.8 a.u., respectively. The snapshots shownin Figs. 3 and 4 thus take place very soon after the initial suddenswitching. This is mainly because we are interested in capturingany transient behavior immediately after the initial switching. Atlater times, the dynamics shown in these snapshots basically re-peats itself.

Let us first consider the deeper triple well system of Fig. 3. Herethe interpretation is straightforward: there are two distinct andspatially separated excitation processes, one taking place withinthe first, broadest well, and the other one between the secondand the third well. The TDM maps thus contain two distinct re-gions in which these excitations are localized, and they do not ex-hibit any coherence with each other as can be seen from theabsence of any signatures in the upper left and lower right parts

of the maps. Another way of putting it is to say that one can definetwo separate, nonoverlapping coherence regions for each of thetwo excitations. We also see that the interacting and the noninter-acting calculations give essentially identical results; the reason isthat in the deeper wells the quantum confinement is dominantover the Coulomb interactions.

The shallower wells offer a somewhat richer picture, see Fig. 4.Here we find that the excitations are much more delocalized andall three wells talk to each other. This in and by itself is not verysurprising; but the time-dependent TDM shows how initially (at0.72 a.u.) the excitations are still very localized, with nonoverlap-ping coherence regions extending within the first well and be-tween the second and third well, respectively. At 4 a.u., a featurein the upper left corner indicates that the excitation that begunin the first well has now spread over the entire system.

We also find that the excitation between the second and thirdwell (see the features on the upper right part of the maps) isnow much more spread out horizontally, but the resulting elec-tron–hole pair is not more delocalized than previously for the dee-per well. An explanation can be given by referring to Fig. 2: a

Fig. 4. Same as Fig. 3, but for the less deep triple-well potential, see Fig. 2 (b).

162 Y. Li, C.A. Ullrich / Chemical Physics 391 (2011) 157–163

substantial portion of the second single-particle wave function is inthe third well; therefore, if an electron–hole pair is established inwells 2 and 3, there is a substantial probability that the electronoriginated in the third well.

Again, we find that the fully interacting results, following froman essentially exact time-dependent CI calculation, agree very wellwith the noninteracting results. Apparently the confinement by thequantum wells of 2 a.u. depth is still relatively strong compared tothe Coulomb interaction. The excitations seem a bit more pro-nounced in the interacting case; due to the electron–electronrepulsion, the two electrons tend to be more separated, and thusare a bit more strongly affected by the initial linear potential.

Fig. 5. Snapshots of the time-dependent TDM for two spinless noninteractingelectrons on a 100-point lattice with 12 quantum wells of depth 0.5 a.u., indicatedby the faint lines. The system undergoes a sudden initial excitation that is localizedin the leftmost well.

4.2. Multiple wells

In the next example we consider two spinless noninteractingelectrons on a one-dimensional 100-point lattice featuring 12identical, equidistant quantum wells of depth 0.5 a.u. The systemis initially in its ground state. At time t0 we switch on a localizedperturbation that affects only the leftmost well, reducing its depthabruptly to 0.495 a.u. The system is then propagated in time for

400 a.u.; the characteristic time scale for this system (defined asin the previous section) is T = 595 a.u.

Ten snapshots of the time-dependent TDM for the multiple wellsystem are shown in Fig. 5. Initially, as expected, the excitation isconfined to within the left end of the lattice. However, we observethat the excitation very rapidly spreads out: around t = 100 a.u. itappears as if an electron–hole pair begins to form at the otherend of the chain (see upper right portion of the panels). The dipolemoments associated with the two electron–hole pairs are aligned.At t = 350.4 a.u., the electron–hole pair at the right end seems toturn around and is fully reversed in the following panel. It then be-

Y. Li, C.A. Ullrich / Chemical Physics 391 (2011) 157–163 163

gins to merge back with the original electron–hole pair at the leftend.

From the panel at time 153.6 a.u. one sees very clearly that thecoherence of the excitation extends over the entire chain. This isindicated by the presence of off-diagonal features in the TDM inthe upper left and lower right corners. As a result, one can definetwo coherence regions which overlap; this is in contrast to thetwo separate excitations that were visible in Fig. 3.

From the time-dependent TDM maps one can identify two waysin which an excitation travels along the chain: (1) A given elec-tron–hole pair increases in size, i.e., its coherence length grows.This shows up as changes along the vertical direction of the TDMmaps (see, for instance, the left portions of the first few panels).(2) Structures along the diagonal direction (see, e.g., the panels att = 59.2 or 400 a.u.) indicate that a disturbance propagates alongthe chain, with a fixed coherence length. The mechanism of prop-agation is that the presence of a localized electron–hole pair willinduce the formation of another pair in the neighboring region,and so on.

The electronic system we consider here is still very much con-fined; truly excitonic effects, such as exciton diffusion, would beginto manifest only for much longer chains [13,20,21]. The requiredelectron–hole attraction is of course only present in systems withsufficiently long-ranged electron interactions, as discussed in theintroduction. We would then expect the TDM to be more concen-trated along the diagonal, depending on the strength of the excitonbinding.

5. Conclusions

In this paper we have introduced a new concept for visualizingelectronic excitation processes: the time-dependent TDM. The def-inition of the time-dependent TDM involves the many-bodyground state and the time-dependent many-body wave functionfollowing an excitation process triggered by a time-dependent per-turbation. In the appropriate limit, the time-dependent TDM re-duces to the well-known stationary TDM.

In practice, fully correlated time-dependent many-body statesare only available for very simple systems, such as two electronsin one dimension, where a time-dependent CI can be carried outor where the two-particle Schrödinger equation can be solvednumerically. Alternatively one could obtain the time-dependentTDM by solving the Kadanoff–Baym equation for the nonequilibri-um Green’s function [29,30] or from time-dependent density-ma-trix functional theory [31]. For more than two electrons thisbecomes increasingly difficult.

In TDDFT, the time-dependent TDM belongs to a class of observ-ables which are implicit density functionals. It is, however,straightforward to express the TDM in terms of TDKS Slater deter-minants, as we have done here, but this approximation is by nomeans guaranteed to be successful, even if the exact TDKS orbitalswere known. There exist other situations in TDDFT where wave-function based approximations have been carried out for implicitdensity functionals, with mixed results. For instance, certain highlycorrelated double ionization processes cannot be described by cal-culating the ionization probabilities directly from the TDKS Slaterdeterminants [32,33].

It is therefore important to carry out benchmark calculations forrelatively simple systems to assess the validity of the TDKS approx-imation for the time-dependent TDM. Here, we have seen that forone-dimensional quantum-confined systems, with weak perturba-

tions, even a completely noninteracting description seems to workreasonably well. However, this is certainly no longer the case inmore extended systems where electron–hole interaction is neededto form excitons. Extensive TDDFT studies and comparisons withhigh-quality benchmarks will be necessary to get a more completepicture.

The main emphasis of this paper, however, was a pedagogicalone. Our goal was to visualize and interpret excitation processesin real time, to see how fast an excitation forms and spreads out,and how different spatial regions of a system are coherently con-nected. The time-dependent TDM appears to be nicely suited forthis purpose, and allows us to read off relevant information suchas coherence lengths and delocalization lengths of electron–holepairs in a relatively simple and intuitive manner.

The true usefulness of the time-dependent TDM is likely toemerge when we apply the concept to simulations of ultrafast exci-tation processes in more realistic molecular chains or donor–acceptor systems, using real-time TDDFT propagation methods.The time-dependent TDM can then be expected to yield valuableinformation on excitonic or charge-transfer excitation processeswhich cannot be obtained from more conventional observablessuch as dipole moments or density fluctuations. TDDFT simulationson molecular chains are currently in progress and will be reportedelsewhere.

Acknowledgement

This work was supported by NSF Grants DMR-0553485 andDMR-1005651.

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