PHYSICAL REVIEW E 85, 016705 (2012)

Solution of the time-dependent Schrodinger equation using time-dependent basis functions

Kalman VargaDepartment of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA

(Received 11 September 2011; revised manuscript received 8 December 2011; published 9 January 2012)

The time-dependent Schrodinger equation is solved by using an explicitly time-dependent basis. This approachallows efficient reflection-free time propagation of the wave function. The applicability of the method is illustratedby solving various time-dependent problems including the calculation of the above threshold ionization of a modelatom and the optical absorption spectrum of a sodium dimer.

DOI: 10.1103/PhysRevE.85.016705 PACS number(s): 02.60.−x, 03.65.−w

I. INTRODUCTION

With the advance of time-dependent probes [1], theinvestigation of the dynamical behavior of matter at thenanoscale has become a very important research direction[2–11]. Comprehensive knowledge of the dynamic behaviorof electrons and ions in condensed-matter systems is pertinentto the development of many modern technologies, such assemiconductor and molecular electronics, optoelectronics, in-formation processing, and photovoltaics. A theoretical descrip-tion of experiments where the quantum system interacts withtime-dependent fields, e.g., photons, involves a Hamiltonianthat explicitly depends on time, and to obtain dynamicalinformation, one has to solve the time-dependent Schrodingerequation (TDSE).

Due to its paramount importance, many different methodshave been developed for efficient numerical solutions of theTDSE [12–27]. These approaches are not restricted to theTDSE, but also are applicable to related equations, suchas the time-dependent Kohn-Sham [25] or time-independentGross-Pitaevskii equations [17,28].

The TDSE most often is solved by time propagating thewave functions. In the time propagation, the wave function isexpanded in a suitable basis,

ψ(r,t) =∑

k

ck(t)φk(r), (1)

where the basis functions are time independent and the linearcombination coefficients are time dependent. By substitutingthis ansatz into the TDSE,

ihψ(r,t) − Hψ(r,t) = 0, (2)

one obtains the following equation:∑k

(ihck − ckH )φk(r) = 0. (3)

Various basis functions are employed in numerical calcula-tions, including plane waves, atomic orbitals, and real spacegrids. In solving quantum mechanical problems, one oftenadopts basis functions, that is, conforms to some properties ofthe potential, e.g., plane wave basis for periodic potentials oratomic orbitals to tame Coulomb potentials of the ionic cores.

The goal of this paper is to demonstrate that time-dependentbasis (TDB) functions are natural and efficient tools to solvethe TDSE. We will show that, by assuming that both thebasis functions and the linear expansion coefficients dependon time, one can derive a time propagation scheme in which

the coefficients are time developed by the potential energywhile the basis functions are evolved with the kinetic energy.This separation allows faster and more efficient computations.The most important advantage of the TDB is that, as the basisfunctions are known analytically in space and time, there is noartificial reflection on the boundaries, and smaller simulationcells can be used.

II. FORMALISM

We will assume that the Hamiltonian can be divided intotwo parts H = H0 + H1. The wave function will be expandedin terms of TDB functions φk(r,t),

ψ(r,t) =∑

k

ck(t)φk(r,t), (4)

where we assume that the linear combination coefficients aretime dependent as well. It will be shown that allowing time de-pendence of both the coefficient and the basis functions greatlyenhances the flexibility of the expansion. By substituting thisansatz into the TDSE, one has∑

k

(ihck − ckH1)φk(r,t) (5)

+∑

k

ck[ihφk(r,t) − H0φk(r,t)] = 0. (6)

In this equation, the first term is similar to the TDSE for thetime-independent basis function Eq. (3), while one obtainsa TDSE for the basis functions in the parentheses of thesecond term. With this separation, the basis functions are timedeveloped by a Hamilton operator H0 while the wave functionis time propagated by H1 on the TDB. By using a basis functionthat satisfies the TDSE with Hamiltonian H0, one ends up withan equation, ∑

k

(ihck − ckH1)φk(r,t) = 0, (7)

and our freedom in splitting H into two parts can now be usedto choose H1 to facilitate an efficient solution of Eq. (7). Apossible choice is

H0 = T , H1 = V (t), (8)

where T is the kinetic energy and V is the potential. In thiscase, the basis functions are propagated by the interaction-free Hamiltonian, and the wave function is propagated by theinteraction on the TDB. This splitting is somewhat similar tothe interaction picture where the state vectors are propagated

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KALMAN VARGA PHYSICAL REVIEW E 85, 016705 (2012)

by the interaction-free Hamiltonian and the time evolution ofstates is governed by the interaction.

Equation (7) can be solved by short time propagation inthe same way as in conventional TDSE calculations [12–27].By choosing a sufficiently short time step �t , during whichthe change in the potential V (t) and the basis functionφk(r,t) is negligibly small, multiplying Eq. (7) by φj (r,t),and integrating over the coordinates, we have∑

k

(ihckSjk − ckVjk) = 0, (9)

where

Sjk = 〈φj |φk〉, Vjk = 〈φj |V |φk〉. (10)

From this equation, ck(t + �t) can be determined by theRunge-Kutta method or by using the evolution operator,

C(t + �t) = e−(i/h)S−1V �tC(t), (11)

where

C(t) = [c1(t),c2(t),. . .]. (12)

Once the coefficients are determined, the wave function is timepropagated by using the expression,

ψ(r,t + �t) =∑

k

ck(t + �t)φk(r,t + �t). (13)

A. Basis functions

One either can use basis functions that satisfy the TDSEwith Hamiltonian H0 = T or can generate a TDB by timepropagation. In the latter case, each basis function has tobe time propagated independently as shown by Eq. (6).Alternatively, one can adopt basis functions whose timeevolution with the free particle Hamiltonian is known.

In this paper, we use a localized Gaussian as the basis,

φk(r,t) =[

σ√π (1 + iσ 2t)

]3/4

exp

[−σ 2

2

(r − sk)2

1 + iσ 2t

], (14)

where sk is the position of the center and σ is the width of thebasis function. The centers of the Gaussian basis are distributedequidistantly on a grid, for example,

sk = a + (k − 1)�s, k = 1,. . .,N, (15)

in one-dimensional problems. This can be generalized straight-forwardly for three-dimensional cases. The centers of theGaussians are distributed in a finite region of space, andsometimes this basis is called the distributed Gaussian basis[29]. Time-dependent Gaussian basis functions also have beenused in the boundary-free propagation approach as describedin Ref. [30].

III. NUMERICAL RESULTS AND DISCUSSION

A. Propagation of a Gaussian wave packet

As a first test example, we have used the TDB to propagatea one-dimensional wave packet. The time development of a

-20 -10 0 10 20x (a.u.)

-0.2

0

0.2

0.4

ψ(x

,t)

FIG. 1. (Color online) Time propagation of a Gaussian wavepacket. A Gaussian wave packet (with parameters σ = 0.5 and k = 1,centered at −2 a.u.) is propagated for t = 2 a.u. with time step�t = 0.1 a.u. The solid and dashed lines show the real and imaginaryparts of the propagated wave packet.

Gaussian wave packet is given by

�(x,t) =[

σ√π (1 + iσ 2t)

]1/4

exp

[−σ 2

2

(x − s − kt)2

1 + iσ 2t

]

× exp

[ik

(x − s − kt

2

)], (16)

where k is the momentum and s is the initial center of the wavepacket. To test the TDB approach, we expand �(x,0) using thebasis functions defined in Eq. (14) and time propagate it usingEqs. (11) and (13). N = 100 basis functions, equidistantlyplaced in the [−a(t = 0),a(t = 0)] interval, are used to expandthe wave packet [a(t = 0) = 5 a.u.]. The interval [−a(t),a(t)],covered by the basis functions, grows as the basis functions aretime propagated, but the centers of the basis functions are fixedin the original [−a(t = 0),a(t = 0)] region. In other words, theboundary of the simulation cell [−a(t),a(t)] is time dependentand increases with time. The calculated wave function usingthese basis functions is shown in Fig. 1, and it is in perfectagreement with the analytical form.

The time propagation of the wave packet for a long timeperiod t = 1000 is shown in Fig. 2. In this case, the originalwave packet, which is confined to the [−a(t = 0),a(t = 0)]region, is propagated for t = 1000, and it spreads out to aboutb(t = 1000) = 5000 a.u., where b(t) is the front of the wavepacket at time t . The agreement of the calculated and analyticalwave packets is perfect. At t = 0, the basis functions Eq. (14),whose centers are equidistantly distributed, are only nonzeroin the [−a(t = 0),a(t = 0)] interval. At t = 1000, the basisfunctions spread out up to a(t = 1000), far beyond the regionwhere their center is located and accurately represent thetime-propagated wave function without any reflection. In thisexample, the basis functions spread out faster than the wavepacket b(t) < a(t), and the time-propagated basis covers aninterval that is larger than the spread of the wave packet. Thatis why there is no reflection. Obviously, in the opposite case, ifthe time-propagated basis would not cover a sufficiently largeregion, the spreading wave packet would reach the boundary,and it would be reflected. This can be avoided by a suitablechoice of basis parameters.

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-4 -2 0 2 40

0.5

1

1.5

2|ψ

(x)|2

-5000 0 5000x (a. u.)

0

1×10-4

2×10-4

|ψ(x

)|2

FIG. 2. (Color online) Time propagation of a Gaussian wavepacket. A Gaussian wave packet with σ = 0.5, k = 1, and s = 0is propagated for t = 1000 a.u. with time step �t = 0.1 a.u. Theupper panel shows two successive basis functions (in the left) andthe wave packet at t = 0 (in the middle). The bottom panel showsthe numerically propagated (solid line) and analytical (circles) wavepacket. The basis functions are multiplied by 1/4.

B. Calculation of bound states by imaginary time propagation

For the next example, we show that the TDB also can beused in imaginary time propagation. By letting t = −iτ in theTDSE, one obtains the imaginary time Schrodinger equation,

h∂ψ

∂τ= −Hψ, (17)

which has the formal solution,

ψ(x,τ ) = e−Hτ/hψ(x,0). (18)

Using the complete set of states formed by the eigensolutionsof H, Hϕn = Enϕn, an arbitrary initial state ψ(x,0) can beexpanded as

ψ(x,0) =∑

n

cnϕn, cn = 〈ϕn|ψ(x,0)〉. (19)

By substituting this expansion into the formal solution of theimaginary time Schrodinger equation, we get

ψ(x,τ ) = e−Hτ/h∑

n

cnϕn =∑

n

cne−Enτ/hϕn. (20)

This equation shows that, for large τ , each eigenfunctionrelaxes to zero at a rate that is proportional to its eigenvalue.This means that the ground state, which relaxes most slowly,persists. After a time τ , the component of the eigenfunction ϕn

is reduced relative to the ground state by the ratio e−(En−E0)τ/h.Propagating an arbitrary initial wave function in imaginarytime, therefore, projects out the ground state to desiredaccuracy.

As an example, we have calculated the ground state wavefunction of a one-dimensional potential well (see Fig. 3). AGaussian,

ψ(x,0) = �(x,0), (21)

with k = 0 and σ = 0.1, is used as an initial trial function.This initial trial wave function is propagated in imaginarytime using the TDB functions with t = −iτ . The number of

-5 0 5x (a. u.)

-6

-5

-4

-3

-2

-1

0

ψ(x

)

FIG. 3. (Color online) Ground state eigenfunction (multiplied by20) of a potential well calculated by imaginary time propagation(solid line) and analytically (dots). The dashed line shows the initialtrial function. The potential well also is shown. Atomic units are used.

TDB functions used in the calculation is N = 20, a = −5 a.u.,and �s = 0.5 a.u. in Eq. (15). Figure 3 shows that thetime-propagated wave function converges to the exact groundstate wave function proving the applicability of the TDB inimaginary time propagation.

C. Calculation of above threshold ionization

In the next example, we will calculate the above thresholdionization (ATI) of a model hydrogen atom. This is a verychallenging problem because the description of ionizationrequires a very extended spatial basis. The time-dependentHamiltonian for a one-dimensional model atom in a laser fieldis

H (x,t) = H ′(x,t) − xE(t), H ′(x,t) = −1

2

d2

dx2+ V (x),

(22)

where E(t) is the time-dependent laser field,

E(t) ={E0sin2

(πt2Tc

)sin ωt, if 0 � t � Tc,

E0sin ωt, t > Tc.(23)

The parameter Tc determines the duration of the sine-squaredramp and is chosen to be a multiple of the optical cycle fre-quency T = 2π/ω. The laser parameters are E0 = 0.1, ω =0.148, and Tc = 3T (atomic units are used). This choicecorresponds to laser intensity I = 3.5 × 1014 W/cm2. At thisfrequency, it takes about five photons to ionize the model atom.These parameters are the same as used in Ref. [31].

To avoid the singularity of the Coulomb potential, a modelsoft Coulomb potential,

V (x) = − 1√1 + x2

(24)

will be used in the calculations [32]. This potential asymptot-ically is equal to the Coulomb potential, has a long Coulombtail, and the high-lying bound eigenstates have a Rydberg seriesstructure. The spectrum of H ′ contains bound states ϕn andcontinuum states ϕE .

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The ATI spectrum [33] can be obtained by projecting thetime-dependent wave function at t = Tfinal onto the field-freecontinuum states ϕE(x),

P (E) = |〈ϕE(x)|ψ(x,Tfinal)〉|2. (25)

This probability distribution gives the energy spectra of theejected electrons.

To simulate photoionization, first we have to calculate theeigenstates of the Hamiltonian H ′. To describe the ionizationprocess, we need to calculate the continuum eigenstates of theHamiltonian as well. In principle, to calculate the continuumstates, one has to solve the Schrodinger equation for energyE > 0 with proper scattering boundary conditions. In thepresent paper, we approximate the continuum states by con-tinuum discretized pseudostates obtained by diagonalizationof the Hamiltonian on a (L2 integrable) real space grid basis.The pseudostates obtained by this approach will have discreteenergies Ek > 0 and, at those energies, will approximate theproper continuum states. The size of the calculational celldetermines the density (the distance between Ek and Ek−1) ofthe pseudostates. Large computational domains have a denserenergy distribution as the eigenstates gradually converge tothe particle-in-a-box eigenstates. The calculation of continuumstates using bound state basis functions has been introducedin Refs. [34,35] and has been used successfully in manyapplications (see, e.g., Refs. [36,37]).

The ground state of H ′, ϕ1(x), will be time propagatedunder the effect of the laser field. This initial wave function,

ψ(x,0) = ϕ1(x) (26)

will be evolved by (1) using a real space grid basis combinedwith the Taylor time propagation method and (2) by theTDB approach. The results of the two approaches will becompared to check the accuracy of the TDB approach. In thefinite difference grid calculation, the computational region ischosen to be −500 < x < 500 (in atomic units), and the realspace grid contains 4000 points. In the TDB calculation, thebasis function defined in Eq. (14) is used with parameters [seeEq. (15)] N = 200, �s = 0.5 a.u., and a = −50.

During the time propagation, the wave function, whichoriginally was confined to the center of the system, graduallyextends into the whole space, the electron is ejected, and theatom is ionized. The time-propagated wave function ψ(x,t)of the ground state, after t = Tfinal = 16T , is shown in Fig. 4.One can see that the wave function almost has reached theboundary.

The ATI spectrum is shown in Fig. 5. This is calculated byaveraging over the population of the odd and even continuumstates [31],

P

(Ei−1 + Ei + Ei+1 + Ei+2

4

)

= |〈ϕi(x)|ψ(x,Tfinal)〉|2Ei+1 − Ei−1

+ |〈ϕi+1(x)|ψ(x,Tfinal)〉|2Ei+2 − Ei

. (27)

The spectrum calculated by using the real space grid and theTDB functions are in very good agreement.

-400 -200 0 200 400x

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Pro

babi

lity

FIG. 4. Squared amplitude of the time-propagated wave functionafter 16 cycles of the laser field. The results of the grid and the TDBcalculations are indistinguishable in the resolution of the figure.

D. Calculation of the optical absorption spectrum of Na2

In this three-dimensional example, we will calculate the op-tical absorption spectrum of a sodium dimer in the frameworkof the time-dependent density functional theory (TDDFT)[38]. Two Na atoms are placed at a distance of 3.07 A awayfrom each other along the x axes. The initial wave orbitalsϕi(r) are obtained as ground state solutions of the Kohn-ShamHamiltonian. The Kohn-Sham Hamiltonian HKS is the sumof the kinetic energy operator, the Hartree potential, and theexchange-correlation potential,

HKS = − h2

2m∇2

r + VH[ρ](r) + VXC[ρ](r). (28)

To represent the exchange-correlation potential V XC, weemployed the adiabatic local density approximation with theparametrization by Perdew and Zunger [39]. The ionic corepotential was taken as a sum of norm-conserving pseudopo-tentials, by Troullier and Martins [40], centered at each ion.

To calculate the absorption spectrum, we use the approachdescribed in Ref. [41]. In this approach, the orbitals at t = 0are perturbed by an instantaneous dipole kick,

ψi(r,t) = e−iλrαϕi(r). (29)

0 2 4 6 8 10 12 14 16 18 20E/ω

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

P(E

)

FIG. 5. (Color online) ATI spectrum of a model atom calculatedby TDB (solid line) and by using a numerical grid (dashed line).Atomic units are used.

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0 1 2 3 4 5 6 7 8 9 10E (eV)

0

0.5

1

1.5

2

2.5

3α(

Ε)

FIG. 6. (Color online) Optical absorption cross section of the Na2

molecule calculated by the TDB (solid line) and by using a real spacegrid (dashed line).

These perturbed orbitals are time propagated by the Kohn-Sham Hamiltonian using the TDB. The dipole moment of thesystem can be calculated using the time-dependent density,

d(t) =∫

ρ(r,t)r dr. (30)

From the time-dependent dipole moment, one can calculatethe optical absorption cross section using [41]

α(E) = e2

λh

∫[d(t) − d(0)]eiEt/hg(t)dt, (31)

where g(t) is a damping function taken to be g(t) = e−i δt .The calculated optical absorption spectrum shown in Fig. 6 isin very good agreement with a previous calculation based ona real space grid [42]. Unlike the real space grid calculationwhere one needs a large computational cell (a 20 A × 20 A ×20 A cube is used with 512 000 grid points) to avoid reflectionsfrom the boundary, in the TDB case, a smaller cell (13 A ×10 A × 10 A) with 10 400 basis functions is sufficient to obtainaccurate results.

During the time propagation, the Gaussian TDB functionsspread out in space. In the numerical calculations, we havefound it to be advantageous to stop the propagation at certaintime Tr and to expand the wave function in terms of the t = 0basis functions,

ψ(r,Tr ) =∑

k

ck(0)φi(r,0). (32)

Then, the calculation can be restarted by propagating the newcoefficients by using Eq. (11), and the wave function can becalculated by using

ψ(r,Tr + t + �t) =∑

k

ck(t + �t)φk(r,t + �t). (33)

E. Multiphoton ionization rate of the H atom

In this section, we calculate the ionization rate of the Hatom in three dimensions. This problem has been solved withvarious approaches, Refs. [43–46], and we compare our resultsto the results given in these references. In Sec. III C, theionization of the H atom was treated in a one-dimensional

TABLE I. Ionization rates of the hydrogen atom exposed tointense laser fields with peak intensity I = 1.75 × 1014 W/cm2. Thelaser frequency ω is given in atomic units. The results of the presentapproach are compared to those of the k-space [44], Floquet [45], andr-space [43] approaches.

Present k-space Floquet r-spaceω approach approach approach approach

0.6 0.003 13 0.003 14 0.003 14 0.003 140.3 0.001 60 0.001 62 0.001 61 0.001 630.2 0.003 47 0.003 49 0.003 50 0.003 50

model framework using the soft Coulomb potential, and theone-dimensional grid and the TDB calculation were compared.As the grid calculation was not able to handle the singularityof the Coulomb potential, a soft Coulomb potential was usedin the calculation.

In the present example, we use the three-dimensionalCoulomb interaction, and the Hamiltonian is

H = −1

2∇2

r − 1

r− xE(t), (34)

where E(t) is given by Eq. (23). The matrix elements of the 1/r

potential and the TDB functions can be calculated analytically[47], and the singularity of the Coulomb potential does notcause any problems. The laser field is ramped according toEq. (23). In the present case, the parameters of Refs. [43–45]are used (Tc = 10Toc, where Toc = 2π/ω is the optical cycleof the laser field). The basis functions are distributed on athree-dimensional grid. The grid points are equidistant, andthe distance between the grid points is �s = 1 a.u. in thex, y, and z directions. The laser is oriented in the x direction,there are 50 basis functions in the x direction, and there are10 basis functions in both the y and z directions. The totaldimension of the basis is N = 5000.

The calculated ionization rates are presented in Table I.The results of the present calculation is in very goodagreement with the results of the momentum (k space) [44],Floquet [45], and coordinate (r space) [43] approaches. In ther-space approach, the wave function is expanded in terms ofLegendre polynomials depending on the angular variable andis multiplied by a radial wave function. In the Floquet andk-space approaches, the wave function is represented usinga partial wave expansion. In these calculations, therefore,one has to calculate the wave function for each partial waveseparately. The calculation has to be repeated for each partialwave, but one only has to solve a one-dimensional problemin each case. In this way, these calculations are faster forthe H atom than the TDB approach, which relies on a three-dimensional product basis. At the same time, the applicationof the TDB basis is simpler, and as we have shown in theprevious examples, it also can be used for multiatom systems.The present calculation takes about 6 h on a single processor.

F. Double-ionization probability of the He atomin a strong laser field

The electron dynamics of the He atom in a laser field hasbeen studied with many different approaches [48–58]. Thenonperturbative time-dependent description of the ionization

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KALMAN VARGA PHYSICAL REVIEW E 85, 016705 (2012)

1015 1016

I (W/cm2)

10-4

10-3

10-2

10-1

100

P2+

FIG. 7. (Color online) Double-ionization probability as a functionof laser field intensity for the He atom. The dots show the resultsof the present calculation; the diamonds are the results fromRef. [62].

process is not simple because one needs a flexible represen-tation of the various possible configurations. Powerful wavefunction expansions are proposed to describe the two electrondynamics [49,55,58], but the applicability of these approachesis likely to be limited to few-electron systems. Similar tothe previous case, we will use the TDDFT to describe theelectron dynamics. The TDDFT Hamiltonian is less accuratethan the few-body Hamiltonian used in Refs. [49,55,58], butthe TDDFT easily can be applied for larger systems. TheTDDFT has been used successfully to study atoms in a laserfield [59–61].

In the present paper, a laser field pulse defined by

E(t) ={E0sin2

(πtTc

)sin ωt, if 0 � t � Tc,

0, t > Tc

(35)

is added to HKS. Whereas, in Eq. (23), note that the sinesquare is used to ramp the laser field; here, the sine square isthe envelope function of the laser profile. In the calculations,we used a laser with a wavelength of 800 nm and modulatedby a 20-cycle-long sine-squared envelope. The size of thesimulation cell is 50 A × 10 A × 10 A, and a 5000 (�s =1) basis function is used. The calculated double-ionizationprobability for various laser intensities is shown in Fig. 7. Theionization probability is calculated by integrating the chargedensity away from the center of the He atom. The calculated

curve is in good agreement with other TDDFT calculations[62,63]. The solution of this problem using real space gridapproaches would require a much larger basis set. To avoidthat, this problem often is solved by using a one-dimensionalHamiltonian [62,63] or by introducing prolate spheroidalcoordinates [60] to decrease the computational complexity.

IV. SUMMARY

To summarize, we have shown that TDB functions canbe used efficiently to solve the TDSE. In the case of thetime-dependent basis, the time evolution of the basis functionsis known, whereas, the expansion coefficients have to bepropagated. The most important advantage of the TDB isthat, as the basis functions are known analytically in spaceand time, there is no artificial reflection on the boundaries, andsmaller simulation cells can be used. This significantly reducesthe computational cost because much smaller computationalcells and, thus, the basis can be used. In addition, in the TDBmethod, the time propagation is governed by the potentialenergy matrix and not by the Hamiltonian. Using localizedbasis functions, the potential energy matrix is much sparserthan the Hamiltonian matrix, and that makes the computationof the time propagation step faster.

We have used Gaussian basis functions as the TDB in thecalculations. The Gaussian TDB functions spread out duringthe calculation. To keep the accuracy of the expansion, oneoccasionally has to restart the calculation. One can try to usea Gaussian basis with complex width parameters to avoid thisstep. Further tests and applications of the TDB approach areunder way to investigate their applicability and to optimizetheir performance.

The present paper is restricted to one-dimensional andthree-dimensional problems where TDB or the tensor productof the TDB can be used. The tensor product of the TDBscombined with the TDDFT can be used to solve varioustime-dependent problems. A further and more difficult stepis to construct appropriate products of the TDB to buildcorrelated few-particle basis functions to describe atoms andmolecules in laser fields with few-electron Hamiltonians.

ACKNOWLEDGMENT

This work was supported by NSF Grant No.CMMI0927345.

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