ARTICLE IN PRESS

0168-9002/$ - se

doi:10.1016/j.ni

�CorrespondE-mail addr

andrian.lenglet@

Nuclear Instruments and Methods in Physics Research A 577 (2007) 343–348

www.elsevier.com/locate/nima

Wave Packet Molecular Dynamics to study atom electronsin strong fields

Andrian Lenglet�, Gilles Maynard

Laboratory of Plasmas and Gaseous Physics, LPGP, CNRS-UMR 8578, Universite Paris-Sud XI, F-91405 Orsay, France

Available online 22 February 2007

Abstract

In order to study the dynamical behavior of atomic electrons interacting with strong fields, a semi-classical model is built from the Wave

Packet Molecular Dynamic model. In our calculations, the Gaussian Wave Packets approach has been improved by introducing

Hermite–Gaussian functions. Results are presented for laser–atom interaction at intensities where tunnel ionization plays an important role.

r 2007 Elsevier B.V. All rights reserved.

PACS: 31.15.Qg; 34.10.þx

Keywords: Many body problem; Non-perturbative method; Collective electronic effects

1. Introduction

For several applications related to high energy density inmatter, one has to get an accurate description of thedynamics of atomic electrons interacting with strong fields.One case concerns laser–atom interaction as for examplewhen considering soft X-ray laser sources produced byfocusing an intense laser beam inside a Xenon gas inRef. [1]. Another case is related to high energy collisionsbetween partially ionized heavy ions as encountered inheavy ion fusion [2]. In both cases the strength and theduration of the time dependent fields are so large thatperturbative theories cannot yield accurate results, there-fore non-perturbative models are required. Several non-perturbative calculations have been proposed in which thetime dependent Schrodinger equation (TDSE) is solvedeither using a grid in real or in Fourier space or also byusing an atomic basis. In our calculations the starting pointhas been the semi-classical approximation in which theperturbation potential can be large but its space variationon a scale length given by the wave function is assumed to

e front matter r 2007 Elsevier B.V. All rights reserved.

ma.2007.02.027

ing author.

esses: andrian.lenglet@pgp.u-psud.fr,

u-psud.fr (A. Lenglet).

be small. When considering only first and second orderderivative of the potentials, Gaussian Wave Packet (GWP)become exact solution of the TDSE. In Ref. [3], thermo-dynamic properties of dense plasmas were studied using atime dependent variational principle (TDVP) [4] togetherwith GWP leading to Wave Packet Molecular Dynamic(WPMD) calculations. The objective of the present work isto apply WPMD for studying dynamical properties ofatomic electrons. Results presented below show thatWPMD with GWP yield realistic results when the semi-classical approximation is valid, as expected. It corre-sponds to very strong perturbation, for which the energygain by the electron is large. To extend the validity domainof the GWPMD calculations we have introduced a sum ofHermite–Gaussian functions. In the general case WPMDwith Hermite Gaussian Wave Packet (HGWP) lead tocalculations that are difficult to solve. We show that thenumerical complexity can be strongly reduced by consider-ing only a reduced ensemble of HGWP (R-HGWP). In thepresent work we will compare the results obtained usingeither spherical GWP (S-GWP), GWP or R-HGWP forcalculating ground state properties of atom and moleculesand their dynamical evolution when interaction with a highintensity laser field or with an energetic ion. Atomic unitswill be used, excepted when specified.

ARTICLE IN PRESSA. Lenglet, G. Maynard / Nuclear Instruments and Methods in Physics Research A 577 (2007) 343–348344

2. Time dependent variational principle

Let us consider a physical system characterized by aHamiltonian H and a wave function j, that depends onseveral parameters fqig. Following Ref. [4] we define aLagrangian through

Lðqj ; _qjÞ ¼X

j

pjðqjÞ _qj �HðqjÞ (1)

where pj ¼ hjjiq=qqjjji is the conjugated momentum ofthe introduced parameter qj, and H ¼ hjjHjji is the semi-classical Hamiltonian. Our quantum problem originallydriven by the TDSE could now be treated by classicalLagrange equations on Lðq; _qÞ.

From the Lagrange equations onL, one gets the dynamicalequations of the coordinates fqjg with the help of the so-callednorm-matrix N defined by Nj;k ¼ qpj=qqk � qpk=qqj:X

j

Nj;k _qj ¼qHqqk

) _qj ¼X

k

N�1k;j

qHqqk

. (2)

N invertibility depends on j choice. If N is invertible, weobtain a set of coupled differential equations. N is easilyinvertible when conjugated moments are by constructionparameters, which is the case with GWP jG:

jGð~q;~xÞ ¼Y

j¼x;y;z

oj

p

� �1=4e�ðoj=2þigjÞðxj�rjÞ

2

eipjðxj�rj Þ. (3)

We have studied the different following shapes:

�

SGWP: Width variations are identical in each direction.8j;oj ¼ o; gj ¼ g. We have 8 degrees of freedom. Thisshape was precedently used in Refs. [3,5]. � GWP, jG as written above in Eq. (3). The problem

dimension is 12.

�

Fig. 1. Minimum energy by minimization of H obtained with HGWP

wave function ansatz. The wave function is written as a linear combination

of ci;j;k where i þ j þ kpn. Note that with n ¼ 0 we retrieve the GWP

HGWP: jHG. The wave function jHG is written as alinear combination of Hermite–Gauss functions:

jHGðci;j;k;~q;~xÞ ¼X

i;j;k2A

ci;j;kci;j;kð~q;~xÞ (4)

where the Hermite–Gauss functions cnx;ny;nzare

cnx;ny;nzð~q; ~xÞ ¼

Yj¼x;y;z

hnjðffiffiffiffiffiojpðxj � rjÞÞ

�e� oj=2þigjð Þðxj�rj Þ2

eipj ðxj�rjÞ. ð5Þ

The hnjare the normalized Hermite polynomials of degree

nj and A the set of allowed integer triplet ði; j; kÞ. In most ofthe presented results we have use a reduce ensemble of ci;j;k

in order to reduce the numerical effort in the N inversion.The basic idea is to limit the dependency of the conjugatedmoments of ~r, ~p, gxi

and oxion the coefficients ci;j;k. It has

been obtained by choosing the set A such that

ci;j;kci�1;j;k ¼ ci;j;kci�2;j;k ¼ 0

ci;j;kci;j�1;k ¼ ci;j;kci;j�2;k ¼ 0

ci;j;kci;j;k�1 ¼ ci;j;kci;j;k�2 ¼ 0. ð6Þ

For example, for ði; j; kÞ 2 ½0; 2�3, the following table liststhe triplet for the non-zero coefficient:

assertion.

i

j k i j k

0

0 0 2 2 0 1 1 0 2 0 2 1 0 1 0 2 2 0 1 1

When condition (6) is satisfied, the norm-matrix becomeseasily invertible. The problem dimension for the HGWPcalculations becomes 12þ card(A).

3. Ground state properties

Let us first consider the case of a spherical atomicpotential. The difference in the results obtained by thedifferent models are nearly independent of the specific formof the atomic potential so that we choose the simple onecorresponding to atomic hydrogen. The stationary state ofthe hydrogen atom is studied from semi-classical Hamilto-nian HðqjÞ written as

HðqjÞ ¼ jðqj;~xÞ �1

2

q2

q~x2�

1

j~xj

��������jðqj;~xÞ

� �. (7)

Minimization of HðqjÞ yields the (1s) ground state,which has been calculated using either the GWP or theHGWP representation. Fig. 1. shows the minimum energyvs. the wave function width o obtained with HGWP fordifferent sets An ¼ fði; j; kÞji þ j þ kpng.From this figure, we can observe that GWP (n ¼ 0)

yields a minimum energy of �0:424 obtained at o�0:57,i.e. 16% higher than the exact value. By adding HGWP ofhigher degrees, we get closer to the exact result of �0:5.

ARTICLE IN PRESSA. Lenglet, G. Maynard / Nuclear Instruments and Methods in Physics Research A 577 (2007) 343–348 345

With a linear combination of HGWP with n ¼ 4, we get aminimum energy of �0:474 for o ¼ 0:841, i.e. 5% error.With n ¼ 20, the minimum energy becomes �0:4993 ato ¼ 1:183. Therefore using 1000 HGWP, a number smallenough to perform dynamical calculations, the error in theinitial state properties can be of the order of 0.1%. Whenusing only R-HGWP, the wave function does not convergeto the exact value when increasing the value of n. Theasymptotic wave function is already obtained at n�10corresponding to 300 R-HGWP with an error reductionfactor of two in the binding energy when compared to theGWP result.

In the case of molecular atoms GWP and S-GWP yielddifferent results. Here also we investigate the simplest caseof Hþ2 molecule. In Fig. 2 is reported the minimum totalenergy vs. the length of the bond as obtained using GWP,S-GWP or R-HGWP.

From Fig. 2 similar conclusions as for the hydrogenatom can be done. We can first remark the good behaviorof the TDVP models and the rather good agreement withthe exact energy values (20% error, for S-GWP, 16% forGWP, 10% for R-HGWP).

From the above two examples we can deduce that theorder of magnitude of the error for the ground state energy ofatomic and molecular system for one active electron if 20%for S-GWP, 15% for GWP and 10% for R-HGWP. In thecase of R-HGWP, an asymptotic wave function is obtainedfor 300 function, whereas using the full HGWP ensemble, anoptimized wave function with an accuracy better than 1%can be constructed using no more than 1000 functions.

Starting from the ground state configurations asobtained by GWP, S-GWP and R-HGWP we haveanalyzed the dynamical evolution of the electron wavefunction when interacting with a strong field. We haveconsidered two cases:

�

+

Fig

and

Interaction with a laser beam at intensities for whichtunnel ionization is important. Tunneling is a pure

-0.6

-0.4

-0.2

0.0

0.2

0.4

108642

H2

en

erg

y (

at.

u.)

Length d(H-H) (at.u.)

Exact result GWP spherical

GWP

HGWP restricted

. 2. Minimum total energy of Hþ2 by minimization of H with GWP

restricted HGWP wave function ansatz.

Fin

al E

nerg

y (

ato

mic

units)

Fig

lase

line

quantum effect, therefore it provides an interestingbenchmark of our semi-classical model.

� Collision with an energetic ion in the strong interaction

regime, for which it can be expected that semi-classicalapproximation can provide reliable results.

4. Tunnel ionization of hydrogen

Tunnel ionization of atoms during their interaction witha high intensity short pulse laser beam is a well documentedprocess, in particular for applications such as soft X-raylaser [6]. In order to get ‘exact’ results to be compared with,we have made calculations using the numerical QPROPcode [7], in which the TDSE is solved on a grid. Our resultshave also been compared with the ones of the widely usedADK model [8].The Hamiltonian of the hydrogen electron interacting

with an intense laser field ~EðtÞ is given by

H ¼ �1

2

q2

q~x2�

1

j~xj� ~EðtÞ � ~x

with

~E ¼ E0sin2 oLt

2Nc

� ðcosðoLtÞ~i þ sinðoLtþ fÞ~jÞ (8)

where oL is the laser pulsation, Nc the number of opticalcycles within the sin2 envelope and ~i;~j are two normalizedorthogonal vectors. Results are presented in the case of awavelength 800 nm and Nc ¼ 4 corresponding to a pulseduration (FWHM) of 7 fs. Intensities will be expressed interms of I14 ¼ 1014 Wcm�2. In Fig. 3 is reported the finalenergy of the electron after interaction with a linearlypolarized beam at intensities from 2I14 up to 103I14, atwhich relativistic effects can still be neglected.In Fig. 3, S-GWP, GWP and R-HGWP results are

compared to Qprop calculations. We can observe thatcalculations yields an energy that depends only on alogarithm scale on the intensities over a large range of

6

5

4

3

2

1

0

2 3 4 5 6 7 8 9

1015

2 3 4 5 6 7 8 9

1016

2 3 4 5 6 7 8 9

1017

Linear polarization GWP spherical GWP

HGWP restricted

Qprop

Intensity W/cm²

. 3. Final energy of the hydrogen electron after interacting with a 70-fs

r pulse, for GWP, SGWP, and restricted HGWP. Laser polarization is

ar.

ARTICLE IN PRESSA. Lenglet, G. Maynard / Nuclear Instruments and Methods in Physics Research A 577 (2007) 343–348346

values. S-GWP, GWP and R-HGWP yield similar results,the corresponding curves oscillate around the QPROP one.At particular values of the intensity the electron gain muchenergy, highlighting a resonance phenomena. Thoseresonances are related to re-scattering of the electron withthe nucleus, which can lead to High Harmonic Generation[9]. The final energy of the electron is directly related to theintensity at the time at which the electron makes a tunnelionization. Therefore a sharp resonance in energy is anindication that the wave packet remains compact, leavingthe atom at a specific time, whereas in QPROP the rate ofionization is not so large compared to the laser frequencyso that there is a broadening of the energy spectra.

An order of magnitude of the ionization threshold I t canbe calculated by looking at the value of the intensity atwhich the electron final energy becomes positive. I t value isabout 5I14 for S-GWP, whereas for GWP, and R- HGWPit is about 4I14, in agreement with the QPROP results.

To get a closer look at the ionization process, we havestudied more specifically the intensity region between I14and 10I14. The electron energy spectra pðEÞ after interac-tion has been calculated with a window-operator method[10]. The ionization degree DI can then be determinedthrough DI ¼

RE40 pðEÞdE. In Fig. 4 we have reported the

values of DI calculated with our WPMD models, which are

1.0

0.8

0.6

0.4

0.2

0.0

1.41.21.00.80.60.40.2x1015

Linear polarization ADK QPROP

GWP spherical

GWP

HGWP (6x6x6)

Ioniz

ation D

egre

e o

f H

ydro

gen

Laser Intensity W/cm2

1.41.21.00.80.60.40.2x1015

Laser Intensity W/cm2

1.0

0.8

0.6

0.4

0.2

0.0

Circular polarization

ADK

GWP spherical

GWP

HGWP (6x6x6)

Ioniz

ation D

egre

e o

f H

ydro

gen

Fig. 4. Ionization degree DI of the hydrogen electron after interacting

with a 70-fs pulse. Laser polarization is linear on the first figure, and

circular on the last.

compared with the QPROP results and the ones of theADK model.Defining the ionization when DI40:5, in the case of

linear polarization, we retrieve the values derived fromFig. 3: ionization occurs at about 5I14 for S-GWP, 4I14 forR-HGWP and 3:8I14 for QPROP, whereas within theADK one gets a value of 4:8I14. For circular polarization,the values of the ionization threshold are I t ¼ 6I14 usingGWP and R-HGWP which is rather close to the ADKvalue I t ¼ 6:8I14. With GWP one gets a slightly largervalue I t ¼ 8I14. We can therefore conclude that ourWPMD models yield rather realistic values for theionization threshold.In Fig. 4 we can observe that the main difference

between WPMD curves and the two QPROP and ADKones is on the slope of the curves close to the ionizationthreshold, the value of the slope being much higher for theWPMD calculation. We retrieve here the same phenomenaas exhibited in Fig. 3: in WPMD calculations, the wavefunction remains compact, being either bound to thenucleus or ionized.The main conclusion on our calculations of tunnel

ionization is that the WPMD models can yield a goodestimate of the ionization threshold but not on phenomenathat depend heavily on the form of the wave packet. Thefact that R-HGWP results are close to the GWP ones is anindication that high order derivatives of the potential donot have a strong contribution on the tunnel ionizationprocess. It is well known that to determine the tunnelionization rate it is necessary to describe accurately thewave function at large distances from the nucleus. Theconditions imposed by Eq. (6) are too restrictive toincreasing significantly the accuracy in that case. GWP isthen more appropriate than R-HGWP for doing fastcalculations of tunnel ionization, when considering atomicpotential with spherical symmetry. When consideringinteraction with a laser beam the main interest of theR-HGWP model is that it can be easily applied to morecomplex molecular structure.

5. Collisions

Ion–atom interaction at high energies is quite similar tolaser–atom interaction at high intensities. It is thusinteresting to compare the two cases within the WPMDmodels.Let us consider the case of a fully ionized ion of charge Z

and velocity Vk1 interacting with an hydrogen atom. It iswell established that the strength of the perturbation can beestimated from the value of the parameter Z ¼ Z=V [11].For Z51 we are in the perturbation regime where first orsecond order calculations are appropriate. In the otherhand, for ZX1 we are in the strong interaction regimewhere non-perturbative calculations are required. More-over at V ’ 1 we are close to the maximum of the stoppingcross-section. Here we present results for a 100 keV/n He2þ

(V ¼ 2, Z ¼ 1) that is in the interesting domain for

ARTICLE IN PRESSA. Lenglet, G. Maynard / Nuclear Instruments and Methods in Physics Research A 577 (2007) 343–348 347

applications of strong coupling together with a largestopping cross-section. The Hamiltonian of the hydrogenelectron wave function is

HðqjÞ ¼ j �1

2

q2

q~x2�

1

j~xj�

Z

j~x� ~RðtÞj

��������j

� �(9)

where ~RðtÞ is the trajectory of the projectile ion of charge Z

of impact parameter b: ~RðtÞ ¼ Vt~ux þ b~uy.For the considered case, the cross-sections for excitation,

ionization and charge transfer have similar amplitudes. Itmeans that after the collision, the wave function should besplit into several parts, each of them corresponding to oneof the three processes. In Fig. 5 we have reported thecontour map of the electron density in the plane of thecollision for an impact parameter value of 2. Fig. 5acorresponds to the initial state before collision, the crossat the origin representing the hydrogen nucleus, while in

a

b

c

Fig. 5. He2þ–H collision at 100 keV/nucleon at impact parameter b ¼ 2.

The projectile comes from the left with y ¼ b ¼ 2. (a) Shows the electron

density jj at the initial state at t ¼ 0, (b) jj at the final state after

interaction for GWP, (c) jj at the final state for restricted HGWP.

Crosses are the ions at the origin the hydrogen nucleus, at about x�60 and

y ¼ 2, the projectile ion for (b) and (c). Contour lines are at

ð0:5; 0:2; 0:1; 0:05Þ of the maximum density.

Fig. 5b and c is represented the final state of the electronthat is determined using either GWP of R-HGWPfunctions.We observe a significant difference between Fig. 5b

and c. The GWP wave packet in Fig. 5b corresponds to anearly free electron expanding into the free space. On theother hand in Fig. 5c we clearly see that the wave functioncalculated by the R-HGWP was sheared by the interactioninto three parts, with similar contribution: one still boundto the hydrogen, a second part around the helium nucleuscorresponding to a charge transfer, and the third part atthe top of the figure representing the contribution ofionization. We can remark also that the angle of theionized part is well oriented in the direction of the projectilewhich is characteristic of strong coupling.An important quantity is the stopping cross-section

S ¼R2pbEðbÞdb, with b the impact parameter and EðbÞ

the energy transferred to the electron during the collision.In Fig. 6 are compared the R-HGWP results with theresults obtained from Quantum-Classical TrajectoryMonte-Carlo (CTMC) calculations [11] for the differentialstopping cross-section. We have also presented in Fig. 6the results of a calculation made with Hermite Gaussianfunction of fixed width and position. These calculationscorrespond to standard Close Coupling Calculations,using Hermite Gaussian functions to describe the wavefunction.The agreement between the CTMC curve and the

R-HGWP one is rather impressive. For fixed HG theagreement is good at large values of the impact parameterbut 30% too small at the maximum of the curve. Thecalculation has been performed using 500 functions, amuch larger number is required to get an accurate value atthe maximum of the curve. However, it is not possible todescribe ionization and charge transfer using HermiteGaussian functions, which are too much localized.

Fig. 6. He2þ–H collision at 100 keV/nucleon. The stopping cross-section

vs. the impact parameter, for CTMC calculations, TDVP models with

restricted HGWP of 280 Hermite–Gauss functions and fixed HGWP of

500 Hermite–Gauss functions.

ARTICLE IN PRESSA. Lenglet, G. Maynard / Nuclear Instruments and Methods in Physics Research A 577 (2007) 343–348348

6. Conclusion

The validity of WPMDs approaches for determining thedynamic of bound electrons interacting with either a highintensity laser beam or an energetic ion has been checked.Using different forms of the wave packet, each of themleading to fast calculation. Our objective is to find amethod that is robust enough for doing parametric studies.One interesting point in the presented models is that theycan easily be applied even for relatively complex targetstructure. Our conclusion is that for tunnel ionizationGWP and HGWP satisfying Eq. (6) yield good predictionof the threshold intensity for ionization but not on thedistribution of energy of the electrons. To increase theaccuracy it will be require to relax some constraints inEq. (6). On the other hand for collision in the strong interactionregime, the R-HGWP model yield unexpected good results.However, it has to be confirmed by considering more cases.

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