Born–Oppenheimer approximation and beyond for time-dependent electronicprocessesL. S. Cederbaum

Citation: The Journal of Chemical Physics 128, 124101 (2008); doi: 10.1063/1.2895043 View online: http://dx.doi.org/10.1063/1.2895043 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/128/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electronic currents and Born-Oppenheimer molecular dynamics J. Chem. Phys. 137, 084109 (2012); 10.1063/1.4747540 Erratum: “Controlling electron transfer in strong time-dependent fields: Theory beyond the Golden Ruleapproximation” [J. Chem. Phys. 113, 11159 (2000)] J. Chem. Phys. 115, 3969 (2001); 10.1063/1.1389850 Controlling electron transfer in strong time-dependent fields: Theory beyond the Golden Rule approximation J. Chem. Phys. 113, 11159 (2000); 10.1063/1.1326049 A formulation and numerical approach to molecular systems by the Green function method without theBorn–Oppenheimer approximation J. Chem. Phys. 111, 6171 (1999); 10.1063/1.479921 A timedependent formulation of the Born–Oppenheimer approximation to describe vibronic nonlinear opticaleffects J. Chem. Phys. 101, 4474 (1994); 10.1063/1.467435

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Born–Oppenheimer approximation and beyond for time-dependentelectronic processes

L. S. Cederbauma�

Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 229,D-69120 Heidelberg, Germany

�Received 17 January 2008; accepted 18 February 2008; published online 24 March 2008�

Explicit computations of electronic motion in time and space are gradually becoming feasible andavailable. The knowledge of this motion is of relevance by itself but is also important forunderstanding available and predicting future experiments on the electronic time scale. In electronicprocesses of interest, usually several and even many stationary electronic states participate and theobvious question arises on how to describe the accompanying quantum nuclear dynamics at least onthe time scale of the process. In this work, we attempt to study the nuclear dynamics in theframework of a fully time-dependent Born–Oppenheimer approximation. Additionally, we attemptto go beyond this approximation by introducing the coupling of several electronic wavepackets bythe nuclear wavepackets. In this context, we also discuss a time-dependent transformation todiabatic electronic wavepackets. A simple but critical model of charge transfer is analyzed in somedetail on various levels of approximation and also solved exactly. © 2008 American Institute ofPhysics. �DOI: 10.1063/1.2895043�

I. INTRODUCTION

The Born–Oppenheimer approximation1,2 is one of thebasic approximations in molecular science. It allows us toseparate the fast electronic motion from the usually muchslower motion of the nuclei and to visualize the molecule asa set of nuclei moving on the potential energy surface pro-vided by the electrons in a specific electronic state. The totalmolecular wavefunction is then a product of the electronicwavefunction of this state � and a nuclear wavefunction �which obeys the stationary Schrödinger equation,

�TN + V�� = E� , �1�

for the nuclei alone. Here, TN is the kinetic energy operatorof the nuclei and E is the total energy of the molecule.

The major result �Eq. �1�� of the traditional Born–Oppenheimer adiabatic approximation has been widely usedto interpret and to analyze numerous experiments and hasserved as the basis for approaching molecular spectroscopy.Later on, the approximation has been extended to visualizethe quantum nuclear motion as a function of time, first, forthe sake of interpretation. Then, it became evident that atime-dependent approach to the nuclear dynamics also bearstechnical advantages, in particular, as it suffices to investi-gate the dynamics on a short time scale in order to under-stand many processes. With the emergence of femtosecondlasers and, with it, of femtosecond chemistry, the treatmentof quantum nuclear dynamics has become necessary and,thus, indispensable, see, e.g., Refs. 3 and 4. In analogy to thestationary Born–Oppenheimer approximation discussedabove, one can view the total molecular wavefunction as a

product of the electronic wavefunction � in a specific stateand a time-dependent nuclear wavepacket �N�t�. This leadsto the time-dependent version of Eq. �1�,

i���N

�t= �TN + V��N, �2�

for the nuclear motion. Here, we briefly mention that, inparticular, in polyatomic molecules, it is likely that thenuclear motion proceeds over the potential surfaces of two oreven several electronic states which are coupled by thenuclear motion. In order to describe the resulting nuclearmotion, one may introduce a group, Born–Oppenheimer ap-proximation for the manifold of coupled electronic states andobtain equations generalizing Eqs. �1� and �2�.2,5–9

In recent years, the ab initio investigation of the time-dependent behavior of several electrons has gained muchpopularity,10–19 mainly due to advances in computationalabilities. This was first driven by theoretical curiosity andmore recently motivated by the advent of subfemtosecondand attosecond experimental technologies.20–23 These tech-nologies bear the promise to visualize the electronic motion,e.g., by using pump-probe measurements, similarly as suc-cessfully done for the nuclear motion using femtosecondlaser pulses.

There are several approaches to compute the dynamicsof the electrons at a given geometrical arrangement of thenuclei. These include the treatment of a few fully correlatedelectrons—usually in an external laser field—with variantsof the multiconfiguration time-dependent Hartree–Fock�MCTDHF� method where the orbital basis used is itselftime dependent.11–15,18 This method is based on the MCTDHwidely used in quantum nuclear dynamics24–27 and adoptedfor fermions. In another approach, the multielectron wave-packet of more-electron systems is directly propagated usinga�Electronic mail: lorenz.cederbaum@pci.uni-heidelberg.de.

THE JOURNAL OF CHEMICAL PHYSICS 128, 124101 �2008�

0021-9606/2008/128�12�/124101/8/$23.00 © 2008 American Institute of Physics128, 124101-1

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many-body Green’s function techniques and until nowmainly applied to study the electron dynamics taking placeafter the ionization of atoms, molecules, andclusters.10,16,17,19 Finally, we mention that for large systems,the time-dependent density functional theory is often used inconnection with a classical treatment of the nuclei.28–31

How to compute the quantum nuclear dynamics accom-panying the dynamics of the electrons? For completeness, weare reminded that there are first attempts to compute the dy-namics of all the particles, i.e., nuclei and electrons,simultaneously.32,33 Clearly, such approaches are interestingbut currently of limited applicability for systems with severalelectrons and nuclei. The natural question arises whether onecan utilize the Born–Oppenheimer approximation similarlyas traditionally done for stationary electronic states. In thispaper, we attempt to answer this question.

II. FULLY TIME-DEPENDENT BORN–OPPENHEIMERAPPROXIMATION

As usual in the Born–Oppenheimer approximation, let usseparate the fast electronic motion from the slower motion ofthe nuclei and consider a product wavefunction �e�t��N�t� ofthe electrons and of the nuclei. Clearly, the electronic wave-packet �e�t� which now depends on time is expected to obeythe time-dependent Schrödinger equation for the electronicHamiltonian He

i��

�t�e = He�e. �3�

The electronic Hamiltonian that is obtained from the fullHamiltonian H of the system by

H = TN + He �4�

of course depends on the nuclear geometry R parametricallyand may also depend on external fields, e.g., laser pulse, andthus, on time.

Certainly, we are interested in solving for the nuclearwavepacket �N�t� by inserting the product ansatz into thetime-dependent Schrödinger equation of the full Hamiltonianassuming that we have solved Eq. �3� and determined �e�t�.For pedagogical reasons, let us first consider a stationaryelectronic state of energy E�R�, i.e., �e�t�=e−iEt/��e�0�. Ofcourse, we would expect to recover an equation such as Eq.�2� for the nuclear wavepacket with V=E�R� because wehave the same physical situation of a specific stationary elec-tronic state. However, inserting �e�t��N�t� into theSchrödinger equation of H, multiplying as usually from theleft with �

e*�t�, and integrating over the electronic coordi-

nates leads to a very lengthy equation for �N�t� with severalunphysical terms. It becomes evident that the ansatz�e�t��N�t� for the total wavefunction is ill defined.

To remedy the situation, we are reminded that �e�t� is acomplex function and that the Schrödinger equation �3� is aprescription to propagate �e�t� starting from a given initialvalue, say, �e�0�. Consequently, neither the initial value of�e�t� nor the physics in this wavepacket changes if we mul-tiply it by a phase ei�t. However, since Eq. �3� is for anarbitrary but fixed nuclear geometry R, this phase can be-

come a function of R and, hence, �e�t� is dressed by a topo-logical time-dependent phase which, in turn, can play a cen-tral role in the nuclear dynamics. Our ansatz for the fullytime-dependent Born–Oppenheimer approximation, thus,reads

��r,R,t� = ei��R,t�/��e�r,t;R��N�R,t� , �5�

where r stands collectively for the electronic coordinates and�e depends as usually parametrically on the nuclear coordi-nates R; i.e., �e is obtained for each value of R as the solu-tion of the electronic equation �3�. As usual, we consider �e

and �N to be normalized to unity at any time t. In particular,

� dr �e*�r,t;R��e�r,t;R� = 1 �6�

for each value of R. The time-dependent topological phase��R , t� will be determined below.

We now insert the ansatz �5� into the Schrödinger equa-tion for the full Hamiltonian H in Eq. �4� and project on thenuclear space by multiplying from the left by e−i�/��

e*�r , t ;R�

and integrating over r. This readily leads to

i���N

�t= �T̃N +

��

�t��N, �7a�

where the dressed kinetic energy operator reads

T̃N =� dr e−i�/��e*TN�ee

i�/�. �7b�

To proceed, we have to make use of the form of thekinetic energy operator TN of the nuclei. Using convenientlyscaled rectangular nuclear coordinates, we may express TN as

TN = −�2

2M� · � = −

�2

2M� , �8�

where � is the gradient in nuclear space and the dot denotesthe usual scalar product. M is an averaged nuclear mass. Ofcourse, one can choose more involved nuclear coordinates,e.g., curvilinear, and the Laplacian � will have a more in-volved appearance.34

With Eq. �8�, one readily obtains for the dressed kineticenergy operator

T̃N = TN −�2

2M�ge +

i

��� +

2i

����� · fe −

1

�2 ����2

+ 2� i

����� + fe� · �� , �9a�

where we have introduced the time-dependent electron-nuclei couplings,

fe�R,t� =� dr �e*���e� ,

�9b�

ge�R,t� =� dr �e*���e� .

Interestingly, one can cast T̃N in a more appealing form,

124101-2 L. S. Cederbaum J. Chem. Phys. 128, 124101 �2008�

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T̃N = −�2

2M�� + fe +

i

������2

+�2

2M��� · fe� + fe · fe − ge� . �9c�

The term which makes the difference T̃N−TN between thedressed and original kinetic energy operators cumbersome isthe last one on the right hand side of Eq. �9a�. Without this

term, T̃N and TN differ only by a function of R and time.

The nuclear momentum coupling operator in T̃N could beeliminated by choosing

�� = i�fe, �10a�

leading to the highly simplified expression for the dressedkinetic energy operator,

T̃N = TN +�2

2M��� · fe� + fe · fe − ge� . �10b�

With this choice of the time-dependent topological phase��R , t�, the equation of motion for the nuclear wavepacketgiven in Eq. �7a� takes on the following appealing appear-ance:

i���N

�t= �TN + W�R,t���N, �10c�

where TN is the usual kinetic energy operator of the nucleiand W�R , t� is the effective time-dependent potential govern-ing the motion of the nuclei. This potential surface reads

W =��

�t+

�2

2M��� · fe� + fe

2 − ge� . �10d�

The time-dependent topological phase � determined fromthe condition �10a� thus allows us to formulate a fully time-dependent Born–Oppenheimer approximation describing atime-dependent electronic wavepacket which gives rise to apotential energy surface on which the nuclear wavepacketmoves.

To make the physical meaning of the proposed phase �clearer, we may consider the dressed nuclear momentum op-

erator � / i�̃=�dr e−i�/��e*� / i��ee

i�/�, which is defined inanalogy to the dressed kinetic energy operator in Eq. �7b�. Itis illuminating to see that the choice of phase in Eq. �10a�eliminates all nonadiabatic terms, thus, restoring the usual

nuclear momentum operator: �̃=�.It is obvious that one can fulfill Eq. �10a� exactly, at least

in the case of a single nuclear coordinate R. Then,

��R,t� = i��R

dR� fe�R�,t� , �11�

where fe=�dr �e*���e /�R�. Since the time-dependent phase

� is a real function, it is important to note that the electron-nuclei couplings fe are indeed imaginary quantities as can beshown by using the normalization condition �6�. We discusshere only briefly the mathematical conditions under whichEq. �10a� can be solved exactly in more dimensions. Sincethe curl of a gradient vanishes, Eq. �10a� implies that fe

should be curl-free in order to be removable by the introduc-

tion of the phase �. In case fe is not curl-free, it will have aremovable and a nonremovable part which will remain as aresidual part in the equations. If at all, we expect the curl offe and, hence, the residual part to be small.

Toward the end of this section, let us investigate ouransatz for the case of a specific stationary real electronicstate � of energy E�R�,

�e�r,t;R� = e−iE�R�t/���r;R� . �12�

Then, remembering that � is normalized, one immediatelyobtains the electron-nuclei coupling fe from Eq. �9b�,

fe�R,t� = −i

���E�R��t . �13a�

It is gratifying to see that it is easy to fulfill Eq. �10a� andobtain

��

�t= E�R� . �13b�

Finally, we determine the effective time-dependent potentialenergy surface and find

W = E�R� −�2

2M� ��r;R�����r;R��dr , �13c�

which exactly reproduces the traditional Born–Oppenheimerapproximation for the nuclear motion on the potential sur-face E�R� of a single specific electronic state �. The tradi-tional Born–Oppenheimer adiabatic approximation �see Eq.�2�� is obtained as usual by neglecting the nonadiabatic cou-pling element ������dr.35,36 For further interesting work onthis nonadiabatic coupling element, see Ref. 37.

As a final remark, we mention the issue of atoms andmolecules being exposed to a periodic continuous laser field.Although this is, of course, a time-dependent problem, theuse of Floquet theory38 can help us to define a time-independent potential for the translational motion of the at-oms and the translational and rotational motions of themolecules.39–42 This is accomplished by studying the Born–Oppenheimer approximation starting from the electronic Flo-quet Hamiltonian instead of the usual electronic Hamil-tonian. Here, the so called quasienergy appears as a phasesimilar to the case of stationary electronic states in Eq. �12�and this phase becomes the potential for the translational androtational motions as found in Eq. �13c� for the nuclear mo-tion in stationary electronic states. Accordingly, we advocatehere to make use of Floquet theory as the starting point ofthe fully time-dependent Born–Oppenheimer approximationwhen studying molecules in periodic continuous fields.

III. FULLY TIME-DEPENDENT GROUP-BORN–OPPENHEIMER APPROXIMATION

The traditional Born–Oppenheimer approximation hasbeen successful in many cases but clearly has its shortcom-ings whenever potential energy surfaces of different elec-tronic states come close in energy. In particular, this approxi-mation breaks down in situations where the surfaces exhibita conical intersection5 and it should be clear that such inter-sections are ubiquitous in polyatomic molecules.5–9,43,44 To

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remedy the situation, one includes the manifold of participat-ing electronic states in the treatment of the nuclear dynamicsand separates this group of states from other states, introduc-ing the group-Born–Oppenheimer approximation.5–9,36 Whenstudying the time dependence of electronic processes, bydefinition, two or more stationary electronic states are in-volved. In some of the studies mentioned in the Introduction,hundreds or even thousands of stationary electronic statesparticipate in the process.16,17 It is, thus, not surprising at allthat at least some of the energy surfaces of these states arecoupled by the nuclear motion rendering the treatment of thedynamics more intricate. In spite of the difficulties involved,many processes of interest are sufficiently fast justifying toinvestigate them by the fully time-dependent Born–Oppenheimer approximation and in more involved cases atleast by its group version which we attempt to introduce inthis section.

For simplicity of presentation, we consider two time-dependent electronic wavepackets �e1�r , t ;R� and�e2�r , t ;R�, each being a solution of the time-dependent elec-tronic Schrödinger equation �3� for different initial condi-tions at some time t, say, t=0. For Hermitian Hamiltonians,the time-evolution operator is unitary, i.e., �e�t�=U�t , t���e�t�� with U†�t , t��U�t , t��=1. As a consequencerelevant to the following, we notice that if we choose �e1 and�e2 at t=0 to be orthogonal to each other, they will stayorthogonal at all times,

� �e1* �t��e2�t�dr =� �

e1* �0�U†�t,0�U�t,0��e2�0�dr = 0.

�14�

A similar statement can be made on the normalization ofeach of these functions.

In analogy to the preceding section, our ansatz for thewavefunction in the fully time-dependent group-Born–Oppenheimer approximation reads

��r,R,t� = ei�1�R,t�/��e1�r,t;R��N1�R,t�

+ ei�2�R,t�/��e2�r,t;R��N2�R,t� , �15�

where a time-dependent topological phase is attached to eachelectronic wavepacket. Here, a remark on the phase seems tobe appropriate. In the traditional group-Born–Oppenheimeransatz for stationary electronic states, the real electronicwavefunction is not uniquely defined in the presence of aconical intersection.45 One can remedy the situation andmake single-valued complex functions by introducing a to-pological phase.46 In the case of electronic wavepackets, thesituation is different as these wavepackets are inherentlycomplex and all quantities including the topological phasesare dynamic quantities depending on time. The fact that sucha phase had to be introduced even in the case of a singleelectronic wavepacket discussed in the preceding section un-derlines the difference.

Inserting the ansatz �15� into the Schrödinger equation ofthe full Hamiltonian leads to two coupled equations for �N1

and �N2 if we multiply from the left once by e−i�1/��e1* and

once by e−i�2/��e2* and integrate over the electronic coordi-

nates r. These equations can be cast into the following forms:

i���N1

�t= �T̃N1 +

��1

�t��N1 + tN12�N2,

�16�

i���N2

�t= �T̃N2 +

��2

�t��N2 + tN21�N1.

Here, the dressed kinetic energy operators are defined incomplete analogy to that in the preceding section �Eq. �7b��but separately for �e1 and �e2 and the respective two time-dependent topological phases �1 and �2. With the choice ofscaled rectangular nuclear coordinates as used in Eq. �8�, the

Eqs. �9a�–�9c� are also valid for T̃N1 and T̃N2 when insertingthe respective quantities �e1 ,�1 and �e2 ,�2. The kinetic cou-pling operator tN12 reads

tN12 =� dr e−i�1/��e1* TN�e2ei�2/�. �17a�

Using Eq. �8�, one readily obtains

tN12 = −�2

2Mei��2−�1�/��ge12 +

2i

����2� · fe12 + 2fe12 · �� ,

�17b�

where the time-dependent electron-nuclei couplings,

fekl�R,t� =� dr �ek* ���el� ,

�17c�

gekl�R,t� =� dr �ek* ���el� ,

with k and l=1,2 are defined in analogy to those in thepreceding section.

If we may now choose

��k = i�fek, k = 1,2, �18a�

the coupled equations for the nuclear dynamics take on theexplicit appearance,

i���N1

�t= �TN + W1�R,t���N1 + tN12�N2,

�18b�

i���N2

�t= �TN + W2�R,t���N2 + tN21�N1.

As in the preceding section, each nuclear wavepacket �Nk

moves on its own time-dependent potential energy surfaceWk�R , t� but now, the motion on one surface is coupled nona-diabatically to the motion on the other surface by the opera-tor given explicitly in Eq. �17b�. The potential surfaces are

Wk =��k

�t+

�2

2M��� · fek� + fek

2 − gek� . �18c�

We briefly mention that if we choose stationary elec-tronic states �ek�r , t ;R�=e−iEk�R�t/��k�r ;R�, k=1,2, the set ofEqs. �18b� reduces to that of the traditional group-Born–Oppenheimer approximation.5–9,36 Explicitly, one obtainsWk=Ek�R�−�2 /2M ��k�r ;R����k�r ;R��dr and the kineticcoupling operator becomes

124101-4 L. S. Cederbaum J. Chem. Phys. 128, 124101 �2008�

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tN12 = −�2

2M�� �1���2�dr + 2� �1���2�dr · �� , �19�

as can be easily verified from Eqs. �17b� and �17c�.

IV. ON TIME-DEPENDENT DIABATIC ELECTRONICWAVEPACKETS

When investigating nuclear dynamics on coupled poten-tial surfaces of stationary electronic states, diabatization is ahelpful tool. In particular, in the cases of strong nonadiabaticcouplings, e.g., in the presence of conical intersections, thecomputation of the nuclear dynamics using the eigenvaluesand eigenstates of the electronic Hamiltonian He is cumber-some. Transforming these adiabatic states to diabatic stateswhich do not diagonalize He but which reduce or even elimi-nate the nonadiabatic couplings ��k���l�dr simplifies thecomputations substantially.5–9

In many cases, in particular, when the dynamics is mul-tidimensional, diabatization is necessary in order to make theproblem tractable. If the dynamics proceeds in a single di-mension, the nonadiabatic couplings can be eliminatedstrictly to zero and one may speak of strictly diabatic elec-tronic states. In more dimensions, these couplings can bemade rather small but cannot, in general, be strictlyremoved;36,47,48 one may speak of quasidiabatic states.36,48

In this work, we investigate the combined electronic andnuclear dynamics and we may ask whether it is possible tointroduce diabatic electronic wavepackets in order to sim-plify the calculation of the nuclear dynamics. We argue thatthis should be, indeed, possible in cases where only a fewstates strongly participate which may change in time, e.g.,states which become dressed states in a varying electric field.Even in those cases where the electronic wavepacket com-prises many electronic states, we expect that a fully time-dependent group-Born–Oppenheimer approach and diabati-zation are useful concepts able to describe properly manyprocesses, in particular, fast ones.

We consider n electronic wavepackets �ek�r , t ;R� and nnuclear wavepackets �Nk�R , t�. The total wavefunction in thefully time-dependent group-Born–Oppenheimer approxima-tion reads �see Eq. �15� for two wavepackets�

��r,R,t� = �Nt ei�/��e, �20a�

where � is the column vector of wavepackets,

�e =�e1

�e2

]

�en

, �20b�

and �Nt �“t” stands for transposed� is the row vector

��N1 , . . . ,�Nn�, and � is the diagonal matrix with elements�k. Let us transform the ei�k/��ek to a new set of electronicwavepackets el�r , t ;R� by an unitary transformation.Explicitly

�e = A*ei�/��e, �21a�

where A*=A*�R , t� is a unitary matrix which depends ontime t and nuclear geometry R. The corresponding nuclearwavepackets,

�N = A�N, �21b�

leave the total wavefunction �=�Nt ei�/��e=�N

t �e invariantbecause A†A=AA†=A*At=AtA*=1.

Inserting �=�Nt �e into the Schrödinger equation for the

full Hamiltonian H, multiplying from the left by �ek* , and

integrating over the electronic coordinates give rise to a setof n coupled equations which can be cast into a matrix equa-tion

i���N

�t= �N + He − i����N, �22a�

where, N, He, and � are matrices with elements Nkl

= �ek�TN�el , Hekl= �ek�He�el , and �kl= �ek � �� /�t�el ,respectively, and the integration is over the electronic coor-dinates r only. He and � are, thus, time-dependent functionsof R and N the time-dependent dressed matrix kinetic energyoperator,

Nkl = TN�kl −�2

2M��ek��el + �ek���el� · �� . �22b�

Now, we make use of the fact that �e obeys the time-dependent electronic Schrödinger equation i���e /�t=He�e,which allows us to eliminate terms in Eq. �22a�. Using thedefinition �21a�, we can rewrite the electronic part ofEq. �22a�,

He − i�� = − i�A�A†

�t+ A

��

�tA†, �23a�

and the dressed kinetic energy operator becomes

N = Ae−i�/�N� ei�/�A†, �23b�

where N� = ���lk�TN��el � is the dressed operator in the origi-

nal electronic wavepackets. Note that N� is an operator acting

also on A†=A†�R , t� to its right.Since ei�/� is unitary as is A, we may combine them into

a single unitary matrix B with B=Ae−i�/� and obtain

N = BN� B†. �23c�

The main idea of diabatization in our context is to choose, inanalogy to the traditional case of stationary electronic states,the matrix B such that N=TN may be up to some smallcorrections. The equations for B follow directly from Eqs.�23c� and �22b�. Here, we do not attempt to discuss howthese equations can be solved and how small the remaining�unremovable� corrections will be. This will depend on thesituation at hand and much more work and experience areneeded to answer these relevant questions. In this work, weonly want to show how the problem of coupled dynamicscan be attacked in principle. Assuming that a B can be foundsuch that N=TN up to some small corrections, Eq. �22a�reduces to

124101-5 Born–Oppenheimer approximation and beyond J. Chem. Phys. 128, 124101 �2008�

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i���N

�t= �TN − i�B

�B†

�t��N. �24�

The matrix equation �24� provides the set of coupledequations of n nuclear wavepackets moving in the potentialprovided by the n electronic wavepackets, each obeying thetime-dependent electronic Schrödinger equation.

V. A CRITICAL TEST MODEL: CHARGE TRANSFERBETWEEN TWO COUPLED SITES

In this section, we discuss the charge transfer from theleft site �L to the right site �R of a system with two sym-metric sites coupled by a harmonic string of frequency �.The Hamiltonian of this model reads

H = h��L �R� + �R �L�� +�

2�−

�2

�R2 + R2� , �25a�

where the electronic energy of each of the two equivalentsites is chosen to be the zero of the energy scale, and thecoupling between the sites,

h = h�0� + h�1�R , �25b�

is taken to be a linear function of the dimensionless distor-tion R of the string from its equilibrium.

We compute the electronic density,

��t� = ���t���̂���t� , �26a�

as a function of time after the system is prepared at t=0 withthe electron at the left site, i.e.,

���t� = e−iHt/��L . �26b�

The density operator is given by

�̂�r� = ��L*�r��L + �

R*�r��R ���L��L�r� + �R��R�r�� , �26c�

where ��L�r��2 and ��R�r��2 are the electron densities at thenoninteracting left and right sites of the system, respectively.

We have chosen this model because it can be solvedanalytically exactly as well as within the fully time-dependent Born–Oppenheimer approximation. Although themodel is simple, it provides a critical test for the approxima-tion since the exact solution describes diabatic dynamics andis, thus, on the other extreme of the Born–Oppenheimer ap-proximation which describes adiabatic dynamics. Further-more, the problem of ultrafast charge transfer that takes placealready in the absence of nuclear dynamics �see below� is ofcurrent interest. Finally, we mention that since only two elec-tronic states are involved, the group-Born–Oppenheimer ap-proximation provides the exact solution of the model.

As a first step, the electronic Hamiltonian is diagonal-ized by introducing the two electronic states,

� =1�2

��L �R � , �27a�

leading to a separable total Hamiltonian,

H = h� + �+ � − h�− �− � +�

2�−

�2

�R2 + R2� . �27b�

Noting that �L = �1 /�2���+ + �− �, one can immediatelyevaluate the density of the charge �e�R , t� at fixed nucleargeometry,

�e�r,t;R� = ��el�t���̂��el�t� ,

�27c���el�t� = e−iHet/��L ,

where ��el�r , t ;R� is the electronic wavepacket at this fixednuclear geometry. Using the representation of He in Eq.�27b�, we obtain

��el�t� =1�2

�e−iht/�� + + eiht/��− �e−i�/2R2t/�, �27d�

and with �̂ in Eq. �26c� transformed to the � representa-tion,

�̂�r� = ��+*�r�� + + �−

*�r��− ���+ ��+�r� + �− ��−�r�� ,

�27e�

� =1�2

��L �R� ,

the result reads

�e�r,t;R� = ��L�r��2 − ���L�r��2 − ��R�r��2�sin2�ht/�� .

�27f�

It is seen that the electronic density oscillates with the fre-quency h�R� /� between the left and right sites.

We now proceed to include nuclear dynamics. To makecontact to the Born–Oppenheimer approximation in Sec. II,we first calculate the time-dependent couplings fe and ge inEq. �9b� using the electronic wavepacket given in Eq. �27d�,

fe = − i�Rt/� ,

�28a�

ge = − i�t/� − �2R2�t/��2 − 2� �h

�R�2

�t/��2.

From Eqs. �10a� and �11�, one can now easily determine thetime-dependent topological phase,

��R,t� = ��R2/2�t , �28b�

and ��� /�t� is seen to be the potential energy of the har-monic spring in our model �see Eq. �25a��. The equation ofmotion of the nuclear wavepacket �10c� now reads

i���N

�t= ��−

1

2

�2

�R2 +R2

2+ 2� �h

�R�2

�t/��2��N. �28c�

In our model, �h /�R=h�1� �see Eq. �25b�� and, hence, the lastterm on the right hand side of the above equation can beabsorbed as a trivial phase into �N by writing �N�t�=e−i�2/3�h�1��t / ��3

�̃N�t�. The resulting equation reads

124101-6 L. S. Cederbaum J. Chem. Phys. 128, 124101 �2008�

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i���̃N

�t= ��−

1

2

�2

�R2 +R2

2��̃N. �28d�

Thus, starting at t=0 with the ground state �0 � of the har-monic spring gives rise to the nuclear wavepacket,

�N�R,t� = �−1/4e�−1/2�R2e−i�t/�2��e−i�2/3�h�1��t/��3

. �28e�

Inserting �=ei�/��el�N with the phase � in Eq. �28b�,the electronic wavepacket in Eq. �27d�, and the nuclearwavepacket in Eq. �28e� into the electronic density expres-sion �26a� immediately gives the appealing relationship,

��r,t� = ��0��e�0 � = �−1/2� dR e−R2�e�r,t;R� , �28f�

between the electronic density in the Born–Oppenheimer ap-proximation and that computed without nuclear motion. Theformer is just the average of the latter in the stationaryground state of the harmonic spring of the model. Using theexplicit expression �27f� derived above, we obtain

��r,t� = ��L�r��2 +1

2���L�r��2 − ��R�r��2�

��cos�2h�0�t/��e−�h�1�t/��2− 1� . �28g�

While at fixed geometry, the electron density oscillates be-tween ��L�2 and ��R�2 for all times, its average over the sta-tionary ground state of the spring, finally after longer times,equilibrates the density to ���L�2+ ��R�2� /2.

We turn now to evaluate the exact solution of the model.We first separate the full Hamiltonian �27b� into two,H+=h�+ �+�+H0 and H−=−h�− �−�+H0, where H0=� /2�−�2 /�R2+R2� is a harmonic oscillator. The exact time-dependent wavefunction in Eq. �26b� can easily be expressedas

���t� = e−iHt/��L =1�2

�e−iH+t/�� + + e−iH−t/��− � . �29a�

With the representation of the electron density operator �̂�r�in the electronic space spanned by ��� ,�� � �see Eq. �27e��,the exact electronic density follows straightforwardly,

��r,t� = ���t���̂���t�

= ��L�r��2 +1

2���L�r��2 − ��R�r��2��C + C*

2− 1� ,

�29b�

where

C�t� = ��0�eiH+t/�e−iH−t/��0 �. �29c�

Left-right correlation functions C�t� have been evaluated inRef. 49 for coupling between the sites as in our model. Thefinal result reads

��r,t� = ��L�r��2 +1

2���L�r��2 − ��R�r��2�

��cos�2h�0�t/��e−2�h�1�/��2�1−cos��t/��� − 1� . �29d�

Expanding the exponent to the leading order in t, we recover

−�h�1�t /��2 which is the exponent found in the fully time-dependent Born–Oppenheimer approximation �see Eq.�28g��.

The purely electronic time scale of the charge transfer isdetermined by h�0�. In those cases where the period Te

=�� /h�0� of the charge oscillations from the left to the rightsite is short, the charge transfer is ultrafast; i.e., several oreven many oscillations can take place before the nuclear dy-namics starts to play a relevant role. The fully time-dependent Born–Oppenheimer approximation is seen to bevalid as long as t��−1�. It is not rare that Te is of the orderof 1–4 fs,10 much faster than vibrational periods of vibra-tions, typically relevant for coupling different sites. The re-sults in Eqs. �29d� and �28g� can serve to estimate the timerange of validity of the Born–Oppenheimer approximation.Finally, let us mention that if the nuclear dynamics is com-puted classically for the above model and the ensuing elec-tronic motion, there will be no impact at all; i.e., the elec-tronic density will be that found in Eq. �27f� for fixed nuclei.

VI. SUMMARY

The study of the quantum nuclear dynamics accompany-ing electronic processes is, on the one hand, a rather intricateissue because of the several and sometimes even many par-ticipating electronic states. On the other hand, it might be ofhelp that many electronic processes are ultrafast and one may“only” need to compute accurately the nuclear dynamics forthe duration of the process and later on, when the electronicprocess is essentially over, be hopefully able to resort to lesscumbersome descriptions of the nuclear dynamics. TheBorn–Oppenheimer approximation provides an importantconcept in the case of a single stationary electronic state, andit is natural to introduce a similar approach for the nucleardynamics accompanying electronic processes. In this work,we attempt to study a fully time-dependent Born–Oppenheimer approximation. It turns out that this is not sostraightforward as in the traditional approach, but we hope tohave found the key ingredient for making the fully time-dependent approach senseful. This is done by introducing atime-dependent topological phase. The resulting equationsare analyzed and discussed.

As several or even many electronic states participate, itmight be of relevance to improve the description and to lettwo or more electronic wavepackets interact through thenuclear motion. This leads to the fully time-dependentgroup-Born–Oppenheimer approximation in analogy to whatis done for the nuclear dynamics on coupled stationary elec-tronic states. Again, however, it is necessary to introducetime-dependent topological phases, one for each of the par-ticipating electronic wavepackets. In the traditional case ofcoupled stationary electronic states, the concept of diabaticelectronic states has been amply found to be of great advan-tage for solving the resulting coupled equations. We have,therefore, also investigated the principle possibility to intro-duce time-dependent diabatic electronic wavepackets. If suchpackets can be found to a good approximation, the equationgoverning the quantum nuclear dynamics would simplifyconsiderably. The introduction of general unitary transforma-

124101-7 Born–Oppenheimer approximation and beyond J. Chem. Phys. 128, 124101 �2008�

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tions within the ansatz for the total wavefunction opens, atleast in principle, the door to investigate new physical con-cepts to study the nuclear dynamics accompanying electronicprocesses.

To illuminate how the fully time-dependent Born–Oppenheimer approximation operates, a simple model ofcharge transfer is studied. In this model, charge transfer cantake place purely electronically at frozen nuclei and we in-vestigate the impact of the nuclear motion. The electronicdensity is computed as a function of time for various ap-proximations and compared with the exact result which isalso derived. The model is a critical test for the Born–Oppenheimer approximation since this approximation de-scribes adiabatic dynamics, whereas the exact solution of themodel describes diabatic dynamics. It is found that the Born–Oppenheimer approach provides a systematic short-time ex-pansion of the terms which describe the impact of thenuclear dynamics in the exact result. The purely electroniccontributions are not affected.

ACKNOWLEDGMENTS

The author thanks Alexander Kuleff for fruitful discus-sions. Financial support by the DFG is acknowledged.

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