R. González-Férez and P. Schmelcher- Electric-field-induced adiabaticity in the rovibrational...

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Electric-field-induced adiabaticity in the rovibrational motion

of heteronuclear diatomic molecules

R. González-Férez1,* and P. Schmelcher2,3,†

1 Departamento de Física Moderna and Instituto “Carlos I” de Física Teórica y Computacional, Universidad de Granada,

E-18071 Granada, Spain2

Theoretische Chemie, Physikalisch-Chemisches Institut, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany3

Physikalisches Institut, Universität Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany

Received 26 November 2004; published 24 March 2005

We investigate the rovibrational dynamics of heteronuclear diatomic molecules exposed to a strong externalstatic and homogeneous electric field. We encounter in the presence of the field the effect of induced adiabaticcoupling among the vibrational and hybridized rotational motions. Exact results are compared to the predic-tions of the adiabatic rotor approach as well as to the previously established effective rotor approximation. Adetailed analysis of the impact of the electric field is performed: the hybridized and oriented rotational motion,the mixing of angular momenta, and the squeezing of the vibrational motion are observed. It is demonstratedthat these effects can well be accounted for by the adiabatic rotor approximation.

DOI: 10.1103/PhysRevA.71.033416 PACS numbers: 33.80.Ps, 33.55.Be, 33.20.Vq

I. INTRODUCTION

Molecules exposed to external fields represent, in spite of its substantial history, a very active and promising researcharea with several intriguing perspectives. Due to the key rolethat external fields play in the cooling, trapping, and guidingof atoms and molecules, the availability of molecular Bose-Einstein condensates 1–3 has stimulated further studies. In-deed, the long-range anisotropic dipole-dipole interaction be-tween ultracold polar molecules will give rise to interestingphysics, such as cold molecule-molecule collision dynamics4–7, molecular collective quantum effects 8,9, ultrahighresolution and high-precision spectroscopy, chemical reac-tions 10,11, molecular optics and interferometry, and po-tentially also to quantum computing 12. Moreover, theavailability of ultracold molecules might provide an im-provement of the precision of several experiments involvingmolecules, such as the measurement of the dipole moment of the electron 13,14, the measurement of the time variationof the fine-structure constant 15, or the study of weak in-teraction effects in chiral molecules 16.

In particular, electric fields play an important role in oneof the main experimental techniques to trap molecules. Anarray of time varying inhomogeneous electric fields has beenused to decelerate and trap polar molecules 17,18, provid-ing a different technique which can be used for a large vari-ety of neutral molecules. Recently, the control of the trans-

lational motion of Rydberg states of the H2 molecule hasbeen demonstrated by applying an inhomogeneous electricfield 19. The group of Bohn has introduced an uncommonspecies of molecular states called “field linked” states20,21 which are composed of two ground-state polar mol-ecules held at large internuclear separation under the jointinfluence of electric dipole forces and external electric fields.

They have shown how the main properties of these statesstrongly depend on the external electric field, leading to con-trol over the ultracold scattering properties 20,21.

Initially studies of molecules in electric fields were moti-vated by the possibility to get a deeper insight into chemicalreaction dynamics by controlling the orientation or alignmentof the involved molecules. At the end of the 1970s the ori-entation of symmetric top molecules with permanent electricdipole moments was achieved by using a hexapole electricfield 22,23. Only molecules in a few specific preselectedstates could be oriented this way. In the early 1990s theorientation of molecules in states became possible via thepassage of the corresponding molecules through an electric

field, known as the “brute force” method 24. Strong electricfields have subsequently been used to orient the rotationalmotion of diatomic and also of polyatomic molecules24–29. In the beginning only static electric fields were em-ployed and low-lying rotational states were considered. Sub-sequently also magnetic fields 30 were used to orient para-magnetic molecules 31 and more recently intense laserfields have been employed to align molecules 32–36.

Traditionally the theoretical description of the nuclear dy-namics of molecules in an electric field is based on the rigidrotor approximation 37, neglecting the “coupling” betweenthe vibrational and rotational motions and assuming a perma-nent dipole moment for the molecule. The pendular statesappear for strong electric field when the molecule is orientedalong the field direction and the rotational motion becomes alibrating one, each pendular state being a coherent superpo-sition of field-free rotational states 38. It is only recentlythat the authors provided a full rovibrational descriptionFRV of a heteronuclear diatomic molecule in a homoge-neous electric field including the coupling between the vibra-tional and rotational motions and taking into account that theelectric dipole moment functions depend on the internuclearcoordinate 39. An effective rotor approximation ERAwhich describes the effect of the electric field on rigid di-atomic molecules 39 was developed. The ERA includes

*Electronic address: [email protected]†Electronic address: [email protected]

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main properties of each vibrational state, and it has beenshown to describe the effect of the electric field even forhighly excited rovibrational states of the molecule, being su-perior to the traditional rigid rotor approach.

In the present work we go beyond the regime of validityof the effective rotor approach which represents a crude adia-batic approximation with respect to the separation of the vi-brational and rotational motion. In particular we investigate

the regime for which a fully adiabatic separation of the rovi-brational motion is necessary: the fast vibrational motion de-pends now parametrically on the angular coordinates. Thelatter effect is exclusively due to the presence of the externalelectric field. Since we focus on effects due to the field-induced adiabaticity and their theoretical description in gen-eral we refrain from using potential-energy curves and dipolemoment functions belonging to specific molecules but useparameter-dependent models of them in order to address andcover as many as possible physically different situations. Theuse of these models does not affect the general validity of our results. We address the regime of field strengths forwhich a nonperturbative description of the nuclear motion is

necessary assuming that the effects of the electric field on theelectronic motion can be described perturbatively.The paper is organized as follows. In Sec. II we define our

rovibrational Hamiltonian and we briefly discuss some spe-cifics of our computational method. Here we also present thekey aspects of the adiabatic separation of the vibrational androtational motions in order to obtain the adiabatic rotorHamiltonian. In Sec. III we describe the potential-energycurves and electric dipole moment functions used to model ageneral heteronuclear diatomic molecule. In Sec. IV wepresent the results and their discussion, including a detailedcomparison of the adiabatic rotor approach ARA with thefull rovibrational description. The conclusions and outlookare provided in Sec. V. Atomic units will be used throughout,unless stated otherwise.

II. ROVIBRATIONAL HAMILTONIAN

AND THE ADIABATIC ROTOR APPROXIMATION

We employ the Born-Oppenheimer approximation for theHamiltonian of a heteronuclear diatomic molecule which isassumed to be in its 1

+ electronic ground state. The mol-

ecule is exposed to an external homogeneous and static elec-tric field. In the rotating molecule fixed frame with the coor-dinate origin at the center of mass of the nuclei theHamiltonian describing the nuclear motion takes on the fol-lowing appearance:

H = −2

2 R2

R R2

R +

J2 ,

2 R2 + R − FD Rcos ,

1

where R and , are the internuclear coordinate and Eulerangles, respectively, and is the reduced mass of the nuclei.J , is the orbital angular momentum, R represents theelectronic potential-energy curve PEC of the molecule infield-free space, and D R is the corresponding electronicdipole moment function EDMF. The first and second terms

are the vibrational and rotational kinetic energies, and thelast term provides the interaction with an electric field of strength F , which is oriented parallel to the z axis of thelaboratory frame. As it was stated above, we consider theregime where perturbation theory holds for the description of the electronic structure but a nonperturbative treatment isindispensable for the corresponding nuclear dynamics.

In the field-free case each state is characterized by its

vibrational , rotational J , and magnetic M quantum num-bers. In particular, the exact solution to the Schrödingerequation belonging to the Hamiltonian 1 is a product of apurely R-dependent vibrational wave function that dependsparametrically on the conserved angular momentum J and aspherical harmonic depending exclusively on the angles , .In the presence of an external electric field only the magneticquantum number M is conserved, giving rise to a noninte-grable two-dimensional dynamics in R , space. In order tosolve the corresponding rovibrational equation of motion weuse a hybrid computational approach, which combines dis-crete and basis-set methods. For the angular part a basis-setexpansion in terms of associated Legendre polynomials is

used, taking into account that M is conserved. The vibra-tional degree of freedom is treated by a discrete variablerepresentation. Due to the typical shape of the molecularPEC, we choose the radial harmonic oscillator discrete vari-able representation, where the odd harmonic oscillator func-tions are taken as basis functions. Employing the variationalprinciple, the initial differential equation is finally reduced toa symmetric eigenvalue problem which is diagonalized withthe help of Krylov space techniques.

The theoretical description of the angular motion of a di-atomic molecule in an external field is traditionally based onthe rigid rotor approach for which the so-called pendularHamiltonian reads

H = J2

2 Req2 − FDeqcos , 2

where Req and Deq are the equilibrium internuclear distanceand the corresponding dipole moment, respectively. For di-atomic molecules this rigid rotor Hamiltonian is integrableand it was solved numerically in the seventies by von Mey-enn 37. Recently, the authors went beyond this rigid rotordescription and proposed an effective rotor approximationwhich includes the main characteristics of each vibrationalstate 39. The corresponding effective rotor Hamiltonianreads as follows:

H

R

=

1

2 R−2

0

J2

− F D R 0

cos + E 0

3

with R−2

0=

0 R−2

0, D R

0 =

0 D R

0, and

0 and E

0 are the field-free vibrational wave function andenergy, respectively. The total wave functions for the nuclearmotion are

0 R · where are the eigenfunctions of the Hamiltonian 3. The ERA takes into account the vibra-tional state dependent moment of inertia and the dependenceof the electric dipole moment on the vibrational coordinate,which are of particular importance for highly excited vibra-tional states. It was shown to properly describe both low-

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lying and highly excited rovibrational states exposed to anelectric field. In particular it is superior to the traditionalrigid rotor approach 39.

The effective rotor represents, in the terminology of anadiabatic approach, a crude adiabatic approximation to thetrue wave function thereby neglecting the influence of theelectric field on the vibrational motion. Obviously, this ap-proximation is expected not to hold for arbitrary parameters

and regimes, i.e., for arbitrary excitations, field strengths,and molecular species. In the present work we investigate therovibrational motion in a regime where the vibrational mo-tion is modified by the electric field, i.e., it becomes para-metrically dependent on the slow angular variable . Thisleads to interesting effects in the rovibrational spectra andeigenfunctions which occur typically in the strong-field re-gime.

Following Ref. 39 we exploit the conservation of theangular momentum associated with the angular motion dueto by the ansatz R , , = M R , eiM , with M beingthe magnetic quantum number. We restrict ourselves to thecase M =0. Let us now perform an adiabatic separation of the

vibrational and rotational motion. To this end we assume thatthe vibrational problem has been solved for a specific valueof the rotational coordinate ,

−2

2 R2

R R2

R + R − FD Rcos R;

= E R; , 4

where R ; is a member of the orthonormal vibrationaleigenfunctions labeled by . R ; depends parametricallyon the angle . Making the following ansatz for the rovibra-tional wave function:

R, =

R; ,

and inserting it in the rovibrational equation of motion be-longing to the Hamiltonian 1, after left multiplication with

* R ; , and performing the integral over R, using the or-thonormality of the vibrational adiabatic eigenfunctions aswell as Eq. 4 we arrive at

1

2 R−2 J

2 + E − E +

A 2

2+ A

1 J

+

A 0

2J2

= 0 5

with

R−2 = 0

* R; R; dR ,

A j =

1

0

* R; J j R; dR, j = 0,1,2, 6

where A j are nonadiabatic coupling terms involving differ-

ent vibrational eigenfunctions, E represents a potentialfor the motion in space, and 1/ 2 R−2 J

2 is an effective

rotational kinetic energy. If the nonadiabatic coupling ele-ments are neglected Eq. 5 reduces to a single channel equa-tion and an adiabatic separation and approximation of theangular and radial motions has been achieved.

Let us comment on the vibrational equation of motion 4.For weak electric fields the interaction term −FD Rcos issignificantly smaller than the vibrational spacing due to theelectronic PEC. This allows us to use the field-free vibra-

tional wave functions and to apply first-order perturbationtheory see Ref. 39 with respect to the electric-field termsthereby leading to the effective rotor Hamiltonian 3. How-ever, for sufficiently strong electric fields and/or highly ex-cited vibrational states, the interaction with the electric fieldleads vibrational states that depend parametrically on the an-gular variable . We therefore enter a regime for which theabove adiabatic separation still holds but the vibrational mo-tion is affected by the electric field. In practice one thensolves the vibrational equation of motion 4 on an angulargrid, i, i =1,… , N thereby obtaining R ; i and E i.This allows us to compute the expectation values R ; i R−2 R ; i and in principal also the nonadia-

batic coupling elements A j which both depend on .Here we assume the validity of the adiabatic approxima-

tion to the separation of the vibrational and rotational motionand neglect all nonadiabatic coupling elements. This definesthe “adiabatic rotor approach” and the “adiabatic rotorHamiltonian” describing the rotational motion of the molecu-lar system:

1

2 R−2 J

2 + E − E = 0, 7

where the first term is an effective rotational kinetic energyand the second one represents the interaction with the elec-

tric field. The rotational equation of motion 7 looks differ-ent for each vibrational state since it explicitly contains theexpectation values R−2 and the vibrational energy E ,which have to be computed for each state.

The main difference of the adiabatic rotor approximation7 compared to the previously obtained effective rotor ap-proach 3 is the way the vibrational motion is treated. Theexpression 7 takes into account the influence of the electricfield on the vibrational motion, which might be of impor-tance for very strong electric fields or highly excited vibra-tional levels due to the field-induced adiabatic coupling be-tween the rotational and vibrational motions. In contrast tothis we neglected the influence of the electric field on thevibrational motion in case of the effective rotor approach.Obviously, the effective rotor approach is contained in theadiabatic rotor approach. Both approximations are, however,complementary in the following sense. There is a large num-ber of situations depending on the field strength, the EDMF,and on the part of the spectrum under consideration, whereeither the ERA or the ARA might be sufficient to describe thebehavior and properties in the field.

Let us comment on how to solve the resulting equations4 and 7 of the above scheme. In a first step, and in orderto facilitate the computational procedure, we choose the ze-ros of the N Rth order associated Legendre polynomial as grid

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points for the angular coordinate. The vibrational equation4 is integrated with the help of the discrete variable ap-proach N R times on the grid i, thus we obtain the vibra-tional wave functions R ; i and the corresponding spec-trum E i. The expectation value R−2 is computed. Thesequantities, E i and R−2 , are introduced in the effectiverotational equation of motion 7, which is solved by meansof a basis-set expansion with respect to the associatedLegendre polynomials.

There are two reasons for applying the adiabatic rotorapproximation compared to the full rovibrational descriptionof the problem. First, a fully adiabatic approach describesemerging field-induced effects. Second, from a computa-tional point of view it provides a drastic reduction of thenumerical effort. In both cases we need the same number of grid points N V in the discrete variable representation appliedto the vibrational coordinate, and the same number of angu-lar functions N R in the basis-set expansion performed for therotational coordinate. In the full rovibrational description wehave to diagonalize a N N real symmetric banded eigen-value problem with N = N V · N R. However, in the case of the

ARA we first diagonalize N R times a N V N V real symmetricbanded eigenvalue problem for the vibrational part and sub-sequently for each vibrational state an N R N R real symmet-ric eigenvalue problem to diagonalize the rotational equationof motion.

III. POTENTIAL-ENERGY CURVE AND ELECTRONIC

DIPOLE MOMENT FUNCTION

The electronic potential-energy curves and the electronicdipole moment functions provide valuable information onthe properties of a molecular system including its rovibra-tional spectrum and its response to external fields. Both

quantities can be obtained from ab initio calculations and/orexperimental results in the literature. Since we are interestedin predicting general properties and principal effects with thefocus on their understanding and theoretical description wedo not address a specific molecular system but use modelfunctions for the PEC and the EDMF.

As a PEC we employ the Morse potential

R = d ee−2 R− Re − 2e− R− Re, 8

where Re and d e are the equilibrium internuclear distance andthe depth of the potential well, respectively. Here we choosethe values =0.5, d e =0.02, and Re =6.0 a.u. being motivatedby the shape of the PEC of heteronuclear alkali dimers 40.The latter are characterized by a large equilibrium distance Re 6–7 a.u., and a very shallow potential well. We willalso study the case =0.5, d e =0.02, and Re =2.2 a.u. Weremark that our model potential does not possess the correctasymptotic behavior for large R which is of importance forvery high excitations close to the dissociation threshold. Ourinvestigation therefore focuses on not too high excitationsfor which the asymptotics is irrelevant.

In order to illustrate the difference with respect to theenergy scales of the vibrational and rotational motions wehave computed the vibrational and rotational energy spacingfor a vanishing electric field. The rotational and vibrational

spacing are E r = E ,1 − E ,0 and E

v

= E +1,0− E ,0, respec-tively, with E , J being the field-free energy of the , J state.Figure 1 presents these two quantities as a function of thevibrational quantum number 0 20 for the PECs with

Re =2.2 a.u. and Re =6 a.u. Note that the vibrational spacingis independent of the equilibrium distance Re, and the differ-ence between both sets of parameters is due to the rotationalspacing. For the Morse potential with Re =2.2 a.u. the ratiobetween both quantities takes the initial value E

v

0 / E r 0

53, and it decreases monotonically as is increased, reach-ing for the highest state the value E

v

20 / E r 2039. For the

Morse with Re =6.0 a.u. a similar behavior is observed, al-though now the ratios are E

v

0 / E r 0 396 and E

v

20 / E r 20

319, i.e., almost one order of magnitude larger than in theprevious case. It is clear from these results that the energyscales associated with the rotational and vibrational motionsare well separated.

For the EDMF we take a Gaussian function given by D R = M 0e− A R − Re − 2

, 9

where Re is the equilibrium distance, A provides the width of the Gaussian, is a shift with respect to Re, and M 0 deter-mines the strength of the EDMF at the maximum R = Re +.By choosing this kind of function we ensure a properasymptotic behavior of the EDMF, satisfying bothlim R→0 D R=0 and lim R→ D R = 0.

We have selected three different shapes for the EDMF, themain difference between them being the position of themaximum and the width of the Gaussian function. The cho-sen parameters are given in Table I. For the different appear-

ances of the three functions we refer the reader to Fig. 2: DC is centered at Re , DW is a broader function, its maximum

FIG. 1. Field-free vibrational energy spacings for the Morsepotentials with Re =2.2 a.u. and Re =6.0 a.u. as a function of thevibrational quantum number . Field-free rotational energy spacingsfor the Morse with Re =2.2 a.u.+ and Re =6.0 a.u.Ã as a functionof the vibrational quantum number .

TABLE I. Parameters of the electric dipole moment functions;see text.

Name M 0D A a.u. a.u.

DC 5 2 0

DS 5 2 2

DW 5 0.5 1


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being shifted with respect to Re, and DS is an even furthershifted but again narrower dipole moment function. In manycases the electric dipole moment of heteronuclear diatomicmolecules is a single humped function possessing its maxi-mum at a position close to Re. However, exceptions do occursuch as the double humped dipole momentum function of the

CO molecule that possesses a zero close to Re and a broadouter hump for large internuclear distances.

IV. RESULTS AND COMPARISON OF THE ADIABATIC

ROTOR APPROXIMATION WITH THE FULL

ROVIBRATIONAL DESCRIPTION

We have performed a full rovibrational analysis i.e., wesolved the Schrödinger equation belonging to Eq. 1 of theeffects of an electric field on the six models for molecularsystems, i.e., the Morse with Re =2.2,6 a.u., and the threeEDMFs being centered DC , shifted DS, and widened DW . The reduced mass of the CO molecule

=12 498.102 a.u. has been used. In order to show very wellpronounced effects for our model system we will choose a

field strength F = 10−3a.u. corresponding to 514 MV/mwhich is somewhat stronger than the static fields available inthe laboratory. We emphasize, however, that the observedeffects are expected to be pronounced for significantly lowerlaboratory field strengths for highly excited rovibrationalstates of certain species, such as the alkali dimers.

For each molecule, we have computed the states with0 20 and angular momentum J =0 for M =0 corre-sponding to the first 21 vibrational states in the absence of the field. The expectation values cos , J2, and R, to-gether with density profiles of the eigenfunctions enable usto find and analyze relevant phenomena. Equally we performfor our model systems studies in the framework of the adia-batic rotor approach ARA and the effective rotor approach

ERA. A comparison between these three approaches exact,ARA, ERA allows us to conclude upon the validity of thedifferent approximations. Let us introduce the relative differ-ence between the results obtained in the ERA and ARA com-pared to the full rovibrational description of the problem:

A

X = A

F − A

X

A

F with X = E ,A , 10

where A

F represents one of the expectation values mentionedabove and the upper right index F ,E ,A refers to the exact,ERA, and ARA approach.

In the following we will use and J as labels of therovibrational states. These labels correspond to quantumnumbers conserved quantities only in the case of the ab-sence of the electric field whereas in the presence of the fielda strong rotation and weak vibration mixing of field-freestates takes place.

A. Morse potential for Re=6.0 a.u.

Figure 3a shows the expectation values of cos for therovibrational states with 0 20 emerging from the J = 0states for F =0 as a function of the vibrational number forthe Morse potential with Re =6 a.u. Results for the threeGaussian EDMF are included in this figure. cos provides

a measure of the orientation of the molecule with respect tothe field direction: the closer it is to 1, the stronger is theorientation. For DC all states show a strong orientation,which slightly decreases as the degree of excitation is in-creased, from the rovibrational ground state with cos =0.9827 to the =20 state with cos = 0.9626. The behav-ior of the states in case of DS is completely different. Theeffect of the electric field on the low-lying states is weakcompared to the high-lying states. For the rovibrationalground state we have cos =0.5360. The reason for thisbehavior is obviously the small values of DS in the neighbor-

FIG. 2. The electric dipole moment functions used in our com-putations, centered DC , shifted DS, and wider DW as a function of

R − Re.

FIG. 3. Color online a The expectation values cos , b therelative error cos

E of the effective rotor approximation, c therelative error cos

A of the adiabatic rotor approximation as afunction of the vibrational label 0 20 for states emerging from

J =0 for F =0 for the Morse potential with Re =6.0 a.u. and the threeEDMFs DC , DS +, and DW .


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hood of the equilibrium internuclear distance. The orienta-tion of the states increases as the vibrational number in-creases: It reaches a maximum for =10 being cos =0.9761 and slightly decreases thereafter. For DW , cos increases very little with increasing and reaches a maxi-mum for the state =5 with cos =0.9813. For =20 weencounter cos =0.9766. It is clear from Fig. 3 that theeffects of the electric field on the orientation depend on the

state under consideration but also strongly on the chosenEDMF. For these model systems and due to the strongelectric-field strength all the considered states show a verystrong orientation. In Fig. 3b, the relative error for thisexpectation value computed by using the ERA, cos

E , isshown as a function of the vibrational number for the threeEDMF. The results provided by the ERA for the expectationvalue cos for this molecule using the dipole momentfunctions DC and DW show a good agreement for all thestates under consideration, i.e., we have cos

E 0.01 for

0 20. Note, however, that in case of DS the states for =2–5 possess a relative error larger than 0.01, while for theremaining states cos

E 0.01. Figure 3c presents the

relative error of cos using the ARA,

cos

A

, as a func-tion of . Again all three cases of EDMF are considered. Theresults obtained for cos are excellent, i.e., cos

A2

10−4 for all the states and any EDMF. Apart from a singleexception =10 we have cos

Acos

E demon-strating that the adiabatic rotor approach is superior to theeffective rotor one.

The dependence of the expectation value J2 on the vi-brational label is illustrated in Fig. 4a for the threeEDMF. J2 provides a measure for the mixture of the field-free angular momentum states with fixed M J , i.e., it describesthe hybridization of the rotational motion for F =0. The ef-fects due to the electric field depend not only on the chosen

EDMF but also strongly on the degree of excitation as al-ready indicated when studying the corresponding behavior of cos . For DC , J2 decreases significantly with increasing . All states show, however, a very strong hybridization of the rotational motion, i.e., J2 =28.321 for the rovibrationalground state and J2 =12.872 for the =20 state. For DW weencounter also a strong mixing: In this case J2 slightlyincreases with reaching a maximum for =6 with J2=26.229 and decreasing thereafter. In case of DS the low-lying states exhibit a much smaller hybridization comparedto the previous cases: For the rovibrational ground state wefind J2 =0.527. The reason for this effect is that the low-lying states “do not feel” the EDMF DS, due to the small

overlap of the wave function and the EDMF. In this case J2rapidly increases as increases, reaching a maximum for

=10 with J2 =20.577 and slightly decreases thereafter. Fig-ure 4b shows the corresponding relative errors of this ex-pectation value computed by means of the ERA, J2

E , as afunction of the vibrational number . Note that for eachEDMF there is a significant number of states with J2

E

0.01. Even more, for DS the states with =3–8 showJ2

E 0.1. Therefore we can conclude that the ERA is not

good enough to describe the hybridization of the rotationalmotion taking place in this systems. The relative errors for

the ARA, J2

A, as a function of are presented in Fig.

4c. This figure clearly illustrates the advantage of our pro-posed approximation ARA compared to ERA. For the threeEDMF and all the considered states we obtain relative errorssignificantly smaller than 0.01. We therefore conclude thatthe adiabatic rotor approach accurately describes the proper-ties of the wave function whereas the effective rotor approxi-mation fails to do so, at least in most cases.

To complete the description of the effect of the electricfield on our systems, we illustrate in Fig. 5a the expectationvalue of the vibrational coordinate R as a function of forthe same set of states and EDMF. For comparison also thecorresponding field-free values of R have been included.This expectation value provides a measure of the size of the

molecular state. In the case of DC the states satisfy RF

R0, where F indicates the presence of the field and 0stands for F =0. They are attracted towards the maximum of DC at R =6 a.u. and have a lower energy compared to thefield-free case. For the low-lying states, =0 –2, in the caseof DS we obtain RF R0, the maximum of DS at R

=8 a.u. is too far, and therefore the attraction is not strongenough to modify the vibrational part of the wave function.The states =3–11 show a completely different behavior,i.e., RF R0 being stretched due to the attraction of theEDMF thereby lowering their energy. The states for

FIG. 4. Color online a The expectation values J2, b therelative error J2

E of the effective rotor approximation, c therelative error J2

A of the adiabatic rotor approximation as a func-tion of the vibrational label 0 20 for states emerging from J

=0 for F =0 for the Morse potential with Re =6.0 a.u. and the threeEDMFs DC , DS +, and DW .


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=12– 20 possess RF R0 being also attracted by themaximum of the EDMF. However, for these states a signifi-cant part of their amplitude is at distances larger than 8 a.u.and that is why they are squeezed compared to their field-free extension. A similar behavior is observed when DW isused. The states for =0–6 are stretched compared to theirfield-free counterparts. For the states =7–9 we have RF

R0. Due to the attraction of the EDMF maximum, thestates with 10 are again squeezed. We would like to pointout the importance of these results: they show how the elec-tric field is affecting the vibrational part of the wave functionof the molecule. Indeed, we can identify this effect as anadiabatic coupling between the vibrational and rotational

motion induced by the strong electric field. Figures 5b and5c show the relative error of R computed by using ERAand ARA, R

E and R

A, respectively, as a function of for the three EDMFs. Comparing these two figures, the ARAapproximates much more accurately the FRV results than theERA does. Only for a few states does the ERA provide anadequate description of the effects of the electric field.

B. Morse potential for Re=2.2 a.u.

Figure 6a shows the expectation values cos for thestates 0 20 emerging from the states with J =0 for F

=0 for the Morse potential with Re =2.2 a.u. Results for thethree Gaussian EDMFs DS , DC , and DW are included. Therelative errors obtained by using the ERA and the ARA,cos

E and cos

A, are presented in Figs. 6b and6c, respectively. For DC all the states exhibit a strong ori-entation, which smoothly decreases as the degree of excita-tion increases, from cos =0.9526 for = 0 t o cos =0.8926 for the =20 state. For DS the low-lying statesshow a weak orientation such as cos =0.1092 for =0.The orientation quickly increases with increasing , reachesa maximum for =9 with cos =0.9312 and slowly de-creases thereafter. The orientation of the states belonging to DW increases from =0 to =2 with cos =0.9485 de-creases slowly thereafter, except for the state =19 withcos =0.8910 for which a dip of this expectation valuewith respect to the values of the neighboring levels is ob-served. The corresponding expectation values provided bythe ERA for DC see Fig. 6b agree well with the exact onescos

E 0.01. Similar results are found for DW except

again for the =19 state cos 19E

0.05, which interruptsthe smooth behavior of the ERA relative error for the otherstates. For DS only the states with =2–8 show a relativedifference larger than 0.01. In Fig. 6c we observe how theARA improves the results of the ERA and apart from a fewexceptions we always find cos

E cos

A. The ARArelative error obtained for the =19 state using DW iscos 19

A 0.05 being much larger than the relative errorfor the remaining states.

Figures 7a–7c illustrate the behavior of J2 , J2

E ,and J2

A, respectively, as a function of for the same set

FIG. 5. Color online a The expectation values R, b therelative error R

E of the effective rotor approximation, c therelative error R

A of the adiabatic rotor approximation as a func-tion of the vibrational label 0 20 for states emerging from J

=0 for F =0 for the Morse potential with Re =6.0 a.u. and the threeEDMFs DC , DS +, and DW . For completeness the field-free values of R have also been included.

FIG. 6. Color online Same as in Fig. 3 but for Re =2.2 a.u.


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of states and the three EDMFs. The hybridization of thismodel system is much smaller than the hybridizationachieved by the Morse potential with Re =6 a.u. due to itssmaller rotational constant. For DC , J2 decreases as isincreased, from J2=10.049 for =0 to J2 =4.163 for

=20. For the DS we obtain a small hybridization for low-lying states, J2=0.018 for the rovibrational ground state,

and strong mixing of the rotational motion for the states

3, J2 increases with increasing , reaching a maximumfor =10 with J2=7.523, and decreasing thereafter. For DW

first J2 slightly increases with , reaches a maximum forthe =5 state with J2=9.473, and decreases as the degreeof excitation is further increased. As for the expectationvalue cos , the =19 state shows an anomalous behavior,having a larger J2 than the rest of the neighboring states:J2=11.128. The relative error of the ERA for DC for1 10 and =20 yields J2

E 0.01. For DS we find

that only the states =12, 16, 17, and 18 satisfy J2

E

0.01. For the states 3 9 we encounter an error being

larger than 0.1. In the case of DW we see that only the stateswith 11 18 possess J2

E 0.01. Again the state

=19 is exceptional with J219E 0.37. The ARA relative

errors, see Fig. 7c, improve in most cases but not alwaysthe results obtained by the ERA. For the state =19 for DW

we find a large relative error J219A 0.37.

The expectation value R, the relative errors R

E and R

A of the ERA and ARA are shown as a function of inFigs. 8a–8c, respectively. By comparing Figs. 5a and8a we conclude that the behavior of R for the threeEDMF is similar for the two cases Re = 2.2 a.u. and Re

=6 a.u. This holds both for the dependence of R on thevibrational excitation and on the choice of the EDMF. More-over, the phenomenon of squeezing and stretching of thevibrational wave function can be observed equally for Re

=2.2 a.u. We remark that the behavior of R as a function of is smooth for all the levels and any EDMF. The compari-

son of the relative errors of the ERA and ARA support ourprevious observation that the ARA is superior to the ERA.

Let us comment on the anomalous behavior of the expec-tation values J2 and cos for the state =19 when com-pared to its neighboring states. Closely inspecting higher ro-tational excitations it turns out that the state emerging fromthe field-free state with =18, J = 3, M =0 comes very closein energy to the considered =19 state. For the field strengthF = 10−3 a.u. the energetical spacing of these two states isvery small E = E 19,0 − E 18,3 510−8. For F =0 thesetwo states would not interact. However, we conjecture thatthe electric field induces a strong nonadiabatic mixing be-tween these two states belonging to two different vibrational

bands thereby causing the observed uncommon properties.To study this phenomenon goes however beyond the scopeof the present investigation.

Comparing the relative errors obtained for J2 andcos , one realizes that the latter expectation value is nor-mally better described by the ERA and ARA than J2. J2provides a measure for the error we perform in our approxi-mate approaches, since the computation of it in theseschemes neglects the elements 6. Therefore we can con-sider it as an indicator of the quality of the eigenfunctionscomputed within a certain approximation. We also conclude




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that the adiabatic rotor approach works better for the Morsepotential with Re =6 a.u. compared to the system with Re

=2.2 a.u. We believe that the reason herefore is the separa-tion of the vibrational and rotational energy scales. For theMorse with Re =6 a.u. the ratio of the rotational and vibra-tional spacing is one order of magnitude smaller compared tothe Morse with Re =2.2 a.u. see Fig. 1.

C. Probability densities

From the many data obtained within the present investi-gation let us show some representative examples for thefield-induced changes introduced in the rovibrational prob-ability densities. We compare the exact FRV results with theresults of the ARA and the ERA for three typical states for acertain EDMF. As already discussed in Sec. II the ERA em-ploys the field-free vibrational wave function whereas theARA takes into account the adiabatic coupling of the rota-tional and vibrational motions.

First we consider the rovibrational ground state for theMorse potential with Re =6 a.u. and the centered EDMF. Fig-ures 9a–9c show contour plots in the R , plane of thesquare of the wave function computed with FRV, ARA, andERA, respectively. The orientation achieved for this state iscos =0.983. The position of the maximum of the wavefunction for F =0 at R =6.02 a.u. for =0 is slightly modifiedby the electric field to R =6.01 a.u. and this is reproduced bythe ARA. The value at the maximum R =6.01, =0 for theFRV probability density is 6.01,02=4.90 being repro-duced by the ARA with 6.01,02 =4.90. However, theERA provides a lower value 6.02,02 =4.25. This effectcan be seen in Figs. 9a–9c. The ERA wave function isvibrationally more extended than the FRV wave function,i.e., we observe a compression due to the field. The ARAreproduces this effect, which is again a manifestation of thefield induced adiabatic coupling between the vibrational androtational motions.

As a second example we analyze the rovibrational groundstate of the Morse potential with Re =2.2 a.u. for DW . Thecontour plots of the probability densities for the FRV, ERA,and ARA are presented in Figs. 10a–10c, respectively. Asin the previous case the wave function is strongly orientedalong the field direction, cos =0.948. The ARA wavefunction provides a good approximation to the FRV one,whereas major discrepancies are observed in case of theERA wave function. The maximum of the density for =0 isnow at R =2.36 a.u., i.e., due to the attraction of the EDMF ithas been shifted from its field-free position R =2.22 a.u. The

ARA yields R = 2.34 a.u., whereas the ERA provides thefield-free value. In addition, the ERA yields for the maxi-mum 2.22,02=1.27, compared to the full rovibrationalresult 2.36,02 =1.37, while the ARA gives a much bet-ter approximation 2.34,02=1.38. This effect can be alsoobserved in Figs. 10a and 10c.

As a last example we present the state emerging from thefield-free vibrational quantum number =5 and the rota-tional quantum number J =0 for the Morse with Re

=2.2 a.u. and the shifted EDMF. The contour plots of theprobability densities for the FRV, ARA, and ERA are in-

cluded in Figs. 11a–11c, respectively. Compared to thetwo above analyzed states, this state shows a weaker orien-tation cos =0.894. The probability density possesses sixmaxima. We will concentrate here on the most pronounced

one with the largest value for R. In the field-free case, thismaximum is at R =3.31 a.u. In the presence of the electricfield, the FRV analysis yields R =3.46 a.u., i.e., the wavefunction, attracted by the maximum EDMF shifted at R

=8 a.u., is shifted towards larger values of the vibrationalcoordinate. Using the ARA we obtain R =3.44 a.u., which isa very good approximation to the exact result. Here themaximal density is 3.46,02 =0.4 being slightly overes-timated by the ARA result 3.44,02=0.42. For the ERAwe obtain a much lower value, 3.31,02=0.31. In addi-tion, the ERA is also not able to reproduce the extension of

FIG. 9. Color online Probability density of the rovibrationalground state of the Morse potential with Re =6.0 a.u. and the cen-tered EDMF for the field strength F =10−3 a.u.


033416-9



each hump of the wave function, see Fig. 11c compared toFig. 11a.

V. SUMMARY AND OUTLOOK

The rovibrational motion of diatomic molecules is givenby a product of a J -dependent vibrational and a rotationalwave function many of the Coriolis coupling terms vanishin case of an electronic ground state of 1

+ symmetry and we

neglect the remaining ones. In the presence of a static ho-mogeneous electric field that is weak enough to be treatedperturbatively with respect to the electronic structure of themolecule but strong enough to act nonperturbatively on its

nuclear motion this separability is no longer present. Still, fornot too strong fields and/or rigid molecules, i.e., moleculesfor which the energy scales of the vibrational and rotational

motions are well separated, the dominant effect of the elec-tric field is to hybridize the rotational motion. The effectiverotor approximation developed in a previous work of theauthors describes this hybridization accurately and in par-ticular it provides the correct description of the angular mo-tion depending on the vibrational state. Here the vibrationalmotion is assumed to be not influenced by the electric field.In contrast to this the traditional pendular state approach pro-vides a hybridization that is independent of the vibrationalstate.

In the present work we made a step beyond the above-described regime and investigated stronger fields and/or flop-

FIG. 10. Color online Probability density of the rovibrationalground state of the Morse potential with Re =2.20 a.u. and DW forthe field strength F =10−3 a.u.

FIG. 11. Color online Probability density of the state =5emerging from the field-free J =0 state of the Morse potential with

Re =2.20 a.u. and DS for the field strength F =10−3 a.u.


033416-10



pier systems. Since we are interested in new phenomena andproperties we did not address a specific molecule, i.e.,potential-energy curve and electric dipole moment function,but a model—the Morse potential—possessing variable pa-rameters and we study three models for the electronic dipolemoment function covering typical cases. The field strengthwe have been using F =10−3 a.u. corresponding to514 MV/m is somewhat above the experimentally acces-

sible static field strength. However, we expect that the prin-cipal effects we have found do occur in certain regimes andspecies at significantly lower field strengths. Moreover, ourresults should be equally applicable to the quasistatic regimeof low-frequency electromagnetic fields for which the dy-namical and electromagnetic frequency scales are well sepa-rated. An estimation of the semiclassical tunneling rate dueto the presence of the field F = 10−3 a.u. indicates that theelectronic ionization process can safely be neglected. Fur-thermore, in many cases the interaction with the field due tothe polarizability of the molecule can equally be neglecteddue to its quadratic dependence on the field strength.

Starting from the rovibrational equation of motion in themolecule fixed frame we have developed an adiabatic sepa-ration of the vibrational and rotational motions. It turns outthat the effective rotor approximation is a crude adiabaticapproach in this framework: the fast vibrational motion is thefield-free one and in particular independent on the slow dy-namical variable, i.e., the angular motion. We established thefull adiabatic approximation, the so-called adiabatic rotor ap-proach, to the rovibrational motion for which the vibrationalwave function depends parametrically on the angular vari-able. Subsequently studies for several model systems includ-ing many vibrational excitations have been performed. Ourfocus has been on the states emerging from the field-freeground rotational states J =0 for M =0. It turns out that ap-plying the above field strength we indeed enter the regime

where the ERA description is no longer adequate and field-induced effects and full adiabaticity occur. The ARA accu-rately describes the energies and physical properties, i.e., to-

tal wave functions in this regime. We have analyzed theorientation effects, the mixing of angular momenta, and thevibrational stretching and squeezing effects due to the exter-nal field in detail. In the case of a shifted electric dipolemoment function we observed a distortion of the wave func-tion specifically a squeezing effect towards the maximum of the EDMF. We remark that these properties and effects de-pend on the state under consideration which adds to the va-

riety of possible behavior and properties in the presence of the field.

Paradigms of heteronuclear diatomic molecules with alarge equilibrium internuclear distance, a small dissociationenergy, and a large maximal electric dipole moment are thealkali dimers. The latter are currently in the focus of ultra-cold molecular physics since they constitute the prototype of an ultracold quantum gas with long-range dipole-dipole in-teractions. Several experimental groups are working in thisdirection 41–43. However, there is still a need for accuratepotential-energy curves and electric dipole moment functionsfor these species, naming specifically the LiCs dimer.

The vibrational state dependence of the adiabatic rotor

equation together with the field-induced adiabaticity of thevibrational and rotational motion open interesting perspec-tives for the application of the external field to hybridize therovibrational motion. Natural extensions of the present workwould include the investigation of the hybridization of statesemerging from higher rotational excitations and/or studies of states with nonzero magnetic quantum numbers M .

ACKNOWLEDGMENTS

The authors thank H. D. Meyer and M. Weidemüller forvery helpful discussions. R.G.F. gratefully acknowledges thesupport of the Junta de Andalucía under the program Re-torno de Investigadores a Centros de Investigación An-

daluces, the Alexander von Humboldt Foundation, and fi-nancial support for traveling by Spanish Project No.BFM2001-3878-C02-01 MCYT.

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