Rovibrational dynamics of LiCs dimers in strong electric fields

Chemical Physics 329 (2006) 203–215 www.elsevier.com/locate/chemphys

Rovibrational dynamics of LiCs dimers in strong electric fields R. Gonza´lez-Fe´rez a

a,*

, M. Mayle b, P. Schmelcher

b,c

Instituto ‘Carlos I’ de Fı´sica Teo´rica y Computacional and Departamento de Fı´sica Moderna, Universidad de Granada, E-18071 Granada, Spain b Theoretische Chemie, Physikalisch-Chemisches Institut, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany c Physikalisches Institut, Universita¨t Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany Received 26 April 2006; accepted 15 June 2006 Available online 28 June 2006

Abstract We investigate the effects of a strong electric field on the rovibrational dynamics of LiCs in its 1R+ electronic ground state. Using a hybrid computational technique combining discretisation and basis set methods, the rovibrational Schro¨dinger equation is solved. Results for energy levels and various expectation values are presented. The validity of the previous developed effective and adiabatic rotor approaches is investigated. The electric field-induced hybridization is analyzed up to high rotational excitations and for a large range of magnetic quantum numbers.  2006 Elsevier B.V. All rights reserved. PACS: 33.20.t; 33.55.Be; 33.20.Vq Keywords: Heteronuclear alkali dimers; Stark effect; Rovibrational spectra

1. Introduction The production of cold and ultracold systems and particularly of molecular Bose–Einstein condensates [1–3] is nowadays almost routinely possible. The experimental techniques to cool, trap, manipulate and guide molecules are based on the application of external fields. For a recent review we refer the reader to the special issue on ultracold polar molecules [4] and in addition the reviews [5,6]. The significant experimental efforts are motivated by the large variety of possible applications, such as control and manipulation of ultracold chemical reactions [7–9], ultracold molecular collision dynamics [10–13], quantum computing [14,15], and experimental realization of few-body quantum effects such as Efimov states [16]. One ultimate goal of this field is to prepare molecules in definite quantum states with

*

Corresponding author. Tel.: +34 958 24 00 29; fax: +34 958 24 28 62. E-mail addresses: [email protected] (R. Gonza´lez-Fe´rez), Michael. [email protected] (M. Mayle), [email protected] (P. Schmelcher). 0301-0104/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2006.06.023

respect to all motions, i.e., the center of mass, electronic, rotational and vibrational motions. Special efforts focus on the achievement of ultracold polar molecular samples, where the long-range dipole– dipole interaction will give rise to new interesting physical phenomena. Indeed, the photoassociation of heteronuclear alkali dimers was recently reported for RbCs [17] and KRb [18]. Even more, the combination of photoassociation with a two-step stimulated emission pumping process has yielded the formation of ultracold RbCs molecules in their absolute vibronic ground state [19]. Theoretical estimations for one color photoassociation and molecular formation rates in cold heteronuclear alkali pairs have been provided as a basis for future experiments with other species [20]. The group of Weidemu¨ller reported the formation and spectroscopic investigation of two cold alkali dimers, LiCs and NaCs, on Helium nanodroplets in their triplet ground state [21]. Moreover, Feshbach resonances have been observed for KRb [22–25] and LiNa [26], and an efficient conversion of the Feshbach-resonance-related state, by an stimulated Raman transition, into the rovibrational ground state of the 1R+ electronic ground state for several

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heteronuclear alkali dimers [27] has been proposed. In view of the experimental progress, a better knowledge of the properties of these molecular systems is of utmost importance. The group of Tiemann is currently performing high resolution spectroscopic analysis of various heteronuclear alkali, such as NaCs [28,29], NaRb [30,31] and LiCs [32], which is providing highly accurate potential energy curves of several electronic states. In addition, the permanent dipole moment functions of the electronic ground and lowest triplet states of all mixed alkali pairs have been computed by ab initio methods [33]. Particularly interesting is the study of the influence of external fields on these molecular systems. External electric and magnetic fields can manipulate and control the interaction between atomic and molecular pairs in chemical reactions or molecular collisions, see for example the recent review of Krems [11]. In a theoretical analysis performed for the Li and Cs atoms, it has been proven that static electric fields can be used to manipulate elastic collisions of ultracold atoms via the instantaneous electric dipole moment of the formed heteronuclear collision complex [13]. The interaction of polar molecules, such as KRb and RbCs dimers, with optical lattices and microwave fields have been theoretically investigated [34]. The group of Bohn has studied the ultracold scattering properties of polar molecules in strong electric field-seeking states [35]. They have found that the spectrum is dominated by a quasiregular series of potential resonances as a function of the field strength, which might provide detail information about the intermolecular interaction. In addition, the internal structure, specifically the rovibrational dynamics, of the molecules is strongly modified by an applied electric field. One of the major effects for strong fields is the appearance of pendular states for which the molecule is oriented along the electric field axis. For the rigid rotor, this has been studied in detail already in the early 1990s and we refer the reader for a deeper insight to the references [36–39]. The rotational motion becomes a librating one, each pendular state being a coherent superposition of the field-free rotational states. Recently, the authors went beyond the traditional rigid rotor approximation [40] and performed a full rovibrational study of a diatomic heteronuclear system in a homogeneous static electric field, including the coupling between the vibrational and rotational motions and the dependence of the electric dipole moment on the internuclear distance [41,42]. For those molecular systems whose energy scales associated to the rotational and vibrational motions differ by several orders of magnitude, the effective rotor and adiabatic rotor approximations (ERA and ARA) were derived. Both approaches characterize a vibrational state dependence of the angular momentum hybridization, while ARA describes a regime of strong fields where the mixing of angular momenta is accompanied by the local squeezing and stretching of the vibrational motion which can be very pronounced for highly excited vibrational states [42]. Even more, the electric field induces avoided crossings among

states of the same symmetry, where a strong mixing and interaction of the rovibrational states take place leading to strongly distorted and asymmetric molecular states including well-pronounced localization effects of their probability density [43]. Motivated by the experimental interest in alkali heteronuclear dimers and due to the large absolute value of the permanent electric dipole moment of the LiCs molecule, as well as the availability of the corresponding potential energy curve [44] and electric dipole moment function [33], we have performed a theoretical investigation of the rovibrational dynamics of the electronic ground state of this dimer exposed to a strong static electric field. Specifically, we consider a large set of states, vibrationally lowlying (m 6 10) but rotationally highly excited (J 6 30) levels with three different azimuthal symmetries, M = 0, 5 and 10, and the field strength regime F = 5.14 · 105–5.14 · 108 V/m. The effect of increasing field strengths on the energy levels and on the expectation values hcos hi, hJ2i and hRi will be analyzed. In particular, we show that for moderate field strengths, low-lying rotational excitations present a significant orientation and angular momentum hybridization. Whereas, stronger fields are needed in order to compensate and overcome the rotational kinetic energy of highly excited rotational levels which are only weakly affected by the field. Moreover, we have proved that for the regime of field strength and the set of states under consideration the effective and adiabatic rotor approximations provide excellent descriptions of this dimer in an electric field. The paper is organized as follows. In Section 2, we present our full rovibrational Hamiltonian and some specific aspects of the effective and adiabatic rotor approaches. The potential energy curve and the electric dipole moment function of the LiCs molecule together with the results and their discussion are described in Section 3. This section also includes a comparison of the effective and adiabatic rotor approaches with the full rovibrational description. The conclusions and outlook are provided in Section 4. Atomic units will be used throughout, unless stated otherwise. 2. The rovibrational Hamiltonian, the adiabatic and effective rotor approximations In the framework of the Born–Oppenheimer approximation, the Hamiltonian of the nuclear motion of a heteronuclear diatomic molecule in its 1R+ electronic ground state, exposed to an external, homogeneous and static electric field, can be written as   h2 o J2 ðh; /Þ 2 o R H ¼ þ eðRÞ  FDðRÞ cos h þ 2 oR 2lR2 2lR oR ð1Þ where the rotating molecule fixed frame with the coordinate origin at the center of mass of the nuclei has been used, with R and h, / being the internuclear distance and Euler angles, respectively. l is the reduced mass of the

R. Gonza´lez-Fe´rez et al. / Chemical Physics 329 (2006) 203–215

nuclei, J(h, /) is the orbital angular momentum and e(R) represents the electronic potential energy curve (PEC) of the molecule in field-free space. The last term provides the interaction between the electric field of strength F and the molecule via its permanent electronic dipole moment which is represented by the electronic dipole moment function (EDMF). The field is thereby oriented parallel to the z-axis of the laboratory fixed frame. We consider the regime where perturbation theory holds for the description of the electronic structure but a nonperturbative treatment is indispensable for the corresponding nuclear dynamics. In the field-free case, each state is characterized by its vibrational m, rotational J and magnetic M quantum numbers. In the presence of an external electric field, only the magnetic quantum number M is conserved giving rise to a non-integrable two-dimensional dynamics in (R, h)-space. The corresponding rovibrational equation of motion is solved by means of a hybrid computational approach which combines a radial harmonic oscillator discrete variable representation for the vibrational coordinate and a basis set expansion in terms of associated Legendre functions for the angular coordinate. Employing the variational principle, the initial differential equation is reduced to a symmetric eigenvalue problem being diagonalized by a Krylov type technique [41]. The theoretical description of the angular motion of a diatomic molecule in an external field is traditionally based on the rigid rotor approach [40] which neglects the coupling between the vibrational and rotational motions and assumes a constant permanent dipole moment for the molecule. The so-called pendular Hamiltonian reads H¼

J2  FDeq cos h 2lR2eq

1 2 ð0Þ 2 ð0Þ hR im J  F hDðRÞið0Þ m cos h þ E m 2l

a fully adiabatic separation of the rovibrational motion has been performed allowing the external field to affect the fast vibrational motion which depends now parametrically on the angular coordinates. The latter effect is exclusively due to the presence of the external electric field. The adiabatic rotor approach is composed by a vibrational equation of motion   1 o 2 o R þ eðRÞ  FDðRÞ cos h wm ðR; hÞ ¼ Em ðhÞwm ðR; hÞ 2lR2 oR oR ð4Þ and the corresponding rotational Hamiltonian H ARA ¼ m

1 2 2 hR im J þ Em ðhÞ  E 2l

ð5Þ

with hR2im = hwmj R2jwmi. The total wave function for the nuclear motion is WARA ðR; h; /Þ ¼ wm ðR; hÞvARA ðhÞeiM/ m ARA where vm ðhÞ are the eigenfunctions of the adiabatic rotor Hamiltonian (5). We remark that Em(h) represents a potential which includes the field interaction and introduces the vibrational state dependency for the angular motion. Both approximations take into account the main properties of each vibrational state therefore describing a vibrational state-dependent hybridization of the angular motion. These effects will be of particular importance for higher excited vibrational states. Both ERA and ARA were shown to properly describe both highly excited vibrational and low-lying rotational states exposed to an electric field, being complementary to each other, and superior to the traditional rigid rotor approach [41,42]. 3. Results

ð2Þ

where Req and Deq are the equilibrium internuclear distance and the corresponding dipole moment, respectively. Recently, the authors went beyond this rigid rotor description and performed an adiabatic separation of the rotational and vibrational motions, assuming that the energy scales associated to them differ by several orders of magnitude. Assuming further that the influence of the electric field on the vibrational motion is very small, one can employ perturbation theory. The effective rotor approach has then been derived [41], and the Hamiltonian describing the rotational motion of the molecular system takes the form H ERA ¼ m

205

ð3Þ

ð0Þ ð0Þ ð0Þ with hR2 imð0Þ ¼ hwmð0Þ jR2 jwð0Þ m i, hDðRÞim ¼ hwm jDðRÞjwm i ð0Þ ð0Þ and wm and Em are the field-free vibrational wave function and energy with J = 0, respectively. In the presence of the field, the rovibrational wave function reduces to ERA WERA ðR; h; /Þ ¼ wð0Þ ðhÞeiM/ where vERA ðhÞ are the m m ðRÞvj eigenfunctions of the effective rotor Hamiltonian (3). The effective rotor approach represents a crude adiabatic approximation with respect to the separation of the vibrational and rotational motion. In a second study [42],

We have performed a full rovibrational study of the influence of an external electric field on the rovibrational spectrum of the LiCs molecule. The PEC and EDMF of its 1R+ electronic ground state are plotted as a function of the vibrational coordinate R in Fig. 1a and b, respectively. The EDMF is negative and of large absolute value at the equilibrium distance Re  6.8 a.u. It reaches a minimum at larger R values, displaced by 1.5 a.u. with respect to Re, increasing and finally approaching zero thereafter. One should, therefore, expect that the effect of the electric field is more pronounced for (not too highly) excited states compared to the ground state. The PEC and EDMF are taken from theoretical studies of the group of Allouche [44] and of Dulieu [33], respectively. Both computations have been performed by means of the package CIPSI (configuration interaction by perturbation of a multiconfiguration wave function selected iteratively) of the ‘‘Laboratoire de Physique Quantique de Toulouse (France)’’ [45]. The short range behaviour of the PEC is provided by a Morse potential fitted to these data. The long-range behaviour, given by the van der Waals interaction [46], has not been included, therefore our study is restricted with respect to the proper description of not too highly excited vibrational levels. The EDMF was linearly extrapolated to short

R. Gonza´lez-Fe´rez et al. / Chemical Physics 329 (2006) 203–215

206 0.05

0

0.04 -0.5

EDMF (a.u.)

PEC (a.u.)

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18

4

6

8

R (a.u.)

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Fig. 1. Electronic potential energy curve (a) and electric dipole moment function (b) of the electronic ground state of the LiCs molecule. Both quantities are given in atomic units.

-0.0172 -0.0174 -0.0176

Energy (a.u.)

distances. The reduced nuclear mass of the LiCs molecule is l = 10490 a.u. Considering the states within vibrational bands m 6 10, angular momenta J 6 30 and magnetic quantum numbers M 6 10, we have analyzed the effect of the electric field on the spectrum and the expectation values hcos hi, hJ2i and hRi. We focus on the regime of field strengths F = 106–103 a.u., i.e., F = 5.14 · 105–5.14 · 108 V/m, containing the experimentally accessible static field strengths. In view of the large number of obtained states, within the present study we will only present the results of the levels of the vibrational band m = 10 for the magnetic quantum numbers M = 0, 5 and 10. These states will be labeled by their rotational and magnetic quantum numbers using (J, M). Please note that in the following we will frequently use the field-free vibrational and rotational quantum numbers, m and J, to label the electrically dressed states, although the only good quantum number in the presence of the field is the magnetic quantum number M.

-0.0178 -0.018 -0.0182 -0.0184 -0.0186 0

5

10

15

20

25

30

field-free rotational quantum number J Fig. 2. The energy as a function of the corresponding field-free rotational quantum number J, for a field strength F = 104 a.u., for the levels with field-free vibrational quantum number m = 10 and magnetic quantum number M = 0 (d), 5 (s) and 10 (+). The dotted line represents the values for the field-free case.

3.1. Rovibrational excitations in an electric field Fig. 2 illustrates the behaviour of the energy as a function of the rotational quantum number J for the states with m = 10, J 6 30, M = 0, 5 and 10, and a fixed field strength of F = 104 a.u. For comparison, the M-degenerate fieldfree energies are also included. For these three azimuthal symmetries the energy shows qualitatively a similar but quantitatively different behaviour as a function of J. The presence of the electric field breaks the M-degeneracy and for fixed J the energy of the levels decreases as the magnetic quantum number is increased. The fully angular momentum polarized levels become more bound in the presence of the field in comparison with their field-free counterparts. The (0, 0) state is shifted around 1% to lower values, while for the (5, 5) and (10, 10) levels this effect is less pronounced. For fixed M, the energy monotonically increases as the rotational number is enhanced, and they reach and

surpass the corresponding field-free energies at J = 7, 12 and 19 for M = 0, 5 and 10, respectively. For highly excited rotational states, the influence of the field is rather weak, and their behaviour is dominated by the highly excited rotational character whereas the field interaction is no longer of importance. Indeed, the J J 20 states are almost degenerate again with respect to the magnetic quantum number and they follow the field-free asymptote. For F = 0, there are 30 rotational levels of the m = 10 band before the m = 11 band is reached, whereas in the presence of the field only 25 are remaining. Of course, the vibrational spacing between the m = 10 and m = 11 bands is still significantly larger than the rotational spacing within the m = 10 band and the coupling between adjacent vibrational levels is very small. The mixing of states with different vibrational quantum numbers m becomes relevant only for very highly excited states and very strong fields.

R. Gonza´lez-Fe´rez et al. / Chemical Physics 329 (2006) 203–215

The behaviour of hcos hi as a function of J for the same set of states and field strength is shown in Fig. 3. Theffi corqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

responding results for D cos h ¼ hcos2 hi  hcos hi are presented in Fig. 4. For comparison, the field-free values are included in both figures. hcos hi and Dcos h characterize the orientation and alignment of the states, respectively. The closer Dcos h is to zero the stronger is the alignment, and the closer the absolute value of hcos hi is to one, the stronger is the orientation of the state along the external field. Let us momentarily focus on the description of hcos hi. Please, note that due to the negative sign of the EDMF, the states will be oriented in the opposite direction of the field. The fully angular momentum polarized states already show a significant orientation with hcos hi = 0.954, 0.727 and 0.518 for M = 0, 5 and 10, respectively. For fixed J, the orientation decreases as the magnetic quantum number is increased. For the M = 0 levels,

0.6 0.4

〈cosθ〉

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

5

10

15

20

25

30

field-free rotational quantum number J Fig. 3. The expectation value hcos hi as a function of the corresponding field-free rotational quantum number J, for the same field strength and states as in Fig. 2. The dotted line represents the values for the field-free case.

207

hcos hi monotonically increases, i.e., the orientation towards the field decreases, augmenting the rotational degree of excitation. The smallest absolute value of hcos hi is achieved for the J = 10 level, and states with J P 10 are oriented in the field direction. hcos hi is maximal for the J = 14 state, and decreases thereafter, approaching the field-free behaviour but without reaching it. A similar picture holds for the other two azimuthal symmetries, although the major differences are that hcos hi does not achieve very large and positive values and only broad maxima as a function of J centered around the J = 18 and 26 states, respectively, are displayed. For highly excited rotational states, the deviation of hcos hi from zero is very small and the M degeneracy is again approached. Dcos h, see Fig. 4, clearly indicates that the spreading of the field-hybridized angular motion increases strongly with increasing the degree of rotational excitation for any azimuthal symmetry. States being close to fully polarized and low rotational excited levels are strongly aligned. Comparing the three different magnetic quantum numbers, we notice that for low rotational excitations and fixed quantum number J the alignment is stronger the larger M is. As J is increased, the corresponding field-free values are approached. Fig. 5 presents the expectation value hJ2i as a function of the rotational excitation for the same set of states and field strength as in Fig. 2. The parabolic shaped curve for the field-free results, hJ2i = J(J + 1), is also included. hJ2i provides a measure for the mixture of field-free states with different rotational quantum numbers J but the same value for M, i.e., it describes the hybridization of the field-free rotational motion. The effects due to the field depend not only on the degree of rotational excitation but also strongly on the azimuthal symmetry, where major differences are observed. The range of J values contributing to each state becomes larger as the magnetic quantum number decreases. The (0, 0) state shows a strong angular

1000

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800 700

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Δcosθ

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500 400

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0 0

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field-free rotational quantum number J Fig. 4. Dcos h as a function of the corresponding field-free rotational quantum number J, for the same field strength and states as in Fig. 2. The dotted lines represent the values for the field-free case.

0

5

10

15

20

25

30

field-free rotational quantum number J Fig. 5. The expectation value hJ2i as a function of the corresponding fieldfree rotational quantum number J, for the same field strength and states as in Fig. 2. The dotted line represents the values for the field-free case.

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208

momentum hybridization, J2h ð0; 0Þ ¼ hJ2 ið0; 0Þ  J ðJ þ 1Þ ¼ 10:40, J being the corresponding field-free rotational quantum number J = 0. For the M = 0 states, hJ2i increases as the rotational excitation is enhanced, it passes through a maximum and a minimum for the J = 11 and 13 levels, respectively. With a further increase of J, this expectation value increases while approaching the field-free behaviour. In particular, for low-lying rotational states the field causes an increase of hJ2i with respect to its field-free value, i.e., J2h ðJ ; 0Þ > 0 for J 6 11. Whereas, J2h ðJ ; 0Þ < 0 for the J P 12 levels, i.e., for these states the mixing with lower rotational excitations is dominant. Please, note that the transition hcos hi < 0 and hcos hi > 0 takes place at approximately the same degree of rotational excitation. For the other two azimuthal symmetries, the (M, M) states show a strong angular momentum hybridization with J2h ð5; 5Þ ¼ 41:41 and J2h ð10; 10Þ ¼ 45:55. In both cases hJ2i monotonically increases as J is enhanced. Again, in the presence of the field the contribution of higher rotations for low-lying rotational states increases hJ2i with respect to its field-free value, i.e., J2h ðJ ; 5Þ > 0 for J 6 13 and M = 5, and J2h ðJ ; 10Þ > 0 for J 6 19 and M = 10. Whereas, the field-induced mixing of rotations for high-lying rotational levels reduces hJ2i with respect to its field-free value while approaching its field-free parabola; consequently the states become M-degenerate again. Accidentally, hJ2i for the J = 10 levels with M = 0, 5 and 10 have a very similar value. The expectation value hRi as a function of the rotational excitation for the same set of states and field strength is presented in Fig. 6, the corresponding field-free results being also included. On a first glance, hRi behaves qualitatively similar to hJ2i. By comparison with Fig. 5 one realizes that for any azimuthal symmetry – the levels with J2h ðJ ; MÞ > 0 are stretched, i.e., in the presence of the field the expectation value of the internuclear distance, hRi, is larger than its field-free value. Contrariwise, for those lev-

els with J2h ðJ ; MÞ < 0, hRi decreases with respect to its fieldfree value. As expected, for highly excited rotational states hRi approaches its F = 0 asymptote and the M degeneracy is almost achieved. For M = 5 and 10, hRi monotonically increases as J is enhanced, while for M = 0 the behaviour is more complicated: A maximum and a minimum is reached for the J = 10 and 13 states, respectively. Accidentally, hR i for the J = 9 levels with M = 0, 5 and 10 have a very similar value. The striking similarity between the courses of hJ2i and hRi is evident: The mixing of the rotational states determines the amount of the molecular stretching due to the centrifugal force. 3.1.1. The role of the Stark potential A deeper physical insight is gained if these results are analyzed by means of the Stark potential [37,39]. Let us assume that the response of the LiCs molecule to an external electric field can be explained by the effective rotor approximation (3). Of course, the validity of this assumption depends on the field strength and on the set of states under consideration. However, in the final part of this section we will show that indeed this approach holds for the considered regime of field strength and set of analyzed states. The effective rotor approximation provides an effective Stark potential (see Eq. (3)), associated to each vibrational band, given by ð0Þ

V F ;eff ðhÞ ¼ Eð0Þ m þ F jhDðRÞim j cos h ð0Þ hDðRÞim

ð0Þ hwð0Þ m jDðRÞjwm i,

ð6Þ wmð0Þ

¼ and and Emð0Þ being with the field-free vibrational wave function and energy, respectively. Note that the negative sign of the EDMF has been already taken into account. In Fig. 7 VF,eff(h) is plotted as a function of the angular coordinate for the m = 10 vibrational level and F = 104 a.u. To elucidate the following discussion, the energies of the rotational states within this vibrational band and with M = 0 as well as the field-free energy of the m = 10,

7.435

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Energy (a.u.)

〈R〉 (a.u.)

7.425

7.415 7.41 7.405

-0.018 -0.0181 -0.0182 field-free groundstate

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7.395 0

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10

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30

field-free rotational quantum number J Fig. 6. The expectation value hRi as a function of the corresponding fieldfree rotational quantum number J, for the same field strength and states as in Fig. 2. The dotted line represents the values for the field-free case.

-0.0185 0

π

2π

θ Fig. 7. Effective Stark potential VF,eff(h) and energies of the first rotational states with M = 0 and m = 10, for the field strength F = 104 a.u.

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J = M = 0 state are also included. Please, note that these eigenvalues are exact in the sense that they have been computed with the full rovibrational Hamiltonian (1). A classification of these states can be performed according to the energetical position in the Stark well. There are levels which do not lie within the Stark potential well, i.e. ð0Þ E > E10 þ F jhDðRÞið0Þ 10 j, and their angular motion does not suffer any restriction. Whereas, for those levels lying ð0Þ ð0Þ within the Stark barrier E < E10 þ F  jhDðRÞi10 j, the possible values of the angular coordinate is restricted as indicated in Fig. 7. For this field strength there are 14 levels, i.e., states with J 6 13, satisfying this condition. These are pendular states and they can be further classified considering their position with respect to the field-free rotað0Þ tional ground state energy E10 . The levels with energies ð0Þ below this value, i.e. E < E10 , are situated in the attractive well with 12 p 6 h 6 32 p; therefore their angular motion is restricted to the interval h ¼ p  p2, i.e., they are oriented ð0Þ against the field direction. The states with E > E10 , are trapped within the repulsive barrier, and the confinement of the librational amplitude decreases with increasing degree of excitation. The above picture can easily be translated to the behaviour of the expectation value hcos hi. Let us consider the (0, 0) rotational ground state. Due to its low energy within the Stark potential, the molecule experiences very strong constraints with regard to the possible values of h, which corresponds to the strong orientation given by hcos hi = 0.9541. Considering higher rotations, the restrictions of the Stark potential ease and therefore the orientation decreases, which is represented in the rise of hcos hi seen ð0Þ in Fig. 3. Finally, we have the levels with E > E10 : Some of them reside mainly in the repulsive regions of the potential where hcos hi changes sign and gets positive. These states are still of pendular character while the projection of the dipole moment on the field axis shows no longer in the opposite direction but on average now is parallel to the field. The maximum of this kind of orientation is reached for J = 13 which is the last level bound by the Stark potential. For states with even higher angular momenta, the rotational kinetic energy becomes larger than the electric field interaction and therefore they hardly feel the field effect: The states become pinwheeling and hcos hi approaches the already mentioned field-free asymptote. The hybridization of the angular motion also changes according to the position of the states within the effective Stark potential. The contribution of higher rotations is dominant for those levels with hcos hi < 0, having therefore J2h ðJ ; MÞ > 0. While for those lying above the field-free rotational ground state, i.e. having hcos hi > 0, lower rotations become dominant and therefore hJ2i is lower than the corresponding field-free value. Even more, the J = 10 level, for which hJ2i is very close to the extremal value at J = 11, possess at the same time the smallest absolute value of hcos hi, whereas the maximum of hcos hi coincides with the minimum of hJ2i. A similar explanation holds for the

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expectation value hRi as the contribution of higher (lower) rotations provokes a stretching (squeezing) of the molecule, i.e., an increasing (decreasing) hRi with respect to the fieldfree value. Again, the minimum of hRi coincides with the minimum of hJ2i and the maximum of hcos hi. For highlying rotations, i.e. states lying significantly above the effective Stark well, the effect of the field is rather weak and therefore hJ2i and hRi approach the corresponding fieldfree parabola. The results for J > M states with M 5 0 distinctly differ from those for the levels with M = 0. One of the major differences is the expectation value hcos hi for the pendular states with energies above the field-free rotational ground state. Indeed, it does not reach very large positive values and does not present such pronounced maximum as in the M = 0 case. The explanation for this phenomenon is found in the range of allowed values of h, which is related to the tilt angle between the angular momentum J with respect to the field axis, as discussed in [39]. In the semiclas1 sical vector pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi model this angle is given by a ¼ cos ðjMj= J ðJ þ 1ÞÞ [47]. For J  jMj large we get a near 90, i.e., the electric field direction lies almost in the plane of the rotating dipole. Thus, the dipole accelerates as it swings through the attractive range of the Stark potential and decelerates again as it reaches the repulsive barrier. On average, the dipole in such states points in direction of the field, i.e. h < p2 or cos h > 0. For fixed J, increasing the magnetic quantum number M up to its maximum M = J decreases a and therefore will consequently diminish this effect. Indeed, in Fig. 3 we observe that the maximum of hcos hi is more distinct the smaller the magnetic quantum number is. Of course, this argumentation holds only if the energy of the corresponding states is large enough such that all h directions are possible. 3.2. Evolution of selected states with increasing field strength An analysis of the evolution of the properties of the LiCs molecule as the field strength is varied from F = 106 a.u. to F = 103 a.u. has been performed. We have considered nine levels with different symmetries, all with vibrational quantum number m = 10, magnetic quantum numbers M = 0, 5 and 10, and possessing field-free rotational quantum numbers J = M, M + 5 and M + 10. The behaviour of the corresponding energies and expectation values hcos hi, Dcos h, hJ2i and hRi as a function of the electric field strength are illustrated in Figs. 8–12, respectively. Let us first study the evolution of the energies, see Fig. 8. In the weak field regime, the field-free symmetry of the levels determines the quadratic Stark effect [40] and therefore the initial trend of the energy to increase (low field-seekers) or decrease (high field-seekers) as the field becomes stronger. Within the scale of the figure, all the levels keep their energy almost constant up to F = 105 a.u. Increasing further the field strength, the M-degeneracy breaks down and the spectrum clearly divides into high and low field-seeking states. The first group is composed by the fully angular

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210

1000 -0.018

600

〈J 〉

-0.019

2

Energy (a.u.)

800

400

200

-0.02

10

-6

10

-5

10

-4

10

0 -6 10

-3

10

-5

10

-4

10

-3

field strength (a.u.)

field strength (a.u.) Fig. 8. Energy as a function of the field strength for states with M = 0 (+, ·, ), 5 (h, s, ) and 10 (n, d, ,) and J = M, M + 5 and M + 10 within the tenth vibrational band.

Fig. 11. The expectation value hJ2i as a function of the field strength for the same set of states as in Fig. 8.

7.44

0.6 0.4

7.43

〈R〉 (a.u.)

0.2

〈cosθ〉

0 -0.2

7.41

-0.4 -0.6

7.4

-0.8 -1 10-6

10-6 10-5

10-4

10-3

field strength (a.u.) Fig. 9. The expectation value hcos hi as a function of the field strength for the same set of states as in Fig. 8.

0.8 0.7 0.6

Δcosθ

0.5 0.4 0.3 0.2 0.1 0 10-6

7.42

10-5

10-4

10-3

field strength (a.u.) Fig. 10. Dcos h as a function of the field strength for the same set of states as in Fig. 8.

10-5

10-4

10-3

field strength (a.u.) Fig. 12. The expectation value hRi as a function of the field strength for the same set of states as in Fig. 8.

momentum polarized levels, which are most affected by the field, and, in our case, also the (15, 10) state. For F = 103 a.u., the position of the (M, M) states is shifted energetically by about 10% in the spectrum, and the field effect is stronger the lower is M. The low field-seeking set is formed by the J > M levels with M = 0 and 5, and the (20, 10) state. With a further increase of the field, these states become also high field-seekers, i.e., their energies reach maxima as a function of the field and decrease thereafter. The position of these extrema with respect to the field depends on the corresponding state, varying from F = 5 · 105 a.u. to F = 2 · 104 a.u. and for fixed M being lower the smaller field-free rotational quantum number is. For very strong electric fields, the energies approach the pendular regime [39,40] which in the framework of the effective rotor approximation is given by E ! x þ ð2J  pffiffiffiffiffiffi jMj þ 1Þ 2x, with x ¼ F hDðRÞimð0Þ 2l=hR2 imð0Þ . Therefore, states with different field-free symmetry but the same 2J  jMj value will have the same pendular energy for

R. Gonza´lez-Fe´rez et al. / Chemical Physics 329 (2006) 203–215

x  1. This phenomenon is observed for F > 104 a.u. between the (10, 10) and (5, 0) pair of levels, and also between the (15, 10) and (10, 0) states. The high field-seeking states increase their orientation as the field is enhanced having always hcos hi < 0, see Fig. 9. The (0, 0) state presents a significant orientation even for very weak fields with hcos hi = 0.525 and hcos hi < 0.8 for F = 106 a.u. and F P 6 · 106 a.u., respectively. The (5, 5) and (10, 10) states are of pinwheeling character for very weak fields F [ 105 a.u., show a moderate orientation for weak fields, and the strong orientation limit, hcos hi < 0.8, is achieved for higher fields F J 3 · 104 a.u. and F J 7 · 104 a.u., respectively. Although hcos hi for the (15, 10) level monotonically decreases as F is increased, the influence of the field is very weak and has a negligible orientation up to F [ 104 a.u. Indeed, the changes on this expectation value are very smooth while its absolute value is very small. For stronger fields, its orientation towards the field axis rapidly increases, reaching hcos hi = 0.698 for F = 103 a.u. For the J > M states with M = 0 and 5 and the (20, 10) level, hcos hi increases as the field is enhanced, passes through a maximum, and decreases thereafter. In particular, hcos hi shows for the (5, 0) and (10, 0) levels pronounced maxima as a function of F, with hcos hi  0.36, for F  1.6 · 105 a.u. and hcos hi  0.42 for F  6 · 105 a.u., respectively. These field strengths lie very close to the value needed for the states to be bound within the effective Stark potential as discussed in the previous section. The influence of the tilt angle of the angular momentum provokes that these humps are less pronounced and shifted to larger field strength for the (10, 5), (15, 5) and (20, 10) states. The larger the difference J  jMj is, the closer lies the plane of the rotating dipole to the field axis, and hence benefits the orientation of the dipole parallel to the field direction, giving rise to positive values of hcos hi. However, the influence of the field is still very weak, hence the states are still mostly of pinwheeling character. At the same time, for a higher J value a stronger field is needed in order to compensate and overcome the corresponding rotational kinetic energy, i.e., to trap the level within the Stark barriers. Therefore, for higher M but the same J  jMj value a less pronounced maximum is obtained. In the strong field regime, all the considered states exhibit a pronounced orientation with hcos hi < 0.5 for F = 103 a.u. and the two pairs of states {(10, 10), (5, 0)} and {(15, 10), (10, 0)} have the same pendular limit for hcos hi. The alignment of the levels monotonically increases as the field strength is increased, see Fig. 10. This effect is most pronounced for the fully angular momentum polarized states. In particular, Dcos h rapidly decreases for the (0, 0) state, while for the (5, 5) and (10, 10) states minor changes are observed for weak fields, i.e. still being pinwheeling like in the field-free case, and thereafter Dcos h smoothly decreases. A similar behaviour is observed for the J > M states for which Dcos h initially stays constant and a strong field is needed in order to provide a large variation of the alignment. However, in the high field regime a significant

211

alignment is achieved for all the states with Dcos h < 0.3 for F = 103 a.u. The description of hcos hi provides an excellent basis for the analysis of hJ2i, see Fig. 11. For the high field-seeking states hJ2i increases as the field strength is enhanced therefore increasing the contribution of higher rotations. For the (M, M) states, a strong mixing of rotational levels takes place even for moderate fields and increases monotonically as the field is enhanced. For the (15, 10) state and weak fields hJ2i keeps almost constant, the rotational hybridization being very weak with J2h ð15; 10Þ < 10 for F [ 7 · 105 a.u. For the J > M levels with M = 0 and 5 and the (20, 10) state hJ2i smoothly decreases as the field is increased, reaches a minimum, and rapidly increases thereafter. hJ2i for the (5, 0) and (10, 0) levels shows minima at field strengths strongly correlated to the maxima of hcos hi. For the states with (10, 5), (15, 5) and (20, 10) these minima are very shallow, in some cases hardly seen on the scale of the figure, and also related to the maxima of hcos hi. A similar analysis holds for the expectation value hRi, see Fig. 12. Again, the field-free symmetry of the states determines the behaviour of the corresponding hRi as the field is varied. The fully angular momentum polarized states are stretched as the field is enhanced, and the stretching is more pronounced the lower M is. Whereas the J > M states are squeezed or stretched according to the decrease or increase of hJ2i with respect to its field-free value. 3.3. The effective and adiabatic rotor approximations versus the full rovibrational description In [41] the effective rotor approach was developed and studied for the specific case of carbon monoxide considering the rotational ground state within any vibrational excitation, i.e., levels with spherical symmetry J = M = 0 for F = 0, and field strengths varying from F = 106 a.u. to F = 104 a.u. For this molecular system it was shown that the agreement with the exact calculation is excellent. A comparison of the adiabatic rotor approach with respect to the full rovibrational description and the effective rotor approximation was carried out in Ref. [42]. This analysis was restricted to parameter dependent models, namely, a Morse potential and Gaussian functions as electric dipole moment function. Only low-lying vibrational excited states with J = M = 0 and the strong field F = 103 a.u. were considered. In particular, some physical situations were found where ERA was no longer adequate and a more elaborated model like ARA is needed. In the present work, we have extended and completed these comparisons by studying highly excited rotational states with different azimuthal symmetries. We have analyzed the spectrum of the LiCs molecule in an external electric field by means of the effective and adiabatic rotor approximations, and compared the results with the full rovibrational description. We have considered states with field-free vibrational quantum numbers m 6 10, rotational quantum numbers J = M, M + 5 and M + 10, and

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relative error energy

10

-4

10

-5

10

-6

10

-1

10

-2

10

-3

10

-4 -6

10

-5

10

-4

10

-3

field strength (a.u.) Fig. 14. Relative errors of ARA and ERA of the expectation value hcos hi as a function of the field strength for the same set of states as in Fig. 13.

10

-2

10

-3

10

-4

10

-5

10

-6

10

-6

10

-5

10

-4

10

-3

field strength (a.u.) Fig. 15. Absolute differences of ARA and ERA of the expectation value hcos hi as a function of the field strength for the same set of states as in Fig. 13.

-3

10

10

10

absolute error 〈cosθ 〉

jAF  Av j v ¼ E; A ð7Þ AF where Av represents the energy or one of the expectation values hcos hi, hJ2i and hRi, and the indices F, E and A stand for exact, ERA and ARA calculations, respectively. Here, we only present the analysis done for the states with field-free m = 10, J = 10, 15 and 20, and magnetic quantum number M = 10. The comparison for levels with lower vibrational, rotational, and magnetic quantum numbers yields similar results. Fig. 13 shows DEE and DEA as a function of the field strength for this set of states. The ERA and ARA relative errors for any of the considered states and field strengths are always bellow 0.1%. For both approximations, DEv is larger the higher the rotational quantum number is. For fixed J, both relative errors keep constant in the weak field regime with very similar values. For F J 2 · 105 a.u., DEv of the (10, 10) level increases as the field is enhanced, and the errors start to differ satisfying DEE > DEA . A similar picture holds for the (15, 10) and (20, 10) states. Due to their highly rotational excited character, stronger fields are needed in order to affect the states, therefore the rise of the relative error is shifted to larger F values. The relative error for hcos hi is presented as a function of the field strength in Fig. 14. For completeness, in Fig. 15 the absolute deviations of both approximations with respect to the exact calculations are included. Unlike for the energy, we notice that the relative errors for ERA and ARA slightly deviate from each other even for weak E A fields. The Dhcos hi and Dhcos hi deviations are larger DAv ¼

relative error 〈cosθ 〉

magnetic quantum numbers M = 0, 5 and 10, for field strengths varying from F = 106 a.u. to F = 103 a.u. In order to perform such a comparison the following relative error has been defined

-6

10

-5

10

-4

10

-3

field strength (a.u.) Fig. 13. Relative errors of ARA and ERA of the energy as a function of the field strength for states with magnetic quantum number M = 10, rotational quantum number J = 10 (s and + for ERA and ARA, respectively), 15 (h and ·) and 20 ( and ), and vibrational quantum number m = 10.

for larger J. Furthermore, major differences between both approximations can now be observed. For the (10, 10) level, both relative errors show a constant behaviour for F [ 105 a.u., smoothly decreasing as the field strength is augmented while satisfying Dhcos hiv < 0.002. A similar behaviour is observed for the (15, 10) level, but for weak fields, F < 6 · 105 a.u. and F < 8 · 105 a.u., we observe Dhcos hiv > 0.01 for v ¼ E and A, respectively. A further increase of the field leads to a decrease of both relative E A errors. For the (20, 10) state, Dhcos hi and Dhcos hi keep constant and larger than 0.01 in the weak field regime. Increasing the field strength, they reach a maximum for F = 104 a.u. with Dhcos hiE ¼ 0:67 and Dhcos hiA ¼ 0:74, decrease thereafter, and in particular they are smaller than 0.01 for F = 103 a.u. The large relative errors are due to the small absolute values of hcos hi. In fact, the absolute error, see Fig. 15, monotonically increases as the field is enhanced reaching a plateau in the strong field limit. Even

R. Gonza´lez-Fe´rez et al. / Chemical Physics 329 (2006) 203–215

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

10

-8

2

relative error 〈J 〉

more, for F [ 4 · 105 a.u. the ERA and ARA absolute differences are of the same order of magnitude for all considered levels. Both approximations perform not as bad as their relative errors suggest, but they are unable to correctly describe the angular motion at the crossover from the (Stark) unbound states to the bound ones. The same effect is observed for low field-seeking states with M = 0 and M = 5, and m 6 10. The (15, 10) state is high field-seeking and therefore shows a similar behaviour as the fully polarized state. E A In Fig. 16 DhJ2 i and DhJ2 i are displayed as a function of the field strength for the same set of states. For low fields, lower rotational levels expose a larger relative error, which is inverted for strong fields. For these three states the relative errors monotonically increase as the field is enhanced, and only a change on the slope is observed at larger fields which coincides with the field strengths at which the angular motion is affected by the restrictions of

10

-6

10

-5

10

-4

10

-3

field strength (a.u.)

relative error 〈R〉

Fig. 16. Relative errors of ARA and ERA of the expectation value hJ2i as a function of the field strength for the same set of states as in Fig. 13.

10

-2

10

-3

10

-4

10

-6

-5

10 10 field strength (a.u.)

-4

10

-3

Fig. 17. Relative errors of ARA and ERA of the expectation value hRi as a function of the field strength for the same set of states as in Fig. 13.

213

the Stark potential. The results for both approximations are indistinguishable on the weak field regime, and only minor differences are observed for strong fields. QuantitaE tively, only DhJ2 i for the (20, 10) state at F = 103 a.u. is insignificantly above 0.01, the remaining ones are all beneath. Finally, the ERA and ARA relative errors for the expectation value of the internuclear distance are presented in Fig. 17. Qualitatively, DhRiE and DhRiA behave similar to the ones of the energy. Once again, both quantities are equal at constant values at low fields. At certain field strengths the two different approximations are splitting apart while their relative errors rise monotonically. On the strong field regime the adiabatic rotor performs slightly better than the effective one. 4. Conclusions We have investigated the rovibrational properties of the LiCs molecule in a static homogeneous electric field by solving the fully coupled rovibrational equation of motion and taking into account the dependence of the electric dipole moment on the internuclear distance. We hereby assumed that the external electric field affects the rovibrational dynamics in a nonperturbative way whereas the influence on the electronic structure can be described by means of perturbation theory. The Schro¨dinger equation for the rovibrational motion was solved by a highly efficient and accurate combination of a discrete variable representation method and a basis set expansion technique applied to the vibrational and rotational coordinates, respectively. We have considered several rotational excitations of the LiCs molecule within the tenth vibrational band and with magnetic quantum numbers M 2 {0, 5, 10}, and analyzed the energies and expectation values hcos hi, hJ2i and hRi for field strengths F = 106– 103 a.u. = 5.14 · 105–5.14 · 108 V/m, which includes the regime of experimental interest. For fixed field strength, we have shown how the influence of the field on the spectrum strongly depends on the field-free rotational quantum number. Three different kind of states could be found: First, there are low-lying rotational excitations whose behaviour is dominated by the Stark interaction and which become more bound in the presence of the electric field. They have a strong pendular signature and acquired a significant alignment and orientation which due to the negative sign of the electric dipole moment function is antiparallel to the field direction. We encounter a strong angular momentum hybridization dominated by the contribution of higher rotational excitations, which provokes a stretching of the molecular states. These effects were especially pronounced for the fully angular momentum polarized levels. Secondly, there are states with large rotational quantum number but still of pendular character whose rotational kinetic energy is of the same order of magnitude as the Stark interaction. They are shifted upwards in the spectrum with respect to their field-free position and are oriented parallel to the field,

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R. Gonza´lez-Fe´rez et al. / Chemical Physics 329 (2006) 203–215

their alignment being reduced. The angular momentum hybridization is now dominated by the mixing with lower rotational excitations, which therefore reduces hJ2i with respect to its field-free value, and the states are slightly squeezed. Finally, the behaviour of highly excited rotational states is dominated by the rotational kinetic energy: They are pinwheeling and resemble the field-free ones. A detailed explanation of these phenomena in terms of the effective Stark potential that the states on the m = 10 vibrational band suffer has been given. The evolution of a certain set of states belonging to different symmetries has been also analyzed with varying field strength. Two different types of behaviour were observed. The fully angular momentum polarized states and the (15, 10) level are high field-seekers, i.e., they become more bound as the field is enhanced while increasing their orientation, alignment and angular momentum hybridization; in addition, they are slightly stretched towards the minimum of the electric dipole moment function. The states with J > M for M = 0 and 5, and the (20, 10) level are for weak fields low field-seekers, i.e., their energy increases as the field is increased. With a further increase of the field they become trapped by the effective Stark potential and finally show a high field-seeking character, i.e., their energy decreases as the field strength is increased. The hcos hi of these states is initially positive and increases as the field is enhanced; when they are trapped by the Stark barrier it passes through a maximum, decreases thereafter and they are finally oriented antiparallel to the field axis. In particular, in the strong field regime all the states present a pronounced orientation. Likewise, hJ2i and hRi initially decrease with respect to their field-free value, reach a minimum which is correlated to the maximum of hcos hi and increase thereafter. Once the states are of pendular character, the number of rotational excitations contributing to the angular momentum hybridization increases as the field is enhanced. We have been focusing on the study of levels within the tenth vibrational band. Due to the particular shape of the electronic dipole moment function the effect of the electric field is somewhat more pronounced for highly excited vibrational levels. The larger extension of these states provides a larger overlap with the electric dipole moment function and therefore a stronger influence of the electric field. Regarding lower vibrational excitations, we found that they are slightly less affected by the field. Indeed, less states are trapped within the Stark potential, their orientation is less pronounced, and the angular momentum hybridization is weaker but the general qualitative behaviour remains unaltered. The validity of the effective and adiabatic rotor approaches to describe highly excited rotational states and M 5 0 levels was demonstrated for the LiCs molecule. In particular, the effective and adiabatic rotor results of the energy and expectation values hcos hi, hJ2i and hRi have been compared with the results obtained from the solution of the rovibrational Schro¨dinger equation. In general, both

approximations perform very well and relative errors below 1% were found. The present study is restricted to moderate vibrational excitations up to m = 10 of the LiCs molecule. A next step would be to investigate the properties of rotational levels within highly excited vibrational bands lying close to the dissociation limit. The radial extension of the wave function reaches significantly beyond the minimum of the electronic dipole moment and therefore a squeezing of the states due to the interaction with strong fields should be expected. Obviously, the coupling between the vibrational and rotational motions will be more pronounced, therefore the effective and adiabatic rotor description should be revisited. Even more interesting is the study of radiative decays and lifetimes of vibrationally and rotationally excited states of the LiCs molecule in the presence of the external field. The group of Stwalley has performed a detailed study of transition probabilities and radiative lifetimes for all levels of the electronic ground state of the KRb molecule in the field-free case [48]. In the presence of an external electric field the selection rule related to the rotational quantum number breaks down, and therefore a new variety of transitions becomes possible. Furthermore, the prevailing transitions should be different in the weak compared to the strong field regime. Acknowledgements R.G.F. acknowledges the support of the Junta de Andalucı´a (under the program of Retorno de Investigadores a Centros de Investigacio´n Andaluces). Financial support by the Acciones Integradas Hispano-Alemanas HA20050038 (MEC and DAAD) and of the Spanish project FIS2005-00973 (MEC) is gratefully appreciated. We thank Matthias Weidemu¨ller for fruitful discussions and Olivier Dulieu for providing us with the electric dipole moment function for LiCs. References [1] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag, R. Grimm, Science 302 (2003) 2101. [2] M. Greiner, C.A. Regal, D.S. Jin, Nature 426 (2003) 537. [3] M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, S. Gupta, Z. Hadzibabic, W. Ketterle, Phys. Rev. Lett. 91 (2003) 250401. [4] Topical Issue on Ultracold Polar Molecules: Formation and Collisions, Eur. Phys. J. D 31 (2004). [5] J. Doyle, B. Friedrich, R.V. Krems, F. Masnou-Seeuws, Eur. Phys. J. D 31 (2004) 149. [6] K.M. Jones, E. Tiesinga, P.D. Lett, P.S. Julienne, Rev. Mod. Phys. 78 (2006) 483. [7] N. Balakrishnan, A. Dalgarno, Chem. Phys. Lett. 341 (2001) 652. [8] E. Bodo, F.A. Gianturco, A. Dalgarno, J. Chem. Phys. 116 (2002) 9222. [9] T. Rom, T. Best, O. Mandel, A. Widera, M. Greiner, T.W. Ha¨nsch, I. Bloch, Phys. Rev. Lett. 93 (2004) 073002. [10] M. Kajita, Eur. Phys. J. D 23 (2003) 337; M. Kajita, Eur. Phys. J. D 31 (2004) 39. [11] R.V. Krems, Int. Rev. Phys. Chem. 24 (2005) 99.

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