Rovibrational dynamics of LiCs dimers in strong electric fields

Transcription

1 Cheical Physics 329 (26) Rovibrational dynaics of LiCs diers in strong electric fields R. González-Férez a, *, M. Mayle b, P. Schelcher b,c a Instituto Carlos I de Física Teórica y Coputacional and Departaento de Física Moderna, Universidad de Granada, E-1871 Granada, Spain b Theoretische Cheie, Physikalisch-Cheisches Institut, I Neuenheier Feld 229, D-6912 Heidelberg, Gerany c Physikalisches Institut, Universität Heidelberg, Philosophenweg 12, D-6912 Heidelberg, Gerany Received 26 April 26; accepted 15 June 26 Available online 28 June 26 Abstract We investigate the effects of a strong electric field on the rovibrational dynaics of LiCs in its 1 R + electronic ground state. Using a hybrid coputational technique cobining discretisation and basis set ethods, the rovibrational Schrödinger 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 agnetic quantu nubers. Ó 26 Elsevier B.V. All rights reserved. PACS: t; Be; 33.2.Vq Keywords: Heteronuclear alkali diers; Stark effect; Rovibrational spectra 1. Introduction The production of cold and ultracold systes and particularly of olecular Bose Einstein condensates [1 3] is nowadays alost routinely possible. The experiental techniques to cool, trap, anipulate and guide olecules are based on the application of external fields. For a recent review we refer the reader to the special issue on ultracold polar olecules [4] and in addition the reviews [5,6]. The significant experiental efforts are otivated by the large variety of possible applications, such as control and anipulation of ultracold cheical reactions [7 9], ultracold olecular collision dynaics [1 13], quantu coputing [14,15], and experiental realization of few-body quantu effects such as Efiov states [16]. One ultiate goal of this field is to prepare olecules in definite quantu states with * Corresponding author. Tel.: ; fax: E-ail addresses: rogonzal@ugr.es (R. González-Férez), Michael. Mayle@pci.uni-heidelberg.de (M. Mayle), Peter.Schelcher@pci.uniheidelberg.de (P. Schelcher). respect to all otions, i.e., the center of ass, electronic, rotational and vibrational otions. Special efforts focus on the achieveent of ultracold polar olecular saples, where the long-range dipole dipole interaction will give rise to new interesting physical phenoena. Indeed, the photoassociation of heteronuclear alkali diers was recently reported for RbCs [17] and KRb [18]. Even ore, the cobination of photoassociation with a two-step stiulated eission puping process has yielded the foration of ultracold RbCs olecules in their absolute vibronic ground state [19]. Theoretical estiations for one color photoassociation and olecular foration rates in cold heteronuclear alkali pairs have been provided as a basis for future experients with other species [2]. The group of Weideüller reported the foration and spectroscopic investigation of two cold alkali diers, LiCs and NaCs, on Heliu 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 stiulated Raan transition, into the rovibrational ground state of the 1 R + electronic ground state for several 31-14/$ - see front atter Ó 26 Elsevier B.V. All rights reserved. doi:1.116/j.chephys

2 24 R. González-Férez et al. / Cheical Physics 329 (26) heteronuclear alkali diers [27] has been proposed. In view of the experiental progress, a better knowledge of the properties of these olecular systes is of utost iportance. The group of Tieann is currently perforing high resolution spectroscopic analysis of various heteronuclear alkali, such as NaCs [28,29], NaRb [3,31] and LiCs [32], which is providing highly accurate potential energy curves of several electronic states. In addition, the peranent dipole oent functions of the electronic ground and lowest triplet states of all ixed alkali pairs have been coputed by ab initio ethods [33]. Particularly interesting is the study of the influence of external fields on these olecular systes. External electric and agnetic fields can anipulate and control the interaction between atoic and olecular pairs in cheical reactions or olecular collisions, see for exaple the recent review of Kres [11]. In a theoretical analysis perfored for the Li and Cs atos, it has been proven that static electric fields can be used to anipulate elastic collisions of ultracold atos via the instantaneous electric dipole oent of the fored heteronuclear collision coplex [13]. The interaction of polar olecules, such as KRb and RbCs diers, with optical lattices and icrowave fields have been theoretically investigated [34]. The group of Bohn has studied the ultracold scattering properties of polar olecules in strong electric field-seeking states [35]. They have found that the spectru is doinated by a quasiregular series of potential resonances as a function of the field strength, which ight provide detail inforation about the interolecular interaction. In addition, the internal structure, specifically the rovibrational dynaics, of the olecules is strongly odified by an applied electric field. One of the ajor effects for strong fields is the appearance of pendular states for which the olecule is oriented along the electric field axis. For the rigid rotor, this has been studied in detail already in the early 199s and we refer the reader for a deeper insight to the references [36 39]. The rotational otion becoes 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 approxiation [4] and perfored a full rovibrational study of a diatoic heteronuclear syste in a hoogeneous static electric field, including the coupling between the vibrational and rotational otions and the dependence of the electric dipole oent on the internuclear distance [41,42]. For those olecular systes whose energy scales associated to the rotational and vibrational otions differ by several orders of agnitude, the effective rotor and adiabatic rotor approxiations (ERA and ARA) were derived. Both approaches characterize a vibrational state dependence of the angular oentu hybridization, while ARA describes a regie of strong fields where the ixing of angular oenta is accopanied by the local squeezing and stretching of the vibrational otion which can be very pronounced for highly excited vibrational states [42]. Even ore, the electric field induces avoided crossings aong states of the sae syetry, where a strong ixing and interaction of the rovibrational states take place leading to strongly distorted and asyetric olecular states including well-pronounced localization effects of their probability density [43]. Motivated by the experiental interest in alkali heteronuclear diers and due to the large absolute value of the peranent electric dipole oent of the LiCs olecule, as well as the availability of the corresponding potential energy curve [44] and electric dipole oent function [33], we have perfored a theoretical investigation of the rovibrational dynaics of the electronic ground state of this dier exposed to a strong static electric field. Specifically, we consider a large set of states, vibrationally lowlying ( 6 1) but rotationally highly excited (J 6 3) levels with three different aziuthal syetries, M =, 5 and 1, and the field strength regie F = V/. The effect of increasing field strengths on the energy levels and on the expectation values hcoshi, hj 2 i and hri will be analyzed. In particular, we show that for oderate field strengths, low-lying rotational excitations present a significant orientation and angular oentu hybridization. Whereas, stronger fields are needed in order to copensate and overcoe the rotational kinetic energy of highly excited rotational levels which are only weakly affected by the field. Moreover, we have proved that for the regie of field strength and the set of states under consideration the effective and adiabatic rotor approxiations provide excellent descriptions of this dier in an electric field. The paper is organized as follows. In Section 2, we present our full rovibrational Hailtonian and soe specific aspects of the effective and adiabatic rotor approaches. The potential energy curve and the electric dipole oent function of the LiCs olecule together with the results and their discussion are described in Section 3. This section also includes a coparison of the effective and adiabatic rotor approaches with the full rovibrational description. The conclusions and outlook are provided in Section 4. Atoic units will be used throughout, unless stated otherwise. 2. The rovibrational Hailtonian, the adiabatic and effective rotor approxiations In the fraework of the Born Oppenheier approxiation, the Hailtonian of the nuclear otion of a heteronuclear diatoic olecule in its 1 R + electronic ground state, exposed to an external, hoogeneous and static electric field, can be written as H ¼ h2 o o 2lR 2 or R2 or þ J2 ðh; /Þ þ eðrþ FDðRÞ cos h 2lR 2 ð1þ where the rotating olecule fixed frae with the coordinate origin at the center of ass of the nuclei has been used, with R and h, / being the internuclear distance and Euler angles, respectively. l is the reduced ass of the

3 R. González-Férez et al. / Cheical Physics 329 (26) nuclei, J(h,/) is the orbital angular oentu and e(r) represents the electronic potential energy curve (PEC) of the olecule in field-free space. The last ter provides the interaction between the electric field of strength F and the olecule via its peranent electronic dipole oent which is represented by the electronic dipole oent function (EDMF). The field is thereby oriented parallel to the z-axis of the laboratory fixed frae. We consider the regie where perturbation theory holds for the description of the electronic structure but a nonperturbative treatent is indispensable for the corresponding nuclear dynaics. In the field-free case, each state is characterized by its vibrational, rotational J and agnetic M quantu nubers. In the presence of an external electric field, only the agnetic quantu nuber M is conserved giving rise to a non-integrable two-diensional dynaics in (R,h)-space. The corresponding rovibrational equation of otion is solved by eans of a hybrid coputational approach which cobines a radial haronic oscillator discrete variable representation for the vibrational coordinate and a basis set expansion in ters of associated Legendre functions for the angular coordinate. Eploying the variational principle, the initial differential equation is reduced to a syetric eigenvalue proble being diagonalized by a Krylov type technique [41]. The theoretical description of the angular otion of a diatoic olecule in an external field is traditionally based on the rigid rotor approach [4] which neglects the coupling between the vibrational and rotational otions and assues a constant peranent dipole oent for the olecule. The so-called pendular Hailtonian reads H ¼ J2 2lR 2 eq FD eq cos h where R eq and D eq are the equilibriu internuclear distance and the corresponding dipole oent, respectively. Recently, the authors went beyond this rigid rotor description and perfored an adiabatic separation of the rotational and vibrational otions, assuing that the energy scales associated to the differ by several orders of agnitude. Assuing further that the influence of the electric field on the vibrational otion is very sall, one can eploy perturbation theory. The effective rotor approach has then been derived [41], and the Hailtonian describing the rotational otion of the olecular syste takes the for H ERA ¼ 1 2l hr 2 i ðþ J 2 F hdðrþi ðþ cos h þ E ðþ with hr 2 i ðþ ¼hw ðþ jr 2 jw ðþ i, hdðrþi ðþ ¼hw ðþ jdðrþjw ðþ i and w ðþ and E ðþ are the field-free vibrational wave function and energy with J =, respectively. In the presence of the field, the rovibrational wave function reduces to W ERA ðr; h; /Þ ¼w ðþ ðrþv ERA j ðhþe im/ where v ERA ðhþ are the eigenfunctions of the effective rotor Hailtonian (3). The effective rotor approach represents a crude adiabatic approxiation with respect to the separation of the vibrational and rotational otion. In a second study [42], ð2þ ð3þ a fully adiabatic separation of the rovibrational otion has been perfored allowing the external field to affect the fast vibrational otion which depends now paraetrically on the angular coordinates. The latter effect is exclusively due to the presence of the external electric field. The adiabatic rotor approach is coposed by a vibrational equation of otion 1 2lR 2 o or R2 o þ eðrþ FDðRÞ cos h w or ðr; hþ ¼E ðhþw ðr; hþ ð4þ and the corresponding rotational Hailtonian H ARA ¼ 1 2l hr 2 i J 2 þ E ðhþ E ð5þ with hr 2 i = hw j R 2 jw i. The total wave function for the nuclear otion is W ARA ðr; h; /Þ ¼w ðr; hþv ARA ðhþe im/ where v ARA ðhþ are the eigenfunctions of the adiabatic rotor Hailtonian (5). We reark that E (h) represents a potential which includes the field interaction and introduces the vibrational state dependency for the angular otion. Both approxiations take into account the ain properties of each vibrational state therefore describing a vibrational state-dependent hybridization of the angular otion. These effects will be of particular iportance 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 copleentary to each other, and superior to the traditional rigid rotor approach [41,42]. 3. Results We have perfored a full rovibrational study of the influence of an external electric field on the rovibrational spectru of the LiCs olecule. The PEC and EDMF of its 1 R + 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 equilibriu distance R e 6.8 a.u. It reaches a iniu at larger R values, displaced by 1.5 a.u. with respect to R e, increasing and finally approaching zero thereafter. One should, therefore, expect that the effect of the electric field is ore pronounced for (not too highly) excited states copared to the ground state. The PEC and EDMF are taken fro theoretical studies of the group of Allouche [44] and of Dulieu [33], respectively. Both coputations have been perfored by eans of the package CIPSI (configuration interaction by perturbation of a ulticonfiguration 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

4 26 R. González-Férez et al. / Cheical Physics 329 (26) PEC (a.u.).2.1 EDMF (a.u.) (a) (b) R (a.u.) R (a.u.) Fig. 1. Electronic potential energy curve (a) and electric dipole oent function (b) of the electronic ground state of the LiCs olecule. Both quantities are given in atoic units. distances. The reduced nuclear ass of the LiCs olecule is l = 149 a.u. Considering the states within vibrational bands 6 1, angular oenta J 6 3 and agnetic quantu nubers M 6 1, we have analyzed the effect of the electric field on the spectru and the expectation values hcoshi, hj 2 i and hri. We focus on the regie of field strengths F = a.u., i.e., F = V/, containing the experientally accessible static field strengths. In view of the large nuber of obtained states, within the present study we will only present the results of the levels of the vibrational band = 1 for the agnetic quantu nubers M =, 5 and 1. These states will be labeled by their rotational and agnetic quantu nubers using (J,M). Please note that in the following we will frequently use the field-free vibrational and rotational quantu nubers, and J, to label the electrically dressed states, although the only good quantu nuber in the presence of the field is the agnetic quantu nuber M Rovibrational excitations in an electric field Fig. 2 illustrates the behaviour of the energy as a function of the rotational quantu nuber J for the states with = 1, J 6 3, M =, 5 and 1, and a fixed field strength of F =1 4 a.u. For coparison, the M-degenerate fieldfree energies are also included. For these three aziuthal syetries the energy shows qualitatively a siilar 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 agnetic quantu nuber is increased. The fully angular oentu polarized levels becoe ore bound in the presence of the field in coparison with their field-free counterparts. The (,) state is shifted around 1% to lower values, while for the (5, 5) and (1, 1) levels this effect is less pronounced. For fixed M, the energy onotonically increases as the rotational nuber is enhanced, and they reach and Energy (a.u.) field-free rotational quantu nuber J Fig. 2. The energy as a function of the corresponding field-free rotational quantu nuber J, for a field strength F =1 4 a.u., for the levels with field-free vibrational quantu nuber = 1 and agnetic quantu nuber M =(d), 5 (s) and 1 (+). The dotted line represents the values for the field-free case. surpass the corresponding field-free energies at J = 7, 12 and 19 for M =, 5 and 1, respectively. For highly excited rotational states, the influence of the field is rather weak, and their behaviour is doinated by the highly excited rotational character whereas the field interaction is no longer of iportance. Indeed, the J J 2 states are alost degenerate again with respect to the agnetic quantu nuber and they follow the field-free asyptote. For F =, there are 3 rotational levels of the =1 band before the = 11 band is reached, whereas in the presence of the field only 25 are reaining. Of course, the vibrational spacing between the = 1 and =11 bands is still significantly larger than the rotational spacing within the = 1 band and the coupling between adjacent vibrational levels is very sall. The ixing of states with different vibrational quantu nubers becoes relevant only for very highly excited states and very strong fields.

5 R. González-Férez et al. / Cheical Physics 329 (26) The behaviour of hcoshi as a function of J for the sae set of states and field strength is shown qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in Fig. 3. The corresponding results for D cos h ¼ hcos 2 hi hcos hi 2 are presented in Fig. 4. For coparison, the field-free values are included in both figures. hcoshi and Dcosh characterize the orientation and alignent of the states, respectively. The closer Dcosh is to zero the stronger is the alignent, and the closer the absolute value of hcoshi is to one, the stronger is the orientation of the state along the external field. Let us oentarily focus on the description of hcoshi. 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 oentu polarized states already show a significant orientation with hcoshi =.954,.727 and.518 for M =, 5 and 1, respectively. For fixed J, the orientation decreases as the agnetic quantu nuber is increased. For the M = levels, cosθ field-free rotational quantu nuber J Fig. 3. The expectation value hcoshi as a function of the corresponding field-free rotational quantu nuber J, for the sae field strength and states as in Fig. 2. The dotted line represents the values for the field-free case. hcos hi onotonically increases, i.e., the orientation towards the field decreases, augenting the rotational degree of excitation. The sallest absolute value of hcos hi is achieved for the J = 1 level, and states with J P 1 are oriented in the field direction. hcoshi is axial for the J = 14 state, and decreases thereafter, approaching the field-free behaviour but without reaching it. A siilar picture holds for the other two aziuthal syetries, although the ajor differences are that hcos hi does not achieve very large and positive values and only broad axia 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 fro zero is very sall and the M degeneracy is again approached. Dcosh, see Fig. 4, clearly indicates that the spreading of the field-hybridized angular otion increases strongly with increasing the degree of rotational excitation for any aziuthal syetry. States being close to fully polarized and low rotational excited levels are strongly aligned. Coparing the three different agnetic quantu nubers, we notice that for low rotational excitations and fixed quantu nuber J the alignent is stronger the larger M is. As J is increased, the corresponding field-free values are approached. Fig. 5 presents the expectation value hj 2 i as a function of the rotational excitation for the sae set of states and field strength as in Fig. 2. The parabolic shaped curve for the field-free results, hj 2 i = J(J + 1), is also included. hj 2 i provides a easure for the ixture of field-free states with different rotational quantu nubers J but the sae value for M, i.e., it describes the hybridization of the field-free rotational otion. The effects due to the field depend not only on the degree of rotational excitation but also strongly on the aziuthal syetry, where ajor differences are observed. The range of J values contributing to each state becoes larger as the agnetic quantu nuber decreases. The (, ) state shows a strong angular Δcosθ field-free rotational quantu nuber J J field-free rotational quantu nuber J Fig. 4. Dcosh as a function of the corresponding field-free rotational quantu nuber J, for the sae field strength and states as in Fig. 2. The dotted lines represent the values for the field-free case. Fig. 5. The expectation value hj 2 i as a function of the corresponding fieldfree rotational quantu nuber J, for the sae field strength and states as in Fig. 2. The dotted line represents the values for the field-free case.

6 28 R. González-Férez et al. / Cheical Physics 329 (26) oentu hybridization, J 2 h ð; Þ ¼hJ2 i ð; Þ JðJ þ 1Þ ¼ 1:4, J being the corresponding field-free rotational quantu nuber J =. For the M = states, hj 2 i increases as the rotational excitation is enhanced, it passes through a axiu and a iniu 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 hj 2 i with respect to its field-free value, i.e., J 2 h ðj; Þ > for J Whereas, J2 hðj; Þ < for the J P 12 levels, i.e., for these states the ixing with lower rotational excitations is doinant. Please, note that the transition hcos hi < and hcos hi > takes place at approxiately the sae degree of rotational excitation. For the other two aziuthal syetries, the (M, M) states show a strong angular oentu hybridization with J 2 h ð5; 5Þ ¼41:41 and J2 h ð1; 1Þ ¼45:55. In both cases hj2 i onotonically increases as J is enhanced. Again, in the presence of the field the contribution of higher rotations for low-lying rotational states increases hj 2 i with respect to its field-free value, i.e., J 2 hðj; 5Þ > for J 6 13 and M =5,andJ 2 hðj; 1Þ > for J 6 19 and M = 1. Whereas, the field-induced ixing of rotations for high-lying rotational levels reduces hj 2 i with respect to its field-free value while approaching its field-free parabola; consequently the states becoe M-degenerate again. Accidentally, hj 2 i for the J = 1 levels with M =, 5 and 1 have a very siilar value. The expectation value hri as a function of the rotational excitation for the sae 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 siilar to hj 2 i. By coparison with Fig. 5 one realizes that for any aziuthal syetry the levels with J 2 hðj; MÞ > 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 levels with J 2 hðj; MÞ <, hri decreases with respect to its fieldfree value. As expected, for highly excited rotational states hri approaches its F = asyptote and the M degeneracy is alost achieved. For M = 5 and 1, hri onotonically increases as J is enhanced, while for M = the behaviour is ore coplicated: A axiu and a iniu is reached for the J = 1 and 13 states, respectively. Accidentally, hr i for the J = 9 levels with M =, 5 and 1 have a very siilar value. The striking siilarity between the courses of hj 2 i and hri is evident: The ixing of the rotational states deterines the aount of the olecular stretching due to the centrifugal force The role of the Stark potential A deeper physical insight is gained if these results are analyzed by eans of the Stark potential [37,39]. Let us assue that the response of the LiCs olecule to an external electric field can be explained by the effective rotor approxiation (3). Of course, the validity of this assuption 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 regie of field strength and set of analyzed states. The effective rotor approxiation provides an effective Stark potential (see Eq. (3)), associated to each vibrational band, given by V F ;eff ðhþ ¼E ðþ þ F jhdðrþi ðþ j cos h ð6þ with hdðrþi ðþ ¼hw ðþ jdðrþjw ðþ i, andw ðþ and E ðþ being 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 V F,eff (h) is plotted as a function of the angular coordinate for the = 1 vibrational level and F = 1 4 a.u. To elucidate the following discussion, the energies of the rotational states within this vibrational band and with M = as well as the field-free energy of the = 1, R (a.u.) field-free rotational quantu nuber J Fig. 6. The expectation value hri as a function of the corresponding fieldfree rotational quantu nuber J, for the sae field strength and states as in Fig. 2. The dotted line represents the values for the field-free case. Energy (a.u.) field-free groundstate Fig. 7. Effective Stark potential V F,eff (h) and energies of the first rotational states with M = and = 1, for the field strength F =1 4 a.u. π θ 2π

7 R. González-Férez et al. / Cheical Physics 329 (26) J = M = state are also included. Please, note that these eigenvalues are exact in the sense that they have been coputed with the full rovibrational Hailtonian (1). A classification of these states can be perfored according to the energetical position in the Stark well. There are levels which do not lie within the Stark potential well, i.e. E > E ðþ 1 þ F jhdðrþiðþ 1 j, and their angular otion does not suffer any restriction. Whereas, for those levels lying within the Stark barrier E < E ðþ 1 þ F jhdðrþiðþ 1 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 rotational ground state energy E ðþ 1. The levels with energies below this value, i.e. E < E ðþ 1, are situated in the attractive well with 1 p 6 h 6 3 p; therefore their angular otion is 2 2 restricted to the interval h ¼ p p, i.e., they are oriented 2 against the field direction. The states with E > E ðþ 1, are trapped within the repulsive barrier, and the confineent of the librational aplitude decreases with increasing degree of excitation. The above picture can easily be translated to the behaviour of the expectation value hcoshi. Let us consider the (,) rotational ground state. Due to its low energy within the Stark potential, the olecule experiences very strong constraints with regard to the possible values of h, which corresponds to the strong orientation given by hcos hi = Considering higher rotations, the restrictions of the Stark potential ease and therefore the orientation decreases, which is represented in the rise of hcoshi seen in Fig. 3. Finally, we have the levels with E > E ðþ 1 : Soe of the reside ainly 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 oent on the field axis shows no longer in the opposite direction but on average now is parallel to the field. The axiu 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 oenta, the rotational kinetic energy becoes larger than the electric field interaction and therefore they hardly feel the field effect: The states becoe pinwheeling and hcos hi approaches the already entioned field-free asyptote. The hybridization of the angular otion also changes according to the position of the states within the effective Stark potential. The contribution of higher rotations is doinant for those levels with hcos hi <, having therefore J 2 hðj; MÞ >. While for those lying above the field-free rotational ground state, i.e. having hcos hi >, lower rotations becoe doinant and therefore hj 2 i is lower than the corresponding field-free value. Even ore, the J = 1 level, for which hj 2 i is very close to the extreal value at J = 11, possess at the sae tie the sallest absolute value of hcos hi, whereas the axiu of hcos hi coincides with the iniu of hj 2 i. A siilar explanation holds for the expectation value hri as the contribution of higher (lower) rotations provokes a stretching (squeezing) of the olecule, i.e., an increasing (decreasing) hri with respect to the fieldfree value. Again, the iniu of hri coincides with the iniu of hj 2 i and the axiu of hcoshi. For highlying rotations, i.e. states lying significantly above the effective Stark well, the effect of the field is rather weak and therefore hj 2 i and hri approach the corresponding fieldfree parabola. The results for J > M states with M 5 distinctly differ fro those for the levels with M =. One of the ajor 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 axiu as in the M = case. The explanation for this phenoenon is found in the range of allowed values of h, which is related to the tilt angle between the angular oentu J with respect to the field axis, as discussed in [39]. In the seiclassical pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vector odel this angle is given by a ¼ cos 1 ðjmj= JðJ þ 1ÞÞ [47]. For J jmj large we get a near 9, i.e., the electric field direction lies alost 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 < p or cosh >. For fixed J, increasing the 2 agnetic quantu nuber M up to its axiu M = J decreases a and therefore will consequently diinish this effect. Indeed, in Fig. 3 we observe that the axiu of hcos hi is ore distinct the saller the agnetic quantu nuber is. Of course, this arguentation holds only if the energy of the corresponding states is large enough such that all h directions are possible Evolution of selected states with increasing field strength An analysis of the evolution of the properties of the LiCs olecule as the field strength is varied fro F =1 6 a.u. to F =1 3 a.u. has been perfored. We have considered nine levels with different syetries, all with vibrational quantu nuber = 1, agnetic quantu nubers M =, 5 and 1, and possessing field-free rotational quantu nubers J = M, M + 5 and M + 1. The behaviour of the corresponding energies and expectation values hcos hi, Dcosh, hj 2 i 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 regie, the field-free syetry of the levels deterines the quadratic Stark effect [4] and therefore the initial trend of the energy to increase (low field-seekers) or decrease (high field-seekers) as the field becoes stronger. Within the scale of the figure, all the levels keep their energy alost constant up to F =1 5 a.u. Increasing further the field strength, the M-degeneracy breaks down and the spectru clearly divides into high and low field-seeking states. The first group is coposed by the fully angular

8 21 R. González-Férez et al. / Cheical Physics 329 (26) Energy (a.u.) -.19 J Fig. 8. Energy as a function of the field strength for states with M = (+,, ), 5 (h, s, ) and 1 (n, d,,) and J = M, M + 5 and M +1 within the tenth vibrational band. Fig. 11. The expectation value hj 2 i as a function of the field strength for the sae set of states as in Fig cosθ R (a.u.) Fig. 9. The expectation value hcoshi as a function of the field strength for the sae set of states as in Fig. 8. Δcosθ Fig. 1. Dcosh as a function of the field strength for the sae set of states as in Fig Fig. 12. The expectation value hri as a function of the field strength for the sae set of states as in Fig. 8. oentu polarized levels, which are ost affected by the field, and, in our case, also the (15, 1) state. For F =1 3 a.u., the position of the (M,M) states is shifted energetically by about 1% in the spectru, and the field effect is stronger the lower is M. The low field-seeking set is fored by the J > M levels with M = and 5, and the (2,1) state. With a further increase of the field, these states becoe also high field-seekers, i.e., their energies reach axia as a function of the field and decrease thereafter. The position of these extrea with respect to the field depends on the corresponding state, varying fro F = a.u. to F =2 1 4 a.u. and for fixed M being lower the saller field-free rotational quantu nuber is. For very strong electric fields, the energies approach the pendular regie [39,4] which in the fraework of the effective rotor approxiation p ffiffiffiffiffiffi is given by E! xþð2j jmjþ1þ 2x, with x ¼ F hdðrþi ðþ 2l=hR 2 i ðþ. Therefore, states with different field-free syetry but the sae 2J jmj value will have the sae pendular energy for

9 R. González-Férez et al. / Cheical Physics 329 (26) x 1. This phenoenon is observed for F >1 4 a.u. between the (1, 1) and (5, ) pair of levels, and also between the (15,1) and (1,) states. The high field-seeking states increase their orientation as the field is enhanced having always hcoshi <, see Fig. 9. The (, ) state presents a significant orientation even for very weak fields with hcoshi =.525 and hcoshi <.8 for F =1 6 a.u. and F P a.u., respectively. The (5,5) and (1, 1) states are of pinwheeling character for very weak fields F [ 1 5 a.u., show a oderate orientation for weak fields, and the strong orientation liit, hcoshi <.8, is achieved for higher fields F J a.u. and F J a.u., respectively. Although hcoshi for the (15, 1) level onotonically decreases as F is increased, the influence of the field is very weak and has a negligible orientation up to F [ 1 4 a.u. Indeed, the changes on this expectation value are very sooth while its absolute value is very sall. For stronger fields, its orientation towards the field axis rapidly increases, reaching hcoshi =.698 for F =1 3 a.u. For the J > M states with M = and 5 and the (2, 1) level, hcoshi increases as the field is enhanced, passes through a axiu, and decreases thereafter. In particular, hcos hi shows for the (5,) and (1,) levels pronounced axia as a function of F, with hcoshi.36, for F a.u. and hcoshi.42 for F 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 oentu provokes that these hups are less pronounced and shifted to larger field strength for the (1, 5), (15, 5) and (2, 1) 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 hcoshi. However, the influence of the field is still very weak, hence the states are still ostly of pinwheeling character. At the sae tie, for a higher J value a stronger field is needed in order to copensate and overcoe the corresponding rotational kinetic energy, i.e., to trap the level within the Stark barriers. Therefore, for higher M but the sae J jmj value a less pronounced axiu is obtained. In the strong field regie, all the considered states exhibit a pronounced orientation with hcoshi <.5 for F =1 3 a.u. and the two pairs of states {(1, 1), (5, )} and {(15, 1), (1, )} have the sae pendular liit for hcoshi. The alignent of the levels onotonically increases as the field strength is increased, see Fig. 1. This effect is ost pronounced for the fully angular oentu polarized states. In particular, Dcos h rapidly decreases for the (, ) state, while for the (5, 5) and (1, 1) states inor changes are observed for weak fields, i.e. still being pinwheeling like in the field-free case, and thereafter Dcos h soothly decreases. A siilar 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 alignent. However, in the high field regie a significant alignent is achieved for all the states with Dcos h <.3 for F =1 3 a.u. The description of hcos hi provides an excellent basis for the analysis of hj 2 i, see Fig. 11. For the high field-seeking states hj 2 i increases as the field strength is enhanced therefore increasing the contribution of higher rotations. For the (M, M) states, a strong ixing of rotational levels takes place even for oderate fields and increases onotonically as the field is enhanced. For the (15,1) state and weak fields hj 2 i keeps alost constant, the rotational hybridization being very weak with J 2 hð15; 1Þ < 1 for F [ a.u. For the J > M levels with M = and 5 and the (2,1) state hj 2 i soothly decreases as the field is increased, reaches a iniu, and rapidly increases thereafter. hj 2 i for the (5, ) and (1, ) levels shows inia at field strengths strongly correlated to the axia of hcos hi. For the states with (1, 5), (15, 5) and (2, 1) these inia are very shallow, in soe cases hardly seen on the scale of the figure, and also related to the axia of hcoshi. A siilar analysis holds for the expectation value hri, see Fig. 12. Again, the field-free syetry of the states deterines the behaviour of the corresponding hri as the field is varied. The fully angular oentu polarized states are stretched as the field is enhanced, and the stretching is ore pronounced the lower M is. Whereas the J > M states are squeezed or stretched according to the decrease or increase of hj 2 i with respect to its field-free value The effective and adiabatic rotor approxiations versus the full rovibrational description In [41] the effective rotor approach was developed and studied for the specific case of carbon onoxide considering the rotational ground state within any vibrational excitation, i.e., levels with spherical syetry J = M = for F =, and field strengths varying fro F =1 6 a.u. to F =1 4 a.u. For this olecular syste it was shown that the agreeent with the exact calculation is excellent. A coparison of the adiabatic rotor approach with respect to the full rovibrational description and the effective rotor approxiation was carried out in Ref. [42]. This analysis was restricted to paraeter dependent odels, naely, a Morse potential and Gaussian functions as electric dipole oent function. Only low-lying vibrational excited states with J = M = and the strong field F =1 3 a.u. were considered. In particular, soe physical situations were found where ERA was no longer adequate and a ore elaborated odel like ARA is needed. In the present work, we have extended and copleted these coparisons by studying highly excited rotational states with different aziuthal syetries. We have analyzed the spectru of the LiCs olecule in an external electric field by eans of the effective and adiabatic rotor approxiations, and copared the results with the full rovibrational description. We have considered states with field-free vibrational quantu nubers 6 1, rotational quantu nubers J = M, M + 5 and M + 1, and

10 212 R. González-Férez et al. / Cheical Physics 329 (26) agnetic quantu nubers M =, 5 and 1, for field strengths varying fro F =1 6 a.u. to F =1 3 a.u. In order to perfor such a coparison the following relative error has been defined DA v ¼ jaf A v j v ¼ E; A ð7þ A F where A v represents the energy or one of the expectation values hcoshi, hj 2 i 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 = 1, J = 1, 15 and 2, and agnetic quantu nuber M = 1. The coparison for levels with lower vibrational, rotational, and agnetic quantu nubers yields siilar results. Fig. 13 shows DE E and DE A 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.1%. For both approxiations, DE v is larger the higher the rotational quantu nuber is. For fixed J, both relative errors keep constant in the weak field regie with very siilar values. For F J a.u., DE v of the (1,1) level increases as the field is enhanced, and the errors start to differ satisfying DE E > DE A. A siilar picture holds for the (15, 1) and (2, 1) 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 hcoshi is presented as a function of the field strength in Fig. 14. For copleteness, in Fig. 15 the absolute deviations of both approxiations with respect to the exact calculations are included. Unlike for the energy, we notice that the relative errors for ERA and ARA slightly deviate fro each other even for weak fields. The Dhcos hi E and Dhcos hi A deviations are larger relative error energy Fig. 13. Relative errors of ARA and ERA of the energy as a function of the field strength for states with agnetic quantu nuber M = 1, rotational quantu nuber J =1 (s and + for ERA and ARA, respectively), 15 (h and ) and 2 ( and ), and vibrational quantu nuber = 1. relative error cosθ Fig. 14. Relative errors of ARA and ERA of the expectation value hcoshi as a function of the field strength for the sae set of states as in Fig. 13. absolute error cosθ Fig. 15. Absolute differences of ARA and ERA of the expectation value hcoshi as a function of the field strength for the sae set of states as in Fig. 13. for larger J. Furtherore, ajor differences between both approxiations can now be observed. For the (1, 1) level, both relative errors show a constant behaviour for F [ 1 5 a.u., soothly decreasing as the field strength is augented while satisfying Dhcoshi v <.2. A siilar behaviour is observed for the (15, 1) level, but for weak fields, F <6 1 5 a.u. and F <8 1 5 a.u., we observe Dhcoshi v >.1 for v ¼ E and A, respectively. A further increase of the field leads to a decrease of both relative errors. For the (2,1) state, Dhcos hi E and Dhcos hi A keep constant and larger than.1 in the weak field regie. Increasing the field strength, they reach a axiu for F =1 4 a.u. with Dhcos hi E ¼ :67 and Dhcos hi A ¼ :74, decrease thereafter, and in particular they are saller than.1 for F =1 3 a.u. The large relative errors are due to the sall absolute values of hcoshi. In fact, the absolute error, see Fig. 15, onotonically increases as the field is enhanced reaching a plateau in the strong field liit. Even

11 R. González-Férez et al. / Cheical Physics 329 (26) ore, for F [ a.u. the ERA and ARA absolute differences are of the sae order of agnitude for all considered levels. Both approxiations perfor not as bad as their relative errors suggest, but they are unable to correctly describe the angular otion at the crossover fro the (Stark) unbound states to the bound ones. The sae effect is observed for low field-seeking states with M = and M = 5, and 6 1. The (15,1) state is high field-seeking and therefore shows a siilar behaviour as the fully polarized state. In Fig. 16 DhJ 2 i E and DhJ 2 i A are displayed as a function of the field strength for the sae 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 onotonically 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 otion is affected by the restrictions of relative error J Fig. 16. Relative errors of ARA and ERA of the expectation value hj 2 i as a function of the field strength for the sae set of states as in Fig. 13. relative error R Fig. 17. Relative errors of ARA and ERA of the expectation value hri as a function of the field strength for the sae set of states as in Fig. 13. the Stark potential. The results for both approxiations are indistinguishable on the weak field regie, and only inor differences are observed for strong fields. Quantitatively, only DhJ 2 i E for the (2,1) state at F =1 3 a.u. is insignificantly above.1, the reaining 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, DhRi E and DhRi A behave siilar 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 approxiations are splitting apart while their relative errors rise onotonically. On the strong field regie the adiabatic rotor perfors slightly better than the effective one. 4. Conclusions We have investigated the rovibrational properties of the LiCs olecule in a static hoogeneous electric field by solving the fully coupled rovibrational equation of otion and taking into account the dependence of the electric dipole oent on the internuclear distance. We hereby assued that the external electric field affects the rovibrational dynaics in a nonperturbative way whereas the influence on the electronic structure can be described by eans of perturbation theory. The Schrödinger equation for the rovibrational otion was solved by a highly efficient and accurate cobination of a discrete variable representation ethod and a basis set expansion technique applied to the vibrational and rotational coordinates, respectively. We have considered several rotational excitations of the LiCs olecule within the tenth vibrational band and with agnetic quantu nubers M 2 {, 5, 1}, and analyzed the energies and expectation values hcoshi, hj 2 i and hri for field strengths F = a.u. = V/, which includes the regie of experiental interest. For fixed field strength, we have shown how the influence of the field on the spectru strongly depends on the field-free rotational quantu nuber. Three different kind of states could be found: First, there are low-lying rotational excitations whose behaviour is doinated by the Stark interaction and which becoe ore bound in the presence of the electric field. They have a strong pendular signature and acquired a significant alignent and orientation which due to the negative sign of the electric dipole oent function is antiparallel to the field direction. We encounter a strong angular oentu hybridization doinated by the contribution of higher rotational excitations, which provokes a stretching of the olecular states. These effects were especially pronounced for the fully angular oentu polarized levels. Secondly, there are states with large rotational quantu nuber but still of pendular character whose rotational kinetic energy is of the sae order of agnitude as the Stark interaction. They are shifted upwards in the spectru with respect to their field-free position and are oriented parallel to the field,