Changes in Dipole Moments and Polarization of a Two-Level System with Intramolecular Coupling #

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1 Internet Electronc Journal of Molecular esgn 00,, Changes n pole Moments and Polarzaton of a wo-level System wth Intramolecular Couplng # José Lus Paz and eresa Cusat epartamento de Químca, Unversdad Smón Bolívar, Apartado 89000, Caracas 080A, Venezuela Receved xxx; Preprnt publshed xxx; Accepted xxx ; Publshed xxx Internet Electron. J. Mol. es. 00,, Abstract Motvaton. Polyatomc molecules can be descrbed by a two level system that can be represented by two crossed harmonc potental curves, ncludng the vbratonal levels, n order to study the nonlnear sgnal response of them. he varaton n some couplng parameter has an mportant nfluence on the behavor of the dpole moments and, consequently, on the Polarzaton n the new coupled states bass. Method. he calculaton method employed s based n the Louvlle formalsm for the densty matrx, where the Conventonal Optcal Bloch Equatons are solved n the Fourer space, to obtan the densty matrx elements, assocated wth the Polarzaton of the system. Changes n the couplng parameters V 0 (energetc dfference between the mnma of the potental curves), S (heght to whch the vbronc couplng occurs) and v (couplng parameter), produce modfcatons n the behavor of the dpole moments and Polarzaton of the coupled system. In ths sense, bascally two studes were done: a) varaton n the parameters S and v, keepng V 0 constant and b) varaton n the parameters S and V 0, keepng v constant. In order to study the effects of these changes, the Rotatng-Wave Approxmaton s neglected, fact that permts us to analze the processes that occur out of the resonance regon. Also, the permanent dpole moments of the states n the uncoupled bass are ncluded. Results. Study A (varaton n S and v): mportant ncrement of dpole moments and Polarzaton magntude values s observed for low values of v and hgh values of S. Study B (varaton n S and V 0 ): an abruptly ncrement n the Polarzaton magntude s presented to hgh values of S and low values of V 0. Conclusons. Changes observed are assocated to the modfcatons n the overlappng ntegral and consequently n the dpole moments of the states n the new coupled bass. Consequently, the nonlnear response suffers changes n ts behavor. Keywords. dpole moments; Polarzaton; crossed harmonc potental curves. Abbrevatons and notatons COBE, Conventonal Optcal Bloch Equatons FWM, Four Wave Mxng RWA, Rotatng Wave Approxmaton INROUCION In nonlnear optcs, the atoms and molecules nteractng wth a classc electromagnetc feld are conceved lke two-level models, where the nteracton s represented by the couplng between the transton dpole moments of the molecular system and the electromagnetc feld. hs model can be represented by levels wthout nternal structure. In other hand, such states n a polyatomc molecule can be thought as vbratonal states belongng to one or two potental energy surfaces. Models lke these, whch consst of two degenerate or quas-degenerate electronc level curves n a small range n the nuclear coordnates, have been appled to Jahn-eller and pseudo Jahn-eller couplng n # edcated to Professor anal Bonchev on the occason of the 65 th brthday. Correspondence author; phone: ; fax: /96; emal: jlpaz@usb.ve. Copyrght 00

2 Internet Electronc Journal of Molecular esgn 00,, molecules and to vbronc couplng on degenerate excted states of dmmers [-]. In ths sense, the ntramolecular couplng arses from the couplng between nuclear and electronc motons n a molecule. It s a very mportant phenomenon, specally to some physcal chemstry processes []. he modfcatons n some couplng parameters modfy the characterstcs and propertes of the crossed two-level systems. he vbronc couplng terms are mportant molecular parameters whch descrbe n a quanttatve way the phenomenon of vbratonal couplng of electronc states. he curve crossng problems have receved specal attenton n the last years and they have been appled n dfferent scence felds [4]. For nstance, chemcal reactons nvolve nucle moton of reactant speces and, n some cases, changes n ts electronc structure. Charge transfer reactons are among the problems of chemcal nterest n chemstry where two or more close potental energy curve crossng occurs. hen, n ths partcular case, we have modeled the molecule as a system consstng of two electronc harmonc potental states, each one ncludng ts lowest vbratonal energy level; these curves are horzontally dsplaced by R o and vertcally by V 0, as depcted n Fg.. Incluson of a resdual perturbaton, whch may arse from a resdual electron-electron correlaton and/or spnorbt couplngs terms n the amltonan of the system, may couple the above electronc states, causng the separaton of the two curves accordng to the avoded-crossng rule. In the present contrbuton, the prncpal am s to show the effects of changes n the couplng parameters v, S and V 0, on the dpole moments of the coupled states, and consequently, on the Polarzaton to the Four Wave Mxng (FWM) sgnal frequency. hese modfcatons are related wth the structure of the two-levels model employed. One of the changes (parameters S and v) mples horzontal dsplacement between the two potental curves, mantanng the value V 0 constant (near degenerate case). he other case (changes n the parameters V 0 and S) mples a vertcal dsplacement of the mentoned curves, mantanng a value of v constant. MAEMAICAL ASPECS. Vbronc Couplng Model In ths case, the model employed s based n a two-level system descrbed by two crossed harmonc potental curves, whch have the same apertures (same force constant δ). hese curves are dsplaced horzontally n nuclear coordnate (R) and vertcally n energy (V 0 ), and only nclude the fundamental vbratonal energy level n ther structure. As t s depcted n the Fgure, the most mportant parameters consdered n our study, whch characterze our model are the followng : the couplng parameter (v), the energy dfference between the mnma of the potental energy curves

3 Internet Electronc Journal of Molecular esgn 00,, (V 0 ), the energy heght at whch the couplng occurs (S). he amltonan of the system s descrbed by two terms, that nclude the molecular amltonan 0, and the contrbuton of the spn-orbt nteracton, whch s ncluded as a perturbaton, provokng the couplng between the electronc states, and causng the separaton or symmetry break of the potental curves, accordng to the avoded-crossng rule. Moreover, the molecular systems nteract wth classc electromagnetc felds treated as plane waves, where we have ncluded a relaxaton mechansm assocated to the presence of the solvent, whch s treated as transparent to the radaton and t s ntroduced n a phenomenologcal way. a) V(R) b) V(R) E 0 v E - E E 0 S Vo Vo -Ro R -Ro R Fgure. (a) abatc and (b) adabatc representaton of the two crossed harmonc potental curves. Regardng to the mathematcal aspects nvolved, each electronc (ψ (R;r)) and vbratonal (ϕ (R;r)) state s descrbed by ts respectve wave functon and ts correspondng energy value. Consderng a lnear combnaton of the egenfunctons of each level and solvng the respectve secular determnant, t s possble to obtan the egenfunctons and egenvalues of the coupled states (consderng () the low (hgh) new coupled states): Ψ ( r; R) = [ V ψ ( r; R) Φ ( R) ( E E ) ψ ( r; R) Φ ( R) ] C k () E = ( ) [ ] / E0 E0 E 4V00 () where the vbratonal energes are E 0 = 0.5 and E 0 = 0.5δ V 0, E = ( E 0 E 0 ). Also, we have V = ; C = V ( E E ) 00 v ϕ0 ϕ 0 0 [ ] / 00 0 ()

4 Internet Electronc Journal of Molecular esgn 00,, and [ ( δ ) ] / 4 ( 4δ ) S (4) ϕ0 ϕ0 = exp / ( δ ) (overlap ntegral) m ~ 0 ~ S = R0 obtaned by the Pekaran formula [5], where δ = 0 / 0 and h. ere, m ~ represents the reduced mass assocated wth the vbratonal modes as descrbed by the molecular coordnate R 0 and frequency 0.. pole Moments n the Coupled Bass Because of the employ of dpole radaton matter nteracton n our model, the permanent dpole moments of states n the uncoupled bass are also ncluded. It has been demonstrated that they contrbute sgnfcantly to the photonc processes that take place outsde the resonant regon of the spectrum [6-]. he general effects on changng the couplng parameter on the global FWM sgnal spectra were shown n prevous work [-]. hey ndcated that the crtcal quanttes on the study of the FWM response n a coupled bass were the transton and the permanent dpole moments when the rotatng wave approxmaton (RWA) was not taken nto account. In ths sense, n the new bass of coupled states, there exst a new set of dpole moments that are dfferent to those n the uncoupled bass. It has been shown that the zero values of the dpole moments n the uncoupled bass do not mply the nullty of the dpole moments n the new generated bass [4]. he mathematcal expresson of the dpole moments for the coupled states are obtaned from the ntegral: m j ( R) = ψ ( r; R) mˆ ψ ( r; R) d r (5) j Solvng the Eq. (5) for each case, the transton and permanent dpole moments can be descrbed by the followng equatons, respectvely: / V00 00 E = 4V E v ( m m ) m (6) aa = V 00 V 00 E a ( E E ) 0 m m a a ( E0 E ) m ( E0 E ) Em ( a v V E E E ) 00 0 (7)

5 Internet Electronc Journal of Molecular esgn 00,, where m, m and m represent the permanent and transton dpole moments of states n the uncoupled bass, respectvely. As we shall show below, these quanttes are very mportant to determne the behavor of the nonlnear sgnal because of ther dependence to the Macroscopc Polarzaton, and consequently wth the ntensty of the sgnal studed.. FWM Sgnal In the present work, we are nterested n a partcular case of nonlnear response: the FWM sgnal spectroscopy. hs known nonlnear technque s related to the mxng of two ncdent beams nteractng wth a medum to generate a thrd beam. Specfcally, the pump beam (hgh ntensty, frequency and propagaton vector k ) nteracts wth the probe beam (less ntense, frequency and propagaton vectork ) at a very small angle θ between them, to generate a thrd beam or sgnal beam wth a frequency = and propagaton vector k k k. hs sgnal s the most ntense and easer to detect. At ths sgnal frequency, we are gong to obtan the Macroscopc Polarzaton values..4 Polarzaton Expresson Polarzaton s related to the response of a system n front of ts nteracton wth an ncdent electromagnetc feld [5]. Moreover, the ntensty of the nonlnear sgnal response s proportonal to the squared Polarzaton. hen, knowng the expresson correspondng to the Polarzaton of the molecular system, t s possble to know the behavor of the nonlnear sgnal to ths system. In order to obtan a Polarzaton expresson, accordng to the specfc characterstcs of our system, the most common mathematcal method employed s the Louvlle formalsm. Startng wth t, t s possble to obtan the Conventonal Optcal Bloch Equatons (COBE), whch permt the study of the temporal evoluton of the system and ts nteracton wth the electromagnetc feld. Among the aspects taken nto account to do ths calculaton we can menton the use of a solvent transparent to the radaton, ncluded n the calculaton n a phenomenologcal way, by the longtudnal and transversal relaxaton tmes, parameters that descrbe the radatonless mechansm of relaxaton. Also, we have consdered the secular approxmaton and we have neglected the Rotatng Wave Approxmaton (RWA), whch, as we alreary menton t, allows us to study the processes that occur out of the resonance frequency. Fnally, the explct ncluson of the permanent dpole moments of states n the uncoupled bass has been consdered. For the present study, and consderng the facts mentoned above, the COBE can be wrtten n the followng general way: 4

6 Internet Electronc Journal of Molecular esgn 00,, [ ] = 0 dt d h h (8) [ ] = 0 dt d h h (9) ( ) [ ] 0 dt d = h (0) hese dfferental equatons represent the temporal evoluton of the coherence - and populatons, and they nclude the term related to the molecular system (assocated wth the resonance frequency 0 ), the term related to the feld system nteracton (descrbed by the dpole amltonan) and the term related to the relaxaton process (characterzed by the presence of the longtudnal and transversal relaxaton tmes and respectvely). Consderng the COBE wrtten above, takng nto account the general expresson to obtan the Macroscopc Polarzaton: ( ) ( ) ( ) ( ) [ ] d c N N N P = = = () and consderng the characterstc of the system under study, t s possble to obtan a general equaton that s the product of two terms, one of them related to the felds nvolved and the densty of molecules present nto the medum, and the second one, related to the dfferent resonant processes nvolved n ths case. hen, n a factorzed way, the followng expresson s obtaned: ( ) ( ) ( ), ;,, ; ξ Υ = P () where ( ) ( ) ( ) ( ) 0, E E N = Υ () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Γ = , ; d c β β λ ξ (4)

7 Internet Electronc Journal of Molecular esgn 00,, ere, N represents the chemcal concentraton of the absorbent molecules, represents the dpole moments and? represents the densty matrx elements, all of them correspondng to the coupled bass. E and E are the pump and probe electromagnetc felds, respectvely, and r r r d = ; = ; = Γ = ( ) ; β = ; λ = (5) j = ( ) 0 j ; ( 0 ) ; 5 = = 0 ( ) RESULS AN ISCUSION In ths secton we show two dfferent knds of results, consderng changes n the parameters v, V 0 and S. he value of the parameters nvolved n ths study are: 0 = s = 680 cm (resonance frequency of an organc molecule.e. Malachte Green), the longtudnal and transversal relaxaton tmes = =. 0 s, S = 0., m = and m =. respectvely, for the permanent dpole moments of the uncoupled states (gvng as a result d = 0. for the dfference n permanent dpolar moments), and m = m = 0. for the transton dpole moments of the uncoupled states. he parameters under study change n the followng range: S = ; v = ; V 0 = In ths partcular case, we have consdered only the term ξ( ;, ) consderng that the term (, ) constant f all varables nvolved are not changed. to be analyzed, Y can be taken as a gven value n a partcular case, and t s In prevous works, a global spectrum characterzed for the presence twelve resonances s obtaned n the frequency space [,,6]. hese peaks present symmetry n ntensty and poston n the spectrum, under the changes n frequency for. Because of ths symmetry fact, we have consdered only the frst sx peaks for the analyss descrbed below. Moreover, each peak can be represented by a reduced polarzaton expresson that permts, n some cases, to explan the orgn and behavor of the nonlnear response of each resonance n the spectrum [6]. In the other hand, the behavor of the transton and permanent dpole moments of the states n the new coupled bass s very mportant due to the dependence of the Polarzaton magntude wth ths quanttes (see Eq. (4)). In the Fgure, these quanttes are depcted as three-dmensonal graphs vs. a) v and S and b) V 0 and S. It s possble to denote, for the frst case (varaton n v and S), that the maxmal values n the dpole moments are obtaned for hgh values of S and low values of v. owever, for the second case (varaton n S and V 0 ), the results are opposte, namely, the dpole 6

8 Internet Electronc Journal of Molecular esgn 00,, moments reach ts maxmal values for hgh values of S, but hgh values of V 0 too. hs aspect have a very mportant nfluence n the behavor of the squared Polarzaton of the system under study, as we wll see bellow. a) b) - - d c d c FIGURE. ranston and permanent dpole moments representaton, n the coupled bass, to the followng cases: a)varaton n the parameters S and v, takng V 0 = 0.0 (near-degenerate case); b) Varaton n the parameters S and V 0, takng v = Study A: Varaton n S and v In the frst case, we have studed the modfcaton of the squared modulus of Polarzaton ξ ( ) ;, wth the changes n the parameters S and v, takng V 0 as a constant value equal to 0.0 (namely, near degenerate case of the potental curves). In the Fgure, three-dmensonal graphs of ξ( ) ;, vs. S and v are depcted, for the sx peaks belongng to the general spectrum. ere, t s possble to observe a smlar behavor for the peaks,, 4 and 5, but peaks and 6 look lke dfferent. For the frst ones, maxmal values of the magntude of ξ( ) ;, are obtaned for low values of v, for almost all values of S. owever, ths values dmnshng for the lowest value of v = 0 and hghest value of S = 0. Peak ncreases ts magntude wth the ncreasng of the parameter v, untl to obtan an asymptotc value for the hghest values n ths parameter. Peak 6 ncrease ts magntude wth the decreasng of the parameters S, presentng a local maxmum for low values of v. 7

9 Internet Electronc Journal of Molecular esgn 00,, It s mportant to menton that the responses of the peaks and 6 are governed for the transton dpole moments, but the other peaks arse when the permanent dpole moments are ncluded n the formalsm. Peak s related wth the coherence densty matrx term, and peak 6 s related wth the populaton densty matrx elements [6]. hs fact provoke the dfferent aspects n the topology of the Polarzaton values. PEAK PEAK PEAK PEAK 4 PEAK 5 PEAK 6 FIGURE. Modfcaton of the factor ξ ( ) ;, (related to Macroscopc Polarzaton), wth the changes n the couplng parameters S and v, wth Vo = 0.0 (near-degenerate case).. Study B: Varaton n S and Vo In the second case, the Fgure 4 depcted the three-dmensonal graphs correspondng to the modfcatons of ξ( ) ;, vs. S and V 0, consderng constant the parameter v = 0.5. Smlar behavor respect to the prevous case s observed. Peaks,, 4 and 5 present dentcal shape, wth a partcular peak of hgh values of ξ ( ) ;, n the regon of low values of V 0 and hgh values of S, whereas peaks and 6 present a dfferent behavor. o the case of resonance (peak ), an abruptly ncreasng of the magntude of ξ( ) ;, s observed for low values of V 0, for almost all values of S. In the other case (peak 6), maxmal values are found for low values of V 0 and S, whch dmnsh for the hghest value of S = 0 and V 0 = 0.0 (lke the prevous case). Agan, ths marked dfference between the peaks and 6, and the other ones (,, 4 and 5) obeys to the fact that ths resonances appear due to the presence of the transton dpole moments of the 8

10 Internet Electronc Journal of Molecular esgn 00,, molecular system. he other ones correspond to the ncluson of the permanent dpole moments, that, for the case of coupled states s always dfferent to zero, even though the permanent dpole moments of the uncoupled stated could be null. PEAK PEAK PEAK PEAK 4 PEAK 5 PEAK 6 FIGURE 4. Modfcaton of the factor ξ ( ) ;, (related to Macroscopc Polarzaton), wth the changes n the couplng parameters S and V 0, takng v = FINAL REMARKS Consderng the results showed n ths work, t s possble to menton the mportance of the characterstc of molecular systems descrbed as two-level systems ncludng the ntramolecular couplng phenomenon. ependng on the shape of the harmonc potental curves employed to descrbe the electronc levels of the system, t wll respond n a dfferent way when ts nteracton wth an electromagnetc feld s studed. Specfcally, the nonlnear response of dye solutons changes when dfferent characterstcs of the potental curves are consdered. In the present work, the modfcatons of the Polarzaton at the frequency, consderng FWM spectroscopy, are studed, varyng the ntramolecular couplng v, S and V 0. Changes n S and v, nvolve an horzontal dsplacement of the potental curves, for a fxed values of V 0, whle changes n S and V 0 nvolve a vertcal dsplacement of the curves, for a fxed values of v. In general, the mportant varatons are observed partcularly for low values of V 0 and v, and hgh values of S, prncpally. hs behavor arses from the modfcatons that suffer the dpole moments n the 9

11 Internet Electronc Journal of Molecular esgn 00,, coupled bass, due to the dependence of the Macroscopc Polarzaton, and consequently the FWM ntensty, wth the mentoned quanttes. At the same tme the dpole moments are quanttes that depend drectly (explct dependence) and ndrectly (by the overlap ntegral dependence) on the couplng parameters under consderaton S, v and V 0. Moreover, the peaks and 6, that depend only on the transton dpole moments between the coupled states, show a partcular behavor that dffers to the other resonances. Whle the peaks,, 4 and 5 present a maxmal values peak n a specfc regon of the graph (hgh values of S and low values of v and V 0 ), the peaks and 6 do not show ths shape, and ther changes nvolve a more extended regon n the graph, namely, the modfcatons come from a wde range of values of S, v and V 0. Acknowledgment he present work was supported by the Fondo Naconal de Cenca, ecnología e Innovacon (FONACI) (grant G ) and by the ecanato de Investgacones de la Unversdad Smón Bolívar (grant GI-). 5 REFERENCES [] M. García-Sucre, F. Goychman and R. Lefebvre, Adabatc Correctons n a Smple Model of wo Interactng Electronc-Potental Curves, Phys. Rev. A 970,, [] J. L. Paz, M. García Sucre, E. Squter, and V. Mujca., he Effect of Intramolecular couplng on the Optcal Susceptbltes of a wo Level Molecule, Chem. Phys. Lett. 994, 7, 4. [] A. Mebel, M. ayash, and S.. Ln, Ab Into Calculatons of Vbronc Couplng, Applcatons to Symmetry Forbdden Vbronc Spectra and Internal Converson n Ethylene, Chem. Phys. Lett. 997, 74, 8 9. [4] V.A. Yurovsky and A. Ben-Reuven, J. Phys. B: At. Mol. Opt. Phys. 998,,. [5] Bartolo, B., 980, Radatonless Processes (New York: Plenum Press) [6] M. A. Kmetc and W. J. Meath, Perturbatve Correctons to the Rotatng Wave Approxmaton for wo Level Molecules and the Effects of Permanent poles on Sngle Photon and Multphoton Spectra, Phys. Rev. A 990, 4, [7] R. Bavl,. F. eller, and Y. B. Band, Nonlnear Optcal Propertes of wo Level Systems wth Permanent pole Moments, Phys. Rev. A 990, 4, [8] J. P. Lavone, C. oerner, and A. A. Vllaeys, Effects of Permanent pole Moments n egenerate Four Wave Mxng processes, Phys. Rev. A 99, 44, [9] R. Bavl and Y. B. Band, Nonlnear Absorpton and sperson n a wo Level System wth Permanent pole Moments, Phys. Rev. A 99, 4, [0] R. Bavl and Y. B. Band, Sum and fference Frequency generaton n a wo Level System wth Permanent pole Moments, Phys. Rev. A 99, 4, []. Cusat, J. L. Paz, M. C. Salazar, and A. J. ernández, Intramolecular Couplng Study of the Resonances n the Four Wave Mxng Sgnal, Phys. Lett. A 000, 67, 8. [] J.L. Paz,. Cusat, M.C. Salazar and A.J. ernández, Intramolecular couplng study on nonlnear sgnals, heor. Chem. Acc. 000, 04, -6. [] J.L. Paz and.cusat, Modfcatons n the Maxmal Intensty of Four-Wave Mxng Sgnal under the Changes of Vbronc Couplng Condtons, Internet Electron. J. Mol. es. 004,, [4] M. García Sucre, E. Squter, J. L. Paz, and V. Mujca, Absorptve and spersve Processes n a wo Level Molecule wth Intramolecular Couplng and Non Zero Permanentd pole Moment, J. Phys. B: At. Mol. Opt. Phys. 994, 7, [5] N. Bloembergen, 996, Nonlnear Optcs (London: World Scentfc). [6] J. L. Paz,. Cusat, M. C. Salazar, and A. J. ernández, Study and Characterzaton of resonances n the Four Wave Mxng Sgnal of a wo Level System wth Intramolecular Couplng, J. Mol. Spect. 00,,

12 Internet Electronc Journal of Molecular esgn 00,, Bographes José Lus Paz s tular professor of Physcal Chemstry at the Unversdad Smón Bolvar. Prof. Paz obtaned the Ph.. degree n Physcal Chemstry at the Unversdad Central de Venezuela (988). r. Paz has nterested on problems of solvent effects and stochastc methodology n four wave mxng. More recently, r. Paz s workng on projects n topcs assocated to algebrac methods appled to Quantum Optcs and Nonlnear Spectroscopy. eresa Cusat s currently Ph.. student at the Unversdad Smón Bolívar. After obtanng a M.Sc. degree n heoretcal Chemstry at the Smón Bolívar Unversty, she worked as engaged assstant professor of general chemstry by conventonal tme, and after that she was collaboratng n a group project between the Unversdad Smón Bolívar and Unversdad Central de Venezuela. Actually, she works as an nvted researcher n the Quantum Optcs group of the Physcs epartment at the Unverstà degl Stud d Palermo, developng a research work related wth entanglement n mesoscopc systems.