AAYSIS OF QUATUM EIGESTATES I A 3-MODE SYSTEM SRIHARI KESHAVAMURTHY AD GREGORY S. EZRA Depatment of Chemisty, Bake aboatoy Conell Univesity, Ithaca, Y 14853, USA. Abstact. We study the quantum eigenstates of a thee degee of feedom spectoscopic Hamiltonian fo the H 2 O molecule. Using the classical esonance zones as a template, we ae able to undestand and oganize the enegy level spectum, and povide quantum numbe assignments fo the eigenstates. 1. Intoduction The poblem of the natue of highly excited ovibational states of polyatomic molecules is of cental significance fo molecula spectoscopy and chemical kinetics [1]. At high enegies, the taditional appoach of molecula spectoscopy, based on on a hamonic oscillatoigid oto model [2], beaks down due to stong intemode couplings. These mode couplings, which esult fom anhamonicities in the potential and/o otation-vibation inteaction, lead to intamolecula vibational enegy tansfe [3] and complicated enegy specta. A typical classical polyatomic molecule Hamiltonian is inheently a multi ( 3) mode, nonintegable system. Moeove, at high levels of intenal excitation, the semiclassical egime is appoached, and the geneal poblem of the classical-quantum coespondence fo nonintegable systems must be faced in ode to elate classical and quantum aspects [4]. Fo both = 2- and 3-mode systems, we can ecognize seveal mechanisms leading to complicated enegy level pattens. One possibility involves stong mixing of nea-degeneate manifolds of states by a single esonant coupling, so that the coesponding classical dynamics is still integable. Anothe possibility is stong state mixing due to the pesence of two o moe esonant couplings, so that the coesponding classical dynamics is nonintegable and we have the possibility of chaos. Finally, thee is the possibility of state mixing due to the phenomenon of dynamical tunneling [5], associated with accidental nea-degeneacies of levels. Model spectoscopic Hamiltonians descibing two modes coupled by a single esonant tem have been widely studied [6]. Fo such systems thee exists an additional constant of motion apat fom the enegy, and the educed phase space is a 2-dimensional sphee, the so-called polyad phase sphee. Kellman and cowokes have studied the bifucation stuctue of fixed points of the educed Hamiltonian on the polyad phase sphee, which coespond to the peiodic obits in the full phase space, and have investigated in detail the
2 elation between the classical phase space stuctue and the associated quantum mechanics [6]. Fo the case of a single esonant coupling, Rose and Kellman have extended this analysis to a multimode plana model of C 2 H 2 [7]. Multimode Hamiltonians with two o moe esonant couplings have not been studied in a systematic fashion. One eason is that, fo multidimensional systems, the stable and unstable manifolds of unstable peiodic obits cannot patition phase space into disjoint zones [8], theeby peventing staightfowad genealization of peiodic obit based appoaches. The classical mechanics of multimode systems is of couse consideably iche than that of 2-mode Hamiltonians: thee is the possibility of Anold diffusion [9], and, of moe immediate elevance to molecula poblems, the existence of intesections between esonance zones. The implications of this latte featue of the phase space stuctue of multi-mode systems have begun to be exploed fo molecula systems [10]. In the pesent pape we outline a qualitative appoach to analysis of enegy level pattens and wavefunctions in multimode systems, which is based on Chiikov esonance analysis [9]. The system studied is a 3-mode spectoscopic vibational Hamiltonian fo H 2 O due to Baggot [11]. We establish a diect coelation between featues of the associated eigenstates with the location of esonance zones in phase space. In paticula, we obseve pogessions of esonant eigenstates along classical esonance zones leading to mixed states in the vicinity of the esonance channel intesections. The classical esonance analysis theefoe povides a template fo analyzing the quantum states, and theeby facilitates an undestanding of the level spectum. 2. Resonance analysis of the Baggot Hamiltonian We conside the classical analog of the spectoscopic Hamiltonian fo H 2 O deived by Baggot [11]. This 3 degee of feedom Hamiltonian descibes the two local OH stetching modes and the HOH bend, and includes two 2:1 stetch-bend esonant tems H 2:1, =1,2, a 1:1 stetch-stetch esonant tem H 1:1 and a 2:2 stetch-stetch esonant tem H 2:2 : H = H 0 + H 1:1 + H 2:2 + H 2:1, (1) whee H 0 is the zeoth ode Hamiltonian H 0 =Ω s I +Ω b I b +α s I 2 + α b Ib 2 + ɛ ss I 1 I 2 + ɛ sb I b I, (2) =1,2 and the esonant coupling tems ae =1,2 =1,2 =1,2 H 1:1 = β 12(I 1 I 2 ) 1/2 cos(θ 1 θ 2 ) H 2:2 = β 22 I 1 I 2 cos[2(θ 1 θ 2 )] (3) H 2:1 = β sb (I Ib 2 ) 1/2 cos(θ 2θ b ). In the above equations, (I, θ) ae the canonical action-angle vaiables fo the two stetches and the bend mode, espectively, with β 12 β 12 + λ (I 1 + I 2 )+λ I b. Values of the vaious paametes ae taken fom Baggot s fit to the vibational levels of H 2 O (Table 2 of efeence [11]). The 2:2 esonance coupling does not significantly affect the classical
phase space stuctues in the egime of inteest fo this pape and will be neglected in the classical esonance analysis. The complete Hamiltonian H, although nonintegable, has a constant of the motion in addition to the enegy: =2 I + I b, (4) =1,2 which is the classical analog of (twice) the polyad numbe, P = =1,2 n s + n b /2. Thus the dynamics is constained to the plane in action space = constant. Chiikov esonance analysis fo H 0 is staightfowad, and in standad fashion one can detemine the location of the esonance zones in action space [9]. Both the 1:1 and 2:1 esonance zones ae shown in Figue 1, pojected onto the (I 1,I 2 ) action plane (P = 8). It is impotant to note that, fo the Baggot Hamiltonian, as β 12 < 0andλ >0 the 1:1 esonance width actually deceases as the actions I 1,I 2 get lage. ote also that we do not explicitly conside seconday esonances since we ae inteested only in the lage-scale stuctue of the classical phase space (i.e., intesection of the lowest ode esonance zones). 3. Analysis and assignment of eigenstates and specta The quantum eigenfunctions and eigenenegies wee obtained by diagonalizing the Baggot Hamiltonian in the numbe basis n 1,n 2,n b, whee n 1,n 2 ae the quantum numbes fo the anhamonic local O-H stetch modes and n b denotes the bend quantum numbe. The Hamiltonian is block diagonal in the polyad numbe P, and the total numbe of states at fixed P is (P +1)(P+2)/2. To analyse the eigenstates, it is natual to poject them onto the (n 1,n 2 ) quantum numbe (action) plane. The physical points in the (n 1,n 2 ) lattice ae those points fo which P n 1 n 2 is nonnegative. The eigenstates ae epesented by plotting at evey physical lattice point a cicle with adius equal to the squae of the coefficient of the coesponding zeoth ode basis state in the eigenstate of inteest. In this pape we concentate on the P = 8 case. Thee ae a total of 45 states and fo this case the 1:1 and 2:1 esonance lines do not intesect in the physical action space. Moeove, the 1:1 esonance inteaction dominates in the sense that an appopiate zeothode pictue of the spectum is one in which only the 1:1 coupling is pesent, i.e., the bend quantum numbe n b is conseved. In this (classically integable) limit we have essentially a stack of independent 2-mode systems, labelled with the quantum numbe n b. States fo the 1:1 coupled local modes ae classified as local o nomal, accoding to whethe amplitude is situtated outside o inside the 1:1 esonance zone, espectively. In the absence of 2:1 couplings, each eigenstate has nonzeo amplitude only along the line I 1 + I 2 = constant. The nomal mode states fom pogessions along the 1:1 esonance line, whee all states in a given sequence have eithe a node o antinode along the diagonal. Figue 1a shows one paticula sequence in the case when only the 1:1 inteaction tem is pesent. Of key inteest is the effect of the 2:1 esonant couplings on this zeoth-ode pictue; the 2:1 couplings couple togethe stacks of states with diffeent values of n b, theeby manifesting the tue multidimensional natue of the system. Figue 1b shows a sequence of states fo the fully coupled system coesponding to those of Fig 1. It is clea that, in the nonintegable case, one of the states in the sequence is significantly distoted by the 2:1 esonant tem (this state is maked with thick cicles). Fo lage values of the polyad numbe P, we obseve pogessions of esonant states along the 2:1 esonance channels. 3
4 8 a b 6 I 2 4 2 0 0 2 4 6 8 I 1 0 2 4 6 8 I 1 Figue 1. case (b). omal mode sequence along the 1:1 esonant line fo the integable case (a) and nonintegable E, cm 1 25650 25600 25550 25500 25450 25400 25350 25300 25250 25200 0 10 20 30 40 β 0 2 4 6 8 10 n b Figue 2. (a) Evolution of the enegy spectum as β β sb is tuned on and (b) associated states gouped by n b., and stand fo nomal mode, nonesonant local mode doublets and esonant local mode doublets espectively.
5 26360 E, cm 1 26310 26260 26210 26160 26110 0 10 20 30 40 β 0 2 4 6 8 10 n b Figue 3. Same as figue 2 with a diffeent egion of the enegy spectum. We now examine the changes in the level spectum fo the 1:1 coupling case when the 2:1 couplings ae tuned on. Figues 2 and 3 show the pats of the spectum which undego the most significant changes fo P = 8. ocal mode (±) doublets situated in the vicinity of the phase space egion coesponding to the 2:1 esonance zone become 2:1 esonant doublets unde the influence of the 2:1 couplings. Most of the mixing occus between the n b =2,4 and 6 manifolds of states, which coelates well with location of the 2:1 esonance zone. All of the avoided cossings seen in Figues 2 and 3 ae simple two state cossings. Fo P = 8, one can theefoe undestand the mixing of states fom diffeent n b manifolds via simple two state analysis. The assignment of the spectum is staightfowad, due to the appoximate consevation of n b, and states ae assigned as nomal mode, nonesonant local mode doublets, o 2:1 esonant local mode doublets. The simple two state scenaio beaks down at highe values of P whee one sees complicated mixings involving thee o moe states. In such cases it becomes moe useful to oganize the spectum on the basis of the quantum numbes associated with esonant sequences as opposed to the n b gouping. The details of the analysis fo highe values of P will be epoted in a futue publication [12]. 4. Acknowledgements This eseach was suppoted by SF gant CHE-9403572. Refeences 1. C. E. Hamilton, J.. Kinsey and R. W. Field, Ann. Rev. Phys. Chem. 37, 493 (1986); M. E. Kellman, in Molecula Dynamics and Spectoscopy by Stimulated Emission Pumping, edited by H.-. Dai and R. W. Field, (Wold Scientific, 1995).
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