classical mechanics of dipolar asymmetric top

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International Journal of Bifurcation and Chaos, Vol. 18, No. 4 (2008) 1127–1149c© World Scientific Publishing Company

CLASSICAL MECHANICS OF DIPOLARASYMMETRIC TOP MOLECULES IN COLLINEAR

STATIC ELECTRIC AND NONRESONANTLINEARLY POLARIZED LASER FIELDS:

ENERGY-MOMENTUM DIAGRAMS,BIFURCATIONS AND ACCESSIBLE

CONFIGURATION SPACE

CARLOS A. ARANGO∗ and GREGORY S. EZRA†Department of Chemistry and Chemical Biology,

Baker Laboratory, Cornell University,Ithaca, NY 14853, USA

†[email protected]

Received November 21, 2006; Revised May 23, 2007

We study classical energy-momentum (E-m) diagrams for rotational motion of dipolar asymmet-ric top molecules in strong external fields. Static electric fields, nonresonant linearly polarizedlaser fields, and collinear combinations of the two are investigated. We treat specifically themolecules iodobenzene (a nearly prolate asymmetric top), pyridazine (nearly oblate asymmetrictop), and iodopentafluorobenzene (intermediate case). The location of relative equilibria in theE-m plane and associated bifurcations are determined by straightforward calculation, with ana-lytical results given where possible. In cases where analytical solutions cannot be obtained, weresort to numerical solutions, while keeping a geometrical picture of the nature of the solutionsto the fore. The classification we obtain of the topology of classically allowed rotor configurationspace regions in the E-m diagram is of potential use in characterization of energy eigenstates ofthe corresponding quantum mechanical problem.

Keywords : Classical mechanics; molecular rotation; relative equilibria; bifurcations.

1. Introduction

The study of rigid body motion is one of the mostimportant topics in classical and quantum mechan-ics [Klein & Sommerfeld, 1965; Casimir, 1931;Deprit, 1967; Arnold, 1978; Zare, 1988; Arnoldet al., 1988; Deprit & Elipe, 1993; Goldstein et al.,2002]. Of particular importance are the integrablecases of the rigid body problem [Oshemkov, 1991;Bolsinov & Fomenko, 2004], which include the freeasymmetric top (Euler top) [Klein & Sommerfeld,

1965; Arnold, 1978; Arnold et al., 1988], thesymmetric top in a uniform external gravita-tional field (Lagrange top) [Arnold et al., 1988;Cushman & Bates, 1997], and the Kovalevskaya top[Arnold et al., 1988; Perelemov, 2002]. The theory ofrigid body motion also provides the basis for anal-ysis and interpretation of the rotational dynamicsand spectra of semi-rigid molecules [Casimir, 1931;Zare, 1988; Townes & Schawlow, 1975; Harter, 1988;Kroto, 1992; Bunker & Jensen, 1998]. In molecular

∗Current address: Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada

1127

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1128 C. A. Arango & G. S. Ezra

terms the Euler top is simply a free asymmetric topmolecule [Zare, 1988; Harter, 1988; Kroto, 1992],the Lagrange top models a symmetric top moleculewith a dipole moment in an electric field [Kroto,1992; Kozin & Roberts, 2003], while there does notappear to be an obvious molecular analogue for theKovalevskaya top. Breaking the symmetry of themoment of inertia tensor in the Lagrange top resultsin the nonintegrable problem of an asymmetric topin a static field. Molecular examples of these twocases are iodobenzene (near prolate) [Peronne et al.,2003, 2004] and pyridazine (near oblate) [Li et al.,1998] in a static electric field. The rotational con-stants of the molecule iodopentafluorobenzene makeit a more generic example of an asymmetric rotor[Poulsen et al., 2004].

The general problem of classical-quantum cor-respondence [Gutzwiller, 1990; Child, 1991] is ofgreat interest for both integrable and nonintegrablerotor systems. Various aspects of the classical-quantum correspondence have been studied fordiatomic molecules in tilted fields, i.e. noncollinearstatic electric and nonresonant linearly polarizedlaser fields [Arango et al., 2005]. The integrablecollinear case exhibits the phenomenon of mon-odromy [Cushman & Bates, 1997] both classicallyand quantum mechanically [Arango et al., 2004].For the nonintegrable case of tilted fields the rotormotion tends to be integrable in both low-energy(pendular) and high-energy (free-rotor) limits, andchaotic at intermediate energies, with the degree ofchaos controllable by variation of the angle betweenthe fields [Arango et al., 2005]. For collinear fieldsthe system is integrable, with both the energy Eand the projection of the angular momentum intothe space fixed z-axis, m, as constants of motion.The effective potential Veff(θ;m) for a given valueof m exhibits extrema in the θ (polar) coordinate,which define the relative equilibria [Arnold, 1978;Arnold et al., 1988; Smale, 1970]. Plotting the loca-tion of these extrema in the E-m plane gives theenergy-momentum diagram for the system [Arango,2005]. The E-m diagram provides a useful globalclassification of the rotor dynamics, as distinctregions of the E-m plane are associated with dif-ferent allowed types of motion of the diatomic. Forsymmetric tops in electric fields similar diagramscan be constructed [Kozin & Roberts, 2003], andanalysis of classical symmetric top E-m and E-kdiagrams helps understand the organization of thequantum level spectrum [Kozin & Roberts, 2003].

The problem of a dipolar asymmetric top in astatic external field is nonintegrable [Arango, 2005;Grozdanov & McCarroll, 1996], as is the problem ofa polarizable asymmetric top in a nonresonant laserfield [Arango, 2005]. For asymmetric tops in eitherstatic or laser fields or collinear superpositions ofthe two, the angular momentum projection m is aconstant of motion. Although the complicated formof the kinetic energy does not allow us to separatean effective potential as straightforwardly as in thediatomic or the symmetric top case, it is still pos-sible to define an effective or amended potential[Arnold et al., 1988; Bolsinov & Fomenko, 2004;Smale, 1970] for the class of motions in whichthe asymmetric top molecule rotates with constantEuler angles θ and ψ (the third Euler angle φis an ignorable coordinate). The energies of theextrema of this potential for given values of m againdefine an energy-momentum diagram [Arnold, 1978;Smale, 1970; Iacob, 1971; Katok, 1972; Tatarinov,1974; Artigue et al., 1986], which can be used toclassify the motions of the asymmetric top.

The asymmetric top in an external static elec-tric field is an example of a dynamical system withsymmetry [Smale, 1970; Arnold et al., 1988; Mars-den, 1992; Marsden & Ratiu, 1999]. The potentialenergy and Hamiltonian in this case are invariantwith respect to rotations about the space-fixed fielddirection. There is therefore an associated constantof the motion m, the projection of the angularmomentum vector j onto the external field axis,in addition to the energy E. The mapping of thesystem phase space onto the E-m plane is calledthe energy-momentum map [Arnold et al., 1988;Smale, 1970; Marsden, 1992; Marsden & Ratiu,1999]. Critical points of this mapping define thebifurcations sets in the E-m plane, which form theboundaries of regions of qualitatively different typesof classical motion [Bolsinov & Fomenko, 2004;Smale, 1970]. Following the fundamental work ofSmale [1970], applications of these concepts weremade to integrable rotor problems [Iacob, 1971],asymmetric rotors in an external gravitational field[Katok, 1972], and symmetric [Tatarinov, 1973] andasymmetric [Tatarinov, 1974] rotors in more com-plicated potentials. Relative equilibria have alsobeen studied for rotating semi-rigid molecules inthe absence of external fields [Montaldi & Roberts,1999; Kozin et al., 2000] and for transition statesin rotationally inelastic collisions [Wiesenfeld et al.,2003].

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Classical Mechanics of Dipolar Asymmetric Top Molecules 1129

In the present paper we study E-m diagramsof relative equilibria for molecular asymmetric topsfor several molecule-field configurations of physicalinterest [Stapelfeldt & Seideman, 2003; Seideman &Hamilton, 2005]: a static electric field; a nonres-onant linearly polarized laser field; both fields ina collinear combination. For the field strengthsconsidered, the free asymmetric top motion isstrongly perturbed. The associated E-m diagramsare obtained for the most part analytically. Ouraim here is to understand the nature of the rotormotions associated with different regions and curvesin the E-m diagrams for physically relevant val-ues of the field parameters. We also study thetopological classification of the allowed θ-ψ con-figuration space of the system in terms of their(multivalued) genus [Arnold et al., 1988]. An exten-sion of these results, to be discussed in a futurepaper, involves comparison of computed quantummechanical eigenstate probability densities with theboundaries of classically allowed regions in θ-ψ con-figuration space. The comparisons indicate that theclassical mechanical methods developed here pro-vide a promising foundation for the difficult taskof classifying the quantum levels of the complexsystem consisting of an asymmetric rotor in exter-nal fields [Block et al., 1992; Moore et al., 1999;Kanya & Oshima, 2004].

We mention that the approach adopted heremay be thought of as the analogue for perturbedrotor systems of the methods applied by Kellmanand coworkers [Kellman, 1995; Rose & Kellman,2000] to vibrational problems exhibiting a singleconserved vibrational (superpolyad) quantum num-ber (see also [Cooper & Child, 2005]).

There have been many studies of the classi-cal dynamics of a rigid asymmetric top rotatingabout a fixed point in a gravitational field (thisis the classic heavy top problem); see, for exam-ple [Klein & Sommerfeld, 1965; Arnold et al., 1988;Iacob, 1971; Katok, 1972; Galgani et al., 1981;Artigue et al., 1986; Chavoya-Aceves & Pina, 1989;Lewis et al., 1992; Broucke, 1993; Gashenenko &Richter, 2004]. Katok [1972] and Gashenenko andRichter [2004] obtained E-m diagrams and ana-lyzed bifurcations of relative equilibria in the E-mplane. These authors also classified the topology ofaccessible system configuration space. We presenthere a similar analysis for several molecular exam-ples of an asymmetric top molecule possessing adipole moment in a static electric field. In the

spirit of Katok’s analysis [Katok, 1972], we usestraightforward analytical and geometric methodsto build the E-m diagram for the molecules ofinterest.

Katok’s treatment was generalized by Tatari-nov [1974] (see also [Arnold et al., 1988]) to includemore complicated gravitational perturbations ofthe rotational dynamics of an asymmetric top.Although given for a specific potential, Tatarinov’sanalysis can be mapped directly onto the molec-ular case of an asymmetric top in collinear fields[Arango, 2005]. Following Tatarinov, we obtainE-m diagrams for asymmetric molecules of phys-ical interest, and study bifurcations and other char-acteristic features of the problem, including thetopology of the θ-ψ configuration space and its clas-sification according to the genera of the (connected)allowed regions [Arnold et al., 1988].

This paper is organized as follows. First inSec. 2 we derive the Hamiltonian for asymmetrictops in the general tilted fields case, with the aim ofintroducing our notation and conventions. In Sec. 3we treat the molecules in a static electric field, whilein Sec. 4 we study the same molecules in a non-resonant linearly polarized laser field. In Sec. 5 weanalyze these molecules in collinear fields. Section 6provides the conclusion.

Finally we mention that, the advantages ofalternative approaches notwithstanding [Cushman,2005], all calculations reported here have been car-ried out using polar coordinates.

2. Hamiltonian for the AsymmetricTop in Combined Fields

2.1. General case: Tilted fields

We use the y-convention [Zare, 1988] to define thethree Euler angles (θ, φ, ψ) describing the orienta-tion of the body-fixed frame with respect to lab-fixed frame. In the body-fixed frame the kineticenergy of the free asymmetric top can be writ-ten in terms of the components of the angularmomentum j = (j1, j2, j3) and the three compo-nents of the (diagonal) moment of inertia tensorI = diag(I1, I2, I3)

T =j212I1

+j222I2

+j232I3

. (1)

For an asymmetric top, I1 = I2 = I3. In termsof the Euler angles and their conjugate momenta

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(pθ, pφ, pψ) the body-fixed components of j are

j1 = pθ sinψ − pφcosψsin θ

+ pψ cosψ cot θ, (2a)

j2 = pθ cosψ + pφsinψsin θ

− pψ sinψ cot θ, (2b)

j3 = pψ (2c)

so that

j2 = j21 + j22 + j23

= p2θ +

1sin2 θ

(pφ − pψ cos θ)2 + p2ψ. (3)

From (1), the kinetic energy T = T (θ, φ, ψ, pθ, pφ,pψ) is then

T =1

2I1

[pθ sinψ +

cosψsin θ

(pψ cos θ − pφ)]2

+1

2I2

[pθ cosψ +

sinψsin θ

(pφ − pψ cos θ)]2

+p2ψ

2I3.

(4)

Immediately we see that φ is an ignorable coor-dinate and pφ = m, the projection of j into thespace fixed z-axis, is a constant of the motion forthe free top.

If the polarizability in the molecule-fixed frameis given by the diagonal tensor α = diag(α1, α2, α3),the interaction with a nonresonant laser field polar-ized along the space-fixed z-axis is [Stapelfeldt &Seideman, 2003]

VL = −ε2L

4[α1 + (α2 − α1) sin2 θ sin2 ψ

+ (α3 − α1) cos2 θ], (5)

with ε2L is proportional to the intensity of the laserfield. Omitting the angle-independent term (whichproduces a constant shift in energy) we obtain

VL = −∆ω2 sin2 θ sin2 ψ − ∆ω3 cos2 θ, (6)

with ∆ω2 = (α2 − α1)ε2L/4 and ∆ω3 = (α3−α1)ε2L/4. Assuming the dipole moment to lie alongthe molecule-fixed z-axis, the interaction with astatic electric field tilted through an angle β withrespect to the space-fixed z-axis and lying in thespace-fixed xz-plane is

VS = −d0εS(cos β cos θ + sinβ cosφ sin θ), (7)

where d0 is the magnitude of the electric dipolemoment and εS is the strength of the static field.

The Hamiltonian of the asymmetric top intilted fields can be written from Eqs. (4), (7) and(6) as

H = T + VS + VL. (8)

Note that the potential VS + VL is a function ofthe angle φ, so that m is not conserved. The asym-metric top in tilted fields is therefore a physicallysignificant rotor problem with three degrees of free-dom. For collinear fields, β = 0, the φ angle is notpresent in H and pφ = m is then a constant of themotion. In the remainder of this paper we considerthe collinear case only.

2.2. Collinear fields

The Hamiltonian for an asymmetric top moleculein collinear fields can be rewritten as

H =12(I−1j) · j + V (θ, ψ), (9)

with

V (θ, ψ) = −ω cos θ − ∆ω2 sin2 θ sin2 ψ

−∆ω3 cos2 θ. (10)

In terms of the Euler angles and their time deriva-tives the body-fixed components of the angularmomentum are

j1 = I1(θ sinψ − φ sin θ cosψ), (11a)j2 = I2(θ cosψ + φ sin θ sinψ), (11b)j3 = I3(φ cos θ + ψ). (11c)

The projection of j onto the space-fixed unit vectorez = (0, 0, 1) is given in terms of the direction cosinematrix C as

m = C3 · j, (12)

where

C3 = (C31, C32, C33)= (−sin θ cosψ, sin θ sinψ, cos θ), (13)

is the vector of body-fixed components of ez.

2.3. Determination of relativeequilibria

As the angle φ is ignorable, we consider relativeequilibria defined by the conditions θ = 0, and ψ =0 [Katok, 1972]. These relative equilibria in generaldefine periodic orbits in the full rotor phase space.

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Using these conditions in Eqs. (11), and rewritingin terms of C3 we obtain

j = φ(IC3). (14)

The equation for m, (12), can be rewritten as

m = φ(IC3) · C3, (15)

which can be used to express φ in terms of m andC3. Substituting the resulting expression for j,

j =m(IC3)

(IC3) · C3

, (16)

into the Hamiltonian (9) gives the effective oramended potential [Arnold et al., 1988; Bolsinov &Fomenko, 2004; Katok, 1972; Tatarinov, 1974] foran asymmetric top molecule in collinear fields

Vm(θ, ψ) =m2

2(IC3) · C3

+ V (θ, ψ). (17)

Relative equilibria with θ = 0, ψ = 0 arefound from the effective potential (17) solving theequations

∂Vm∂θ

= − m2(IC3)[(IC3) · C3]2

· ∂C3

∂θ+∂V

∂θ= 0, (18a)

∂Vm∂ψ

= − m2(IC3)[(IC3) · C3]2

· ∂C3

∂ψ+∂V

∂ψ= 0, (18b)

with

∂C3

∂θ= (−cosψ cos θ, sinψ cos θ,−sin θ), (19a)

∂C3

∂ψ= (sinψ sin θ, cosψ sin θ, 0) (19b)

and∂V

∂θ= ω sin θ − 2∆ω2 sin θ cos θ sin2 ψ

+ 2∆ω3 sin θ cos θ, (20a)

∂V

∂ψ= −2∆ω2 sinψ cosψ sin2 θ. (20b)

In general the solutions of Eq. (18) depend onm. For a given m the zero set of ∂Vm/∂θ (respec-tively, ∂Vm/∂ψ) in the θ-ψ space gives the solu-tion set for Eq. (18a) (respectively, (18b)). Thesolution set typically consists of (possibly disjoint)curves in the θ-ψ plane. In practice (see below), weare able to give a natural parametrization of eachof these solution curves, so that the general solu-tion of Eqs. (18) is obtained by evaluating ∂Vm/∂ψ(respectively, ∂Vm/∂θ) along each of the particular

solution curves. Zeroes of the relevant functions arefound by interpolation along the curve, yielding val-ues of θ and ψ that solve (18) for given m. Finally,the effective potential (17) is evaluated at each ofthese solution points, which in general have differentenergies for a given m. The calculation is repeatedfor different values of m to obtain the complete E-mdiagram.

To clarify the general procedure just outlined,consider the Euler problem (free asymmetric top),with ω = ∆ω2 = ∆ω3 = 0. Equations (18) in thiscase are

∂Vm∂θ

= −sin θ cos θ(I1 cos2 ψ + I2 sin2 ψ − I3)m2

[(IC3) · C3]2

= 0, (21a)

∂Vm∂ψ

= −sinψ cosψ sin2 θ (I2 − I1)m2

[(IC3) · C3]2= 0. (21b)

The common denominator is never zero, and is suf-ficient to find the zeros in the numerators. The solu-tions θ = 0, π satisfy both equations simultaneouslygiving the solution set A = 0, π × [0, 2π). In thecartesian product the first set gives the possible val-ues of the θ coordinate, the second set the possiblevalues of the ψ coordinate.

Setting ψ = 0, π/2, π, or 3π/2 solves Eq. (21b);substituting these values into (21a) gives

sin θ cos θ(I3 − I1)m2 = 0, ψ = 0, π, (22a)

sin θ cos θ(I3 − I2)m2 = 0, ψ =π

2,3π2. (22b)

For these equations θ = 0, π/2, π, are solutions. Inthe same notation used before the new solutions areB = π/2 × 0, π, and C = π/2 × π/2, 3π/2.

The complete solution set for the Euler problemis SE = A∪B ∪C. For each of these sets is possibleto obtain a Vm-m curve simply by evaluating theeffective potential on each of the solution sets

Vm(A) =m2

2I3, (23a)

Vm(B) =m2

2I1, (23b)

Vm(C) =m2

2I2. (23c)

The E-m diagram consists of three parabolasand the regions enclosed between them. This dia-gram is shown in Fig. 1 for the three molecules con-sidered here (cf. Sec. 3.4 of Arnold [1988]; Ch. 14 of

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0

2

4

6

8

10

E

0

2

4

6

8

10

E

(a) (b)

0

2

4

6

8

10

4 3 2 1 0 1 2 3 4

E

m

(c)

Fig. 1. E-m diagrams for free asymmetric top molecules (Euler tops). (a) Iodobenzene; (b) Pyridazine; (c) Iodopenta-fluorobenzene.

Bolsinov and Fomenko [2004]). The different linesrepresent the three parabolas (23) and correspondphysically to rotations of the top about the bodyfixed axes (in a positive or negative sense). Theregion below the red curve is physically inaccessible.In the axis convention used to obtain the asymmet-ric top Hamiltonian the red curve, Eq. (23b), cor-responds to stable rotation about the body fixedx-axis; the green parabola, Eq. (23c), to an unsta-ble rotation about the y-axis, and the blue curve toa stable rotation about the z-axis.

For a given (E,m) point in Fig. 1, the effec-tive potential (17) gives an equation to solve in θor ψ. The solutions are found as contours of con-stant Vm = E in the θ-ψ configuration space, thePoisson sphere S2. These contours define the classi-cally accessible (Vm ≤ E) and forbidden (Vm > E)regions for given E and m. The topology of the dif-ferent solutions on the (θ, ψ) sphere is characterized

by the (multi-valued) genus defined as in Arnold[1988]: the connected region B has genus if B isdiffeomorphic to the sphere S2 with disjoint discsremoved. If the classically accessible region is dis-connected, then it is assigned a multivalued genus1, 2, . . . , where each k is the genus of one of thecomponents.

Figure 2 shows the classically accessible regions(in black) of S2 for iodopentafluorobenzene. Panel2(a) represents an (E,m) point located between thered and green parabolas in Fig. 1(c); the multival-ued genus is 1,1 since there are two disjoint classi-cally accessible regions and from the point of viewof each there is one white disc removed from thesphere. In the same way, panel 2(b) is the acces-sible region for a (E, m) point located betweenthe green and blue parabolas of 1(c), but now thegenus is 2 since two white discs are removed. Finallythe region above the blue parabola in 1(c) has 0

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2

1.5

1

0.5

0

10.750.50.250

ψ/π

θ/π10.750.50.250

θ/π10.750.50.250

θ/π

(a) (b) (c)

Fig. 2. Regions of classically allowed motion in θ-ψ configuration space for iodopentafluorobenzene for the different regionsof Fig. 1(c).

genus since all the sphere is classically accessible(panel 2(c)).

3. E-m Diagram for Asymmetric TopMolecules in Static Electric Fields

Energy-momentum diagrams for the asymmetric topin an external gravitational field havebeen studied byKatok [1972] andby Gashenenko and Richter [2004].An important conclusion in these works is that, inthe study of relative equilibria and their bifurca-tions in asymmetric tops, the relevant parametersare the ratios between two of the moments of inertiaand the third one, e.g. I1/I3 and I2/I3.

In this section we treat as examples threeasymmetric top molecules of recent theoreticaland experimental interest [Kozin & Roberts, 2003;Stapelfeldt & Seideman, 2003]: the near-prolatetop iodobenzene (C6H5I) [Poulsen et al., 2004;Bulthuis et al., 1997], the near-oblate top pyridazine(C4H4N2) [Kozin & Roberts, 2003; Li et al., 1998],and the intermediate case iodopentafluorobenzene[Poulsen et al., 2004]. Relevant physical parametersfor these molecules are given in Table 1.

In Fig. 3 we show the definition of the bodyfixed frame for these molecules. The moment of iner-tia Ii is related to the rotational constant Bi byBi = (2Ii)−1. The rotational constants Bi, the fieldparameter ω, and the energy are all scaled by B3.For iodobenzene an electric field of εS = 25kVcm−1

[Bulthuis et al., 1997] gives ω/B3 = 4.52; for pyri-dazine a field of strength εS = 56kVcm−1 [Liet al., 1998] gives ω/B3 = 19.04; in iodopentaflu-orobenzene, an electric field of εS = 25kVcm−1

gives ω/B3 = 18.93 [Kozin & Roberts, 2003;Poulsen et al., 2004; Bulthuis et al., 1997].

For the asymmetric top molecule in an elec-tric field, the relative equilibria are found by solv-ing the equations (setting ∆ω2 = ∆ω3 = 0 inEqs. (18)):

∂Vm∂θ

= −sin θ cos θ(I1 cos2 ψ + I2 sin2 ψ − I3

)m2

[(IC3) · C3]2

+ω sin θ = 0, (24a)

∂Vm∂ψ

= −sinψ cosψ sin2 θ(I2 − I1)m2

[(IC3) · C3]2= 0, (24b)

Table 1. Molecular parameters.

Molecule B3a B1/B3 B2/B3 d0

b α1c α2 α3 Point Group

C6H5Id 0.189 0.117 0.132 1.70 10.2 15.3 21.5 C2v

C4H4N2e 0.208 0.490 0.970 4.14 5.84 10.29 10.35 C2v

C6F5If 0.034 0.264 0.359 1.54g 10.5 17.9 23.8 C2v

aRotational constant in cm−1

bDipole moment in DebyecPolarizabilities in A3

dIodobenzene [Poulsen et al., 2004; Bulthuis et al., 1997]ePyridazine [Innes et al., 1988; Hinchliffe & Soscun, 1994]f Iodopentafluorobenzene [Poulsen et al., 2004]gab-initio 3-21G

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z

y

x

z

y

x

z

y

x

Fig. 3. Definition of body-fixed coordinate frames foriodobenzene (upper panel), pyridazine (middle panel), andiodopentafluorobenzene (lower panel). Hydrogen and fluorineatoms are represented by white circles, carbon atoms by blackcircles, and nitrogen and iodine atoms by grey circles, respec-tively. In each case the x-axis is perpendicular to the plane ofthe molecule, and the dipole moment points along the body-fixed z-axis.

where

(IC3) · C3 = I1 sin2 θ cos2 ψ + I2 sin2 θ sin2 ψ

+ I3 cos2 θ. (25)

Again we have the solution set A= 0, π× [0, 2π).There are two solution subsets A1 ⊂A, and A2 ⊂A,

given by A1 = 0× [0, 2π), and A2 = π× [0, 2π).The Vm-m curves for these subsets are

Vm(A1) =m2

2I3− ω, (26a)

Vm(A2) =m2

2I3+ ω. (26b)

The solutions ψ = 0, π/2, π, 3π/2 to (24b) mustbe substituted into (24a). After rearranging anddividing by I3, this gives

(1 − i1)m2 cos θ = −ωI3[i1 + (1 − i1) cos2 θ]2,(27a)

(1 − i2)m2 cos θ = −ωI3[i2 + (1 − i2) cos2 θ]2,(27b)

for ψ = 0, π and ψ = π/2, 3π/2 respectively, andi1 ≡ I1/I3, i2 ≡ I2/I3. For the case I1 > I2 > I3,i.e. i1 > i2 > 1, the right-hand sides (RHS) of theseequations are functions with maxima at θ = 0, πand a minimum at θ = π/2; the left-hand sides(LHS) are functions with a minimum at θ = 0, amaximum at θ = π, and a fixed zero at θ = π/2.In Fig. 4, the LHS and RHS of these equations areplotted in order to show the nature of the solutions.As m is increased the amplitude of variation of thesolid curve about zero gets larger until it intersectsthe dashed line at θ = 0, at which point a solu-tion of the equation is obtained. Since the curvesdo not change their shape, there can exist only onesolution for each of the Eqs. (27) for a given m.This unique intersection gives a value of θ, whichtogether with the respective ψ gives the solution ofEqs. (27). The form of the curves indicates that theintersection occurs initially at θ = 0 and then movesto larger values of θ as |m| increases, i.e. the solu-tion A1 bifurcates twice, first for the ψ = 0, π curveand then for the ψ = π/2, 3π/2 curve. Since thesolid line is always zero at θ = π/2, the intersectionof the two curves cannot go beyond this point, thismeans that for large values of m the asymmetrictop in an electric field behaves like an Euler top.

The bifurcations in the E-m and E-θ diagramsare shown in Fig. 5. For iodobenzene we set ω/B3 =10, for pyridazine ω/B3 = 20, and for iodopentaflu-orobenzene ω/B3 = 20.

In the E-m diagrams there are four regionsdelimited by different color curves. The allowed con-figuration space corresponding to the regions delim-ited by the red and green curves have genus 1,1;those between the green and the blue curves havegenus 2; between the blue and magenta curves the

May 15, 2008 13:22 02087


0

10.750.50.250

θ/π

−ω I3

−ω I3 i12

(1−i1) m2

0

10.750.50.250

θ/π

−ω I3

−ω I3 i22

(1−i2) m2

Fig. 4. Plots of RHS (dashed), and LHS (solid) of Eqs. (27) for m values at which curves do not intersect. Left panel:Eq. (27a); right panel: Eq. (27b).

4

3

2

1

0

−1

20100

10−1

E

10.750.50.250

(a) (b)

6

4

2

0

40200

10−2

E

10.750.50.250

(c) (d)

1086420

20100

10−1

E

m

10.750.50.250

θ/π(e) (f)

Fig. 5. E-m and E-θ bifurcation diagrams for dipolar asymmetric top molecules in a static electric field. (a), (b) iodobenzene(ω/B3 = 10); (c), (d) pyridazine (ω/B3 = 20); (e), (f) iodopentafluorobenzene (ω/B3 = 20).

May 15, 2008 13:22 02087


genus is 1, and above the magenta curve the genus0. It can be seen that the only difference betweenthese diagrams and those for the Euler top is thepresence of the region with genus 1.

The curves themselves represent rotations in φat constant θ and ψ. Associated θ values for givenenergy are shown using the curve colors in panels5(b), 5(d) and 5(f). The red curve is a stable rota-tion with ψ = 0, π and θ; the green curve is unstablewith ψ = π/2, 3π/2 and θ in the RHS panels. Theblue and magenta curves are associated with degen-erate equilibria in the ψ coordinate and θ = 0 (sta-ble, blue) or θ = π (unstable, magenta). Physicallythe blue and magenta curves in Fig. 5 representthe situation when the molecule’s dipole is orientedwith the field (blue) and against it (magenta), whileat the same time the molecule is rotating aboutits own z-axis with direction and angular velocitygiven by m.

4. E-m Diagrams for AsymmetricTop Molecules in NonresonantLaser Fields

The effective potential for interaction of a moleculewith a nonresonant linearly polarized laser field(ω = 0) is

Vm(θ, ψ) =m2

2(IC3) · C3

− ∆ω2 sin2 θ sin2 ψ

−∆ω3 cos2 θ. (28)

The relative equilibria are obtained by solving

∂Vm∂θ


[(IC3) · C3]2

+∂V

∂θ= 0, (29a)

∂Vm∂ψ


[(IC3) · C3]2+∂V

∂ψ

= 0, (29b)

where (IC3) · C3 is again given by Eq. (25) and

∂V

∂θ= −2∆ω2 sin θ cos θ sin2 ψ


∂V


From (29)–(30) it is seen that θ = 0, π are againsimultaneous solutions and the first solution set is

A = 0, π × [0, 2π). In contrast with the electricfield case, the symmetry of the laser interactiongives only one Vm-m curve for A,

Vm(A) =m2

2I3− ∆ω3. (31)

It is straightforward to see that θ = π/2 is a solutionof (29a), which after substitution into (29b) gives

−sinψ cosψ(I2 − I1)m2

I1 cos2 ψ + I2 sin2 ψ− 2∆ω2 sinψ cosψ = 0,

(32)

with immediate solutions ψ = 0, π/2, π, 3π/2. Thesecond solution set is therefore B = π/2 ×0, π/2, π, 3π/2. There are two subsets, B1 ⊂ Band B2 ⊂ B, given by B1 = π/2 × 0, π andB2 = π/2 × π/2, 3π/2. For these the Vm-mcurves are

Vm(B1) =m2

2I1, (33a)

Vm(B2) =m2

2I2− ∆ω2. (33b)

Now, ψ = 0, π/2, π, and 3π/2 are particularsolutions of Eq. (29b); substituting these values intoEq. (29) and dividing by I3 produces

(1 − i1)m2 = −2∆ω3I3[i1 + (1 − i1) cos2 θ]2, (34a)

(1 − i2)m2 = 2(∆ω2 − ∆ω3)I3[i2 + (1 − i2) cos2 θ]2,(34b)

for ψ = 0, π, and ψ = π/2, 3π/2 respectively, and asbefore i1 ≡ I1/I3 and i2 ≡ I2/I3, with i1 > i2 > 1.The nature of the solutions depends on the valuesof ∆ω2 and ∆ω3. For iodobenzene and pyridazine∆ω3 > ∆ω2 > 0, since for these molecules α1 <α2 < α3 (cf. Table 1). This is however not alwaysthe case: for pyridine with the same axis conventionα1 < α3 < α2 [Hinchliffe & Soscun, 1994], whichimplies ∆ω2 > ∆ω3 > 0.

For a physically reasonable laser intensity1012 Wcm−2 applied to iodobenzene we have∆ω2/B3 = 284.6 and ∆ω3/B3 = 630.5; the samefield applied to pyridazine produces ∆ω2/B3 =225.7 and ∆ω3/B3 = 228.9; iodobenzene gives∆ω2/B3 = 2280.06 and ∆ω3/B3 = 4097.94. Thesedimensionless ratios are much larger than the corre-sponding energy ratios for physically relevant valuesof the interaction with a static electric field, ω = 10and ω = 20. As we wish to investigate the interest-ing dynamical regime in which the effects of bothfields are of similar magnitude (see next section),

May 15, 2008 13:22 02087


Table 2. Hamiltonian parameters and asymmetries with respect to y-axis.

Molecule ω/B3 ∆ω2/B3 ∆ω3/B3 2(∆ω2 − ∆ω3)/B3 (α3 − α2)/(α1 − α2) (I3 − I2)/(I1 − I2)

C6H5I 10 28.46 63.05 −69.88 −1.22 −6.61C4H4N2 20 22.56 22.88 −0.64 −0.014 −0.045C6F5I 15 228.0 409.8 −363.58 −0.797 −1.785

we reduce the intensity of the laser by a factor of10 to obtain the values listed in Table 2.

The plot of the RHS and LHS of Eqs. (34)is shown in Fig. 6. The situation is different fromthe static electric field case. Now the LHS of theequations, the solid horizontal line, moves downas m increases. This line intersects the RHS curve(dashed) at θ = 0, π simultaneously for m values

M1 = ±[−2∆ω3I3

1 − i1

]1/2

, (35a)

M2 = ±[2(∆ω2 − ∆ω3)I3

1 − i2

]1/2

, (35b)

for ψ = 0, π and ψ = π/2, 3π/2, respectively. Asthe value of m increases the points of intersectionapproach the value θ = π/2 symmetrically, i.e.π/2 − θleft = θright − π/2. Finally, at m values

M1 = ±M1i1, (36a)M2 = ±M2i2, (36b)

for ψ = 0, π and ψ = π/2, 3π/2, respectively, theintersection occurs exactly at θ = π/2. For largervalues of m there is no solution.

Considering only positive values of m, Eqs. (34)can be solved for cos2 θ to get

cos2 θ =m−M 1

M1 −M 1

, (37a)

cos2 θ =m−M 2

M2 −M 2

, (37b)

for ψ = 0, π and ψ = π/2, 3π/2, respectively. Thetwo solutions of each of these equations togetherwith the respective ψ values give the solution setsC for ψ = 0, π and D for ψ = π/2, 3π/2. The Vm-mcurves for these solutions are

Vm(C) =(M1 −M 1)m2

2[I3(m−M1) + I1(M1 −m)]

−∆ω3m−M1

M1 −M1

, (38a)

Vm(D) =(M2 −M 2)m2

2[I3(m−M2) + I2(M2 −m)]

−∆ω2 + (∆ω2 − ∆ω3)m−M 2

M2 −M 2

, (38b)

0

10.750.50.250

θ/π

−2∆ω3 I3

−2∆ω3 I3 i12

(1−i1) m2

0

10.750.50.250

θ/π

2(∆ω2−∆ω3) I3

2(∆ω2−∆ω3) I3 i22

(1−i2) m2

Fig. 6. Plots of RHS (dashed), and LHS (solid) of Eqs. (34) for m values at which curves do not intersect. Left panel: Eq. (34a)(i1 = 1.25, (1 − i1)m

2 = −5, 2∆ω3I3 = 10); right panel: Eq. (34b) (i2 = 1.2, (1 − i2)m2 = −4, 2(∆ω2 − ∆ω3)I3 = 7).

May 15, 2008 13:22 02087


with m ∈ [M1,M1] and m ∈ [M2,M2] for thefirst and second equations respectively. The valueof Vm(C) at m = M1 is equal to that of Vm(A) atm = M1, while its value at m = M1 is equal toVm(B1) at m = M1; similarly, the value of Vm(D)at m = M2 equals Vm(A) at m = M2, and atm = M2 equals Vm(B2) at m = M2. These resultsindicate that in the E-m diagram the solution setC is a bridge connecting solutions A and B1. In likefashion the solution set D connects the solutions Aand B2.

Finally, the last solution set is obtained fromEqs. (29) after removing all the common factors

− (I1 cos2 ψ + I2 sin2 ψ − I3)m2

(I1 sin2 θ cos2 ψ + I2 sin2 θ sin2 ψ + I3 cos2 θ)2

− 2∆ω2 sin2 ψ + 2∆ω3 = 0, (39a)

− (I2 − I1)m2

(I1 sin2 θ cos2 ψ + I2 sin2 θ sin2 ψ + I3 cos2 θ)2

− 2∆ω2 = 0. (39b)

Solutions of these equations for given m areobtained by finding the intersections of the zero con-tours of the LHS of these equations in the θ-ψ space.For the case of the laser field, these equations canbe rearranged to get m2 in terms of θ and ψ

m2 =2(∆ω2 sinψ2 − ∆ω3)

I3 − I1 cos2 ψ − I2 sin2 ψ[(IC3) · C3]2, (40a)

m2 =2∆ω2

I1 − I2[(IC3) · C3]2, (40b)

where the definition (25) is used. With the equa-tions written in this form is clear that we must findψ such that

2(∆ω2 sin2 ψ − ∆ω3)I3 − I1 cos2 ψ − I2 sin2 ψ

=2∆ω2

I1 − I2. (41)

Rearranging and simplifying gives the condition

1 − ∆ω3

∆ω2=I3 − I2I1 − I2

, (42)

which can be written in terms of the polarizability

α3 − α2

α1 − α2=I3 − I2I1 − I2

. (43)

In general, this equality is not satisfied for phys-ical parameter values. For example, Table 2 lists

10

8

6

4

2

0

−2

−4

6040200

10−2

E

m

0

2

1,1

2

2,2 1,1,1,1

41,1

Fig. 7. E-m diagram for the asymmetric top molecule iodo-pentafluorobenzene in a nonresonant laser field: ∆ω2/B3 =228.006 and ∆ω3/B3 = 409.794.

the LHS and RHS of Eq. (43) for the molecules ofinterest.

As condition (42) is not fulfilled for themolecules considered here, the last solution set isempty, E = ∅. The complete solution for the asym-metric top rotor in a linearly polarized laser field isthen given by SL = A∪B∪C∪D. The Vm-m curvesfor these solutions in the case of iodopentafluo-robenzene, iodobenzene and pyridazine are shownin Figs. 7–9, respectively. In these figures the solu-tion sets are given by different types of curves. Thesolution set A is given by the red curve, B1 by thegreen curve and B2 by the blue. The bridge solu-tion sets C, connecting A with B1 are magenta; thebridge D, connecting A with B2, is the cyan curve.

4

3

2

1

0

−1

6040200

10−2

E

m

0 2

1,1

2

2,21,1,1,1

4

1,1

1,1

Fig. 8. E-m diagram for the asymmetric top moleculeiodobenzene in a nonresonant laser field: ∆ω2/B3 = 28.456and ∆ω3/B3 = 63.05.

May 15, 2008 13:22 02087


5

4

3

2

1

0

−1

−2

107.552.50

10−1

E

m

0

1,1

2

2,2

4

2

1,1

1,1,1,1

1,1

Fig. 9. E-m diagram for the asymmetric top molecule pyridazine in a nonresonant laser field: ∆ω2/B3 = 22.56 and ∆ω3/B3 =22.88.

For iodopentafluorobenzene and iodobenzenethere are nine different regions delimited by dif-ferent curve types with their corresponding generaindicated with arrows in Fig. 8. The classicallyaccessible θ-ψ configuration space for each of theseregions is shown in Fig. 10. The simplest regionis characterized by genus 0; in this region themolecule is free to move in any possible configu-ration in θ-ψ as can be seen in Fig. 10(i). Thereare two regions with genus 2: one for high energyand large m, the other for low energy and smallm. In the high energy region 10(a) shows thatthe molecule is localized in the equatorial regionand the poles are forbidden. The second regionwith genus 2 is shown in 10(c); in this case theforbidden regions are on the equator with ψ =0, π. For genus 4 there is only one region, shownin panel 10(f); the molecule can access neither

the poles nor the equatorial regions with ψ =0, π. In the case of genus 2,2, panel 10(d), theaccessible region is delocalized in ψ and restrictedin θ to a region close, but not on, the poles.Finally, for genus 1,1,1,1 (10(f)) the motion ishighly localized near θ = π/4, 3π/4 and ψ = π/2,3π/2.

The fact that both the inertia and polarizabil-ity tensors of pyridazine have near-oblate symme-try makes it harder to observe the different regionsin the E-m diagram, as can be seen from Fig. 9.As I2 ≈ I3 and ∆ω2 ≈ ∆ω3, the Vm-m curves forthe solution sets A and B2 are very similar. Themagenta bridging solution C connecting A and B1

is easily seen, but the cyan D bridge is much harderto see. As for iodopentafluorobenzene and iodoben-zene, there are nine different regions which have thesame distribution of genus.

May 15, 2008 13:22 02087


2

1.5

1

0.5

0

ψ/π

(a) genus= 2 (b) genus= 1, 1 (c) genus=2

2

1.5

1

0.5

0

ψ/π

(d) genus=2, 2 (e) genus= 1, 1, 1, 1 (f) genus= 4

2

1.5

1

0.5

0

10.750.50.250

ψ/π

θ/π10.750.50.250

θ/π10.750.50.250

θ/π

(g) genus= 1, 1 (h) genus= 1, 1 (i) genus= 0

Fig. 10. Classically allowed (red) and forbidden (green) θ-ψ configuration space and associated genera for the various regionsof Figs. 7 and 8.

5. E-m Diagrams for AsymmetricTop Molecules in Collinear Fields

We now consider the rotational dynamics of dipolarasymmetric tops in combined static (Sec. 3) andnonresonant laser (Sec. 4) fields. The polarizationof the laser field is taken to be collinear with thestatic field, so that m is a conserved quantity.

A related problem has been analyzed by Tatari-nov [Arnold et al., 1988; Tatarinov, 1974], who stud-ied the problem of the rotation of a rigid body abouta fixed point with a potential

V = P C3 · RCM +ρ

2(IC3) · C3, (44)

where P and ρ are constants, and RCM is the loca-tion of the center of mass relative to the fixed point.In Tatarinov’s work E-m diagrams and genera areobtained for the effective potential.

In molecular terms, the first term of the poten-tial (44) corresponds to the interaction of an elec-tric field along the space-fixed z-axis with thedipole moment of the molecule, where the dipolemoment vector can point in an arbitrary directionin the molecule-fixed frame, R, not only along themolecule fixed z-axis as in the case studied here.The second term of (44) has no obvious molecu-lar analog, but the dependence on Euler angles isexactly the same as for the laser interaction. In fact,the potential energy for an asymmetric top in tilted(noncollinear) fields can be written as

V = −Ω · C3 − (∆ΩC3) · C3, (45)

with the definitions Ω ≡ (ω sin β, 0, ω cos β), and∆Ω ≡ diag(0,∆ω2,∆ω3).

The effective potential for the dipolar asym-metric top in collinear fields, where the dipolemoment points along the molecule-fixed z-axis, is

May 15, 2008 13:22 02087


(cf. Eq. (17))

Vm(θ, ψ) =m2

2(IC3) · C3

− ω cos θ − ∆ω2 sin2 θ sin2 ψ

−∆ω3 cos2 θ. (46)

The relative equilibria are obtained by solving theequations

∂Vm∂θ


[(IC3) · C3]2

+∂V

∂θ= 0, (47a)

∂Vm∂ψ


[(IC3) · C3]2+∂V

∂ψ= 0,

(47b)

with

(IC3) · C3 = I1 sin2 θ cos2 ψ + I2 sin2 θ sin2 ψ

+ I3 cos2 θ, (48)

and∂V

∂θ= ω sin θ − 2∆ω2 sin θ cos θ sin2 ψ


∂V


In both Eqs. (47) the common factor sin θ givesthe simultaneous solution θ = 0 or π, so the firstsolution set is A = 0, π× [0, 2π). As for the static

electric field case, this solution generates two sub-sets A1 = 0× [0, 2π) and A2 = π× [0, 2π). TheVm-m curves for these solutions are

Vm(A1) =m2

2I3− ω − ∆ω3, (50a)

Vm(A2) =m2

2I3+ ω − ∆ω3. (50b)

As for the laser interaction case, Eq. (47b) has solu-tions ψ = 0, π/2, π and 3π/2, which after substitu-tion into (47a) give

cos θ(1 − i1)m2 = −I3(ω + 2∆ω3 cos θ)× [i1 + (1 − i1) cos2 θ]2, (51a)

cos θ(1 − i2)m2 = −I3[ω − 2(∆ω2 − ∆ω3) cos θ]× [i2 + (1 − i2) cos2 θ]2. (51b)

Figures 11–13 show plots of the LHS and RHSof Eqs. (51) for iodopentafluorobenzene (ω/B3 =20, ∆ω2/B3 = 228.006 and ∆ω3/B3 = 409.794),iodobenzene (ω/B3 = 10, ∆ω2/B3 = 28.456 and∆ω3/B3 = 63.05) and for pyridazine (ω/B3 = 20,∆ω2/B3 = 22.56, and ∆ω3/B3 = 22.88).

Figures 11 and 12 show the LHS of (51) foriodopentafluorobenzene and iodobenzene (respec-tively) with m = 0 as a dotted line in both cases;it is clear that on each figure that for m = 0 thereis only one intersection at θ > π/2. As the valueof m increases, the dashed curve will intersect thesolid line at additional points. The RHS of (51a)gives −I3(ω + 2∆ω3) at θ = 0 and −I3(ω − 2∆ω3)

0

10.750.50.250

LH

S, R

HS

θ/π10.750.50.250

θ/π(a) (b)

Fig. 11. RHS (solid line) and LHS for m = 20 (dashed line) of Eqs. (51) for iodopentafluorobenzene. (a) Eq. (51a);(b) Eq. (51b). Dotted line: m = 0.

May 15, 2008 13:22 02087


0

10.750.50.250

LH

S, R

HS

θ/π10.750.50.250

θ/π(a) (b)

Fig. 12. RHS (solid line) and LHS for m = 10 (dashed line) of Eqs. (51) for iodobenzene. (a) Eq. (51a); (b) Eq. (51b). Dottedline: m = 0.

0

10.750.50.250

LH

S, R

HS

θ/π10.750.50.250

θ/π

(a) (b)

Fig. 13. RHS (solid line) and LHS for m = 4 (dashed line) of Eqs. (51) for pyridazine. (a) Eq. (51a); (b) Eq. (51b). Dottedline: m = 0.

at θ = π; for these two molecules in the fields spec-ified above |ω+ 2∆ω3| > |ω− 2∆ω3|, which impliesthat the LHS curve intersects the RHS curve first atθ = π, and then at θ = 0. For larger m only the left-most intersection remains with the two intersectionswith θ > π/2 disappearing simultaneously. This sit-uation is repeated for the second Eq. (51), for whichthe RHS at θ = 0, π gives −I3[ω − 2(∆ω2 − ∆ω3)]and −I3[ω + 2(∆ω2 − ∆ω3)] respectively; for thefields employed we have for both molecules that|ω − 2(∆ω2 − ∆ω3)| > |ω + 2(δω2 − ∆ω3)|.

For pyridazine the situation is different. FromFig. 13(a) it is clear that the RHS curve of (51a)is intersected first at θ = π, but the two solutionswith θ > π/2 will now disappear before the dashedcurve intersects the solid curve at θ = 0. For (51b)there is only one solution, which moves from θ = 0to larger values of θ as the value of m increases.

Figures 11 to 13 explain the observed bifurca-tion structure of the solutions of Eqs. (51). The twopanels in Figs. 11 and 12 and panel 13(a) show theexistence of a nonbifurcating solution and a pair of

May 15, 2008 13:22 02087


solutions emerging from a saddle-node bifurcation.A global view of the bifurcations is obtained rear-ranging Eqs. (51) to

cos θ(1 − i1)m2 + I3(ω + 2∆ω3 cos θ)× [i1 + (1 − i1) cos2 θ]2 = 0, (52a)

cos θ(1 − i2)m2 + I3[ω − 2(∆ω2 − ∆ω3) cos θ]× [i2 + (1 − i2) cos2 θ]2 = 0, (52b)

and then plotting the zero contours of the LHS inthe θ-m space. These contours are shown in Fig. 14for the molecules treated here. The contours indi-cate that a solution branch for both Eqs. (52) beginsat θ = 0, branching out of the A1 solution. On the

other hand, the right branch from the saddle-nodepair begins at θ = π, which means that this solu-tion emerges from the A2 solution. The left branchof the saddle-node pair is connected only to its rightpartner.

The last solution set is obtained after removingall common factors in Eqs. (47), and rearranging toobtain m2 as function of θ and ψ,

m2 =2(∆ω2 sin2 ψ − ∆ω3) cos θ − ω

cos θ(I3 − I1 cos2 ψ − I2 sin2 ψ

) [(IC3) · C3]2,

(53a)

m2 =2∆ω2

I1 − I2[(IC3) · C3]2, (53b)

40

30

20

10

0

m

(a) (b)

40

30

20

10

0

m

(c) (d)

60

40

20

0

10.750.50.250

m

θ/π10.750.50.250

θ/π

(e) (f)

Fig. 14. Zero contours of the LHS of Eqs. (52). Iodobenzene: (a) Eq. (52a), (b) Eq. (52b). Pyridazine: (c) Eq. (52a);(d) Eq. (52b). Iodopentafluorobenzene: (e) Eq. (52a), (f) Eq. (52b).

May 15, 2008 13:22 02087


10

8

6

4

2

0

−2

−4

6050403020100

10−2

E

m

0

21

1,1

1,1,1,1

2

1,11

1,14

3

−250

−200

−150

−100

−50

22181410

10−2

E

1,2

1,1,1

1,1,2

1,1

2

Fig. 15. E-m diagram for the dipolar asymmetric top molecule iodopentafluorobenzene in collinear static electric and non-resonant laser fields. ω/B3 = 20, ∆ω2/B3 = 228.006 and ∆ω3/B3 = 409.7945.

with (IC3) · C3 given by Eq. (48). The solution isobtained by finding all the values of θ and ψ suchthat

2(∆ω2 sin2 ψ − ∆ω3) cos θ − ω

cos θ(I3 − I1 cos2 ψ − I2 sin2 ψ

) =2∆ω2

I1 − I2, (54)

which can be rearranged and simplified to get

2 cos θ(I1 − I3I1 − I2

− ∆ω3

∆ω2

)=

ω

∆ω2. (55)

In terms of the polarizability this gives

cos θ =ω

2∆ω2

(I1 − I3I1 − I2

− α3 − α1

α2 − α1

)−1

. (56)

In contrast to Eq. (43), Eq. (56) depends on thefields. For the fields used in iodobenzene we obtain

θ = 0.4896π, while for pyridazine there is nosolution.

The range of m for this solution is obtainedfrom Eq. (53b). The smallest value of m occurs atψ = π/2, 3π/2, the largest at ψ = 0, π, giving thevalues

Mi =(

2∆ω2

I1 − I2

)1/2

×[Ii +

ω2(I3 − Ii)4∆ω2

2

(I1 − I3I1 − I2

− ∆ω3

∆ω2

)−2],

(57)

where i = 1 for ψ = 0, π, and i = 2 for ψ =π/2, 3π/2. A solution of (53) therefore exists form ∈ [M2,M1]. Noting Eqs. (51), and keeping in

May 15, 2008 13:22 02087


2

1.5

1

0.5

0

ψ/π

(a) genus= 2 (b) genus=1 (c) genus=1, 1

2

1.5

1

0.5

0

ψ/π

(d) genus= 1, 1 (e) genus= 4 (f) genus= 2

2

1.5

1

0.5

0

ψ/π

(g) genus=1, 1, 1, 1 (h) genus= 1, 1 (i) genus= 1

2

1.5

1

0.5

0

ψ/π

(j) genus= 3 (k) genus= 1, 1, 1 (l) genus= 1, 2

2

1.5

1

0.5

0

10.750.50.250

ψ/π

θ/π10.750.50.250

θ/π10.750.50.250

θ/π

(m) genus= 1, 1, 2 (n) genus=1, 1, 1, 1 (o) genus=2

Fig. 16. Classically allowed (red) and forbidden (green) θ-ψ configuration space and associated genera for the regions inFig. 15. Genus 0 not included.

May 15, 2008 13:22 02087


3

2

1

0

−1

6050403020100

10−2

E

m

0 2

1 1,1

1,142

1,1,1,1

1,1

1

31,1,11,2

1,1,2

Fig. 17. E-m diagram for the dipolar asymmetric top molecule iodobenzene in collinear static electric and nonresonant laserfields. ω/B3 = 10, ∆ω2/B3 = 28.456 and ∆ω3/B3 = 63.05.

mind Eq. (54), we conclude that this solution coin-cides with the solution of (53a) at m = M2, andwith the solution of (53b) at m = M1.

Using Eqs. (53b) and (57) we obtain

sin2 θ sin2 ψ =M1 −m

I1 − I2

(I1 − I22∆ω2

)1/2

. (58)

This equation, together with (55) and (53b) can besubstituted into the effective potential (17) to getthe Vm-m curve

Vm =(m−M1

2

)(2∆ω2

I1 − I2

)1/2

− ω cos θ

−∆ω3 cos2 θ, (59)

where the last two terms are to be evaluated usingEq. (56). The first term shows a linear dependenceon m. In fact, as seen above, this solution is a bridgebetween the solutions of (51) at ψ = π/2, 3π/2 and

ψ = 0, π. Although it has not been considered, fornegative values of m the solutions are symmetricwith respect to reflection on the m = 0 line.

The E-m diagram for iodopentafluorobenzeneis shown in Fig. 15. There are 16 different regionsdelimited by the Vm-m curves. In the figure thereare two parabolas corresponding to the solution setsA1 (red) and A2 (green), the A1 parabola is alwaysbelow A2 as should be evident from Eqs. (50).Emerging from the A1 parabola at m ≈ 10 thereare two curves, one blue the other magenta. Thesecurves correspond to the independent branch solu-tions of Eq. (52), the blue curve is the solutionfor ψ = 0, π, the magenta one for ψ = π/2, 3π/2.These two curves intersect near m ≈ 50, and nearthis intersection they are connected by the bridgesolution (59) (cyan). Emerging from the A2 curve,near m ≈ 10, there are also two curves with thesame colors as above; these are the two saddle-node

May 15, 2008 13:22 02087


3

2

1

0

−1

2520151050

10−2

E

0

1,1

1

2

23

4

3

2

1

0

−1

−2

−3

302520151050

10−2

E

m

0

1,12

2

2,2

4

3

1

(a) (b)

Fig. 18. E-m diagram for the dipolar asymmetric top molecule pyridazine in collinear static electric and nonresonant laserfields. (a) ω/B3 = 20, ∆ω2/B3 = 22.56, and ∆ω2/B3 = 22.88; (b) same electric field but a laser field ten times more intense.

bifurcation solutions of Eq. (52) with the blue curvefor the ψ = 0, π solution and the magenta curve forthe ψ = π/2, 3π/2 solution. The saddle-node bifur-cation are observed as cusps near m ≈ 40 for theblue curve and m ≈ 25 for the cyan curve. Thecomplete E-m diagram is obtained when negativevalues of m are considered, the diagram is sym-metric about the m = 0 axis. For the full dia-gram the bifurcation partners form “smiles” thatare typical features of this type of system [Tatari-nov, 1974] (for analogous structures in the case ofdiatomic molecules in collinear fields, see [Arangoet al., 2004]).

The accessible θ-ψ configuration space for thedifferent regions of Fig. 15 is shown in Fig. 16. Com-paring with the figure for the laser interaction only(Fig. 10), we see that the genus 2,2, panel 10(d),is not present when the electric field is turned on;instead, there is a genus 1,2 region, panel 16(l). Thisindicates that the effect of the electric field is rela-tively strong compared to the laser and that it tendsto align the molecule dipole along the space fixedz-axis. This is also the case for the genus 3 regionof Fig. 16(j), which is absent in Fig. 10. Note alsothat the genus 1,1,1,1 configurations are different inboth figures.

For iodobenzene the E-m diagram, Fig. 17,is simpler than the one for iodopentafluoroben-zene, Fig. 15. Since the rotational constants forthis molecule are considerably larger than those foriodopentafluorobenzene, the E-m diagram extendsover a smaller energy range. There are only fewqualitative changes with respect to Fig. 15. The

most important difference is that the lower smilelies completely within the 2 region and does not goover to the 1,1 region as for iodopentafluorobenzene.This eliminates the lower 2 region observed in thelower panel of Fig. 15 and also the 1,1 region in thesame panel.

The E-m diagrams for pyridazine are shown inFig. 18. In panel (a), where ω/B3 = 20, ∆ω2/B3 =22.56, and ∆ω3/B3 = 22.88, we obtain a very simpleE-m diagram with only six regions. There is onlyone “smile” with genre 2 and 3. The second smiledoes not form since there is only one branch for theψ = π/2, 3π/2 solution. In panel (b) we show theE-m diagram for the same value of electric field butfor a laser field ten times more intense. In this case,the smile has grown bigger, overlapping the regionswith genre 1, 2 and 1,1. It is observed that belowand above the overlapped regions the genus is thesame as for the weaker laser field case, while insidethe smile the genus can vary according to the E-mvalues.

6. Summary and Conclusions

In this paper we have studied the classical mechan-ics of rotational motion of dipolar asymmetric topmolecules in strong external fields. Static electricfields, linearly polarized nonresonant laser fields,and collinear combinations of the two were inves-tigated. The particular asymmetric top moleculesiodobenzene, pyridazine and iodopentafluoroben-zene have been treated for physically relevant fieldstrengths.

May 15, 2008 13:22 02087


Following Katok [1972] and Tatarinov [1974],we have computed diagrams of relative equlibria inthe E-m plane; the relative equilibria correspondphysically to periodic motions with the two Eulerangles θ and ψ constant [Katok, 1972; Arnoldet al., 1988]. We have also examined the classicallyallowed θ-ψ configuration space for different regionsof the E-m diagrams, and have classified the con-figuration space topology according to their genus[Arnold et al., 1988]. We anticipate that this clas-sical mechanical investigation will be useful in thedifficult problem of assigning quantum mechanicaleigenstates and energy levels for asymmetric tops inexternal fields [Moore et al., 1999; Kanya & Oshima,2004].

Acknowledgments

We are grateful to Mr. Michael Zukovsky for provid-ing partial translations of references [Katok, 1972]and [Tatarinov, 1974].

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