marko härtelt and bretislav friedrich- directional states of symmetric-top molecules produced by...

8/3/2019 Marko Hrtelt and Bretislav Friedrich- Directional states of symmetric-top molecules produced by combined static a

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Directional states of symmetric-top molecules produced by combinedstatic and radiative electric fields

Marko Hrtelt and Bretislav Friedricha

Fritz-Haber-Institut der Max-Planck-Gesellschaft, D-14195 Berlin, Germany

Received 28 March 2008; accepted 25 April 2008; published online 12 June 2008

We show that combined electrostatic and radiative fields can greatly amplify the directionalproperties, such as axis orientation and alignment, of symmetric top molecules. In ourcomputational study, we consider all four symmetry combinations of the prolate and oblate inertiaand polarizability tensors, as well as the collinear and perpendicular or tilted geometries of the twofields. In, respectively, the collinear or perpendicular fields, the oblate or prolate polarizabilityinteraction due to the radiative field forces the permanent dipole into alignment with the static field.Two mechanisms are found to be responsible for the amplification of the molecules orientation,which ensues once the static field is turned on: a permanent-dipole coupling of the opposite-paritytunneling doublets created by the oblate polarizability interaction in collinear static and radiativefields and b hybridization of the opposite parity states via the polarizability interaction and theircoupling by the permanent dipole interaction to the collinear or perpendicular static field. Inperpendicular fields, the oblate polarizability interaction, along with the loss of cylindricalsymmetry, is found to preclude the wrong-way orientation, causing all states to become high-field

seeking with respect to the static field. The adiabatic labels of the states in the tilted fields dependon the adiabatic path taken through the parameter space comprised of the permanent andinduced-dipole interaction parameters and the tilt angle between the two field vectors. 2008 American Institute of Physics. DOI: 10.1063/1.2929850

I. INTRODUCTION

Directional states of molecules are at the core of allmethods to manipulate molecular trajectories. This is be-cause only in directional states are the body-fixed multipolemoments available in the laboratory frame where they canbe acted upon by space-fixed fields.

In the case of polar molecules, the body-fixed permanentdipole moment is put to such a full use in the laboratory bycreating oriented states characterized by as complete a pro-

jection of the body-fixed dipole moment on the space-fixedaxis as the uncertainty principle allows.

The classic technique of hexapole focusing selects pre-cessing states of symmetric top molecules or equivalentwhich are inherently oriented by the first-order Stark effect,1

independent of the strength of the electric field applied atlow fields. For the lowest precessing state, only half of thebody-fixed dipole moment can be projected on the space-fixed axis and utilized to manipulate the molecule. Pendularorientation is more versatile,2,3 applicable to linear mol-ecules as well as to symmetric and asymmetric tops, butworks well only for molecules with a large value of the ratioof the body-fixed electric dipole moment to the rotationalconstant. However, for favorable values of this ratio, on theorder of 100 D /cm1, the ground pendular state of a 1 mol-ecule utilizes up to 90% percent of the body-fixed dipolemoment in the space-fixed frame at feasible electric fieldstrength, of the order of 100 kV /cm.

Pendular states of a different kind can be produced bythe induced-dipole interaction of a nonresonant laser fieldwith the anisotropic molecular polarizability.47 The resultingdirectional states exhibit alignment rather than orientationand can be used to extend molecular spectroscopy,8 suppressrotational tumbling,911 focus molecules,12 or to decelerateand trap molecules.13

The reliance on special properties of particular mol-ecules has been done away with by the development of tech-niques that combine a static electric field with a nonresonantradiative field. The combined fields give rise to an amplifi-cation effect which occurs for any polar molecule, as only ananisotropic polarizability, along with a permanent dipole mo-ment, is required. This is always available in polar mol-ecules. Thus, for a number of molecules in their rotationalground state, a very weak static electric field can convertsecond-order alignment by a laser into a strong first-orderorientation that projects about 90% of the body-fixed dipolemoment on the static field direction. If the polar molecule is

also paramagnetic, combined static electric and magneticfields yield similar amplification effects. So far, thecombined-field effects have been worked out for linear polarmolecules and corroborated in a number of experiments byBuck and co-workers1416 and Poterya et al.17 on linear rare-gas clusters, as well as by Minemoto and co-workers whoundertook a detailed femtosecond-photoionization study onoriented OCS.18,19

Here, we analyze the amplification effects that combinedelectrostatic and nonresonant radiative fields bring out insymmetric-top molecules. In our analysis, we draw on ouraElectronic mail: [email protected].

THE JOURNAL OF CHEMICAL PHYSICS 128, 224313 2008

0021-9606/2008/12822 /224313/19/$23.00 2008 American Institute of Physics128, 224313-1

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previous work on linear molecules20,21 as well as on the workof Kim and Felker,22 who have treated symmetric-top mol-ecules in pure nonresonant radiative fields.

For linear molecules, the amplification of the orientationarises from the coupling by the electrostatic field of themembers of quasidegenerate opposite-parity tunneling dou-blets created by the laser field. Thereby, the permanent elec-tric dipole interaction creates oriented states of indefinite

parity.16

We find that this mechanism is also in place for aclass of precessing symmetric top states in the combinedfields. However, another class of precessing symmetric topstates is prone to undergo a sharp orientation via a differentmechanism: The laser field alone couples states with oppo-site parity, thereby creating indefinite parity states that canthen be coupled by a weak electrostatic field to the fieldsdirection.

In Sec. II, we set the stage by writing down the Hamil-tonian in reduced, dimensionless form for a symmetric-topmolecule in the combined electrostatic and linearly polarizednonresonant radiative fields and consider all four symmetrycombinations of the polarizability and inertia tensors as well

as the collinear and perpendicular tilted geometries of thetwo fields. We derive the elements of the Hamiltonian matrixin the symmetric-top basis set and discuss their symmetryproperties. We limit ourselves to the regime when the radia-tive field stays on long enough for the system to developadiabatically. This makes it possible to introduce adiabaticlabeling of the states in the combined fields, sort them outsystematically, and define their directional characteristics,such as orientation and alignment cosines.

In Sec. III, we introduce an effective potential thatmakes it possible to regard the molecular dynamics in thecombined fields in terms of a one-dimensional 1D motion.The effective potential is an invaluable tool for making sense

of some of the computational results obtained by solving theeigenproblem in question numerically.

In Sec. IV, we present the results proper. First, we con-struct correlation diagrams between the field free states of asymmetric top and the harmonic librator, which is obtainedat high fields. Then, we turn to the combined fields and con-sider collinear fields and perpendicular fields in turn. It iswhere we discuss the details of the two major mechanismsresponsible for the amplification of the orientation by thecombined fields.

Section IV discusses the possibilities of applying thecombined fields to a swatch of molecules representing thesymmetry combinations of the polarizability and inertia ten-sors and of making use of the orientation achieved in aselection of applications. The main conclusions of thepresent work are summarized in Sec. VI.

II. SYMMETRIC-TOP MOLECULES AND THEIRINTERACTIONS WITH STATIC AND RADIATIVEELECTRIC FIELDS

We consider a symmetric-top molecule which is bothpolar and polarizable. The inertia tensor I of a symmetrictop molecule possesses a threefold or higher axis of rotationsymmetry the figure axis and is said to be prolate or oblate,depending on whether the principal moment of inertia about

the figure axis is smaller or larger than the remaining twoprincipal moments which are equal to one another. Theprincipal axes a, b, c ofI are defined such that the principalmoments of inertia increase in the order IaIbIc.

The symmetry of the inertia tensor is reflected in thesymmetry of the polarizability tensor in that the principalaxes of the two tensors are collinear. However, a prolate oroblate inertia tensor does not necessarily imply a prolate oroblate polarizability tensor. Four combinations can be distin-guished: i I prolate and prolate, i.e., I

aI

b=I

cand

ab =c with a the figure axis; ii I prolate and oblate,i.e., IaIb =Ic and ab =c, with a the figure axis; iii Ioblate and prolate, i.e., Ia =IbIc and a =bc with cthe figure axis; and iv I oblate and oblate, i.e., Ia =IbIc and a =bc, with c the figure axis. The body-fixedpermanent electric dipole moment of a symmetric-topmolecule is bound to lie along the molecules figure axis.The symmetry combinations of I and are summarized inTable I in terms of the rotational constants

A 2

2Ia, B

2

2Ib, C

2

2Ic1

and polarizability components parallel, , and perpendicu-lar, , to the figure axis.

The rotational Hamiltonian Hr of a symmetric top mol-ecule is given by

Hr = BJ2 + BJz

2, 2

with J the angular momentum operator, Jz its projection onthe figure axis z z a for prolate and z c for oblate top,and

A/B 1 0 for I prolateC/B 1 0 for I oblate.

3The symmetric-top molecule is subject to a combinationof a static electric field, S, with a nonresonant laser field, L.

The fields S and L can be tilted with respect to one anotherby an angle, . We limit our consideration to a pulsed planewave radiation of frequency and time profile gt such that

L2t =

8

cIgtcos22t, 4

where I is the peak laser intensity in cgs units. We assumethe oscillation frequency to be far removed from any mo-lecular resonance and much higher than either 1 with thepulse duration or the rotational periods. The resulting effec-tive Hamiltonian Ht is thus averaged over the rapid oscil-

TABLE I. Symmetry combinations of the inertia and polarizability tensorsfor a polarizable symmetric top molecule. See text.

case iI prolate, prolate

case iiI prolate, oblate

case iiiI oblate, prolate

case ivI oblate, oblate

IaIb =Ic IaIb =Ic Ia =IbIc Ia =IbIcAB =C AB =C A=BC A=BCab =c ab =c a =bc a =bc

224313-2 M. Hrtelt and B. Friedrich J. Chem. Phys. 128, 224313 2008



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lations. This cancels the interaction between and L seealso Refs. 23 and 24 and reduces the time dependence ofLto that of the time profile,

L2t =

4

cIgt . 5

Thus the Hamiltonian becomes

Ht = Hr+ V + Vt, 6

where the permanent, V, and induced, V, dipole potentialsare given by

V = Bcos S, 7

Vt = Btcos2 L Bt, 8

with the dimensionless interaction parameters defined as fol-lows:

S

B, 9a

,t = ,gt, 9b

, 2,I

Bc, 9c

, 9d

t = t t gt. 9e

The time-dependent Schrdinger equation correspondingto Hamiltonian 6 can be cast in a dimensionless form,

i

B

t

t=

Ht

Bt , 10

which clocks the time in units of /B, thus defining a shortand a long time for any molecule and pulse duration. Inwhat follows, we limit our consideration to the adiabatic re-gime, which arises for /B. This is tantamount to gt1, in which case the Hamiltonian 6 can be written as

Ht

B=

Hr

B cos S cos

2 L + . 11

Hence, our task is limited to finding the eigenproperties ofHamiltonian 11. For ==0, the eigenproperties be-come those of a field-free rotor; the eigenfunctions then co-incide with the symmetric-top wavefunctions JKM and theeigenvalues become EJKM/B, with K and M the projectionsof the rotational angular momentum J on the figure andspace-fixed axis, respectively. In the high-field limit, and/or , the range of the polar angle is confinednear the quadratic potential minimum, and Eq. 11 reducesto that for a two-dimensional angular harmonic oscillatorharmonic librator, see Sec. IV A.

If the tilt angle between the field directions is nonzero,the relation

cos S = cos cos + sin sin cos 12

is employed in Hamiltonian 6, with L and L, seeFig. 1.

If the static and radiative fields are collinear i.e., =0or , M and K remain good quantum numbers. Note thatexcept when K=0, all states are doubly degenerate. While

the permanent dipole interaction V mixes states with J=1 which have opposite parities, the induced dipole in-teraction mixes states with J=02 which have sameparities but, when MK0, also states with J=1 whichhave opposite parities. Thus, either field has the ability tocreate oriented states of mixed parity.

If the static and radiative fields are not collinear i.e.,0 or , the system no longer possesses cylindrical sym-metry. The cos operator mixes states which differ by M=1 and so M ceases to be a good quantum number.

A schematic of the field configurations and moleculardipole moments, permanent and induced, is shown in Fig. 2.

The elements of the Hamiltonian matrix in thesymmetric-top basis set JKM possess the following sym-metries:

JKMHJKM = 1MMJ K MHJ K M,

13

J KMHJ KM = 1MMJK MHJK M.

14

Since the Hamiltonian has the same diagonal elements forthe two symmetry representations belonging to +Kand K, itfollows that they are connected by a unitary transformation

FIG. 1. Illustration of the angles used in Eq. 12 for arbitrary fielddirections.

224313-3 Directional states of symmetric-top molecules J. Chem. Phys. 128, 224313 2008



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U. On the other hand, complex conjugation K of a symmet-

ric top state yields

KJKM = 1MKJ K M, 15

and so we see from Eqs. 13 and 14 that the +K and Krepresentations are related by the combined operation UK,which amounts to time reversal. The two representationshave the same eigenenergies and are said to be separabledegenerate.25 We note that the +K and Krepresentations areconnected by time reversal both in the absence and presence

of an electric field see also Ref. 26; the time reversal of agiven matrix element is effected by a multiplication by1MM. The symmetry properties, Eqs. 13 and 14, aretaken advantage of when setting up the Hamiltonian matrix.In what follows, we concentrate on the case of collinear i.e.,=0 or and perpendicular fields i.e., =/2.

We label the states in the field by the good quantum

number Kand the nominal symbols Jand M which designatethe values of the quantum numbers J and M of the free-rotorstate that adiabatically correlates with the state at 0

and/or 0, JJ and MM. We condense our notationby taking K to be nonnegative, but keep in mind that each

state with K0 is doubly degenerate, on account of the +Kand K symmetry representations. For collinear fields, M

=M.In the tilted fields, the adiabatic label of a state depends

on the order in which the parameters , , and are turnedon, see Sec. IV C. As a result, we distinguish among

J, K,M ;,,, J, K,M ;,,, J, K,M ;,,,

and J, K,M ;,, states, or J, K,M ;p for short, withp any one of the parameter sets ,, or ,, or,, or ,,.

The J, K,M ; p states are recognized as coherent lin-ear superpositions of the field-free symmetric-top states,

J,K,M;p = J,M

aJMJKMJ,K,M, 16

with aJMJKM the expansion coefficients. For a given state, these

depend solely on p,

aJMJKM = aJM

JKMp . 17

The orientation and the alignment are given by the

direction-direction two-vector correlation27 between the di-pole moment permanent or induced and the field vector Sor L. A direction-direction correlation is characterized by asingle angle, here by the polar angle S for the orientation ofthe permanent dipole with respect to S or L for the align-ment of the induced dipole with respect to L. The distribu-tion in either S or L can be described in terms of a series inLegendre polynomials and characterized by Legendre mo-ments. The first odd Legendre moment of the distribution inS is related to the expectation value of cos S, the orienta-tion cosine,

cos S

JM JMa

JM

JKM*aJM

JKM2J+ 11/22J + 11/2

1MK J 1 J K 0 K

cos J 1 J M 0 M

+ sin 12 J 1 J

M 1 M

J 1 J M 1 M

= J

,K,M

;pcos SJ,K,M;p , 18

and the first even Legendre moment of the distribution in Lto the expectation value of cos2 L, the alignment cosine,

cos2 L JM

J

aJM

JKM*aJMJKM

JJ13 +2

32J+ 11/22J + 11/2

1MK J 2 J M 0 M

J 2 J K 0 K

= J

,K,M

;pcos2 LJ,K,M;p . 19

The states with same K and same MK have both the sameenergy,

EJ, + K,M;p = EJ, K, M;p , 20

and the same directional properties, as follows from theHellmann-Feynman theorem,

cos S = E/B

21

and

FIG. 2. Color online Schematic of the configurations of the fields anddipoles. An electrostatic, S, and a linearly polarized radiative, L, field areconsidered to be either collinear or perpendicular to one another. While thepermanent dipole of a symmetric-top molecule is always along the figureaxis a or c for a prolate or oblate tensor of inertia, the induced-dipolemoment L is directed predominantly along the figure axis for anoblate anisotropy of the polarizability tensor, , and perpendicular toit for a prolate polarizability, . See Table I and text.




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cos2 L = E/B +

. 22

The angular amplitudes of the permanent and induced di-poles are given, respectively, by

S,0 = arccoscos S 23

and

L,0 = arccoscos2 S1/2. 24

The elements of the Hamiltonian matrix, evaluated in thesymmetric-top basis set, are listed in the Appendix. The di-mension of the Hamiltonian matrix determines the accuracyof the eigenproperties computed by diagonalization. For thecollinear case, the dimension of the matrix is given as Jmax+1maxK , M , with J ranging between maxK , M and Jmax. For the states and field strengths considered here, a1212 matrix yields an improvement of less than 0.1% ofE/B over a 1111 matrix. However, the convergence de-pends strongly on the state considered. For the perpendicularcase, the dimension of the matrix is Jmax +1

2 K2, with J

ranging between K and Jmax and M between J and J. Con-vergence within 0.1% can be achieved for the states consid-ered with Jmax=10.

The J, K,M ; p states are not only labeled but alsoidentified in our computations by way of their adiabatic cor-relation with the field-free states. The states obtained by thediagonalization procedure cannot be identified by sorting ofthe eigenvalues, especially for perpendicular fields when thestates undergo numerous crossings, genuine as well asavoided. Therefore, we developed an identification algorithmbased on a gradual perturbation of the field-free symmetric-top states. Instead of relying on the eigenvalues, we compare

the wavefunctions, which are deemed to belong to the samestate when their coefficients evolve continuously through acrossing as a function of the parameters p. If 0 is thestate to be tracked, one has to calculate its overlap k0with all the eigenvectors k that pertain to the Hamiltonianmatrix with the incremented parameters. The state with thelargest overlap is then taken as the continuation of0. Thismethod has been used for the general problem of tilted fields.

III. EFFECTIVE POTENTIAL

In order to obtain the most probable spatial distribution

geometry, wavefunctions, such as

J,K,M;p =J,K,M;p, , 25

obtained by solving the Schrdinger equation in curvilinearcoordinates, need to be properly spatially weighted by a non-unit Jacobian factor, here sin . Alternatively, a wavefunc-tion,

2 = 2 sin , 26

with a unit Jacobian can be constructed which gives themost probable geometry directly; such a wavefunction is aneigenfunction of a Hamiltonian

H

B=

d2

d2+ U, 27

where U is an effective potential.20,21 Equation 27 showsthat corresponds to the solution of a 1D Schrdinger equa-tion for the curvilinear coordinate and for the effectivepotential U. Since can only take significant values withinthe range demarcated by U, one can glean the geometry fromthe effective potential and the eigenenergy. In this way, onecan gain insight into the qualitative features of the eigen-problem without the need to find the eigenfunctions explic-itly. Conversely, one can use the concept of the effectivepotential to organize the solutions and to interpret them ingeometrical terms.

For collinear fields, the effective potential takes the form

U= M2 14sin2

1

4 K2 + K2 2MK cos

sin2 cos

cos2 + . 28

Its first term, symmetric about =/2, arises due to the cen-trifugal effects and, for M0, provides a repulsive contri-bution competing with the permanent fourth term and in-duced fifth term interactions. For K0, the second term

just uniformly shifts the potential, either down when 0prolate top or up when 0 oblate top. The third termprovides a contribution which is asymmetric with respect to=/2 for precessing states, i.e., states with MK0. It isthis term which is responsible for the first-order Stark effectin symmetric tops and for the inherent orientation their pre-cessing states possess. The fourth term, due to the permanentdipole interaction, is asymmetric with respect to =/2 forany state and accounts for all higher-order Stark effects. Thefifth, induced-dipole term, is symmetric about =/2. How-ever, it gives rise to a single well for prolate anda double well for oblate . This is of key impor-tance in determining the energy level structure and the direc-tionality of the states bound by the wells.

IV. BEHAVIOR OF THE EIGENSTATES

A. Correlation diagrams

In the strong-field limit, a symmetric-top molecule be-comes a harmonic librator whose eigenproperties can be ob-tained in closed form. The eigenenergies in the harmoniclibrator limit are listed in Tables II and III and used in con-structing the correlation diagrams between the field-free andthe harmonic librator limits, shown in Fig. 3 for the perma-nent dipole interaction, and Fig. 4 for the prolate,, and oblate, , induced-dipole interactions.

TABLE II. Eigenenergies for the permanent dipole interaction in the har-monic librator limit. See Refs. 35 and 37.

EN,K,M

B=K2 +N21/2 +KM+ 183KM

2 3N2

N= 2J K+M + 1




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The correlation diagram for the permanent dipole inter-

action, Fig. 3, reveals that states with K=0 split into J+1

doublets, each with the same value of M. The other states,on the other hand, split into J+ K at least doubly degenerate

states, each of which is characterized by a value of K+M

for a given J. For KJ, some states are more than doublydegenerate. In the harmonic librator limit, the levels are in-finitely degenerate and are separated by an energy differenceof21/2.

The correlation diagrams for the induced dipole-

interaction reveal that states with J M + K shown byblack

full

lines in Fig. 4

for oblate and all states for

prolate which have same MK form degenerate doublets. In

contradistinction, states with J M + K shown by reddouble-thick grey lines and green thick grey lines inFig. 4 for K=0 and K0, respectively bound by V oblateoccur as tunneling doublets. This behavior reflects a crucialdifference between the prolate and oblate case, namely,that the induced-dipole potential V is a double well potentialfor oblate and a single well potential for prolate.

The members of a given tunneling doublet have same

values of KM and K but Js that differ by 1. The tunnel-ing splitting between the members of a given tunneling dou-

blet decreases with increasing as expa b1/2

, witha , b0, rendering a tunneling doublet quasidegenerate at a

TABLE III. Eigenenergies for the induced dipole interaction in the har-monic librator limit. See also Ref. 22.

prolate: oblate:

EN,K,M

B=K2

+2N+11/2 +M2 +K2

142N2+ 2N+ 3

ENK,M

B=K2

+2N+11/2 +M2

2+

K2

2

N+12

2

12

N=J

M for M KN=J K for M K

for J M+ K

N= 2J K M for KM0

for J M+ K

N=J1 for KM0 when J K+M is odd

N=J for KM0 when J K+M is evenfor K or M= 0

N=J1 when J M or J K is odd

N+ =Jwhen J M or J K is even

FIG. 3. Correlation diagram, for the permanent dipoleinteraction, between the field-free 0 symmetric-top states JKM and the harmonic librator states Nobtained in the high-field limit , see also Ref.40. At intermediate fields, the states are labeled by

JKM.




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sufficiently large field strength. In the harmonic libratorlimit, the quasidegenerate members of a given tunneling dou-blet coincide with the N+ and N states, see Table III. The N+

and N states with N+ =N for J M + K always pertain to

the same Jbut different KM and so are precluded from form-ing tunneling doublets as they do not interact. In the pro-late case, the formation of any tunneling doublets is barredby the absence of a double well. Note that in both the prolate

and oblate cases, the harmonic librator levels are infinitelydegenerate and their spacing is equal to 21/2.

The correlation diagrams of Figs. 3 and 4 reveal anotherkey difference between the permanent and induced dipoleinteractions, namely, the ordering of levels pertaining to the

same J. The energies of the levels due to V increase withincreasing K+M. For V prolate, they decrease with in-creasing M for levels with M K while states with

FIG. 4. Color online Correlation diagram, for the induced-dipole interaction, between the field-free 0 symmetric-top states JKM center and theharmonic librator states NKM obtained in the high-field limit for the prolate, on the left, and oblate, on the right cases. The harmoniclibrator states are labeled by the librator quantum number N and the projection quantum numbers K and M. At intermediate fields, the states are labeled by

JKM. States that form tunneling doublets only for have J K + M and are shown in color: red double-thick grey lines for doublets with KM= 0

and green thick grey lines for doublets with K,M0. Members of the degenerate doublets have J K + M and are shown in black thick full lines. Seetext.




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M K have the same asymptote. The energy level patternbecomes even more complex for V oblate, which gives rise

to K +1 asymptotes. If K =J

, the energy decreases withincreasing M, while for KJ it only decreases for levels

with M + KJ. All other levels connect alternately to the

asymptotes with N =J or J1. This leads to a tangle ofcrossings, avoided or not, once the two interactions are com-bined, as will be exemplified below.

B. Collinear fields

Figures 5 and 6 display the dependence of the eigenen-ergies, panels ac, orientation cosines, panels df, and

alignment cosines, panels gi, of the states with 0J

3, 1MK1, and K=1 on the dimensionless param-eters and that characterize the permanent and induceddipole interactions. These states were chosen as examplessince they well represent the behavior of a symmetric top inthe combined fields. The two figures show the dependenceon for fixed values of and vice versa. Note that nega-tive values of correspond to prolate and positive val-ues to oblate. The plots were constructed for I prolate withA /B =2 but, apart from a constant shift, the curves shown areidentical with those for I oblate. Thus Figs. 5 and 6 representthe entire spectrum of possibilities as classified in Table I.

The left panels of Figs. 5 and 6 show the eigenenergies

and the orientation and alignment cosines for the cases ofpure permanent and pure induced dipole interactions,

respectively.For an angle

arccosKM

JJ+ 129

such that 0/2, the pure permanent dipole interac-tion, 0 and =0, panels a, d, g of Fig. 5, producesstates whose eigenenergies decrease with increasing fieldstrength i.e., the states are high-field seeking at all values of and their orientation cosines are positive, which signifiesthat the body-fixed dipole moment is oriented along Sright-way orientation. For states with /2, the

eigenenergies first increase with i.e., the states are low- field seeking and the body-fixed dipole is oriented oppo-sitely with respect to the direction of the static field S the socalled wrong-way orientation. For states with K=0, theangle becomes the tilt angle of the angular momentumvector with respect to the field direction, which, for J0, isgiven by

0 arccosM

JJ+ 11/2. 30

At small , states with 031/2 are low-field seeking and

exhibit the wrong-way orientation, while states with 031/2 are high-field seeking and right-way oriented.

FIG. 5. Color online Dependence, in collinear fields, of the eigenenergies, panels ac, orientation cosines, panels df, and alignment cosines, panels

gi, of the states with 0J3, 1MK1, and K=1 on the dimensionless parameter which characterizes the permanent dipole interaction with theelectrostatic field for fixed values of the parameter which characterizes the induced-dipole interaction with the radiative field; 0 for prolate

polarizability anisotropy and 0 for oblate polarizability anisotropy. The states are labeled by JKM. Note that panels a, d, and g pertain to thepermanent dipole interaction alone. See text.




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At large-enough values of, all states, including those

with K=0, become high-field seeking and exhibit right-wayorientation. In any case, the dipole has the lowest energywhen oriented along the field. Since the asymmetric effectivepotential 28 disfavors angles near 180, and increasingly so

with increasing KM, the eigenenergies for a given Jdecreasewith increasing KM. The ordering of the levels pertaining to

the same Jis then such that states with higher KM are alwayslower in energy.

The eigenenergies and orientation and alignment cosinesfor a pure induced-dipole interaction, =0 and 0, areshown in panels a, d, and g of Fig. 6. The eigenenergiesare given by

E

B=

H

B= JJ+ 1 + K2 cos2 , 31

cf. Eqs. 2 and 11. However, Figs. 6a6c and 12 onlyshows E/B +, which increase with increasing laserintensity for prolate and decrease for oblate. Note thatboth prolate and oblate eigenenergies, E/B =, decreasewith increasing laser intensity, and so all states created by apure induced-dipole interaction are high-field seeking as aresult. Note that

cos2 =

, 32

and thus the alignment cosines are given by the negativeslopes of the curves shown in Fig. 6a.

In panels d and g, one can see that only the precess-ing states are oriented, and that their orientation shows adependence on the parameter which qualitatively differsfor oblate and prolate: For 0, the orientation is

enhanced for states with J M + K and suppressed for

states with J M + K, while for 0 it tends to be sup-pressed by the radiative field for all states. This behaviorfollows readily from the form of the effective potential, Eq.28, as shown in Fig. 7. For the prolate case 0, thepotential becomes increasingly centered at /2 with in-creasing field strength and therefore tends to force the body-fixed electric dipole and thus the figure axis into a directionperpendicular to the field. On the other hand, for the oblatecase 0, the effective potential provides, respectively,a forward 0 and a backward well for the N+

and N states of a tunneling doublet for J M + K or of

a degenerate doublet for J M + K. However, only forthe degenerate-doublet states does the increasing fieldstrength result in an enhanced orientation at S =0. This dis-tinction is captured by the effective potential, Fig. 7, whoseasymmetric forward for MK0 or backward MK0

FIG. 6. Color online Dependence, in collinear fields, of the eigenenergies, panels ac, orientation cosines, panels df, and alignment cosines, panels

g-i, of the states with 0J3, 1MK1, and K= 1 on the dimensionless parameter which characterizes the induced-dipole interaction with theradiative field; 0 for prolate polarizability anisotropy and 0 for oblate polarizability anisotropy for fixed values of the parameter which

characterizes the permanent dipole interaction with the radiative field. The states are labeled by JKM. Note that panels a, d, and g pertain to theinduced-dipole interaction alone. See text.




10/19

well lures in the J M + K states. The J M + K statesbecome significantly bound by the V oblate potential at values large enough to make them feel the double well,which makes the two opposite orientations nearly equiprob-able.

Figure 8 shows the dependence of the J-state paritymixing on the parameter at =0 panel a and =10

panel b. The mixing is captured by a parameter

J=2n+1

aJMJKM2, 33

with n an integer. In the absence of even-odd J mixing,

=0 for J even and =1 for J odd; for a perfect J-parity

mixing, = 12 for either J even or odd. We see that for the

sampling of states shown, the nonprecessing states becomeparity mixed only when 0. However, all precessing statesare J-parity mixed as long as 0.

The eigenenergies as well as eigenfunctions in the har-monic librator limit for both the prolate and oblate polariz-

ability interactions have been found previously by Kim andFelker,22 and we made use of the former in constructing thecorrelation diagram in Fig. 4. We note that in the prolatecase, the eigenenergies and alignment cosines, as calculatedfrom KimFelkers eigenfunctions, agree with our numericalcalculations for the states considered within 4% already at=50. The prolate harmonic librator eigenfunctions ren-der, however, the orientation cosines as equal to zero, whichthey are generally not at any finite value of.

For the oblate case, the eigenenergies in the harmoniclibrator limit agree with those obtained numerically for thestates considered within 5% at =50. For 350, thedifference between the alignment cosines obtained numeri-

cally and from the analytic eigenfunctions is less then 3%.

We note that for J M + K, only one eigenfunction exists,pertaining either to N+ for KM0 or to N for KM0, cf.Table III. This eigenfunction is strongly directional, lendinga nearly perfect right-way orientation to an N+ state and a

nearly perfect wrong-way orientation to an N state. Theseeigenfunctions pertain to the degenerate doublets. The orien-tation cosines can be obtained from the analytic wavefunc-

tions within 0.01% for =100. For J M + K includingthe cases when either K=0 or M=0, both the N+ and N

analytic solutions exist, cf. Table III, and pertain to the tun-neling doublets. The degeneracy of the doublets in the + limit precludes using the analytic eigenfunctions incalculating the orientation cosines. In contrast to the numeri-cal results, the analytic solution predicts a strong orientation,which, in fact, is not present, as shown in Fig. 7. The linearcombination of the analytic eigenfunctions f+ and f, whichis also a solution, given as f

1,2=1 /2f+f, does not ex-

hibit any orientation, just alignment.The above behavior of symmetric-top molecules as a

function ofat =0 sets the stage for what happens oncethe static field is turned on and so 0. For 0, thecombined electrostatic and radiative fields act synergistically,making all states sharply oriented. For 0, the orienta-tion either remains nearly zero for states with KM=0 ortends to vanish for states with KM0 with increasing.

The synergistic action of the combined fields arises in

two different ways, depending on whether J M + K or

J M + K.

FIG. 8. Color online Dependence of the J-parity mixing parameter onthe parameter at = 0 panel a and = 10 panel b. Note that thebetter the J-parity mixing, the closer is the parameter to 1 /2. See text.

FIG. 7. Effective potential U for KM=1 and K=1 along with the eigenen-

ergies of states with J

=1,2 , ,5 horizontal lines and their alignment dia-monds and orientation circles amplitudes L,0 and S,0. The gray lineshows the induced-dipole potential V. See Eqs. 28, 23, and 24 andtext.




11/19

For J M + K, the orientation is due to a coupling ofthe members of the tunneling doublets e.g., the 2, 1 , 1 and3, 1 , 1 states by the permanent dipole interaction. The tun-neling doublets occur, and hence this mechanism is in place,for 0. The coupling of the tunneling-doublet membersby the permanent dipole interaction is the more effective thesmaller is the level splitting between the doublet memberswhich correlate with the N+ and N in the harmonic libratorlimit. Since the levels of a tunneling doublet can be drawn

arbitrarily close to one another by the induced dipole inter-action, the coupling of its members by even a weak electro-static field can be quite effective, resulting in a strong orien-tation of both states. The wrong-way orientation of the uppermembers of the tunneling doublets can be converted into aright-way orientation. Such a conversion takes place at suf-ficiently large where the permanent dipole interaction pre-vails over the induced dipole interaction. As noted in previ-ous work,21 the coupling of the tunneling doublets by V alsoarises for the nonprecessing states, in which case one canspeak of a pseudo-first-order Stark effect in the combinedfields. The precessing states show, in addition, the wellknown first-order Stark effect in the electrostatic field alone,

which relies on the coupling of states with the same K anddoes not involve any hybridization of J.

As noted above, the J M + K states are strongly ori-ented by the induced-dipole interaction alone. Since themembers of a degenerate doublet that correlate with the N+

and N states e.g., the 1, 1 , 1 and 1,1,1 states havedifferent values ofKM, adding an electrostatic field does notlead to their coupling, as collinear V and V can only mixstates with same KM. However, the static field skews the

effective potential U, Eq. 28, that enhances the orientationof the right-way oriented states N+ and, at sufficiently high, reverses the orientation of the initially wrong-way ori-ented states N, see below.

The molecular-axis orientation by which the synergismof the static and radiative fields for oblate manifests itselfcan be best seen in Fig. 5f and interpreted with the help ofthe effective potential, Figs. 9 and 10. Figures 6d6f pro-vide additional cuts through the same , parameterspace. Conspicuously, all states, for oblate, become right-way oriented at a sufficiently large , cf. Fig. 5f. However,some of the states either become e.g., 2, 1 , 0 or are e.g.,1,1,1 wrong-way oriented first. As increases, the

FIG. 9. Color online Effective potential Ufor K=1 and MK= 1 left panels, MK=0 center panels, and MK= 1 right panels along with the eigenenergies

and orientation amplitudes shown by dots and squares of the wavefunctions for states with J

=1. The columns are comprised of panels pertaining toincreasing values of. Red grey curves correspond to =15, green light grey curves to =0, and blue full curves to =15. See text.




12/19

2,1,1 state is even seen to become right-way oriented,then wrong-way oriented, and finally right way orientedagain. This behavior is a consequence of the different typesof coupling that the states in question are subjected to. Wewill discuss them in turn.

The 2, 1 , 0 state is the upper member of a tunnelingdoublet whose lower member is the 1, 1 , 0 state, cf. Fig. 4.At =0, the 1, 1 , 0 and 2, 1 , 0 states are not oriented, as isthe case for any states with KM=0. However, the value of=15 is large enough to push the levels into quasidegen-eracy, see Fig. 6a, in which case the static field can easily

couple them. However, at , such a coupling results inlocalizing the wavefunctions of the 1, 1 , 0 and 2, 1 , 0 pairin the forward and backward wells, respectively, of the ef-fective potential U, which, for , are mainly due to thepolarizability interaction. As becomes comparable to ,the effective potential becomes skewed. The forward wellgrows deeper at the expense of the backward well and thewrong-way oriented 2, 1 , 0 state is flushed out into the for-ward well as a result, thus acquiring the right way orienta-tion. The blue full effective potentials and wavefunctions inthe middle panels of Figs. 9 and 10 detail this behavior. We

note that the lower and upper member of a given tunnelingdoublet is always right- and wrong-way oriented, respec-tively, at . This is because the coupling by V makesthe states to repel each other, whereby the upper level ispushed upward and the lower level downward. The notedorientation of the two states then immediately follows fromthe Hellmann-Feynman theorem.

The 1,1,1 state has J M + K and thus, in the ra-diative field alone, is a member of a degenerate doublet,

along with the 1, 1 , 1 state, cf. the blue full effective po-tentials and wavefunctions in the upper left and right panelsof Fig. 9. While the 1, 1 , 1 state is always right-way ori-ented, the 1,1,1 state is wrong-way oriented even at =0, thanks to the asymmetry of the effective potential due toits angle-dependent third term, proportional to KM, cf. Eq.28. An increase in removes the degeneracy of the doubletand causes the wrong-way oriented 1,1,1 state to have ahigher energy than the right-way oriented 1, 1 , 1 state. As increases, V deepens the forward well and, as a result, thewavefunction of the 1,1,1 state rolls over into it, thusmaking the state right-way oriented.

The 2,1,1 state exhibits an even more intricate be-

FIG. 10. Color online Effective potential U for K=1 and MK= 1 left panels, MK= 0 center panels, and MK=1 right panels along with the

eigenenergies and orientation amplitudes shown by dots and squares of the wavefunctions for states with J

=2. The columns are comprised of panelspertaining to increasing values of. Red grey curves correspond to = 15, green light grey curves to = 0, and blue full curves to = 15. See text.




13/19

havior. Instead of a wrong-way orientation at low , en-hanced by the radiative field, the state becomes right-way

oriented first, due to an avoided crossing with the 1,1,1state whose behavior, sketched above, is, of course, alsoaffected by the same avoided crossing. This is followed, atincreasing , by a native wrong-way orientation that, at20, is reversed by virtue of the deepening forward wellof the effective potential, which confines the state.

We note that the tangle of level crossings seen in Figs. 5and 6 that complicate the directional properties of symmetrictops in the combined fields is caused by the reversed order-ing of the energy levels due to the permanent and induceddipole interactions: As the static and radiative fields arecranked up, the levels comb through one another.

C. Perpendicular fields

For a tilt angle 0 or between the static, S, andradiative, L, fields, the combined-field problem loses its cy-lindrical symmetry and M ceases to be a good quantum num-ber. This greatly contributes to the complexity of the energylevel structure and the directional properties of the statesproduced. At the same time, the M-dependent effective po-tential, Eq. 28, so useful for understanding the directional-ity of the states produced by the collinear fields, cannot beapplied to the case of perpendicular fields, as M is not de-fined.

Figures 11 and 12 show the dependence of the eigenen-

ergies and of the orientation and alignment cosines on thefield strength parameters and for a similar set of states

as in Figs. 5 and 12 for the collinear fields.Figures 12c and 12d capture well the main patterns ofthe behavior. For prolate, the interaction with the radiativefield L aligns the body-fixed electric dipole along theperpendicular static field S, cf. Fig. 2. For oblate, Laligns perpendicular to S. As a result, in perpendicularfields, prolate yields a strong orientation whereas oblatea vanishing one. This is the inverse of the situation in col-linear fields. However, since V prolate is a single well po-tential, the levels lack the patterns found for collinear fieldsfor V oblate. Due to the multitude of avoided crossings, thestates often switch between the right- and wrong-way orien-tation, even over tiny ranges of the interaction parameters.Therefore, a much finer control of the parameters is neededin the case of perpendicular fields in order to preordain acertain orientation. For oblate, the coupling of the differentstates is weak, and the avoided crossings that abound in theparallel case, Fig. 6, are almost absent, see Fig. 12.

The states are essentially all high-field seeking in theradiative field for 0 and low-field seeking for 0.This reflects the repulsive and attractive character of the po-larizability interaction in the prolate and oblate case, respec-tively. In the oblate case, the states shown are essentiallyhigh-field seeking in the static field. This means that for0, much of the wrong-way orientation seen, e.g., in

FIG. 11. Color online Dependence,on perpendicular fields, of the

eigenenergies, panels a and b, ori-entation cosines, panels c and d,and alignment cosines, panels e and

f, of the states with 0J3, 1MK1, and K=1 on the dimen-sionless parameter that character-izes the permanent dipole interactionwith the electrostatic field for fixedvalues of the parameter that char-acterizes the induced-dipole interac-tion with the radiative field; 0for prolate polarizability anisotropyand 0 for oblate polarizabilityanisotropy. The states are labeled by

JKM. The orientation cosinescos S are calculated with respect to

the electrostatic field and the align-ment cosines cos2L with respect tothe laser field. See text.




14/19

Figs. 5d and 5f, can be eliminated, see Fig. 11d. Unfor-tunately, the elimination of the wrong-way orientation hap-

pens at the expense of the magnitude of the orientation,which remains small.

Depending on the relative strength of the two fields, theeffects of one can dominate those of the other. This contrastswith the behavior in the collinear fields where even a tinyadmixture of the static field can dramatically change the be-havior of the states due to the radiative field such as thecoupling of the tunneling doublets.

A detailed comparison of the effect the two fields haveon a given state is complicated by the dependence of thestate label on the sequence in which the parameters , ,and are varied. This behavior is further analyzed below.

As the dependence of the orientation cosine on the parameter indicates, see Fig. 11c, the largest orientation fora prolate polarizability is attained for the 1, 1 , 0 state. Otherstates, such as the 1,1,1 state, become right-way orientedonly for sufficiently large field strengths. At large, theorientation of all states becomes substantially greater thanwhat is achievable with collinear fields, cf. Fig. 5e. For theoblate polarizability, the states behave similarly, exhibiting auniform behavior. On the other hand, in dependence on ,Fig. 12, the 1, 1 , 1 stateinstead of the 1, 1 , 0 stateexhibits the strongest right-way orientation over most of therange shown. This is an example of the sequence dependenceof the state label. The 2,1,1 state shows the most dra-

matic variations with . It has three rather sharp turn-around points, where the direction of the dipole moment

changes. Another example of the sequence dependence of thelabel is the 2, 1 , 1 state in Fig. 11c, which is wrong-wayoriented at 10, whereas none of the states shown in Fig.12d is wrong-way oriented for ranging from 20 to 0.

When , quasidegenerate states are formed, simi-lar to the tunneling doublets. However, the electrostatic fieldis not able to couple them as a result of which the states areonly aligned.

In tilted fields, the label of a given state may depend onthe sequence in which the fields are switched on and the tiltangle spanned by the field vectors is varied. Figure 13 shows

the adiabatic evolution of the states with J=1, K=1, and

M

=1,0,1 for three different sequences leading to crossingpoints in Figs. 11 and 12 at =10, =15, and =/2.The three sequences are the following: 1 The electrostaticfield is turned on to a value such that =10; then, the laserfield is switched on to a value such that =5105; then,the tilting of the fields is carried out to =/2; and finally, is raised in steps of 5105 up to =15. The corre-

sponding states are labeled as J, K,M ;,,. 2 Thelaser field is turned up to a value such that =15; then,the electrostatic field is switched on to a value such that =5105; the fields are tilted to =/2; and is raised insteps of 5105 to =10. The states are labeled as

FIG. 12. Color online Dependence,on perpendicular fields, of the

eigenenergies, panels a and b, ori-entation cosines, panels c and d,and alignment cosines, panels e and

f, of the states with 0J3, 1MK1, and K=1 on the dimen-sionless parameter that character-izes the induced-dipole interactionwith the radiative field; 0 forprolate polarizability anisotropy and0 for oblate polarizability aniso-tropy for fixed values of the param-eter that characterizes the perma-nent dipole interaction with theelectrostatic field. The states are la-

beled by JKM. The orientation co-sines cos S are calculated with re-

spect to the electrostatic field and thealignment cosines cos2L with re-spect to the laser field. See text.




15/19

J, K,M ;,,. 3 The laser field is turned up to a valuesuch that . Subsequently, the electrostatic field is turnedon to a value such that =10; finally, the fields are tilted to=/2. The corresponding states are labeled by

J

, K,M

;,,. We note that the statesJ, K,M ;,, yield identical results with those obtainedfor sequence 3.

As one can see in Fig. 13, there are no discontinuities forany of the calculations that might suggest that our state-tracking algorithm has failed. However, the three sequencesyield different labels for the same state at the crossing point.The 1, 1 , 0 ;,, and the 1, 1 , 1 ;,, stateschange their labels for sequence 2 as compared with se-quences 1 and 3. We note that the absence of genuinecrossings for sequences 1 and 2 precludes the possibilitythat the tracking algorithm jumped an avoided crossing. In

the middle panel of sequence 2, the M

=1 and M

=1 statesappear degenerate, but they are not. The very weak electricfield, necessary to give a meaning to the mutual tilting of thefields, introduces a small splitting. The wavefunctions andthus the orientation are different for these two states. Theorientation cosines for all three states of the middle panel forsequence 2 are shown in the panels inset.

To our knowledge, the above phenomenon of labelswitching has not been previously described. We find that it

occurs not only for states with the same J and different M

but also for states with different J.Two mechanisms seem likely to be responsible for label

switching: i symmetry breaking and ii chaotic behavior ofthe underlying classical system and the concomitant singu-larities of the classical phase space.

The first mechanism is at hand whenever several inter-actions are present, only some of which break the symmetryof the system completely. Whereas the symmetry-conservinginteractions give rise to a block structure of the Hamiltonianmatrix and so allow for genuine crossings of states belongingto different blocks, the symmetry breaking interaction leadsto just a single block and makes all crossings of the statesavoided. If the symmetry-conserving interactions are turnedon first, the system may have undergone a genuine crossingbefore the symmetry-breaking interaction is turned on sec-ond. However, the opposite is not true, as the symmetry-breaking interaction, when turned on first, makes all cross-ings avoided once and for all.

However, in the case of our system, there seems to be a

symmetry left, which prevents the system from possessingavoided crossings only. For sufficiently high densities ofpoints in the parameter space, one would expect to make anyavoided crossings, if present, visible. Nevertheless, even atthe highest point densities, many of the crossings have beenfound to be genuine. For a closed path in the parameter spaceat 0, the evolution of states exhibits label switching aswell. This is the case for closed paths defined in terms ofand at fixed as well as in terms ofand at fixedin which case the symmetry does not change. Closedpaths defined in terms of and at fixed do not leadto label switching, which suggests that the permanent dipoleinteraction plays a major role. From this, we infer that label

FIG. 13. Color online Comparison of the evolution ofstates for different paths through the parameter space.

From top to bottom, these are JKM ;,,,

JKM ;,, and JKM ;,,. In the middle

panel, the 110 label is interchanged with the 111 labelin the other two panels. We dub this effect labelswitching.




16/19

switching is not only due to symmetry breaking. We checkedwhether the linear rotor in the combined tilted fields alsoexhibits label switching and found that it does.

Arango et al.28 and Kozin and Roberts29 have shown thata diatomic and a symmetric top in an electrostatic field aloneexhibit classical and quantum monodromy.30 In quantum me-chanics, monodromy reveals itself as a defect in the discretelattice of states.

For tilted fields, M ceases to be a good quantum number,and its classical analog, m, is no longer a constant of motionfor a classical diatomic or a symmetric top in tilted fields. Asa result, the problem becomes nonintegrable. For tilt angles/4, Arango et al. found that the diatomic exhibits ex-tensive classical chaos,28,31 while Kennerly mentions quan-tum chaos for symmetric tops.32 In order to see monodromyfor the tilted fields problem, one would wish to construct aquantum lattice. However, it is not clear how to go about it inthe absence of good quantum numbers apart from K. Apossibility is to make use of the root-mean-square expecta-tion value of M, M21/2, and construct a quasimonodromydiagram with its aid. Such a diagram can be then comparedwith classical results, since m21/2 is well defined in classicalcalculations.28 The quasimonodromy diagrams that we soconstructed strongly depend on the ratio of the two fieldstrength parameters. The structure of the diagrams for thetwo types of polarizability anisotropy differs markedly, al-though it is the electrostatic field alone that is supposed todetermine the behavior. The quasimonodromy diagrams arefurther complicated by the dependence on the representation

of the angular variables, as M21/2 differs in the two repre-sentations. Only M is well defined and independent of therepresentation. The quasimonodromy diagrams can be obtainon request.

Hence, we found that in titled fields, switching of theadiabatic labels of states for linear molecules as well as sym-metric tops takes place, where it arises in connection with thechange of the character of a crossing, from genuine toavoided. Monodromy, however, is present for both collinearand tilted fields. We note that whether label switching couldbe experimentally observed and utilized is not clear at thispoint.

V. EXAMPLES AND APPLICATIONS

Table IV lists a swatch of molecules that fall under thevarious symmetry combinations of the and I tensors, asdefined in Table I. The table lists the rotational constants,dipole moments, polarizability anisotropies, as well as thevalues of the interaction parameters and attained, re-spectively, at a static field strength of 1 kV /cm and a laserintensity of 1012 W /cm2.2 The conversion factors are alsoincluded in the table. While the field strength of the electro-static field of a kV/cm is easy to obtain or sometimes evendifficult to avoid, a laser intensity of a petawatt per cm2 issomewhat harder to come by. However, pulsed laser radia-tion can be easily focused to attain such an intensity, and ananosecond pulse duration is generally sufficient to ensureadiabaticity of the hybridization process.

We note that the directional properties displayed in Figs.512 should be attainable for most of the molecules listed inTable IV.

The amplification of molecular orientation in the com-bined fields may find a number of new applications.

In molecule optics,12 a combination of a pulsed nonreso-nant radiative field with an inhomogeneous electrostatic fieldcan be expected to give rise to temporally controlled, state-specific deflections. In an inhomogeneous static field pro-duced by an electrode micro-array,33 and a nanosecond laserpulse of 1012 W /cm2, deflections on the order of a millira-dian appear feasible for light molecules. Since the synergisticeffect of the combined static and radiative fields is only in

place when both fields are on, the deflection of the moleculeswould be triggered by the presence of the nanosecond laserpulse. The sensitivity of the deflection process to the magni-tude of the space-fixed electric dipole moment and its direc-tion right-or wrong-way would simultaneously enable stateselection.

The directional properties of symmetric tops in the com-bined fields may come handy in the studies of rare-gas mol-ecules of the type found by Buck and Farnik. Some of thespecies identified by these authors may, in fact, possess athreefold or higher axis of rotation symmetry,17,34

Drawing on the analogy with previous work on spectral

TABLE IV. Values of parameters and for choice symmetric top molecules whose properties were takenfrom Refs. 38 and 39. The conversion factors are =503.2 D S kV/cm /B MHz and =3.1658107 I W /cm2 3 /B MHz. Numbers in parentheses are order-of-magnitude estimates.

MoleculeB

MHz

D

3 at 1 kV /cm

at 1012 W /cm2

Acetonitrile 9199 3.92 1.89 0.22 65.0Ammonia 298500 1.47 0.24 0.0025 0.24Benzene-Ar 1113 0.1 6.1 0.05 1735

Bromomethane 9568 1.82 1.95 0.10 64.5Chloromethane 13293 1.89 1.69 0.07 40.2Fluoromethane 10349 1.85 0.84 0.09 25.7Iodomethane 7501 1.64 2.15 0.11 90.7Trichlormethane 3302 1.04 2.68 0.16 257Trifluoromethane 7501 1.65 0.18 0.11 7.60




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effects in single electrostatic35,36 or radiative fields,4 we alsonote that the combined fields can be expected to dramaticallymodify the spectra of symmetric top molecules: This is dueto a change of both energy levels and the transition dipolemoments. The latter are strongly affected by the directional-ity of the wavefunctions, which give rise to their widelyvarying overlaps. However, detailed simulations of such ef-fects in the combined fields are still wanting.

Last but not least, molecules in tilted electrostatic andradiative fields can serve as prototypical systems for thestudy of quantum chaos. The absence of good quantum num-bers and the multitude of unstable equilibria suggest thispossibility. Indeed, as pointed out in the work of Ezra andco-workers,28,31 linear molecules in noncollinear fields ex-hibit chaotic dynamics. We hope that our present study ofsymmetric tops in the combined fields will provide an impe-tus for further work on quantum chaos and monodromy ex-hibited by such nonintegrable molecular systems.

VI. CONCLUSIONS

In our theoretical study of the directional properties ofsymmetric top molecules in combined electrostatic and non-resonant radiative fields, we saw that collinear perpendicu-lar fields force permanent dipoles of molecules with oblateprolate polarizability anisotropy into alignment with thestatic field. We found that the amplification mechanism thatproduces highly oriented states for linear molecules K=0 is

also in place for precessing states of symmetric tops with J

K + M. This mechanism is based on the coupling by acollinear electrostatic field of the tunneling doublet statescreated by the interaction of the oblate polarizability with alinearly polarized radiative field. The efficacy of this cou-pling is enhanced by an increased strength, 0, of theanisotropic polarizability interaction that traps the tunnelingdoublets it creates deeper in a double well potential anddraws the members of the doublets closer to one another.Apart from this synergistic effect of the combined fields,

there is another effect in place, but for states with J K+ M. Such states occur as exactly degenerate doublets in theradiative field alone, whose both members are strongly butmutually oppositely oriented along the polarization plane ofthe field. This orientation can be manipulated by adding anelectrostatic field, which easily couples the right-way ori-ented states to its direction, whether this is parallel or per-pendicular to the radiative field. At a sufficiently large

strength of the permanent dipole interaction, the wrong-way oriented member of the no longer degenerate doubletbecomes also right-way oriented. The above patterns of theenergy levels are complicated by numerous avoided cross-ings, themselves unavoidable, due to the opposite ordering ofthe energy levels for the permanent and induced dipole in-teractions.

The absence of cylindrical symmetry for perpendicularfields is found to preclude the wrong-way orientation for theoblate polarizability, causing all states to become high-fieldseeking with respect to the static field. The changes of thesystems parameters , , and cause genuine crossings tobecome avoided and vice versa. This, in turn, causes the

eigenstates to follow different adiabatic paths through theparameter space and to end up with adiabatic labels that de-pend on the paths taken.

The amplification of molecular orientation by the syner-gistic action of the combined fields may prove useful in mol-ecule optics and in spectroscopy. Monodromy and quantumchaos lurk behind the combined-field effect whose furtheranalysis may thus be expected to shed new light on both.

ACKNOWLEDGMENTS

We thank Dmitrii Sadovskii, Boris Zhilinskii, and IgorKozin for discussions related to the issue of label switching.We are grateful to Gerard Meijer for discussions and support.

APPENDIX: MATRIX ELEMENTS USED

It is expedient to represent the elements of the Hamil-tonian matrix in the basis of the symmetric-top wavefunc-

tions, JKM. These are related to the Wigner matrices DMKJvia

JKM = 1MK2J+ 182

1/2DMKJ ,, . A1The field operators can be expressed in terms of the Leg-

endre polynomials PJcos and spherical harmonicsYJM,, which, in turn, are related to the Wigner matricesby

DM0J ,, = 4

2J+ 11/2YJM* ,, A2

D00J ,, = PJcos . A3

The evaluation of the matrix elements of the Hamil-tonian then only requires the application of the triple prod-uct over Wigner matrices theorem,

DM3K3J3 RDM2K2J2 RDM1K1J1 Rd

= 82 J1 J2 J3M1 M2 M3

J1 J2 J3K1 K2 K3

,where R denotes ,,.

A For the collinear case, the matrix elements to be

determined are

JKMHBJKM = JKMJJ+ 1 + K2

JKM

JKMcos JKM

JKMcos2 JKM,

A4

where

cos = P1cos = D001 ,, , A5




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cos2 = 13 2P2cos + 1 =13 2D00

2 ,, + 1 . A6

The properties of the 3j symbols eliminate interactions between states with different M and K. As a result, states belonging tocertain M and K values can be treated in a separate calculation,

B For the case of tilted fields, the matrix elements can be expressed either in terms of the static field or the laser field.The simpler case is keeping the laser field fixed in space and tilting the electric field relative to it. Then, the cos operatoris replaced by

cos s = cos cos + sin sin cos = cos D001 + sin 1

2 D101 D101 .

Different K states do not mix. The matrix elements are

If, on the other hand, the laser field is tilted relative tothe fixed static field, the cos2 L operator becomes

cos2 L = cos2 cos2 + 2 sin cos sin cos cos

+ sin2 sin2 cos2 =1

3+

2

3cos2 D00

2

+ 216

sin cos D102 D10

2 1

3D00

2 sin2

+ 16

sin2 D202 + D20

2 .

The selection rules then become J=J,J1,J2 and M

=M,M1,M2. The equivalence of the two choices ofexpressing the fields has been numerically checked, but onlythe first, simple choice has been used in the calculations.

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