i.n. kozin, r.m. roberts and j. tennyson- relative equilibria of d2h^+ and h2d^+

8/3/2019 I.N. Kozin, R.M. Roberts and J. Tennyson- Relative equilibria of D2H^+ and H2D^+

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MOLECULAR PHYSICS, 2000, VOL. 98, NO. 5, 295307

Relative equilibria of D2H1

and H2D1

I. N. KOZIN1*, R. M. ROBERTS1 and J. TENNYSON2

1Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK2Department of Physics and Astronomy, University College London,

London WC1E 6BT, UK

(Received 29 July 1999 ; revised version accepted 14 September 1999)

Relative equilibria of molecules are classical trajectories corresponding to steady rotationsabout stationary axes during which the shape of the molecule does not change. They can beused to explain and predict features of quantum spectra at high values of the total angularmomentum Jin much the same way that absolute equilibria are used at low J. This paper givesa classication of the symmetry types of relative equilibria of A B2 molecules and computes therelative equilibria bifurcation diagrams and normal mode frequencies for D2H

and H2D .

These are then fed into a harmonic quantization procedure to produce a number of predic-tions concerning the structures of energy level clusters and their rearrangements as Jincreases.In particular the f ormation of doublet pairs is predicted for H 2D

from J 26.

1. Introduction

Relative equilibria play a central role in classicalmechanics as organizing centres for more complexdynamics. They typically take the form of steady rota-tions about stationary axes during which the `shape ofthe system does not change. In applications to molecularspectroscopy they have previously been referred to as`stationary points [1, 2] or J-dependent referencepoints [3]. Unstable relative equilibria have also been

studied in the guise of `rotational or `centrifugal bar-riers [4, 5] and `transition states [6]. Symmetry aspectsrelated to relative equilibria have been discussed in thecontext of rotational energy surfaces [7, 8] and a detailedgroup theoretical analysis of relative equilibria of Ryd-berg states of atoms and molecules has been given in [9].Dynamical symmetry breaking in H2X molecules hasalso been formulated in terms of a modication ofLonguet-Higgins concept of feasible symmetry opera-tions [10].

In a previous paper [11] we suggested that relativeequilibria could be used systematically to explain andpredict f eatures of the quantum spectra of moleculesat high values of the total angular momentum J inmuch the same way that ordinary (absolute) equilibriumpoints are used for J 0 and, combined with perturba-tion theory, at low values of J. We also presented acomplete bifurcation diagram of all the relative equili-

bria of H3 and used this to predict a number of featuresin its energy spectrum.

In this paper we give a general analysis of AB2 molecules and, in particular, of the two H3 isotopomersD2H

and H2D . Our interest in J-dependent relative

equilibria of these ions was stimulated by their possibleimportance in the context of predissociation spectra [12]

in which very high rotational excitations are possibleClassical interpretations can also be of assistance in

understanding the results of quantum variational calcu-lations.In section 2 of the paper we present the classical equa

tions of motion of a molecule (in the BornOppenheimer approximation) and use these to show tharelative equilibria are precisely the critical points oeective potential energy functions obtained by addingappropriate `centrifugal potentials to the molecularpotential energy function. We then give the linearizedequations of motion near a relative equilibrium. Theseare used later to compute normal mode frequencies andto determine stability properties. The modications thaare needed f or linear congurations of molecules arealso described. The nal part of the section summarizesthe harmonic quantization procedure introduced in [11]

For each relative equilibrium this yields estimates of theenergies and symmetries of quantum states with wavefunctions localized near the relative equilibrium and itssymmetry-equivalent counterparts.

In section 3 we give a complete classication of alpossible symmetry types of relative equilibria of AB2molecules, distinguishing them by both their `shapeand the orientation of their total angular momentum

Molecular Physics ISSN 00268976 print/ISSN 13623028 online # 2000 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals/tf/00268976.html

* Author for correspondence. Permanent address: Instituteof Applied Physics, R ussian Academy of Sciences, UlyanovSt. 46, 603600 Nizhnii Novgorod, Russia.


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vectors. The symmetry types of their normal modes andground state are also listed. This is the data that isneeded to determine the symmetries of localizedquantum states in the harmonic quantization procedure.

These methods are applied to D2H in section 4 and

to H2D in section 5. In both cases we have computed

all the relative equilibria and the frequencies of theirnormal modes. The energies of the relative equilibria

as functions of the total angular momentum J are pre-sented in gures 1 and 4 up to J 70. The normal modefrequencies for some of the linearly stable relative equi-libria are presented in the other gures in sections 4 and5. These results are then used to draw some conclusionsabout the existence of energy level clusters with particu-lar multiplicities and symmetry types for specic rangesof total angular momentum values.

2. Classical equations of rovibrational motion

This section describes the classical equations ofmotion for molecules and derives from these the equa-tions for its relative equilibria and the linearizations ofthe equations of motion at the relative equilibria. Themethods used to analyse the stability of relative equili-bria and to obtain harmonic quantization estimatesfrom them are outlined.

2.1. Nonlinear congurationsWith the standard choice of Euler angles [13], the

kinetic energy of a nonlinear molecule with N nucleitakes the form

T 12 xTIx x

Tg _q 12 _qTa _q; 1

where q is the 3N 6 dimensional vector of internaldisplacements, _q the corresponding vector of velocities,x the angular velocity, I the inertia tensor, a the reducedmass matrix and g the Coriolis interaction. This expres-sion can be rearranged to give

T 12 x A _qTIx A _q 12 _q

Tb

1_q; 2

where A I1g and b1 a gTI1g a ATIA. Thissplitting of the kinetic energy into two parts is gaugeinvariant, in other words it is independent of thechoice of internal coordinates q [14]. The rst term isthe `vertical kinetic energy and can be taken to be agauge independent denition of the rotational energyof the molecule while the second is a gauge independentdenition of the `horizontal vibrational energy. In thelanguage of [14] A is a gauge potential.

The canonical coordinates conjugate to x and q are

J @T

@x Ix A _q; 3

p @T

@_q ATJ b1 _q: 4

Here J is just the angular momentum in body coordi-nates. Substituting into equation (2) gives the Hamiltonian form of the kinetic energy

T 12

JTI1J 12

p ATJTbp ATJ; 5

where again the rst term is the rotational kinetic energyand the second the vibrational kinetic energy. The fulHamiltonian is obtained by simply adding the potentiaenergy function Vq to this. Note that the Hamiltoniancan also be expressed as the sum of the vibrationakinetic energy and the eective potential

VJ V 12

JTI1J: 6

In this context the rotational kinetic energy term i

sometimes referred to as the centrifugal potential [14].To obtain this form for the kinetic energy both the

inertia tensor I and the matrix a gTI1g a ATIAare required to be invertible. The former conditionholds precisely for nonlinear congurations of the mol-ecule. Since a is always invertible the latter conditionholds for congurations which are suciently close toone for which the internal coordinates satisfy the Eckartcondition g 0, or equivalently A 0. Such internacoordinates can be chosen for any nonlinear conguration. Hamiltonians near linear congurations are dis

cussed in the next subsection.The equations of motion for the Hamiltonian

H T V with T given by equation (5) are

_J J @H

@J J I1J Abp ATJ;

_q @H

@p bp ATJ;

_p @H

@q

1

2

@

@q p AT

JT

bp AT

J

@VJ@q : 7

Relative equilibria are just equilibrium points of theseequations and these are easily seen to be the solutions ofthe following system of equations

I1J J;

@VJ@q

0;

p T J; 8

296 I. N. Kozin et al.


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for arbitrary . The rst two equations involve only Jand q. Any solution of these determines a relative equi-librium by taking p to be given by the third equation.Thus we can conclude that a relative equilibrium alwaysrotates about a principal axis of its conguration, andthe conguration is a critical point of the eective poten-tial for the corresponding value of J. Note that for atriatomic molecule the principal axes, and hence the

rotation axes and angular momentum vectors, musteither lie in the molecular plane or be perpendicular toit.

This suggests a very simple recipe of nding molecularrelative equilibria. Take the potential function, nd theinertia tensor from geometrical properties of the mol-ecule, form the eective potentials with J parallel tothe principal axes of the inertia tensor and search fortheir stationary points.

The absolute value J of the total angular momentumJ is an integral of motion and so the number of inde-pendent equations in (7) is actually 6N 10 where Nis number of nuclei in the molecule. Therefore thenumber of degrees of freedom in the reduced ro-vibra-tional problem is 3N 5. The value J of the totalangular momentum can be regarded as a parameter.

To obtain the linearized equations at a relative equi-librium we assume that internal coordinates have beenchosen so that the Eckart condition A 0 is satised atthe relative equilibrium. The values of the variablesJ ;q;p at the relative equilibrium are denotedJe ;qe ;0 and innitesimal perturbations from thesevalues are written J Je Je , q qe s and

p r, where is a vector in the plane perpendicularto Je. Let Je denote the 2 3 matrix which satisesJe Je . Let AJ J

TA. Then with respect to thevariables ;s ;r the second derivative, or Hessian, ofthe Hamiltonian is given by:

JTe l

1Je JTe C 0

CTJe U Ba1

0 a1BT a1

0BBB@

1CCCA; 9

where l1 and

C @2VJ@J@q

; U @2VJ@q2

; B @AJ@q

are all evaluated at the relative equilibrium. The linear-ized equations of motion are:

_

_s

_r

0BB@

1CCA

l1 eI

Je C 0

0 a1BT a1

CTJe U Ba1

0BB@

1CCA

s

r

0BB@

1CCA;

10

where C is the projection of C to the plane perpendicularto Je , I is the identity matrix and e is the principamoment of inertia of the conguration about the axiparallel to Je. Equivalently these equations can bewritten as:

_ l1 eI

Je Cs ; 11

as BT B_s U Ba1BTs CTJe: 12

Note that BT B is skew-symmetric while U Ba1BT

is symmetric. A calculation shows that U Ba1BT isthe Hessian of the modied potential V JTlJ at therelative equilibrium, where l is the inverse of l ga1gT

A relative equilibrium is linearly stable if the f ournormal mode frequencies determined by equation (10are all real. Note that in general the precessional motionis coupled to the internal vibrations though the Coriolisinteraction and it does not always make sense to distin-guish the precessional mode from the vibrational modesIn some cases symmetries force C 0, in which case the

precessional mode is decoupled from the others.To each normal mode of equation (10) we can assign

an index 1, 1 or 0 as follows. A stable normal mode forwhich the energy increases as its amplitude grows ilabelled by 1, one f or which the energy decreases by1, and an unstable normal mode by 0. A relative equi-librium is a local minimum of the energy if its normamodes all have the index 1. In this case it is Liapunovstable: all trajectories with initial conditions close to therelative equilibrium remain close to it for all time. Arelative equilibrium which is not a local minimum may

still be linearly stable, though typically it will not beLiapunov stable as nearby classical trajectories mayslowly drift away from the relative equilibrium as aresult of Arnold diusion [15].

2.2. L inear congurationsIn the case of perturbations from linear molecules the

transformation to internal coordinates described abovebecomes singular because the inertia tensor I has a zeroeigenvalue at the linear reference conguration. More-over an extra coordinate is required to describe theinternal displacements of the molecule, making q into

a 3N 5 dimensional vector. To compensate for thisonly two independent Euler angles are used [16]. Wechoose the z axis for the molecular coordinate systemto be the principal axis direction parallel to the linearreference conguration. The two Euler angles are theangles which describe this direction in space. Thisrestricts x to be two dimensional, xlin !x ;!y , andwe can replace the inertia tensor I by

Ilin Ixx Ixy

Ixy Iyy

Relative equilibria of D2H and H2D

297


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and the 3 3N 6 Coriolis interaction matrix g by a2 3N 5 matrix glin. We may now proceed as in thenonlinear case. The matrix Ilin is invertible and thekinetic energy (2) becomes [16]

T 12

xlin Alin _qTIlinxlin Alin _q

12

_qTb1 _q ; 13

where Alin I1lin glin and b

1 a ATlinIlinAlin. As in thenonlinear case the internal coordinates can be assumedto satisfy the Eckart condition Alin 0 at the linearconguration and so b is well dened.

Momentum coordinates Jlin and p are dened byequations (3) and (4) with x, I and A replaced by xlin ,Ilin and Alin. The vector Jlin Jx ;Jy continues to givethe x and y components of the angular momentum inmolecular coordinates. The z component is equal to theprojection of the `vibrational angular momentum ontothe z axis, Jz ga

1pz. This is known as the Sayvetzcondition. The Hamiltonian form of the kinetic energybecomes

T 12 JTlinI

1lin Jlin

12

p ATlinJlinT

bp ATlinJlin: 14

The rst term may still be regarded as the vertical kineticenergy and the second as the horizontal kinetic energy.The full Hamiltonian Hlin is obtained by adding thepotential energy V to this and may therefore be writtenas the sum of the horizontal kinetic energy and an eec-tive potential VJlin dened as in (6) with Jlin and Ilinreplacing J and I.

The equations of motion for the internal variablesq ;p are completely analogous to those given in (7).Those for Jlin Jx ;Jy become:

_Jx

_Jy Jz 0

1

1 0 @Hlin@Jx

@Hlin@Jy

0BBB@

1CCCA

ga1pz0 1

1 0 l1lin Jlin Alinbp A

TlinJlin: 15

It is easily seen that equilibrium points of the full setof equations of motion must satisfy either Jx Jy 0or Jz 0. In the rst case they must also be criticalpoints of the original potential V, and so can only beequilibrium points of the full molecular Hamiltonian.However the second case yields non-trivial relative equi-libria, the shapes of which are given by critical points ofthe eective potential VJlin . For these relative equilibriathe axial symmetry of the eective potential implies thatthe direction of the total angular momentum J can beanywhere in the xy plane.

When the equations of motion are linearized at alinear relative equilibrium it is found that _Jx _Jy andthe 3N 5 normal mode frequencies are all given by thelinearized equations for q and p:

_s

_r a1BT a1

U Ba1 s

r 16or, equivalently

s BT B_s U Ba1BTs 0; 17

where

U @2VJlin@q2

and B @JTlinAlin

@q;

evaluated at the relative equilibrium.

2.3. Harmonic quantizationThe energies of quantum states which are localized

near linearly stable relative equilibria are given approximately by the harmonic quantization formula [11]:

En1 ...nd EreJ Xdl 1

kl!lJ nl 1

2

; 18

where EreJ is the energy of the relative equilibrium, nis the number of quanta in the lth normal mode, !lJ isits frequency, kl its stability index and d 3N 5 is thenumber of degrees of freedom. A better correspondencewith the quantum energy levels is obtained by replacingJ by JJ 11=2. This estimate ignores anharmonic

corrections and also the splitting due to tunnellingbetween symmetry equivalent relative equlibria. It wilclearly only be useful for wave functions which arelocalized near to the relative equilibrium.

Approximate wave functions localized near a relativeequilibrium can be labelled according to how they transform under the molecular symmetry group of the rela-tive equilibrium [11]. A wave function with nl quanta inthe lth normal mode will have symmetry type

Glocn1...nd

G0 Gn11 . . . G

ndd ; 19

whereG

0 is the symmetry of the ground state of therelative equilibrium, i.e. the wave function with zeronormal mode quanta, Gl is the representation of thesymmetry group on the lth normal mode, denotesthe (tensor) product of representations and Gnll is theproduct of Gl with itself nl times. From the correspondence principle in the classical limit the ground state ofthe relative equilibrium must be the product of groundstate wave functions in each normal mode and the sym-metric top wave function jJ K J Mi, where the quantization axis is chosen along the axis of steady rotationof the relative equilibrium [17]. Since ground state wave



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functions of normal modes are locally symmetric, thelocal symmetry type is determined by the actions ofequivalent rotations [18] of the molecular symmetrygroup of the relative equilibrium on jJ K J Mi andin general will vary with J.

Symmetry equivalent relative equilibria have sym-metry equivalent wave functions which interact witheach other via tunnelling. The resulting wave functions

of the quantum states of the molecule given by harmonicapproximation are linear combinations of the localizedwave functions and can be labelled by how they trans-form under the full molecular symmetry group of themolecule. The representation of this group on thespace of all quantum states obtained in this way froma single localized wave function of type Glocn1 ...nd is thecorresponding induced representation. These are mostconveniently calculated using Frobenius reciprocity[11] or, equivalently, the reverse correlation technique[18]. In general these representations are reducible andthe energies of the irreducible components are split bytunnelling, giving rise to energy level clusters with char-acteristic symmetry properties. The structures of someof these clusters for relative equilibria of AB2 moleculesare described in the next section.

3. Symmetries of relative equilibria of AB2 molecules

In this section we give the symmetry groups of allpossible types of relative equilibria of AB2 moleculesand describe the appropriate symmetry labelling ofapproximate wave functions localized near each type.Our notation for molecular symmetry groups willfollow that of [18] except where explicitly stated other-wise.

3.1. The group C2vM

In molecular coordinates the Hamiltonian of an AB2molecule is invariant under the permutation-inversiongroup C2vM = fE;12;E

;12g where (12) denotes

the permutation of the two identical nuclei, E is inver-sion and (12) is their composition. The representationsof this group can be found elsewhere [18]. Occasionally,for example when considering linear congurations, it is

more convenient to interpret this group as the iso-morphic C2hM but we will use the C2vM grouponly. We denote the subgroups of order two ofC2vM by

C2M fE;12g;

CsM fE;Eg;

CiM fE;12g:

To simplify notation we always denote the totally sym-metric representation s of these groups by A and the

other representations by B. In table 1 we show the repre-sentations of C2vM that are obtained by inductionfrom A and B for each of the order two subgroupsThe nal column shows the representation obtained byinduction from the trivial representation of the triviagroup fEg.

3.2. EquilibriaThere are four possible types of equilibrium cong-

urations of an A B2 molecule, symmetric linear (SL)asymmetric linear (AL), isosceles triangle (IT) andasymmetric triangle (AT). The SL and IT equilibriaare both invariant under the full symmetry groupC2vM while the AL and AT equilibria are only invar-iant under CsM. It follows that AL and AT equilibriaform symmetry equivalent pairs. The normal mode sym-metries for each of these equilibrium types are

SL : A1 A2 2B2 ; AL : 3A B;

IT : 2A1 B2 ; AT : 3A:

Note that the doubly degenerate bending modes of SLand AL equilibria `split into A2 B2 and A B representations of their molecular symmetry groups. The

degeneracy is therefore due to spatial symmetries andnot permutation-inversio n symmetries. The eects othese extra symmetries can be captured using `extendedmolecular symmetry groups D1 h(EM) and C1 v (EM[18], however these symmetries disappear when we con-sider relative equilibria and so we do not need to useextended symmetry groups in this paper.

3.3. Relative equilibriaThe possible congurations of relative equilibria are

the same as those for equilibria, namely SL, AL, IT and

AT. However two relative equilibria with the same con-guration may rotate about dierent molecular axesand thus have dierently oriented total angularmomentum vectors. For linear congurations theangular momentum vectors must be perpendicular tothe linear axis and so there is only one type of symmetriclinear and one type of asymmetric linear relative equi-librium. We continue to denote these by SL and ALrespectively.

We saw in section 2.1 that a relative equilibrium witha nonlinear conguration must rotate about one of itsprincipal axes. Accordingly the IT relative equilibria


299

Table 1. The representations ofC2vM obtained by inductionfrom its subgroups.

C2M CsM CiM fEg

A A1 A2 A1 B2 A1 B1 A1 A2 B1 B2B B1 B2 A2 B1 A2 B2 7


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come in three dierent symmetry types, IT, where x;y or z denotes the rotation axis. For AT cong-urations there are no symmetries to distinguish betweendierent directions in the molecular plane and so an ATrelative equilibrium can either have the rotation axis

perpendicular to the molecular plane, ATy , or lying inthe molecular plane, ATxz.

The local permutation-inversion symmetry groups ofeach of the dierent types of relative equilibria are givenin the second column of table 2. The table also lists therepresentations of these groups on the vibrationalnormal modes, the precessional normal modes of non-linear congurations and on the ground states of therelative equilibria (see section 2.3). Note that in eachcase the ground state is totally symmetric when J iseven, but has non-trivial symmetries when J is odd.

The third column in the table shows the number ofdistinct symmetry-related relative equilibria that mustaccompany a relative equilibrium of each type. In eachcase except ATy the number is equal to 4, the order ofC2vM, divided by the order of the local symmetrygroup. To understand the exceptional case note thatthe molecular Hamiltonian is also invariant under thetime-reversal operation which multiplies all momenta by1. This operation maps each SL and AL relative equi-librium to itself, but in all other cases maps a relativeequilibrium to a dierent relative equilibrium. Weinclude relative equilibria which are mapped to eachother by time-reversal symmetry in the denition of`symmetry-related relative equilibria. In the IT andATxz cases time reversal maps a relative equilibrium toanother relative equilibrium that can also be obtainedfrom a permutation-inversion symmetry, so no extrarelative equilibria are obtained. However in the A Ty

case time reversal maps a relative equilibrium to a dif-ferent relative equilibrium than those that are obtainedfrom the permutation-inversion symmetries.

Note that the three dierent types of relative equili-bria with CsM local permutation-inversion symmetry

groups, namely AL, ITy and ATy , can be distinguishedby their time-reversal symmetry properties. The AL relative equilibria are mapped to themselves, the ITy relativeequilibria to distinct permutation-inversio n symmetry-related relative equilibria and ATy relative equilibria

to otherwise unrelated relative equilibria. The quantumconsequences of these dierent time-reversal symmetryproperties will be explored elsewhere. In this paper ourlabelling of quantum levels will make use of the C2vMpermutation-inversion symmetry group only.

3.4. Cluster symmetry typesTables 1 and 2 together give all the information that is

needed to determine the possible symmetry types of clus-ters of energy levels predicted by harmonic quantizationat relative equilibria. The symmetry types of wave func-

tions localized near a single relative equilibrium aregiven by equation (19) with the appropriate entriefrom table 2 substituted for the Gj. The induced clusterrepresentations are then given by table 1. The dimensionof the cluster representation is always equal to thenumber of symmetry-related relative equilibria. Thuthe energy levels associated to the SL relative equilibriawill be non-degenerate, while the others will be eitherdoublets (AL and IT relative equilibria) or quadruplets(AT relative equilibria).

As an example we consider clusters associated to ITy

relative equilibria. The local symmetry type of anapproximate wave function with all n quanta in asingle symmetric normal mode is A if J is even and Bif J is odd. The corresponding induced cluster types areA1 B2 and A2 B1 , respectively. If all n quanta are inthe (asymmetric) precessional mode then the local sym-metry type is A if J n is even and B if J n is odd andthe cluster types are again A1 B2 and A2 B1, respect-ively.

If time-reversal symmetry is ignored the local symmetry types for ATy relative equilibria are the same asthose for ITy relative equilibria and so the induced


Table 2. Symmetry types of relative equilibria of AB2 molecules.

GroundLocal Vibrational Prec. state

symmetry normal norm.Label group No. modes mode J even J odd

SL C2vM 1 A1 A2 2B2 7 A1 B1AL CsM 2 3A B 7 A BITy C

s

M 2 3A B A B ITx C2M 2 2A B B A B ITz CiM 2 2A B B A B

ATy CsM 4 3A B A B ATxz fEg 4 7 7 7 7


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cluster symmetry types are also the same. The eect ofthe time-reversal symmetry is to `double-up the eigen-spaces. Thus for symmetric (resp. asymmetric) normalmodes the cluster representations are 2A1 2B2 and2A2 2B1 for J (resp. J n) even and odd. The AT

xz

quadruplets always have symmetry type A1 A2B1 B2 .

4. Relative equilibria of D2H1

In the next two sections we describe the relative equi-libria of the two isotopomers D2H

and H2D of H3

and make some predictions about associated spectralpatterns. The numerical calculations assume an isolatedground electronic state, use the potential function ofDinelli et al. [19] and neglect non-adiabatic corrections.Thus the only dierences between the Hamiltoniansused here for D2H

and H2D and in [11] for H3 are

the masses of the nuclei. For both D2H and H2D

therelative equilibria are found by solving for the criticalpoints of appropriate eective potentials V

Jand V

Jlinusing the MATHEMATICA computer package [20]. Plotsof the energies of the relative equilibria as functions of Jare given as gures 1 and 4. The normal mode frequen-cies for each of the relative equilibria were calculatedusing equations (10) and (16), again with MATHEMA-TICA. Plots of the normal mode frequencies for the lin-early stable relative equilibria are shown in a series ofgures in this section for D2H

and in the next sectionfor H2D

. The rest of this section describes the bifurca-tions of relative equilibria that occur for D2H

and theirassociated spectral consequences, while the next section

does the same for H2D .

4.1. Bifurcations of relative equilibriaThe relative equilibrium bifurcation diagram for

D2H up to J 70 is shown in gure 1. At low energies

for J< 50 the only relative equilibria are isosceles tri-angles rotating about each of their three principal axes.The ITy relative equilibria have minimum energy and sohave stability index (1111) and are both linearly andnonlinearly stable. The ITz relative equilibria are alsolinearly stable, but with stability index (1111) and

are likely to be unstable over long time periods as aresult of Arnold diusion. The ITx relative equilibriahave stability index (1110) and so are both linearlyand nonlinearly unstable. As J increases centrifugalforces elongate the ITz relative equilibria, reducing theangle between the two HD bonds, until at J 50 theynally disappear in a bifurcation with branches of `rota-tional barrier relative equilibria of the same symmetrytype which come in from innity. However before thishappens the ITz relative equilibria lose stability in aHamiltonian Hopf bifurcation [11, 21] at J 37. Inother words the frequencies of two normal modes

become equal at the bifurcation point and thereafterform a complex conjugate pair. This changes the stability index to

(1100

).

For both the ITx and ITy relative equilibria the centrifugal forces act to increase the angle between the twoHD bonds until, at J 50 and J 59 respectively, theybecome linear. These two bifurcations stabilize the SLrelative equilibria which start out at low values of Jwithstability index (1100), along with branches of AL relative equilibria. The SL go on to lose stability again in abifurcation to AL relative equilibria at J 90 (noshown in gure 1) before eventually colliding withmatching rotational barriers.

At J 39 there is a fold bifurcation which gives birth

to a pair of branches of ATxz relative equilibria withstability indices (1111) and (1110). The linearly stablebranch loses stability soon afterwards in a HamiltonianHopf bifurcation and is unlikely to have any observableinuence on spectra. The unstable branch collides withthe A L relative equilibria, raising their stability, andthese are then stabilized in a bifurcation with ATy aJ 61. However both the AL and ATy relative equilibria quickly disappear in collisions with matching rota-tional barriers and again no observable eects on thespectrum are likely. The energies of the ATy relative


301

Figure 1. The energies of linearly stable () and unstable( ) relative equilibria of the D2H ion as functions o

the total angular momentum J. The linearly stable congurations are labelled by IT for isosceles triangle, AT forasymmetric triangle, SL for symmetric linear and AL forasymmetric linear. The superscripts indicate the directionof the angular momentum vector. The horizontal dottedline denotes the rst dissociation energy of the ion to twoproducts.


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equilibria are so close to those of the AL relative equili-bria that they are indistiguishable in gure 1.

4.2. Normal modes and spectral clustersThe normal mode frequencies of ITz relative equili-

bria are shown in gure 2. The two key features are aHamiltonian Hopf bifurcation involving the anti-sym-metric precessional and stretching modes at J 37

and a close avoided crossing between the two symmetricvibrational modes at J 25. It follows from the discus-sion of section 26 that as the ITz relative equilibria havesymmetry group CiM the associated quantum energylevel doublets will be of types A1 B1 and A2 B2.Excited symmetric modes will always be of typeA1 B1 if J is even and type A2 B2 if J is odd. Theavoided crossing will result in many resonances betweenthese around J 25. Excited anti-symmetric modes willgive doublets of type A1 B1 for even numbers ofquanta and A2 B2 for odd numbers when J is evenand the opposite for J odd. Since the precessionalmode has stability index 1 the corresponding energylevels will decrease in energy as the numbers of quantaincrease. The eects of the Hamiltonian Hopf bifurca-tion on these patterns remain to be investigated.

The normal mode frequencies of the ITy relative equi-libria are given in gure 3 up to J 60, where therelative equilibria `become linear. All the vibrationalmodes are symmetric with respect to the local symmetrygroup CsM while the precessional mode is anti-sym-metric. There is therefore no interaction between theprecessional and vibrational modes. The most notable

feature is that the lowest frequency mode changes fromprecessional to a bending vibrational mode at J 41.The frequency of the latter goes to zero at the bifurca-tion point from the SL relative equilibria. The normalmode frequencies of SL relative equilibria are given inthe same gure for the range J 60 90 in which theyare stable. The A1 mode is the symmetric stretchingmode, the A2 mode consists of bending oscillations inthe plane containing the total angular momentum vectorand the B2 modes are linear combinations of asymmetricstretching and bending in the plane perpendicular to the

total angular momentum vector. The lowest frequencyB2 mode changes from a bending oscillation near J 60to a stretching oscillation near J 90.

The CsM symmetry group of ITy relative equilibria

implies that the associated energy level doublets are oftypes A1 B2 and A2 B1. For low values of J the lowenergy spectrum will consist of excited precessionalstates with the alternating sequence of A1 B2 andA2 B1 doublets for J even, reversed for J odd, typicalof rigid rotors. This should change for J> 40 when thelowest frequency becomes vibrational and consequentlythe associated doublets will all have the same type as the


Figure 2. Classical normal mode frequencies for the ITz REof the D2H

ion. Solid lines correspond to modes whichare symmetric with respect to CiM, large dashed lines tomodes which are anti-symmetric and the small dashed lineshows the real part of the complex f requency after theHamiltonian Hopf bifurcation at J 37. Short dashlines represent estimates of classical frequencies fromquantum calculations [23]. Circles indicate experimentavibrational frequencies [22, 23].

Figure 3. Classical normal mode frequencies for the ITy andSL relative equilibria of the D2H

ion. Solid lines correspond to symmetric ITy modes, the large dashed line tothe anti-symmetric ITy precessional mode and individually labelled dotted lines to the SL normal modes. Shortdash lines represent estimates of classical frequencies f romquantum calculations [23]. Circles indicate experimentavibrational frequencies [22, 23].


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ground state. However, the precessional frequency isstill very close and the next excited doublet should beprecessional.

The frequencies of the ITy relative equilibria trans-form continuously into the frequencies of the SL relativeequilibria. However, the changes to the quantum spec-trum will be more pronounced. The symmetry groups ofthe SL relative equilibria are the full permutation-inver-

sion group of the molecule and there are no distinctsymmetry equivalent relative equilibria. It follows thatthe energy levels should be singlets and we expect to seea splitting of energy level doublets as J passes through60. The symmetry of the ground state f or SL relativeequilibria is A1 for J even and B1 for J odd. Thelowest frequency normal mode is B2 and so for J> 60the two lowest energy states will have symmetries A1 andB2 when J is even and B1 and A2 when J is odd.

5. H2D1

5.1. Bifurcations of relative equilibriaThe energies of the relative equilibria of H2D

areshown as functions of J up to J 70 in gure 4. Thebifurcation diagram is rather more complicated thanthat for D2H

and we concentrate on describing someof the key dierences. First note that at low J it is theITx relative equilibria that are linearly stable, instead ofthe ITz relative equilibria. These lose stability in a bifur-cation to ATxz relative equilibria at J 20 beforebecoming linear at J 36. The ATxz relative equilibria

themselves lose stability in a Hamiltonian Hopf bifurca-tion at J 33 and then disappear in a collision withrotational barriers. The unstable ITz become stable atJ 24 as a result of a bifurcation to another branch of(unstable) AT xz relative equilibria. The ITz relative equilibria go on to lose stability again in a HamiltonianHopf bifurcation at J 41 and then collide with rota-tional barriers at J 55.

In contrast to D2H

the AL relative equilibria havelower energies than the SL relative equilibria. The ALrelative equilibria are stabilized as Jincreases by a bifurcation to ATxz relative equilibria at J 41, followed byan exchange of stabilities with the minimal energy ITy

relative equilibria at J 51. This exchange also involvea branch of ATy relative equilibria that is not visible inthe gure. As a result the ITy relative equilibria losestability and the AL relative equilibria become the minimum energy congurations. To begin with the AL rela-tive equilibria give the absolute minimum of the energybut from J 53 they are only a local minimum abovethe dissociation energy. Both the A L and the ITy relativeequilibria go on to collide with matching rotational bar-riers, at J 68 and J 63, respectively.

The higher energy SL relative equilibria are also sta-bilized by a pair of bifurcations, rst to ITx relativeequilibria at J 34 and then to another branch of ITy

relative equilibria at J 54. The SL relative equilibrialose stability again in a bifurcation to AL relative equi-libria at J 61 while just before this the ITy relativeequilibria become stable as a result of a bifurcation toATy relative equilibria. The ITy relative equilibria lose

stability in a Hamiltonian Hopf bifurcation at J 74(not shown) before colliding with its rotational barrierThese are the (linearly) stable relative equilibria withboth the highest energy and the highest angularmomentum.

5.2. Normal modes and spectral clustersThe analysis of energy level clusters associated to lin-

early stable relative equilibria of H2D proceeds exactly

as for D2H . Here we concentrate on features associated

with the ITx to ATxz and the ITy to AL bifurcations of

relative equilibria.The normal mode frequencies for the ITx relativeequilibria in their stable range J 0 20 and for theATxz relative equilibria in the range J 20 49 aregiven in gure 5. The ITx local symmetry group iC2M and so the associated energy level doublets willbe of types A1 A2 and B1 B2 . The four-fold multi-plicity of ATxz relative equilibria implies that they wilhave energy level quadruplets of symmetry typeA1+ A2+ B1+ B2. Thus as J passes through 20 thereshould be a merging of pairs of doublets of complementary symmetry type to form quadruplets.


303

Figure 4. The energies of linearly stable () and unstable( ) relative equilibria of the H2D

ion as functions ofthe total angular momentum J. The labelling is the sameas in gure 1.


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The normal mode frequencies of the ITy

relativeequilibria in their stable range J 0 50 are givenin gure 6, followed by the normal mode frequenciesof the AL relative equilibria in their stable rangeJ 50 68. A s f or D2H

, symmetries prevent theITy precessional mode from interacting with thevibrational modes in harmonic approximation. Thereis also a similar switch in the lowest frequency modefrom precessional to vibrational, and this will beaccompanied by a similar rearrangement of A1 B2and A2 B1 doublets. In this case it is likely to bemore distinct as there is a greater separation betweenthe precessional frequency and the lowest vibrationalfrequency. However, the main dierence between thetwo cases comes when the ions `go linear. In this casethe transition is to AL relative equilibria which appearin pairs and so will continue to be accompanied bydoublets. Moreover the local symmetry groups of ITy

and A L relative equilibria are the same, CsM, andso their clusters will have the same symmetry types.It is therefore likely that the transition from ITy toAL structure will not be observable in the quantumspectrum.

6. Summary and discussionWe have described a general method for computing

the relative equilibria of a molecule and their normamodes. The symmetry classication of all the possiblerelative equilibria and their associated `harmonicquantum states has been given f or an arbitrary AB2molecule. The possible applications of these results tospecic molecules are illustrated using the examples ofD2H

and H2D . All the relative equilibria types listed

in table 2 exist for each of these examples, and almost alof them are linearly stable for some range of angularmomentum values. The only exceptions are ITx forD2H

and ATy for both D2H and H2D

. In generawe expect a linearly stable relative equilibrium to act asan organizing centre for families of quantum states withwave functions which are localized near it and its sym-metry equivalent counterparts. Tunelling will createclusters of energy levels of symmetry types which arecharacteristic of the symmetry type of the relative equi-librium and bifurcations of the relative equilibrium maylead to rearrangements of these clusters.

Despite having the same potential function, the relative equilibrium bifurcation diagrams of H3 , D2H

and


Figure 5. Classical normal mode frequencies of the ITx andATxz relative equilibria of the H2D

ion. Dash-dot linescorrespond to the symmetric modes of ITx , large dashedlines to the anti-symmetric modes, solid lines to the nor-mal mode frequencies of ATxz and the dotted line showsthe real part of the complex frequency of ATxz after theHamiltonian Hopf bifurcation at J 34. Short dash linesrepresent estimates of classical frequencies from quantumcalculations [23, 24]. Circles indicate experimental vibra-tional frequencies [23, 25].

Figure 6. The classical normal mode frequencies for ITy andAL relative equilibria of the H2D

ion for the values of Jfor which they are linearly stable. The dashed lines correspond to symmetric ITy , the dotted line to the anti-symmetric ITy precessional mode, the solid lines to thsymmetric AL modes and the dash-dot line to the anti-symmetric AL mode. Short dash lines represent estimateof classical frequencies from quantum calculations [2324]. Circles indicate experimental vibrational frequencies[23, 25].


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H2D are quite dierent. The dierences are dictated to

a large extent by the symmetries of the ions and the roleof the centrifugal f orces. Yet there are some commonfeatures, for example the stabilization of linear cong-urations as J increases. This is well known [4, 26] and isperhaps a general phenomenon of AB2 molecules. Herewe have shown that both symmetric and asymmetriclinear congurations can be stabilized and that either

can occur as the global minimum of the total energy.In the case of D2H

the transition to a symmetric linear(SL) global minimum occurs at J 59. For H2D

it isthe AL congurations which become global minima ofthe energy, in this case at J 51. Their energy increasesabove the dissociation level at J 53. Linearly stableSL relative equilibria exist above dissociation forJ 54 61. Moreover in this case, as for H3 [11],there are also linearly stable relative equilibria with tri-angular congurations above dissociation.

In [11] we predicted that f or H3 the transition tolinear geometry will be accompanied by a rearrangementof energy level doublets associated to equilateral trianglerelative equilibria to form triplets associated with sym-metric linear congurations. The results of this papershow that for D2H

the transition will be accompaniedby a splitting of the energy level doublets of typesA1 B2 and A2 B1 associated with the minimalenergy ITy relative equilibria to give singlets associatedwith the SL geometry. The dierence is due to the lowerpermutational symmetry group of D2H

. Since the ALand ITy relative equilibria have the same symmetrygroup the corresponding transition for H2D

may

have little eect on the spectrum.Another common feature of all the ions is that the

lowest excited level changes from rotational to vibra-tional as J increases. It happens most quickly in H3where the precessional frequency exceeds the smallestvibrational normal mode at J= 30 [11]. This value is35 for H2D

. The highest value is 42 for D2H and

the eect here is probably the least likely to be observedbecause the precessional and vibrational frequencies aretoo close.

Perhaps the most likely event to be detected in the

quantum spectrum is the bifurcation from IT

x

to AT

xz

relative equilibria in H2D . This should be accompanied

by a merging of pairs of energy level doublets of typesA1 A2 and B1 B2 to form quadruplets. This bifurca-tion is somewhat unusual for AB2 molecules. In mostcases rotation about the x axis is unstable and the ITx

relative equilibrium can simply `become linear withoutany change in stability as the BAB bond angle increases.However, because of the initially small bond angle, rota-tion about the x axis is stable for H2D

at low values ofJ, but becomes unstable as J and the angle increase.Similar bifurcations, though with decreasing bond

angles, do occur from ITz relative equilibria for anumber of AB2 molecules [27]. In this case theoreticapredictions [1, 2, 28, 29] have been conrmed by fulquantum mechanical calculations [30, 31] and experimentally for H2Se [32], H2Te [33] and H2S [34]. Thesebifurcations are also accompanied by merging of pairsof doublets. The value of Jfor which the ITx bifurcationoccurs in H2D

is higher than that for the ITz bifurca

tions in H2Se, H2Te, comparable with that in H2S andlower than expected in H2O [35]. For H2D

the ITz

relative equilibria also undergo a bifurcation but themechanism underlying this is not so easily explainedand must be a more subtle interaction between thepotential and centrifugal eects.

The harmonic approximation used above is ratherpoor in terms of high resolution spectroscopy when esti-mating pure vibrational frequencies since the anharmonic eects are very strong in D2H

and H2D : the

normal modes at J= 0 overestimate the experimentafrequencies by 79% (see circles in gures 2, 3, 5 and6). However, it is still instructive to compare the presentcalculations with available experimental and calculateddata. For instance, it is clear f rom gure 2 that theremust be a strong DKa= 1 resonance at low J in D2H

between the levels of the 2 mode (the lowest symmetricone) and the levels of the asymmetric 3 mode havingone additional precessional quantum (recall that the pre-cessional mode in ITz descends in energy). This is notthe case in H2D

for which 2 and 3 always diverge (seegures 5 and 6). Indeed, Foster et al. [22] noted in theexperimental analysis of D2H

that they had to add a

Ka-dependent interaction term and drop a J-dependentterm compared to their H2D

model [25] when ttingsimilar data. Similarly one can predict a number of otherresonances of which the most notable is the avoidedcrossing again in ITz of D2H

between 1 and 3 aJ 25. Interestingly this feature resembles the crossingin H2D

at slightly lower values of J (see gure 5).It is often straightforward to correlate the classica

normal mode frequencies to a quantum spectrumFor example, the energy dierences EKa JEKa J 1 correlate with the precessional frequency

of the IT

x

relative equilibrium of H2D

andEKc J 1 EKc J with the precessional frequency of ITy. These J dependences of characteristicfrequencies present another way of comparing our classical calculations with experiment. Unfortunately theassigned experimental data are very limited J46[22, 25]. The observed predissociation spectra of D2H

and H2D should have very high J, especially when

kinetic energy release is high [12] but these spectra stilremain unassigned in detail. The only extensive sourceof assigned levels known to us is [24] where the assignment of the ground vibrational state of H2D

, obtained


305


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from quantum variational calculations, has been givenup to J 30. Another source is the calculation of energylevels in the ground and 1 states of H2D

and D2H

based on the eective Hamiltonian tted to experi-mental data [23].

The quantum data obtained in this way are presentedin the respective gures of the normal modes. One cansee that apart from the strong anharmonic shifts the J

dependences are reproduced rather well. For preces-sional modes the classical frequency again gives anupper boundary for quantum levels. The relative errorcomes down to less than 6% for ITy relative equilibriumof H2D

at J 30. For ITx the classical estimates arealso close to the quantum but become lower startingfrom J 14, `foreseeing the bifurcation. This is how-ever not unusual since the quantum dierence is some-what more `inertial. Quantum mechanically themerging of two doublets is due to the tunnelling eectbetween equivalent relative equilibria. But the tunnellingtakes place only if the height of a separation barrier ishigher than at least the ground state of the relative equi-librium. We can estimate the ground state energy as halfof the precessional frequency (see equation (18)) andcompare it with the barrier height. The barrier exceedsthe ground state for J> 26. When this happens in H2Seor H2S the ratio between the cluster width and the spaceto the next level is 1:4 [35]. Therefore we can hope thatJ 26 gives us an estimate of the point above whichcluster features may be observed in the spectrum ofH2D

. However, there is a complication due to theHamiltonian Hopf bifurcation at J 33. Energetically

this happens below the barrier which suggests thatcluster features may be perturbed but persist. To verifythis a detailed analysis is required.

The analogous comparisons of classical and quantumdata for D2H

illustrate mostly how little we knowabout this ion. With recent advances in variationalmethods it is becoming feasible to carry out more exten-sive calculations which should clarify the quantumstructure of high ro-vibrational levels in H2D

andD2H

and particularly show if the formation of four-fold clusters of levels takes place in H2D

and what

happens to them in the Hamiltonian Hopf bifurcation.

We are grateful to Boris Zhilinskii for a number ofvery useful discussions. This work was supported by aUK EPSRC research grant. INK also thanks partialsupport from the Russian Fund for FundamentalResearch through grant No 97-02-16593.

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i.n. kozin, r.m. roberts and j. tennyson- relative equilibria of d2h^+ and h2d^+

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