M.S. Child, T. Weston and J. Tennyson- Quantum monodromy in the spectrum of H2O and other systems:...

8/3/2019 M.S. Child, T. Weston and J. Tennyson- Quantum monodromy in the spectrum of H2O and other systems: new insi

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MOLECULAR PHYSICS, 1999, VOL. 96, NO. 3, 371379

Quantum monodromy in the spectrum of H 2 O and other system

new insight into the level structure of quasi-linear molecules

M. S. CHILD1, T. WESTON1 and J. TENNYSON2,1Physical and Theoretical Chemistry Laboratory, South Parks Road,

Oxford OX1 3QZ, UK2Department of Physics and Astronomy, University College London, Gower Street,

London WC1E 6BT, UK

(Received 1 April 1998; revised version accepted 16 June 1998 )

The concept of quantum monodromy is introduced to give insight into the energy levels ofsystems with cylindrically symmetrical potential energy barriers. The K structure of bendingprogressions of bent molecules and the pendular states of dipolar molecules in strong electricelds are taken as molecular examples. Results are given for a two-dimensional champagnebottle model, for six computed bending progressions for water taken from the compilation ofPartridge and Schwenke and for the quantal spherical pendulum. Sharp changes in the energy

level patterns around the barrier energy are related to changes in the forms of the relevantclassical trajectories. Analytical insight into the mathematical origin of the monodromy arealso given.

1. Introduction

The concept of monodromy (meaning `once round) iswell established in the mathematical literature[1, 2]. Onetype of physical application occurs in any time periodic

system, where for example the stabilities of xed pointsin the period map (or Floquet map) are characterized bythe eigenvalues of the monodromy matrix [3]. Anothertype of application comes from the classical mechanicalliterature [4], where it is used to demonstrate a grosstopological obstruction to the global construction ofangleaction variables. Translated into quantummechanical language, this implies the absence of anysmoothly valid set of quantum numbers for the entirespectrum. The quantum monodromy of the presentpaper is employed in the latter sense. Specically it isshown, by semiclassical arguments, to be a generic prop-

erty of any system with a cylindrically symmetricalpotential barrier. A graphical illustration of the lack ofconnectivity is shown in gure 1 below. Molecular illus-trations are given for the bendingrotating states of H2Oand for the pendular states of dipolar molecules instrong electric elds [5].

It has of course been known since the early papers ofDixon [6] and Johns [7] on the spectroscopy of quasi-linear molecules that the states below the barrier to lin-earity are conveniently labelled by bent moleculequantum numbers v 2, K and those above by linear

molecule ones, say n 2, ; but it is less often nthat the algebraic relations K and n 2 2v 2between the two sets imply that both designationequally valid for any particular energy state. The m

dromy provides an elegant test for the smoothneeither of these individual labelling schemes (and ofother scheme) by tracing out a circle in energy angular momentum space around a `critical pointhe barrier energy with K 0. At the same timbrings out a sudden change in the energy dependenceither v 2,K or n 2, when crossing the barrier toearity, which is re ected for example in the abhalving of the apparent bending vibrational frequin the photoelectron spectrum of H2S [8].

The existence of such abrupt changes is not of conew. In particular Dixon [6] drew attention to the pence of a sharp dip in the local vibrational level spaat K 0 at the barrier to linearity, which is smooaway as K increases. The resulting local convergenthe low K levels certainly implies a reorganization oenergy level pattern around the barrier maximum,

00268976/99 $12.00 1999 Taylor & Francis Ltd.

n 2 in this notation corresponds to vlinear2 of Johns [7]. T

is also a spectroscopic convention to use k for the signed of K, but in the absence of similar conventions for andm (for the pendular states) it is less confusing in the prcontext to treat K , and m as signed quantities.


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the precise consequences for the energy variation atxed v 2 or n 2 have not previously been discussed. Norhas the generality of the phenomenon been determinedby reference to the equally abrupt changes in the natureof the classical trajectories.

Previous work on the vibrationalrotational struc-tures of RennerTeller systems may also be mentioned.For example Jungen and Merer [9] show that the com-

putational equivalence of the K and

bases is broken bythe presence of RennerTeller coupling. The in uence ofthis coupling on the rate of switch-over from bent tolinear patterns has also been discussed [10, 11], withparticular reference to the spectrum of H2S [12]. Thepresent paper deals however only with motions on asingle potential energy surface. Its novel conceptualaspects relate (a) to the absence of connectivity ineither the v 2, K or n 2, labelling schemes and (b) tothe semiclassical origin of abrupt changes in the energylevel patterns around the barrier maximum.

The monodromy is introduced in section 2 by refer-

ence to a two-dimensional champagne bottle model thatwas presented in mathematical detail elsewhere [13]. Themain purpose of the paper is however to demonstratethat it also persists in the fully coupled vibrational spec-trum of H2O even when the stretching modes are alsoexcited. Results are presented for several calculatedbending progressions up to total energies of28000cm 1 on the accurate potential energy surface ofPartridge and Schwenke [14]. The monodromy is clearlyapparent for v 1, v 2,v 3 bending progressions withv 1 v 3 2 although there are some assignment ambi-

guities at high v 2, particularly for (v 1,v 3 2,0 and(1,1). The generality of the eect is further demonstratedby giving results for the spherical pendular states of adipolar molecule in a strong electric eld. The relevantangular momentum is then the projection m in the elddirection while a quantum number v, which counts thenodes in the direction, plays the role of v 2. The con-venient labelling schemes are n , m , with n 2v m ,for the low-lying near harmonic states and j,m with

j v m , for the high-lying near free rotor states.Each shows the same qualitative monodromic patternas the previous bendingrotating states.

Section 3 summarizes the main conclusions of an ear-lier semiclassical analysis of the champagne bottle model[13] and indicates their extensions to the spherical pen-dulum. The conclusions are summarized and discussedin section 4. Finally interested readers are referred to theappendix on analytical aspects of the monodromy.

2. Quantum monodromy

The origin of the term monodromy is illustrated ingure 1 which shows the central part of a lattice ofquantum eigenvalues for the scaled champagne bottle

Hamiltonian described in the appendix. Each poilabelled by the conserved angular momentum K avibrational quantum number v which increases fzero up each vertical (constant K ) ladder. The mdromy is associated with the non-connectivity oflabelling scheme, due to the presence of a so-called

tical point with zero-angular angular momentum atop of the potential maximum (E 0 in this moConsider for example the arrowed unit cell with DK , Dv 1,0 and 0,1 in the lower part of

gure, and follow the smooth evolution of this cean anticlockwise circuit around the critical point. vector 0,1 returns to itself but 1,0 returns to 1,both being related to their initial forms by

DK

Dv

1 0

1 1

DK

Dv.

Equation (1) or its inverse for clockwise transport be veried to apply to the circuit of any vector arothe critical point. The conclusion is that (v ,K may vide local labels for individual quantum states but the resulting scheme is not smoothly connected ovewhole lattice.

Further insight is obtained by noting that pointhe upper part of gure 1 are roughly interleavesuccessive vertical columns, which means thatenergy dierences corresponding to Dv 1 roughly half those corresponding to DK 1

372 M. S. Child et al.

-10 0

K

-5

0

5

E

The quantum monodromy

(1,0)

(0,1)

Figure 1. The central part of the computed quantal specfor the scaled champagne-bottle model with b 0.00The scaling relations are given after equation (A 2).


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other words there is an approximate 2:1 resonancewhich may be characterized by the quantum numbern 2v K , which is of course the linear moleculequantum number of Johns [7]. It is also normal toreplace K by in the labelling of linear states. Figure2 demonstrates some interesting symmetries betweenthese two v , K and n , (with K ) labellingschemes, by use of continuous and dashed curves to

join points with common v and common n values re-spectively. The smooth roughly quadratic energy vari-ation with K at xed v in the lower part of the guregoes over to a succession of sharp kinks around K 0for energies above the critical point. Conversely the

dashed curves are roughly quadratic in or K forE > 0 but become kinked for E < 0. These features arerelated in section 3 to the forms of the positive andnegative energy trajectories, by use of the Bohr quanti-zation rule.

Similar features are shown in gure 3 for the energiesof the K a J levels of a variety of computed bendingprogressions of water. These energy levels were takenfrom Partridge and Schwenke [14] who computedthem on a very accurate spectroscopically determinedpotential energy surface and also gave a very large

number of computer generated assignments. K a isof course a rigorous quantum number for H2O indeed is v 2 except as a counting number) but csymmetric top like degeneracies are expected for J and K a . Moreover even the 110 to 111 rotatienergy splittings in the known bands are too smabe resolved on the scale of gure 3. The plotted powere therefore taken as Partridge and Schwenkes

JJ0 energy levels for the relevant bending progressiThe lines drawn in gure 3 link the solid points common v 2 (solid lines) and n 2 (dashed lines) agure 2. In some cases it was necessary to reassigncounting number v 2 and these points have a supeposed cross (+ ). The open points at relatively levels of vibrational excitation, which are not joby lines, fall outside the v 2, K a pattern and are ususupernumerary to it. We believe that these levels appear here due to problems with the automatic asment procedure employed by Partridge and Schw[14]; in many cases they are alternatives to those g

by the solid circles. These problems with automassignment have been previously discussed [15,16should also be noted that possible high energy anolies for the 2, v 2,0 and 1,v 2,1 progressions haveomitted by truncating the diagrams.

Nevertheless, the qualitative similarity between 2 and the various panels of gure 3 is remarkable. simple quartic minus quadratic champagne bottle mis of course inadequate to account even for a bencut through the full potential energy surface, buqualitative features of the diagram depend solely

the existence of a cylindrically symmetric potemaximum at the barrier to linearity, which leads tabrupt change from smooth to kinked variations inquantized (continuous lines) and n quantized (dalines) energy levels around the position of this mmum. Quantitative dierences between gures 2 ain the vibrational energy level spacings and in the shof the smooth continuous and dashed curves depencourse on the nature of the potential surface.

A third example, illustrated in gure 4, is taken the spherical pendulum model for dipolar pendstates. The discussion in the appendix shows that tpossible labelling schemes may be employed in caseeach involving the conserved angular momenprojection, m , onto the z axis. Points in each verladder may be ascribed successive polar quantum nbers v 0,1,2, . . . , but the degenerate oscillator ln 2v m is strongly preferred for small amplioscillations of the pendulum. The smooth quadvariation with m at given n is then seen to go ovethe characteristic kinked shape at energies abovebarrier to end-over-end rotation; the dashed cuwhich are kinked at low energies and smooth at ene

Quantum monodromy in H2O and other spectra

-40 -20 0 20 40-10

-5

0

5

10

-40 -20 0 20 40

K

-10

-5

0

5

10

E

champagne bottle

Figure 2. The computed quantal spectrum f or the scaledchampagne-bottle model with b 0.00625. Points indi-cate the computed eigenvalues. Solid and dashed lines

join those with common values of the bent molecule quan-tum number v and the linear molecule number n v Krespectively. Note the abrupt transition from smooth tokinked variations in both sets of curves around the criticalpoint at K 0, E 0.


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-40 -20 0 20 40-40 -20 0 20 40

Ka

0

10000

20000

30000

E(cm

-1

)

0 v2

0

-40 -20 0 20 400

10000

20000

30000

-40 -20 0 20 400

10000

20000

30000

-40 -20 0 20 40

Ka

0

10000

20000

30000

E(cm

-1)

0 v2 1

0

10000

20000

30000

-40 -20 0 20 400

10000

20000

30000

-40 -20 0 20 400

10000

20000

30000

-40 -20 0 20 40

Ka

0

10000

20000

30000

E(cm

-1)

1 v2

0

-40 -20 0 20 400

10000

20000

30000

-40 -20 0 20 400

10000

20000

30000

-40 -20 0 20 400

10000

20000

30000

-40 -20 0 20 400

10000

20000

30000

-40 -20 0 20 40

Ka

0

10000

20000

30000

E(cm

-1)

0 v2

2

-40 -20 0 20 400

10000

20000

30000

-40 -20 0 20 400

10000

20000

30000

-40 -20 0 20 400

10000

20000

30000

1 v2 1

-40 -20 0 20 400

10000

20000

30000

E(cm

-1)

-40 -20 0 20 40

Ka

0

10000

20000

30000

E(cm

-1)

-40 -20 0 20 40

K

0

10000

20000

30000

E(cm

-1)

2 v2

0

Figure 3. Monodromy plots of various computed bending progressions of water taken from the data of Partridge and Schw[14]. The rotational energies are taken for the JJ0 states; hence K a J. The solid circles, which are joined by solid and dalines as in gure 2, display the monodromic pattern. Those with superimposed crosses (+ ) have been reassigned with respv 2 (see text). Open circles indicate reported, usually extra, states [14] that do not conform with the pattern.


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above the barrier, are those with a common rotationallabel j v m . The main qualitative features againcarry over from gures 2 and 3.

3. Semiclassical interpretation

The semiclassical origin of the monodromy in gures1 and 2 may be derived from the Bohr quantizationcondition for the champagne bottle model (seeappendix).

1h

R m a x

R min

p R E, K ; R dR v E, K1

2

p, 2

where R min and R max are the classical turning points,given by zeros of the radial momentum

p R E, K ; R 2m E K 2h 2 /2m R 2 A R 2 B R 4 1 /2.3

The partial derivative identity

E /K v v /K E / v /E K 4

is rst employed to examine the derivatives of the con-tinuous curves in gure 2 at K 0, taking care to recog-

nize that the function v E,K is non-analytic atcritical point K , E 0,0 , due to conuence oinner turning point R min with the singularity at R(see the appendix for a more detailed discussion).

The energy derivative at K 0 reduces to

v

E K 0

2h

R max

R min

m dR

p R E,0; R1h

dRR

T vib

h,

where T vib is the classical time period of the vibratCorrections to the BohrSommerfeld rule due to a lrithmic divergence of T vib as E 0 are shown where [13] to diminish rapidly away from the cripoint. It was also found that T vib is approximately metric with respect to the sign of E over a wide enregion.

The angular momentum derivative

v

K E

K h

p

R max

R min

dR

R2 2m E K 2h 2 /2m R 2 A R 2 B R 4

must be treated with care. It is zero for K 0 and Ebecause the integral is nite. However the intdiverges as K

1for E > 0 because R min 0

K 0 and dierent branches of the square root mbe taken according to the sign ofK . The presence obranch point of v (K ,E at E 0, K 0 lies at the h

of the monodromy. Its physical origin may be demstrated by use of the semiclassical identity K h mwhere is the angular velocity; thus equation (6) mare-expressed as

v

K E

1

2p

R max

R min

RdR

1

2pdt

D

2p,

whereD is the polar angle change during a vibraticycle. The limiting value as K 0 is readily dedfrom the forms of the negative and positive energy

jectories in gures 5(a ) and (b ) respectively. D 0E < 0 and D p for E < 0 according to the sigK . Hence

v

K E

0, E < 0,12 , E > 0, K > 0,12 , E > 0, K < 0.

Equation (8), which explains the sharp transition frosmooth continuous curve at K 0,E < 0 in gure 2kink when E > 0, is seen from the trajectory diagradepend solely on the presence of a cylindrically


-15 -5 5 15

m

-1.0

0.0

1.0

2.0

E/me

spherical pendulum

Figure 4. The computed quantal spectrum for the sphericalpendulum model with /B 100. Points indicate thecomputed eigenvalues, which are labelled upwards fromzero in each column by v . m is the z component of angu-lar momentum. Solid and dashed lines join those withcommon values of the pendulum quantum numbern 2v m and the rotational quantum number jv m respectively. Note the abrupt transition fromsmooth to kinked variations in both sets of curves aroundthe critical point at m 0, E / 1.


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metric maximum in the potential function; the precisefunctional form is irrelevant.

The behaviour of the dashed curves, linking pointswith linear molecule quantum numbers K andn 2v is also readily explained. Their derivativesare given by

E / n n / E / n /E , 9

where n /E 2 v /E K ; while in accordance with(8)

n

E

K2v K

1, E < 0, K 0, E > 0.

The kinks in the continuous curves at 0 and Ehave been transferred to kinks in the dashed oneE < 0. It is also evident from equations (4), (5),(9) and (10) that the slopes of the two sets of kincurved (dashed and continuous) have a mirror metry around the barrier maximum to the extent T vib E T vib E .

The positive and negative curvatures of the couous (constant v ) and dashed (constant n ) curverelated elsewhere [13] to the positive and negative cessional rates of the negative and positive energy

jectories in gure 5.

4. Discussion and conclusions

Monodromy would appear to have important coquences for the structure of the energy levels of qulinear molecules, of which we have taken water possible example. To observe monodromy in watwould be necessary to assign spectra with at leasquanta of bending excitation. Published spectra go half this far [17], although recent work has extenthe assignment to the 0,6,0 band [18]. Furthermanalysis of sunspot spectra [15] and near infrlaboratory spectra [16] has extended the assignmto levels with 7 quanta of bending excitation in com

nation with excited stretching states. As there remmany unassigned transitions, particularly in the sunspectra, it seems likely that higher bending states wanalysed soon.

Other systems may also be considered, because whas a relatively high barrier to linearity and maytherefore seem the most likely candidate in a sefor monodromy. Moreover the selection rules for trtions within the electronic ground state are muchfavourable for the observation of long progressthan those involving electronic excitation. Onother hand many well known excited states of qlinear molecules are complicated by electronic degencies and the consequent RennerTeller non-adiabcoupling. The present single-surface approach toanalysis is therefore inappropriate. Other candidinclude single-surface, quasi-linear molecules, sucC3O2 [19] and HNCO [20], for which bending sphave been recorded. However it would appear thasystems analysed to date either have nonlinear mintoo shallow to support a signicant number of benstates below the monodromy point, as is the caseC3O2 and HNCO [21], or that the observed spectr


-10.0 -5.0 0.0 5.0 10.0-10.0

-5.0

0.0

5.0

10.0

(a)

-10.0 -5.0 0.0 5.0 10.0-10.0

-5.0

0.0

5.0

10.0

(b)

Figure 5. Classical trajectories for the scaled champagne-bottle model with b 0.00625, K 1 and (a ) E

7.5, (b ) E 7.5. Note that the total angular changefor one vibrational cycle reduces to zero in the limitK 0 in panel (a ), but to p, according to the sign ofK in panel (b ). Secondly the opposite senses of precessionin the two panels should be noted.


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not yet extend to the monodromy point, as is the casefor H2O. No rm experimental evidence for this inter-esting eect is therefore yet available.

As indicated in the introduction, the plots in gures24 are in one sense simply presentations of knownresults in quasi-linear molecule theory [6, 7]. We havehowever stressed the generality of the abrupt change inlevel structure about the monodromy point, for all

systems with cylindrically symmetric potential barriers.The relationship between these abrupt changes and theforms of relevant classical trajectories was strictlydemonstrated only for the two-dimensional champagnebottle and spherical pendulum models but the similaritywith the fully coupled computed results for water wasstrikingly evident. It remains to be seen whether this ismerely evidence of a rather weak interaction betweenthebending and stretching modes in this particular mol-ecule, but since the freedom to pass through linearityis associated with the vanishing of one moment ofinertia, it seems likely that the monodromy will prove

to be a persistent eect. No doubt perturbations willoccur in most molecules at energies as high as25000cm 1 but use of the double v 2, K 2 , n 2,2 gridemployed in gures 24 is expected to be a powerful aidto assignment.

The authors are grateful to Professor RichardCushman for drawing their attention to this interestingproblem and to CCP6 for support. Helpful discussionswith G. G de Polavieja, B. P. Winnewisser and M. Win-newisser are gratefully acknowledged. TWis grateful for

an EPSRC research assistantship.

Appendix: analytical considerations

The mathematical origin of the monodromy is illu-strated below by reference to the champagne bottlesystem. Analytical aspects of the more complicatedspherical pendulum system are also addressed.

A.1. Champagne bottleThe K structure associated with high amplitude

bending states of a bent molecule is modelled by the`champagne bottle [13] Hamiltonian, with the classicalform

H1

2mP

2R

P2u

R2 A R

2B R

4. A 1

Angular momentum is conserved, with P u K h inquantum mechanics, and the quantum mechanicaleigenvalues are readily obtained from an expansion indegenerate harmonic oscillator states [22]. The present

analytical discussion starts by transforming equatioto the scaled form adopted in the earlier paper

v K ,1

2

1

2p

z 1

z 2

z1 2z z 2 2bz 3 K 2

1 /2

in which z 2m A/

h2 1/2

R2, m

/2A h 2

b h 2 /8m A 3 1 /2B while z 1 and z 2 are the turpoints. The monodromy arises from a conuencthe inner turning point with the singularity as Kand it follows from equation (54) of [13] that vcontains a turning point contribution of the form

f K , 12p

K arctan

K

2ln

2

K2

4

K

1

2p ImK

i

ln

K i

2

K p

2

It is evident from equation (A 3) that v K , has a lrithmic singularity at K 0.

The nature of the monodromy in gure 1 can alstraced to the multivalued nature of the arctangent tribution to f K , . The numbering given in the diagcorresponds to choice of the principal vp/2 < arctan /K < p/2, but equation (A 3) im

that v K , v K , K if the arctangent is alloto pass smoothly to the next sheet after an anticlockcycle around the origin. Conversely the energy muincreased to that of the original point K , v K , the quantum number v K , is conserved duringcycle. Consequently the vector DK ,Dv betwpoints labelled K i, v i and Kj, vj in gure 1 evolvthe vector DK ,Dv DK around such a v K ,serving cycle, which accords exactly with equation

Equation (A 3) also has implications for the deriv(v /K as K 0, which was analysed pictorialsection 3. Thus

LimK 0

f

K

1

2parctan

K

p

2.

The terms on the right hand side cancel for < 0combine to give 12 for > 0. Hence the limit(v /K implied by equations (A 3) and (A4) coinwith those given by equation (8).


These scaling relations are unfortunately incorrect iearlier paper [13].


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A.2. Spherical pendulumThe pendular motions of a diatomic molecule with

dipole and rotational constant B in an electric eldare governed by the Hamiltonian

H B J2

cos

B p2

p2u

sin2 cos. A 5

The angular momentum component p u m h is againconserved and the quantum mechanical Hamiltonian isreadily diagonalized in a spherical harmonic basis. Thesemiclassical situation has an additional complication,compared with the earlier model, that the followingBohrSommerfeld quantization condition may involveconuences between either of the classical turningpoints, 1 and 2, and the singularities at p and 0 respectively:

v1

2

1h

p d

2

h B1/2

1

2

E cos m 2h 2 / sin2

1 /2 d.

A 6

The coalescence at p is responsible for the mono-dromy in gure 4, while that at 0 has consequences

for the limiting behaviour of (v

/m

asm

0. Theanalysis is simplied by the substitutions a E / ,b h B /

1/2 and z cos, which reduce equation(A 6) to the form

v1

2

1pb

z 1

z 2

z3

a z2

z b2

m2 1 /2

1 z 2dz . A 7

Thus

v

m

b m

2p I I , A 8

where

I

z 1

z 2

dz

z 1 z 3 a z 2 z b 2m 21 /2

2

z 1 1 z 1 z 31/2

z 1 z 2

z 1 1,

z 1 z 2

z 1 z 3

1/2

.

A 9

The notation in equation (A 9) has z 1 > z 2 > z 3 foroots of the cubic polynomial, whileP a2, k is the cplete elliptic integral of the third kind, in the notatioGradsteyn and Ryzhik [23]. A careful examinatiothe behaviour of the roots z i as m 0 , coupled the identity [24]

a2, k k 2 /a2, k K k

p

2a

2

1 a2 a2 k 2

1 /2

A

leads to the conclusion that I is nite for E < diverges as p/b m for E > , while I divergep/b m for all E. Consequently

Limm 0

v

m E

12 , f or E < ,1, f or E > .

A

It follows that the labels n 2v m and j v used for the continuous and dashed lines in gure spectively are dened in such a way that (n /m Efor E < and (j /m E 0 for E > . The angues of the partial derivative identity in equationtherefore ensure that (E /m n 0 for E < (E /m j 0 for E > .

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M.S. Child, T. Weston and J. Tennyson- Quantum monodromy in the spectrum of H2O and other systems:...

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Transcript of M.S. Child, T. Weston and J. Tennyson- Quantum monodromy in the spectrum of H2O and other systems:...