kevin a. mitchell and robert g. littlejohn- the rovibrational kinetic energy for complexes of rigid...

8/3/2019 Kevin A. Mitchell and Robert G. Littlejohn- The rovibrational kinetic energy for complexes of rigid molecules

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MOLECULAR PHYSICS, 1999, VOL. 96, NO. 9, 13051315

The rovibrational kinetic energy for complexes of rigid molecules

KEVIN A. MITCHELL* and ROBERT G. LITTLEJOHN

Department of Physics, University of California, Berkeley, California 94720, USA

(Received 13 August 1998; accepted 24 November 1998)

The rovibrational kinetic energy for an arbitrary number of rigid molecules is computed. Theresult has the same general form as the kinetic energy in the molecular rovibrationalHamiltonian, although certain quantities are augmented to account for the rotational energyof the monomers. No specic choices of internal coordinates or body frame are made in orderto accommodate the large variety of such conventions. However, special attention is paid tohow key quantities transform when these conventions are changed. An example system isanalysed explicitly as an illustration of the formalism.

1. Introduction

The rovibrational kinetic energy of a molecular (rigid

body) complex has previously been computed explicitlyfor a number of specic cases. Examples include calcula-tions for the atommonomer system, consisting of apoint particle and a rigid body, by Brocks and vanKoeven [1], van der Avoird [2], and Makarewicz andBauder [3]; the molecular dimer, consisting of tworigid bodies, by Brocks et al. [4] and van der Avoird[5]; the molecular trimer by Xantheas and Sutclie [6]and van der Avoird, Olthof and Wormer [7]; and closelyrelated systems of single molecules with internal rotation[8, 9]. In the present paper we derive the rovibrational

kinetic energy of an arbitrary molecular complex con-taining an arbitrary number of rigid bodies. That is, weexpress the kinetic energy in terms of the total angularmomentum of the complex and the momenta associatedwith the internal degrees of freedom. The term `rovibra-tional is perhaps misleading since it implies small ampli-tude vibrations of the complex, an assumption we donot make. A rovibrational kinetic energy for a generalmolecular complex has been proposed earlier by Makar-ewicz and Bauder [3]; their analysis diers markedlyfrom ours in that they do not impose rigidity conditionson the monomers nor do they distinguish the relativerotational motion of the monomers from the internalvibrations of the monomers.

One method for deriving the rovibrational kineticenergy of a system of rigid bodies would be to beginwith the rovibrational kinetic energy of a system ofpoint particles and then impose rigidity constraintswithin certain subsets of these particles. A general analy-sis of internal constraints on n -body systems has been

given by Menou and Chapuisat [10] and by Gatti et al[11]. These authors observe that such a formalism may

be applied to nd the rovibrational kinetic energy of arigid body complex, though they do not derive such akinetic energy. The constrained systems approach hasthe advantage of naturally allowing the relaxation othe rigidity constraints to include small internal vibra-tions of the monomers. However, within a strictly rigidbody, the positions, masses, and velocities of the consti-tuent particles are irrelevant. Rather, only the overalorientation, moment of inertia, and angular velocityare of interest. Therefore, in our derivation of the rovi-brational kinetic energy, we assume from the outset that

the bodies are rigid and use a kinetic energy consistingof only the translational and rotational energies of therigid bodies. It should be mentioned that whicheverapproach is followed there exists an ambiguous extra-potential term in the quantum kinetic energy. The originof this ambiguity rests in the lack of knowledge aboutthe potential which connes the system to the manifoldof rigid shapes (see, e.g. Kaplan et al. [12]). For simpli-city, we neglect any extrapotential terms arising f romthe constraint process and adopt the standard form(translational plus rotational energies) for the quantumkinetic energy of a rigid body.

Our approach is modelled on the derivation of [13]valid for clusters of point particles, but is augmented toinclude the rotational kinetic energy of the monomersthe current paper therefore generalizes many resultsvalid for point particles to systems of extended rigidbodies.

Also, in deriving the rovibrational kinetic energy, wemake no specic choice of internal coordinates or bodyframe. This allows our results to be applicable to thewide range of coordinate and frame conventions suitable to dierent molecular complexes. We discuss how

Molecular Physics ISSN 00268976 print/ISSN 13623028 online 1999 Taylor & Francis Ltdhttp://www.tandf.co.uk/JNLS/mph.htm

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* Author for correspondence. e-mail: [email protected]
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the various quantities appearing in the kinetic energytransform under changes of internal coordinates andbody frame. We also present two distinct decomposi-tions of the kinetic energy into rotational and vibra-tional (internal) contributions. The rst decompositionis independent of the conventions for internal coordi-nates and body f rame. The second decomposition hasa form more common to rovibrational Hamiltonians in

the literature, though it is not independent of coordinateand frame conventions.

The current paper has been inuenced by recent workexhibiting the importance of gauge theory and geometricphase in the study of rovibrational coupling [1319]. Inaddition to rovibrational coupling, the geometric, orBerrys, phase [20] has important applications inBornOppenheimer theory [21], optics [22, 23], andguiding centre motion [24], to name a few examples.The inuence of gauge theory on our analysis here ismost readily evident in the form of the kinetic energyequation (27) which, owing to the appearance of a gaugepotential, is reminiscent of the kinetic energy of acharged particle in a magnetic eld. Notions of gaugeinvariance and covariance also have inuenced ouranalysis of transformation properties presented in sec-tion 2.3. Although our development is inuenced by thetechniques and concepts of gauge theory and geometricphase, this paper requires no specic background ineither eld.

2. The classical kinetic energy of a molecular complex

2.1. The principal derivation

In this paper, a molecular complex is modelled by acollection of n rigid bodies or monomers. No constraintsare placed on the positions or orientations of themonomers or on the symmetry of the moment of inertiatensors. In particular, we allow the monomers to bepoint particles (atoms), collinear bodies, or non-colli-near bodies. However, for mathematical simplicity, initi-ally we assume that each monomer is non-collinear.Then, after deriving the rovibrational kinetic energy,we comment on the straightforward generalization ofallowing complexes containing collinear monomers

and point particles.We begin by introducing three classes of frames whichare important in our derivation. The space frame (SF) isthe inertial, laboratory-xed frame. There are n individ-ual body frames (IBF), one for each body. IBFa,a = 1, . . . , n , is xed to rigid body a and rotates withthe body. Finally, there is a single collective body frame(CBF), which is xed to the complex as a whole. TheCBF diers from the other frames in that it must bespecied for each shape or internal conguration ofthe complex. In general, this specication produces sin-gularities in the CBF, a fact which has been studied

in three- and four-atom systems [25, 26]. There is nocanonical way of choosing the CBF, though there areseveral methods commonly used in such Hamiltonianssuch as xing the CBF to the principal axes of the com-plex or to one of the IBFs. In this paper we will make nospecic choice of CBF.

We employ the following notation to denote theframe to which the components of a vector are referred

For an arbitrary vector v, an s superscript, that is vs

indicates components in the SF; an ia superscript indicates components in the IBF of body a; and a c super-script indicates components in the CBF. (For notationalsimplicity, at a certain point we will drop the c superscript, leaving it understood thereafter that a vectowithout a superscript is implicitly in the CBF. ) The components of v in the various frames are related by properorthogonal 3 3 matrices, which we dene by

vs = Rvc, ( 1)

vs

= Ssav

ia

, ( 2)

vc = Scavia. ( 3)

The matrix Ssa determines the orientation of body a, inthe SF; the matrix R determines the orientation of theentire complex in the SF; and the matrix Sca determinesthe orientation of body a in the CBF.

The conguration of the complex of n rigid bodies isspecied fully by the centre of mass position of eachbody and the orientation of each body. To eliminatethe overall translational degrees of freedom, we x the

centre of mass of the entire complex at the origin. Thecentre of mass positions of the n bodies are thendetermined by n - 1 Jacobi vectors [27, 13] rsaa = 1, . . . , n - 1. The orientations of the rigid bodieare specied by the n matrices Ssa 2 SO( 3)a = 1, . . . , n . Taken together, rsa, a = 1, . . . , n - 1 andSsa, a = 1, . . . , n specify a lab description of the conguration. To shift to an internalexternal, or shapeorientation, description of the conguration, we introduce 6n - 6 internal, or shape, coordinates q

= 1, . . . , 6n - 6. The q may be separated into 3n - 6

coordinates parametrizing the distances between therigid bodies and 3n coordinates (Euler angles) parametrizing the orientations of the bodies in the CBFHowever, here we allow the q to be completely arbi-trary, so long as they are invariant under rotations ofthe complex. The Jacobi vectors rca, referred to thecollective body frame, and the matrices Sca are bothfunctions of the internal coordinates q. In practiceone may dene the CBF by specifying the f unctionrca( q) and S

ca( q) , where q without a superscript refer

to the collection of all coordinates q. The orientationR 2 SO( 3) (dened in equation (1)) of the complex and

1306 K. A. Mitchell and R. G. Littlejohn


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the shape q fully determine the conguration in the SF,as may be seen by the following equations:

rsa = Rrca

( q) , ( 4)

Ssa = RS

ca( q) . ( 5)

Equation (4) is an application of equation (1), and equa-tion (5) is implied by equations (1)(3). The above equa-tions relate the internalexternal description of aconguration (in terms of R and q ) to the originaldescription (in terms of rsa and S

sa).

We assume that the masses of the n bodies have beenabsorbed into our denition of the Jacobi vectors. Thus,the kinetic energy of the complex, without the overalltranslational contribution, is

T =1

2

n - 1

a= 1

j rsaj2

+1

2

n

a=1

!sa M

sa!

sa, ( 6)

where the dot is used for time derivatives,Ms

a is themoment of inertia of body a in the SF, and !sa is theangular velocity of body a.

In general, an angular velocity is a vector which meas-ures the rotation rate of one frame with respect toanother frame. The components of this vector may bereferred to either of these two frames (or to an arbitrarythird frame for that matter). For example, the angularvelocity !sa measures the rotation rate of the IBF ofbody a with respect to the SF; its components arereferred to the SF.

We introduce the notation v f or the antisymmetric

matrix which maps a vector u into the vector v u.Then,

!sa

= SsaSsTa

, ( 7)

!iaa = S

sTa

Ssa, ( 8)

where the T superscript denotes the matrix transpose.Note that these two formulas are consistent with thechange of basis relation equation (2), that is!

sa = S

sa!

iaa , as may be seen from the general relation

Q( v )QT

= (Qv) , ( 9)

where the vector v is arbitrary and Q 2 S O ( 3) .We proceed by expressing the kinetic energy equation

(6) in terms of the internal velocities q and the totalangular velocity !c. The total angular velocity measuresthe rotation rate of the CBF with respect to the SF. Ithas components in both the SF and the CBF which are

!s

=RRT, ( 10)

!c

= RT R. ( 11)

These equations are analogous to equations (7) and (8)Equation (11) permits the time derivatives of equations(4) and (5) to be expressed as

rsa

=Rrca + Rr

ca,

q = R(!c rca + r

ca,

q) , ( 12)

Ssa =RSca + RS

ca,

q

= R[(!c )Sca + Sca,

q], ( 13)

where the `, subscript denotes the derivative with

respect to the coordinate q

. In the above equationwe have used the convention, which we adopt for theremainder of the paper, that the Greek indices , t, . .are implicitly summed from 1 to 6n - 6 when repeatedHowever, the Greek indices a, b, . . . which label eitherthe Jacobi vectors or monomers are summed explicitlyCombining equations (7) and (13) and using equation(5), we nd

!sa = R[(!

c) + Sca,S

cTa

q]RT. ( 14)

Since Sca,ScTa is antisymmetric, we dene a vector sa

such that

sca = S

ca,S

cTa , ( 15)

siaa = S

cTa S

ca,. ( 16)

By comparison with equations (7) and (8), we see thatqs

ca is the angular velocity of the IBF a with respect to

the CBF. By inserting equation (15) into (14) and usingequation (9), we nd

!sa = R(!

c+ q

s

ca) = !

s+ q

s

sa. ( 17)

The above equation expresses the angular velocity of theIBF of body a with respect to the SF as the sum of the

angular velocity of the CBF with respect to the SF plusthe angular velocity of the IBF of body a with respect tothe CBF. However, this decomposition has no inherentphysical meaning since it depends on the conventionused to dene the CBF. By appropriately changingthis convention, either of these two terms could bemade to vanish.

Using equations (12) and (17) in equation (6), wewrite the kinetic energy as

T = 12!M!+ a!q + 12h tq q t, ( 18)where

M=n -1

a= 1

( r 2aI- rarTa) +

n

a= 1

Ma =n -1

a= 1

( r 2aI- rarT

a)

+

n

a= 1

SaMiaa S

Ta, ( 19)

a =n -1

a= 1

ra ra, +n

a= 1

Masa =n -1

a=1

ra ra,

+

n

a= 1

SaMiaa s

iaa, ( 20)

Rovibrational kinetic energy for complexes of rigid molecules 1307


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ht =

n -1

a=1

ra,ra,t +n

a= 1

saMaat =n - 1

a= 1

ra,ra,t

+

n

a=1

siaaMiaa siaat, ( 21)

and where henceforth we suppress the c superscript onvectors and tensors referred to the CBF. We have also

used I for the identity matrix and Ma = RT

MsaR and

Miaa = SsTa M

saS

sa for the moment of inertia of body a

in the CBF and IBF a respectively. In the above equa-tions, we present two expressions for each of the quan-tities M, a, and ht. The rst expression involvesquantities referred entirely to the CBF. The secondexpression is slightly more complex but has the advan-tage that all of the dependence on Sa( q) and ra( q) isshown explicitly by Sa, ra, ra,, and s

ia

a. Note that Mia

a

is the moment of inertia of body a in its own IBF and assuch is a constant matrix independent of both the shapeand orientation.

By rearranging the terms in equation (18), we put thekinetic energy in the form

T = 12(!+ Aq)M(!+ At q t) + 12gtq q t, ( 22)

where

A = M- 1a, ( 23)

gt = ht - aM-1at. ( 24)Converting the velocities to momenta, we nd

J =T

!= M(!+ A q) , ( 25)

p =T

q= gt q

t+ AJ. ( 26)

The vector J is the total angular momentum of the com-plex. It satises the usual body-referred `anomalouscommutation relations f Ji, Jjg = - k eij k Jk , where eij kis the usual LeviCivita symbol. The classical kineticenergy is now expressible in terms of momenta as

T = 12JM- 1J + 12( p - AJ) gt( p t - AtJ) , ( 27)where gt is the inverse of gt.

Equation (27) decomposes the kinetic energy into twoterms. The rst term is the kinetic energy the complexwould have if it were a rigid body of xed shape. Weregard this term as the rotational kinetic energy of thecomplex. The second term we regard as the internal, orvibrational, kinetic energy. Often in such rotationvibration decompositions a dierent rotational termappears which contains a modied moment of inertiatensor and a modied angular momentum vector. Wewill relate the above decomposition to such alternative

decompositions in the next section. For now, howevernote that the appearance of A in the internal kineticenergy couples the internal degrees of freedom to theangular momentum. For this reason, we call A theCoriolis potential. Furthermore, we call gt the internametric because it acts to square p - AJ. Note that theinternal kinetic energy has the same `jp - e Aj 2 form asthe kinetic energy of a particle in a magnetic eld, where

the role of the vector, or gauge, potential is played bythe Coriolis potential and the role of the electric chargeis played by the angular momentum.

The kinetic energy given in [13] for a collection ofpoint particles has exactly the same f orm as equation(27). However, for collections of point particles, thequantities dened in equations (19)(21) do not containthe terms with Miaa . The appearance of these terms, andof course the introduction of an extra 3n internal coordinates, are the sole modications necessary to augmentthe kinetic energy of a system of point particles to

include the rotational kinetic energy of the monomersThe fundamental reason why the f orm of the kineticenergy is the same for these two cases is the rotationalsymmetry of the kinetic energy operator; equation (27is in fact a general result, valid for any SO( 3) invariantmetric. We will discuss this matter further in a futurepublication.

We comment now on how the preceding results aregeneralized to include collinear monomers and poinparticles. First, the number of coordinates changes. Anon-collinear body requires three Euler angles to specify

its orientation fully, whereas a collinear body requireonly two spherical coordinates to specify its orientation(the direction of its collinear axis). A point particle, ofcourse, requires no orientational coordinates. Thereforeinstead of 6n - 3 coordinates as before, there are3n + 3n n + 2n c - 3 coordinates parametrizing thecentre of mass system. Here, n = n n + n c + n p is thetotal number of monomers, where n n is the number ofnoncollinear monomers, n c is the number of collinearmonomers, and n p is the number of point particles.

We next describe the form of the rovibrational kineticenergy when incorporating collinear bodies and pointparticles. First, the computation leading from equation(6) to equation (27) is essentially unchanged by theinclusion of collinear bodies and point particles, solong as one takes the moment of inertia tensor Ma oa point particle to be 0. The moment of inertia tensor ofa collinear body is explicitly Ma = a( I- nan

Ta) , where

na is a unit vector pointing along the collinear axis anda is the single nonzero principal moment. The quantityna is a more natural measure of the orientation of acollinear body than Sa, since Sa overparametrizes theorientations. Therefore we rewrite equations (19)(21



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in the following form, more appropriate for a generalcomplex:

M=n - 1

a=1

( r2aI- rar

Ta) +

n n

a=1

SaMiaa S

Ta

+

n n+ n c

a= n n+ 1

a( I- nanTa) , ( 28)

a =n - 1

a=1

ra ra, +n n

a= 1

SaMiaa s

iaa

+

n n+ n c

a= n n+ 1

ana na,, ( 29)

ht =

n - 1

a=1

ra,ra,t +n n

a= 1

siaaMiaa siaat

+

n n+ n c

a= n n+ 1

ana,na,t, ( 30)

where we have ordered the monomers with the non-col-linear bodies rst, the collinear bodies second, and thepoint particles last. We omit the straightforward proofof these equations relying instead on the following twoobservations. First, the orientational contribution fromthe point particles has dropped out since Ma is 0 forsuch particles. Next, the orientational contributionfrom a collinear body is identical to the contributionof a single Jacobi vector. This fact is easily understoodby modelling a collinear body as two point particles

connected by a Jacobi vector. The above equationsmay be combined with equations (23) and (24) toobtain the kinetic energy equation (27) of a general mol-ecular complex. The only note of caution occurs if theentire complex should become collinear, in which caseM is not invertible and singularities may arise.

2.2. An alternative form of the kinetic energyWe rearrange the kinetic energy equation (27) to place

it in a form which is more common in the literature onrovibrational Hamiltonians. We dene the modiedmoment of inertia tensor ~M by

~M= M- h taa

Tt, ( 31)

where ht is the inverse matrix of ht, and the vector K,often called the angular momentum of vibration, by

K = htap t. ( 32)

We use these denitions to place the kinetic energy in theform

T = 12( J - K)~M

- 1( J - K) + 12ph

tp t. ( 33)

The above equation may be veried by comparing theterms of order J2, J1, and J0 in equations (33) and (27)The equality of the respective terms is apparent from thefollowing identities:

gtM- 1at = h

t ~M- 1

at, ( 34)

gt = h

t + hs

( as ~M- 1

a) ht, ( 35)

~M

- 1= M- 1 + gtAA

Tt, ( 36)

which follow from equations (31) and (24).Equation (33) provides an alternative rovibrationa

decomposition of the kinetic energy, in a formcommon in the literature for rovibrational Hamilto-nians. For example, the WilsonHowardWatson mol-ecular Hamiltonian [28, 29] is expressed in this mannerusing the Eckart conventions, for which various simpli-cations occur.

2.3. Changing the internal coordinates and the collectivebody frame

When the conventions for the internal coordinates orthe CBF are changed, the quantities dened in thipaper do not, in general, remain invariant. Insteadthey transform via a precise set of rules. Although theanalysis of these rules is not critical to the logical ow othis paper, we include an account of them for the following two reasons. An understanding of transformation rules facilitates conversion between dierent set

of conventions. This is important in actual problemswhere it is not uncommon to utilize more than onecoordinate or frame convention, especially for largeamplitude motions. Also, knowledge of transformationproperties leads to the denition and consideration ofquantities which transform in a simple manner (that isinvariantly or covariantly). Such quantities often havespecial geometric or physical signicance. The review ofLittlejohn and Reinsch [13] contains an in depth discus-sion of transformation properties specialized for systemsof point particles. Since essentially these results are

unchanged by the generalization to include rigidbodies, we simply summarize here the key results from[13] and comment on how they are to be extended.

We rst introduce the important concept of q tensorsWe consider a new set of coordinates q 0 which arefunctions of the old coordinates q. A q tensor trans-forms under such a change in cordinates by contractingeach lower Greek index with q/ q 0 t and each upperGreek index with q 0 t/ q. The rank of the tensor isthe total number of such indices, whether upper olower. For example, sa is a rank one q tensor becauseit transforms via



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s0

a =q

t

q 0 sat, ( 37)

where s 0a is computed from equation (15) using the newcoordinatesq 0 and sa is computed using the old coor-dinates q. Other rank one q tensors are siaa, a, A, p,and q . Rank zero q tensors, also called q scalars, areinvariant under coordinate transformations. Theyinclude Ssa, Sa, R, r

sa, n

sa, !

sa, !

s, Msa, Ms,

~M

s, T , Js,

and Ks, as well as these same quantities referred (whereappropriate) to the CBF or IBF a. Rank two q tensorsinclude ht, h

t, gt, and gt.

Next, we dene the concept of an R tensor, which isimportant when changing the CBF. We consider a newCBF such that the orientation matrix R0 with respect tothe new frame is related to the old orientation R by

R0 = RU( q) , ( 38)

where U( q) 2 SO( 3) is a smooth function of q . Thecoordinatesq are held xed. The quantities ra and Sa

transform via

r 0a = UTra, ( 39)

S0a = UTSa, ( 40)

where we have omitted the q dependence. We call ra arank one R tensor because it has one Latin index whichtransforms with one copy ofUT. In general, an R tensortransforms by contracting each Latin index with UT asin equation (39). The rank of the R tensor is the numberof Latin indices it possesses. Other rank one R tensors

include na, !a, and J. Rank zero R tensors, also calledR scalars, do not depend on the choice of CBF andinclude T and q . Rank two R tensors include M andMa.

The quantity A is not an R tensor. Instead, it has amore complicated transformation property

A 0 = UTA - c, ( 41)

where c = UTU,. An interpretation of this transfor-

mation property as the gauge transformation law of anon-Abelian gauge potential is given in [13]. We simplynote that such an analysis motivates the introduction ofthe Coriolis eld strength,

Bt = At, - A,t - A At. ( 42)

The Coriolis eld strength is a rank two q tensor anda rank one R tensor. We will use the Coriolis eldstrength only brie y in section 5. However, it plays acentral role in the gauge theoretical approach torovibrational coupling.

One should be aware that other quantities which havebeen introduced also are not R tensors. These includeht and

~M. However, it should be noted that both of

these quantities have counterparts, gt and M, respectively, which are R tensors. For further insight into therelationship between these two pairs of quantities, see[13]. Various other quantities which are not R tensorinclude sa, a, K, !, and p.

An important observation is that the two decompositions of the kinetic energy, equations (27) and (33), dierin their transformation properties. Specically, equation

(27) decomposes the kinetic energy into two terms whichare q and R scalars. Equation (33), on the other handdecomposes the kinetic energy into two terms which areq scalars but not R scalars. Thus, the latter decomposi-tion is dependent on the choice of CBF, whereas theformer is not. For this reason, we view equation (27as the fundamental rovibrational decomposition othe kinetic energy. However, depending on the CBFconvention, the decomposition of equation (33) verywell may be easier to compute (see section 5)Reference [13] provides further discussion of thesedecompositions.

3. General expressions for the quantum kinetic energy

3.1. The unscaled kinetic energyTemporarily we abandon the specic system of a mol-

ecular complex in order to present expressions for thequantum kinetic energy of a general system. Largely theresults of this section are similar to previous work oNauts and Chapuisat [30] which, in turn, relies on sev-eral earlier references. We note also that van der Avoirdet al. [7] have employed a similar formalism for the

water trimer. Here we will summarize relevant aspectof these results to x notation and to lay the foundationfor section 4. Also, we apply the formalism to the simpleexample of a single rigid body, which will be of futureuse. In section 3.2 we present a new approach to scalingthe wavefunction.

We denote the classical kinetic energy by

T = 12p a Ga b ( x) p b , ( 43)

where x stands for a collection of generalized positionvariables x a , a = 1, . . . , d (d is the number of degrees of

freedom),p

a ,a

= 1,. . .

, d, are generalized momentaand Ga b are the components of the inverse metric

tensor G-1. When repeated, the indices a , b , c , . . . areassumed to be summed from 1 to d, both in equation(43) and the subsequent development. The momenta p aare linear combinations of the canonical momenta,

p a = Cab ( x) p b , ( 44)

where p a is the momentum canonically conjugate to x aand the Ca

b are components of the change of basimatrix. Nauts and Chapuisat [30] call the more generamomenta p a quasi-momenta and reserve the term



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momenta for what we call the canonical momenta p a .The kinetic energy equation (43) is expressed in terms ofthe canonical momenta p a by

T = 12 p c Cc

aG

a bC

d

bp

d= 12 p a

~G

a b

pb

, ( 45)

where~

Ga b

are the components of the inverse metric~G

-1= CTG-1Cwith respect to the canonical momenta.

From now on, we omit the explicit x dependence.The quantum kinetic energy T is often expressed in

the Podolsky form [31]

^T =

1

2

1~G

1/ 2^p a

~G

1/ 2 ~G

a b ^p b , ( 46)

where~

G = det~G and where ^p a is the momentum

operator conjugate to x a ,

^p a = - i

x a, ( 47)

setting h = 1. The operator ^T may be expressed also

using the adjoints of the momentum operators:^T = 12

^py

a

~G

a b ^p b ( 48)

or, more generally,

^T = 12p

y

aG

a bp

b, ( 49)

where p a is the operator corresponding to the classicalmomentum p a ,

p a = Cab ^p b . ( 50)

As observed by van der Avoird et al. [7], even though theadjoint form of the kinetic energy represents the samedierential operator as the Podolsky form, equations(48) and (49) are convenient for evaluating matrixelements since the adjoint of the momentum operatoreectively acts on the bra to the left.

Equations (48) and (49) are straightforward conse-quences of the denition of the adjoint. For an arbitraryoperator Â , the matrix element of Â

y

with respect towavefunctionsU and U0 is

h Uj^

Ay

U0i = h

Â Uj U

0i , ( 51)

where the inner product is dened via

h Uj U0i = dv U U0 , ( 52)

and the volume element is

dv =~

G1/ 2

dx 1 . . . dx d =G

1/ 2

jdetCjdx 1 . . . dx d , ( 53)

where G = detG. From this denition, one nds thatthe momenta ^p a are not in general Hermitian butrather satisfy the relation

^py

a =1

~G

1/ 2^p a

~G

1/ 2. ( 54)

More generally, the momenta p a satisfy

py

a = p a +1

~G

1/ 2[p b

~G

1/ 2Ca

b], ( 55)

where the square bracket notation indicates that ^p b acts

only on the terms inside the brackets. Equations (46)(54), and (50) combine to prove equations (48) and (49)

An important illustration of the preceding formalismand one which we shall need later is that of a single rigidbody. We dene Euler angles [x 1, x 2, x 3 ] = [a, b, g] in theusual way by R(a, b, g) = Rz (a)Ry ( b)Rz (g) , where Rrotates the space frame into the body frame and Ri ia rotation about the ith space axis. The body-referredangular momenta [p 1, p 2, p 3] = [J1, J2, J3] are noncanonical momenta related to the canonical momenta[p 1, p 2, p 3] = [p a, p b, p g] via [32]

J1

J2

J3

= C

pa

p b

pg

, ( 56)

where

C=

-cos g

sin bsing cosg cotb

sing

sin bcos g - sing cotb

0 0 1

. ( 57)

The classical kinetic energy in terms of the angulamomenta is T = JM-1J/ 2 where M= G is the bodyreferred moment of inertia tensor, which is independenof the Euler angles. The volume element is computedfrom equation (53) to be

dv = ( detM) 1/ 2 sin bda dbdg = 8p 2( detM) 1/ 2 dR,

( 58)

where dR= sin bdadbdg/ ( 8p 2) is the normalized Haarmeasure on SO( 3) . The quantum kinetic energy i

expressed in terms of the operators

^

Ji

^J1^

J2^

J3

= C

- i/ a

- i/ b

- i/ g

. ( 59)

Using the volume element dv , one may verify that ^Ji iHermitian. (On a deeper level, ^Ji is Hermitian because iis a symmetry of the kinetic energy.) Thus, the quantumkinetic energy equation (49) acquires the familiar form

^T = 12

^JM- 1^J. ( 60)



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3.2. The scaled kinetic energyOften it is useful to multiply the original wavefunction

U by some real positive function S ( x) to form a newwavefunctionW,

W = S U. ( 61)

Such a scaling produces a new kinetic energy operatoracting on the new wavefunction W. In this section, we

derive the form of this new kinetic energy operator.Similar discussions are given by Nauts and Chapuisat[30] and Chapuisat, Belafhal and Nauts [33]. The mostnotable distinction between our approach and theseearlier accounts is our introduction of a new adjoint,shown in equation (64). This adjoint allows for adierent form for the scaled kinetic energy operatorshown in equation (67) and the associated extrapotentialterm in equation (68).

We note that equation (61) induces a new innerproduct on the scaled wavefunctions. We denote this

new inner product with anS

subscript and dene it via,

h Wj W0i

S=

1

S

W1

S

W0

= W1

S2

W0

=

~G

1/ 2

S2 dx

1 . . . dx d W W 0 . ( 62)

Thus,~

G1/ 2

/ S 2 dx 1 . . . dx d is the volume element associ-ated with the scaled wavefunctions. The operatoradjoint taken with respect to this new inner productwill be dierent in general from the adjoint taken withrespect to the old inner product. To avoid confusion we

will denote the new adjoint by A y( S ) . These two adjointsare related by the following computation

h Wj^

Ay( S )

W0i

S= h

Â Wj W

0i

S=

Â W

1

S2

W0

= W^

Ay 1

S2 W

0= W S

2 Â

y 1

S2 W

0

S

,

( 63)

which summarizes as

Ay ( S ) = S 2A y 1

S2 . ( 64)

Equation (64) combines with equation (55) to yield

py( S )a = p

y

a- 2[p a ln S ]. ( 65)

The scaling of the wavefunction transforms thekinetic energy operator into ^T

S= S

^T ( 1/ S ) . Combining

equations (49) and (64), we nd

^T

S=

1

2

1S

py( S )a S G

a bS p b

1S

. ( 66)

By more or less straightforward commutation of opera-tors in the above equation and using equations (55) and(65), we arrive at the main result of this section

^T

S= 12 p

y( S )a G

a bp b + V S , ( 67)

where

VS

= - 12( Ga b

[p a ln S ][p b lnS ] + [py ( S )a G

a b[p b ln S ]])

= 12( Ga b [p a ln S ][p b ln S ] - [p

y

a Ga b [p b ln S ]]) . ( 68)

Comparing equation (67) with the unscaled expression(49), we note that the two operators dier by the addi-tional scalar term in equation (67) and the dierenadjoints which are used. Thus, scaling the wavefunctionmay be used to place the adjoint of the momenta in analternative, perhaps more attractive, form, but only athe expense of introducing an extrapotential term intothe kinetic energy.

4. The quantum kinetic energy of a molecular complexWe quantize the classical kinetic energy equation (27using equation (49) derived in the previous section. Thisapproach requires the operators ^p ,

^Ji, and their

adjoints. Since the classical momentum p is canonicallyconjugate to q, the quantized operator ^p has the usuaform of equation (47)

^p = - i

q. ( 69)

The quantized angular momenta ^Ji satisfy thestandard ànomalous commutation relations [^Ji,

^J

j] =

- i k eij k Jk and of course commute with all rotationallyinvariant operators, for example,

[^

Ji, M] = [^

Ji,~M] = [

^Ji, A] = [

^Ji, a] = [

^Ji, gt]

= [^

Ji, ht] = [^

Ji,^p] = 0. ( 70)

To compute the volume element of equation (53) werequire explicit coordinates covering all directions oconguration space. This means dening three Eulerangles [1, 2, 3] = [a, b, g], describing the collectiveorientation R, which complement the d - 3 internacoordinates q. Here, d = 3n + 3n n + 2n c - 3 is thedimension of the centre of mass system. We adopt theEuler angle conventions used in section 3. The noncanonical momenta p i =

^Ji and p = ^p -

^JA are

expressed in terms of the canonical momenta^p i = - i/

i and ^p by

^J

i

^p -^JA

=

Cj

i0

-

k

Ak

Cj

kd

t

^p j

^p t, ( 71)

where C ji

are the components of the matrix in equation(57) and the sum over j is implicit. The full d d matrix



9/11

in equation (71) corresponds to the matrix Cab in equa-

tion (50). Its determinant is equal to the determinant ofthe upper left block alone, that is det C= - 1/ sin b. Themetric Ga b with respect to the momenta p i = Ji andp = p - JA is seen from equation (27) to have de-terminant

G = gdetM, ( 72)

where g = detgt. The volume element is therefore

dv =G

1/ 2

j detCjda dbdgdq 1 . . . dq d-3

= ( gdetM) 1/ 2 sin bdadbdgdq 1 . . . dq d- 3

= 8p 2( gdetM) 1/ 2 dRdq 1 . . . dq d-3. ( 73)

An identity we will use later is the following alternativeexpression for G:

G = h det~M, ( 74)

where h is the determinant of ht. This identity followsfrom the fact that the change of basis connecting equa-tion (27) with equation (33) is orthogonal.

Since detM and g are rotationally invariant, theirpresence in dv is irrelevant for the computation of ^J

y

i .Therefore, the computation of ^J

y

i reduces to the case of asingle rigid rotor examined in section 3, from which werecall that ^J

y

i =^Ji. Therefore, from equations (27) and

(49), we nd^T = 12

^JM- 1^J + 12( ^p y - A

^J) gt( ^p t - At^J) . ( 75)

The ordering of the operators Ji with respect to theother factors is irrelevant, on account of equation (70).The ordering of the ^p with respect to g

t and A, how-ever, is essential. Note that the ^p are not in generalHermitian but rather satisfy

^py

=1

G1/ 2^p G

1/ 2 =1

( gdetM) 1/ 2^p( gdetM)

1/ 2, ( 76)

as seen from equation (54) and the fact that~

G = G/ ( detC) 2 = G( sin b) 2.We now scale the wavefunction by a factor

S = ( 8p 2) 1/ 2G1/ 4 = ( 8p 2) 1/ 2( detM) 1/ 4g1/ 4 ( 77)

to obtain a new form of the kinetic energy. First, wenote that the transformed volume element is

dv

S2 = dRdq

1 . . . dq d-3. ( 78)

The angular momenta ^Ji are still Hermitian with respect

to this new volume element, that is ^Jy( S )

i=

^J

i, as may benoted from equation (65) and the fact that S is rotation-ally invariant. However, since the new volume element

contains no q dependence in the Jacobian prefactor, wehave the added benet that ^p is now Hermitian, that is

^py( S ) =

^p. ( 79)

Therefore, the transformed kinetic energy of equation(67) takes the simple form

^T

S= 12

^JM-1^J + 12( ^p - A

^J) gt( ^p t - At^J) + V S ,

( 80)

where the extrapotential term may be reduced to

VS

= 12G- 1/ 4

qgt

qt

G1/ 4 . ( 81)

We observe that VS

is an R scalar, but not a q scalarTherefore, V S depends on the choice of internal coordi-nates, but not on the choice of CBF. Further discussionof this matter is given in [13].

The quantum kinetic energy also may be placed in aform analogous to equation (33). The unscaled kinetic

energy equation (75) becomes^T = 12(

^J -

^Ky) ~M

-1(

^J -

^K) + 12

^py

ht

p t, ( 82)

where

^K = hta

^p t. ( 83)

Similarly, the scaled kinetic energy becomes

^T

S= 12(

^J -

^Ky ( S )) ~M

- 1(

^J -

^K) + 12

^pht

p t + V S . ( 84)

5. Example: a monomeratom complexWe compute the rovibrational kinetic energy expli

citly for a system containing a single non-collinearrigid monomer with moment of inertia M1 and asingle atom, for example, ArNH3. The kinetic energyof such systems has been studied by Brocks and vanKoeven [1], van der Avoird [2], and Makarewicz andBauder [3]. Our presentation is mainly designed to illus-trate the formalism of the preceding sections, althoughwe believe that the derivation of the Coriolis potentiaA

i

, Coriolis eld strength Bi

t, and internal metric gt

is new.We dene the CBF by xing it to the rigid monomerThis implies that the matrix S, dening the orientationof the monomers IBF in the CBF, is constant. We takethis constant to be the identity,

S( q) = I. ( 85)

Since there is only one rigid body and one Jacobi vector,we drop all `a subscripts, except on M1, where the `1serves to distinguish the moment of inertia of themonomer from the total moment of inertia M of thecomplex. The Jacobi vector r locates the atom with



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respect to the monomer, and therefore its componentsmay be chosen as the internal coordinates, that is

[q 1, q 2, q 3] = [r 1, r 2, r 3]. ( 86)

Thus,

r i, = di. ( 87)

Furthermore, from equations (15) and (85), we have

s = 0. ( 88)

Inserting equations (85), (87), and (88) into (28)(30), weobtain

M= r 2I- rrT + Mi1, ( 89)

ai = ( r r,) i =

j

eijrj, ( 90)

ht = r,r,t = dt, ( 91)

where the i superscript on Mi

1 indicates that it is referredto the monomers IBF (which agrees here with the CBF)and hence is a constant matrix.

We proceed rst to construct the kinetic energyequation (33), which results here in a simpler formthan equation (27). Using equations (31) and (32) wecompute

~M= Mi1, ( 92)

K = r p, ( 93)

where p = [p1, p

2, p

3]. These are particularly simple

results and, together with equation (91), yield theclassical kinetic energy

T = 12( J - r p)(Mi1) -1( J - r p) + 12pp. ( 94)The quantum kinetic energy requires the further result

G = h det~M= detMi1, ( 95)

which follows from equation (74) and shows that G isconstant. Hence, from equation (76) it is clear that ^p isHermitian with respect to the original inner product.

Since

^

K = r

^

p we nd that

^

K also is Hermitian withrespect to both the original and the scaled innerproducts. Furthermore, the extrapotential term of equa-tion (81) arising in the scaled kinetic energy vanishes.Thus, both the original and the scaled quantum kineticenergies are identical and each is formed simply byreplacing p and J in equation (94) with p and ^J, respect-ively. Our results agree with earlier derivations byBrocks and van Koeven [1] and van der Avoird [2].

To simplify the algebra in constructing the kineticenergy equation (27), we assume the rigid body is aspherical top with Mi1 = I. The total moment of inertia

tensor given in equation (89) may be inverted explicitlywith the form

M- 1 =1

r 2 + I+

1

( r 2 + )rrT, ( 96)

and combined with equations (23) and (24) to yieldexplicit forms for the Coriolis potential and the internametric,

Ai

=j

1r

2 + eijrj, ( 97)

gt =1

r2 +

(dt + rr t) . ( 98)

The inverse of the internal metric is

gt =

1

[( r

2+ ) dt - rr t]. ( 99)

Equations (96), (97), and (99) combine with (27) to yieldan explicit form for the classical kinetic energy. As

earlier, the quantum kinetic energy, both original andscaled, is obtained simply by replacing p and J by theiroperator counterparts, without the need for Hermitianconjugates or an extrapotential term.

It is interesting to compute the Coriolis eld strengthdened by equation (42),

Bi

t = ets1

( r 2 + )2

( r irs + 2dis) . ( 100)

As the separation r of the atom from the monomer goesto innity, the Coriolis eld strength Bit ! etsr irs/ r

4

We change the CBF, as in equation (38), via a matrixU( r) which rotates z into r/ r . Then, since Bt is a rankoneR tensor, as r goes to innity, the new eld strengthtensor B 0t changes as

B 0t = UTBt ! ets

rs

r 3z. ( 101)

The above asymptotic form is that of a (non-Abelianmonopole eld [34]. A similar monopole eld is alreadyknown to exist in the three-body problem [15, 16], a facwhich has led to several useful applications [15, 16, 1935]. We remark that the above asymptotic form is valid

even if the monomer is an asymmetric top.

6. Conclusion

We have computed the kinetic energy of an arbitrarymolecular complex f or arbitrary coordinate and bodyframe conventions. In so doing, we have tried to providean ecient framework in which explicit Hamiltoniansmay be computed for specic choices of coordinatesand frames. We have provided a discussion of transfor-mation properties to facilitate the changing of these con-ventions. Our formalism is illustrated with the example



11/11

of a monomeratom system, and more complexsystems may be handled with similar ease within ourframework.

One of the more novel and intriguing aspects of ourderivation is the appearance of the Coriolis potentialand the various insights which are possible by adoptinga gauge theoretical viewpoint. We briey cite two areasof current research which are based on this perspective.

First, using gauge theoretical reasoning we havemanaged to generalize the Eckart conditions, so oftenemployed for small vibrations in molecules, to systemsof rigid bodies. Much of the formalism for small vibra-tional analysis in molecules then can be ported overreadily to study small amplitude vibrations in clustersof rigid molecules. Second, we have been able to under-stand rotational splittings in molecules with internalrotors as a sort of Coriolis AharonovBohm eect.These applications and others are planned for futurepublications.

The authors gratefully acknowledge stimulating dis-cussions with Professor R. J. Saykally and the astutereview of the manuscript by Dr M. Muller. This workwas supported by the US Department of Energy underContract No. DE-AC03-76SF00098.

References[1] Brocks, G., and va n Koeven, D., 1988, Molec. Phys.,

63, 999.[2] van der Avoird, A., 1993, J. chem. Phys., 98, 5327.[3] Makarewicz , J., and Bauder, A., 1995, Molec. Phys.,

84, 853.[4] Brocks, G., van der Avoird, A., Sutcliffe, B. T., and

Tennyson, J., 1983, Molec. Phys., 50, 1025.[5] v a n der Avoird, A., Wormer, P. E. S., and

Mosz ynski, R., 1994, Chem. Rev. , 94, 1931.[6] Xantheas, S. S., and Sutcliffe, B. T., 1995, J. chem.

Phys. , 103, 8022.[7] van der Avoird, A., Olthof, E. H. T., and Wormer,

P. E. S., 1996, J. chem. Phys. , 105, 8034.[8] Lister, D. G., Macdonald, J. N., and Owen, N. L.,

1978, Internal Rotation and Inversion (New York:Academic Press).

[9] Gordy, W., and Cook, R. L., 1984, MicrowaveMolecular Spectra (New York: Wiley).

[10] Menou, M., and Chapuisat, X., 1993, J. molecSpectrosc. , 159, 300.

[11] Gatti, F., Justum, Y., Menou, M., Nauts, A., andChapuisat, X., 1997, J. molec. Spectrosc. , 181, 403.

[12] Kaplan, L., Maitra, N. T., and Heller, E. J., 1997Phys. Rev. A, 56, 2592.

[13] Littlejohn, R. G., and Reinsch, M., 1997, Rev. modPhys., 69, 213.

[14] Guichardet, A., 1984, Ann. Inst. Henri Poincare, 40329.

[15] Iwai, T., 1987, J. math. Phys., 28, 964.[16] Iwai, T., 1987, J. math. Phys., 28, 1315.[17] Iwai, T., 1988, J. math. Phys., 29, 1325.[18] Shapere, A., and Wilczek, F., 1989, Amer. J. Phys., 57

514.[19] Montgomery, R., 1996, Nonlinearity, 9, 1341.[20] Shapere, A., and Wilcz ek, F., 1989, Geometric Phases

in Physics (Singapore: World Scientic).[21] Mead, C. A., 1992, Rev. mod. Phys., 64, 51.[22] Chiao, R. Y., and Wu, Y.-S., 1986, Phys. Rev. L ett., 57

933.

[23] Tomita,A., and Chiao, R. Y., 1986, Phys. Rev. L ett., 57937.[24] Littlejohn, R. G., 1988, Phys. Rev. A, 38, 6034.[25] Littlejohn, R. G., Mitchell, K. A., Aquilanti, V.

and Cavalli, S., 1998, Phys. Rev. A, 58, 3705.[26] Littlejohn, R. G., Mitchell, K. A., Reinsch, M.

Aquilanti, V., and Cavalli, S., 1998, Phys. Rev. A58, 3718.

[27] Aquilanti, V., and Cavalli, S., 1986, J. chem. Phys.85, 1355.

[28] Kroto, H. W., 1975, Molecular Rotation Spectra(Chichester: Wiley).

[29] PapouSek, D., and Aliev, M. R., 1982, MolecularVibrational-Rotational Spectra (Amsterdam: Elsevier).

[30] Nauts, A., and Chapuisat, X., 1985, Molec. Phys., 551287.

[31] Podolsky, B., 1928, Phys. Rev. , 32, 812.[32] Biedenharn , L. C., and Louck, J. D., 1981, Angular

Momentum in Quantum Physics (Reading, MAAddison-Wesley) p. 64.

[33] Chapuisat, X., Belafhal, A., and Nauts, A., 1991J. molec. Spectrosc. , 149, 274.

[34] Goddard, P., Nuyts, J., and Olive, D., 1977, NuclPhys. B, 125, 1.

[35] Mitchell, K. A., and Littlejohn, R. G., 1997, PhysRev. A, 56, 83.

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