Phase-space invariants for aggregates of particles:Hyperangular momenta andpartitions of the classical kinetic energyVincenzo Aquilanti, Andrea Lombardi, and Mikhail B. Sevryuk Citation: The Journal of Chemical Physics 121, 5579 (2004); doi: 10.1063/1.1785785 View online: http://dx.doi.org/10.1063/1.1785785 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/121/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phase space geometry of dynamics passing through saddle coupled with spatial rotation J. Chem. Phys. 134, 084304 (2011); 10.1063/1.3554906 Multiple free energies from a single simulation: Extending enveloping distribution sampling to nonoverlappingphase-space distributions J. Chem. Phys. 128, 174112 (2008); 10.1063/1.2913050 Phase-space reaction network on a multisaddle energy landscape: HCN isomerization J. Chem. Phys. 123, 184301 (2005); 10.1063/1.2044707 Machine failure forewarning via phase-space dissimilarity measures Chaos 14, 408 (2004); 10.1063/1.1667631 Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics Chaos 7, 159 (1997); 10.1063/1.166252

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Phase-space invariants for aggregates of particles: Hyperangularmomenta and partitions of the classical kinetic energy

Vincenzo Aquilanti and Andrea LombardiDipartimento di Chimica, Universita` di Perugia, Via Elce di Sotto 8, I-06123 Perugia, Italy

Mikhail B. Sevryuka)

Institute of Energy Problems of Chemical Physics, The Russia Academy of Sciences, Leninski� prospect 38,Bldg. 2, Moscow 119334, Russia

~Received 25 May 2004; accepted 2 July 2004!

Rigorous definitions are presented for the kinematic angular momentumK of a system of classicalparticles~a concept dual to the conventional angular momentumJ), the angular momentumLj

associated with the moments of inertia, and the contributions to the total kinetic energy of thesystem from various modes of the motion of the particles. Some key properties of these quantitiesare described—in particular, their invariance under any orthogonal coordinate transformation andthe inequalities they are subject to. The main mathematical tool exploited is the singular valuedecomposition of rectangular matrices and its differentiation with respect to a parameter. Thequantities introduced employ as ingredients particle coordinates and momenta, commonly availablein classical trajectory studies of chemical reactions and in molecular dynamics simulations, and thusare of prospective use as sensitive and immediately calculated indicators of phase transitions,isomerizations, onsets of chaotic behavior, and other dynamical critical phenomena in classicalmicroaggregates, such as nanoscale clusters. ©2004 American Institute of Physics.@DOI: 10.1063/1.1785785#

I. INTRODUCTION

This paper continues a search forindicatorsof dynami-cal features of a system of particles, as arising out of mo-lecular dynamics simulations. The need for such indicatorsoriginates from the fact that the exploration of a variety ofprocesses in an aggregate of particles at the nanoscale levelusually requires extensive classical trajectory calculations~exact quantum mechanics being currently prohibitive!, andit is often very important to be able to characterize the ag-gregate structure and dynamics by focusing on a few keyquantities. More detailed motivations for a search for suchquantities and applications to dynamics of classical clustersare presented in our works.1,2 Our viewpoint pursued in theseworks was inspired by progress in the study of few-bodyprocesses in hyperspherical and related coordinates3,4 in mo-lecular quantum mechanics, so far mainly developed forthree-body bound states and reactions,5–8 leading to recentbenchmark computational results for triatomic reactions9,10

~first of all, the F1H2 exchange reaction! via the so calledhyperquantization algorithm.11–15 Advances for systems offour ~or more! particles have to be recorded,6,16–20 and al-though efficient numerical implementations require furtherstudy, the role played byhyperspherical harmonics, as eigen-functions of hyperangular momenta, appears decisive.

The hyperspherical parametrization of theN-body prob-lem has a long history dating back to the 1930s, but the basicevents were the introduction by Smith, at the end of the

1950s and at the beginning of the 1960s, of three interrelatedkey concepts.

~i! Kinematic rotations, which follow from a generaliza-tion of the notion of reaction skewing angle, familiar inchemical kinetics. The skewing angle had been introducedfor three-center reactions by Eyring and Polanyi21 at the be-ginning of the 1930s, while Smith generalized the underlyingalgebraic concept to four- and five-atom systems in 1959.22

This extension was subsequently detailed in our paper.3 Ref-erence 3 gave explicitly the algorithm for discrete transfor-mations among particle coupling schemes as sequences of‘‘rotations’’ in the abstract kinematic space and pointed at theuse of such rotations for the so-calledsymmetricor demo-cratic representation ofN-body systems for anyN>3. Theseprocedures involve an analysis in terms of continuous grouptransformations, well classified by now forN53,23 N54,24

and also forN>5.25

The concepts of thekinematic spaceandkinematic rota-tions are crucial for our study, so we recall here very brieflythe main idea of these notions. The positions ofN particlesare described byn vectorss1 ,..., sn wheren5N or n5N21 ~one may think of, e.g.,N radii vectors of the particles orN21 Jacobi vectors, or the Radau and the orthogonal localvariants!.3 Each vectorsa consists of three componentss1,a ,s2,a , s3,a (1<a<n). A rotation of theN-particle system inthe physical space~an ‘‘outer motion’’! corresponds to re-placing s1 ,..., sn with Rs1 ,..., Rsn , respectively, whereRPO(3) is a 333 orthogonal matrix. But there is an alterna-tive ~in fact, dual! way of describing the system of particles.Instead of consideringn vectors of length 3

s15~s1,1,s2,1,s3,1!, ..., sn5~s1,n ,s2,n ,s3,n!,a!Author to whom correspondence should be addressed. Electronic mail:sevryuk@mccme.ru

JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 12 22 SEPTEMBER 2004

55790021-9606/2004/121(12)/5579/11/$22.00 © 2004 American Institute of Physics

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one may use three vectors of lengthn,

S15~s1,1,s1,2,...,s1,n!, S25~s2,1,s2,2,...,s2,n!,

S35~s3,1,s3,2,...,s3,n!.

The vectorSi (1< i<3) consists of thei th components of allthe n vectorss1 ,..., sn . The vectorsS1 , S2 , S3 lie in then-dimensionalkinematic space. A kinematic rotationof thesystem in question, i.e., a rotation in this abstract space~an‘‘inner motion’’ !, means replacingS1 , S2 , S3 with QS1 ,QS2 , QS3 , respectively, whereQPO(n) is an n3n or-thogonal matrix.

In practice, instead ofSi5(si ,1 ,si ,2 ,...,si ,n), the kine-matic n-vectors are defined asSi5(c1si ,1 ,c2si ,2 ,...,cnsi ,n)wherec1 , c2 ,..., cn are certain fixed scaling coefficients thatdepend on the masses of the particles, see Sec. II A below.

~ii ! Hyperangular momenta, and specifically thegrandangular momentumwhich appeared in early hypersphericalapproaches to theN-body problem as a ‘‘generalized’’ angu-lar momentumlike operator acting in a 33(N21) hyper-space. Smith himself26 defined the grand angular momentumas a tensor in classical mechanics consisting of elements ofthe form

si ,asj ,b2sj ,bsi ,a .

However, his work was fully devoted to quantum mechanics,starting a search, which is still continuing, for the eigenfunc-tions of the corresponding operator, the hyperspherical har-monics ~see the reviews27,28 and the papers29–32 for recentachievements!. These functions can be easily constructed inthe asymmetric parametrization,4 where the scheme isadopted of choosing first two particles, then adding sequen-tially the other ones. In thesymmetricparametrization, whichtreats all particles ‘‘democratically,’’ one must look for har-monics which are eigenfunctions of the various hyperangularmomenta.

In the present paper, we define the angular momentumfor kinematic rotations, the so-calledkinematic angular mo-mentum, for classical mechanics. This notion is dual to theconventional angular momentum in the same sense as kine-matic rotations themselves are dual to conventional rotationsin the physical space, see Sec. II B below. Therefore, thekinematic angular momentum generalizes the correspondingquantum mechanical operator whose eigenfunctions consti-tute a basis set for collective motions of the particles.30,32

~iii ! Symmetric hyperspherical coordinates, which ex-ploit both kinematic rotations and hyperangular momentaand were introduced first for three bodies: in a plane bySmith33 and in the full three-dimensional space byZickendraht34 ~see also Ref. 35! and by Whitten and Smith36

~all variants of symmetric and asymmetric hyperspherical co-ordinates for three bodies are reviewed in Ref. 5, while Refs.2 and 37 discuss theelliptic ones, which establish a bridgebetween symmetric and asymmetric coordinates!. Four-bodysystems were handled by Zickendraht38 and by Ohrn andLinderberg.39 The symmetric coordinates can be introducedfor a general case ofN particles and any underlying spacedimension d by an approach that uses a singular valuedecomposition40–42 ~see Ref. 3, where a generalization was

presented of the previous work43 which had treated the caseof N.4 andd53). By this approach, the singular values aresimply related to the principal moments of inertia of thesystem~accordingly they are often referred to as principalaxis coordinates!, while the remaining coordinates are relatedto ordinary rotations in the physical space and kinematicrotations, see Sec. II C below.

Following the techniques justified in Refs. 1 and 2 andgeneral ideas of the hyperspherical parametrization, wepresent in this paper the precise definitions and basic prop-erties of several new instantaneous phase-space invariantsfor a classicalN-body system. In fact, some of these invari-ants have already been described in naive terms and ex-ploited in our note2 where their usefulness as sensitive andeasily calculated indicators of phase transitions and onsets ofchaotic behavior in classical nanoaggregates has been shown~for more traditional invariants, see Refs. 2 and 44–48, andreferences therein!. Two of the phase-space invariants de-fined in the present work are the kinematic angular momen-tum mentioned above and the angular momentum associatedwith the moments of inertia, other invariants are terms ofvarious partitions of the kinetic energy. These terms corre-spond to different modes of the motion of the particles. Theword invariant here means that the quantity under consider-ation is fixed under all the orthogonal coordinate transforma-tions in both the physical space and the kinematic space. Theword instantaneousindicates that the quantity in question isby no means an integral of the motion. For instance, there isno conservation law for the kinematic angular momentum, incontrast to the ordinary angular momentum.

The paper is organized as follows. Section II begins withan introduction of fundamental concepts such as positionmatrices, allowed coordinate frames, and privileged coordi-nate frames~Sec. II A!. The kinematic angular momentumKof a system of particles is defined in Sec. II B in a wayemphasizing the parallelism amongK, the conventional an-gular momentumJ, and Smith’s grand angular momentumL. Section II C is devoted to the main mathematical appara-tus for defining and evaluating partitions of the kinetic en-ergy of the system, namely, the singular value decompositionof matrices. In Sec. III, we define three different kinetic en-ergy partitions—thehyperspherical partition~Sec. III A!, theprojective partition~Sec. III B!, and thesingular value ex-pansion ~Sec. III C!—and point out their basic properties.Section IV summarizes the invariance features of the angularmomenta and of the terms of the energy partitions and con-cludes with remarks on particular cases, extensions, and ap-plications. The Appendix lists the values of all the quantitiesconsidered in the simplest two-body problem.

II. CLASSICAL HYPERANGULAR MOMENTUMTHEORY

In this section we discuss some key concepts whichserve as a background for the introduction of the energypartitions in Sec. III. We first define theposition matrixandthe important concepts ofallowed coordinate frameandprivileged coordinate frame. Then we considerhyperangularmomenta J, K, and L, pointing out their invariance underorthogonal coordinate transformations. Thesingular value

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decompositionis also presented as a mathematical tool for aclearer understanding of the hyperspherical coordinates, ofthe grouping of coordinates corresponding to different kindsof motion, and as a convenient practical way to calculatevarious quantities involved in the energy partitions.

A. Position matrices Z

Our concern will be systems ofN>2 classical particleswithout external forces. Denote bym1 ,..., mN the masses ofthe particles and byr1 ,..., rN , their radii vectors with re-spect to the center of mass. It is expedient to introduce themass-scaledradii vectors qa5(ma /M )1/2ra (1<a<N)where

M5 (a51

N

ma

is the total mass of the system. Using the vectorsqa insteadof ra simplifies the expression for the total kinetic energyTof the particles:

T51

2 (a51

N

mau rau25M

2 (a51

N

uqau2. ~1!

Note that

(a51

N

ma1/2qa5M 21/2(

a51

N

mara50. ~2!

Consider any Cartesian coordinate frame whose origin coin-cides with the center of mass of the system. The 33N matrixZ whoseath column (1<a<N) is constituted by the com-ponentsq1,a , q2,a , q3,a of the vectorqa in this coordinateframe will be called theposition matrix:

Z5S q1,1 q1,2 ¯ q1,N

q2,1 q2,2 ¯ q2,N

q3,1 q3,2 ¯ q3,N

D .

The choice of another Cartesian coordinate frame corre-sponds to the coordinate changeZ85RtZ where an orthogo-nal matrix RPO(3) describes the transformation of theframes and the superscript t designates transposing. On theother hand, one can also consider orthogonal transformationsof coordinate framesin the kinematic space. If the orthogo-nal transformation of the original Cartesian frame is per-formed in the kinematic space by a matrixQPO(N) thenthe corresponding coordinate change isZ85ZQ.

In the sequel, all the Cartesian coordinate frames withthe origin at the center of mass of the system of particles andall the coordinate frames obtained from these Cartesianframes by orthogonal transformations in the kinematic spacewill be referred to asallowed coordinate frames. The posi-tions of particles in any allowed frame are described by the33N position matrixZ, the position matrices in differentallowed frames are connected by coordinate changes of theform Z85RtZQ with RPO(3), QPO(N).

Among all the allowed coordinate frames, consider thoseobtained from conventional Cartesian frames by orthogonaltransformationsQ in the kinematic space of the form

Q5S Q11 ¯ Q1,N21 ~m1 /M !1/2

Q21 ¯ Q2,N21 ~m2 /M !1/2

¯ ¯ ¯ ¯

QN1 ¯ QN,N21 ~mN /M !1/2

D PO~N!.

We will call such framesprivileged coordinate frames. Thelast column of the position matrixZ in a privileged frame iszero due to Eq.~2!. While using a privileged coordinateframe, it is therefore more suitable to deal only with the firstN21 columns of the position matrix. The 33(N21) matrixconstituted by these columns will be referred to as there-duced position matrix. In contrast, the 33N position matri-ces in allowed coordinate frames will be sometimes calledthe full position matrices.

Thus, we have two alternative descriptions of a systemof N particles: by the 33N full position matrix in an arbi-trary allowed coordinate frame and by the 33(N21) re-duced position matrix in a privileged coordinate frame. Theprototypes of these two descriptions are, respectively, thecollection ofN radii vectors of the particles and the collec-tion of N21 Jacobi vectors. The mass-scaled radii vectorsconstitute the full position matrix in a Cartesian frame; simi-larly, the Jacobi vectors~for any Jacobi coupling scheme!,mass scaled in an appropriate way, constitute the reducedposition matrix in a suitable privileged frame. ForN52, the331 reduced position matrix is just

~m1m2!1/2r /M ,

wherer is the vector columnr12r2 ~or r22r1) joining thetwo particles. ForN53, properly mass-scaled Jacobi vectorsconstituting the 332 reduced position matrix are exempli-fied by

@~m11m2!m3#1/2

M S m1r11m2r2

m11m22r3D ,

F m1m2

~m11m2!M G1/2

~r12r2!.

However, it is not our intention to emphasize the differ-ence between the two descriptions of systems of classicalparticles outlined above. The reason is that all the importantmechanical quantities which will be defined in the sequel~the angular momenta and the terms of various kinetic en-ergy partitions! are independent of the particular choice of anallowed coordinate frame and~in the case of a privilegedframe! of whether they are determined on the basis of the fullposition matrix or the reduced position matrix. Such invari-ance properties will be summarized in Sec. IV below. So, wewill usually utilize the same term ‘‘position matrix’’ and thesame notationZ for both cases. For the number of the col-umns of this matrix, we will adopt the notationn, so thatn5N in the case of the full position matrix in an allowedcoordinate frame andn5N21 in the case of the reducedposition matrix in a privileged coordinate frame.

B. The angular momenta J , K , and L

The totalkinetic energyof a system ofN>2 classicalparticles described by the 33n position matrixZ is

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T5M

2Tr~ ZZt!5

M

2 (a51

n

(i 51

3

Zia2 ,

where Tr denotes the trace of a square matrix. Indeed, thisexpression is valid for the full position matrices in Cartesiancoordinate frames@see Eq.~1!# and is invariant not onlyunder orthogonal coordinate transformationsRPO(3) in thephysical space, but also under orthogonal coordinate trans-formationsQPO(n) in the kinematic space. The Newtonianequations of motion read

MZia52]U/]Zia , 1< i<3, 1<a<n,

whereU5U(Z) is the potential energy of the system.The quantity

r5@Tr~ZZt!#1/25F (a51

n

(i 51

3

Zia2 G1/2

is called thehyperradiusof the system of particles underconsideration. To hyperradiusr, there is conjugated the lin-ear momentum

Pr5M r5M

rTr~ZZt!5

M

r (a51

n

(i 51

3

ZiaZia .

The totalangular momentum Jof the system in questionis given by the formulas

J25 (1< i , j <3

Ji j2 , Ji j 5M (

a51

n

~ZiaZj a2Zj aZia!. ~3!

One may also callJ thephysicalangular momentum becauseit accounts forexternal rotationsof the system~rotations inthe physical three-dimensional space!. It is clear from Eq.~3!that each of the three componentsJ12, J13, J23 of J and,consequently, the angular momentumJ itself are invariantunder orthogonal coordinate transformationsQPO(n) in thekinematic space. Curiously, the trivial fact thatJ is also in-variant under orthogonal coordinate transformationsRPO(3) in the physical space is not so obvious from Eq.~3!.However, this fact follows immediately from another expres-sion for J,

J25M2 (a,b51

n

@Gab(1)Gab

(3)2Gab(2)Gba

(2)#, ~4!

where

Gab(1)5(

i 51

3

ZiaZib , Gab(2)5(

i 51

3

ZiaZib ,

Gab(3)5(

i 51

3

ZiaZib .

The dual concept is thekinematic angular momentum Kof the system which accounts for kinematic rotations. Wewill define it as

K25 (1<a,b<n

Kab2 , Kab5M(

i 51

3

~ZiaZib2ZibZia!,

~5!

or equivalently as

K25M2 (i , j 51

3

@D i j(1)D i j

(3)2D i j(2)D j i

(2)#, ~6!

where

D i j(1)5 (

a51

n

ZiaZj a , D i j(2)5 (

a51

n

ZiaZj a ,

D i j(3)5 (

a51

n

ZiaZj a .

It is evident from Eq.~5! that each of then(n21)/2 compo-nentsK12, K13,..., Kn21,n of K and, consequently, the kine-matic angular momentumK itself are invariant under or-thogonal coordinate transformationsRPO(3) in thephysical space. Equation~6! implies thatK is also invariantunder orthogonal coordinate transformationsQPO(n) in thekinematic space.

The notion of the kinematic angular momentum was firstintroduced in our paper1 but with another formula which isnot invariant under orthogonal coordinate transformations inthe physical space@and not dual to Eqs.~3! and ~4! for thephysical angular momentum#.

The grand angular momentumL of the system of par-ticles was defined by Smith26 as

L25M2 (i , j or i 5 j ,a,b

~ZiaZj b2Zj bZia!2

5M2

2 (a,b51

n

(i , j 51

3

~ZiaZj b2Zj bZia!2

5M2~V1V32V22!, ~7!

where

V15Tr~ZZt!5r2, V25Tr~ZZt!5rr5rPr /M ,

V35Tr~ ZZt!52T/M .

The grand angular momentumL is also invariant under or-thogonal coordinate transformations in both the physicalspace and the kinematic space. From the relation

L252Mr2T2r2Pr2

it follows immediately that

T5TL1Tr , TL5L2

2Mr2 , Tr5Pr

2

2M5

M r2

2

~cf. Ref. 26!. We will call TL the grand angular energyandTr the hyperradial energy.

Finally, following our previous work,1 introduce thequantities

TJ5J2

2Mr2 , TK5K2

2Mr2 .

The termTJ measures the contribution of the external rota-tions of the system~outer motions! to the total kinetic energyT. This term will be therefore called theouter angular en-ergy. Analogously, the termTK connects with the contribu-

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tion of the kinematic rotations~inner motions! to the totalkinetic energyT and will be called theinner angular energy.

C. Singular value decomposition of the positionmatrix

As a matter of fact, the outer angular energyTJ under-estimates the kinetic energy associated with external rota-tions of the system. For instance, suppose that the pairwisedistances between the particles are kept fixed, i.e., the systemrotates around its center of mass as a rigid body. ThenTr

50 andT5TL , but J is generically still smaller thanL inthis case~as the simplest examples show49! and, conse-quently,TJ is generically smaller thanTL , the opposite ofwhat one would probably expect. Similarly, the inner angularenergyTK underestimates the kinetic energy associated withkinematic rotations. Besides, the hyperradial energyTr issensitive to changes in the hyperradiusr, i.e., the ‘‘overallsize’’ of the system, but the three principal moments of iner-tia of the system may well vary in such a way thatr will befixed andTr will therefore vanish. For a more precise ac-count of the energy contributions from different kinds ofmotion in the system, an adequate mathematical apparatus isthe singular value decomposition~SVD! of matrices.40–42

Introduce the notationm5min$n,3%, i.e., let m53 forn>3 andm5n for n51 or n52. Any 33n matrix Z can bedecomposed as the product of three matrices

Z5DYXt, ~8!

whereDPO(3) is a 333 orthogonal matrix,XPO(n) is ann3n orthogonal matrix, and all the entries of the 33n ma-trix Y are zeroes, with the possible exception of the diagonalentries

Y115j1 , Y225j2 ,...,Ymm5jm , j1>j2>¯>jm>0.

It is the representation of Eq.~8! which is called the SVD ofmatrix Z.40–42The numbersj1 , j2 ,...,jm are called thesin-gular valuesof matrix Z and are determined uniquely al-though the factorsD andX in Eq. ~8! arenot. If n>3 thenthe singular values ofZ are the square roots of the eigenval-ues of the 333 symmetric matrixZZt. If n<3 then thesingular values ofZ are the square roots of the eigenvaluesof the n3n symmetric matrixZtZ.

The principal moments of inertiaI 1>I 2>I 3 of the sys-tem in question with respect to its center of mass can beexpressed in terms of the singular values of the position ma-trix Z in a remarkably simple way. Namely, ifn>3 then

I 15M ~j121j2

2!, I 25M ~j121j3

2!, I 35M ~j221j3

2!;

if n52 then

I 15M ~j121j2

2!, I 25Mj12 , I 35Mj2

2 ;

and if n51 then

I 15I 25Mj12 , I 350.

These formulas imply, in particular, that for systems of twoor three particles, the largest principal moment of inertia isthe sum of the two other moments. This relation is in factvalid for any flat body.33

If the numberN of particles is no greater than 3 andZ isthe full 33N position matrix then the last singular valuejN

of Z is necessarily zero. All the preceding singular valuesj1 ,...,jN21 of Z are generically positive, and in the sequel,we will assumefor simplicity that they are positiveindeed. Inall the other cases, all the singular values of the positionmatrix Z are generically positive, andwe will assume them tobe positive.

In all the cases except for reduced position matrices forsystems of four particles (N54, n53), the factorsD andXin the SVD of Eq.~8! can be chosen to bespecial orthogo-nal: DPSO(3),XPSO(n). In the caseN5n1154, a SVDof the 333 matrix Z with special orthogonal factorsD andX exists if and only if detZ.0. This fact is closely related tothe chirality issues:50 two configurations of four particleswith different signs of detZ are of different chiralities.

For n.3, the lastn23 columns of matrixX in Eq. ~8!are irrelevant (Z does not depend on them!, and the so-calledthin SVD of Z is more suitable. To be more precise, the 33n position matrixZ with n>3 can be decomposed as

Z5DJBt, ~9!

where DPO(3), J5diag(j1,j2,j3) is the diagonal 333matrix with the diagonal entriesj1 , j2 , j3 , andB is an n33 matrix with orthonormal columns~i.e., the three col-umns ofB are pairwise orthogonal and have unit length asvectors in then-dimensional Euclidean space!. One says thatthe matrixB belongs to theStiefel manifoldVn,3 .51–53 Therepresentation of Eq.~9! is called thethin SVD ~Ref. 42! ofZ. The practical recipe for obtaining this decomposition is asfollows. An orthogonal matrixDPO(3) is chosen to diago-nalize the 333 symmetric matrixZZt:

D tZZtD5diag~j12 ,j2

2 ,j32!.

The rowsv1 , v2 , v3 of the 33n matrix D tZ are pairwiseorthogonal~as vectors in then-dimensional Euclidean space!and their lengths arej1 , j2 , j3 , respectively. Ifj1>j2

>j3.0 then B is the n33 matrix with columnsv1 /j1 ,v2 /j2 , v3 /j3 . The situation wherej1>j2.0 and j350corresponds to the casen5N53. In this case, the first twocolumns of a 333 matrixB arev1 /j1 andv2 /j2 , while thethird column is chosen to ensure the orthogonality ofB.

For n,3, the last 32n columns of matrixD in Eq. ~8!are irrelevant, and it is again better to use the thin SVD ofZ.Namely, the 33n position matrixZ with n<3 can be de-composed as

Z5AJXt, ~10!

whereXPO(n), J5diag(j1,...,jn) is the diagonaln3n ma-trix with the diagonal entriesj1 ,..., jn , and A is a 33nmatrix with orthonormal columns~so thatA belongs to theStiefel manifoldV3,n , see Refs. 51–53!. The representationof Eq. ~10! is also called thethin SVD ~Ref. 42! of Z and canbe computed as follows. An orthogonal matrixXPO(n) is tobe chosen to diagonalize then3n symmetric matrixZtZ:

XtZtZX5diag~j12 ,...,jn

2!.

The columnsu1 ,..., un of the 33n matrix ZX are pairwiseorthogonal ~as vectors in the three-dimensional Euclidean

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space! and their lengths arej1 ,..., jn , respectively. Ifj1

>¯>jn.0 then A is the 33n matrix with columnsu1 /j1 ,..., un /jn . The situation wherej1>¯>jn21.0andjn50 corresponds to the casesn5N52 andn5N53.In these cases, the firstn21 columns of a 33n matrix A areu1 /j1 ,..., un21 /jn21 , while the last column is to be chosento be orthogonal tou1 ,..., un21 and to have unit length.

For square matrices, the thin SVD and the SVD are thesame, so that forn53, Eqs.~8!, ~9!, and ~10! represent thesame decomposition ofZ.

III. PARTITIONS OF THE KINETIC ENERGY

In this section, we show how to obtain energy partitions,following a representation of theN-particle system based onthe hyperangular momenta introduced in Sec. II. Partitions ofthe kinetic energy are particularly important towards a com-plete understanding of the different kinds of motion fromboth a qualitative and a quantitative point of view. Our par-titions come from the separation of degrees of freedom in-duced by the symmetric hyperspherical coordinaterepresentation.3,5,33–36,38,39,43Such a separation is expected tobe more physically motivated, in distinguishing differentkinds of motions, than the usual separation of molecular vi-brations and rotations, when dynamics is considered for clus-ters or reactions.

A. The hyperspherical partition of the kinetic energy

The hyperspherical partition, to be proposed in the fol-lowing, is strictly related to the grouping of hypersphericalcoordinates, each of them rigorously describing a particularkind of motion. So we will have thehyperradial motion,accounting for the ‘‘overall system size’’ variations,externalrotations~or outer angular motions!, andkinematic rotations~or inner angular motions!, see Sec. II B. On the other hand,as mentioned in Sec. II C, the three principal moments ofinertia may well vary in such a way thatr50, while theinertia is redistributing with respect to the three principalaxes. In order to be complete in our description, we will alsointroduce a term of the partition measuring the total varia-tions of the inertia obtained as the sum of two contributions,the hyperradial term for the total inertia, and a term account-ing for the redistribution.

The hyperradiusr of the system is expressed in terms ofthe singular valuesj1 ,..., jm of its 33n position matrix as

r25j121¯1jm

2 51

2M~ I 11I 21I 3!,

where I 1 , I 2 , I 3 are the principal moments of inertia. Thelinear momentumPr conjugated tor is therefore

Pr5M

r~j1j11¯1jmjm!

and the hyperradial energyTr is

Tr5M

2r2 ~j1j11¯1jmjm!2.

This quantity accounts for variations ofr. If the singularvalues of the position matrix of the system are changing in

such a way thatr50, i.e., j1j11¯1jmjm50, then Tr

50. The energy measuring arbitrary variations of the singu-lar values, i.e., arbitrary variations of the principal momentsof inertia, is

TI5M

2~ j1

21¯1 jm2 !.

We will call TI the inertial energy. The difference

Trot5T2TI

corresponds to the kinetic energy associated with all thekinds of rotation~both external and kinematic! in the systemand will be called therotational energy. There hold the in-equalities

0<Trot<TL<T, 0<Tr<TI<T.

In addition to the angular momentaJ, K, andL, definethe angular momentum Lj associated with the singular val-ues~or associated with the moments of inertia! as

Lj25 (

1<k,s<mLj,ks

2 , Lj,ks5M ~jkjs2jsjk!. ~11!

This quantity is also invariant under orthogonal coordinatetransformations in both the physical space and the kinematicspace. Let theshape energycorresponding toLj be

Tj5Lj

2

2Mr2

~cf. Ref. 1!. The shape energy measures the redistribution ofthe inertia around the three principal axes. It is easy to seethat

Tj5TI2Tr5TL2Trot.

Introduce also the difference

Tac5TL2TJ2TK2Tj5Trot2TJ2TK .

We will call Tac the angular coupling energy. Finally, wewill say that

T5TL1Tr5TJ1TK1Tj1Tac1Tr ~12!

is the hyperspherical partitionof the kinetic energyT ~seeFig. 1!.

The inequalities

J21Lj2<L2, K21Lj

2<L2

hold, so that

TJ1Tj<TL , TK1Tj<TL .

On the other hand, it is not hard to construct examples~forN>3) where

J21Lj25L2 and K.0

or

K21Lj25L2 and J.0.

Both situations may occur forLj50.54 Consequently, theangular coupling energyTac ~and even the sumTac1Tj) canbe negative.

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Usually the momentumLj and accordingly the energyTj are relatively small because the singular values are sub-ject to the conditionj1>j2>¯>jm ~generically j1.j2

.¯.jm for every moment of time! and the vector(j1 ,j2 ,...,jm) cannot rotate much.

B. The projective partition of the kinetic energy

The quantities that characterize adequately the energycontributions from the external and kinematic rotations takenseparately can be defined as follows. In the space of all the33n matricesZ, there act two orthogonal groups: the exter-nal rotation group SO(3) which acts as (R,Z)°RZ for anyRPSO(3), and thekinematic rotation group SO(n) whichacts as (Q,Z)°ZQt for any QPSO(n). Let Se(Z) be theorbit of Z under the action of the first group, i.e., the mani-fold of all the matrices of the formRZ, RPSO(3). LetalsoSk(Z) be the orbit ofZ under the action of the second group,i.e., the manifold of all the matrices of the formZQ, QPSO(n). Denote byPe(Z) the tangent space toSe(Z) atpoint Z and byPk(Z), the tangent space toSk(Z) at pointZ. The spacePe(Z) is constituted by all the matrices of theform Z1RZ with skew-symmetricRPso(3). The spacePk(Z) is constituted by all the matrices of the formZ1ZQ with skew-symmetricQPso(n).

Equip the space of all the 33n matrices with the scalarproduct according to the formula

^Za,Zb&5Tr@Za~Zb! t#5 (a51

n

(i 51

3

Ziaa Zia

b .

For a given position matrixZ, consider theorthogonal pro-

jections Ze and Zk ~in the sense of this scalar product! of itstime derivativeZ on Pe(Z) andPk(Z), respectively. Then

Text5M

2Tr@ Ze~ Ze! t#5

M

2 (a51

n

(i 51

3

~ Ziae !2

is the energy of the external rotation of the system at thegiven moment of time and will be called theexternal energy.Similarly,

Tint5M

2Tr@ Zk~ Zk! t#5

M

2 (a51

n

(i 51

3

~ Ziak !2

is the energy of the kinematic rotation of the system and willbe called theinternal energy. The difference

Tres5Trot2Text2Tint

will be called theresidual energy. Finally, we will call

T5Trot1TI5Text1Tint1Tres1TI ~13!

the projective partitionof the kinetic energyT ~see Fig. 1!because the termsText and Tint are defined via orthogonalprojections in the space of 33n matrices.

If the system rotates around its center of mass as a rigidbody, thenTI50 and

Text5Trot5TL5T,

as expected. The external energyText vanishes if and only ifJ50, and one can prove the inequalities

TJ<Text<Trot.

Moreover, letj* denote theminimal positivesingular valueof the position matrixZ (j* 5j3 in the case ofN>4 par-ticles andj* 5jN21 for systems ofN52 or N53 particles!.Then

Text5J2

2Mj*2

for N52,

Text<J2

2Mj*2

for N53, and

Text<J2

4Mj*2

for N>4.Analogously, the internal energyTint vanishes if and

only if K50, and the inequalities

TK<Tint<Trot

hold. Moreover,

Tint50

for N52,

Tint<K2

4Mj*2

for N53 or N54, and

Tint<K2

2Mj*2

FIG. 1. A schematic representation for the hyperspherical partition, the pro-jective partition, and the singular value expansion of the kinetic energy of asystem of particles. The names of the terms that can be negative are pre-sented initalics.

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for N>5.Finally, sinceText>TJ andTint>TK , one has

Trot2Text2Tint5Tres<Tac5Trot2TJ2TK .

In particular, the residual energyTres can be negative. Forinstance, if the system rotates around its center of mass as arigid body, thenTres52Tint, and genericallyTres,0.

The time derivatives of the singular values ofZ neededfor the computation of the hyperradial energyTr and theinertial energyTI can be found very easily. In terms of thethin SVD of Eq.~9! for n>3, the derivatives of the singularvalues ofZ are just the diagonal entries of the 333 matrixW5D tZB:41,55–60

j i5Wii , 1< i<3. ~14!

In terms of the thin SVD of Eq.~10! for n<3, the derivativesof the singular values ofZ are the diagonal entries of then3n matrix V5AtZX:41,55–60

ja5Vaa , 1<a<n. ~15!

The works41,55,56contain also formulas for thesecondderiva-tives of the singular values of a matrix depending on severalparameters. The second derivatives of the smallest singularvalue were also found in the earlier paper.61

The external energyText and the internal energyTint, inspite of their rather complicated definition, are very easy tobe calculated as well.

Let first n>3 and the 33n position matrixZ be decom-posed as in Eq.~9!. Then

Text5M

2 (1< i , j <3

~j iWji 2j jWi j !2

j i21j j

2 ,

Tint5M

2 (1< i , j <3

~j iWi j 2j jWji !2

j i21j j

2 1T2M

2 (i , j 51

3

Wi j2

~recall thatW5D tZB). The term

T2M

2 (i , j 51

3

Wi j2

in the expression forTint vanishes forn53.Suppose now that we are studying a system ofN53

particles in a privileged coordinate frame, and the corre-sponding 332 reduced position matrixZ is decomposed asin Eq. ~10!. Then

Text5M

2

~j1V212j2V12!2

j121j2

2 1T2M

2~V11

2 1V122 1V21

2 1V222 !,

Tint5M

2

~j1V122j2V21!2

j121j2

2

~recall thatV5AtZX).Finally, for N52 one has

Text5Trot5TL5TJ5J2

2Mr2

and

Tint5TK50

~the complete summary of the kinetic energy partitions forN52 is given in the Appendix!. It is interesting that therelationTint5TK is valid also forN53, although, of course,the internal and inner angular energies are generically posi-tive in this case.

C. The singular value expansion of the kinetic energy

Finally, we propose an expansion of the kinetic energyobtained directly from the SVD of the position matrixZ.Assume that the singular values of the position matrixZ arepairwise distinct. For generic trajectoriesZ(t) this conditionis met for every moment of time~an analogue of the vonNeumann–Wigner noncrossing rule for diatomic moleculeterms of the same symmetry type!.62 Then the factorsD andX in the SVD of Eq. ~8! of Z can be chosen to dependsmoothly on timet, and it becomes possible to define a onemore partition of the kinetic energy of the system. Namely,set

Zout5DYXt, Zin5DYXt.

It is not hard to prove that the matrix-valued functionsZout

andZin are independent of the particular choice of the factorsD(t) and X(t), provided that the singular values ofZ arepairwise distinct. Let

Eout5M

2Tr@ Zout~ Zout! t#5

M

2 (a51

n

(i 51

3

~ Ziaout!2,

Ein5M

2Tr@ Zin~ Zin! t#5

M

2 (a51

n

(i 51

3

~ Ziain !2,

Ecoupl5M Tr@ Zout~ Zin! t#5M (a51

n

(i 51

3

ZiaoutZia

in .

We will call Eout, Ein, and Ecoupl the outer term, the innerterm, and thecoupling, respectively. One can prove that thesum Eout1Ein1Ecoupl is equal to the rotational energyTrot.The relation

T5Trot1TI5Eout1Ein1Ecoupl1TI ~16!

will be referred to as thesingular value expansionof thekinetic energyT ~see Fig. 1! because the termsEout, Ein, andEcoupl are obtained directly from the SVD of the positionmatrix.

If the system rotates around its center of mass as a rigidbody, then

Eout5Trot5TL5T,

as expected. Moreover,Zin50 and henceEin5Ecoupl50 inthis case~note that genericallyTK.0, TacÞ0, Tint.0, andTres,0 for rotations of a rigid body!!. On the other hand, theequalityJ50 doesnot imply Eout50. However, the relations

Ein2Eout5Trot, Ecoupl522Eout

hold wheneverJ50, so thatEin>Eout, Ein>Trot, andEcoupl

<0 for J50. In the case of an arbitrary value ofJ, theinequality

5586 J. Chem. Phys., Vol. 121, No. 12, 22 September 2004 Aquilanti, Lombardi, and Sevryuk

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uEin2Eout2Trotu<J

j*S Eout

M D 1/2

holds forN>4 ~recall thatj* denotes the minimal positivesingular value of the position matrixZ, i.e., j* 5j3 for N>4).

Similarly, the equalityK50 does not implyEin50, butit implies the relations

Eout2Ein5Trot, Ecoupl522Ein,

so thatEout>Ein, Eout>Trot, andEcoupl<0 for K50. In thecase of an arbitrary value ofK, the inequality

uEout2Ein2Trotu<K

j*S Ein

M D 1/2

is valid for N<4 ~note that the minimal positive singularvaluej* of the position matrixZ is jN21 for N<4).

Typically in molecular simulations,Eout@Trot, Ein

@Trot, and, accordingly,Ecoupl,0 and2Ecoupl@Trot. That iswhy we call Eq.~16! an expansionrather than apartition.

The termsEout, Ein, andEcoupl can be calculated as fol-lows. Let firstn>3. Consider the thin SVD of Eq.~9! of theposition matrixZ and introduce the 333 matrix

G5D t~ ZZt1ZZt!D5D tZBJ1JBtZtD.

Then60

~D tD ! i j 5Gi j

j j22j i

2 , 1< i , j <3, iÞ j , ~17!

and

2j i j i5Gii , 1< i<3. ~18!

In fact, Eqs.~17! and ~18! are just a particular case of theso-called abstract Hellmann–Feynman theorem.63 Equation~18! provides an alternative expression for the time deriva-tives of the singular values ofZ, cf. Eq. ~14!. What is im-portant for computing the terms of the singular value expan-sion of T is the relations of Eq.~17!. Since DPO(3) isorthogonal at every moment of time,D tDPso(3) is skew-symmetric. Consequently, Eq.~17! determines the 333 ma-trix D tD completely. Now one can successively find

D5D~D tD !, Zout5DJBt, Zin5Z2Zout2DJBt

and, finally,Eout, Ein, andEcoupl.Let nown<3. Consider the thin SVD of Eq.~10! of the

position matrixZ and define then3n matrix

F5Xt~ ZtZ1ZtZ!X5XtZtAJ1JAtZX.

Then60

~XtX!ab5Fab

jb22ja

2 , 1<a,b<n, aÞb, ~19!

and

2jaja5Faa , 1<a<n. ~20!

Equations~19! and ~20! are again a particular case of theabstract Hellmann–Feynman theorem,63 and Eq. ~20! pro-vides an alternative expression for the time derivatives of the

singular values ofZ as compared with Eq.~15!. Now, sinceXPO(n) is orthogonal at every moment of time,XtXPso(n) is skew symmetric. Therefore, Eq.~19! determinesthen3n matrix XtX completely, so that one can successivelyfind

X5X~XtX!, Zin5AJXt, Zout5Z2Zin2AJXt

and, finally,Eout, Ein, andEcoupl.For alternative expressions for the derivatives of the left

and right factors in the SVD of matrices depending on pa-rameters, see Refs. 41, 55, 57, 58, and 64. The formulaspresented in Ref. 57 essentially coincide with those given inRef. 60 and used here in Eqs.~17! and ~19!. Various subtlemathematical issues on the SVD of parameter-dependent ma-trices are discussed in Refs. 60 and 64–67.

IV. CONCLUDING REMARKS

Summary of the paper. The main goal of this paper hasbeen the rigorous introduction of several new instanta-neously invariant quantities describing the motion of classi-cal particles in a~nano!aggregate. Two of these quantities arethe kinematic angular momentum@Eqs.~5! and ~6!#, dual tothe ordinary physical angular momentum@Eqs.~3! and ~4!#,and the angular momentum of Eq.~11! associated with thesingular values. The other quantities are energy characteris-tics of the motion, they constitute three partitions of the totalkinetic energy of the system and are summarized in Fig. 1.All the invariants defined in the paper depend only on thepositions and velocities of the particles~and not on the po-tential energy!. All our study agrees with~and is in fact in-spired by! the general lines of the hyperspherical parametri-zation of the multibody problem.

We have also given straightforward computational for-mulas for all the invariants introduced and presented someidentities and inequalities these invariants satisfy~the corre-sponding proofs will be given elsewhere!. Being based onthese formulas, we have developed and implemented effi-cient FORTRAN codes68 which calculate all the terms of thepartitions for a given trajectory of anN-body system.

All the analysis of the present paper can be easily carriedover to the case where the dimensiond of the physical spaceis other than 3. Such a generalization will be also developedelsewhere.

Summary of the invariance properties of the angular mo-menta and of the energy partitions. All the characteristics ofa classical system of particles we have considered in thepresent paper are ‘‘instantaneous invariants’’ of the system inthe sense explained in Sec. I. To be more precise, the follow-ing statement holds.

The angular momentaJ, K, L, and Lj given by Eqs.~3!–~7! and ~11!, the singular valuesjk , the total kineticenergyT of the system, the termsTL , Tr , TJ , TK , Tj , Tac

of the hyperspherical partition@Eq. ~12!# of the kinetic en-ergy T, the termsTrot, TI , Text, Tint, Tres of the projectivepartition @Eq. ~13!# of the kinetic energyT, and the termsEout, Ein, Ecoupl of the singular value expansion@Eq. ~16!# ofthe kinetic energyT are invariant under orthogonal coordi-nate transformations in both the physical space and the kine-

5587J. Chem. Phys., Vol. 121, No. 12, 22 September 2004 Aggregates of particles

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matic space. Moreover, all these quantities do not depend onwhether they are determined from the full position matrix inan allowed coordinate frame or the reduced position matrixin a privileged coordinate frame.

The invariance properties of the singular values requiresome comment. If the numberN of particles is less than 4then the corresponding 33N full position matrices possessN singular values while the 33(N21) reduced position ma-trices haveN21 singular values. In this case, the fact thatthe singular values are independent of the type of the posi-tion matrix is to be understood as follows: ifj1 ,..., jN21 arethe singular values of any reduced position matrix describingthe system at a given moment of time then the singular val-ues of any full position matrix arej1 ,..., jN21 , jN withjN50.

Applications. So far, FORTRAN codes68 calculating theenergy partitions have been used for various systems and theresults are encouraging. As was emphasized in Sec. I, someof the phase-space invariants introduced in this paper can beused as indicators ofcritical phenomena in classical~nano!aggregates~in the sense that these invariants undergosharp changes during various critical events—such as phasetransitions—in the evolution of the system! and of differentdynamical regimes. This has been precisely the concern ofour more recent works, where examples of energy partitionsfor trajectories for rare gas clusters, Ar3 ,2 Ar13,2,69 andAr38,69 have been discussed with the aim of finding correla-tions between the features of the partitions and signatures ofphase transitions and chaotic behavior. As a further explana-tory application of our theory, we also recently consideredclassical trajectory simulations of the prototypical reactionF1H2:69 the results obtained regarding the terms of the par-titions allow us to provide a picture of the reactive and in-elastic processes in the collision of a fluorine atom with thehydrogen molecule.

In conclusion, we believe that the sphere of applicabilityof the invariants is wider. Indeed these new invariants areintroduced from a phase-space approach to the dynamics, asubject which has persistently been given careful attention asa valid perspective in the general field of many-bodydynamics70,71 and in chemical reaction theory inparticular.72–74 Hopefully, they can help us to describe en-ergy transfer processes in aggregates of particles and to un-derstand the energy partition among various degrees of free-dom. Since this theory provides a measure of the energycontributions from various kinds of motion, making feasiblea search for approximate conservation laws for the new in-variants, we are also encouraged to consider their quantummechanical analogs, as well as the corresponding energy lev-els and eigenstates, in order to simplify the task of the quan-tum extension of the theory.

ACKNOWLEDGMENTS

The authors are grateful to E. Yurtsever for many stimu-lating discussions. The authors acknowledge the supportfrom the EU through the COST D9 Action. This work hasalso been financed by the Italian MIUR~Ministerodell’Istruzione dell’Universita` e della Ricerca!, ENEA ~Enteper le Nuove Tecnologie, l’Energia e l’Ambiente!, and ASI

~Agenzia Spaziale Italiana!. The work of M.B.S. was sup-ported by the Italian Ministry of Foreign Affairs via a fel-lowship organized by the Landau Network—Centro Voltaand ~in part! by the Council for grants of President of theRussia Federation, Grant No. NSh-1972.2003.1.

APPENDIX: THE CASE OF TWO PARTICLES

It is instructive to list the values of all the quantitiesconsidered in the present paper in the simplest case of twoparticles (N52). Let r be the vector joining the particlesand r its length~the distance between the particles!. Then

r5j15~m1m2!1/2r /M5~m/M !1/2r

wherem5m1m2 /M is the reduced mass of the particles,

Pr5~m1m2!1/2r 5~mM !1/2r ,

J5L5mu@r , r #u,

where@r , r # is the standard vector product of the two vectors,

K5Lj50,

TI5Tr5m r 2

2,

Eout5Text5Trot5TJ5TL5mu@r , r #u2

2r 2 ,

Ein5Ecoupl5Tint5Tres5TK5Tj5Tac50,

and

T5mu r u2

2.

1V. Aquilanti, A. Lombardi, and E. Yurtsever, Phys. Chem. Chem. Phys.4,5040 ~2002!.

2V. Aquilanti, A. Lombardi, M. B. Sevryuk, and E. Yurtsever, Phys. Rev.Lett. ~accepted for publication!. See also: http://bong.chm.unipg.it/CVs/aquila.html/Partitions/prl/.

3V. Aquilanti and S. Cavalli, J. Chem. Phys.85, 1355~1986!.4V. Aquilanti, S. Cavalli, and G. Grossi, J. Chem. Phys.85, 1362~1986!.5V. Aquilanti, S. Cavalli, G. Grossi, and R. W. Anderson, J. Chem. Soc.,Faraday Trans.86, 1681~1990!.

6A. Kuppermann, inNew Methods in Quantum Theory, edited by C. A.Tsipis, V. S. Popov, D. R. Herschbach, and J. S. Avery~Kluwer, Dor-drecht, 1996!, p. 501.

7A. Kuppermann, J. Phys. Chem.100, 2621~1996!; 100, 11202~E! ~1996!.8V. Aquilanti, G. Capecchi, and S. Cavalli, Adv. Quantum Chem.36, 341~2000!.

9V. Aquilanti, S. Cavalli, D. De Fazio, A. Volpi, A. Aguilar, X. Gime´nez,and J. M. Lucas, Phys. Chem. Chem. Phys.4, 401 ~2002!.

10V. Aquilanti, S. Cavalli, D. De Fazio, A. Volpi, A. Aguilar, and J. M.Lucas, Chem. Phys.~to be published!.

11V. Aquilanti, S. Cavalli, and D. De Fazio, J. Chem. Phys.109, 3792~1998!.

12V. Aquilanti, S. Cavalli, D. De Fazio, A. Volpi, A. Aguilar, X. Gime´nez,and J. M. Lucas, J. Chem. Phys.109, 3805~1998!.

13V. Aquilanti, S. Cavalli, D. De Fazio, and A. Volpi, Adv. Quantum Chem.39, 103 ~2001!.

14V. Aquilanti, S. Cavalli, D. De Fazio, and A. Volpi, Int. J. Quantum Chem.85, 368 ~2001!.

15D. De Fazio, S. Cavalli, and V. Aquilanti, Int. J. Quantum Chem.93, 91~2003!.

16V. Aquilanti, L. Bonnet, and S. Cavalli, Mol. Phys.89, 1 ~1996!.17V. Aquilanti and S. Cavalli, J. Chem. Soc., Faraday Trans.93, 801~1997!.18A. Kuppermann, J. Phys. Chem. A101, 6368~1997!.

5588 J. Chem. Phys., Vol. 121, No. 12, 22 September 2004 Aquilanti, Lombardi, and Sevryuk

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

141.209.100.60 On: Sat, 20 Dec 2014 12:10:10

19V. Aquilanti, A. Beddoni, S. Cavalli, A. Lombardi, and R. G. Littlejohn,Mol. Phys.98, 1763~2000!.

20K. Museth and A. Kuppermann, J. Chem. Phys.115, 8285~2001!.21H. Eyring and M. Polanyi, Z. Phys. Chem. Abt. B12, 279 ~1931!.22F. T. Smith, J. Chem. Phys.31, 1352~1959!.23R. G. Littlejohn, K. A. Mitchell, V. Aquilanti, and S. Cavalli, Phys. Rev. A

58, 3705~1998!.24R. G. Littlejohn, K. A. Mitchell, M. Reinsch, V. Aquilanti, and S. Cavalli,

Phys. Rev. A58, 3718~1998!.25K. A. Mitchell and R. G. Littlejohn, J. Phys. A33, 1395~2000!.26F. T. Smith, Phys. Rev.120, 1058~1960!.27J. S. Avery, inConceptual Trends in Quantum Chemistry, edited by E. S.

Kryachko and J. L. Calais~Kluwer, Dordrecht, 1994!, p. 135.28V. Aquilanti, S. Cavalli, C. Coletti, D. De Fazio, and G. Grossi, inNew

Methods in Quantum Theory, edited by C. A. Tsipis, V. S. Popov, D. R.Herschbach, and J. S. Avery~Kluwer, Dordrecht, 1996!, p. 233.

29D. Wang and A. Kuppermann, J. Chem. Phys.115, 9184~2001!.30V. Aquilanti, A. Beddoni, A. Lombardi, and R. G. Littlejohn, Int. J. Quan-

tum Chem.89, 277 ~2002!.31D. Wang and A. Kuppermann, J. Phys. Chem. A107, 7290~2003!.32V. Aquilanti, A. Lombardi, and R. G. Littlejohn, Theor. Chem. Acc.111,

400 ~2004!.33F. T. Smith, J. Math. Phys.3, 735 ~1962!.34W. Zickendraht, Ann. Phys.~Leipzig! 35, 18 ~1965!.35W. Zickendraht, Phys. Rev.159, 1448~1967!.36R. C. Whitten and F. T. Smith, J. Math. Phys.9, 1103~1968!.37V. Aquilanti and S. Tonzani, J. Chem. Phys.120, 4066~2004!.38W. Zickendraht, J. Math. Phys.10, 30 ~1969!.39Y. Ohrn and J. Linderberg, Mol. Phys.49, 53 ~1983!.40R. A. Horn and C. R. Johnson,Matrix Analysis, 2nd ed. ~Cambridge

University Press, Cambridge, 1990!.41G. W. Stewart and J.-G. Sun,Matrix Perturbation Theory~Academic,

Boston, 1990!.42G. H. Golub and C. F. Van Loan,Matrix Computations, 3rd ed.~Johns

Hopkins University Press, Baltimore, 1996!.43W. Zickendraht, J. Math. Phys.12, 1663~1971!.44R. J. Hinde, R. S. Berry, and D. J. Wales, J. Chem. Phys.96, 1376~1992!.45P. V. Coveney and A. N. Chaudry, J. Chem. Phys.97, 7448~1992!.46R. J. Hinde and R. S. Berry, J. Chem. Phys.99, 2942~1993!.47B. M. Smirnov, Phys. Usp.37, 1079~1994!.48Y. Zhou, M. Karplus, K. D. Ball, and R. S. Berry, J. Chem. Phys.116,

2323 ~2002!.49Suppose that a rigid body with the principal moments of inertiaI 1>I 2

>I 3 ~with respect to its center of mass! rotates around its principle axiscorresponding to, say,I 1 with angular velocityv.0. ThenJ5I 1v andT5I 1v2/2, i.e., J252I 1T. On the other hand,L252Mr2TL5(I 11I 2

1I3)TL5(I11I21I3)T in this particular case, since 2Mr25I 11I 21I 3 ~seeSec. III A!. GenericallyI 21I 3.I 1 andL.J.

50P. G. Mezey,Shape in Chemistry: An Introduction to Molecular Shape andTopology~VCH, New York, 1993!.

51I. M. James,The Topology of Stiefel Manifolds~Cambridge UniversityPress, Cambridge, 1976!.

52D. B. Fuks and V. A. Rokhlin,Beginner’s Course in Topology. GeometricChapters~Springer, Berlin, 1984!.

53A. Edelman, T. A. Arias, and S. T. Smith, SIAM J. Matrix Anal. Appl.20,303 ~1998!.

54Let I 15I 21I 3 in the set-up of Ref. 49@this is the case if the body in

question is flat~Ref. 33!#. ThenJ5L5I 1v andLj50. On the other hand,

one can compute thatK52(I 2I 3)1/2v in this particular case. Generically

I 2.0, I 3.0, andK.0.55J.-G. Sun, J. Comput. Math.6, 258 ~1988!.56J.-G. Sun, J. Comput. Math.6, 325 ~1988!.57K. Wright, Numer. Math.63, 283 ~1992!.58M. A. Wicks and R. A. DeCarlo, SIAM J. Matrix Anal. Appl.16, 123

~1995!.59L. Qiu, B. Bernhardsson, A. Rantzer, E. J. Davison, P. M. Young, and J. C.

Doyle, Automatica J. IFAC~International Federation of Automatic Con-trol! 31, 879 ~1995!.

60L. Dieci and T. Eirola, SIAM J. Matrix Anal. Appl.20, 800 ~1999!.61G. W. Stewart, Numer. Linear Algebra Appl.56, 231 ~1984!.62Even if the initial configuration of the particles~at t50) is symmetric~cf.

Ref. 1! and some singular values ofZ(0) therefore coincide, the singular

values ofZ(t) are generically pairwise distinct for anytÞ0.63V. Aquilanti, S. Cavalli, and M. B. Sevryuk, J. Math. Phys.35, 536

~1994!.64A. Bunse-Gerstner, R. Byers, V. Mehrmann, and N. K. Nichols, Numer.

Math. 60, 1 ~1991!.65W. Govaerts and B. Werner, Numer. Math.70, 303 ~1995!.66J.-L. Chern and L. Dieci, SIAM J. Matrix Anal. Appl.22, 772 ~2000!.67M. Baumann and U. Helmke, Future Generation Computer Systems19,

353 ~2003!.68Available from the authors.69V. Aquilanti, E. Carmona Novillo, E. Garcia, A. Lombardi, M. B. Sevryuk,

and E. Yurtsever, Comput. Mater. Sci.~accepted for publication!. See also:http://bong.chm.unipg.it/CVs/aquila.html/Partitions/examples/.

70W. P. Reinhardt, Prog. Theor. Phys. Suppl.116, 179 ~1994!.71R. G. Littlejohn and M. Reinsch, Rev. Mod. Phys.69, 213 ~1997!.72P. Pechukas and J. C. Light, J. Chem. Phys.42, 3281~1965!.73J. C. Light, Discuss. Faraday Soc.44, 14 ~1967!.74S. Wiggins, L. Wiesenfeld, C. Jaffe´, and T. Uzer, Phys. Rev. Lett.86, 5478

~2001!; T. Uzer, C. Jaffe´, J. Palacian, P. Yanguas, S. Wiggins, Nonlinearity15, 957 ~2002!.

5589J. Chem. Phys., Vol. 121, No. 12, 22 September 2004 Aggregates of particles

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