christophe iung et al- vector parametrization of the n-atom problem in quantum mechanics with...

8/3/2019 Christophe Iung et al- Vector parametrization of the N-atom problem in quantum mechanics with non-orthogonal coordinates

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Vector parametrization of the N -atom problem in quantum

mechanics with non-orthogonal coordinates

Christophe Iung,a Fabien Gatti,a Alexandra Viela and Xavier Chapuisatb

a L aboratoire Structure et Dynamique des et Solides (UMR 5636) CC 014,Syste `mes Mole culaires

des Sciences et T echniques du L anguedoc, 34095 Montpellier Cedex 05, FranceUniversite b L aboratoire de Chimie (CNRS, URA 0506) Centre ScientiÐque dÏOrsay 490),T he orique (Ba üt

Paris-Sud, 91405 Orsay Cedex, FranceUniversite

Recei ved 30th April 1999 , Accepted 1st June 1999

This article aims to present a general method that enables one to build kinetic energy matrices in getting rid,for the angular coordinates (internal and Eulerian), of the heaviness of di†erential calculus (for expressingkinetic energy operators) and numerical integration (for calculating matrix elements). Therefore, instead of 3N [ 3 coordinates, only N [ 1 radial distances are to be treated as coordinates. In the present formulation,the system is described by any set of n vectors i \ 1, . . . , nN and the kinetic energy operator is expressed inMR

i,

term of (n [ 1) angular momenta i \ 1, . . . , n [ 1N and the total angular momentum J . The formalismMLi ,proposed is general and gives a remarkably compact expression of the kinetic energy in terms of the angularmomenta. This expression allows one to circumvent the seeming angular singularities.

I. Introduction

In previous articles,1 h 3 the continuous geometrical representa-tion of a deformable N-atom molecular system and the varia-tional determination of its dynamical states, bydiagonalization of the Hamiltonian matrix in an appropriatebasis, have been studied for a triatomic molecule2 or a mol-ecule parametrized by Jacobi vectors.1 This approach avoidsthe di†erential calculus steps usually required for expressingquantum mechanical kinetic energy operators in terms of curvilinear internal coordinates. Indeed, many sets of internalcoordinates can be considered3 h 20 and the larger the mol-ecule, the more diversiÐed the coordinate sets that all include3N [ 6 internal coordinates. Moreover, the rotation of thebody-Ðxed frame (BF) (whose axes rotate in a conventionalmanner when the atoms move), is measured by three Eulerangles in the space-Ðxed frame (SF).21 In ref. 1, a set of nJacobi vectors i \ 1, . . . , n) has been used to describe the(R

i,

molecule (n \ N [ 1). Vector has been taken parallel to theRn

GzBF axis whereas is parallel to the (xz)BF plane (G is theRn~1center of mass of the system). The (n [ 2) other vectors has

been viewed as totally free in BF. It is worth noticing that R

nand played a speciÐc role, being linked either totallyRn~1 (Rn)or partially to the deÐnition of the orientation of BF.(R

n~1)

Consequently, the BF components of the angular momentaassociated with and do not satisfy the usual proper-R

nRn~1ties of angular momenta (commutation relations and

hermiticity) and do not commute with the other angularmomenta associated with the vectors [i \ 1 , . . . , (n [ 2)]R

iand the total angular momentum J . This crucial point will bereferred to several times in this work.

In the present article, we generalize the results obtained inref. 1 by using a set of n vectors which are not Jacobi vectorsand thus could be parametrized by non-orthogonal coordi-nates, such as e.g. valence vectors. Our aim is to present ageneral method that enables, for all angular coordinates

(internal and Eulerian), the construction of kinetic energymatrices without resorting to (i) di†erential calculus forexpressing kinetic energy operators and (ii) numerical integra-

tion to calculate angular matrix element for any selectedvectors i \ 1, . . . , nN. Consequently, only n radial dis-MR

i;

tances must be treated numerically as coordinates instead of 3n coordinates when the angles are explicitly treated.

This work is based on the concept introduced in 1992 in ref.3, where a general expression of the kinetic energy has beengiven in terms of the BF-components of the angular momentaassociated with the vectors i \ 1 , . . . , n [ 1

Nused to

MRi

;describe the system. But the non-hermiticity of the BF-components of the angular momenta associated with andR

nand the non-commutation of them with the otherRn~1angular momenta has not been fully appreciated in ref. 3. For-

tunately, we demonstrate in this article that most terms gener-ated by non-hermiticity and non-commutation cancel eachother. Therefore, the quantization of the kinetic energy for Natoms adds only very few speciÐc terms, compared with theclassical expression of the kinetic energy.

This article also constitutes a generalization of ref. 2 whichhas been dedicated to a vector parametrization of three-atomsystems by valence coordinates. In the latter work thequantum expression of the kinetic energy has been establishedin terms of the angular momenta associated with two valence

vectors only. The quantization has led to one single speciÐcquantum term, i.e. which has no counterpart in the classicalexpression of the kinetic energy.

In Section 2, we present the quantum mechanical back-ground used to express the kinetic energy operator in terms of the momentum vectors i \ 1, . . . , nN conjugated to the nMP Œ

i;

vectors i \ 1, . . . , nN. In Section 3, the expressions of theMRi

;projections in BF of vectors i \ 1, . . . , nN are given, alongMP Œ

i;

with their adjoints, in terms of the BF components of theangular momenta i \ 1, . . . , n [ 1N of the (n [ 1) rotatingML Œ

i;

vectors i \ 1, . . . , n [ 1N and the total angular momen-MRi

;tum In Section 4, the results of Sections 2 and 3 are used forJ Œ.deriving a compact expression of the kinetic energy operatorin terms of i \ 1 , . . . , n [ 1N and This expression isML Œ

i; J Œ.

compared with its classical counterpart; the main di†erencearises from the fact that the frame axes depend on andR

nIn Section 5, we propose a particular basis set whichRn~1

.

Phys. Chem. Chem. Phys., 1999, 1, 3377 È3385 3377



It is worth mentioning that a Euclidian normalization isused,24,29 i.e. the elementary volume is :

dq\ Rn2 sin b dR

nda dbR

n~12 sin h

n~1BF dR

n~1

]dc dhn~1BF <

i/1

n~2Ri2 sin h

iBF dR

id/

iBF dh

iBF (12)

In that case, the operators and are given by(P Œir)s P Œ

ir P Œ

ir \

and:[iÅ LRi

(P Œi

r)s\ P Œi

r [2iÅ

Ri

(13)

Using eqn. (7), the BF components of the self-adjoint oper-ators (i \ 1, . . . , n [ 2) and the operator are given by:P Œ

iP Œn~1

P Œi\ (P Œ

i)s\

P Œir sin h

iBF cos /

iBF

]1

Ri

([sin hiBF sin /

iBFL Œ

izBF] cos h

iBFL Œ

iyBF)

P Œir sin h

iBF sin /

iBF

]1

Ri

([cos hiBFL Œ

ixBF] sin h

iBF cos /

iBFL Œ

izBF)

P Œir

cos hiBF

]1

Ri

([sin hiBF cos /

iBFL Œ

iyBF] sin h

iBF sin /

iBFL Œ

ixBF)

q nt tt tt tt tt tt tt t

t tt ts p

(14)

Because of the non-hermiticity of the y-BF component of [eqn. (10)], the x-BF component of is not self-L Œ

n~1P Œn~1adjoint.

(P Œn~1

)s \ en~1

Pn~1rs ]

L Œn~1s ] e

n~1Rn~1

(15)

A straightforward calculation provides the expression of in terms of (P Œ

n~1)s P Œ

n~1:

(P Œn~1

)s \ P Œn~1

]a [iÅRn~1

sin hn~1BF

0

0 b (16)

just in making use of the following commutators:

[L Œ(n~1)xBF

, cos hn~1BF ] \ 0 ; [L Œ

(n~1)xBF, sin h

n~1BF ] \ 0 (17)

[L Œ(n~1)yBF

, cos hn~1BF ] \ iÅ sin h

n~1BF ;

[L Œ(n~1)yBF

, sin hn~1BF ] \ [iÅ cos h

n~1BF (18)

[L Œ(n~1)zBF

, cos hn~1BF ] \ 0 ; [L Œ

(n~1)zBF, sin h

n~1BF ] \ 0 (19)

Finally, the vector is obtained by substituting by itsP Œn

L Œnexpression in terms of the other angular momenta:

1

Rn

AJ ŒyBF

[ ;i/1

n~1L ŒiyBF

B1

Rn

A[J Œ

xBF] ;

i/1

n~1L ŒixBF

BP Œnr

q nt tt t

P Œn

\ t t (20)t tt ts p

This equation, along with eqn. (8) È(10), leads to the expres-sion of the BF-components of P Œ

ns :

iÅ cot hn~1BF

Rn

P Œn

s \ P Œn

] 0 (21)

[2iÅ

Rn

q nt tt tt t

t tt ts p

Therefore, all the expressions of the BF-components of P Œiand (i \ 1 , . . . , n) are available. We are now able toP Œ

is

propose a compact expression of the kinetic energy operatorin terms of the BF components of and (i \ 1, . . . , n [ 1).J Œ L Œ

i

IV. Quantum expression of the kinetic energy in

terms of angular momenta

First of all, the vectors are substituted in the expression of P Œis

the kinetic energy operator [eqn. (4)] by their expressions

[eqn. (14), (16), (21)] which leads to :

2T Œ \ ;i,j/1

nM

i,jP Œi

Æ P Œj

(22)

] ;i/1

n C[M

i,n~1

iÅP ŒixBF

Rn~1

sin hn~1BF

] Mi,n

AiÅ cot hn~1BF P Œ

ixBF

Rn

[2iÅP Œ

izBF

Rn

BD(23)

The Ðrst term [eqn. (22)] can be identiÐed with the expres-sion of the kinetic energy in classical mechanics (which will becalled the ““classicalÏÏ kinetic energy), except for the BF com-ponents of and which do not commute with the BFP Œ

n~1P Œncomponents of vectors Consequently, the commutatorsP Œ

i

.must be taken into account when[P Œi ,

P Œj] \ P Œi Æ P Œj [ P Œj Æ P Œideriving the kinetic energy. The result is:

2T Œ \ ;i/1

nM

i,i(P Œ

i)2 [

iÅMn~1,n~1

Rn~1

sin hn~1BF

P Œ(n~1)xBF

]iÅM

n,nRn

(cot hn~1BF P Œ

nxBF[ 2P Œ

nzBF) (24)

] 2 ;i/1

n~1;

j;i,j/1

n~1M

i,jP Œi

Æ P Œj

] 2 ;i/1

n~2M

i,nP Œi

Æ P Œn

] 2Mn,n~1

P Œn

Æ P Œn~1

(25)

] ;

i/1

n~2M

i,n~1

A[

iÅP ŒixBF

Rn~1 sin hn~1BF

] [P Œn~1

, P Œi]

B(26)

] ;i/1

n~2M

i,nAiÅ cot h

n~1BF P Œ

ixBF

Rn

[2iÅP Œ

izBF

Rn

] [P Œn

, P Œi]B (27)

] Mn~1,n

AiÅ cot hn~1BF P Œ

(n~1)xBF

Rn

[2iÅP Œ

(n~1)zBF

Rn

[iÅP Œ

nxBF

Rn~1

sin hn~1BF

] [P Œn~1

, P Œn]B

(28)

Therefore, the fact that the operators and areP Œ(n~1)yBF

P ŒnyBF

non-hermitian and that some commutators are not[P Œi, P Œ

j]

equal to zero, generate the last two terms in eqn. (24) È(28). Weshall not examine eqn. (24) because it has been explicitlytreated in ref. 1, dedicated to the description of a molecule byJacobi vectors. We have established that eqn. (24) can berewritten in the following form:

;i/1

nM

i,i(P Œ

i)2 [

iÅMn~1,n~1

Rn~1

sin hn~1BF

P Œ(n~1)xBF

]iÅM

n,nRn

] (cot hn~1BF P Œ

nxBF[ 2P Œ

nzBF) \ ;

i/1

nM

i,i

C(P Œ

ir)2 [

2iÅP Œir

Ri

D] ;

i/1

n~1 AMn,n

Rn2

]M

i,iRi2

BL Œis Æ L Œ

i] ;

i/1

n~2;

j;i,j/1

n~1 AMn,n

L Œi

Æ L Œj

Rn2

B]

Mn,n

(J Œ Æ J Œ[ 2;i/1n~1 J Œ Æ L Œ

i)

Rn2

(29)

We have shown that the order of the operators in the pro-ducts that contain is not immaterial: placing the BFL Œn~1components of on the right hand side of all products isL Œ

n~1




particularly useful for simplifying the expression of the kineticenergy operator and its representation in the angular basis set.Therefore, it is worth noticing that the substitution of the ( P Œ

i)2

by their expression [eqn. (7)] in the classical formula ;i/1n

leads exactly to the quantum expression eqn. (29),Mi,i

(P Œi)2

except for which must be substituted for This isL Œis Æ L Œ

i(L Œ

i)2.

no longer true for i \ n [ 1 and n.Next, the quantization of the terms generated by the o†-

diagonal masses is undertaken. Consequently, terms (25) to(28) have to be calculated. The results are summarized below:

(i) Term (25) corresponds to the classical expression of thekinetic energy generated by the o†-diagonal masses. Vectormust be placed on the right of all scalar products whereP Œ

n~1it appears.(ii) Term (26) comes from the non-hermitian character of

and the non-commutation of with By means of P Œn~1

P Œi

P Œn~1

.the commutators given in Appendix 1, it can be shown thatthis term is equal to zero, which implies that:

P ŒisP Œ

n~1] P Œ

n~1s P Œ

i\ 2P Œ

iÆ P Œ

n~1(30)

Here again, the order of the operators in the scalar product isstrictly Ðxed.

(iii) Term (27) comes from the non-hermitian character of P Œnand the non-commutation of with With the help of theP Œ

iP Œn

.commutators given in Appendix 1, it can similarly be shownthat this term is equal to zero, i.e. :

P ŒisP Œ

n] P Œ

nsP Œ

i\ 2P Œ

iÆ P Œ

n(31)

(iv) Term (28) comes from the non-hermitian character of P Œnand and the non-commutation of with UsingP Œ

n~1, P Œ

n~1P Œn

.Appendix 1, it can be shown that:

Eqn. (28) \2iÅM

n~1,nRn

]G

[P Œn~1r cos h

n~1BF ]

[1 ] sin2(hn~1BF )]L Œ

(n~1)yBF

sin hn~1BF

H(32)

The fact that term (28) is not equal to zero means that non-hermiticity and non-commutation of the two operators gener-ates purely quantum terms.

It should be emphasized that the calculations leading toresults in eqn. (30) È(32) are long and tedious and that theorder of the operators in the scalar product is strictly Ðxed, i.e.is not immaterial.

The following equation establishes the relationship betweenthe ““classicalÏÏ expression of the kinetic energy and its““quantumÏÏ counterpart :

2T Œquant

\ 2T Œclass

]iÅM

n~1,n~1Rn~1

sin hn~1BF

P Œ(n~1)xBF

]iÅM

n,nRn

(cot hn~1BF P Œ

nxBF[ 2P Œ

nzBF) (33)

]2iÅM

n~1,nRn

G[P Œn~1r cos h

n~1BF ]

[1 ] sin2(hn~1BF )]L Œ

(n~1)yBF

sin hn~1BF R

n~1H

(34)

2T Œclass

\ ;i/1

nM

i,i(P Œ

i)2 ] 2 ;

i/1

n~1;

j;i,j/1

nM

i,jP Œi

Æ P Œj

] 2 ;i/1

n~2M

i,nP Œi

Æ P Œn

] 2Mn~1,n

P Œn

Æ P Œn~1

(35)

Consequently, the quantum expression contains only twoadditional terms generated by the diagonal masses and(M

n,nand two other terms coming from the o†-diagonalMn~1,n~1

)mass.M

n,n~1

Another long and tedious calculation that takes intoaccount the non-commutation of the operators, leads to thefollowing expression of the kinetic energy for an N-atom mol-

ecule in terms of BF components of angular momenta and theBF-angles and noted andh

iBF /

iBF h

i/i

(/n~1BF \ 0, h

nBF \ 0) :

T Œ J \ ;i/1

nM

i,i

A(P Œir)2

2[

iÅP Œir

Ri

B] ;

i,j/1‰i:j

nM

i,jMsin(h

i)sin(h

j)cos(/

i[/

j)

] cos(hi)cos(h

j)NP Œ

ir P Œ

jr

] ;i,j/1‰iEj

n~1

Mi,j sin(hi)sin(hj)sin(/i [/j)AP ŒirL Œ

jzRj B

] ;i,j/1‰iEj

n~1P Œir sin(h

i)AM

i,jcos(h

j)

Rj

[M

i,nRn

B]

Ae~iÕiL Œj` [ eiÕiL Œ

j~

2i

B[ ;

i/1

n~1 Mi,n

Rn

P Œir sin(h

i)Ae~iÕiL Œ

i` [ eiÕiL Œ

i~

2i

B[ ;

i,j/1‰iEj

n Mi,j

Ri

P Œjr cos(h

j)sin(h

i)Ae~iÕiL Œ

i` [ eiÕiL Œ

i~

2i

B] ;

i,j/1‰i:j

n~1

GM

i,j

RiRj

[cos(hi)cos(h

j)

] 12

e~iÕisin(hi)eiÕjsin(h

j)]

[M

i,nRiRn

cos(hi) [

Mj,n

RjRn

cos(hj) ]

Mn,n

2Rn2

HAL Œi`L Œ

j~

2

B] ;

i,j/1‰i:j

n~1 GMi,j

RiRj

[cos(hi)cos(h

j)

] 12

eiÕi sin(hi)e~iÕjsin(h

j)]

[M

i,nRiRn

cos(hi) [

Mj,n

RjRn

cos(hj) ]

Mn,n

2Rn2

HAL Œi~L Œ

j`

2

B[ ;

i,j/1‰i:j

n~1 Mi,j

4Ri

Rj

sin(hi)sin(h

j)

] (e~i(Õi`Õj)L Œi`L Œ

j` ] ei(Õi`Õj)L Œ

i~L Œ

j~)

] ;i/1

n~1 GMn,n

2Rn2

]M

i,i2R

i2

[M

i,ncos(h

i)

RiRn

HL ŒisL Œ

i

] ;i,j/1‰i:j

n~1 GMi,j

RiRj

sin(hi)sin(h

j)cos(/

i[/

j) ]

Mn,n

2Rn2

HL Œiz

L Œjz

] ;i,j/1‰i:j

n~1sin(h

j)A

[M

i,jRiRj

cos(hi) ]

Mj,n

RjRn

B]

Ae~iÕjL Œi` ] eiÕjL Œ

i~

2

BL Œjz

] ;i,j/1‰i:j

n~1

sin(hi)A[

Mi,j

RiRj

cos(hj) ]

Mi,n

RiRnB

]Ae~iÕiL Œ

izL Œj`] eiÕiL Œ

izL Œj~

2

B] ;

i/1

n~1 Mi,n

Rn

Pir sin(h

i)Ae~iÕiJ Œ̀ [ eiÕiJ Œ~

2i

B] ;

i/1

n~1 [Mi,n

sin(hi)

RiRn

Ae~iÕiJ Œ̀ ] eiÕiJ Œ~2

BL Œiz

]GM

i,ncos(h

i)

RiRn

[M

n,nRn2

HAJ Œ̀ L Œi~ ] J Œ~L Œ

i`

2

B[ ;

i/1

n~1

AM

n,nJ ŒzL Œiz

Rn2 B]

Mn,n

J Œ2

2Rn2

] ;i/1

n~1 ÅMi,n

RiRn

] sin(hi)Ae~iÕiL Œ

i` [ eiÕiL Œ

i~

2B (36)

3380 Phys. Chem. Chem. Phys., 1999, 1, 3377 È3385



The advantage of this equation compared to some developedexpression previously proposed by other authors in the case of polyatomics is to be as compact and general as possible. Thislater point will be discussed in Section 6. The physicalmeaning of each term is clear because of the presence of theangular momenta. Therefore, if we compare this equation tothose obtained by substituting, in the classical expression

vectors for their expression [eqn.T Œ \ 12;i,j/1n P Œ

iM

i,jP Œj, P Œ

i(7)], there is only one purely quantum term, namely ;i/1n~1

under the condition(ÅMi,n

/ RiRn)sin(h

iBF)(e~iÕiL Œ

i` [ eiÕiL Œ

i~)/2,

that the BF-components of are appropriately placed onL Œn~1the right of each product.In a previous work dedicated to triatomic molecules,2 the

quantization of the classical expression of the kinetic energyprovided an additional term, which(iÅM

1,2/ R

1R

2)sin(h

1BF) L Œ

1y,

corresponds to the extra term obtained in the general formulaif n \ 2 and were denoted in this(/

2BF \ 0). M

1,2, R

1, R

2h

1BF

article as M, r, R and a, respectively. Therefore, the quantumand classical expressions are very similar only if the non-commutating operators are placed in the di†erent products ina prescribed order.

V. Angular basis functions: integration over

angular coordinates, matrix representationWe now focus our attention on the action of the kinetic oper-ator on an angular basis set that described the various rota-tions of the molecule, parametrized by three Euler angles (a\

and the spherical angles of the vectorshnSF , b\/

nSF , c\/

n~1E2 )

(i \ 1, . . . , n [ 1) viewed in BFRi

(hiBF , /

iBF).

Moreover, in the absence of external Ðeld, SF is isotropic,i.e. the orientation of zSF is arbitrary, so that any observablemust be a-independent.30 The overall rotation of the moleculecan thus be described by the following basis set:31 h 32

S(a), b, c o J, 0, XT\ Y JX (b, c)([1)X (37)

where X is the projection of on GzBF.J ŒThe current element of the working angular function basis

for the BF spherical angles of vectors (i \ 1, . . . , n [ 1) isRigiven by:

S(a), b, c, hn~1BF , /

1BF , h

1BF, . . . ,

/n~2BF , h

n~2BF oX, l

n~1, l

1, X

1, . . . , l

n~2, X

n~2TJ

\ Y JX (b, c)([1)X P

ln~1X ~X 1~ÕÕÕ~X n~2[cos(h

n~1BF )]

]Y l1X 1(h

1BF , /

1BF) . . . Y

ln~2X n~2(h

n~2BF , /

n~2BF ) (38)

where is a normalized associated Legendre func-PlX [cos(h)]

tion times ([1)X and /) is a spherical harmonicsY lX (h,

Y lX (h, /) \P

lX [cos(h)](1/ J 2p)eiX Õ.

The current element of this working angular basis set is,hereafter, denoted by MSangles o . . .T

JN.

The action of the kinetic energy operator on the angularbasis functions requires the use of the basic formulae given inAppendix 2 for the angular momenta (i \ 1, . . . , n [ 2) asL

iwell as the speciÐc formula established in ref. 1 [eqn. (73a)],given below for L Œ

(n~1)BBF, L Œ

(n~1)zBF:

L Œ(n~1)BBF

MSangles o . . .TJN

\ ÅcB

(ln~1

, Xn~1

)S(a), b, c o J, 0, XT

Pln~1X ~(:i

n/~

12

X i)B1[cos(hn~1BF )]

] S/1BF , h

1BF , . . . , /

n~2BF , h

n~2BF o l

1, X

1, . . . , l

n~2, X

n~2T

(39)

L Œ

(n~1)

zBFMSangles o . . .T

JN

\ iÅAX[ ;i/1

n~2XiBMSangles o . . .T

JN (40)

Actually, no real difficulties are generated by the fact thatthe operators and (i \ 1 , . . . , n [ 2) do not satisfyL Œ

n~1L Œisimilar relationships. In order to illustrate this important

point, the following calculation shows how one of the morecomplicated terms of the kinetic energy, [M

i,n~1/4R

iRn~1(i \ 1, . . . , n [ 2) acts upon thesin(h

iBF)sin(h

n~1BF )e~iÕiL Œ

i`L Œ

n~1` ]

angular basis function:

sin(hiBF)sin(h

n~1BF )e~iÕiL Œ

i`L Œ

n~1` MSangles o . . .T

JN

\ c`

(ln~1

, Xn~1

)sin(hiBF)sin(h

n~1

BF )

] e~iÕiL Œi`S(a), b, c o J, 0, XT

Pl(n~1)X ~(:i

n/~

12

X i)`1[cos(hn~1BF )]

]S/1BF , h

1BF , . . . , /

n~2BF , h

n~2BF o l

1, X

1, . . . , l

n~2, X

n~2TJ

\ c`

(ln~1

, Xn~1

)c`

(li, X

i)sin(h

iBF)sin(h

n~1BF )

] e~iÕiS(a), b, c o J, 0, XT

Pl(n~1)X ~(:i

n/~

12

X i)`1[cos(hn~1BF )]

]S/1BF , h

1BF , . . . , /

n~2BF , h

n~2BF o l

1, X

1, . . . , l

i, X

i] 1, . . .T

J

\ c`

(ln~1

, Xn~1

)c`

(li, X

i)S(a), b, c o J, 0, XT

] sin(hn~1

BF )Pl(n~1)

X ~(:in/~

12X i)`1[cos(h

n~1

BF )]

] MB`~(l

i, X

i] 1)

]S/1BF , h

1BF , . . . , /

n~2BF , h

n~2BF o l

1, X

1, . . . , l

i] 1, X

i, . . .T

J

] B~~(l

i, X

i] 1)

]S/1BF , h

1BF , . . . , /

n~2BF , h

n~2BF o l

1, X

1, . . . , l

i[ 1, X

i, . . .T

JN

\ c`

(ln~1

, Xn~1

)c`

(li, X

i)

] MB`~(l

i, X

i] 1)B

`~(l

n~1, X

n~1] 1)

]S . . . o . . . , ln~1

] 1, Xn~1

, . . . , li] 1, X

i, . . .T

J

] B`~(l

i, X

i] 1)B

~~(l

n~1, X

n~1] 1)

]S . . . o . . . , l

n~1

[ 1, Xn~1

, . . . , l

i

] 1, Xi

, . . .TJ

] B~~(l

i, X

i] 1)B

``(l

n~1, X

n~1] 1)

]S . . . o . . . , ln~1

] 1, Xn~1

, . . . , li[ 1, X

i, . . .T

J

] B~~(l

i, X

i] 1)B

~`(l

n~1, X

n~1] 1)

]S . . . o . . . , ln~1

[ 1, Xn~1

, . . . , li[ 1, X

i, . . .T

JN (41)

In the latter expressions, the notation introduced in Appendix2 has been used.

We now calculate the matrix representing the kineticT Œ J,energy operator in the angular basis set TheMSangles o . . .T

JN.

integration is over the angles only, [i.e. the matrix elementsare expressed in terms of J, X, M, (i \ 1, . . . , n [ 2], andl

i, X

ion the one hand, and (i \ 1, . . . , n) on the otherln~1

Ri

P Œir

hand. J and M \ 0 are Ðxed, i.e. is a diagonal block of T ŒJ

T Œat constant J. In the basis using an approachMSangles o . . .TJN,

similar to that used for eqn. (41), it is rather easy but long toestablish the matrix elements . . . ,S . . .X

i{ , l

i{ , X

j{ ,

. . . , which are non-zero. Inspec-lj{ . . . o T Œ J o . . .X

i, l

i, X

j, l

j. . . T

tion of eqn. (36) reveals that the non-zero matrix elementsfulÐl the following conditions:

(i) *X\X[X@\ 0, ^1^1 (#i \ 1, 2, . . . , n [ 2)(ii) *X

i\X

i[X

i{ \ 0,

. . . , n [ 2)(iii) *X\*Xi

]*Xj

(#iD j \ 1,^1 (#i \ 1, 2, . . . , n [ 1)(iv) *l

i\ l

i[ l

i{ \ 0,

Consequently, the non-zero matrix elements . . . ,S . . .Xi{ , l

i{ ,

. . . , with i\ j are given by theXj{ , l

j{ . . . o T Œ J o . . .X

i, li, X

j, lj. . .T

following formulae in which only the modiÐed quantumnumbers are mentioned. Therefore, represents either ] or

ei[, and X) \c`

(J, X) \ Å2J J(J ] 1) [X(X] 1) c~

(J,

Å2J J( J ] 1) [X(X[ 1).




(i) Diagonal terms of T Œ J

S . . .Xi, li, . . . , X

j, lj. . . o T Œ J o . . .X

i, li, . . . , X

j, lj

. . .T

\ ;i/1

nM

i,i

A(P Œir)2

2[

iÅP Œir

Ri

B] ;

i/1

n~1Å2l

i(li] 1)

AMn,n

2Rn2

]M

i,i2R

i2

B] ;

i,j/1‰i:j

n~1 Mn,n

Å2XiXj

2Rn2

[ ;

i/1

n~1 Mn,n

Å2XXi

Rn

2]

Mn,n

Å2J( J ] 1)

2Rn

2

Let us note that

Xn~1

\X[ ;i/1

n~2Xi

(42)

(ii) Non-diagonal terms for *X\ 0, *Xi\ 0 ; *l

i\ ^1

S . . .Xi, li] (e

i1 ) . . . o T Œ J o . . .X

i, li, . . .T

\G

[M

i,nP Œnr

2iRi

[M

i,nP Œir

2iRn

]ÅM

i,n2R

iRn

H] Mc

`(li, X

i)Bei~(l

i, X

i] 1) [ c

~(li, X

i)Bei`(l

i, X

i[ 1)N

]

GM

i,nP ŒirP Œ

nr [

Mi,n

R

i

R

n

Å2li(li] 1)

HBei0(l

i, X

i) (43)

(iii) Non-diagonal terms for *X\ ^1, *Xi\ ^1; *l

i\ 1

or [1

SX^ 1, . . .Xi

^ 1, li

] (ei1 ) . . . o T Œ J o . . .X

i, li, . . .T

\G<

Mi,n

P Œir

2iRn

[M

i,nXi

2RiRn

HcB

( J, X)BeiB(li, X

i)

]M

i,n2R

iRn

Bei0(li, X

i^ 1)c

B(li, X

i)cB

( J, X) (44)

(iv) Non-diagonal terms for *X\ ^1, *Xi

\ ^1 ; li

\ 0

SX^ 1, . . .Xi

^ 1, li. . . o T Œ J o . . .X

i, li, . . .T

\ [M

n,n2R

n2 cB(li , Xi)cB( J, X) (45)(v) Non-diagonal terms for *X\ 0, *X

i\ [*X

j\ ^1 ;

*li\*l

j\ 0

S . . .Xi^ 1, l

i, . . . , X

j< 1, l

j. . . o T Œ J o . . .X

i, li, . . . , X

j, lj. . .T

\ cB

(li, X

i)cY

(lj, X

j)

Mn,n

4Rn2

(46)

(vi) Non-diagonal terms for *X\ 0, *Xi\*X

j\ 0 ; *l

i\

or [1^1 ; *lj

\ 1

S . . .Xi, li

] (ei1), . . . , X

j, lj

] (ej1 ) . . . o T Œ J o . . .X

i, li, . . . , X

j, lj. . .T

\ Mij

P ŒirP ŒjrBei0(li , Xi)Bej0(lj , Xj) [ MijP Œjr

2iRi

] c`

(li, X

i)Bei~(l

i, X

i] 1)Bej0(l

j, X

j)

[M

ijP Œir

2iRj

c`

(lj, X

j)Bej~(l

j, X

j] 1)Bei0(l

i, X

i)

]M

ijP Œjr

2iRi

c~

(li, X

i)Bej0(l

j, X

j)Bei`(l

i, X

i[ 1)

]M

ijP Œir

2iRj

c~

(lj, X

j)Bej`(l

j, X

j[ 1)Bei0(l

i, X

i)

]M

ij4R

iRj

Bei~(li, X

i] 1)Bej`(l

j, X

j[ 1)c

`(li, X

i)c

~(lj, X

j)

]M

ij4R

iRj

Bei`(li, X

i[ 1)Bej~(l

j, X

j] 1)c

~(li, X

i)c`

(lj, X

j)

[M

ij4R

iRj

Bei~(li, X

i] 1)Bej~(l

j, X

j] 1)c

`(li, X

i)c`

(lj, X

j)

[M

ij4R

iRj

Bei`(li, X

i[ 1)Bej`(l

j, X

j[ 1)c

~(li, X

i)c

~(lj, X

j)

(47)

(vii) Non-diagonal terms for *X\ 0, *Xi\ [*X

j\ ^1 ;

*li

\ ^1 ; *lj

\ 0

S . . .Xi^

1, li]

(ei1), . . . , Xj< 1, l

j. . . o T Œ J o . . .X

i, li, . . . , X

j, lj. . . T

\ ^P ŒirM

i,n2iR

n

BeiB(li, X

i)cY

(lj, X

j)

[C M

i,n2R

iRn

cB

(li, X

i)cY

(lj, X

j)Bei0(l

i, X

i^ 1)

]M

i,n2R

iRn

ÅXicY

(lj

, Xj)BeiB(l

i, X

i)D

(48)

(viii) Non-diagonal terms for *X\ 0, *Xi\ [*X

j\ ^1 ;

*li

\ 0 ; *lj

\ ^1

S . . .Xi ^ 1, li , . . . , Xj< 1, lj] (e

j1 ) . . . o T Œ J o . . .X

i, li, . . . , X

j, lj. . . T

\G<

P ŒjrM

j,n2iR

n

]M

j,nÅX

j2R

jRn

HcB

(li, X

i)BejY(l

j, X

j)

[M

j,n2R

jRn

cB

(li, X

i)cY

(lj, X

j)Bej0(l

j, X

j< 1) (49)

(ix) Non-diagonal terms for *X\ 0, *Xi\ [*X

j\ ^1 ;

*li

\ ^1 ; *lj

\ ^1

S . . .Xi^ 1, l

i] (e

i1), . . . , X

j< 1, l

j

] (ej1 ) . . . o T Œ J o . . .X

i, li, . . . , X

j, lj. . . T

\ Mij2 GP Œ

irP Œ

jr ^ P ŒirÅXj

iRj

<P ŒjrÅXiiR

i

] Å2XiXjRiRjH

]BeiB(li, X

i)BejY(l

j, X

j)

]M

ij2R

i

G<iP Œ

jr [

ÅXj

Rj

HcB

(li, X

i)Bei0(l

i, X

i^ 1)BejY(l

j, X

j)

]M

ij2R

j

G^iP Œ

ir [

ÅXi

Ri

HcY

(lj, X

j)Bej0(l

j, X

j< 1)BeiB(l

i, X

i)

]M

ij2R

iRj

cB

(li, X

i)cY

(lj, X

j)Bei0(l

i, X

i^ 1)Bej0(l

j, X

j< 1)

(50)

VI. Example and discussion

The conÐguration of an N-atom molecule can be described byN [ 1 relative position vectors after elimination of the center-of-mass motion. Many sets of coordinates commonly used fordescribing molecules can be viewed as spherical coordinatesfor these vectors. The spherical angles are local, i.e. they aredeÐned for frames which change from one vector to another.The coordinates actually consist of (i) the N [ 1 vectorlengths, (ii) N [ 2 planar angles between pairs of vectors, (iii)N [ 3 dihedral angles between two vectors around a thirdone, and (iv) three Euler angles orienting the BF frame with

respect to the SF frame. These 3N [ 3 coordinates are thelocal spherical coordinates for the N [ 1 vectors. This articlehas aimed at exploiting this type of parametrization.




Fig. 1 (AB)CD(EF) system parametrized by 5 valence vectors.

The Ðrst innovation of the method proposed in this paper isto build a particularly compact expression of the kineticenergy. Moreover, in this expression, each term possesses itsown physical meaning and is individually hermitian. Thisapproach is absolutely general and can be applied to anysystem. In order to show how it is applied in a particular case,we shall consider the system (AB)CD(EF) (Fig. 1) because thedetermination of its kinetic energy operator (for J \ 0) hasbeen the subject of a recently published work.20 In that work,Rempe and Watts have followed the method initiated byHandy22 and used valence coordinates to describe the molec-

ular deformations. They have actually applied, with the helpof Mathematica,34 the chain rule twice and obtained the devel-oped expression of the kinetic energy containing more than400 terms, all of them being not individually hermitian. Let usapply our approach to this system with the same coordinates,i.e. the spherical coordinates of the valence vectors rep-R

iresented in Fig. 1. Each vector is parametrized by sphericalRicoordinates and respectively bond lengths, bondR

i, h

i/i,

angles and torsion angles. The correspondence betweenRempe et al.Ïs valence coordinates and the spherical coordi-nates of used in our approach is easily established sinceR

iRnand are andR

n~1CD m CE m .

Fig. 2 The 5 Jacobi vectors used in the calculation of the matrix M .

Fig. 3 Valence vectors and used to parametrize and(R1, R2 R3) CH3Jacobi vectors and describing the motion of A and B in the(R4

R5)

course of an reaction.SN2

(x, y, z) Rempe axes ] (y, x, [z) axes in this paper

h1Rempe , h

2Rempe , h

3Rempe , h

4Rempe , h

5Rempe

]h1, p[h

2, h

3, p[h

4, h

5In order to derive a compact expression of the kinetic energyoperator with the angular momentum coupling terms in eqn.(36), we have to determine the M matrix [eqn. (5)] for(AB)CD(EF). We follow the method proposed in Section 2.First a set of 5 Jacobi vectors i \ 1, . . . , 5) has to be(r

i,

deÐned (Fig. 2). The A matrix relating the valence vectors (Ri ,i \ 1, ..., 5) parametrizing the system to the Jacobi vectorsselected (Fig. 2) is :

R1

[m

Bm

AB

0 [m

EFm

ABEF

[1m

Dm

CD

r 1

R2

0 [m

Em

EF

mAB

mABEF

[1 [m

Cm

CD

r 2

R3

\m

Am

AB

0 [m

EFm

ABEF

[1m

Dm

CD

r 3

(51)

R4

0m

Fm

EF

mAB

mABEF

[1 [m

Cm

CD

r 4

R5 0 0 0 0 1 r 5

q n q nq nt t t tt tt t t tt tt t t tt tt t t tt tt t t tt tt t t tt tt t t tt tt t t tt tt t t tt tt t t tt t

s p s ps p

where and are respectively the massesmA

, mB

, mC

, mD

, mE

mFof A, B, C, D, E and F while m

AB\ m

A] m

B, m

EF\ m

E] m

F,

and Next,mCD

\ mC

] mD

mABEF

\ mA

] mB

] mE

] mF

.from eqn. (5), the following M matrix is obtained.

kAC

01

mC

0 kDF

0

M \1

mC

0 kBC

0 1m

D

0

1

mC

[1

mD

1

mC

qtttttttttts

01

mC

1

mD

[1

mD

01

mC

(52)

kDE

[ 1m

D

[1

mD

kCD

nttttttttttp

where

kij

\1

mi

]1

mj

(i, j \ A, B, C, D, E, F).

Consequently, the expansion including more than 400 termsproposed by Rempe and Watts can be factorized by using eqn.(36), with n \ 5 and the masses given in eqn. (52). TheM

ijcompact character of the present method is thus clearly illus-trated in this particular case.

The second innovation consists in obtaining analyticalexpressions for the action of all angular operators in the basisfunctions in an adequate representation. Simultaneously,getting rid of the angular singularities is a straightforwardmatter. The next step for completing the present formalism isthe demonstration of the fact that the treatment of the angularsingularities can be generalized to the radial singularities.25This generalization is required for studying e.g. the inversionmotion of the ammonia molecule.25 To this large amplitudemotion is associated the vector linking the center of mass of

to the nitrogen atom. The vector length is zero when theH3molecule is planar and the subsequent seeming radial singu-

larity can also be avoided. Moreover, our approach can beimproved by taking into account the permutation symmetries

of the system. A symmetry adapted basis can be deÐned. Allthese improvements have been used for studying the ammoniaspectroscopy.25




If there are no identical particles, the method introducedcan be straightforwardly used. The Ñexibility of our approachis also worth mentioning: any set of vectors can be used. Forinstance, for the study of a reactionSN

2(A ] CH

3B ]ACH

3] B), the combination of Jacobi and valence vectors illus-trated in Fig. 3 is advisable. To achieve the construction of matrix T J radial basis functions are selected. The integralsover the radial coordinates must be numerically calculated.From that point on, the rest of the work is numerical and thenumerical e†ort will clearly impose limits to the size of the

systems that can be actually treated. For systems free in alltheir deformation degrees of freedom, Ðve particles may betreated at present. If larger systems are considered, model con-straints must be introduced, such as freezing a part of thesystem. This subject, is to a large extent, still to be explored.In all cases, it should be emphasized that it is proÐtable tohave a quantum FBR (Ðnite basis representation) in which thekinetic energy matrix is sparse. Combined with a DVR (dis-crete variable representation) for the potential, the FBR thatwe propose constitutes an appropriate framework for futuredynamical studies of more-than-three particle molecules (seeref. 35 and ref. 36 for instance).

Acknowledgements

Claude Leforestier is warmly thanked for many fruitfulProf.discussions.

Appendix 1

In this appendix, the various commutators that are requiredfor the calculations in this article are given. They can be easilyobtained from the deÐnition of the BF-components (i \ 1,L Œ

i. . . , n [ 2).

[L Œix

, sin /i] \ iÅ cos2(/

i)cot h

i;

[L Œix

, cos /i] \ [iÅ cos /

isin /

icot h

i(53)

[L Œix

, sin hi] \ iÅ sin /

icos h

i;

[L Œix

, cos hi] \ [iÅ sin /

isin h

i(54)

[L Œiy

, sin /i] \ iÅ sin /

icot h

icos /

i;

[L Œiy

, cos /i] \ [iÅ sin2(/

i)cot h

i(55)

[L Œiy

, sin hi] \ [iÅ cos /

icos h

i;

[L Œiy

, cos hi] \ iÅ cos /

isin h

i(56)

[L Œiz

, sin /i] \ [iÅ cos /

i; [L Œ

iz, cos /

i] \ iÅ sin /

i(57)

[L Œiz

, sin hi] \ [L Œ

iz, cos h

i] \ 0 (58)

Appendix 2

This appendix puts together the various formulae required forapplication of the kinetic energy operator upon the basis func-tions given in eqn. (41) (see ref. 33).

cos(h)PlX (cos h) \ B

`0 (l, X)P

l`1X (cos h) ] B

~0 (l, X)P

l~1X (cos h)

(59)

sin hPlX (cos h) \ B

``(l, X)P

l`1X `1(cos h) ] B

~`(l, X)P

l~1X `1(cos h)

(60)

sin hPlX (cos h) \ B

`~(l, X)P

l`1X ~1(cos h) ] B

~~(l, X)P

l~1X ~1(cos h)

(61)

dPlX (cos h)

dh \ B0`

(l, X)P

lX `

1(cos h) ] B0~(l, X)P

lX

~1(cos h)

(62)

where

B`0 (l, X) \

S (l [X] 1)(l ]X] 1)

(2l ] 1)(2l ] 3);

B~0 (l, X) \

S (l [X)(l ]X)

(2l [ 1)(2l ] 1)(63)

B``(l, X) \ [

S (l ]X] 1)(l ]X] 2)

(2l ] 1)(2l ] 3);

B~`(l, X) \S (l [X)(l [X[ 1)

(2l [ 1)(2l ] 1)(64)

B`~(l, X) \

S (l [X] 1)(l [X] 2)

(2l ] 1)(2l ] 3);

B~~(l, X) \ [

S (l ]X)(l ]X[ 1)

(2l [ 1)(2l ] 1)(65)

B0`(l, X) \ 1

2J (l ]X] 1)(l [X) ;

B0~(l, X) \ [1

2J (l ]X)(l [X] 1) (66)

The following relationships are obtained for the spherical

harmonics:cos(h)Y

lX (h, /) \ B

`0 (l, X)Y

l`1X (h, /)

] B~0 (l, X)Y

l~1X (h, /) (67)

exp(i/)sin(h)Y lX (h, /) \ B

``(l, X)Y

l`1X `1(h, /)

] B~`(l, X)Y

l~1X `1(h, /) (68)

exp([i/)sin(h)Y lX (h, /) \ B

`~(l, X)Y

l`1X ~1(h, /)

] B~~(l, X)Y

l~1X ~1(h, /) (69)

To specify how the operators andL Œi`BF , L Œ

i~BF , L Œ

izBF(L Œ

i2)BF

(i \ 1, . . . , n [ 2) act upon the spherical harmonics, the follow-ing classical expressions for the kinetic momenta are used:

L ŒiBBFY liX i(hi , /i) \ ÅciB(li , Xi)Y liX iB1(hi , /i)

\ ÅJ li(li] 1) ^X

i(X

i] 1)Y

liX iB1(h

i, /

i) (70)

L ŒizBF

Y liX i(h

i, /

i) \ ÅX

iY liX i(h

i, /

i)

(L Œi2)BFY

liX i(h

i, /

i) \ Å2l

i(li

] 1)Y liX i(h

i, /

i) (71)

whereas, for the operators fulÐlling theJ Œ`BF , J Œ

~BF , J Œ

zBF , (J Œ2)BF

abnormal relationships, we have:

J ŒBBFY

JX (b, c) \ Åc

YY JX Y1(b, c)

\ ÅJ J( J ] 1)<X(X] 1)Y JX Y1(b, c) (72)

J ŒzBF

Y JX (b, c) \ ÅXY

JX (b, c) ;

(J Œ2)BFY J

X (b

,c) \ Å2J( J ] 1)Y

J

X (b

,c) (73)

References

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3 X. Chapuisat and C. Iung, Phys. Rev. A, 1992, 45, 6217.4 B. Podolsky, Phys. Rev., 1928, 32, 812.5 B. R. Johnson, J. Chem. Phys., 1983, 79, 1906.6 R. T. Pack and G. Parker, J. Chem. Phys., 1987, 87, 3888.7 R. T. Pack and G. Parker, J. Chem. Phys., 1989, 90, 3511.8 X. Chapuisat and A. Nauts, Phys. Rev. A, 1991, 44, 1328.9 Y. Justum, M. Menou, A. Nauts and X. Chapuisat, Chem. Phys.,

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