A single continuum description of molecular dissociations

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A single continuum description of molecular dissociations

J. Robert, J. Baudon

To cite this version:

J. Robert, J. Baudon. A single continuum description of molecular dissociations. Journal de Physique,

1986, 47 (4), pp.631-638. �10.1051/jphys:01986004704063100�. �jpa-00210242�

A single ^continuum description of molecular dissociations

J. Robert and J. Baudon

Laboratoire de Physique des Lasers (*),

Université Paris-Nord, Avenue J. B. Clément, ⁹³⁴³⁰ Villetaneuse, ^France

(Reçu le 12 août 1985, révisé le 28 octobre, accepte ^{le 6 decembre} 1985)

Résumé.

On rappelle ^la définition des coordonnées d’Eckart _pour ^un système ^{de N} particules, ainsi que l’expres-

sion hamiltonienne de l’énergie cinétique classique. L’importance ^{du Jacobien dans la} définition du domaine de

configuration ^est soulignée. ^{On montre en outre} que la fonction d’onde ne souffre d’aucune singularité ^{à la frontière}

de ce domaine. Les dissociations du système en p sous-systèmes (2 ~ p ~ N) peuvent ^être décrites, ^en coordonnées d’Eckart, ^dans l’unique continuum r1 ^~ oo, où _r1 est la plus grande ^des coordonnées d’Eckart radiales. L’iden- tification d’une voie de dissociation se fait au moyen d’un ^vecteur unitaire de RN-1, 0393, appelé ^« pointeur chimique» :

dans une dissociation p-aire, ^{r a} pour lieu ^une variété d’ordre p - ^{2. On construit une} algèbre ^{de matrices de Gram}

décrivant les partitions ^du système ^en plusieurs sous-systèmes ; ^un développement ^{en ordre} par rapport ^{à la coor-} donnée du continuum unique permet d’expliciter ^le comportement ^{de 0393 au} voisinage ^{d’un lieu de} dissociation.

Abstract.

The definition of Eckart coordinates for a N-particle system ^{and the} Hamiltonian form of the classical kinetic _energy are given. ^The important ^role of the Jacobian in the definition of the configuration ^{domain is} empha-

sized. It is shown that no singularity ^of the wavefunction occurs at the frontier of this domain. The dissociations of the system ^into p subsystems (2 ~ p ~ N) ^are described, ^{in Eckart} coordinates, ^{within the} single ^continuum

r1 ^~ oo, where r1 ^{is the} largest radial Eckart coordinate. Each dissociation channel is identified by ^{means of a}

unit RN-1 vector 0393 called the « chemical pointer» : ^{in a} p-ary dissociation, the locus of 0393 is a (p - 2) ^manifold.

A Gram matrix algebra ^is constructed to describe the partitions ^{of the} system into several subsystems. Owing ^to

a power expansion ^{on the} single-continuum ^variable, the behaviour of 0393 in the vicinity of its loci is explicited.

Classification

Physics ^Abstracts

34.10

1. Introduction.

In their original ^form [1], ^{the Eckart} coordinates of ^a system ^{of N} particles ^are particularly ^well adapted ^to

the description ^of molecular collisions. This is essen-

tially ^{due to their inertial and canonical characters} (which provide ^{a direct} way of quantization ^and give

a rather simple form of the Hamiltonian), ^{and to}

their ability ⁱⁿ describing simply the dissociation channels.

In a previous ^paper [2], it has been shown that the

use of the nuclear Eckart frame (i.e. ^the principal

inertia axes of the nuclei) ^{as the} rotating ^{frame to}

which the electrons are referred, ^{leads to an} expres- sion of the Hamiltonian of a polyatomic ^molecule

which is _very similar to that commonly ^{used for}

diatomic molecules. Purely nuclear, ^or electro-nuclear

dynamical couplings appearing ^{when a} Born-Oppen-

heimer basis set is used, ^{have been} given explicitly.

(*) L.A. du CNRS no 282.

When the electro-nuclear couplings ^are neglected,

the nuclear motion reduces to an adiabatic motion

on a single potential ^surface.

This problem of the motion of several atoms,

including eventually ^an infinite extension for some of the coordinates, has received much attention for a long

time. Many curvilinear coordinates have been studied

[3]. ^In particular, ^a great deal of attention has been, and is, ^{devoted to the} hyperspherical coordinates [4-5].

For N particles ^{whose centre of mass is at rest these}

coordinates generally consist in one radius _p and 3N-4 angles. ^For any dissociation of the system, ⁱⁿ which some of the mutual distances become infinite,

the hyperspherical ^radius p tends to infinity ^{and all}

dissociation belong ^{to this} unique continuum. As it will be seen further this property ^{is also true for Eckart} coordinates. In addition however these latter coor-

dinates have the great advantage ^to provide ^a very direct _way of identification of the various dissociation channels. This point ^{has been} already studied in [2]

for binary dissociations. The aim of the present paper

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004704063100

632

is to give ^a generalization ^{of this} property ^{to dissocia-} tions of _any order. For this purpose, the key point ^{is the} expression ^{of the} mutual distances between particles.

In the present ^{case, this} expression ^is particularly simple. ^It gives ^a general ^and simple ^{method to} identify

the various dissociations into 2, 3,

subsystems.

Moreover it makes _very direct the obtention of the

relationship existing between Eckart coordinates and

hyperspherical ^ones, and it exhibits quite clearly ^the

role of the kinematic rotations in the rearrangement

processes.

In part ^{2 the} Eckart coordinates and the classical Hamiltonian expression ^of the kinetic energy ^are

briefly ^{recalled. The} important ^{role of the Jacobian}

in the quantization ^is also examined : the relevant

configuration space is delimited and it is shown that

no singularity ^{occurs at} its frontiers. Dissociation and rearrangement problems ^{are studied in} part ^{3. The} link existing ^{between internal and} external coordinates in a dissociation is studied in part ⁴ by ^{means of a Gram}

matrix algebra.

2. Eckart coordinates [1].

2.1 DEFINITION.

Let us consider a system ^{of N} particles ^{whose centre of mass is at rest. The} configu-

ration of the system ^is completely ^defined by ^{means of}

n

N - 1 configurational ^vectors R,,, ^{which can} always ^{be chosen in such a} way that the Lagrangian

form of the kinetic energy ^T is expressed ^{as :}

The Gram matrix of the system ^{is the n x n matrix G} whose elements are :

As no more than 3 configurational ^{vectors are} linearly independent, ^{no more than 3} eigenvalues ^{of G are}

non-zero. In addition these are obviously ^real positive, namely : ri, r2, r3.

G is diagonalized by ^an orthogonal ^{n x n matrix C} entirely ^defined by ^{3 n - 6} independent dimensionless

quantities q,, ^..., q3n-6.

The 3 n - 3 coordinates : _ri, r2, r3, ql, ..., q3n-6

only specify ^{the relative} positions ^{of the} particles.

They ^{need to be} completed by ^{a set of 3 Euler} angles

a, P, y defining the absolute position in space of the instantaneous solid. The body-fixed ^or rotating ^frame

is unambiguously ^defined by considering the 3 ortho- normalized vectors a; (i

1, 2, 3) ^defined by :

ri, r2, r3 ^are the instantaneous radii of gyration ^and

ai, a2, a3 lie along ^the instantaneous principal ^axes

of inertia. In the rotating frame, any configurational

vector can be expressed ^{as :}

The classical Hamiltonian form of T has been given by ^Eckart [1] :

with: Bi = (rk - rj)’I(rk’ ⁺ rJ); ^Ci

^{(rf -} r;)2/

(2 rk rj), ^{where i,} j, k ^{is a} cyclic permutation ^{of 1, 2, 3.}

The momenta are defined ^as follows :

Mi

^M aim

ðT/ðwi ^{where M is the total} angular

momentum and _Wi

w a;, ^w being ^the angular velocity ^vector,

with

2.2 QUANTIZATION. ^IMPORTANT ROLE OF THE JACO- BIAN. - When canonical coordinates Q1, Q2, - - ol Q3n,

such as the Eckart ones, ^are used, ^{the most} straight-

forward quantization ^{is carried out} by applying ^the simple ^rule [6] :

and defining ^{the scalar} product by :

In order to recover the usual Cartesian coordinates

XI,

X3n, ^{in which no} special topological problem

occurs, ^one has to consider Cartesian wavefunctions Oc whose scalar product ^{is :}

where :

For Eckart coordinates :

where a ^{is the} (3 n-6)-th ^order determinant whose k-th row is :

with :

As both scalar products ^{must lead to identical} physical results, ^{the two} wave-functions are necessarily ^related by :

where a is an arbitrary ^constant phase factor. In the

following ^{one takes a}

^{0; as} usual, ^this gives ^{in the}

Hamiltonian an additional ^« extra-potentials ^term [3].

ln Cartesian coordinates, ^the configuration space is R 3". Because of the existence of nodal surfaces J

0,

e.g. ri

r2, etc..., the transformation X -+ Q ^is

not univocal. As a consequence the configuration

space in canonical coordinates cannot be R 3" : in fact R 3" must be quotiented by ^{the Jacobian kernel, i.e. it}

must be reduced to a domain limited by ^{nodal sur-}

faces. In particular, ^for coordinates ’1’ r2, r3, this means

that _any configuration ^{of the} system ^of particles ^is

describable within the restricted domain ri > r2 > r3.

On the nodal surfaces J

0, 0 necessarily ^vanishes identically ^{and no} singularity ^occurs.

Example : ^{the triatomic case.}

The configuration ^{of 3} particles ^{of masses} mi, m2, m3 is specified by ^{2 vectors :} pi2 joining particle ^{1 to} particle ^{2, and} P12,3 joining ^{the centre of mass 1,2 to} particle ^{3. The} configurational ^{vectors are :}

where

G is a 2 x 2 matrix whose eigenvalues ^{are the roots}

of the characteristic equation :

Obviously, RI, R2 being given, ^{one has to choose}

which root r2, r2 is ^the largest ^one.

C is here a 2 x 2 rotation matrix depending ^{on one} angle q :

where

The rotating ^frame { ai } ^{is defined} by :

The Jacobian is : _{J = r1} r2(ri - r2) ^sin p.

The nodal surface (r1

r2) corresponds ^{to the} geometry :

Making ^the choice ri > r2, ^one has to examine the

asymptotic behaviour of 0 when _{ri - r2} ^- 0+.

Let us define :

As shown in ^a previous paper [2] ^{the triatomic}

Hamiltonian is :

where V is the potential energy.

When v -+ 0+, ^the asymptotic behaviours of the relevant quantities ^{are as follows :}

V does not present any singularity ^{and it is} independent of q ^when ri

rZ. It can then be simply ^{written :} V(r l’ ri)

V(u).

The asymptotic ^{form of the} Hamiltonian is :

It is worthnoting ^that variables q ^{and y are} actually

combined into the unique ^{variable 0 = q +} y. In the case ri = r2, there is an ambiguity ⁱⁿ the definition of the Eckart vectors ai, a2 (these ^{are two} orthogonal

unit vectors in the plane ^{of the three} particles) ; ^howe-

ver this ambiguity ^{is of no} consequence, since a’change

in the definition _{of a1, a2} (i.e. ^{a rotation around} a3)

modifies at the same time q ^and y in such a way that 0 remains the same.

The wavefunction 0(u, v, q, a, p, y) ^is separable ⁱⁿ

the form :

634

The separated differential equations ^are readily ^obtai-

ned :

where Jl2, m2 ^E N, ^and E1 ⁺ E2

^E.

The important point ^{here is that G is the} regular

solution of ^an equation containing ^a pure repulsive centrifugal potential, ^{i.e. a} regular ^{Bessel function.}

As v-+O+, ^{G-v’ with 21=} (m2 ⁺ 1)il2 ^{+ 1. On}

another hand, ^J asymptotically ^{behaves as U3 v.}

Therefore, the Cartesian wavefunction GC behaves as

i.e. it is also regular.

3. Dissociations.

3.1 REARRANGEMENTS. - In a binary dissociation,

the system ^{of N} particles breaks into 2 subsystems ^of N1, N2 particles (N1 ⁺ NZ

N, ⁰ N1 N), ^whose

centres of mass are separated by ^an infinitely large

distance. In general, ^several binary dissociations exist,

each of them corresponding ^{to a} specific rearrangement channel. These considerations are readily generalized

to ternary,... p-ary dissociations. The definition of ^a rearrangement channel is intimately ^{related to the}

choice of the configurational vectors {Rex}. ^This

choice itself is related to a specific way of constructing

the centre of mass of ^the system. ^This construction can

be carried out iteratively, ^{in the} following way : ^one takes first the centre of ^mass of 2 particles, ^{then one} joins ^{it to a third} particle (or ^{to the centre of mass of}

several other particles, already determined); ^this

determines a new partial ^{centre of} mass, etc... The relative vectors obtained along ^this construction, ^once conveniently weighted, ^{form a} possible ^{set of confi-} gurational ^{vectors. Let} a(N) ^be the number of different

sets of this kind. One considers here that _any change

of the type : Rex -+ - Rex leaves invariant the configu-

rational set. Using ^{the tree} theory, ^{one can show that}

[7]:

or equivalently ^that a(N) obeys ^{the recurrent rela-}

tion [8] :

Let [’1’ r2, r3, C(qi, ..., Q3n-6), 0152, P, y] ^{be the Eckart}

coordinates defined _upon one of the previous configu-

rational sets, { R,,, 1. ^{One has to examine how these}

coordinates change when another reference set of the

same kind, { R’ I is taken in place ^of { Ra }. ^{The new}

set can be expressed ^{as :}

where the X fJa. depend only ^{on the masses.}

As the kinetic _energy must have the same form in both sets, then :

Therefore :

This means that the matrix X, whose elements ^are Xa,

has the property :

X is then a rotation in R ", depending only ^{on the}

masses. This transformation X is one of the a(N)

kinematic rotations [4].

As already ^{shown in a} previous paper [2] ^{the Eckart}

coordinates transform in the following way :

where the transformation q -+ q’, ^or equivalently

C -+ C’ is one of the kinematic rotations :

Obviously ^{there are more than} a(N) transformations which obey ^to conditions (12) ^and (13). Starting ^from

a set { R. }, ^one may generate ^another possible ^set by setting

where X _{is any} constant orthogonal ^{n x n matrix.}

Generally, ^{a vector} :1B)’ ^{is not obtained} by weighting

a relative vector joining ^two partial ^{centres of mass.}

In fact however, as soon as dissociation problems

are considered, supplementary conditions are imposed

to the configurational ^{vectors : the} specification ^{of a} given fragmentation ^{of the} system into ^two subsystems

or more, implies ^{to relate} directly ^{some of the} configu-

rational vectors to distances between specific partial

centres of _mass, and this relation is automatically

realized when a set of the type Ra is used. A set of this type ^{will be called a} proper ^set. In summary ^{3C corres-}

ponds ^{to an} arbitrary change ^of origin ^on the variables q> whereas X describes a transition between two

rearrangements.

3.2 MUTUAL DISTANCES IN ECKART COORDINATES.

The consideration of mutual distances is evidently important ^{in a} discussion about the dissociations.

It is also important ^{in the} derivation of the expression

of the potential in Eckart coordinates.

The relative vector PJlV joining ^two particles ^{M, v,}

can be expanded ^{over the} configurational ^vectors R0153,

as :

where the quantities Yay depend only ^{on the masses.}

Using (4), equation (17) gives :

where

For a given couple (Jl, v), Zuy depends ^{on the masses}

and on the variables _qk.

It is worthnoting ^that Zffv ^{remains invariant under}

a change of configurational vectors, ^{since this} quantity

is the scalar product ^of p,v by ^{the invariant vector}

ri ai. Obviously ^this invariance is preserved ^{for all} configurational ^sets, proper ^{or not.}

Finally the mutual distance p,, ^is given by :

By ^{means of this} expression ^{of the mutual distances,}

one sees that, ^{when no} particular constraint ^is imposed

to the coefficients ZP’, (i.e. ^no particular ^condition imposed ^{to variables} qk), ^then when, 1 ^{is made} infinite,

r2 and _r3 being ^finite, all relative distances become infinite, ^{i.e. the} system breaks completely, ^{into N} particles. All dissociations of lower orders (dissocia-

tions into N - 1, ^{N -} 2,..., ² subsystems) ^{are also}

obtained within the same unique ^continuum (ri infinite,

r2 and _r3 finite), by imposing ^{a certain number of}

conditions to Z!". ^This point will be examined in the

following paragraph.

3.3 DISSOCIATIONS. CHEMICAL ^POINTER.

Let us

consider a dissociation of the system into p ^sub- systems S1, ..., Sp, containing respectively N1, ..., Np particles, ^with N1 ^{+ ... +} Np

^{N and 2 p N.}

In this p-ary dissociation, ^{it is} assumed that all mutual distances within the subsystems ^remain finite, ^whereas

the mutual distances between the centres of mass

(or ^as well between particles belonging ^{to different} subsystems) ^{tend to} infinity.

To constrain the mutual distances internal to a given subsystem, say Sk, ^{to be} finite, ^{it is} necessary and sufficient that Nk -1= nk ^internal mutual distances PIlV ^remain finite when _ri ^-+ oo (r2, r3 finite).

Consequently :

or :

The number of such equations ^is equal ^{to the number}

of couples ⁱⁿ Sk, ^i.e. 2 nk(nk ⁺ 1). However, only nk equations ^{among them are} independent. ^{Similar equa-}

tions are obtained for all subsystems 5’i,...,Sp.

Let us now introduce the following R "-vectors :

r is a unit vector depending only ^{on the variables} qk.

On another hand, ^{the vectors Y’’’’} (and ^{even their} improper ^version 1JPV) depend only ^{on the masses.}

Then equations (21-22) ^{become :}

There are nl+n2+w+np=n-p+1=N-p

independent equations. They express the fact that

T(qk) ^is orthogonal ^{to n -} p ⁺ 1 fixed vectors. As

1 r 12

1, ^{this means} that, ^{in a} givenp-ary dissociation,

the locus of r on the hypersphere ^{of radius} unity ^is

a (p - 2) ^{manifold, from which} the intersections with the (p - 2) ^manifolds corresponding ^{to the other}

p-ary dissociations ^are removed (these intersections include all dissociations of lower order : p - 1,

p - 2, ..., 2).

Actually, ^{as T, as well as Y#’v, are defined} except ^for

an arbitrary orthogonal transformation X (which

fixes the origin ^of the qk ^{and the} orientation of the

corresponding curvilinear frame), ^{there is a double}

determination for each dissociation channel. For

instance, ^{in the case} p

2, where the loci of rare reduced to simple points, ^one binary dissociation is associated with 2 diametrically opposite points ^on

the hypersphere. This is in fact of no importance ^since

the domains of variation of qk ^can be restricted to

[0, n[ ^without any loss of generality. ^{In the} following,

all vectors will be chosen on a half hypersphere.

The number of (p - 2) manifolds is equal ^{to the}

number of _ways of partitioning ^{a set of N elements} into p non-empty subsets, ^{i.e. the} Stirling ^{number of}

the second kind [9] :

The direction of the unit vector r is the label of a

specific dissociation channel, ^{in the} unique rl-conti-

nuum. For this _{reason, r may} be called the chemical

pointer.

Example : system of 4 particles.

In this case n

3, ^{then r and the 6 vectors Y#,V} belong

to R 3. For each binary dissociation (in ^{2 + 2 or}

3 + 1 particles), ^{r is} perpendicular ^{to 2 vectors Y#’v,}

and thus it points ^{in a fixed direction} (0-manifold).

636

The number of such points ^{on the} half-sphere ^is equal

to S’

⁷ (see points A, B, ..., ⁱⁿ Fig. 1). ^{In the same}

manner, for each ternary dissociation channel (2 + 1 + 1 particles) ^{r is} perpendicular ^{to one vector YIlV and}

its locus is a (halo ^{circle of radius 1, from which the}

intersection points ^{with the other similar} (halo ^circles

must be removed (Fig. 1). ^Note that there exist 83

⁶

ternary dissociation ^channels. Finally, ^{there is} only

one quaternary dissociation (explosion ^{of the} system)

in which r is ^no longer constrained except ^{that it}

cannot belong ^{to the} previous ^loci.

Fig. 1.

Dissociation of a system ^{of 4} particles (1, 2, 3, 4).

(a) Definition of the configurational ^{vectors :} R1

p’/2 ^12, R2

/,11/2 ^34, R3

/11/2 ^{C34 C12’ where C34 is the centre}

of mass 3, 4 ^and C12 ^{that of} 1, 2.

(b) ^{Loci of the chemical} pointer ^{r on the} sphere ^{of radius} 1, for the different dissociations of the system. ^The figure ^is

drawn in the case ml = m2

m3

m4. When r points

on a circle labelled by p, ^v the relative distance p,, ^remains finite when r 1 ^{-+ oo.} The circle p, ^v is in the plane perpendicu- lar to a vector YIlV (see text). Only ^the directions of these ^vec- tors are shown. The intersecting points ^{of these 6 circles cor-} respond ^{to the 7} possible binary dissociations (only ^{a half} sphere, e.g. xi a 0, ^{has to be} considered) : ^{A : 12 +} 34,

B : 23 + 14, C 13 ^{+ 24} (intersections ^{of two} circles);

D : 123 + 4, E : 234 + 1, ^{F : 341} + 2, ^{G : 412} + 3 .(inter-

sections of three circles).

Let us take the following ^{choice of} configurational

vectors :

In such conditions, the components ^{of the vectors} YIl" are as follows :

4. Internal and external Eckart coordinates. Gram matrix algebra.

4.1 INTERNAL AND EXTERNAL COORDINATES. - In order to analyse ^the partitioning ^{of a} system ^into several subsystems, ^or conversely ^the collapse ^of

several subsystems ^{into a} unique ^one, using ^Eckart

coordinates only, ^{it is} necessary ^to examine the link

existing between the coordinates of the subsystems (internal coordinates) plus ^{those defining} their relative

positions (external coordinates) ^{and the} coordinates of the whole system.

Let us consider a system ^{of N} particles ^{as the sum}

of subsystems Sl, S2,

Sp, containing respectively NI, N2, ..., Np particles, ^with given ^relative positions (subsystem Sd). This ^is equivalent ^{to make a choice of}

a proper configurational ^{set of n} vectors { Ra}, ^such

that it can be partitioned into p ^{+ 1 subsets :} nk

configurational ^vectors Rp ^{belonging to subsystem}

Sk (k

1, ..., p), plus ^nd = p - I configurational ^vec-

tors Ry specifying the relative positions ^{of the sub-} systems Sk (i.e. subsystem Sd) :

For each subsystem, either internal or external, ^one defines Eckart coordinates, ^{i.e. a} Gram matrix G k

with characteristic values t1,2,3’ ^an Eckart frame

a1,2,3 ^{and a} rotation matrix Ck, ^{where k}

⁰ (sub- system Sd), 1, ..., p. ^Similar quantities, namely G,

’1,2,3’ a1,2,3 ^and C, ^{are defined for the whole} system.

4. 2 COMBINATION OF THE SUBSYSTEMS.

Each vector

Rf ^{can be} expanded ^{as :}

and :

All vectors e) ^{can be gathered into a} unique family

{ efJ} ^(f3

^{l, ...,} n), using ^{the same} ordering ^{of the} subsystems ^{as that used in} (26). ^This definition of efJ includes also ^cases in which subsystems ^{contain less}

than 4 particles, ^{since in such cases the number of}

non-zero characteristic values reduces to 2, 1, 0, respectively ^{for 3, 2, 1} particles (or ^{centres of} mass).

Let us now define a n x n matrix C’ as being ^the block-diagonal ^{matrix, whose blocks are} Cd, ^C 1,

^{C P.}

By ^{means of} C’, ^{one can} write any configurational

vector as :

From (28), ^{the total} Gram matrix is readily ^{derived :}

or :

where P is the n x n matrix of elements : P rzp

^ea ep.

Taking ^{into account the} definition of e,, ^{the P} matrix takes the following ^{form :}

where crosses and hatched areas correspond ^{to non-}

zero elements (respectively (rt)2 ^and rt rl’ ^ak k’).

Let C p be the rotation which diagonalizes ^{P :}

where A is the diagonalized Gram matrix of the total system.

Using ^the definition of C, ^G

C L1C and (29),

one gets :

and thus :

In summary, knowing ^the partial ^Eckart coordinates of the subsystems Sd, Si,

Sp, ^{one is able} to construct

the coordinates for the total system : ^from (32) ^one

deduces the values of r 2 ^{and the} Eckart frame { a, 1,

i.e. the Euler angles ^{of the total} system. Then equa- tion (31) gives ^{the total rotation matrix C.}

It _may be noticed that this combination of partial

Eckart coordinates ^{into the} total ones does not imply

the choice of _proper configurational sets, neither ^for

subsystems S1, ..., Sp, ^{nor for} Sd. However if one needs

simple asymptotic ^{forms as} ri ^{-+ oo} (cf. § 3.1) ^when

the system ^{breaks into} S1 ⁺ S2 ^{+ ... +} Sp, ^{the confi-} gurational ^set of Sd ^is necessarily ^a proper ^one. More

generally, ^{in a} collision with rearrangement, ^{it is}

necessary ^to take all configurational ^{sets as} proper

ones.

4.3 BEHAVIOUR OF r IN THE VICINITY OF A DISSOCIA- TION LIMIT POINT.

Let R 1 ^{be the} unique (proper) configurational ^{vector of} Sd, along ^{which the} binary

dissociation takes place. ^{From the} expansion (4)

one has :

In particular :

When ri ^{-+ oo,} R1 only becomes infinite and it takes the asymptotic ^{form :}

On another hand, Ra# 1 being finite, Cla ^{tends to zero,}

then ^as expected F -+ { 1, 0 ... 0 1. ^{From the above} expression ^of Cia, ^{one sees that the} vanishing ^corm- ponents ^{of r behave as :}

D1ex ^{is the} projection ^{of an internal} configurational

vector, belonging ^to S1 ^or S2, ^on the direction of the dissociation (Fig. 2), ^{which means that} D1ex actually

become an internal variable in S1 ^or S2.

In fact this property ^{can be} generalized ^to any kind of dissociation. Let us consider a well defined _p-ary dissociation. As shown before, the chemical pointer r, ^{defined on} the proper ^set of configurational ^vectors Rex, belongs ^{to a} given p - ^{2 manifold.} By ^{a convenient}

rotation X in R ", ^{one can} pass ^{to an} improper ^set Sip

such that the components ^{of the} chemical pointer ^are { 1, o ... o }, ^{which means that} ?1 only ^{becomes infinite,}

with 11 r, a 1 (oo ), ^{i.e. that an} improper binary

dissociation occurs. Then the components C1a ^{of the} improper rotation matrix C behave as before, ^{and one}

can deduce the behaviour of the components ^{of the}

638

Fig. ^2.

Binary dissociation of a system ^{into 2} subsystems Sl ⁺ S2, described in a proper ^set of configurational ^{vectors :} R1 (unique ^{vector in} Sd), becoming ^infinite when ’1 ^--+ oo, and Rex (systems S1, S2) remaining ^{finite. To the} first order in

ri 1, the chemical pointer ^{behaves as :}

proper matrix C, by applying the rotation X which

depends ^not solely ^{on the masses, but also on the}

relative asymptotic direction of those _proper vectors

Ra(E Sd) along which the _p-ary dissociation takes

place.

5. Conclusion.

From a dynamical point ^of view, the inertial and canonical characters of the Eckart coordinates sim-

plify significantly ^the expression of the kinetic part ^of the Hamiltonian of ^a N-particles system. These coordinates _{appear a} priori ^as curvilinear coordinates suitable for the description ^of the motion of a many-

body system. ^The difficulties which could come of the

non univocal character of the transformation from Cartesian coordinates into Eckart ones are actually

ruled out as soon as one considers the relevant confi-

guration domain, whose frontiers are the nodal surfaces of the Jacobian. Just as does the r

0 frontier in

spherical coordinates, the nodal surfaces act as

« mirrors » on which the wavefunction vanishes.

Consequently any molecular geometry ^{can be des-} cribed, and _any dynamical problem ^can be solved in

principle, ^within this limited domain, ^without any risk of singularity ^{at the frontiers.}

In addition to these interesting dynamical properties,

the Eckart coordinates have proven ^to be particularly

°

efficient in the description ^{of all} possible rearrangement

channels. Firstly all these channels belong ^{to a} single

continuum (ri ^oo, where r1 ^{is the} largest ^{radial coor-} dinate). Secondly, these coordinates lead to a vectorial

representation ^{of the} rearrangements since ^{the tran-} sition from one rearrangement ^{to another one is}

equivalent ^{to a rotation in} space R n (n

^{N -} 1), depending only ^{on the masses} (kinematic rotation).

This transformation involves only ^the dimensionless Eckart variables _qk, i.e. the elements of C, ^{the rotation}

in R" which diagonalizes ^the Gram matrix of a confi-

gurational vector set. The unit R "-vector r, ^whose components ^are the elements of the first line of C,

allows a straightforward identification of the disso- ciation channels. This vector acts as a chemical

pointer : ^{to a} specific p-ary dissociation corresponds

a locus of r, consisting ^{in a} (p - 2) manifold. In

particular, ^definite points ^{on the} hypersphere ^of

radius 1 correspond ^{to all} possible binary dissociations.

The asymptotic ^behaviour of r in the vicinity ^{of these} points ^{can be} derived, by taking ^{as a} coordinate axis in R" the direction related to a specific binary ^disso-

ciation. To the first order in ri 1, ^the components T a ( a # 1) ^{behave as} D1,)r1’ ^where D 1 a is the pro-

jection ^{of the internal} configurational ^vector R.

on ai(oo), ^{i.e. on} the direction of the separation ^{of the}

2 fragments. ^This property ^{can be extended to disso-} ciations of higher order, ^since formally any given

p-ary dissociation is equivalent ^{to a} binary ^disso-

ciation in a definite improper configurational ^set.

This behaviour provides ^some insight ^{on the proper}

way of expanding ^the potential ^{and the wave function}

over angular ^harmonics.

The main remaining ^{tasks are then to derive the} corresponding system ^of coupled equations ^{and solve}

it in some specific ^cases, using ^{more or less} approxi-

mate methods. From this point ^{of view, the case of a}

diatom-atom collision involving only rotational exci-

tation, appears ^as the simplest ^one. Moreover it is

a test case since such collisions have been already extensively ^{studied in} using standard coordinates

[10,11].

Acknowledgments.

We are grateful ^to Drs. A. Nauts and X. Chapuisat

for providing them with their article prior ^to publi-

cation. We thank Pr. F. Combes and Dr. M. Ducloy

for their interest in this work.

References

[1] ECKART, C., Phys. ^{Rev. 46} (1934) ^383.

[2] ROBERT, ^{J. and} BAUDON, J., J. Phys. ^B (1985), ^{to be} published.

[3] NAUTS, ^{A. and} CHAPUISAT, X., ^Mol. Phys. ⁵⁴ (1985),

to be published ^and references therein.

[4] SMITH, ^F. T., ^{J. Chem.} Phys. ³¹ (1959) ^1352.

[5] FANO, U., Rep. Prog. Phys. ⁴⁶ (1983) ^97.

LAUNAY, ^{J. M. and LE} DOURNEUF, M., Proc. XIIIth

ICPEAC, ^Berlin (1983), ^Invited Papers, p. 635.

[6] DIRAC, ^{P. A.} M., ^The Principles of Quantum ^Mecha-

nics (Clarendon Press) 1957, p. 87.

[7] COMBES, F., private communication.

[8] DUCLOY, M., private communication.

[9] ABRAMOWITZ, ^{M. and} STEGUN, ^I. A., Handbook of

Mathematical Functions (Dover Publications) 1972,

p. 824.

[10] REBENTROST, F. and LESTER Jr., ^W. A., ^{J. Chem.} Phys.

67 (1977) ^3367.

[11] LAUNAY, ^J. M., ^J. Phys. ^{B 9} (1976) ^1823.