Few-Body Systems 2, 127-143 (1987)

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9 by Springer-Verlag 1987

Asymptotically Adapted Three-Body Molecular States A. V. Matveenko I and Y. Abe 2 1 Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Head Post Office Box 79, SU-101000 Moscow, USSR 2 Research Institute for Fundamental Physics, Kyoto University, Yukawa Hall, Sakyo-ku, Kyoto 606, Japan Abstract. For the first time a formal theory for three-body rearrangement scattering processes in the molecular-state approach is formulated removing difficulties with unphysical long-range couplings.

1 Introduction The Born-Oppenheimer adiabatic or molecular-state method has first appeared in atomic physics [ I ] but now is widely used also in nuclear physics [2 ]. Most of the textbooks consider it to some extent [3]. Nevertheless the method is defective, what has not been so important in atomic physics in the beginning but soon became apparent and now is well demonstrated by extensive muonic molecule calculations [4]. Again, it was not so important in nuclear physics as many other uncertainties were involved in that case. To understand the essence of the problem we should return to the simplest case, the three-body molecular states with welldefined interaction, without any additional complications like spin ~tc. So, we have for example the HD § system consisting of three particles with Coulomb interaction, namely p + d + e. Much heavier nuclei are almost fixed at some stationary positions and a valence electron moving with a rather high velocity provides the binding of the system. This clear physical picture gives grounds for the usual adiabatic strategy for the solution of the problem. In a first step the nuclei are considered infinitely heavy (fixed), so that a much simpler three-dimensional problem o f the electron moving in the field of two fixed centers is to be solved. For the particular case of the Coulomb interaction this two-center problem happens to be completely separable in prolate spheroidal coordinates thus providing a comparatively easy way to calculate the eigenvalues and eigenfunctions of the problem both depending on the internuclear distance as a parameter [6]. Next, this parameter is again converted into a dynamical variable. In this treatment the electronic motion (with fixed nuclei) appears in the zeroth order, the vibrational m o t i o n of the nuclei is of the second order, and the rotations are of the fourth order in the expansion parameter (m/M) I/4, where

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A.V. Matveenko and Y. Abe

m and M are typical masses of the light and heavy particles, respectively [7 ]. This approach results in a strongly coupled system of Schr6dinger equations for the radial motion of nuclei, which persists to be coupled even in that part of configuration space, where two particles are bound forming an atom while one of the nuclei is far away. As a result of such a nonphysical asymptotic coupling the boundary conditions for the scattering problem in this approach are not easy to meet. The situation is rather strange-in zeroth order the theory provides extremely good results, but the slightest attempt to improve this approximation destroys the concept. Several attempts were made to overcome this disaster. In the most extensive calculations [8] 52 bound and 792 continuum states of the two-center problem were used. The authors claim that, though the theory is defective, as far as only an almost complete set of solutions was employed, one could be sure of their results. The variational calculations from [9] have disproved this assertion. In another approach [10], see also ref. [11], the infinite system of coupled radial Schr6dinger equations was treated by transforming it to a physically more acceptable form. We shall mention here two other attempts to treat the boundary conditions in the molecular-state framework, namely the introduction of translational exponential factors and of the so-called diabatic (i.e., opposite to molecular or adiabatic) states. It is not an easy task to discuss these approaches, because they have never been defined clearly. Furthermore, when introducing them one usually supposes that some adiabatic states are already available. Nevertheless, the relevant literature is enormous. Here, we shall give only representative references [ 12], see also the recent paper by Macek [5] with the discussion of the problem. So, all the problems that appear when adiabatic states are introduced in the study of the three-body states, namely, the effect of proper mass, asymptotic radial and Coriolis coupling between slow and fast degrees of freedom, are still open to a direct treatment. In what follows we present a formal theory, which is free of all those drawbacks but contains all the advantages of the usual Born-Oppenheimer approach. Accordingly, our method should be very appropriate for studying the scattering problem and also weakly bound states in the framework of the molecular-state approach. Our strategy is rather different from the classical one, though the starting point is the same, i.e. the three-body Hamiltonian in the rotating frame with its z-axis along the internuclear axis. We examine carefully the structure of different coupling terms, which are parts of the total Hamiltonian, and introduce two successive transformations of the coordinate system and also of the wave function arriving at a new much simpler expression for the total Hamiltonian. This new Hamiltonian allows one to redefine the partition of the physical system into its fast (originally electronic) part and rather slow (nuclear) part. A new two-center Hamiltonian appears, which exactly reproduces the spectrum of the residual atom, when one of the nuclei lies infinitely far away. Next, a partial-wave analysis provides the states with exact quantum numbers of the total angular m o m e n t u m J and parity p. After that the two-center Hamiltonian converts into a system of (J + 1) or J coupled Schr6dinger equations in two variables to account for the m o t i o n of a fast subsystem. This program has partly been presented by one of the authors [ 13]. Here it will be reproduced more clearly stressing important

Asymptotically Adapted Three-Body Molecular States

129

physical details. After that we shall construct a formal scattering theory with exact asymptotic states and provide formulae for the amplitude and scattering cross-sections in the laboratory frame in terms of the molecular-state S-matrix. The main idea of our paper is to make the traditional molecular three-body Hamiltonian adequate for a description of three particles in the fragmentation regions. At the same time we should take care of preserving all the advantages of the usual Born-Oppenheimer description. The transformed Hamiltonian appears to be very close to that normally used in the hyperspherical-type approach to a three-body problem. Accordingly, as a by-product, we have established the connection between the Born-Oppenheimer adiabatic approximation and the adiabatic hyperspherical approximation. 2 Molecular Description: Jacobi Coordinates, Recoil Operators In order to describe in a simple way the asymptotic states for the transfer reaction (a + c)o,n + b ~ (b + c)l'm' + a

(1)

we need two sets of relative Jacobi coordinates (Fig. la, b). Here are two cores a and b with masses ma and rnb and a valence particle c with mass rnc, which can bind the whole system. In this case and for low relative velocities, the process (1) should proceed through a molecular state. That is why one usually writes down the internal Hamiltonian in terms of relative Jacobi coordinates (Fig. 1c). The m o t i o n of a valence particle c is quantized onto the R-axis of a rotating frame. The three sets o f Jacobi coordinates (Fig. l a - c ) are interconnected by relations of the type (rRa) =(10

f1) ( 1 e

~)(Rr).

(2)

The coefficients e and f depend on the masses of the particles, FHb

e- - -

ma + m~

f'

/'He

(3)

ma + m e

The transformation matrix (2) is factorized into two matrices of successive shifts.

jc Fig. 1. Jacobi coordinates for a the left-hand andb right-hand sides of the transfer reaction (1). In e the Jacobi coordinates, one proceeds with in the molecular-like approach, are given. Formula (18a) provides the value of co

c

b a)

o

b)

a c

a

R

b ci

130

A.V. Matveenkoand Y. Abe Eq. (2) also provides formulae for the gradients --e

(VRat = (--lf

1 +ef) (;~) "

Vr a ]

(2a)

The change of variables (2) can be treated as a change of the representation by introducing the operator Ta = exp(eRVr) exp(frVR),

(4)

which transforms the Hamiltonian H, or any other operator, to a new representation: Ha = TaHTa 1.

The operator Ta has the properties TaRTa I = Ra, TaVR T~"1 =

TarTa 1 = ra,

VRa ,

Z a V r T a I = Vra ,

(4a)

which follows immediately from the operator identity 1 exp(Y) X e x p ( - Y) = X + [Y, XI + ~ [Y, [Y, X]] + ' " .

(5)

The translational operator (4) allows for recoil effects exactly [14], though it can hardly be recommended for practical use. Difficulties in the numerous attempts [ 12] to describe the asymptotic states in a molecular basis by incorporating some additional exponential factor in front of the molecular wave function stem from its peculiar form (4). The operator (4) does not coincide with the usual kinematic rotation operator of Smith [ 15 ] as it also interchanges the Jacobi vectors. As usual we choose Jacobi coordinates (Fig. l c) with polar coordinates (R, O, (I)}for the vector R. The Hamiltonian of the associated three-body problem will be -

--

+

2M

-~

+ - -

2MR 2

2m

A~ + V

(6)

with the potential energy (6a)

V = v~(ra) + vb(rb) + vc(R)

and the squared angular momentum operator 1 L2 -

~

sin 19 a19

Sin 19

~

1

~19

~)2

sin2 19 ~ep2 "

(6b)

The reduced masses M and m are given by 1

1 -

M

ma

1 + --,

mb

1

m

1

-

1 + - , m c gila + m b

(6c)

and we introduce ~ = (m~ - m a ) / ( r n a + rob) for further use. Next, the operator l)(~b, 19, 0) = exp(-- idPlr exp(-- i191y,)

(7)

Asymptotically Adapted Three-Body Molecular States

13 1

transforms the Hamiltonian

H=

(8)

At this point the vector R is still referring to the initial laboratory frame, but the m o t i o n of the valence particle is described in the rotating coordinate system with

[7] ex, = e|

~5),

ey, = e . ( O , q~),

ez, = eR(O, q~),

(7a)

owing to the specific form o f / 9 , which depends on the angular-momentum projections ly,, lz, o f the particle in the frame (7a). The transformation (8) is usually introduced as a simple coordinate transformation 0 r 0 -1 = r',

/)R/) -1 = R.

(9)

In order to obtain the standard Born-Oppenheimer picture the prolate spheroidal coordinates [6] should be used for the m o t i o n o f the light particle 1 ra + rb ~ - - - , R

r~ - rb r~---, R

(9a)

~o = a r c t g ( y ' / x ' ) ;

thereafter we arrive at what can be called an original Born-Oppenheimer or adiabatic Hamiltonian [4] HBo = h

1

1(1

2 M R 2 r'2A*' -- 2-M

0) 2

R + ~-R

J'l + 2MR= + ~-R

+ ~--R Cl - M - R- 2 9

(10) The squared total angular m o m e n t u m j2-

1 (~ sin 2 (9 ~-~

~)2

1 ~ 0 sin 19 ~O sin O ~|

cos O

02 ~02

(10a)

and the two-center Hamiltonian h for the m o t i o n of the valence particle (fast subsystem) are expressed as usual by h -

1 2m

At' + V,

(10b)

where the Laplace operator is n o w given by At, = A~n + A ,

(1 0c)

with 4 z2x~n _ R 2 ( ~ 2 _ r?2) ~-~ (~2 _ 1) O--~+0n(1

n2)

and

4 02 Ar

2 ~tP2 .

1 All the calculations are done in the {~, 72,~0} coordinates, but r' is sometimes used to simplify the notation

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A.V. Matveenko and Y. Abe

The two last terms o f (10) account for the radial and angular coupling o f the m o t i o n of a valence particle to the frame of R-coordinates. The precise form o f the coupling operators follows from l { - ~ 2 -rl2 (~ -- Ir

Ix, +-ily,

=

e x p ( -+ i~)

8} ~ ,

8 - - 1 ) ~ + 07 --1r

-+ ~2 _ r/2 ( r / - x~) ~-~ -- (~ - ~r~)

8 lz, = - i 3~0

s = [(~2

1)(1

s

i

~2)]1/2

' (10d)

The projections o f J are given by J = ex, s ~ O 8qb

i ctg O

+ ey, -- i ~-~ + ez, -- i

.

(10e)

For the volume element we have Ra d R dr' = R 2 sin O dR dO d,~ --g- (~2 _ 7/2) d~ dr~ de.

(10f)

Because the coupling terms in (10) do not vanish as R ~ 0% the two-center Hamiltonian (10b) does not reproduce the spectrum o f the (a + c) or (b + c) " a t o m s " in the same limit. One can say that the recoil effect is not accounted for properly at this stage. Almost all the matrix elements of the coupling terms from (10) calculated between different eigenstates of (10b) for a pure Coulomb interaction [4] tend to some constant values as R ~ o o . 3 Elimination of the Radial Coupling: Hyperspheroidal Coordinates It has recently been shown [ 16] that the radial coupling is unphysical in the sense that it can be eliminated exactly by an isometric transformation of the total Hamiltonian (10)

HA = e-AHBo eA

(11)

with the generator A = ln(x/p)R

+~

,

( 11 a)

where m

p = 1 + ~-~ (~2 + r/2 _ 1 - 2x~r/+ ~2)

(lib)

is a dimensionless function of relative coordinates. Direct use of the operator identity (5) yields for the transformed Hamiltonian 1 ( 3_~

HA =hA

5 3 ) + RA 8RA

3

J2-2J'l

2MR~ + p 2MR~

p2 R2

2m R~ A

(12)

Asymptotically Adapted Three-Body Molecular States

133

and for the wave function ~A = e-Aq~( R, r') = ~

1

~(x/P R, r').

(12a)

The expression (12a) follows just from the known identity exp[ln(a) rVr] f(r) = f(ar).

(13)

The coordinates and the wave function are thus changed by the transformation (11 ). The new slow radial variable RA is given by [ 17 ] RA = x/~R.

(12b)

In (12) the operator 1 2m P2&~n + V

ha-

(14)

represents the transformed Hamiltonian for the motion of the fast subsystem. Here R2 2x~, = R--~-A~.

(14a)

Now, one can see that the radial coupling term is missing in (12) as the recoil effect is now described exactly by replacing the original Jacobi reduced masses m and M with the coordinate-dependent masses m/p and Mp. In order to see this, R h should be changed back to R by using (12b) in the expressions (12) and (14). These variable reduced masses have the important properties P m

(Mp) -1

_~ Rr[t t ~

(~--~l,w-*-1) --->

1 tTaa

1

Rrct 1~~176 m b

(~--~l,r/~- 1)

+

1 mc

rn2c1

Pa

,

rn

1 + - -

- M a 1 =(MPa) -1,

(15)

1fla + m c

just like it should be for the left-hand side of the reaction (1). In the case of Rrg~ ~ oo limit formulae like (15) hold with a and b being interchanged. Since the transformation (11) leads to the change of the independent variable by (12a), a new collective radial variable R A should be introduced with the very useful asymptotic properties JX/~aR = X / - M - ~ Ra, RA= v ' P R ~ [ v ~ b R v/-M-~Rb,

Rr2a -->oo ( ~ - + 1 , ~ 7 - + - 1 ) , R r ; 1 -+~176(~-+ 1, r~-+ 1),

(15a)

where Ra and R~ are the Jacobi radii that correspond to ra and rb. The formulae (15) indicate that in the region of pair collisions the transformed Hamiltonian has the proper behaviour and (15a) should be used for matching the eigenfunction of HA with the asymptotic solutions that represent the channels of the reaction (1). It can easily be shown [17] that RA coincides with the hyperradius of the system, which is the proper variable to treat Fock's singularity. So, while restoring

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A.V. Matveenko and Y. Abe

the proper pair-collision behaviour o f the total three-body Hamiltonian, we have got the theory that is adequate for treating the region of the triple collision (RA -+ 0). The result o f this section can be summarized in a simple way. The R, ~, coordinates o f the usual Born-Oppenheimer description should be changed to "hyperspheroidal" coordinates RA, ~, 7. Changing the wave function (12a) is also desirable in order to simplify the resulting Hamfltonian. One can also say that the A-transformation simulates in its own way a "radial" part o f the recoil operator (4) resulting in the Hamiltonian (12) with good asymptotic behaviour o f the radial variables. The original Born-Oppenheimer quasiseparation o f variables is substituted by the new one.

4 Rotational Coupling: Inertia Tensor of a Three-Body System The commutation relation [A, J 91] = 0 implies that the structure o f the rotational coupling operator is not changed by the A-transformation. This is an additional indication that RA is a good three-body generalized radial coordinate. On the other hand, the asymptotic properties o f the J 91 operator are known to be poor [4]. Due to this fact we have solved the problem o f the proper asymptotical behaviour o f the theory only for the simplest case of J = 0 as yet. A further transformation is needed and it is almost evident that this should be a rotation. In order to simplify the presentation, this rotation will be accomplished in two steps. The first one is to restore the proper commutation relations o f the components of J that are too complicated until now. Indeed, using Jx', Jy' and Jz, = lz,, from (10e), we have [dx,,Jy,l=-ictgOJx,

-iJz,,

[Jx,,dz,]=[Jy,,J~,]=O.

(16)

To restore a usual algebra for the projections o f J we define the unit vectors el, e2 and ea anew by el = ex, sin ~o -- ey, cos ~,

e 3 = eR,

e2 = ex' cos r + ey, sin q0,

(17)

so that for the components of J = e l J a + e2J2 + eaJa we have [J1, J2] = - iJa and so on. These properties happened to be useful [13] to introduce one more transformation o f the total Hamiltonian (18)

H A ~ = e-S~HA e ~ ,

where ~ = - i w J 1 and co, defined by the expression sin 26o -

m (~/2M p

~:)s

l~l - A '

A-

m s2 M p2 '

(18a)

is the angle between the vector R and the principal axis of the inertia tensor o f the three-body system (Fig. lc). So, (18) provides the new rotational reference frame, in which by using (5) we arrive at HAa = hA~z --

+RA

(19)

Asymptotically Adapted Three-Body Molecular States

135

The operator

1 l {Aj~+iJl[4s163 hart = hA + TR + 2MR~ 1 - A

2s

}

(20)

with

s

~2_r/~

~(~2_1)~+~(1_7/2)~

~

~

(20')

should be referred to as a rotational dynamical two-center Hamiltonian. It contains the operator

TR = ~ \ 11

12

13 ! '

(20a)

which is just the Hamiltonian of an asymmetric top with the classical expressions for the principal inertia moments I1 =I2 +I3 =MR2p,

12 =89

+ x/~ - A),

(21)

so that p already given by (1 lb) is simply defined by I1. The original long-ranged Coriolis coupling term J" 1 from (10d) is now transformed into the rotational coupling term that is given in the curly brackets of (20). Now it has a multiplier in front of it which makes it zero as Rr~,~ -+ oo. So, the transformed//An Hamiltonian is now completely separable in the part of the configuration space defined by the channels of the transfer reaction (1). Three subsequent rotations were involved to arrive at the resulting Hamiltonian (19) with its rotational part given by (20). Finally we give the expression of the Euler angles a,/3, 7 between the space-fixed frame and the one diagonalizing the inertia tensor of the three-body system [ 13, 18 ], sin O ctg(a - ~) = cos O ctg ~o+ ctg co sin ~o cos/3 = cos O cos co - sin 19 sin co cos ~0, sin co tg 3' = - cos co ctg ~o - ctg 19 - sin ~0

(22)

This rotation depending on internal coordinates of the system again can be thought of as an angular part of the total recoil operator (4), but contrary to the A-transformation it diagonalizes the rotational coupling only in the asymptotic region. Now the sequence of the coordinate transformations used in this paper can finally be summarized. The original Born-Oppenheimer three-body center-of-mass coordinates R, if, 19, ~, r/, ~0were substituted by a special choice of hyperspherical coordinates RA, ~, r/, or, /3, 7- There exists a number of papers in the fields of nuclear, molecular, and atomic physics, where authors start by using hyperspherical coordinates in their investigations. We shall indicate here only representative

136

A.V. Matveenko and Y. Abe

references [ 19, 20 ], noting the papers by Johnson [21 ], where there is an interesting discussion of different variants of the Hamiltonian in hyperspherical coordinates, and the papers by Macek who was probably the first one to introduce the Born-Oppenheimer-type description of a three-body system in hyperspherical coordinates [5, 22]. 5 Partial-Wave Analysis of HA~z: States with Good Quantum Numbers The asymptotically adapted three-body molecular Hamiltonian (19) allows one to meet proper boundary conditions for the transfer process (1). In the usual BornOppenheimer description the two-center Hamiltonian (10b) is used to provide the molecular-type eigenfunctions as a basis to expand a total three-body wave function. In our case the operator (20) plays that role. The angular degrees of freedom should be separated at the first step. This procedure can be combined with an introduction into the theory of the states with good quantum numbers of total angular momentum J and total parity p by writing the expansion of a three-body wave function in the form J

xPff(RA, r') = ~

BffK(a,/3, "g)~b~P(ga, ~, r~).

(23)

K=0

This function has to satisfy the Schr0dinger equation HAaqff = Eq~.

(24)

In (23) the summation is over the values of the total-angular-momentum projection onto the rotating z'-axis. The quantum number M is the total-angular-momentum projection onto the original fixed z-axis. The angular part of the wave function has the form BJMP(O~,fl,'Y)=DfM-K(~,fi,')')+p(

. - 1-,j~n] I t/_MK[Ol.,fi,~ ).

(23a)

It contains Wigner /)-functions as defined in [18] and provides good quantum numbers J and p. The number of terms in the expansion (23) is not higher than J + 1. Hence, the projection (24) onto the states (23a) leads to the system of J + 1 or J Schr6dinger equations with the matrix Hamiltonian = hAa -- ~-~ 3R-~ +

RA aR A

2MR]x

,

(25)

which includes the matrix operator of the dynamical two-center problem for the Jp rotational states hAa; this is just the operator (20) averaged over the angular states (23a). The operator of an asymmetric rotator TR couples the states (23a) for K' = K + 2 and the Coriolis-type operator from (20)includes also (K' = K' + 1)type coupling. Both couplings disappear in the asymptotic region where Rra 1 or Rrgl -->.o, and then K starts to be a good quantum number, what is not true in the general case. Again only asymptotically the rotational part of the total wave function decouples into the form (23a) with the arguments of the 1)-function being converted into op, O, ~o--~ due to formulae (22), where co should be put equal to zero. The system of Schr6dinger equations in three variables (RA, ~, 7/}

Asymptotically Adapted Three-Body Molecular States

137

with the Hamiltonian (25), which we rewrite as HSpq'sp(RA, ~, 71) = E~I'SP(RA, ~, rl)

(26)

using ~sp for the column-vector ff~v from (23), is ready for the solution. If a three-dimensional code is available, the solution of (26) is straightforward and provides the spectrum and eigenfunctions of a three-body system. The scattering problem is more complicated, as can be expected. In the next section we shall introduce the Born-Oppenheimer-like approximation for the solution of (26) that will be a tool for treating both the eigenvalue problem and the scattering problem (1). 6 Generalized Born-Oppenheimer Description: Two-Center Problem with Good

Quantum Numbers Now we are ready to make use of a rather general Born-Oppenheimer or adiabatic idea, i.e. to separate approximately the total dynamical system into its fast and slow parts. The starting Hamiltonian is given by (25) in our case, the collective "slow" variable will be RA, SO that a fast subsystem will depend on two intrinsic variables ~, ~ with the generalized two-center rotational Hamiltonian h sp from (25), which is just the operator (20) projected onto rotational states (23a). The associated Schr6dinger equation reads hSV(JnP(~, 7; RA) = eSnP(RA)(OSnP(~, 7; RA).

(27)

Its eigenvalues and eigenfunctions depend on RA, which is a parameter for the moment. The "vibrational" quantum number n can easily be interpreted by inspecting the Rr~,~ ~ oo limit of the Schr6dinger system (27). As it follows from expressions (20) and (20a), hA~

~ ha + -- - Rra~b-+= 2 Ia ' (~-+1,n-++_1)

(27a)

so that the projection of J onto the body-fixed z'-axis starts to be a good quantum number in this limit. For a general spherically symmetric pair potential the orbital m o m e n t u m l of the (a + c) subsystem will be the additional asymptotically good quantum number. Both can be used for the classification of the states. In order to distinguish between the (a + c) and (b + c) asymptotic states the third quantum number ~ = a or b will be introduced. This specification allows one, for example, to consider the limit R A -+ ~ as R r ; ' -+ ~ for a = a or as R r b 1 "+ ~ for oL=b.

One can say that the solutions of (27) diagonalize the rotational coupling exactly but unfortunately only by some numerical procedure. Once Eq. (27) is solved, we can introduce the Born-Oppenheimer-like expansion for the solution of (26) (J, p indices for the solutions of (27) are omitted from now on) ~SP(RA, ~, 7) = ~

~0tK~(~, ~7;RA)XlK~(Ra).

(28)

lKo~

Eq. (26) projected onto the solutions of (27) provides the system of Schr6dinger

138

A.V. Matveenko and Y. Abe

equations for Xn(RA). The index n = (lKa} enumerates the states that should be involved in a particular calculation. These are chosen according to the following criteria. As long as we are interested in some specific two-body asymptotic states for the reaction (1), these define at least two corresponding sets of quantum numbers n. The simplest analysis of the en(RA) behaviour provides one with the physical understanding of their particular role in the process involved. This is just due to the fact that e,,(RA) are the generalized effective potentials and as it is experienced from the Born-Oppenheimer method only non-repulsive terms are really important. On the contrary, only in this case we can speak of the molecularlike dynamics.

7 One-Level Approximation: Classical Rotator Model The success of the Born-Oppenheimer method is mainly due to a rather high accuracy of its simplest one-state approximation. The theory given in this paper should provide more accurate results also on that level of approximation, since it incorporates additional operators into the generalized two-center operator (14). Its eigenfunctions should be chosen for the case o f the zeroth total angular m o m e n t u m J. If J =/=0, the generalized two-center Hamiltonian for the rotational states looks too complicated. Recently the one-level alternative was proposed for the states with normal parity p = ( - 1) s, J > 0 [24]. In that case the last term of the generalized two-center operator (20) is left out and in addition only the K = 0 projection is used in the decomposition (24) of the total wave function. Under these simplifications the system of equations (27) is reduced to a single Schr6dinger equation. The authors of ref. [24] have not given any motivation for this approximation, though o-terms (K = 0) are known to be the most important ones in the Born-Oppenheimer method. The special choice of the rotational part of the total Hamiltonian in the form (20a) finds justification in ref. [25 ], where it is explained that in two cases, where the Coriolis coupling can be eliminated by redefining the body axes, the rotational part of the operator is transformed into the form (20a). The numerical example from [24] also supports this model. 8 Scattering Amplitude: Body-Fixed to Laboratory-Frame Transformation The coupled system of Schr6dinger radial-like equations for X[xP~(RA) that follows from the Born-Oppenheimer ansatz (28) has the usual form

- 2-M

+ ORA

+ eJP(RA)

42~R]~ + ~

Usp + QJp

XsP = E• (29)

Here eSP(Ra) is the diagonal matrix from the generalized two-center problem (27). The matrix elements of UJp and QSp are defined by

i, ] = (IKot},

(30) where ~b[p are the solutions of (27) corresponding to eJip. In some way the matrices

Asymptotically Adapted Three-Body Molecular States

139

U se and QSp restore the radial coupling of the problem, but this is an inherent feature of the Born-Oppenheimer scheme. It is rather straightforward to find the discrete spectrum of (29). It should be close to the one of the total projected Hamiltonian (26), if the redefined Born-Oppenheimer approximation is valid. In the case o f the scattering process (1) we look for a body-frame solution ~tm~ with initial relative m o t i o n in the direction o f the laboratory z-axis, i.e. D - l cal,m,c~,kl.e,] l m ~ t ~ "t exp(ikl,a,R~,) ~Ot'm'(ra'), cblm~(R~, r~) = exp(ikl~ Ra) ~0/m(ra) + ~11 ,.~,

t'm'~'

(31 )

where q0lm(r~)= ~Otm(ra)Ylrn('r~) with ~ = a or b are the wave functions of the b o u n d states for the scattering process (1) in the laboratory frame. The channel wave numbers are defined by

k~,~ = 2Mp,~(E - e/$(oo)).

(32)

This equation follows from the Schr6dinger system (29) using the relation (15a) between R and RA. The equations (29) should be solved in their original form with

(ka,~)2 = 2M(E - e/~(~))

(32a)

.a/z 9 In the expressions for the channel wave numbers, eS~(~,) so that kl~ = 7~a ,~t~,,~ represent the exact values of channel energies, as it was discussed earlier. Our basic solutions qtfflK~(Ra, r') with good q u a n t u m numbers J and p are n o w given by x~ffMJplKctt.l~

= ~

B ~Sp, (c~,

r/, RA)Xt'K'~' (Ra)Ra ,

(33)

K'l'od

where ~sp(~, rT;RA) are the solutions of the generalized two-center problem (27) and the expansion (28) was used. As RA -+ ~ b o t h (0tsxP~and B~v sp simplify ~sff~(~, r/;RA)

:

A

[

~otK(r,~)Ytx(r,~) exp -- iK ~o -

-+ RA__+ ,,~

BMs~:(o~, ~, 3')

~

"2

BMs~ (I,, |

,

~0 -- ~- .

(33a)

R A ~ oo

For the functions ~,JplKces xrK'~' ~I.AJ"t we have chosen the solution of (29) subject to the b o u n d a r y conditions .,JplKo~t-Cl'~ a.vx',~' ~,,J = 0 and

xJPIKc• l'K'~'

(~

) a/2 [6u,6xic6e, c,,p(-1)t e x p ( - ikla,~Ra)

-- SSP(l'K'odllKoO exp(ikff,~,RA) ].

(34)

The phase factor p ( - 1) l should be incorporated into the theory to simplify the presentation o f the free m o t i o n term of (31) w h e n it is transformed to the bodyfixed coordinate system [26], namely,

140

A.V. Matveenkoand Y. Abe i - -

exp(iktaRa) ~Olm(ra) -+

R~

v-~ 1 +p(-- 1)TM 2~ (2L + 1) PL(COS O)

2kt~R,~

2

p,L

X [ ( - li L exp(-iklo, R,~)- exp(ikt~Ro~)]~Otm(r~) _

i

4kl~R,~ Jp

(2J+ 1) [ p ( - 1)t e x p ( - ikt,~Ro~) - exp(ikto, Rc,)]

X B-ram dp, O, ~o-

~tma(~, ~7; c~).

(35)

Now the scattering wave function cblm'~(R,,,, r~) can be found by the expansion ~/m~(Ra,

A(J, p,M, l, m, a)~SPtK~(RA, r'),

r~) = ~

(36)

J,p,M

where A(J, p, M, l, m, a) should be given by i

A(J, p,M, l, m, o0 = ~ ( 2 J + 1)6rng~Kiml(MktAa)-1/2

(37)

in order to provide the asymptotic form (31). A further body-fixed- to spacefixed-frame transformation should be applied [26] to obtain the expression for the scattering amplitude from (31)

ftl'm'a',',,~r mo~ twa'

i(Mc~) 4(ktee~ct,a,)l/2 ~

~m

~

~iS2[rnl[1 + p ( - 1)/'-L' 1

J p L ',.Q~Q'

O, 0).

x (2J +

(38) In this expression the transition matrix

T sp = 5n,6aa, Saa, - S sp

(38a)

and the Clebsch-Gordan coefficients Ca~gb~are involved in the summation. The degeneracy-averaged scattered intensity is given by

I(cdl'lal) = (2/+ 1)- I k/-1kr~, ~

lm'~'( 0 , r [ftmc~

2,

(39)

mm'

and the integrated cross section

o(oe'l'loel) =

drb sin 0 dO I(a'l'lal)

(40)

7r 2 (2J + 1)pSp(oJl'lal),

(41)

o

o

is simplified to

where the average transition probabilities are defined by (42)

Asymptotically Adapted Three-Body Molecular States

141

The scattering amplitude from (38) is the main result of this paper. It was not possible to obtain it without approximations involved in previous attempts [2, 4, 15], because the asymptotically adequate two-center Hamiltonian was not available until now. In the derivation of (38) we followed the paper by Pack [26], where some important details can be found. 9 Summary The formal theory for three-body rearrangement scattering processes in the molecular-states approach was formulated for the first time without any difficulties from unphysical long-range couplings. The main idea used was to transform the traditional molecular three-body Hamiltonian into a form free from radial or angular coupling in the asymptotic region. Hence the wave function and the internal variables were to be changed. The transformed Hamiltonian was projected onto states with good quantum numbers of total angular momentum J and parity p, thus providing a system of Schr6dinger equations in three variables. Two of these variables can be chosen to describe the fast subsystem. The relative two-dimensional Harniltonian is to be referred as dynamical two-center Hamiltonian for rotational states. Its eigenvalues and eigenvectors provide grounds for the Born-Oppenheimer-like approach with original physical intuition but without the traditional asymptotic difficulties. In this way the eigenvalues of the generalized adiabatic Hamiltonian (27) form a family of effective potentials for the radial-like system of Schr6dinger equations (29) in the slow variable. The solution of this system for scattering states in the form (34) defines the body-fixed molecular stage S-matrix S Jp. The specific phase factor in (34) is chosen to simplify the matching of the general body-fixed solution (36) with the scattering solution of the usual form (31 )given in the laboratory frame. The laboratory-frame scattering amplitude (38) follows from this matching. If molecular-type states are really involved in the scattering process only those e~P(RA) that are attractive and capable of supporting bound states are important for the scattering process. A further preliminary information is supplied by the general behaviour of the elements of U sp and QSp matrices, which are sensitive to such specific phenomena as the crossing and quasi-crossing of e~nP(RA), which are very important for the dynamics of reaction (1). As a result, only few matrix elements of S sp for a limited number of {J, p}pairs should be included into the summation (38) in order to derive the molecularstate scattering amplitude. The important kinematical features are completely accounted for by the Clebsch-Gordan coefficients and the coefficients of the irreducible representations of the rotation group. The angular distribution of the products of reaction (4) should be very sensitive to the particular form of the scattering amplitude (38) and we believe that it also can serve for the parametrization of the accurate experimental differential cross-sections, if the molecular description of process (1) is supposed to play a governing role. Expression (38) for the scattering amplitude is a partial-wave decomposition very close in structure to that of the usual theory of elastic two-particle scattering [3]. In our case partial amplitudes are much more complicated and the angular

142

A.V. Matveenkoand Y. Abe

distribution of partial waves is accounted for by Wigner D functions of the two angles that define the vector connecting the scattering products. From the formal point of view we have a multichannel scattering in a non-central field. As usual, the partial-wave representation of the scattering amplitude should be used in the low-energy region, where only few partial waves are really important. The three-body Hamiltonian HAa (19) was derived from the original BornOppenheimer adiabatic Hamiltonian (10) in order to describe exactly the effect of the pair collision in a three-body system. The case of the triple collisions was not taken into account explicitly. Nevertheless, the RA-part of//An coincides precisely with the hyperradial part of a three-body Laplacian, what is a proper way to account for Fock's singularity [20]. It is clear that the rest of the Hamiltonian (19) can be given in the form of the "angular hyperspherical part" of a three-body Laplacian. Due to this fact one can say that we have established the connection between the Born-Oppenheimer and hyperspherical harmonics methods in a threebody problem. At this point it should be mentioned that there exist recent calculations of a molecular-like system in the hyperspherical basis [27]. The convergence of the method happened to be rather poor. In this paper we went into the opposite direction. As the starting point the traditional molecular Hamiltonian was chosen and transformed to account better for the finite masses of the "centers". The one-state Born-Oppenheimer-like calculations of the binding energies of the eee +system [23 ] and the position of the U = 2, p = 1) resonance in tp- + t scattering proved that our approach describes the dynamics of a three-body system much better than the original Born-Oppenheimer method [24]. In 1981 Fano [28] put forward a program for a unified treatment of collisions in a system of few particles. In our paper this program is partially fulfilled. Though the scattering problem in the system of three particles was the main subject of this paper it should be noted that our work is also highly relevant to most chemical physics. In this context we refer to refs. [29] and [30] with regard to the formulation of molecular problems in the body frame of the inertia tensor and to hyperspherical formulations, respectively.

Acknowledgement. The authors highly appreciate the useful comments of Prof. W. Scheid on reading the manuscript of this paper. The fruitful discussions with Dr. P. Fiziev were very helpful. References 1. Born, M., Oppenheimer, J. R.: Ann. Phys. (Leipzig) 84,457 (1927); Born, M., Huang, K.: DynamicalTheory of Crystal Lattice. London-NewYork: Oxford UniversityPress 1954 2. Park, J. Y., Scheid,W., Greiner, W.: Phys. Rev. C20, 188 (1979); Broglia, R. A., Liotta, R., Nitsson, B. S., Winther, A.: Phys. Rep. 29C, 291 (1977) 3. Landau, L. D., Lifschitz, E. M.: Quantum Mechanics.Oxford: Pergamon 1965 4. Vinitsky, S. I., Ponomarev, L. I.: Soy. J. Part. Nucl. 13,557 (1982) 5. Matveenko, A. V.: JETP Lett. 40, 1329 (1984); Pack, R. T.: Phys. Rev. A32, 2022 (1985); Macek, J., Jerjian, K. A.: Phys. Rev. A33,233 (1986) 6. Power, J. D.: Phys. Trans. Soc. Lond. A247, 663 (1973)

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7. Judd, B. R.: Angular Momentum Theory for Diatomic Molecules. New York: Academic Press 1975 8. Gocheva, A., et al.: Phys. Lett. 153B, 349 (1985) 9. Frolov, A. M., Efros, V. D.: J. Phys. B18, L265 (1985) 10. Bunker, P. R., Moss, R. E.: Mol. Phys. 33, 417 (1977) 11. Ponomarev, L. I., Somov, L. N., Vukajlovic, F. R.: J. Phys. B14,591 (1981) 12. Bates, D. R., McCarrol, R.: Proc. R. Soc. A245, 175 (1958); Smith, F. T.: Phys. Rev. 179, 111 (1969); Chen, J. C. Y., Ponce, V. H., Watson, K. M.: J. Phys. 6B, 965 (1973); Delos, J. B.: Rev. Mod. Phys. 53,287 (1981) 13. Matveenko, A. V.: In: Dynamics of Few-Body Systems (Proceedings of the Xth European Symposium, Balatonftired, Hungary, 1985), (Bencze, Gy., et al., eds.), p. 311. Budapest 1986 14. Matveenko, A. V.: Proceedings of the 1983 RCNP International Symposium on Light-Ion Reaction Mechanisms, Osaka, 1983, p. 906 15. Smith, F. T.: Phys. Rev. 120, 1058 (1960) 16. Matveenko, A. V.: Phys. Lett. 129B, 11 (1983) 17. Soloviev, E. A., Vinitsky, S. I.: J. Phys. 18B, L557 (1985) 18. Varshalovich, D. A., Moskalev, A. H., Khersonsky, V. K.: Quantum Theory of the Angular Momentum. Leningrad: Nauka 1975 19. Smirnov, Yu. F., Shitikova, K. V.: Soy. J. Part. Nucl. 8,344 (1977); Ballot, J. L., Fabre de !a Ripelle, M.: Ann. Phys. 127, 62 (1980) 20. Fock, V.: K. Norske Vidensk. Selsk. Forhandl. 31, 138 (1958); Klar, H.: J. Phys. A18, 1561 (1985) 21. Johnson, B. R.: J. Chem. Phys. 73, 5051 (1980); J. Chem. Phys. 79, 1906, 1916 (1983) 22. Macek, J. H.: J. Phys. B1,831 (1968) 23. Kaschiev, M., Matveenko, A. V.: J. Phys. B18, L645 (1985) 24. Kaschiev, M., Matveenko, A. V., Reval, J.: Phys. Lett. 162B, 18 (1985) 25. Hori, S.: Suppl. Prog. Theor. Phys. (Extra Number) 80 (1968) 26. Pack, T. P.: J. Chem. Phys. 60, 633 (1974) 27. Aquilanti, V., Grossi, G., Lagana, A.: Lett. Nuovo Cim. 41,541 (1984); Greene, C. H.: Phys. Rev. A26, 2974 (1982) 28. Fano, U.: Phys. Rev. A24, 2402 (1981) 29. Nakamura, H.: Phys. Rev. A26, 3125 (1982) 30. Manz, J.: Comments At. Mol. Phys. 17, 91 (1985) Received August 26, 1986; revised November 10, 1986; accepted for publication February 3,1987