/ • • -:> r - w z

cnimm ебидшимп

института 1ДМ1ЫХ

исшднаш дубиа

Е4-91-288

V.I.Korobov, I .V.Puzynin, S.I.Vinitsky

BOUND STATES OF HYDROGEN MESIC MOLECULAR IONS: VARIATIONAL APPROACH

1991

During the last decade, the energy levels of the bound states of mesomolecular ions

of hydrogen isotopes were subject of intensive study of the fenomenon of muon catalyzed

fusion [1-6). Characteristics of the bound states with the spatial parity A = ( —1) are

computed there, and especially ones of weakly bound states of mcsic molecular ions dd^i

and dtft defining mainly the kinetics of physical processes. Last year papers studying

mesomolecular s tates with the so-called anomalous parity A = - ( - 1 ) were appeared [7-

9). These states differ essentially from the usual ones in their physical properties. They

are assumed to play a role as hindering factors in the process of mnonic catalysis. Having

captured a muon they withdraw it from a normal catalytic cycle.

However, with the growth of the total angular momentum the amount of energy level

calculations provided with different methods decreases significantly and already for the

F - s t a t e of mesomolecular ion tip there is not any variational calculation and the only one

was carried out with the use of adiabatic approach [ij. In this paper we present the results

of calculation for the energy levels of 32 states of mesomolecular ions of hydrogen isotopes.

It turns out that these states, 22 states of normal parity and 10 states of anomalous parity,

represent the whole list of bound states of mesic molecular ions.

The variational approach to be expounded below was used in our previous papers [2,7].

We extended the method to states of an arbitrary spatial parity and a total angular mo­

mentum J > 2. The form proposed for the trial functions allow us to take into account

2 © Объединенный институт ядерных исследований Дубна, 1991

peculiarities of a behaviour of the wave function components connected with the use of rota­

tional and spatial symmetries anil to compute effectively matrix elements. As computation

shows, the instability doesn't increase dramatically with the number of trial functions and

it allows to carry out calculations with more than 2600 basis functions.

1. T h e o r y

As it is well known, a hydrogen mesic molecular ion is a system of three particles: a

negatively charged muon / / " , and two nuclei a and b of hydrogen isotopes (a, 6 -- p, i/ or / ) .

The nonrelativistic Hamiltonian of the system in the Jacobi coordinates (e — h --- mt- -• 1)

is

1 1 1 1

*1Ч ra rb '<

n i j 1 I (m, t m2) l =m„X + (Л/„ + Мь) ' .

Here R is the position vector of nucleus a relative to nucleus b with the masses Ma r : Мь,

r is the position vector of the muon with the mass m^ relative to the center of mass of

the nuclei. We will connect a moving frame with this system with respect to which the

Kuler angles will be determined. To do that we will direct the z' axis along the nuclei of

the molecule, and the a,f axis will be placed in the plane of three particles in the direction

of the muon.

Since equation ( I ) is invariant under three dimensional rotations, we may separate

the states with a certain quantum number of the total angular momentum ./ and its

projection Л/ onto the с axis. The wave function may be expressed in terms of the Wigner

/^-functions, D[a,lhy), dependent on the Euler angles of the moving frame:

.1

*M(B.,I-)--= Yl 0м.«(»-<' .7)^(Я,г,в) , С!) m- -.1

2/M2

' ' 1 2

,'i

where в is the angle between vectors R and r. The Hamiltonian also commutes with the

operator of spatial parity Яд : (R, r) —> (— R, — r) , having eigenvalues A = 1 and A = - 1.

This operator acts on the Wigner O-functions as follows:

To determine the wave function of a given spatial parity A, we introduce the symmetrized

Wigner D-functions l

^,(» .Лт)=[ 1 ^1^)]М д *" ( а ' Д т ) + А ( ~ 1 ) ' , " " д * - - - ( о ' / ? ' 7 ) ] • which are eigenfunctions of the operators J 2 , J. and Py.

Р)Р^т = \Р^т, А = 1 , - 1 ;

and are normalized so that

jdc Jsin/3d/3 j'dr,\vQM' ( а , / 9 , 7 ) ТУ^'т, ''mm •

The wave function in the representation of the total angular momentum and spatial

parity has the form

J *w(R- ,r )= Y, х > м т ' ( « . / 9 , 7 ) ^ А ( Я , ' - . в ) ;

т=д(Д) (,'J)

/ 0 , for A = < - ! ) ' ; * ' \ l , for A = • ( - 1 ) J ;

This expression is a starting point for futher construction of a variational trial function.

The states of parity A = ( - 1 ) and A = - ( - 1 ) arc called, respectively, the states with

normal and anomalous parity.

Another form of the wave function expansion follows from the Schwartz representa­

tion [10] based on bipolar harmonics. We will present it because it explicitly describes the

behaviour of the wave function of a state in the vicinity of coalescence points of particles

with a given spatial parity and a total angular momentum:

' ' t , J ' J (4a)

for A = ( - 1 ) J ;

4

*#(R, r ) = Y, Y'%-(А,т) Rb rb-GiX(R,r,0),

for A = - ( - I ) - ' ;

where

V J # ( R , r ) = £ ^ „ . ( R J y i ^ f r X / m u b m j I J A f )

and

GflU^r,0) = GflilOfT,$)\l + — i — fi + R - u + 0 ( R 2 ) .

The last expression is the so-called Kato coallescence condition [11,12]. Note that the

number of components in the expansion (4a) is equal to J, whereas in (46) to 7 - 1,

which coincides with the number of components in the expansion of the state with the

corresponding parity in (3).

In the expansion (4b) t\ and ii differ from zero, therefore the asymptotic scattering

states of parity Л = —( — 1) , for Я —» oo, consist of an isolated nucleus and an atom

with nonzero angular momentum. Thus, the energy of dissociation for the molecule with

anomalous parity is equal to the energy of the isolated atom with principal quantum

number n = 2: Ea = - R y m a / * 1 , where rna is the reduced mass of the atom with the

heavier nucleus.

Applications of the Wigner's D-functions to three-body calculation were developed

by many authors [13]. They differ in positioning of the moving frame of references and,

thus, they have different partition of the wave function to the components depending on

intrinsic coordinates of the problem. Our choice is convenient to the adiabatic calculations

involving two heavy particles and one light particle of an opposite charge.

2. T h e wave func t ion

To obtain an accurate variational solution for the Schrodinger equation with the Hamil-

tonian (1) it is necessary to take into account the peculiarities of the behaviour of the wave

function.

Symmetry allows to reduce the equation onto a subspace of states with certain quantum

numbers of the total angular momentum and spatial parity. The wave function of the

state as well as separate components of the wave function representation obeys certain

5

requirements imposed on the behaviour at the vicinity of points of collisions or in tho rase

of axial singularity when a position of a moving frame isn't determined. In particular,

a three-particle system is properly described by the wave function (4) with well-behnvcd

components G j ^ (R,r,0). Unfortunatly, this expansion leads to significant difficulties in

matrix element calculations for high numbers of total angular momentum. We would like

to investigate the behaviour of the components of the expansion (3) to provide necessary

conditions for constructing of a suitable trial function for this approach.

To determine the asymptotics of the components of (3) when R --• 0 we make use of

the Chang-Fano relation [14] between the bipolar harmonics and Wigner 0-functions:

yJjJ(ft,f) = vS Y. 4tVMm (^P'-l) *WM), (•'•) т=„(Л)

where the coefficients Tj ™A are expressed through the Clebsch -Gordan coefficients:

'l + А ( - 1 ) ' 1 + ' Л 2/j + 1 rriJmX (l,0t2m\Jm).

The relation (5) allows us to express the components of the wave function expansion (3)

in terms of the components of expansion (4):

J

Fix{R,r,e) = V2i X ; r , f £ A K , , m 0 9 , O ) R1'rh GfX(R,r,0),

. [ l \ ••-- J h , for A (- \)J, where < ' i

U =J I 1 - l2 '" v ' " J

(«) for A - - ( - 1 Г

From this representation it is seen that the functions F^ have the asymptotics О (1) when

R —> 0 for the states of normal parity and 0{R) for the states of anomalous parity. It is

to be noted that , when R = 0, the components F^ with different m are dependent on

each other and may be expressed through one component., for example / ' j .

The behaviour of the components in the neighborhood of the nuclei coallosceurc point

restricts to the choose of intrinsic variables describing the trial function. The functions

G / J (R,r,0) of the expansion (4) do not depend on the angle 9 at R -- 0 [12|. Hence, the

components of (3) have the angular dependence, when R -* 0, and this dependence is to

be taken into account. It is hardly reasonable to use the distances between particles: ra/,.

6

rafi and гьр (the so-called Hylleraas variables), as intrinsic coordinates with expansion (3)

because, for r a $ — 0, ra(t and rbft arc equal to each other, and it is impossible to deter­

mine the angle variable в in terms of them. On the contrary, the spheroidal coordinates,

£ " (гац +• гър) /R and r\ - (r„M - гьр) / Я , at ft = 0 give, respectively, the radial and

angular dependence of the function F„t . It is quite surprising result because the Hylleraas

coordinates are very useful with the Schwartz expansion (-1) [3 5].

Now consider the behaviour of the components of (3) when three particles lie on the

same line (axial singularity). 'Го do this let us express the wave function in terms of the

coordinates: a , /? , /?,-r,, ?/,, zt, where the first three coordinates are spherical coordinates

of the vector R, and the last three are coordinates of the vector v with respect to the frame

of the spherical coordinates of the veclor R. The singularity occurs when the coordinates

;r, and i/, become equal to zero, In this case the position of the moving frame is indefinite.

It is known that the wave function is infinitely different iablc beyond the coallescencc

points [15], Therefore it may be expanded in a Taylor series around a regular point placed

on the axis:

Ф # ( П , г ) £ ^ / , , K , ? , / f , : ) t / , 4 ^ ( o . . ^ / ? . . r „ . v . , r , ) . M J < J (7)

l , •_• 2 x 1 / 2 where /> [.r, L */, )

Multiplying the function Фд} by '1 м,„ (<>./^i 1) {»i'd integrating over the angular variables.

we obtain the component of expansion (3), / ' , '„- On the other hand, integrating first over

the angular variable 7 and considering that

Jo

we find that F^ (/.',/>,-)) /»'" / / ; ' X (/f>,->) and the functions H^ are hounded in the

neighborhood of the axis.

We now can construct the expansion for I he wave function that defines the Hayleigh

KHz variational procedure for the stationary SVhn'idinger equation. 1'sing (he spheroidal

coordinates we introduce the expansion for the components Ffn of (3):

Fi>{R,s,V) = n"4(e-i)(i-r,i)r/2x x ^ С Я ^ ' ч ' " e x p ( - ( « „ 4 Ц „ { | й ] ,

(8)

and { b S j - ifA = ( - l ) J

I >n > in, if A = - ( - 1) ; that takes into account the asymptotics of the axial singularity since

The inequality establishing admissible values for powers of the variables R and £ provides

the behaviour: F^f —+ const for normal parity and F^ -* R for anomalous parity, when

R —> 0. Note that the condition connecting the components at the coallescence point of

nuclei complicates the computation of the matrix elements and it will be omitted.

For identical nuclei the Hamiltonian commutes with the permutation operator of parti­

cles Раь: (R-,r) —* ( - R , r ) , and the wave function can be decomposed on parts symmetric

and antisymmetric under permutation:

*J

M*« (R,Y)-- (I - Л,*) * # ( R , r ) (9)

The operator P„(, acts on the Wigner functions: Pat^itm -'-• ( - 1 ) OJ

M _m, which deter­

mines a symmetric and antisymmetric parts of the components of expansion (3):

r i ( 1 0 ) F^°(R,i,l)= [FiUfl,Z,v) A ( . . . i r ^ ( R , f , - 4 ) ] .

The components F^ are here either even or odd with respect to the variable 77. Therefore

it is necessary to retain either even or odd powers of this variable under the sum sign in

(8).

To describe the behaviour of the wave function, when R -> 0, we make use of the

representation (4). Symmetric and antisymmetric terms of the components of the wave

function expansion (4) are of the form

o(bW'.«)=[ciiC,'.»l-(-i|''c{iW'1»-»)], (") 0 < tf < jr.

8

When GJ¥(R, rt7j) (t = s or o) is odd with respect to the variable t) = cos в then it

vanishes a t the coallescence point of the nuclei (we recall that component G , ^ does not

depend on angular variable for 7? = 0). Thus the necessary corrections to the behaviour

of the function ^ л ' ( Я , £ , л ) can be done by using the formula (6) for symmetric and

antisymmetric terms respectively.

The representation (8) for the symmetric wave function was for the first time used

in [16] for calculating of the energy level of the mesic molecular ion ppfi in a state with

total angular momentum J = 1 and normal spatial parity.

3 . Calculat ion of matr ix e lements

Substitution of the basis sets of (3 ) , (8) into the functional

(Ф,ЯФ)/(Ф,Ф)

reduces the initial equation (1) to the generalized algebraic eigenvalue problem

Ax = XBx, (12)

where the matrix A is composed of the Hamiltonian matrix elements (Ф^,ЯФ 7 ) , and В is

the overlap matrix of basis functions (Ф<,Ф^), both are symmetric.

To compute the matrix elements in (12), we carry out at first integration over angular

variables, it allows us to obtain the system of differentia] equations in a three-dimensional

domain of intrinsic coordinates describing internal position of particles. As a next step, we

rewrite that system in spheroidal coordinates of the two-center problem: R,(,i}. Then,

inserting the basis functions of the expansion (8), we reduce the problem to the calculation

of certain integrals that can be found analitically.

Before computing the integrals over angles, we rewrite the kinetic energy operator in

a convenient form:

\2' 1

2(xi2 1 д R2d f (J-l)'

R2 dR dR Д 2 ^ Д - <»>

Here J is the operator of total angular momentum, I is the operator of the muon orbital

momentum with respect to the center of mass of nuclei, and the last term is the operator

9

of the muon kinetic energy, where the vector r , is expressed in terms of the cylindrical

coordinates, 7,/>, z», with respect to the frame of reference of spherical coordinates of the

vector R . We recall tha t the position of the reference frame of spherical coordinates is

defined by the Euler angles a,/3,0 (a and /? are spherical angles of the vector R ) .

The first term in (13) acts only on the functions dependent on the intrinsic variables

of the system, whereas the operator of the total angular momentum is expressed in terms

of the Euler angles and its action on the Wigner functions is defined by

*±P®m = - ( ± ^ - iJ,')VJ

M

X

m = ± V J ( J + 1) - m ( m i \)VJ

M\mil,

where J z / , Jyt and Jtt are components of the vector .1 with respect to the moving reference

frame. The position of the moving frame is determined by rotation of the reference frame

of the spherical coordinates of the vector R on an angle 7. The operator 1 in the moving

frame is given by expressions:

Э

4 = -(±','-*V) = ^ ~ д а

Since the components of J and 1 commute [\Ji, U\ - 0 , i - ^ ' ,y ' , ~ ') , we may transform the

second term of (13) as follows:

(J - l ) 2 _ J 2 + l 2 - 2 ( J , 1 ) J 2 _+ l 2 - 2Jjls, ./;_['. l_Jir+ 2цп№ ~ 2/Л2Й 2 " " 2 ^ 1 2 Я 2 ~ ' ~~2дГ 2Я 2 '

The kinetic energy operator of the muon in the cylindrical coordinates may be written

_ J_ /1?L I?- 'L °- X ' 2 М з I P 2 O7 2 + р9ррдР ' a ? J '

Next we insert the expansion (3) into the kinetic energy operator (13) and integrate

over the angular variables. Upon simple calculations, using the expressions obtained for

the operator components and the orthogonality relations for the Wigner functions, we get

the system of equations for the components of the expansion (3):

(I'D E (H-m • - * w 0 * # 0

m ' - | i ( A )

m • / < ( A ) , . . ,J

10

и - - U "mm — \

/'12 l

l f hk#k-h\e + \« + J ( y + l ) - n , 2 + . - 2 m 2 / / > 2 \ . P m

(m J. 1) 1 £ | .

(14')

<m.,m± 1 - . 2/1Л2 [p

Hero r о а 2 ™ 2 i <?_ a_ _ 9 2 _

' " " V2" p ^ p a p a";2

- > ™ H - { [ I ' f„,oA( i ) J ] l ^ ( J - t i ) - r»(n, ь о ; . } 1 2 , . J . \ . . .П / m l 1,MI ' » t , m i I '

«iiicl V(R,p,z) is the potential energy of the Coulomb interaction. So. the problem of

separation of angular variables is completed.

Now the IlainiKonian may he finally written in the spheroidal coordinates of the Uvo

center problem:

1 Г 1 f> ..; <> " ш т ~ 2/7,2 1 /? 2 '<>я" OR'

я2 (e - T' _ 2 K - _ , 2

* p i + - - •«.- K p , ' " ' M "'•«•":

( 1 Л « )

where —2 - (£ 2 + n 2 I I 2<>; I

'2/i и И 2

«'/

»'1„,H1 2,,,2/f2

//„„„• - o; whon |n ; и» | • 2;

II.V.)

(I .VI

V[H.t,,,)

m 2 ( f 2 r,2) i) i) i) 0 P 2 Щ{( ' U ,•»„<' " ' , 7 , , /? 2 (£ 2 i,2

I I

and the parameter к = (Мь ~ Ma) / (Ma -f- Мь) determines the shift of the origin of tin*

vector г from the center of mass of nuclei to their geometric center.

To determine the matrix elements (Ф<, НФ^) it is necessary to insert the basis functions

(8) into (15a-c). Considering that the element of volume in (lie spheroidal coordinates is

equal to R (£ — n 2 ) / 8 , wc find that fractional powers and rational parts of expression

are cancelled out and only a polynomial of variables /?,£ and n multiplied by exponential

remains. In this way the problem reduces to the computation of integrals of the form:

Г , т (<*,/?)= / dR J RlC e x p [ - ( « + / ? 0 R | < / £ , and I'„ / r," d,,. Jo J\ J i

The first integral after integrating by parts is reduced to

Г , л m j n ! ( . ^ m - i ) ! _ Л С - " M i 1) ! / H \' , m ( , ^ = ^»+i"(aT/j ' j j = "^"(' "< I)-''-! \" ' >*) '

The basis functions (8) are either even or odd with respect to the permutation of heavy

particles. To simplify the computation of matrix elements we separate the part linearly

dependent on the asymmetry coefficient к from the Hamiltotiian (15). Then the remaining

part connects basis functions of the same periuutational parity, whereas the separated pari

connects functions of the different parity.

One can see from the formulas (15a-c) of the JJamiltonian, it is well fitted to tin-

calculation of the adiabatic systems. Indeed, the reduced mass of the heavy particles / i ^

in this case is much greater than the reduced mass of the light particle дз> and я ratio

fi = //3/^12 c a n be used as the adiabatic parameter of the system. Off diagonal terms

of equations (14) - (15) and the part connecting symmetric and asymmetric components

of the wave function are proportional to the small parameter /x, and, thus, the quanlnm

numbers m and the eigenvalues of the permutation operator arc approximate quantum

numbers of the adiabatic systems.

12

4. Resu l t s

Here we shall analyse the results of computation of the binding energy levels of me-

somolecular ions of hydrogen isotopes. Partially they were published earlier in [2,7,19]

and they were obtained with the use of some slight modifications of the method we have

expounded in this paper. Without going into details we explain only the strategy of choice

of the sets of the basis functions.

The basis functions (8) depend on the nonlinear parameters a and /?. Search for optimal

nonlinear parameters for every basis function is a cumbersome task. Therefore, in practice,

a set of nonlinear parameters otmu fimt i s used which depends on a component number то

and on a permutation parity of two heavy particles t of a given basis function. The wave

function is a complicate function of the variable Я, and it is convenient to introduce two

groups of nonlinear parameters that provide good c;. i,<;rgence of the expansion at the

vicinity of zero and at long distances. In particular, this trick was successfully applied to

study weakly bound states when a molecule is extended in the coordinate space. Here we

use two sets of nonlinear parameters in all calculations.

The eigenvalue for the certain state of the mesomolecular ion is computed as follows.

For fixed values of nonlinear parameters the problem (12) is solved for a given state, either

ground or excited. Then the eigenvalue is minimized by varying nonlinear parameters.

Since the estimation for the energy derived from (12) provides the upper bound estimate,

the minimization retains this property and only increases the accuracy of the solution.

The nonlinear parameters we have used here are cited in [2,7,19].

The algebraic problem for eigenvalues (12) was solved by inverse iteration method

(it can be found, e.g. in Ref. 17) for the following reasons. First, a particular bound

state is calculated for some values of nonlinear parameters optimal for the given state.

Therefore, it is sufficient to look for only one eigenvalue and the remaining part of the

spectrum is not necessary. Second, this method provides less computational time. We

make use of the symmetric structure of the problem and the number of operations is about

n 3 / 6 multiplications which is significantly lower comparing to other direct methods of the

eigenvalue computation. This is especially important for calculations with a large number

13

of basis functions when the computational time of the problem (12) rapidly grows with

the number of functions and very soon becomes dominant.

As the number of basis functions increases, the algebraic eigenvalue problem (12)

becomes more and more singular. The inverse iteration method is the most stable one for

the solution of that problem. However, it also cannot ensure the stability of eigenvalues for

large matrices. The stability is achieved when we solve the following regularized eigenvalue

problem [18]:

A.x = XB.x, where A. = A + SD, B. = B;

where S is a small parameter, and a diagonal matrix D is composed of elements \ац - A6,,|.

That removes from the spectrum infinite eigenvalues defining the singularity of the problem

and causing instability of the true eigenvalues. To be successful this procedure requires

the following condition:

*№\* + \ \ в \ \ 7 ) 1 / 2 - M —lf7 « U \{x,Ax)2 + (xtBz)2]

where x is an examined eigenvector, and e is a round-off error [18]. When an exponential

expansion [3-5] is used, the latter condition is not satisfied, therefore the only way is

to increase the precision of calculations to quadruple precision. In our calculations this

condition was checked and it was fulfilled even when 2600 basis functions were used. The

parameter 6 was taken in the interval 10"" 1 4 — 1 0 - 1 5 and its variation allowed approximately

estimate the accuracy of a calculated eigenvalue.

The energy levels of mesomolecular ions are reported in Table 1 (normal parity) and

Table 2 (anomalous parity). The values for 5 - and P-states presented in Table 1 have been

computed in Ref. 2. The only exception is the calculation of the weakly bound state of dtfi

with J = 1 for which a more accurate series of computations was performed and listed in

Table 3 below, here щ is the number of basis functions employed. The values of D and F

states obtained with the use of this variational method are published for the first time. The

corresponding series of computations with a growing number of basis functions are reported

in Table 4 (the number of basis functions was pointed out in parantheses). The energy

levels of mesomolecular ions with anomalous parity have been studied in [?,19], here we

present only the results (Table 2). In the calculations we made use of the following values

u

Tabic 1: Binding energies -cj,. (in eV) of mcsomoi -ciilar ions of hydrogen isotopes with spatial parity A — ( 1) .

J и

0 U 0 1

1 О 1 1 2 О 3 О

ррр

253.1523

107.2658

ddp

325.0735 35.843В

226.6815 1.97475 86.4936

tip

362.9097

83.7711

289.1419

•15.2057

172.7022

48.8376

pdp

221.5494

97.4980

Р' / '

213.8402

99.1262

dtp

319.1396

34.8340

232.4711

0.6600

102.6486

T a b i c 2: Binding energies сj,. (in rV) of iiiesuinalrriiiar ions of hyilrogr isotopes with spatial parity A ( I) .

J v

1 0

2 0

3 0

ppp

13.540(1

ddp

22.5912

9.8203

"!• pdp

27.4122 .1.6849

17.7713

5.4187

pip

1.5788

dip

19.1217

7.S990

15

of physical constants: Mp - 1836.1515m f f, A/ d - 3670.481m f, Л/, ----- Г»496.918т е >ш ( 1 -

206.76867n t,Ry = 13.6058Q41eK. The energy levels have heen obtained in Ref. 2 with the

use of muonic mass m^ = 206.769m e here we convert these values to the given mass set

adding the energy shifts computed in Rcf. 20.

Tables 6-8 contains a comparison of muonic molecular calculations with the best liter­

ature results. One can easily see that S- and Я-s ta le energy levels are in a good agreement

with the highly accurate exponential Slater-type basis set [4] results. As far as the states

of higher rotational numbers is concerned the situation changes. The only one result that

is lower than ours was obtained with the use of the same Slater-type basis set method

for the mesornolecular ion dtfi{J = 2) [5]. The reason is that the evaluation of angular

integrals with bipolar harmonics of high angular momentum and /or anomalous spatial

parity (Eq.(4)) becomes very laborious. In any case our results seems to have the errors of

computation comparable with the error dependent on the uncertainty in the measurement

of the mass values.

An important characteristic of a molecule is its mean size. In Table 5 we present

mean sizes of mesomolecular ions of hydrogen isotopes for states with the total angular

momentum J > 2 with normal parity in units e .-_ fi - mh 1. The accurate values

of mean sizes of mesomolecular ions with J < i of normal parity were calculated in

Ref. 21 (in our approach we reproduce usually four figures of their results). Mean sizes of

mesomolecular ions of anomalous parity have been calculated in Ref. 19.

We have also studied the possibility of the existence of other bound states. The op­

erator H - e+E, where s is the energy of the molecule dissociation, has as many negative

eigenvalues as there are bound states of the molecule. Therefore, if we insert a trial function

of high dimension we have the right to expect that a finite-dimensional matrix A - E.B

exactly reproduces the number of negative eigenvalues of the initial operator and thus

determines the number of bound states of a molecular system. This estimation has been

performed for all the systems presented in Tables 1 and 2 and for the mesic molecular ion

ttfi with the total angular momentum J ~ A with the use of about 700-800 basis functions

and this investigation has shown that there are no other bound states for mesomolecular

ions of hydrogen isotopes.

16

Tabic 3 : Series of calculations of the energy level of a weakly bound state of mesic molecular ion dtp.

i n» - £ ] ] ( n ; )

1 1513 0.65923

2 2084 0.65968

3 2174 0.65985

4 2313 0.65990

5 2660 0.65999

Table 4 : Series of calculations of binding energies -£jv (in eV) of meso-molecular ions of norma] spatial parity with the total angular momentum J > 2 .

drf/z(J = 2| ttp(J = 2) dtti[J = 2) ttfi(J = 3)

86.48790 172.69753 102.64146 48.83234

(160) (160) (212) (160)

86.49237 172.70066 102.64422 48.83570

(222) (222) (274) (222)

86.49361 172.70215 102.64861 48.83764

(426) (426) (550) (442)

17

Table 5: Mean sizes of mesir molecular ions of normal spatial parity with the total angular momentum J > 2.

J - e j Я 4 Та Я 2 r\ d •

ddft 2 86.494 3.910 2.688 2.688 16.86 10.03 10.03

tip. 2 172.702 3.298 2.359 2.359 11.76 7.423 7.123 j

dip 2 102.649 3.599 2.634 2.408 14.15 9.376 7.923

Нц 3 48.838 4.040 2.750 2.750 17.77 10.49 10.4П !

Tabic 6 : Comparison of muonic molecular calculalions (normal parilv, J = 0,1).

J V This work Ref. 4 Ref. 1

PPH 0 0 253.1523 253.152332 253.0

dd/t 0 0 325.0735 325.073540 325.0

ddfi 0 1 35.8436 35.844360 35.8

iili 0 0 362.9097 362.909770 363.0

till 0 1 83.7711 83.771216 83.9

pdfi 0 0 221.5494 221.549410 221.5

р1ц 0 0 213.8402 213.840179 214.0

dt)i 0 0 .419.1396 319.130722 319.2

dtii 0 1 34.8340 34.834491 34.9

PPf 0 107.2658 107.265971 107.0

ddp 0 226.6815 226.681678 226.6

ddfi 1 1.97475 1.974817 1.956

tt/t 0 289.1419 289.141783 289.2

lift 4.1.2057 45.205856 45.2

pd/i 0 97.4980 97.498160 97.4

р1ц 0 99.1262 99.126501 99.0

din 0 232.4714 232.471594 232.4

dtp 1 0.6600 (1.660178 0.050

.. I

1 »

Tabic 7: Comparison of uinonir molecular calculations (normal parity, J >2).

ddp

Up

dip

Up

— _ T 1

V This work Rcf. 5 ; Ref. 3 : Ref. 1

0 ЯВ.-1936 86.1311 86.3 j

0 172.7022 172.5204 ! 172.7

0 102.6486 102.61891 102.6318 102.5 J

0 •18.8376 1 -18.7 '

Table 8: Comparison of niuouir molecular calculations (anomalous parily).

This work Ref. 8 !

ddfi

а,, pdl,

pi I,

,ll(i

,l,l,i

III,

dtp

It,,

13.ri.H10 13.5-12

22.51112 \ 22.5!I5

27.1122 : 27.113

3.6S11I j 3.68.]

1.5788 | 1.566

Ml. 1217

11.82113

17.7713 j

7.8(100 !

5.1187

Ml. 121

14

There exists a less complicated but also less rigorous proof of tlir statement. The shift

of energy levels due to the rotational motion of a molecule is proportional to mM/;i|;> J22).

According to this rule we may extrapolate the energy levels given in Tables 1 and 2 to

higher values of J. All the extrapolated values belong to the continuous spectrum which

means that the corresponding bound states of mesomolecular ions do not exist.

The authors would like to express their gratitude to Prof. 1<.I. Ponomarev for collab­

oration and worthful discussions and to Prof. H.J. Monkhorst for the given possibility to

carry out bulky computations of a weakly bound state of mesoinolccular ion tltp. \VV are

also thankful to our colleagues from the LCTA Computing Center helping us in numerous

calculations.

A p p e n d i x A . Variational form in spheroidal coordinates

Matrix elements of the Hamiltouian (Ф,, lhV}) can be determined from the variational

functional:

J J К ;,

7 ,„,,,, l't......

+ 7 ^ m . l ^ , » n < l ' ( / ? , 4 . ' / ) l 0 2 } H- ((' » 2 ) dlididr,;

where elements T compose the operator of kinetic energy and are uf the form:

Г" 1

2м7г [\dF™ 1

an ) ! • • , \ 0FT

Ъ'й

I •> ill'

) 2 / J 3

J(J I I) 2 m 2

2, 0l-\„' Or, M i 4 > - ' - ) i

KnV

? + r,7 - 1 + к 2

mm" 2M 1 2i me n2)[i)n \ 0FLUe ' N - J - ' C 'h^;;)

((C-N^M. *<*£)£] «?4 me - n2)

"'««Л. "2J \к * - ^

Tm.mil '• T m m i l -I 7'„

20

'ptr _. p* — (m ! 1 )/•„,;! . 2 -M а о 47» « W . . ) ^ .

7™*™ и = { [ l + с„, 0 Л(--1) 7 ] [./(./ + I ) - ш ( ш . 1 ) ] } ' / 2 . -,i\Um - •)£)„,,, .

Matrix elements T" connect basis functions Ф; and Ф) of the same permutation parity anil

are equal to zero for functions with opposite parity, whereas matrix elements TK connect

functions with opposite parity and vanish otherwise.

References:

1. S.I. Vinitsky, V.S. Mclezhik, I..I. Ponninarev. I.V. Pnzynin, 1.1'. Puzynina. L.N. So-

mov and N.F. Truskova, Zh. Kksp. Tcor. Kiz. 70 (1980) (W8. jSov. Pliys. JKTP. 52

(1980) 353); S.I. Vinitsky and L.I. Ponomarcv. Klein, ( 'hast. Atomn. Yadr. 13 (11)82)

1336. [Sov. .1. Part. Nucl. 13 (1982) .Г>Г>71; Л.11. (ioclieva el al. Pliys. Lett R. 153

(1985) 349.

2. S.I. Vinitsky, V.I. Korobov. I.V. Puzynin. Preprint .I1NH. PI 85 !H7. Dulina. l!>8.r,

(in Russian); S.I. Vinitsky, V.I. fv.rol.ov. I.V. Piizynin. Zh. Kksp. T o r . Kiz. 01 (198(1)

705, [Sov. Phys. JKTP, 04 ( I98(i) I17|; V.I. Knroliov. I.V. Puzynin. S.I. Vinitsky. Pliys.

Lett. В, 10С (1987) 272.

3. A.M. Krolov, V.I). Kfros, Pis'ma Zh. Kksp. Tec.r. Kiz. 30 (1981) 11!) ;.IKTP Lett. 30

(1984) 545|; A.M. Krolov, V.I). Knos. .1. Pliys. 11, 18 (1985) 1.21(5; A.M. Kn.lov. Ya.l.

Kiz. 44 (1980) 1.Ш7 [Sov. .1. Nucl. Phys. 44 (198(1) SS8].

S.A. Alexander, II..I. Monkhorst, Phys. Uev. A. 38 (1988) 2(1.

A. Scrinzi. K. Szalewich, Phys. Rev. A, 30 (1989) 2855.

M. Kaiiiiinura, Phys. Rev. A, 38 (1988) «21.

V.I. Korobov, S.I. Viuitsky, Phys. Lett. H, 228 (1989) 21.

8. S. Hara, T. Ishiliara, Pliys. liev. A. 40 (1989) 12.12.

9. S.I. Vinitsky, I.V. Pnzynin, .lu.S. Sinirnov, Vail. Kiz. 51 (199(1) Kit;:! (in Russian).

10. С. Schwartz. I'liys. Rev. 133 (I'Mil) 1700.

11. R. T. Park anil W. B. Brown, .1. Cliem. Pbys. 45 (lillili) V.li.

12. V.I. Korobov, Yail. Fiz. 51) (108!)) 15!)-. (in Russian).

13. G. Ureit, Pliys. Rev. 35 (1030) 5«9; Л.К. Ithalia. Л. Temkin. Itcv. Mini. Phys. 3G

(196'1), and references therein.

M. E.S. Chang, U. Fain., Phys. Rev. Л, G (1972) 173.

15. T. Kato, Trans. Amer. Math. Soc. 70 (1951) 195.

16. A. Halpern, Phys. Rev. Lett. 13 (1961) Ш) .

17. B.N. Parfett, The Symmetric Eigenvalue Problem. Prentice Hall. KIIKICWUOII t "litis.

N.I, 1980.

18. V.I. Korol.ov, Zli. vychisl. Mat. mat. l-'iz. 28 ( 1!IS8) 1113, T.S.S.lf. Comput. Maths.

Math. Phys. 28 , No. 5 ( I ' M ) 105;.

19. V.I. Korobov, I.V. Puzynin, S.I. Vinitsky, I'liys. I.ell. I!. 255 (1991) 7.

20. S.A. Alexander, II..I. Miuiklmrst, K. Szalewiiz. AM* СопГегецсе Proceedings. 181

(1989) 216.

21. A. Scrinzi, S.A. Alexander. II..I. Monkhorsl, Pliys. Rev. A. 37 (1.988) 2271).

22. M. Born, I.R. Oppcriheimer, Ann. il. Phys., 84 (1927) -157; see also C"h. IX in L.I. SchifT.

Quantum Mechanics, McCiraw Hill. New York, 19Г|Г>.

KciriMil In Publishing Dcp.n iiiu-iii oii'lune 21, 1991.

22

Коробов В.И., Пузынин И.В., Виницкий С И . Е4-91-288 Связанные состояния мезомолекулярных ионов водорода: вариационный подход

Предложен вариационный метод для расчета связанных состояний трехчастичных систем с произвольным орбитальным моментом J и про­странственной четностью X. Метод применяется для расчета полного на­бора связанных (32 состояний) мезомолекулярных ионов изотопов водорода с J = 0,1,2,3 и четностью X = (-1) и X = -4-1) .

Работа выполнена в Лаборатории теоретической физики ОИЯИ и Лаборатории вычислительной техники и автоматизации ОИЯИ.

Сообщение Объединенного института ядерных исследований. Дубна 1991

Korobov V.I., Puzynin I.V., Vinitsky S.I. E4-91-288 Bound States of Hydrogen Mesic Molecular Ions: Variational Approach

A variational method is proposed for computation of bound states of three-particle systems with an arbitrary total angular momentum J and spatial parity X. It is applied to compute the complete set of bound states (32 states) of mesic molecular ions of hydrogen isotopes with J = 0,1,2,3 and parity A= (—1) and X = - ( - 1 ) J .

The investigation has been performed at the Laboratory of Theoretical Physics, JINR and Laboratory of Computing Techniques and Automation, JINR.

Communication of the Joint Institute for Nuclear Research. Dubna 1991

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