11- * - ' *

V

LAMP/94/1

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

LAMPSERIES REPORT

(Laser, Atomic and Molecular Physics)

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL.

SCIENTIFICAND CULTURALORGANIZATION

SEMICLASSICAL HYPERSPHERICAL MATRIX ELEMENTSFOR HELIUM DOUBLY EXCITED STATES

J. Mahecha Gomez

MIRAMARE-TRIESTE

LAMP/94/1

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

LAMPSERIES REPORT

(Laser, Atomic and Molecular Physics)

SEMICLASSICAL HYPERSPHERICAL MATRIX ELEMENTSFOR HELIUM DOUBLY EXCITED STATES

J. Mahecha Gomez 'International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

A classical description of the two-electron atom, analogous to the quantum adiabatichyperspherical channel approach, is presented. The classical problems, analogue to thequantum eigenvalue problem for the great angular momentum operator, and the separateddynamical systems defined by each of the other constants of the motion of the non-mterac'iing system, are solved, using the Hamilton-Jacobi method. Some matrix elementsof the Coulomb interaction terms of the Hamiltonian for doubly excited helium atom usingthe Heisenberg correspondence principle are calculated.

MIRAMARE - TRIESTE

January 1994

'Permanent address: Departamento de Fisica, Univeraidad de Antioquia, ApartacloAereo 1226, Medellin, Colombia.

Preface

The ICTP-LAMP reports consist of manuscriptsrelevant to seminars and discussions held at 1CTP in the fieldof Laser, Atomic and Molecular Physics (LAMP).

These reports aim at informing LAMP researchers onthe activity carried out at ICTP In their field of interest, withthe specific purpose of stimulating scientific contacts andcollaboration of physicists from Third World Countries.

If you are interested in receiving additionalinformation on the Laser and Optical Fibre activities at ICTP,kindly contact Professor Gallleno Denardo, ICTP.

I, INTRODUCTION

The theoretical and experimental study of atomic doubly excited states is a topicof current interest. Experiments start with apectroscopic measurements in doublyexcited states of helium using synchrotron radiation [1], and in doubly excited statesof H~ ion using UV tunable lasers [2], and continue with recent experiments like thepreparation of two electrons of Ba atoms in states known as "frozen planet orbits"[3, A). From this experiments it has been concluded that atomic doubly excited stateshave the characteristic that is not possible an approximate description of the electronicstates as a whole in terms of its constituents parts. As many of those states are highlyexcited, and located in the region of validity of the correspondence principle, thenclassical models are expected to be valid [5j.

Moreover, some full quantum-mechanical calculations about doubly excited statesshown [6J that the conventional CI methods are very slowly convergent. The adia-batic channel treatment in hyperspherical coordinates for helium atom [7,8], as otherthree-body systems with Coulomb interactions [9j, proves useful, even is not free ofconvergence problems; it separates the coordinates in a set which gives the the shapeand orientation of the system, and a coordinate responsible of the overall size.

Even does not pretends to compete with the quantum adiabatic hyperspheri-cal methods, nor with highly accurate ab mitio calculations [10] in a quantitativeagreement, the present paper shows that some relevant results of the adiabatic hy-perspherical approach in highly excited states can be easily obtained using a simplesemiclassical method.

Section II presents the Hamilton-Jacobi solution of the dynamical systems asso-ciated with the great angular momentum of two particles, and the or! ital angularmomentum of a particle. Section III presents the classical problem in tl.e Hamilton-Jacobi formalism analogous to the adiabatic hyperspherical channel description ofthe two-electron atom developed by Macek, Fano, and Lit! [6-8]. In section IV asemiclassical calculation of matrix elements is presented.

We assume that the system consists of two particles with equal masses of vaJuem, and the nuclear mass is infinity. In the numerical calculations, atomic units wereused.

II. THE HYPERSPHERICAL HAMILTONIAN

The hyperradius and the mock angle are defined respectively by

R = {T\ + 4)1/1, and tana = ^ = u. (1)

Fano [6] uses the following hyperspherical coordinate set to describe the two-particle system.

instead of the independent particle coordinate set,

in which the Hamiltonian can be expressed in terms of one-free-partide Hamiltoniansand the Coulomb interaction potential energy, in the form

• + e-2m 2m 2mr\

From Eqns. 1 can be derived the following expression for Eq. -1.

I l i . p£\ . 1 i\2m

in which V is the Coulomb potential energy,

V(R,a,0u) = ~\ — +rt1 COSQ sino (1 +sin 2

Eq. 5 for H can be expressed in the form H = Hu + V where

(6)

(7)

We see that Ho depends on R, pR and A, and A depends on pa, a and thecoordinates from which depend the squares of the angular momenta.

sin2*/ (8)

A is a constant of the motion for the non-interacting system, as the energy, andthe magnitudes of the angular momenta of the individual j>Hi-ticlesi.

A2 can be identified [11] with the square of the magnitude of a (j x 6 tensor, calledgreat angular momentum, or generalized angular momentum, with components ofthe form A "n = xmpvn - ynpim, where m,n = 1,2 are labels for the particles, withsimilar relations for the xz and yz components. The diagonal components respect tom,n are the usual angular momentum components. A condition for a close (or notexact) 3-body collision is that A2 be small. A has a minimum when the velocities v,of the particles have the same relation as their impact parameters A,. The "maximumsymmetry" is obtained in the case in which not only A is minimum, but also vv = v2

and 6j = i j . A is related with the "correlation degree" betweesi the motions of theparticles through a cross term (O]6j - v^b,)'. IJI the iion-inl-rrat-liiig case, llt l2, A andE axe functions of vlt u2, 6t and b2.

A. Hamilton-Jacobi equation for Ho

As fa do not appear in Ho, it follows that the p^.-are constants of the motion. Alsoare constants U and A. In consequence, the coordinates R and a can be separated,5 = SAB + SAII1I> an<i the corresponding Hamilton-Jacobi equations axe:

(9)n\ da+ ^

cos'a sin Q

As A is a constant, Ho describes the hyperradial motion; A measures the strengthof a "centrifugal" field that keeps a pair of particles from approaching a force centresimultaneously. It can be defined an effective potential for the hyperradial motionhaving a minimum at certain large Ra, in order to consider the motion as periodic;at the end, Ra goes to infinity; compare with Eq. 42.

The solution ft r the hyperradial motion depends on A, Ra and fio = (6J + <

f A [[(fl/flo)2 - 1]' /2 - cos"1 (Ra/R)} if R < Ra

\ A [[(2Ra - Rf/Rl - I ]" 2 - cos-'[fto/(2R(, - fl)]] if ft > «„

The action variable for the R motion is given by

(10)

1ft = ^ f PR dR,

where it follows that

«-fThe "frequency" of the R motion is

dEOIR

The angular variable 8R is

IT A

dlR

from where is obtained the following expression for R in terms of $R or t is:

(11)

(12)

(13)

(14)

(15)

In the quantum case, the Sdirodinger pquation, corresponding to the first ofEqns. 9, can be solved by a Bessel function of integer order depending on the kineticenergy through the product kR, as it represents an unbound motion.

B. Motion on the sub-manifold \ = constant

A is a. constant of the motion in the absence of a force field, such as —Cj ft, whichdepends on the angular variables. Several values of li and /j can be associated witheach A, so the A eigenvalues are highly degenerated; that degeneracy is removed by-C/f l .

Second of Eqna. 9 describes a dynamical system whose "Hamiltonsan" is A.2. The"potential" well is

COS2 Q SU1 Q(16)

If li and li are different from zero, it occurs an oscillatory "motion" around theminimum of V|l)2, located at

tana. (IT)

and the value of V^i, at equilibrium is Vj',2 = (/; + (S)'J.The turning points of the motion are such that n = n+ or a = n_, which can be

expressed by means of A, denned at Eq. 20,

For given values of lt and (2, A must to be larger than /] + l2, and when A attainsits minimum, then a = constant = QC.

The solution to the second of the Hamiltou-Jacobi equations in Eqns. y is

where

a=-t{, 6 = A ' - ( f - ^ , c = ~;^,a ' = a , 6 ' = A 2 + ; 2 - ; 2 , c' = - A 2

b"= A3 - I] + II A = (/, + /, - A)(/2 - I, + A)((, - (, - A)((i + J, + A).

(20)

It is convenient to define a function .V, with ,r = it2 and y = 1 + u1, by

X = ax2 + bi- + c = u'y' +b'y + r. (21)

The action variable Ia is denned analogously as IR in E<|. 11, from which results

A - 2/ a + I, + (,. (22)

6

The "frequency" is wa - dtf/dla = 4A. The angular variable 9a is defined inanalogous way to B/t in Eq. 14. Now, using the exp ession for SAJ,<J> an expressionfor 0a as function of u is obtained, which when is inverted gives u as a function of"time" through 6O:

(23)

The phase curves for different values of A, with fixed I, and /3, show periodicitiesand the absence of separatrices. Also, the variations of u are very fast when A growths,the same as the excursions of ti. From this behaviour can be concluded that the radialcorrelations have an important kinematic component.

The Schrodinger equation corresponding to this problem has as solutions thecalled hyperspherical harmonics [6, 7].

The semiclassical wave functions, WKB approximation, have the form [5]:

(24)

Then, from SA<I/» can be easily calculated the WKB function which approximatesthe "Jacobi polynomial" part of the hyperspherical harmonics [12].

The density, or distribution function in the configuration space respect to u is

<10O A

du n((25)

as X — 0 at the turning points of u, in that points the density is infinite. The functionp(u) satisfies the normalization relation:

2 /*""* p(u)du = 1, (26)

where um;n, umol are easily obtained from Eq. 23. The density function will be usefulto calculate classical average values, like < C(ct,0u) >.

C. Motion on the sub-manifolds I, - constant

Eq. 8 defines a dynamical system with "Hamiltonian " I1. Next we will drop theindexes, to represent any particle described by spherical coordinates {8, <j>).

pt is a constant of the motion, equals to m, which determines a "potential"Vm = m!/sin* 0. The turning points of the 6 motion are determined by sin3 6 = m2//1;/ > m and at equilibrium / = m.

The maximum and minimum values of pt occurs when 0 = TT/2; and 0nMz,min aresymmetrical around that value. When $ varies from TT/2 to 8min, then pt is negative.

The Hamilton-Jacobi equation which is derived from Eq. 8 has as solution (13,14]

Sim = rn<t> — ' t a n ' - + m t a n " 1 — + - ( / - tn),I tx 2

where

(27)

(28)

It is deduced, the following expression for / and in in terms of the action variables:

' = h + h, m = A,. (29)

The "frequency" of this motion is a> = 21,From Sitnt 'he following expression for 0 and pe in terms of the action-angle

variables (/»,&) is deduced,

cos 8 = 7 sin 8e, and pt = —-l~l cos 6e -, where j 2 = 1 —

For the motion along the 0 coordinate it follows that

, + sin OA COS OHt an 4> '•

(1 - - r 2 ) J / 2 sin S» cos

_(1 _ 72)i/asin 8, sin Of, + ros 6t cos (

(30)

(31)

The evolution parameter associated with this "Hamiltoniaii" is not a time. Inthis case it is a parameter describing all the possible realizations of 6s (perihelion lineargument) and 64 (node line argument). In the a motion, whose Hamiltonian is A2,the evolution parameter determine all the realizations of 0^. or the radial correlationvariable u.

From St,m, can be calculated the WKB wave function corresponding to theOj = {Qi,<t>i} dependence [15] in the hypersplieric.il harmonics, after a coupling bysuperposing their products multiplied by Clebscli-Ctoidan coefficients in the classicallimit [16,17].

III. HAMILTON-JACOBI EQUATION FOR THE TWO-ELECTRON ATOM

In this section we will develop the adiabatic byperspherica] channel approxima-tion, within the Hamilton-Jacobi formalism, on a parallel way with the quantumformulation presented in by Fano [6]. A semiclassical treatment of this problem wasgiven before by Peterkop [18], in a plane situation in which L - 0, when the onlyrelevant coordinates are a, R and 0j2, using an expansion near the VVannier saddlepoint on the coulomb potential.

The Hamiltonian in Eq. 5 can be written as

(32)

where it is defined HR by the relation

6H). (33)

Then the Hamilton-Jacobi equation for HR, with R fixed, which involves only theangular coordinates, is

HR = U^R). (34)

The presence of C, which depends on 0i, 6j, and <pi - <fa, makes than /] and l2

are not longer constants of the motion. As also C depends on a, it is concluded thatA is not a constant of the motion. Here the index n denotes three quantities whichreplace the constants of the motion (A,/i,<j), and tends to them when the Coulombinteraction effects are small, that is, when fl-tO (limit in which is dominant thecentrifugal potential energy over the Coulomb potential energy) [19].

A characteristic function £„ generates a canonical transformation to the action-angle variables of HR, with fixed R,

(35)

and is a solution to Eq. 34, which explicitly has the form,

2mR2^~~da- + cos2 a ^10)1' cos2 o sin3 Bx 51 ,dY,u,2 . 1 /dS^i Ze1

sin

Ze*R cos a R sin a

(36)

This equation determines EM up to an R dependent constant like in Eq. 27; whenR makes a complete cycle, that constant can has a change which is not an integermultiple of 2ff, so the WKB wave function acquires a "geometric phase", which couldhas effects on the semiclassical spectrum of adiabatic curves.

The Sehaviour of the adiabatic channel potential curves (/„(/?) is such that

For fl-»oo, UJ.R) -> — and for R -> 0, £/„(«) -• ^TTZ- (37)R 2mR1'

A. Adiabatic approximation

The Hamilton-Jacobi equation for H is

(38)

If we call if = {a,9i,0i,4fi,<f>i}, and assume that, in Born-Oppenheimer approx-imation, E is approximated by

then Eq. 38 obtains the form

The adiabatic approximation consists in taking

lR ^ W'with which Eq. 40 reduces to

2m

(39)

(40)

(40

(42)

whose solution [»Fli = ± /{2 [E -To know the adiabatic channel potential curves ('„ is necessary to solve the an-

gular motion at fixed values of R, that is to fiiiri the act ion-angle variables (/„,<?>„),according to Eq. 36. The potential energy curves (•'„(/?) will be the eigenvalues ofthe matrix corresponding to HR according to tlie equation Eq. 34. On the adiabaticpotential energy curves depend the hyperradUl motion and the values of the totalenergy, according to Eq. 42.

IV. MATRIX OF COULOMB INTERACTIONS

A set of action-angle variables for the two-electron system is given by Eq. 35.This set can be completed with (/«, 0fl), associated to the hyperradial motion. InEq. 35, when ft-td, the quantities {/„,/,>,, /6j. /,,,, !,H,0,^. tie,, f*,,*1*, ,0*,} replacethe set {/^,^tf}. This set corresponds to the uncoupled slates |n,.c/1m|/Jrj!j > inthe quantum description. When the angular momenta of the particles are coupledto give a total angular momentum state, like the quantum \>irJil?LM >, the set ofaction-angle variables is {/o, /$,, /s2, / j , /*,#.»,^s, ,^si-"8i''*}. where 6$ describes therotation of the plane /i — it around L, and 0* is the conjugate to L,, [20].

Action variables can be replaced by the five constants of the motion in which isexpressed the hyperspherical basis in the quantum case, so in principle the functionC(a,6u) has a dependence on all the quantities in Eq. 35, which we will write as,

C = C(A,li,li,ml,mJ;0A, f/,>0^,0,,,,,#,„,). (43)

There are two ways to handle the iv or angular part. One is to solve the Hamilton-Jacobi equation, Eq. 36, that is, to find an expression {'„ for HR as a function of theaction variables /„. The other is to diagonalize the semiclassical matri. associatedwith HR, whose non trivial part according to Eq. 33 is given by the matrix of C.According to the Heisenberg's procedure to associate a matrix to a classical dynamicalvariable [20-22], C must be expressed in a Fourier series as:

10

u ,.. ! •it-akHVM^nn •'

exp >(nA ^A + 1(, &h + " ' J "'a + " » i "m>

and the Fourier coefficients have the expression;

(44)

O^n,, n,2 »„, nmj (A, / „ (,, m,, m4) = ^

C*(A, / i , / j , m i , rn j ; SA , fl;,, Sjj, S m , , #m2)

exp !(-J

d6hh f d6l7 j d6m, f d6mi

(45)

The integrals over the angular variables can be changed by integrals over u,and <j>, with help of the Jacobian

du d6l (46)

That Jacobian, according to before mentioned formulas, Eq. 23, Eq. 30, andEq. 31, is given by

2Au sin 0i sin 0jJ = (47)

wh're X was defined in Eq. 21, and 7, in Eq. 30.J corresponds to the square of the quantum wave function, Gfll2(a)Yiimi

y)2m2(fi5), in the uncoupled representation [7j.The final expression for the Fourier coefficients of C is

2Au

i ( - "A A - " I , 01, - "h 8tl ~ "mi "m, ~ "m, "n, ! -

where

and

. . , fr'(l + U2) + Id _,CO8fl,-

(48)

(49)

(50)

11

t s.s.s

A quantization process leads to (with h = 1, and the Langer collection [2J])

Ia — n,.f + - , /(

with which

(52)

This expression for A in terms of the quantum numbers agrees in the semiclassicallimit with the known expression A = (A+2) s - l / 4 , Ref[6j. It was included the Langercorrection 1/2, which accounts some zero point field effects.

With these Fourier coefficients can be constructed the C matrix in the semiclas-sical limit |17, 21, 22]. For that is required to express the harmonics of the Fourierseries as a difference between quantum numbers:

"A = nrc - n'rc

«/, = 'i - 'i - m, + ni\

"h = '3 - 'j - >"i + " ' i!!„,, = m, - m'inn4j = in, - m'T (53)

The relation between the matrix and the Fourier coefficients then has the form

C n d - m , J - m m, m n' f-m' I ' -m' m< „• =

(54)

It is convenient to perform a canonical transformation in order to give tothe C matrix an identical form to the one in the quantum treatment, likeC-nT<..)i.ii.<num11;n;c,t\,i'1,m\,m'7' Note tha t mus t to he satisfied the relation m , + rn2 =

The before found matrix elements are evaluated between uncoupled states- In thefull quantum mechanical calculation the following formuln WHV found for the generalmatrix element [6], when the angular momenta of elecirijn.s are coupled to form thetotal angular momentum:

< nrJthLMMn'J'&L'M' >= 6LL.6Mht.[- < n, )s a sin r r '

£ « l ^ r ( 7 ) t+i£j cos*+1o ^4 ; s i n M 1 olliiLM\Pk(cos0ll)\l[l'2LM >],

i 6{a--)n'r.>

(55)

12

where 9(x) is a unit step function, and the coupled angular states |/ii3£A* > areobtained from the on >particle angular states, given by the spherical harmonics func-tions, with help of the Clebsch-Gordan coefficients.

The diagonal elements have a classical counterpart which can be easily obtainedwith help of the classical probability density for the u motion, and the followingformula to evaluate the average value of P*(cos 8n) in the coupled states obtained byLeopold et a! [20]:

ft1 =< Pk(cos8n) >9,ttheL6u= [PkWVPkicosx), (56)

where x is the angle between the individual angular momentum vectors, given by.

(57)

This expression guarantees that are valid the inequalities |Ji - l?\ < L < h + h. Theexpression < P >«,, denotes an average of P over the angular variable 6h.

Then the following expression for the matri < elements, diagonal respect to thequantum numbers /,, h, L, M, and diagonal or not respect to the nrc quantumnumbers, is obtained:

ihLM >= -Z < (1 +

(58)

Here < ... >„„„;< denotes an average respect to u using the "probability density"

,(u) cos[(nrc - n'rc]8A], (59)

where p(u) is given by Eq. 25, taking into its dependence of A on n,c an averagebetween nrc and n'rc. The relation between u and the angular variable 0A is given byEq. 23.

Tables show some of the numerical results. In order to obtain a sign agreementwith the quantum mechanical values, it was necessary in Eq. 19 to add the constantIatt/2, like in Eq 27. That give up a phase shift in the semiclassicat wave function,which contributes with an overall phase factor of exp[i(nrc - n'rt)irj2] into the matrixelement defined by Eq. 48.

A more general expression, valid for non diagonal matrix elements respect to(UJi), is obtained from an adequate classical limit for the Percival-Seaton factor,h{l'A,hh\ I ) , given in Ref. [24] by,

(60)

13

which depends on the reduced mat r ix element* <>f l lacali 's lensoi- opera tors C ( * ' and

on the Wigner symbol 6 — j .Dickinson and Richards [25] obtained a simple expression for the classical limit

of the Percival-Seaton factors, Ref. [26]:

(61)

J],(/3) is a m a t r i x e lement associated with a finite ro ta t ion , s = l} - t[, t = l2 - i^and x is given by Ref. [25].

V. CONCLUSION

The dynamical system defined by A = constant is a zeroth approximation tothe real system at constant ii, if are neglected the interactions. That is the reasonwhy the C matrix, in the hyperspherical representation, has larger elements on thediagonal, fact detected by Lin [19). The potential V|,;3 is shallow, and infinite ata = 0 and a = ir/2; the u motion presents hard humps al the turning points; thatbehaviour is ;onserved in the real situation in which R is an adiabatic coordinate. Itwas found a formula for the WKB approximation to the hyperspherical harmonics,and an analytic expression for the semiclassical matrix elements of C; that formulawould be optimized by choosing adeq late intermediate values for nrc between thetwo involved states. Table I shows a sign discrepancy on one of the matrix elements,which can be attributed to that the weight of the quantum effects on small matrixelements is comparable with the classical value; thi behaviour occurs in almost allthe elements which are small, most of them locatod far from the diagonal. Thisformulation also can be useful in order to identify the dominant classical effects, andthe strength of the quantum effects which can be important in some states. A classicaldescription based on adiabatic hyperspheric channels can give satisfactory results inhighly doubly excited states of helium and other two-electron systems.

ACKNOWLEDGMENTS

The support from the Centra de Investigaciones en Ciencias Exactas y Naturalesof the Universidad de Antioquia, CIEN, the International Centre for TheoreticalPhysics, ICTP, and Colciencias under contract No. Co: 1115-05-012-92, is ack IOWI-edged.

1-1

REFERENCES

[I] R. P. Madden, and K. Codling, Astrophys, J. 141, 364-375 (1965); Phy. Rev.Lett, 10, 516-518 (1963).

[2] H. C. Bryant et al, Phy. Rev. Lett. 38, 228-230 (1977).[3] U. Eichman, V. La:lge, and W. Sandner, Phy. Rev. Lett. 64, 274-277 (1990).[4] K. Rkhter, and D. Wintgen, Phys. Rev. Lett. 65, 1965-1965 (1990).[5] I. C. Percival, Adv. Chem. Phys. 36, 1-61 (1977).[6] U. Fano, Rep. Prog. Phys. 46, 97-165 (19S3).[7] J. H. Macek, Phys. Rev. 146, 50-53 (1966); 160, 170-174 (1967).[8] C. D. Lin, Adv. At. Mol Phys. 22, 77-142 (1986).[9] J. Botero, Phys. Rev. A 35, 36-50 (1987).

[10] D. Wintgen, and D. Delande, J. Phys. B 26, L399-L405 (1993).[II] F. T. Smith, Phys. Rev. 120, 1058-1069 (1960).[12] D. L. Knirk, Phys. Rev. Lett. 32, 651-654 (1074).[13] M. Born, The Mechanics of the Atom (Bell, ind Sons ltd., London, 1927), page

137.[14] J. Mahecha, (unpublished), (1979).[15] R. M. More, Ann. Phys. 207, 282-342 (1991), and references cited herein.[H] P J Brussaard, and H. A. Tolhoek, Physica XXIII, 955-971 (1957).[17] S. C. McFarlane, J. Phys. B 25, 4045-4057 (1992); 26, 1871-1884 (1993).[18] R. Peterkop, J. Phys. B 4, 513-521 (1971).[19] C D. Lin, Phys. Rev. A 10, 1986-2001 (1974).[20] J. G. Leopold, I. C. Percival, and A. S. Tworkowski, J. Phys. B 13, 1025-1036

(1980); J. G. Leopold, and I. C. Percival, ibid. 13, 1037-1047 (1980).[21] L. D. Landau, and E. M. Lifshitz, Quantum Mechanics (non retativistic theory)

(Pergamon, London, 1958), page 166.[22] A. B. Migdal, Qualitative Methods in Quantum Theory (Benjamin, New York,

1977), page 166.[23] R. Langer, Phys. Rev. 51, 669-676 (1937).[24] \. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univer-

sity Pres:, Princeton, 2nd. ed. 1974), p»ge 111.[25] A. S- Dickinson, and D. Richards, J. Phys. B 14, 1916-1936 (1974); Dickinson A

S, and Liu W K J. Phys. Chem. 90, 3612-3619 (1986).[26] I. C. Perciva), and M. J. Seaton, Proc. Camh. Phil. Soc. 53, 654-659 (1957).

15

TABLES

TABLE I. Matrix elements of the Coulomb interaction terms of the hypersphericalHwniltonian for helium S states. QM, exact quanta! results; SC, semiclassical approxi-mation. * Given by [19],

(in h)("re, n'rc)

(0,0)(0,2)(0,4)(0,6)(2,2)(2.4)(2,6)(4,4)(4.6)(6,6)

Ci, h)("re, »«)

(0,0)(0,2)(0,4)(0,6)(2,2)(2,4)(2,6)(4,4)(4,6)(6,6)

(0,0)

SC12.094.633.312.89

17.387-715.35

19.859.58

21.46

QM

11.18*3.69"1.98"1.49"

16.86-7.18"4.59"

19.47"9.23"

21.19"

TABLE 11.

(0,

SC11.773.502.261.88

15.585.873.74

17.737.48

19.22

1)

QM

15.982.681.651.07

19.785.293.40

21.997.01

23.54

(1.1

SC

9-852.120.820.64

13.244.021.87

15.215.43

16.61

The same

mC{a,«v.)

QM

9.39*2.12*0.50*0.47

13.06'4.05*1.70

15.10'5.51

16.55

• as Table 1,

< nrc(,/jlAf|C(ii,»:

(1 .

SC

10.171.980.880.61

13.463.881.93

15.415.29

16.81

1)

QM

10.22

1.74

U.C*

0.3*

13.55

3.741.84

15.54

5.2216.97

l)K/ilaO-:

(2,2)SC

9.181.460.280.26

11.782,81o.xy

13.113.91

14.04

but for P

QM

8-79"

1.60"

0.10

0.29

11.65*2.91

0.81

13.35

4.0314.BI!

states,

15)KV2Uf >(1,2

SC

9.921.75o.erj0.4.>

12.71

3.3H1.4X

14.47

4.60

15.77

)

QM

15.1(5

1.18O.w;

0.2417.542.881.56

19.264.17

20.56

(3,3)

SC

8.821.160.070.14

10.992.200.47

12.403.09

13.48

(2,2)

SC

9.48

1.320.350.21

11.972.680.96

13.583.78

14.80

QM

8.47"1.35

-0.070.24

10.892.320.41

12.353.21

13.46

QM

9.511.230.S10.16

12.042.640.97

13.683.78

1 :.92

16

TABLE III. The same as Table 1, but for D states.

Ci, h)

(tire, Kc)

(0,0)(0,2)(0,4)(0,6)(2,2)(2,4)(2,6)(4,4)(4,6)(6,6)

Ci. 'a)

("re. Kc)(0,0)(0,2)(0,4)(0,6)(2,2)(2,4)(2,6)(4,4)(4,6)(6,6)

(0,2)

SC

12.443.452.241.86

15.165.423.44

17.006.81

18.35

QM

11.913.021.661.22

14.855.183.08

16.766.65

18.16

TABLE IV.

(0,3)

SC

13.273.582.371.97

15.125.313.42

16.696.53

17.90

QM

17.393.031.861.35

19.384.813.16

20.986.08

22.22

(1,1)

SC

10.191.970.880.61

13.483.871.94

15.435.28

16.83

The same a

< n rci, MJ

(1.2)

SC

10.101.680.670.44

12.S43.291.50

14.584.53

15.88

*|C(«,«i:

QM

9.S91.890.610.42

13.353.871.78

15.365.34

16.80

s Table 1,

Vf|C(a,«,

QM

14.961.260.630.20

17.432.951.53

19.164.24

20.46

i)\Kchh2l

(1,2)

SC

10.151.670.S70.44

12.873.281.50

14.614.52

15.90

but for F

i)KJlh3>

(1,3)

SC

10.401.600.660.43

12.663.051.38

14.224.16

15.41

QM

14.611.400.570.29

17.223.081.48

18.974.36

20.28

states.

M >

QM

10.501.470.560.31

12.752.981.35

14.334.13

15.54

(1,3)

SC

10.201.660.640.44

12.523.111.36

14.094.23

15.28

(2,2)

SC

9.731.240.370.20

12.132.590.99

13.733.69

H.94

QM

10.041.610.510.33

12.473.131.29

14.074.29

15.29

QM

9.791.150.330.15

12.212.560.99

13.843.69

15.07

17

w9