Self-consistent nuclear calculations with a deformed basis

Self-consistent nuclear calculations with a deformed basis

1.C [ Nuclear Physics A142 (1970) 49--62; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written...

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1.C

[

Nuclear Physics A142 (1970) 49--62; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout writtenpermissionfrom the publisher

SELF-CONSISTENT NUCLEAR CALCULATIONS

WITH A DEFORMED BASIS D. R. TUERPE Lawrence Radiation Laboratory, University of California, Livermore, California t and W. H. BASSICHIS and A. K. KERMAN Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts Received 22 September 1969 Abstract: The eigenfunctions of a cylindrically symmetric, deformed harmonic oscillator(h.o.) potential are used as a basis in which to expand Hartree-Fock orbitals. Self-consistent calculations for nuclei with 4 _--
1. Introduction I n principle, a self-consistent field calculation can be performed by e x p a n d i n g the single-particle wave functions of a d e t e r m i n a n t in a complete set of eigenfunctions a n d d e t e r m i n i n g the e x p a n s i o n coefficients via a variational principle. If a complete exp a n s i o n were possible, all results would then be i n d e p e n d e n t of the choice of the basis set. I n practice, of course, one chooses only a small n u m b e r of basis states, thus destroying the independence, a n d m u s t minimize also with respect to the parameters describing the basis. F o r example, if a h a r m o n i c oscillator basis is chosen, the harm o n i c oscillator c o n s t a n t must be treated as a variational p a r a m e t e r 1,2). W h e n the basis is t r u n c a t e d one m u s t be assured that the results o b t a i n e d with a particular set will n o t be appreciably affected b y enlarging the basis. M o s t calculations presently being performed 1, z) utilize a spherical h a r m o n i c oscillator basis with various n u m b e r s of shells included in the space. The choice of harm o n i c oscillator (h.o.) wave functions is c o n v e n i e n t because of the property that the p r o d u c t of two such functions, q~l (rl)c~2(r2), can be written as a finite n u m b e r of p r o d ucts of h.o. functions of the c.m. coordinate a n d the relative coordinate 3). As is t This work was performed under the auspices of the US Atomic Energy Commission. 49

50

D . R . TUERPE el al.

well known, this transformation greatly simplifies the calculation of the matrix elements of the two-body potential. Another reason for the use of the spherical h.o. basis is because the matrix elements are readily related to those used in many shell-model calculations. It has been found, however, that certain calculable quantities require a very large spherical basis for convergence 4). This is certainly true for the quadrupole moment and probably higher moments would be even more sensitive. Thus, though a space which can accommodate eighty particles yields an energy for 52Fe within 4 % of that obtained in a space capable of holding 252 particles, the quadrupole moments as calculated in the two spaces differ by more than a factor of two for 52Fe. The factor is nearly 1.5 even for a nucleus as light as 2°Ne. It is expected that the necessity of using such large spaces, with the resultant millions of matrix elements necessary for the calculations, can be avoided if the basis states used are themselves deformed. A deformed basis has been used before for light nuclei with effective interactions 5). A parameter characterizing the deformation of the basis is introduced into the calculations and the energy must be minimized with respect to this parameter also. One might imagine that the advantage would therefore be negated. Since, however, the time consuming part of the calculation is the calculation of matrix elements, and since these may be calculated for a large number of deformations simultaneously, in a time essentially independent of this number, the advantage remains. The necessity to minimize with respect to the size and deformation of the h.o. potential for each nucleus and for each different two-body potential has sometimes been overlooked 6), with subsequent erroneous conclusions. In this paper we use the "realistic" potential cf Tabakin 7) because it has been used extensively with a spherical basis 2,4,9). The results are compared to the previous calculations to determine which basis would be more suitable for studying deformations. It is shown that it is extremely advantageous to use a deformed basis for the elements with .4 < 80 and we therefore expect that this method will be necessary for studying the deformed nuclei in the rare-earth region. 2. Details of the calculation

"['he Hartree-Fock equations can be found in a large number of articles 8, 9) and it should be quite sufficient to characterize a particular calculation by simply describing the Hamiltonian, the size and nature of the basis, and the restrictions imposed, such as axial symmetry or an inert core. The Hamiltonian used here is H = T + V,

(I)

where V is the Tabakin interaction 7) and T is the kinetic energy. Calculation with this H will therefore include the kinetic energy of the center of mass and this should be subtracted if one is calculating binding energies. This could be circumvented if one

SELF-CONSISTENT NUCLEAR CALCULATIONS

51

H = T+V-rc.m.

(2)

used instead as is usually done 9, z o) but since this was not done in the spherical calculations that are entering the comparison, it will not be used here. The basis consists of the eigenfunctions of a cylindrically symmetric, i.e. deformed, harmonic oscillator potential V

1

-

h2

1

h2

2 M'~ 2 p2q_ -2 M/¢ 2 -72,

(M = nucleon mass),

(3)

whose energy eigenvalues contain up to a total of four quanta. In the case of zero deformation of the basis, this corresponds to the inclusion of the ls, lp, 2s-ld, and 2p-lf shells. The basis states depend on the projection of orbital angular momentum on the z-axis (m), the number of quanta in the xy-plane and the number of quanta in the z-direction (n=). They are labelled by n, m, and n=, where n is given by

(np)

np =

2n+lml

(here n starts from zero).

(4)

Thus the energy is given by E . . . . = (2n + Im] + 1)h% +(nz

+½)h,:Oz

(5)

and the 09 are related to the harmonic oscillator constants by cop-

h

and

o~= =

M?

h

.

(6)

Mtc

The explicit form of these eigenfunctions in coordinate space is


(7)

L~ml

here is an associated Laguerre polynomial, and H,= is a Hermite polynomial. The normalization constant, N, is given by

.,,

-

A(m) ~/~ \(n+lml)!l - -

(s)

\~/xZ"'n=!/

'

with the phase choice

A(m) =

(-1)"

= 1

m > 0, m < 0,

(9)

which proved convenient in the calculation of matrix elements. The expansion of the single-particle H F orbitals will then in general be given by

I,~> = Y c L . . . . . n'nz "s~

Inmn=>lm~>l*> -

Z C,a. . . . . . n'nz "$z

Inmn=,ms, t>.

(10)

52

D.R. TUERPEet

al.

Here Ims) and Iz) are spin and isospin state vectors and the ½, being c o m m o n to all such vectors in these calculations, can be omitted. The summations were restricted so that the parity, ( - 1 ) nz+m, was a g o o d n u m b e r n u m b e r and the states were either protons or neutrons. Axial symmetry was also assumed so that the projection of total angular m o m e n t u m on the z-axis, m + ms, was a g o o d q u a n t u m number. All o f the particles are treated self-consistently with no other restrictions. The method o f calculation of the matrix elements of the kinetic energy and the twob o d y potential, and a derivation of the relevant cylindrical transformation brackets analogous to the Moshinsky brackets used with spherical h.o. states, is given in the appendix. 3. Minimization of the energy To find the best determinental wave function one must find the m i n i m u m of the energy with respect to 3' and ~: of eq. (3), as well as the expansion coefficients, C. It is perhaps more meaningful to define two other parameters; the average oscillator constant ])av : 1(27+tc) (11) and a deformation parameter for the basis * B-

•--Y

~av. In a spherical h.o. basis ~'avis just the usual oscillator constant and B is zero. In fig. 1, the energy is displayed as a function o f B for various values ofy,v, for 2 ONe . The dotted

I I~

- 3 4 --35

NEON 2 0

I : ~AV=2.2

i "~

\

"rr : ~ A V = 2 , 6

-40

l ~ ~ ~ l a i f~

-4-1 _

Tr

--42 --4.3

,,

--4-4 --4-5

I •1

,

t 0

I .1

I

I .2

,

I ,3

I

I .4.

I

I .5

I

I .6

I

I o7'

B

c -~

Fig. 1. Variation of HF energy (E) with 7av. and B in ZONe. I" B represents the deformation of the basis; it is not the same as the deformation of the density (fl) or the deformation of the potential (6) as defined in ref. 11).

SELF-CONSISTENT NUCLEAR CALCULATIONS

53

vertical line is drawn at B = 0 and the intersections of this line with the curves are the energies that would be obtained with a spherical basis for the corresponding 7. The energy at the optimum 7 in a spherical basis is - 3 8 . 8 MeV. At the optimum 7av. and B in the deformed basis the energy changes to --43.1 MeV. This gain of 0.2 MeV per particle is not very significant but the large value of B at the minimum is important for it indicates non-negligible amounts of what is called "major shell mixing" from shells excluded in the spherical calculation.

4. Comparison with spherical basis In this section the results in the deformed basis are compared to the results in a spherical basis of the same size (small) and a much larger spherical basis (large). The large space consists of six major oscillator shells plus the li~ level. It has been shown [ref. 4)] that the energy is not significantly improved using the larger spherical basis although it is, of course, always lower. In fig. 2, the energy differences between the small and large space (x) and between the deformed and large space are (0) plotted. Note that the gain in energy obtained by greatly expanding the spherical basis is nearly matched, for all nuclei, by deforming the small basis. z~ {MeV)

TRUNCATION

+

DEFORMATION

EFFECTS

--ENERGY

X(~--

E ( I g e sph) - E ( s m sph) E(Ige sph)-E(sm

def)

]O-X

X

9

X

8

X

7 6 :-

X

5

4

X -

X

.

O

® ®

o

--®

o

°

®

°

o

o

e

o

,i,l,J,l,l,J,I,l,l,l,l,l,l,],f,l,l, 4-

8

12

16 20

2@ 28

32

36

40

44

48

52

56

A (MASS NUMBER}

Fig. 2. H F energy differences between calculations in the large spherical space and the small spherical space (x). HE energy differences between the large spherical space and the small deformed space are

denoted by C).

It is more significant to consider the quadrupole moment. The calculated values of in the small and large spherical basis are quite different, except for the lightest nuclei, because of small admixtures of large/-values from the higher shells. In fig. 3, the ratios of the quadrupole moments in the large space and small space (x) and the

D.R. TUERPE et aL

54

ratios of the quadrupole moments in the large space and the deformed space (C)) are shown. In all cases the oscillator parameters employed were the optimum ones. Here the significant advantage of the deformed basis is clear. The energies, quadrupole moments, rms radii and kinetic energies per particle for the small, large and deformed basis are tabulated in table 1. The previously observed convergence of all but the quadrupole moment remains and the truncation effect on the quadrupole moment has been essentially eliminated. A perturbation argument was invoked *) to make plausible the assumption that the large space results for ( Q ) would not be affected by a further expansion of the basis. This is in agreement with the resutts in fig. 3 for most nuclei, though it appears that in some cases small amounts of higher/-values would be admitted if the basis were expanded. QB

TRUNCATION + DEFORMATION EFFECTS

Qs

-- QUADRUPOLE

X

MOMENT

X

2.4- .--

X

X - Q(Ig* sphl

2.2

Q(sm ,ph) "

~'.0



X

X

1.8

(.:) -

q!,=g,

--

Q(srn def)

,ph)

X

1.6 1.4 1.2

-

x

X

o

o

X

®

1.o

~

°8-

"o O

®

O

O

O

.6

l,l,l,l,l,l,ITltl,l,l~Irl,ltl,l,ltlK 4-

S

12 16 2 0 2 4 2 8 32 3 6 4 0 4 4 4 8 5 2

56

A ( M A S S NUMBER)

Fig. 3. Comparison of ratios of quadrupole moments obtained in the large spherical space to those obtained in a small spherical space (x) with the ratios of quadrupole moments obtained with large spherical space to those obtained with the small deformed space. The dotted line indicates convergence, i.e. the quadrupole moments are equal in the various spaces if the ratio is 1.

The resultant wave functions in the deformed basis may be related to shell-model states by projecting the deformed states onto spherical harmonic oscillator functions. The projection amplitudes, (nmnzlNlm), can be evaluated by a straight forward integration. Here Inmn~)is given by (7) and

(rlNlm)

=

~ ~ - t e-~'z/"v'Lls+__~l( r (V--r) LF(N+l+½)J \--•av. /

2)

Ytin(r2).

(13)

This has been done for several cases. For example, convergence of 8Be in small spherical space follows from the fact that the amount of mixing in the deformed basis of states not included in the small spherical basis is small. Quantitatively this admix-

SELF'CONSISTENT NUCLEARCALCULATIONS

55

TABLE 1 Comparison of binding energies, quadrupole moments, rms radii and kinetic energies per particle

'tHe

sin. sph.

SBe

12C

1.74

0.94

--10.1

--0.10

--1.71

--14.5

160

2ONe

2*Mg

2sSi

32S

--34.3

--38.8

--52.0

--73.5

--93.4

--38.4

--46.6

--59.8

--82.8

--98.9

--34.3

--43.13

--57.02

--80.5

--96.3

E

lg. sph.

(MeV)

sin. def.

1.74

0.763 --10.76

sin. sph.

0

0.457

--0.323

0

0.507

0.619

--0.657

0.365

lg. spho

0

0.413

--0.301

0

0.723

0.897

--0.908

0.695

sm. def.

0

0.466

--0.328

0

0.770

0.957

--1.05

0.786

sin. sph.

1.96

2.52

2.40

2.35

2.43

2.55

2.64

2.69

Q(b)

-v/R2

lg. sph.

1.98

2.45

2.37

2.39

2.56

2.61

2.68

2.69

fire)

sin. def.

1.96

2.48

2.36

2.36

2.57

2.63

2.72

2.71

K.E.

sm. sph.

12.27

14.64

17.32

19.04.

21.26

22.70

23.33

23.84

part.

lg. sph.

12.55

15.36

18.21

19.39

20.74

22.45

23.07

24.05

(MeV)

sin. def.

12.27

14.74

18.07

18.99

20.40

22.13

22.72

23.67

4OCa

44Ca

4aCr

4aTi

52Fe

a6A

44Ti

56Fe

sm. sph.

--118.48 --142.8

--151.8

--155.4

--176.8

--171.9

--201.7

--217.5

E

lg. sph.

--123.3

--145.8

--154.8

--160.5

--186.4

--177.8

--209.3

--226.3

(MeV)

sin. def.

--120.6

--142.8

--152.2

--157.7

--183.8

--175.08 --206.9

--224.5

Q

sin. sph.

--0.284

0

0.232

0.345

0.520

(b)

lg. sph.

--0.58

0

0.458

0.619

sin. def.

--0.523

0

0.560

0.724

sin. sph.

2.78

2.79

2.85

lg. sph.

2.77

2.84

sm. def.

2.75

2.79

~/R 2 (fm)

0.419

0.522

0.517

1.416

1.011

1.236

1.335

1.456

0.871

1.494

1.576

2.85

2.90

2.94

2.95

2.98

2.90

2.91

2.96

2.94

2.97

3.02

2.86

2.85

2.92

2.92

2.97

3.00

K.E.

sm. sph.

24.4

24.04

25.11

25.13

26.04

26.02

26.78

27.48

part.

lg. sph.

23.6

23.16

24.36

24.46

25.31

25.41

26.51

26.97

(MeV)

sin. def.

23.88

24.04

25.10

25.09

25.95

25.95

26.75

27.41

Calculations performed in the small spherical basis (sin. sph.), large spherical basis (lg. sph.), and small deformed basis (sm. def.). The second order corrections as well as c.m. and Coulomb corrections have not been included. Of course neither the radii nor the binding energies agree with experiment as has been discussed previously 2, 4).

56

D.R. TUERPEet aL

ture is given by p = 1 ~ [[2 ' a antra

(14)

where A is the mass number of the nucleus and states are excluded from the sum if they are contained in the large space. For 2°Ne, P ' = 0.00017 while for 52Fe P ' = 0.00026. Thus it appears that the small deformed basis and the large spherical one are nearly equivalent.

5. Coulomb effects The purpose of this section is to emphasize the necessity of minimizing the energy to determine the optimum oscillator parameters for each nucleus and for each force. It is presently the practice to carry out H F calculations first with a nuclear interaction and then with a combined nuclear and Coulomb interaction; correcting also for the c.m. motion. The c.m. and Coulomb effects, calculated in this manner, are the topic of a recent paper 6). In that paper a spherical basis was used which is one shell smaller than the small space considered here. Then, though truncation effects will be extremely severe in such a small space, the changes in the radii and quadrupole moments of various nuclei when the Coulomb force is added to the nuclear force were calculated. Furthermore, the calculations were all done with a single oscillator parameter, i.e. the optimum 7 for ~60 with only the nuclear force considered. The conclusion was that the quadrupole moments and radii are altered by less than 2 ~ by the inclusion of the two effects. Consider first the quadrupole moment. In 2°Ne the result of the nuclear calculation of Gunye 6) was = 0.5146 (without Coulomb and c.m.) = 0.5100 (with Coulomb and c.m.). The conclusion was that including the Coulomb and c.m. effects decreased (Q> by less than 1 ~. (The two calculations were done with the same oscillator parameter.) In the present work the quadrupole moments were also calculated with and without the Coulomb force included. Care was taken, however, to determine the optimum oscillator parameters for each force, which indeed were altered significantly when the

SELF-CONSISTENT NUCLEAR CALCULATIONS

57

Coulomb force was included. For 2°Ne, for example, ( Q ) -- 0.770 b (no Coulomb) for 7av -- 2.6, B = 0.34, ( Q ) = 0.816 b (with Coulomb) for ?av = 2.8, B = 0.46. The conclusion is the opposite of ref. 6). The quadrupole moment, at least for this case, is increased by more than 5 ~o rather than decreased by less than 1 ~ . It is illustrative to note that if the same oscillator constants, as determined without the Coulomb force, are used with Coulomb present, one obtains ( Q ) = 0.743 b (with Coulomb) ?av. = 2.6, B = 0.345. Since this result does not correspond to the optimum parameters it is not meaningful. The fact that the quadrupole moment increases when the proper parameters are used will be further reinforced by the c.m. correction. It has been shown that proper treatment of the c.m. motion sometimes increases and sometimes decreases the calculated value of ( Q ) [ref. 4)]. In 2°Ne, however, the c.m. correction results in an increase of ( Q ) by approximately 3 ~o. If this result were added to the decrease in ( Q ) obtained with the wrong parameters, one would indeed find that the net effect of Coulomb and c.m. corrections was a decrease in ( Q ) of about 1 ~o. If, indeed, the effects are additive the correct conclusion is that the net effect is an increase of 8 ~ . Similar, though somewhat less dramatic, statements can be made about the radius. There the effects were found to be in the same direction (increased radii) as in ref. 6), but at least twice as large. 6. Conclusions

The results of this paper clearly indicate that a deformed harmonic oscillator basis is superior to a spherical basis if one wishes to consider non-closed shell nuclei. This is true regardless of the way in which the results of the H F calculations are interpreted. According to one approach, an effective interaction is used instead of a realistic one s). The parameters of the force are then determined by requiring the H F energy to agree with experimental binding energies. Though the meaning of quantities, other than the energy, calculated with the H F determinant is unclear; quadrupole moments, radii, form factors and other calculated quantities are compared with experiment. It is then crucial that truncation effects be minimized. Furthermore, since one expects to fit binding energies with high precision, the small gain in the energy alone should require the use of a deformed basis. Another approach is to interpret Hartree-Fock calculations as a means of obtaining a basis in which to carry out a perturbation expansion. This approach is followed in the present paper. It is then advantageous to place as small a burden as is possible on higher order corrections. Here again the small gain in energy and the large gain in deformation will be significant. For example, in 2°Ne, the higher order corrections

D . R . TUERPE e t aL

58

to the energy would be 4.3 MeV higher for a small spherical basis and a higher order correction to ( Q ) of 50 ~ would be required just to reach the value obtained in first order with the small deformed basis. In the light of the results presented here, calculations are presently being undertaken in a deformed basis of about the same size as the large spherical basis referred to here. In these calculations, c.m., Coulomb and second order corrections will be included and it is hoped that useful results will be obtained for nuclei in the rare-earth region,

Appendix A In this appendix the expressions for the matrix elements of the kinetic energy and the transformation brackets from individual particle states to c.m. and relative coordinate states are derived. 1. The expectation value of the kinetic eneroy is: p2

(n'm'n'~[ p~ + ~ 2M

2M

2

+ P-Y--~[nmnz) 2M = (n'm'[ p 2 + p_~Zy[ n m ) 6 , , . . + ( n ' l 2M 2M

p2 Inz)6,',6m'm.

(A.1)

For the two-dimensional part of this problem, it is convenient to go to a new representation I~, fl), which has a one-to-one correspondence with the states Inm). This is achieved as follows. Consider the two-dimensional Hamiltonian H = 2M 1 ,tPx2 ..t_ Py2 +

k 2 , ' X 2 .q_ 2",,

t

y ))

where k

-1 and h

1.

(A.2)

Raising and lowering operators are defined by Q+ = - i Q x + Q , ,

Q - = iQx+Q~,,

with Qx = P x - ky,

Qr = Pr + k x

and 7r+ = i~x+rCy,

re_ ---- --iZx+rCr,

(A.3)

with 7z~ = p~ + ky,

ny = p r - kx.

We then define the states le, fl) = (Q+)'(rc+)al00), where 100) is the lowest state defined by Q_I00) = rc_]00) = 0. It follows that Q+I~, fl) = ~/4k(a+ I)1~+ 1, fl), n+le, fl) = x/4k(fl+ 1)1c¢ f l + l ) , Q_ le, fl) = ~/4kc~

Ic~- 1, flS,

zr_ I~, fl) = ~/4kfl

I~, f l - 1).

(A.4)

SELF'CONSISTENT NUCLEAR CALCULATIONS

59

The eigenvalues of the Hamiltonian are then given by HI~, B> = k ( ~ + f l + l ~ ,

p>

(A.5)

and of the z-component of angular momentum by Lz[~, fl) = (~-fl)l~, fl).

(A.6)

Now we can make the correspondence between the representation I~t, fl) and the representation In, m). = n+m, = n

,

fl = n

for m > 0 ,

fl = n - m

for m < 0.

(A.7)

The states let, fl) have the normalization: (0t, fll~, fl> = (4k)=+a0dflt---~. 4k

(A.8)

In terms of these raising and lowering operators, the kinetic energy operator is given by: pZ + P ~ _

2~t

2~t

(A.9)

1 (½{Q+,Q_}+½{~+,~_}+Q+~++Q_x_),

8M

where {s, t} stands for the anti-commutator of s and t. Utilizing equations (A.1), (A.4) and (A.9), the matrix elements of the kinetic energy operator between deformed harmonic oscillator states are given by: ( n ' m' n'=[K.E.lmnn=> -

1 (2n+lml+l)+

2M---~

1

2Mx (n,+½),

n' = n, m' = m, n: = n=,

1 ~/(n+lml+l)(n+l),

n'=

n+l,

x/(nz+2 )(n = + 1 ) ,

n'=

n, m' = m, n~' =

m'=

m, n'= = n=,

2M7

= - 21M

K

n~+2.

(A.10)

Here the off-diagonal matrix elements for the one-dimensional part, which are well known, have been put in for completeness. 2. The transformation brackets between individual particle states with coordinates rl, r 2 and states with the c.m. coordinate R = (rl +r2)/x/2 and relative coordinate r = ( r l - r 2 ) / , j 2 are factorable, and are defined by (pilnl ml>(zlln=l>(p2[n2 m2>(z2[nz2) = ~ (plnm)
m i n2 mE) ~, (zln~>(ZlN=)(n= N=ln~l nz2),

.=N=

(A

II)

D . R . TUERPE et al.

60

where r 1 = Pi+ zi and p~ is the relative coordinate vector in the xy-plane, R o is the c.m. coordinate in the xy-plane and small letters designate relative quantum numbers, capital letters designate c.m. quantum numbers. The two-dimensional part of the transformation beackets will be derived in the le, fl) representation. Since there exists a one-to-one correspondence between this basis and the In, m) representation, there is no difficulty in going from the brackets in the 1c¢,fl) representation to those in the In, m) representation. Under the coordinate transformation from r 1, r2 to r, R, the raising and lowering operators for the two particles become 1

e+i

.... +

+ Q,~,+),

Q+2

1

1 ,]~(Q ....+-Q,~+) 1

~+2 - ~ 0

g+l = ~(ff ....+ +grel+),

z....+-zc,¢,+)

(A.12)

1

where the subscripts c.m. and rel refer to creation operators in the c.m. and relative states respectively. The desired transformation brackets are defined by [o~1fll)[~2 [32) = Z lab)laB)(abAB[~z~ [31 o~2f12),

(A.13)

ab AB

where lab) = (Qrel+ )a(Tr,rel+ )b[oo)rel

and l A B ) = (Qc.m.+)a(rc¢.m.+)B[00).... .

(A.14)

It follows from eqs. (A. 12) and (A. 13) that the transformation conserves energy and the z-component of the angular momentum, i.e., ~+~2

=

a+A,

(A.15)

~1+B2 = b + B,

the two-dimensional part of the brackets is obtained by noting that the two-particle state lu,/~)l=zB2) -- (Q+aY'(rc+,)'flOO)~(Q+z)'2(~+z)#fiO0)2, (A.16) can be written, using eq. (A.12) as

,1 L [~I[31)[~2f12) =i

j~0

, . . ,

l~0

1

1

(Qc.m.+) (11: ....+)

x (Q.o,+)~"-'+'2-"(..o,+)#'-J+#~-'(-

ly.-k+#.-,ioo>.,

100).... '

(A.17)

here (}) is the usual binomial coefficient. A term by term comparison of the two expansions, eqs. (A.17) and (A.13) then yields the transformation brackets. Combining these with the one-dimensional brackets, which are derived elsewhere 5) and properly taking into account the nor-

SELF-CONSISTENTNUCLEARCALCULATIONS

61

malization condition, one finally obtains



=V

~l_l_Igl+tl2+fll+fl2+n~l+/Iz2

a!B!a!b!Nz!nz[ iX1 !~X2 !ill !fl2 !llzl !nz2]

min(A, ~ cq)

X i ....

(0, A-~2)

\x/2/

( ~ x) ( ) 0~2

A--i ( -

min(N.... 1)

[nzl~

1)~2-a+i

( )(eft2 ) mi,(B, a,) X fll (_l)a2-B+j j=m~x(O,B-~) J J

nz2 ~

x l = max(0,~ N=- n=2) \l]\Nz-1/(-1)''~-N'+'" Capital letters stand for c.m. quantum number and small case letters refer to relative quantum numbers. The symbols max (a, b) and min (a, b) stand for the maximum and minimum of a and b, respectively.

Appendix B For completeness, the matrix elements of a general non-local two-particle potential between deformed h.o. states are given in this appendix. If we consider a potential which depends on the relative coordinates r = r 1 - r 2 and r' = r3-r4 and the total spin S and total isospin T of the two particles, then we can write the matrix elements of this potential between deformed h.o. states with spin and isospin wave function

Ims~5 a s : =

~,, ( n 1 m 2 n z l , nmnz NMN= n'm'n'= ST

n2m 2

nz21nmnz; NMNz) (n'm'n', NMNdn3 m 3 nz3, n4 rn4 nz4)

x (½2-tmsl ms2lSMs)(½½ms3ms, lSMs)(½1zl "~21TMT)(½½z3z41TMT)

× f f (nmnz[r)(r[V(S,T)lr')(r'ln'm'n')dard at'.

(B.1)

(For the Tabakin potential, V(S, T) contains a sum over certain quantum numbers which do not appear in the basis states, namely the relative total and orbital angular momenta.) The first two brackets above are the transformation brackets, the next four are Clebsch-Gordan coefficients which couple spin and isospin and (rl V(S, T)[r') represents the non-local potential in configuration space. If the non-local potential is of the separable type, so that we can write

(rlV(S,

T)lr') = E m, m'

Vm(r,O, S, T)Vm,(r', O', S, T)e-'m*eim''b',

(B.2)

62

D.R. TUERPEet al.

the six-dimensional integrals in (B. 1) can be written as products of t w o - d i m e n s i o n a l integrals, since the integration over ~b a n d ~b' can be performed immediately. These t w o - d i m e n s i o n a l integrals were evaluated o n the c o m p u t e r by a sixteen-point G a u s s i a n integral routine. The matrix elements of a local potential can easily be o b t a i n e d by the replacement (rlVir') = V(r)6(r-r'). The authors wish to acknowledge m a n y stimulating discussions with Prof. E. Teller a n d Drs. A. G o l d b e r g a n d M. Weiss. The authors are especially indebted to Dr. Ben Carter for his assistance in deriving the t r a n s f o r m a t i o n brackets.

References 1) K. T. R. Davies, S. J. Krieger and M. Barangcr, Nucl. Phys. 84 (1966) 545 2) A. K. Kerman, J. P. Svcnne and F. H. M. Villars, Phys. Rev. 147 (1966) 710; W. H. Bassichis, A. K. Kerman and J. P. Svenne, Phys. Rev. 160 (1967) 746 3) I. Talmi, Hclv. Phys. Acta 25 (1952) 183; M. Moshinsky, Nucl. Phys. 13 (1959) 104 4) W. H. Bassichis, B. A. Pohl and A. K. Kerman, Nucl. Phys. Al12 (1968) 360 5) L. A. Copley and A. B. Volkov, Nucl. Phys. 84 (1966) 417; M. Muthukrishnan, Nucl. Phys. A93 (1967) 417 6) M. R. (3unye, Nucl. Phys. All8 (1968) 174 7) F. Tabakin, Ann. of Phys. 30 (1964) 51 8) F. Villars in: Rcndiconti Della Schvola Internazionale E. Fermi, XXIII Corso (Academic Press, New York, 1963); M. Baranger in 1962 Lectures in theoretical physics, ed. M. Long, (W. A. Benjamin, Inc., New York, 1963) 9) A. K. Kerman, Nuclear forces and HF calculations, Summer School of Cargdse/Corse, September, 1968 10) B. F. Gibson, A. (3oldberg and M. S. Weiss, Nucl. Phys. Al18 (1968) 225 11) S. (3. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 2 (1955)