The quantum theory of nucleon correlation in finite nuclei. II. Higher order correlation effects and additive pair approximation

(2.19)

.

It is easy to introduce (2.16) into this formula. First we note that E(2) = EF + ,??(‘J). Substituting this into (2.19) we get matrix components of H-E, which we compute from (I, 2.20). Naturally, in doing that we write $c2) = $F + g(2) and for g(2) and gt3) which appear in (2.19) we substitute their form from (1.7). We obtain by simple calculation ~(3) = c -+$-E’2,(N3;3N2). SijlNijt ijl N3 In this formula Cijl is the solution and X3 are defined as follows:

(2.20)

of (2.13), Nij, is given by (2.14) and N, , N3

N, = 3 ! ( #‘2’ 1 t,V”‘),

(2.21)

N3 = 3! (c+V3) 1 #‘“‘),

(2.22)

and X,=3!‘2(g’2)IH-EFIg’3)j+ 1

cc (f(ij)jH-EFlf(sfu)). ii1 stu (ijl# stu)

(2.23) I

As we see the formula for E”c3) is very similar to the formula for ,!?c2)given in (1.16). Apart from the last term which is proportional to e(z) and very small because of the structure of (N3 - N,), E(3) consist of a term additive in triplets plus the term containing the off-diagonal matrix components X3 . We can compute eC3) as follows. First the 3-particle correlation functions diil and the correlation energies CiiE can be computed from (2.13), one at a time. Besides the &‘s we need the matrix components N, , N3 and X3. These can be computed using the $ifl’s obtained from (2.13). The Et2) appearing in (2.20) assumed to be known from preceding calculations. Using the same arguments as in the case of the 2-particle approximation we can state that if we add Et3), computed as described, to the previously computed Ec2) we get an upper limit to the exact energy: E’3’ = EF + ,??2) + Z?‘3) > E. 595/76/1-10

(2.24)

144

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(B) Full Orthogonality We have seen in developing the 2-particle approximation that we can develop two different modeIs depending on the orthogonality condition used. The use of partial orthogonality, which does not restrict the generality of the wave function, led to the Eq. (1.14) for the $ij and to the Eq. (1.16) for Et3). The full orthogonality, which restricts the generality of the wavefunction slightly (as we discussed in Section II-B of I) leads to Eq. (1.15) and (1.17), respectively. Comparing these two sets of equations we see that while the two equations for & are not much different the two energy expressions differ considerably. Although (1.16) has the same form as (1.17) the two expressions are different because 2 is a much simpler expression than X. Since the two orthogonality conditions led to two different models in the 2-particle approximation, we can expect the same to be the case in the 3-particle approximation, and since one of the models was much simpler than the other it is worthwhile to explore whether this will be the case in-the 3-particle approximation. Let us start again with the “independent triplet” wavefunction (2.2) but les us now replace the orthogonality condition (2.5) with the following: (4s I 6,t> = 0,

(s = I,..., A; (ijl> included).

(2.25)

We do not have a normalization condition like (2.6) in this case. We get for iij, , the correlation energy of the triplet with an arbitrary $ij2 : (2.26) The variation of iij, with respect to & is carried out exactly in the same way as in the 2-particle approximation (Sec. IV-D of I). We get for dijl the equation

iHijL + p123°ijll Ail = zijl+ijl 9

(2.27)

where PlZ3 is a projection operator which eliminates the “basic set,” including & , r#~, & from the function on which it operates. cijl is the minimum of gijl . As we see this equation is quite different from (1.15). The term -P120ijoij present in (1.15) is missing from (2.27). The reason is that the full orthogonality eliminates the matrix component from which this term would come. With full orthogonality we have <$ijL I Oijl I Pijl) = O. Also, for both orthogonality

(2.28)

conditions,
(2.29)

THE

QUANTUM

Let us multiply We get

THEORY

OF NUCLEON

CORRELATION

(2.27) from the left by (&,

IN

1, and subtract

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145

the result from (2.26).

It is clear that iijl should become equal to Cijl if the solution of (2.27) is substituted into (2.26). But (2.30) shows that Eni,l- CijCwill be zero only if +rjl

s 0.

(2.31)

The result is that (2.27) is not compatible with (2.26); therefore (2.27) has only the trivial solution (2.31). Since (2.28) is valid for any higher order correlation function, the same result would be obtained for higher order approximations. In the 2-particle approximation we do not have this problem because the corresponding matrix component is not zero. It is clear therefore that, in the case of full orthogonality, instead of (2.2) we have to start with the following function: * = #F + g’2’ + f($)

= $/J(2)+ f(i)).

(2.32)

Using (1, 2.20) we get by simple calculation

where all symbols are familiar

except the following

<+ijt I Bijz) =
matrix component:

H i gt2’),

(2.34)

which means that we have compressed all terms depending on the 2-particle functions into c&,, , i.e., this function does not contain the 3-particle functions &.r . Variation of (2.33) with the full orthogonality as subsidiary condition gives

where Ci,il is the minimum

of gijl and

$ij, = &Jij, .

(2.36)

The equation for the best 3-particle function now contains all the 2-particle functions in $,, , which is, of course, the consequence of the presence of gt2) in (2.32). Next let us calculate Cijc from (2.35) and compare it with (2.33). The iij, of

146

SZASZ

AND

SCHROEDER

(2.33) must be equal to gija if the &, occurring in (2.33) is the solution of (2.35). From the condition II = ) (2.37) Ejl

Eijl

we get by simple calculation (2.38) where fiz = ( I,P / #(“J>. W e note that (2.38) is exactly analogous (I, 4.33) obtained for cii in I: Eij = &($ij 1Oij j /Jdij)* To show the analogy we use (I, 2.20) to transform

to the result

(2.39) (2.39) back into the form

T&<~G I Oij I ~1) = I H I #F>-

(2.40)

Now let us recall that in our notation I&

(2.41)

EC l/(l),

so we can write Rl = (Ip

j p)

= (#,

I #F) = 1)

(2.42)

and using (2.40) and (2.42) we can write (2.39) in the form

Gj = Wij)l H I VW% .

(2.43)

The analogy between (2.43), for the 2-particle case and (2.38) for the 3-particle case is complete, thereby showing that our choice for the starting trial function of the 3-particle approximation, Eq. (2.32), is the right one. Let us consider now the complete trial function of the 3-particle approximation: #(3)

=

#‘2’

+

g’3’.

Using our standard formula (I, 2.20) for the matrix components simple calculation, for the 3-particle correlation energy the formula

(2.44) we get by

where Iv3 = (Ip

I $h’“‘),

(2.46)

THE

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IN

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NUCLEI

147

and

23= 1231cstu IIf If(stuD.

(2.47)

(ijZ #stu)

Next we introduce Simple calculation

the correlation gives

energies Cijl , the solutions of (2.35), into (2.45).

where Nijl = (i,P) + f(ijZ) / t,V2) + f(i$)).

(2.49)

Our result is that we can compute the correlation functions &it from (2.35), one at a time, if the l-particle functions and the 2-particle functions are known from a -previous calculation. Using the correlation energies Cijr , obtained from the “independent triplet” equations we can compute the total correlation energy ,!?c3)from (2.48). This formula is completely analogous to (1.17), the formula for Ec2). Applying the same arguments which led to (2.24), we obtain again E’3’

= EF +

E(3)

+

E(3)

>, E,

(2.50)

where ,??t2)is obtained from (1.17), E(3) from (2.48) and E is the exact energy.

III.

THE ~-PARTICLE

APPROXIMATION

In the treatment of the higher order effects beginning with the 4-particle approximation, we have to consider now the problem of interacting and non-interacting groups. The quadruple excitations which represent the 4-particle effects may describe the collision of 4 particles (or rather the property of the wave function making such collisions very unlikely in case of a repulsive interaction) or they may describe the simultaneous collision of two pairs of particles. In the former case we may talk about an interacting group which was called a “linked cluster” in the earlier literature; in the latter case we talk about a noninteracting group (“unlinked cluster”). Let us write the full 4-particle correlation function in the form +ijkl = 4&t + +F.L 7

(3.1)

where the superscripts L and U refer to “linked” and “unlinked,” respectively. It is plausible to assumethat the unlinked terms are built from the 2-particle functions +ij , already known from the 2-particle approximation. Therefore our

148

SZASZ

AND

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task here is to determine the linked term. We will proceed in such a way that first we shall put c$” = 0 and determine 4”. Then with the 4” known, we shall enIarge the model to include 4”. It is obvious from the preceding discussion that we can again develop two different models according to the orthogonality condition. In connection with this we note that the energy expression (2.20) is much more complex than the energy expression (2.48). It is clear that as we proceed to higher approximations we should try to keep the formulas as simple as possible. When we consider a trial function like (3.1) which is rather complex first because it is a 4-particle term, second because it consists of two parts, we must try to work with that energy expression which is the simpler; from the S-particle approximation we see that the full orthogonality leads to a simpler energy expression. For this reason, we shall develop the model with partial orthogonality with a trial function containing only c#+; the model containing 4” as well as 4” will be developed only with full orthogonality. (A) Partial

Orthogonality

For the starting point we choose, in analogy with (2.2), the trial function # = #F + f”fijkl>,

(3.2)

where fL contains only the “linked” correlation function. Since this function is completely analogous to the 3-particle case, we do not need to repeat the details of the calculations here. For the best 4-particle linked correlation functions we get the equation

in complete analogy with (2.13). For the 4-particle correlation energy we get ~(4)

=

c

GjkfijkL 4

+

+

-

(e(2)

+

E’(3))

( N4

N,

N3),

(3.4)

ilk2

where Zijkl is the solution of (3.3) and

NiikL = , N4 = 4! ( i,b4)j a,+b’“‘),

(3.5) (3.6)

and X4 is the collection of off-diagonal matrix components between 4-particle functions of difficult indices, as well as between 4-particle and lower order functions.

THE

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(B) Full Orthogonality Let us start now with the following

trial function:

5) = $bF+ g(2) + g(3) + fyijkz)

= lp3) + fyijkz).

(3.7)

In this trial function we have only the “linked” term. The mathematical procedure is therefore completely analogous to the 3-particle case. We get first the correlation energy of the quartet ([jkl):

where pi = E(2) + Et31 and iijk, contains all lower order correlation functions. Variation of (3.8) with respect to the 4-particle function yields the equation

lHift2 +

- al +i”jkL= Eijkl$kkl- P1234Jijkl 3

p1234°ijk,

where Jij,, is the antisymmetrized form of &,, (3.9) and (3.8) gives the relationship hkl

= <<+EkL I = ((f%‘jkZ)

. The “compatibility

BijkC>im3>

(3.9)

test” between

(3.10)

I H I P’

+ gc3’)/fi3),

where fi3 = (ip

/ zp).

(3.11)

The 4-particle linked terms can be computed from (3.9), one at a time, similarly as in the 3-particle case. As we see, this equation contains all the lower order functions through the presence Of Jijl;, . Let us proceed now to the full 4-particle approximation which contains all linked terms for all quartets:

$A’“’= 3p”’+ c fyijkr). ijkl

(3.12)

With (3.12) we can build up the expression for the 4-particle correlation energy. Using the standard formula for the matrix components, Eq. (I, 2.20), we get

where iv4 = (fp

I tp”‘),

(3.14)

and 24

=

C ijkl

C sptu

(f’-(ijkl)

I H

IfYvW

KijkO

#

(wu>>.

(3.15)

150

SZASZ

AND

Using (3.9) we get for the correlation j34)

=

SCHROEDER

energy the formula ^ ~m$-hlcz I 2 ,

c

ijkl

4

(3.16)

4

where N
/ I+ht3)+ f=(ijk/)).

(3.17)

The philosophy behind the procedure described here is of course the same as the general philosophy of the independent pair model. Writing (3.7) we assumed that the linked correlation function 4” can be accurately computed in the absence of other linked as well as unlinked correlation functions. Having determined the linked terms, one at a time, from (3.9) we must now include the unlinked terms in the total 4-particle correlation energy. In order to do that let us consider now the complete 4-particle approximation including all linked and unlinked terms: (3.18)

p4) = p3) + i;z (fL(ijkZ) + f U(ijkz)).

We get the energy with (3.18) easily from (3.13). Let us introduce the following notations: (3.19) Riikz E Hiikz + Oijkz a; Ah

E

$

<~&cz

1 &kz

1 #hkz)

A%Lz

=

$

<$ikz

1 &kz

1 +zkz);

A&Z

=

&

<$zkZ

I &cl I 4&z)

24L = 2;”

c

c

ijkZ (ijkZ#

sntu sptu)

c

c

=

(fL(ijkZ)

+

2(&k~

(3.20)

1 ‘$iiikz);

(3.21)

+ ~<#‘L%c I $ijkz);

(3.22)

I H If”(sptu));

(3.23)

I H If”(sptu));

(3.24)

(f u(ijkZ) I H I f “(sptu)).

(3.25)

2(fL(ijkZ)

ijkl sptu (ijkl #s&u)

z4”

=

c

c

ilk1 sptu (ijkl #sptu)

Using these notations we get, for the correlation energy in the 4-particle approximation, E(4)

=

c

A&z

iikl

where &a is the normalization

+

Ai’iz

+

Tc4L

A&z

+

xy

+

T4”

(3.26)

+ fi4

integral of (3.18): N4 = (ip 1 zp).

iv4

’

(3.27)

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151

The Z?L4)can be computed as follows. The linked terms $kkl are the solutions of (3.9), and the unlinked terms #,, can be built up from the 2-particle functions & and (fikl . (See th e discussionbelow.) The expression (3.26) is arranged in such a way that it is easy to extract from it the energies related to the linked or the unlinked terms separately. For example, let us assumethat r#~”< 4” and let us put 4” == 0 in (3.18). Then (3.26) becomes E’(4)

=

c

Afkl

ijkl

j

$L.

4

(3.28) 4

From (3.8) and (3.9) we seethat

where N;,, = (#(3) + f”(ijkl)

1yY3) + fL(i$l)).

(3.30)

On putting (3.29) into (3.28) we get back (3.16). On the other hand if we assume that 4”
=

c ijkl

Z4”

&& N+ 4

(3.31)

K’

Regardlesswhether we use (3.26) or (3.28) or (3.31) we have again the relationship E’4’

=

,5’3’

+

E(4)

3

E,

(3.32)

where E is the exact energy.

IV. THE HIGHER ORDER EFFECTS: FULL VARIATIONAL ENERGYOF THE SYSTEM It is clear from the preceding two sections that we are able to write down the correlation energy of the n-particle approximation, for n = 5, 6,..., A, simply by inspecting the formulas for the 3-, and 4-particle cases.The sameis true for the equations for the n-particle functions. Also, a general procedure can be developed for the unlinked correlation functions as we have demonstrated in the case of 4-particle unlinked functions. The procedure is first to write down the energy with an unspecified general n-th order correlation function, and then write this as the sum of linked and unlinked terms. The result is an expression like (3.26) in which we have “linked, ” “unlinked,” and “mixed” matrix elements. We ar’e now in the position to compare the two different orthogonality conditions. Comparing first the equations for the correlation functions, we see that

152

SZASZ

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the equations are much simpler with partial orthogonality. To demonstrate this let us look at (2.13) and (2.35) which are the equations for &, . Writing #ii1 = PC1+ $221we have from (2.13) &l#ijC

=

cijl#zjl

(4.1)

3

where Sijl = Hzj, + f’l2,Oijl *

(4.2)

On the other hand, we get from (2.35) szjl&il

=

Gijl

+

ec2))

+ifl

-

p123Jijt

-

(4.3)

Equation (4.1) is much simpler than (4.3) mainly because&, contains all the lower order correlation functions and this complicated expression is missing from (4.1). A comparison of the energy expressions, however, shows that the formulas there are much simpler with full orthogonality than with partial orthogonality; the reason for this being that the off-diagonal matrices are much simpler with full orthogonality. Faced with the dilemma of a choice between the two models we note that one possibleprocedure would be to determine the correlation functions with partial orthogonality first, then orthogonalize them to the HF functions &. , $,i , & ,..., etc., thereby making them fully orthogonal, and then use them in the energy expressionswith full orthogonality. The energy obtained this way is still an upper limit to the exact energy. How can the equations for $ij , & , etc., be solved? No direct (exact) method of solution exist at the present time. Approximate solutions can be obtained by using the CT (configuration interaction) method or by using correlated functions. (We refer to the method developed by Hylleraas [3] and by Clark and Feenberg [4]). We are able now to write down the variational solution of the Schroedinger equation. On the basisof the preceding discussionwe get, using full orthogonality:

+ E(5) +

... +

E(A),

(4.4)

THE QUANTUM

THEORY OF NUCLEON CORRELATION

IN FINITE NUCLEI

153

where the higher order terms can be constructed as outlined. We note that the energies related to the linked correlation functions consist always of a term additive in pairs, triplets, etc. plus a nonadditive term. In order to achieve a certain desired accuracy, it may not be necessary to go beyond the first few terms of the series (4.4). There is evidence in the atomic calculations [5] as well as in the calculations for nuclear matter [6] which seem to show that the 2-particle correlation effects are much more important than the higher order effects. (Actually, for atoms the HF calculations generally give 99 % of the total energy (except for He) which showsthat even the 2-particle correlations are rather small. This is not true, however, in nuclei where the HF calculations gave only about 30-50 % of the total energy per particle [7]). It is plausible therefore to consider an approximation in which only the 2-particle effects are considered. This will be done in the next section.

V. THE “ADDITIVE

PAIR" APPROXIMATION

In the reaction-matrix theory of the nuclear matter developed by Brueckner, Bethe and Goldstone [8] the basic assumption is to consider the interaction of one particle pair in detail while neglecting the interaction of the other particles. This is particularly clear in the formulation given by Gomes, Walecka and Weisskopf (GWW) [9], which they called the independent pair model. In addition to computing the interaction energy of individual pairs independently from the other particles, GWW have also assumed that the total energy of the system is equal to the sum of the energies of the individual (independent) pairs. We shall call this assumption the “additive pair approximation.” We note that a generalization of the Brueckner theory to finite systemssuch as developed by Rodberg [lo] also leads to an additive pair approximation. The importance of pair correlations was early recognized by Macke [l 11. Investigating the properties of an electron gas Macke observed that the second order perturbation energy diverges: a convergent expression can be obtained, however. if one summed up the energies of 2-particle unlinked clusters of all orders. Investigating finite atomic systems Sinanoglu has shown [12] that the second order perturbation theory leads to an “additive pair” formula. In a theory developed for the treatment of electron correlation in atoms [13] Sinanoglu assumedthat the 3-, 4-, and higher order linked terms are negligibly small in atoms becausethe Pauli principle keeps the electrons spatially separated; important are only the 2-particle correlations plus the 2-particle unlinked clusters.

154 According to Sinanoglu

SZASZ

AND

SCHROEDER

[13] for a atom one can write in good approximation (5.1)

where the gij’s are the energies of the independent pairs. Equation (5.1) represents the sum of ,!V?@) plus the energies of all 2-particle unlinked clusters. In the present work we shall adopt the basic idea of Sinanoglu which is that the summation of the unlinked 2-particle terms results in an additive pair approximation. Adopting this idea we shall present here an “additive pair approximation” which will be a generalized and improved version of Sinanoglu’s theory. The generalization will consist of our theory being applicable to finite Fermion systems with a Hamiltonian given by (1.1) and not only to Coulomb interactions. The improvement will consist of some of the approximations being investigated in more detail here than in Sinanoglu’s work. Actually our goal will not be to show that (5.1) is a good approximation. We shall try to answer the question: What exactly are the approximations which lead to (5.1)? While we shall answer this question clearly we will not be able to prove that (5.1) is a good approximation. Let us assume that we omit from the general variational energy expression (4.4) all linked correlation terms beginning with the 3rd order. What is left is the sum of all 2-particle terms linked plus unlinked:

where &?, Et), etc., are the contributions of 2-particle unlinked terms to the higher order correlation energies. We shall assume that these are small and concentrate on the 4th order term. The wavefunction underlying this approximation is (3.18) with the linked terms omitted:

z)(4)= t,P) + ; C C f “(ijkl), 13 kl

where we have written the (ij) and (kl) summations as independent (hence the 4). For the 4-particle unlinked correlation function we put [13]

We evaluate Agkl by putting (5.4) into (3.22). Trying to separate the important

THE

QUANTUM

THEORY

OF

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terms from the smaller ones we shall proceed as follows. in (3.22):

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Consider

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155

the first term

(5.5) where we have used the fact that R is symmetric. Now we select from A($ij+ki) that term in which the position of the coordinates is the same as in the function on the left of (5.5) assuming that that term is the leading term. We put therefore

where the word “Exchange” indicates those terms which result from the antisymmetrization of 4&l . The argument underlying this step is the assumption that if we would use a nonantisymmetrized correlation function in (3.22)-although this would violate the Pauli principle-it would not lead to a too large numerical error. A nonantisymmetrized function would yield the first “diagonal” term of (5.6) and in the final analysis we shall keep only this term neglecting the “Exchange” terms. An argument in favor of this method is presented by the HF calculations for atoms. In that case the “diagonal” terms (the Coulomb integrals) are always much larger than the “exchange” terms (the “exchange” integrals). We assume that this argument can be carried over to the case of general nonlocal potentials. We separate similarly the “direct” and “exchange” terms in the second term of (3.32) as well as in fld . In order to compare the 4-particle unlinked correlation energy to the 2-particle correlation, we introduce the Eij’s into the former. This is accomplished by rewriting (3.19) as follows:

where Hij and Oij are the 2-particle operators defined by (I, 4.16) and (I, 4.14) and Bijkl is the remainder of Oiilcl after Oij and O,, are extracted. Putting (5.7) into (3.22) it is easy to seethat we shall get terms of the form of (I, 4.20) which lead directly to cij through (I, 4.22). The lengthy but straightforward calculation gives finally E = EF + ,f@ + j3~4,= & + c %Lii i
l

R L f L’ ’

(5.8)

i.e., the total energy of the 2-particle correlations plus the 2-particle unlinked term is given by EC=+%+&. i
L

i-

L’

156

SZASZ

AND

SCHROEDER

Here Eii is the solution of (1.15) and (5.10) (5.11) (5.12)

i
k
+

2(4i,

/ $h’)
1 u 1 $k,)

+

2(4jk

-

2<$jL

I $ij)
I v 1 +k,>

+

(%k

+

4&(i2)

- 2
i %3 i 4di2)

4kt(34)

j +ij)+il +

%Z +

‘%k

+

%)

,%j~kl

$k1(34))

hcz(34)l Ui(3) + Uj(3) + U,(l) &(34))

/ 2’ I ‘$k,)

+ Wl)

+ -& + -%a” + exchange terms.

(5.13)

In the above expressions the prime on the summations means that (kl) # (ij); the 2-particle functions are the solutions of (1.15); l ilz is defined by (1.12) and Ui is the HF potential (I, 2.21), xz and xdu are the off-diagonal terms in (5.2). Finally L’ is the “exchange” part of the normalization integral fld . It is easy to seefrom (5.9) that if we putfU = 0 in (5.3) then (5.9) becomesequal to EQ). We are now in the position to answer the question: What approximations lead to (5.1)? The expression (5.9) is exact with (5.3). We get the additive pair formula from (5.9) by assumingthat (1) R and L’ are negligibly small, and (2) L,JL can be replaced by 1. As we have indicated before we cannot prove that these approximations are permissible. We can present a semiquantitative argument, however, showing that the above approximations are not unreasonable. Consider first the L,,/L. Replacing pij by its average /3 we get from (5.10) and (5.12) for a system with A particles Lij _L -

1 + gp + ;/?(A - 2)(A - 3) + gl”(A - 2)(A - 3) 1 + &&4

- 1) + &@A(A

- l)(A - 2)(/t - 3) .

(5.14)

In I we have presented arguments showing that /I must be a small number. We have evaluated (5.14) for the following cases:6 = k (constant) or p = kA-” for n = 1, 2, 3. Setting k = 0.01 we get the curves shown in Diagrams 1 and 2.

THE

QUANTUM

THEORY

OF

NUCLEON

CORRELATION

1.0 ‘T-.‘-Lij I ‘Ice ..‘. .-\ i 0.8 L ‘,‘.\\‘\ I\r .’ ‘.\\.\ i \ o.b!-

-,

\

0.4

c

\

JO.94

1.

J \

\\

\

\ 50

100

I57

0.96

\

\\

------l.\\i.

NUCLEI

IO.98 1 \

0.2 ;o.,L.

FINITE

\\

\

/

IN

I 150

I 200

0.92 '\ '\\I! *

o.go

FIG. I. The quantity L,JL from Eq. (5.14) as a function of A. The full curve, plotted on the left hand scale, is for the case ,!?= 0.01. The dotted curve, plotted on the right hand scale, is for the case /3 = 0.01/A.

FIG. 2. The quantity L,/L from Eq. (5.14) as a function of A. The full curve is for the case p = 0.01/,42. The dotted curve is for the case p = 0.01/M.

As we seefrom (5.14) the limit of L,j/L is 1 for A = 2 asit should be. The diagrams show that for ,8 = 0.01 the curve goes to zero at about A - 90; for k = 0.01/A the curve goes to zero for A + a3 but it is still about 0.9 for A - 230. For /3 = 0.01/A2 and p = 0.01/A3 the curves go to 1 for A + co and the deviation

158

SZASZ

AND

SCHROEDER

from 1 is only about 1%. Therefore if /3 = 0.01 and constant the Lij/L cannot be replaced by 1; if /3 = 0.01/A the error introduced would be about 10 “/,; for ,/?I= 0.01/A2 and jI = 0.01/A3 the error introduced would be less than 1%. A different choice of k leads to similar curves. It is clear that in order to get a meaningful approximation in (5.1) or indeed in (5.9) /3 must be proportional to A-” with n > 1. If this condition is satisfied then L,JL can be replaced by 1 in good approximation. Looking at L’ and R we can neglect L’ as an exchange term and we can neglect the similar terms in R. In R we have the off-diagonal terms xZ and $“. We have seen in I that &, can be considered small relative to the diagonal term Cij Aij [I, Eq. (4.19)]. Also, calculations for Coulomb interactions [5] have shown that 8, is small compared to the diagonal term. While the numerical values will be very different for nuclear calculations, in a semiquantitative discussion we can assume that the relative magnitude of these terms will be the same for a nonlocal nucleon-nucleon interaction as it is for Coulomb interaction. The same argument applies to the relative magnitude of X4” and A& (See (5.2)). In the first four terms of R the matrix (pFLjl/ v I $,,) has the same magnitude as Zij [I, 4.331 but they are multiplied by a term of the magnitude of /3; also there will be considerable cancelation between these four terms. The fifth term is proportional to j3”. Finally, no argument could be formulated for the 6th and 7th terms of R. Summarizing, we may say that the approximations leading from (5.9) to (5.1) are plausible, although we cannot ascertain their accuracy at this stage. Our attitude will be that (5.1) is certainly a useful starting point because of its simplicity; should it prove to be inaccurate, one can always fall back on the exact formuIa (5.9).

VI.

DISCUSSION

We consider Eq. (4.4) the main result of this paper. Looking at both parts of the article our main result is that the variational solution of the many-particle Schroedinger equation is a finite sum of A terms as given by (1.3); the explicit form of that expression is given by (4.4). We have also shown that the terms of (1.3) can be calculated in successive approximations. At each step a system of independent equations must be solved for the correlation functions. The energy is an upper limit to the exact energy at each stage of the approximations. We have also seen that the unlinked correlation terms can be incorporated into the energy as in (5.9); plausible approximations led us to the “additive pair” formula given in (5.1). We can discuss our results from several different points of view. From the point of view of general mathematical physics, our results provide useful insight into

THE

QUANTUM

THEORY

OF

NUCLEON

CORRELATION

IN

FINITE

NUCLEI

159

the structure of the many-particle Schroedinger equation with fairly generally defined particle-particle interactions. For the nuclear physics our method provides a scheme on the basis of which highly accurate calculations can be made. It is clear that calculations with, e.g., Tabakin potentials or with other similar phenomenological nucleon-nucleon potentials can now be made utilizing our scheme; such calculations can be based, e.g., on the two-particle approximations or on the additive pair approximation. The accuracy of such calculations will certainly surpass the accuracy of the HF calculations. We note that while our method is developed for finite systems, the generalization of the method to arbitrarily large systems will be straightforward, opening the way for the development of a Thomas-Fermi type model and for nuclear-matter calculations. Work in this direction is in progress. Finalty the question might be asked: Is it possible to test the applicability of phenomenological nucleon-nucleon potentials to many-body problems with this method? What we have in mind is whether on the basis of a calculation made with our method one could decide if a given phenomenological potential accurately describes the nucleon-nucleon interaction in the nucleus? A clear answer could only be given if the exact solution of the Schroedinger equation would be available. Since that is not the case, what can be said at this stage is that e.g. using the 2-partitle approximation we can get a more reliable answer to the question above than what would be obtained from a HF or perturbation theory calculation.

ACKNOWLEDGMENT The

authors

are indebted

to Dr.

J. Shapiro

for useful

discussions.

REFERENCES 1. 2. 3. 4. 5.

L. SZASZ AND J. SCHROEDER, Ann. Whys. (N.Y.) 68 (1971), 1. F. TAIBAKIN, Anu. Phys. (N.Y.) 30 (1964), 51. E. A. HYLLERAAS, Z. Phys. 54 (1929), 347; 60 (1930), 624; 63 (1930), 291. J. W. CLARK AND E. FEENBERG, Phys. Rev. 113 (1959), 388. R. E. WATSON, Php. Rev. 119 (1960), 170; D. F. TUAN AND 0. SINANOGLU, .I. Chem. Phys. 41 (1964), 2677; R. K. NESBETT, Phys. Rev. 175 (1968), 2; L. SZASZ AND J. BYRNE, Phys. Rev. 158 (1967), 34; H. F. SCHAEFER AND F. E. HARRIS, Phys. Rev. 167 (1968), 67. 6. H. A. BETHE, Phys. Rev. B 138 (1965), 804. 7. A. K. KERMAN, J. P. SVENNE, AND F. M. VILLARS, Phys. Rev. 147 (1966), 710; W. H. BASSICHIS, A. K. KERMAN AND J. P. SVENNE, Phys. Rev. 160 (1967), 746; A. K. KERMAN AND M. K. PAL, Phys. Rev. 162 (1967), 970. 8. K. A. BRIJECKNER AND C. A. LEVINSON, Phys. Rev. 97 (1955), 1344; H. A. BETHE, Phys. Rev. 103 (1956), 1353; H. A. BETHE AND J. GOLDSTONE, Proc. Roy Sot. A 238 (1957), 551. 595/76/1-

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C. GOMES, J. D. WALECKA S. RODBERG, Ann. Phys. (iV. MACKE, Z. Naturforschung SINANOGLU, Proc. Roy. Sot. SINANOGLU, J. Chem. Phys.

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241