Self-consistency in the application of Brueckner's method to doubly-closed-shell nuclei

Nuclear Physics A174 (1971) 1--25; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

SELF-CONSISTENCY IN THE APPLICATION OF BRUECKNER'S METHOD TO DOUBLY-CLOSED-SHELL NUCLEI L. SCH,~FER and HANS A. WEIDENMOLLER

Institut fiir Theoretische Physik der Universitiit und Max-Planck-lnstitut fiir Kernphysik, Heidelberg, Germany Received 3 June 1971 Abstract: The relationships between various self-consistency conditions for the sinsie-particle potential U are investigated. These conditions are the Brueckner-Hartree-Fock condition, the requirement that the single-particle orbitais be maximum overlap orbitais, or generalized natural orbitals. The implications of the use of the latter two sets of single-particle functions are examined in detail. It is shown that these two different choices lead to the cancellation of different sets o f diagrams in the linked-cluster expansions of various quantities of physical interest. Both sets of basis functions lead to definitions of the particle-hole matrix elements of U which in lowest order coincide with each other, and with that obtained from the Brueckner-Hartree-Fock condition. The definition of the hole-hole matrix elements of U, on the other hand, as obtained from the Brueckner-Hartree-Fock condition is not related to these two sets.

1, Introduction. Discussion of the results l.l. FORMULATION OF THE PROBLEM

In the application of Brueckner's method to nuclear matter, one often chooses the single-particle potentials in a self-consistent way. This leads to the mutual cancellation of a large number of diagrams which appear in the Brueckner-Goldstone expansion of the ground state energy E o. It is believed that the first few terms (in the number of hole lines) of the linked-cluster expansion of Eo give a good approximation to the ground state energy, if the requirement of self-consistency is imposed. In nuclear matter, translational invariance alone suffices to determine many properties of the unperturbed system. The single-particle states, for instance, are plane waves. The single-particle potential U is diagonal in momentum representation. Hence, selfconsistency is only needed to determine the diagonal elements of U or, equivalently, the single-particle energy as a function of momentum. In a doubly closed-shell nucleus, on the other hand, no preferred 'set of singleparticle states is given a priori. Therefore, self-consistency is here employed for two different purposes: (i) To determine a "best" set of single-particle functions. (ii) Once this set is found, self-consistency is used to determine the diagonal elements of U, or the single-particle energies. While the second problem is usually solved in complete analogy to nuclear matter, several different prescriptions have been proposed for solving problem (i). Among these, we mention the choice of maximum overlap orOctober 1971

1

2

L. SCH.g,FER AND H. WEIDENMOLLER

bitals 1), of natural orbitals 2), and the principle of maximum cancellation of facIorizable insertions 3, 4) which in turn is a generalization of the well-known BruecknerHartree-Fock prescription. It is the purpose of this paper to elucidate the physical content and the implications of choosing maximum overlap orbitals or natural orbitals, and to compare the self-consistency conditions emerging from these two choices with each other, and with the Brueckner-Hartree-Fock condition. In subsects. 1.2 and 1.3 we define the maximum overlap orbitals and the generalized natural orbitals, respectively. We summarize our findings and discuss various relations which are satisfied by these choices of single-particle functions as well as their implications for the linked-cluster expansions of various quantities of physical interest. In sect. 2, we use the projection operator technique to derive exact relations for maximum overlap orbitals. We pay special attention to the equivalence of various conditions. We show that in general maximum overlap orbitals are different from generalized natural orbitals. Results concerning linked-cluster expansions are derived in sect. 3. Subsect. 3.1 is devoted to maximum overlap orbitals, subsect. 3.2 to generalized natural orbitals. These two different sets of functions lead to different definitions of the single-particle potential U. In subsect. 3.3, we compare these definitions, and discuss their relation to the one obtained from the Brueckner-Hartree-Fock condition. In appendix A, we prove the existence of maximum overlap orbitals and discuss the question whether they are uniquely determined. Appendix B is devoted to some explanatory remarks concerning details of our diagram expansions.

1.2. MAXIMUM OVERLAP ORBITALS Let q)o denote the unperturbed ground state wave function (a Slater determinant of A single-particle wave functions ~ol, i = i , . . . , A), and let ~Po denote the true wave function of the ground state. We use the words hole (particle) states to label singleparticle states which are occupied (empty) in the unperturbed ground state. Following in part a recent paper by Kobe ~), we show in sect. 2 that the following three conditions are strictly equivalent (if Eo is non-degenerate): (i) The particle-hole matrix elements U~ are chosen according to the BrillouinBrueckner condition 5, 6), for brevity denoted by BBC. Various equivalent formulations of this condition are given in subsects. 2.2 and 3.1 below. (The index M stands for maximum overlap orbitals.) (ii) The ground state wave function ~Po does not contain any i p-I h states (defined with respect to q~o as the vacuum state). Equivalently, the perturbation expansion of ~Po does not contain such states as intermediate states or final states. Equivalently, the linked-cluster expansions of ~Yo a n d E o do not contain such states. (iii) The functions ~oi contained in q% are chosen in such a way that I(~Polq~o)l 2 is stationary with respect to variations 6(pl of the tpi. The variation must leave the normalization J" Itpil2dax = l of the single-particle wave functions unchanged.

DOUBLY CLOSED-SHELL NUCLEI

3

One particular way to satisfy condition (iii) which seems very reasonable on physical grounds is to choose the functions ~0~ in such a way that I<~Vo]Oo)l z attains its maximum value (global maximum). In quantum chemistry, the functions ~Pi corresponding to this choice are termed "maximum overlap orbitais" 7). They were introduced by Brenig s). It can be shown that a set of maximum overlap orbitals always exists. The proof is given in appendix A. There, we use the assumption that kuo is square-integrable. The proof shows that without this assumption, maximum overlap orbitals cannot be defined, since ]<~Pol~o)l 2 is not bounded from above unless kUo is square-integrable. The Hamiltonian H of the full nuclear problem is translationally invariant, the eigenfunction ~vo, therefore, not square-integrable, since it contains a plane wave for the c.m. motion. This points to the need to modify H by adding a term which binds the centre of mass 9), if one wishes to use maximum overlap orbitals in the calculation (subsect. 2.1). This remark holds probably also for the BBC although the latter is only a necessary (and not a sufficient) condition for determining the maximum overlap orbitals. We believe that the c.m. problem has not received due attention in applications of the Brueckner method to light nuclei [except in ref. 10)]. The maximum overlap orbitals need not be uniquely defined since it is possible that several different Slater determinants yield the same global maximum for Il 2 (see the example in appendix A). The existence of the maximum overlap orbitals shows that the conditions (i) to (iii) can always be satisfied. However, even if the global maximum of I( ~uo1~ o)12 uniquely determines 4~o (except for a phase), the single-particle orbitals are not determined uniquely through conditions (i) to (iii), for two reasons. Firstly, since the conditions (i) to (iii) are only necessary, and not sufficient, conditions for maximum overlap it is possible that sets of orbitals exist which are not maximum overlap orbitals and yet obey conditions (i) to (iii). This possibility is demonstrated in appendix A. Secondly, the maximum overlap condition as well as conditions (i) to (iii) only specify the Slater determinant ~o and not the single-particle orbitals themselves. This is clearly shown by condition (iii). The expression I(~'o1¢,o)12 remains unchanged under any unitary transformation restricted to the space of hole states, and under any unitary transformation restricted to the space of particle states. Thus condition (iii) only defines a decomposition of the space L 2 of (square-integrable) single-particle functions into two orthogonal subspaces containing the particle states and the hole states, respectively. Because of the equivalence of conditions (i) to (iii), the same statement holds for the Briliouin-Brueckner condition, and we will show this explicitly in subsect. 2.2. The decomposition of L 2 into the two orthogonal subspaces just mentioned is thus the full physical content of the BBC. According to condition (ii) this decomposition has the property that there are no lp-lh states contained in ~o. The choice for U~ implied by the BBC can be made for any choice (self-consistent or not) advocated for Upp, or for U~. This is also shown in subsect. 2.2. Condition (ii) is of interest in its own right since it can be shown that Kobe's for-

4

L. SCH,~FER AND H. WEIDENMIL)LLER

mulation 1) of this condition is equivalent to the following one. Consider the sum of all linked Goldstone diagrams, starting with the vacuum (~o) and terminating with an (outgoing) particle-line and an (incoming) hole-line when read from bottom to top. Such diagrams are termed particle-hole diagrams, in what follows. An equivalent formulation of the BBC is then (subsect. 3.1)

E --0. If a diagram can be separated into two parts by cutting a particle-line and a hole-line at the same horizontal level in such a way that one of the two parts is a particle-hole diagram, then this part is called a particle-hole insertion. Condition (1.1) implies that in the expansions of ground state expectation values of operators and in the expansions of Eo and tP o certain classes of diagrams containing particle-hole insertions cancel mutually. In particular diagrams containing lp-lh intermediate states do not occur. This aspect of the BBC will be considered in detail in subsect. 3.1. We emphasize that in practical calculations U~ as determined from the BBC changes with the number of terms taken into account in the Bethe-Goldstone series. This reflects the fact that the approximation for ~o (and hence the maximum overlap condition) change with the number of terms taken into account. We demonstrate this in sect. 3 by giving a method to calculate the diagrammatic expansion of U~ up to any desired order and by giving explicitly a few diagrams of first and second order (in the number of hole-lines, see fig. 3). The Brillouin-Brueckner condition js formally very similar to the Briliouin condition familiar from Hartree-Fock theory. Yet the physical origin and content of the two conditions are quite different. While the Briilouin condition results from minimizing the expectation value of H with respect to a Siater determinant, the BBC determines two subspaces o f L 2 from properties of the exact ground state wave function ~oThis indicates that the BBC has little to do with minimizing the energy (sect. 2). 1.3. GENERALIZED NATURAL ORBITALS The (square-integrable) eigenfunctions of the single-particle density matrix are called "natural orbitals". If ~o is square-integrable, these functions form a basis in the space L 2 of single-particle functions. The unperturbed wave function ~o is defined in terms of those A natural orbitals which correspond to the A largest eigenvalues of the density matrix. If some of the eigenvalues are degenerate there may be some arbitrarinessin thechoice of ¢/'o- It follows from this definition of ~o that the sum of the occupation numbers of the hole states attains its maximum value (global maximum), N< = ~ (~olai+all~o) = maximum.

(1.2)

i<=A

Here, the creation (destruction) operator corresponding to the natural orbital tp~ is denoted by a,.+ (ai). We use the term "generalized natural orbitals" to denote any set of single-particle wave functions tp,, r~ ct = I . . . . . A which satisfies condition (i.2).

DOUBLY CLOSED-SHELL NUCLEI

5

Just as in the case of maximum overlap orbitals, the condition (1.2) is invariant under a unitary transformation of the set of hole states, or the set of particle states. Therefore, condition (1.2) only determines t a decomposition of the space L 2 into two orthogonal subspaces, the space spanned by the set of hole states {¢p~, i ~ A}, and the one spanned by the set of particle states {¢p~, a > A}. Here and in the following, the index N stands for generalized natural orbitals. We thus expect that condition (1.2) provides us with a functional relation between U~ and the matrix elements Up~ and U~ of the single-particle potential U, and leaves the choice of UpNpand U~ at our disposal. This is, indeed, what we shall find. The interest in condition (1.2) and, hence, in the generalized natural orbitals is due to the fact that condition (1.2) guarantees that the expansion of ~o in terms of n-particle-n-hole excitations out of ~o is optimized. For further details on this statement and on other properties of the natural orbitals we refer to the work by Kobe z) and by Coleman 11). If we consider diagram expansions, the generalized natural orbitals are associated with the concept of dangling insertions. Following Kirson 3), we define a dangling insertion as that part of a diagram which can be separated from the rest by cutting one particle-line and one hole-line at the same horizontal level. We show in subsect. 3.2 that the use of generalized natural orbitals is equivalent to the condition that the sum of all dangling insertions vanishes *t. This implies that in the linked-cluster expansion of the ground state expectation value of any operator, including the energy of the system, all diagrams containing dangling insertions cancel mutuallytt*. They do not cancel, however, in the linked-cluster expansion of the ground state wave function 9%. In particular, ~o contains lp-lh components. If we use generalized natural orbitals, the matrix elements Up~ are given by the sum of all linked diagrams which begin with the vacuum, which have one open (outcoming) particle-line and one open Ongoing) hole-line and which do not contain any dangling insertions. In fig. 5 we show the first few terms of the hole-line expansion of UNh . The use of generalized natural orbitals thus leads to considerations and relations which bear a close formal analogy to those associated with the use of maximum overlap orbitals. This situation changes if we use the natural orbitals themselves, since now we determine not only two subspaces of the space L 2, but rather a complete basis of single-particle functions (up to a possible degeneracy). The natural orbitals, therefore, determj.'ne all of the non-diagonal elements of the single-particle potential. We show, however, that it is not possible to express UpSpor UN by an expansion in powers of the two-body interaction, or in the number of hole lines. The use of natural orbitals seems to have little practical interest. However, it is possible to determine the natural orbitals within an approximation which takes into account all diagrams up to a certain order in the number of hole lines (subsect. 3.2). This statement contradicts one made by Kobe 2). '? Similar results have been obtained by Brandow ts), although the terminology used in this paper is somewhat confusing. '*~ This contradicts a statement made by Kirson a).

6

L. S C H A F E R A N D H. W E I D E N M O L L E R

2. Maximum overlap orbitais, and the Brillouin-Brueckner condition 2.1. T H E C E N T R E - O F - M A S S

PROBLEM

It was pointed out in sect. 1 that maximum overlap orbitals can only be defined if 7'0 is square-integrable. We therefore add 9) to the exact translationally invariant Hamiltonian H a potential Vc.m.which binds the centre of mass. We believe that this procedure is also necessary if the Bethe-Goldstone series is to converge. The resulting Hamiltonian H ' is written in the form H'

=

H -J- I/c.m. =

I .... "t- Vc.m. "~- Hint,

(2.1)

where t.... denotes the kinetic energy of the c.m. motion, and where Hin t is the intrinsic Hamiltonian which depends only upon the relative coordinates of the A particles. The wave function ~o obeys the equation H ' ~ 0 = Eo ~o.

(2.2)

In order to determine Eo from eq. (2.2), one separates H ' into an unperturbed part Ho with normalized ground state eigenfunction ~o and a perturbation V and uses the Bethe-Goldstone expansion. The unperturbed Hamiltonian Ho has the form A

A

A

H o = ~. [t(i)+u(i)] = Z ho(i) = • t(i)+ U, i=l

i=1

(2.3)

i=1

and V is given by V = Zv(i,j)+V~.~.-U.

(2.4)

i
Here, t(i) and u(i) denote kinetic energy and potential, respectively, of the ith particle. The choice of U is discussed below. The two-body interaction is denoted by v (L j). In practical applications, the term Vc.m. in eq. (2.4) is usually neglected. We believe that this approximation deserves further investigation. To obtain from E o the energy Ein t of intrinsic motion, one has to subtract from Eo the energy of the centre of mass. It is convenient to use for Vc.,,. a harmonic-oscillator potential. It is then reasonable to assume that the difference E o - E i n t is given by the lowest eigenvalue of H~.,,. = t .... + V¢.,,.. Alternatively, one may use eq. (2.2) to determine ~uo and calculate Ein t as the expectation value of H i n t with respect to q'o [ref. to)].

2.2. E Q U I V A L E N C E

OF THE CONDITIONS

(i) T O (iii)

As shown in the appendix there existsaset of maximum overlap orbitals ~Pi, i = !, . . . , A with the properties that

~o(Xl . . . . . xA)

q)*(xi i=

dx.i, = global maximum =

]

(2.5)

DOUBLY CLOSED-SHELLNUCLEI

7

and

f

l~,(x)12dx = 1,

i=

1. . . . .

(2.6)

A.

Unless ~o vanishes identically (a case of no interest to us), the maximum value (2.5) is different from zero. Because of the antisymmetry of ~o, the condition (2.5) is equivalent to the condition I<~o1~o>1 ~ = global maximum used in sect. 1, if 4% is the Slater determinant of the functions rpi, i = 1. . . . , A. It follows from the definition of the maximum that the left-hand side of eq. (2.5) is stationary with respect to variations 6tp~ of the single-particle wave functions q~i subject to the constraints (2.6). This is condition (iii) of subsect. 1.2. The Euler-Lagrange equations of this variational principle have the form 7) (iii)

~I~K(XK)= N - 1

f

~o(X 1 . . . . . xA) 11 rp*(x,)dx i,

N =

K = 1. . . . .

A,

(2.7)

tgK

. . . . . x.) 1-I i=l

I1 dxj.

(2.8)

j=l

Because of the antisymmetry of ~0, the solutions of eqs. (2.7) form an orthonormal set,

f

f

"

j=l

i~K

(2.9)

= 6,K.

In the following, the A solutions ~i(x) of eq. (2.7) are called hole states. To these A functions, we add a set of orthonormal functions q~o, a > A, all orthogonal to all ~i, i < A, called particle states, to obtain a complete orthonormal basis in L 2. Multiplying eqs. (2.7) with ~a, * a > A, and integrating we obtain (iia)

0 =

f

~o(Xt . . . . .

A

A

i¢:K

j= 1

xA)qJ *.(XK) 11 q~(X~) I-[ d x j .

(2.10)

We have thus found a set {rpi, q~o} of single-particle states with the property that teo does not contain any lp-lh states (defined with respect to ~o as vacuum). This is part of condition (ii) of subsect. 1.2. By multiplying eq. (2.10) with q~,(x), summing over all a > A, and using the completeness relation as well as the antisymmetry of ~o, we can derive eqs. (2.7)and (2.8). This shows that conditions (iia)and (iii) are equivalent. For the following discussion it is convenient to introduce the projection operators p(n), n > 0 which project onto n-particle-n-hole states, and to normalize ~o according to the condition

P(°~l~'o> = I~'o>.

(2.11)

8

L. S C H A F E R A N D H. W E 1 D E N M O L L E R

We introduce the notation .4

P = E P("),

(2.12)

n=2

.4

Q = Z P(")"

(2.13)

n=|

We now are in a position to formulate the various equivalent conditions. Let ~o be the solution of the Schr6dinger equation (2.2) corresponding to the nondenerate eigenvalue Eo. The following conditions for the choice of the set {q~;} of single-particle orbitals are equivalent: (ia)

P t ' ) r [ ~ o ) = 0.

(2.14)

T -- H' + H'Q(Eo - Q H ' Q ) - IQ H'.

(2.15)

Here, T is defined by

This is Kobe's form ~) of the BBC.

(ib)

P(')RI4~o) -- 0,

(2.16)

I~ = H ' + H ' P ( E o - P H ' P ) - ' P H '.

(2.17)

P(')] ~Uo> = 0.

(2.18)

where (iia)

This is the same as eq. (2.107 and is equivalent to (iii), as shown above. (ii b)

[~o) = ]tPo) + P(Eo - P H ' P ) - 'PH'] ~Po).

(2.19)

Eq. (2.19) shows that the perturbation expansion of 7/o does not contain lp-lh states as intermediate states. This is one particular formulation of condition (ii) stated in subsect. 1.2. We prove the equivalence of (ia 7, (ib), (iib) with (iia): (ia) ~ (iia). Eliminating QJ~o) from the left-hand side of eq. (2.2) and using the definition (2.157, we obtain the identity TI~P0> = Eo]~o). By projecting with P(~) on this equation the result follows. (iia) ~ (ib) ~ (lib) ~ (iia). We eliminate Pl~'o) from the left-hand side of eq. (2.2) and use the definition (2.17). We get ~(e(O) +P('))I ~Po) = Eo] ~Uo).

(2.20)

Multiplying eq. (2.20) from the left with p(1) and using condition (iia) we obtain statement (ib). To show that (ib) implies (lib) we state that [~Po) is eigenfunction to ~ with the eigenvalue Eo where /~ = (p(O)+ p(,))/~(p(O) + p(,)). (2.21 )

DOUBLY CLOSED-SHELL NUCLEI

9

This can be seen by multiplying eq. (2.20) from the left with prO), using the hermitian adjoint ofeq. (2.16), and adding the result to eq. (2.16). It is easy to prove, however, that if 14%) is an eigenfunction o f / / w i t h eigenvalue E o, then [1 + P ( E o - P H ' P ) - 1 PH']]4%) is an eigenfunetion of H' with eigenvalue Eo. But the eigenvalue Eo of H ' was assumed to be non-degenerate and we get the result I~o) = [1 + P(E o - PH'P)- I P n ' ] [ ~ o ) ,

(2.22)

which is condition (iib). From condition (iib) the condition (iia) follows trivially by multiplying from the left with pt~). 2.3. IMPLICATIONS OF THE BBC 2.3.1. The energy expansion. use eq. (2.11). This yields

We multiply eq. (2.20) from the left with (~ol and

Eo = (0o]/~f¢o + Ptt)~Yo).

(2.23)

The use of condition (iia) reduces this equation to

Eo -- <~ol//1~o).

(2.24)

From the definition (2.17) of/7I we see that lp-lh states do not appear as intermediate states in the perturbation expansion of E o because/7 only contains the operator P. The cancellation of all diagrams containing lp-lh intermediate states in the energy expansion alone does not imply the BBC. Following Kirson a) we will show explicitly below that these diagrams are also cancelled, if we use the generalized natural orbitals. We can show explicitly that the calculation of Eo from eq. (2.24) depends only on the decomposition of the space of single-particle functions into two orthogonal subspaces. Using eq. (2.4) in the form V = W - U we get from eq. (2.24) A

Eo = ~ (q~,lt@l)+(C'olW+

WP(Eo-PH'P)-~PWI~o).

(2.25)

t=l

The operator H' has the form H ' = ~ = 1 t(i)+ IV. Once 4o is known, P is uniquely determined. Any unitary transformation among the hole states (or among the particle states) leaves eq. (2.25) unchanged. The choice of these states is relevant only inasmuch as it yields arguments which allow us to terminate the series expansion of Eo after the first few terms. 2.3.2. Choice of the particle-hole matrix elements of U. In order to determine the single-particle potentials from eq. (2.16), we must relate Ho to the single-particle orbitals ~ , q~o. We do this by imposing the condition (ic)

(q~,lholq~,) = O.

(2.26)

[We do not require that the qh and ~o are eigenfunctions of ho.] Eq. (2.26) implies PtI)H o P = 0 = ptl~Hol~o).

(2.27)

10

L. SCHAFER AND H. WEIDENMIJLLER

Using this in eq. (2.16), we obtain (ic)

+ e(eo-eH'P)-'PW}l,t,o>

=

P"W{l +e(Eo-el4'P)-'ew}l%>.

(2.28)

The right-hand side o1" this equation as well as the curly bracket and the state 14'o) on the left-hand side depend only upon the decomposition of L 2 into the space of hole states and the space of particle states. Once this decomposition is made, the expressions just mentioned are uniquely determined, except for a phase in Iq~o) which cancels. Eq. (2.28) contains the expressions P(1)UlCbo) and P(1)UP as unknowns. Both expressions only contain the particle-hole matrix elements (~pilukoa) of U and their hermitian adjoints. Eq. (2.28) constitutes a system of linear inhomogeneous equations for these matrix elements. This system is obtained by multiplying eq. (2.28) from the left with a set of particular l p-lh states 4~. The solutions (q~lU]~oQ) depend, of course, on the particular representation chosen, i.e. on the choice of the states q~,., ~%. A unitary transformation among the states ~p~, or the states ~pi, induces the equivalent transformation among the matrix elements (~o;Iul~o~). Usually one requires the functions ~0g,q~ to be eigenfunctions of ho. Because of eq. (2.26), the matrix elements Uph are needed in the construction of the states q~,.,q~,. For this reason, the choice (2.28) is termed self-consistent. We note that eqs. (2.26) and (2.28) together are equivalent to eq. (2.16). Thus, they represent together another version of the BBC. We have seen that the BBC does not yield a unique definition of U. It only determines two orthogonal subspaces and provides us with a relation for Uvh. 2.3.3. A variationalprinciple. The operator/~ defined in eq. (2.17) depends upon P. Let us choose P in accordance with the BBC, and let us keep it fixed. Eqs. (2.16) and (2.21) then show that

6~,,(q~olB(P fixed)lq~o)lsBc = 0.

(2.29)

A similar stationarity property of T [see eq. (2.15)] for the BBC was emphasized by Kobe 1). 2.3.4. Comparison with the Brillouin condition. Let ]Xo) be the solution of the Hartree-Fock problem for H', so that (Xoln'lXo) = minimum.

(2.30)

(q~olH'lq~o) > (XolH'lXo) > Eo --- (q~olH'l~Po).

(2.31)

It follows that

We see that 4% is not the "best" Slater determinant as far as the expectation value of H ' is concerned. The reason why Eo is lower than (XoIH'IXo) although (q%lH'14%) is bigger than this value, lies in the correlation contained in ~o, i.e. in the occurrence of the term (q%lH'[P~Uo).

DOUBLY CLOSED-SHELLNUCLEI

I!

2.4. MAXIMUM OVERLAP ORBITALS AND GENERALIZED NATURAL ORBITALS From the definition of the generalized natural orbitals given in subsect. 1.3 we find that they satisfy the following system of equations: 0 = (qJola~'aol~o)

for all

i ~ A,

all

a > A.

(2.32)

(For a discussion of eq. (2.32) see subsect. 3.2 below.) In the same notation the BBC takes the form 0 = (~olal+aalkUo)

for all

i ~ A,

all

a > A.

(2.33)

The conditions (2.32) and (2.33) differ from each other and are in general incompatible [ref. 7)]. Indeed, writing qJo = O o + P ( l ) q / o + P ~ o and using eq. (2.33), we obtain from eq. (2.32) 0 = (P~ola~a,lP~o)

for all

i < A,

all

a > A.

(2.34)

The BBC and the condition for generalized natural orbitals are compatible only if eqs. (2.33) and (2.34) can simultaneously be fulfilled. Eq. (2.33) defines the set of hole states and the set of particle states. Given these sets, eq. (2.34) imposes a restriction on PWo. It is easy to construct functions ~o which violate eq. (2.34) once eq. (2.33) is satisfied. Thus, eqs. (2.33) and (2.32) can simultaneously be satisfied only, if we impose restrictions upon H', or upon the system. This is the case, for instance, if we put V = 0, or if we consider nuclear matter, where translational invariance guarantees that eqs. (2.32) and (2.33) are both satisfied, if the operators a2 create plane wave states. 3. Comparison of different choices of U and of their implications for linked-cluster expansions 3.1. CONSEQUENCES OF THE BBC We first show that the BBC implies eq. (1.1) and that the converse is also true. Hence, eq. (1.1) is another, equivalent formulation of the BBC. Then we examine the implications of the BBC for the expansions of the ground state expectation values of operators different from the energy and finally we discuss the calculation of Uph in terms of diagrams. To show that the BBC implies eq. (1.1), we write the denominator occurring in eq. (2.15) in the form [ E o - Q H ' Q ] -1 = [ < ¢ , o l H o I ~ o ) - Q H o Q + f E - Q V Q ] -1,

(3.1)

where we have defined 6E = E o - (OolnolOo>.

(3.2)

From now on, we assume that c/,o is an eigenstate of Ho, i.e. that the eqs. (2.26) and (2.27) are fulfilled. We expand eq. (2.15) in powers of {fiE-QVQ} and represent the terms occurring in the expansion by Goldstone diagrams, neglecting the exclusion

12

L. SCH.~FER AND H. WEIDENMOLLER

principle in intermediate states. Each diagram thus obtained contains a linked part which begins with the vacuum state and terminates with one particle-line and one hole-line. This is called the main part in what follows. In addition, the diagram may or may not contain one or several linked parts, each beginning and terminating with the vacuum state, and one or several fie insertions. In what follows, linked parts different from the main part and 6E insertions are jointly termed "disconnected pieces". If a diagram contains no disconnected pieces, i.e. consists only of the main part without fiE insertions it is called a linked diagram. All other diagrams are called unlinked. We have to show that the sum of all unlinked diagrams vanishes. In doing so, we follow an argument due to Brandow 4). The set of all unlinked diagrams appearing in the perturbation expansion of eq. (2.14) is decomposed into two distinct classes in the following way. Given any such diagram and reading from bottom to top, we denote by A that disconnected piece in the diagram which terminates first (it cannot be the main part). By definition, A must be either a 6E insertion, or a linked part without fiE insertions. In the first case, the diagram belongs to class one, otherwise to class two. We now perform a partial summation over the diagrams in class two. We sum over all relative time orderings between A and the rest of the diagram, keeping the time orderings in the rest of the diagram as well as the position of the last interaction in A fixed. As a result of this procedure, A becomes an on-shell insertion into the rest of the diagram at the point where the last interaction in A takes place ~2). In the expansion of eq. (2.14), we have to sum over all topologically different Goldstone diagrams. Moreover, we recall from the linked-cluster expansion of the ground state energy Eo that there is a one-to-one correspondence between the set of all fie insertions and the set of all linked parts (different from the main part) without fiE insertions. This means that there is a one-to-one correspondence between the unlinked diagrams in the two classes defined above. Since hE insertions and unlinked parts always carry opposite signs, the diagrams in the two classes cancel pairwise. We have thus shown that in the perturbation expansion of eq. (2.14), only linked diagrams survive. According to eq. (2.14), the sum of these diagrams vanishes. This is the content o f e q . (1.I). To show that eq. (1.1) implies the BBC we recall that a particle-hole insertion (PHI) is defined as that part of a diagram which can be separated from the rest by cutting a particle-line and a hole-line at the same horizontal level right above (below) the highest (lowest) interaction in the insertion. The cut is to be performed in such a way that the interaction just referred to belongs to the PHI, while the propagators for particle and hole do not. Special attention has to be paid to signs, since we cut both a hole-line and a closed loop (see appendix B). The PHI extends either downwards ( P H I D ) or upwards (PH1U) from the place of the cut. We have to distinguish carefully between these two cases. We first show that the linked-cluster expansions of Eo and of ~o do not contain any diagrams with PHID, if eq. (1.1) is fulfilled. From this we can deduce the BBC.

DOUBLY CLOSED-SHELLNUCLEI

13

We examine the linked-cluster expansion of Eo and of ~Uoand consider all diagrams which contain at least one PHID. These diagrams are grouped into classes. The diagrams in a given class differ only in the topological structure of that PHID which has the highest interaction relative to the other PHID in the diagram (termed the highest PHID in what follows), the rest of the diagram being topologically equal within each class. [The highest PHID may itself contain an unrestricted number of PHI. ] According to the definition just given, the level of the highest interaction of the highest PHID is fixed within each class. We now sum over all diagrams within a given class which differ only in the relative time ordering of the highest PHID and the rest of the diagram. This puts the highest PHID on the energy shell, and eq. (1.1) implies that the sum of all diagrams in a given class vanishes. Hence, the linked-cluster expansions of E o and of ~'o only contain diagrams without PHID. [For what follows it is important to note that the argument just given does not exclude the occurrence of PHIU in the linked-cluster expansions of Eo and ~o, provided these PHIU do not in turn carry PHID.] This implies that lp-lh states cannot appear as intermediate states in the expansion of Eo, and as intermediate or final states in the expansion of ~Fo,since otherwise the part of the diagram below these states would be a PHID. It follows that the perturbation expansion of ~0 (which contains both linked and unlinked diagrams) cannot contain lp-lh states as intermediate or final states either, since these would also have to appear in the linked-cluster expansion of either ~'o or Eo. This implies eq. (2.18), q.e.d. The argument given in the last paragraph has not ruled out the occurrence of PHIU in the linked cluster expansions for ~vo and for Eo, provided these PHIU do not in turn carry PHID. It is trivially clear that in the expansion of ~o, such PHIU must, indeed, occur since they cannot be put on the energy shell. In the expansion of Eo, these insertions can be put on shell. However, their sum is not related to the Hermitian adjoint of relation (1.1) since this adjoint consists of a sum over PHIU, each PHIU carrying an arbitrary number of PHI, while in the expansion of Eo only such PHIU survive which do not carry any PHID. Equivalently, we could determine E o as the sum of all diagrams having PHID but no PHIU. This can be done in analogy to the reasoning given in the last paragraph by ordering the diagrams according to the lowest PHIU. We thus see that the BBC does not imply the vanishing of all PHI, but only of some of them, in the expansion of E o (see fig. 1). We now consider the linked-cluster expansion of the ground state expectation value of some operator different from H'. It is well known 4. 13) that this expansion is given by the sum of all linked diagrams which contain the operator just once. We call the main part of the diagram that part which contains the operator and which cannot be divided into a PHI and a part containing the operator. We group together all diagrams which differ only in the structure of that highest PHID which is an insertion into the main part. Repeating the argument given above we see that the sum of all diagrams having at least one PHID in the main part vanishes. We are left with those diagrams which have no PHID in the main part. These can be grouped according to the lowest

14

L. S C H ~ , F E R A N D H. W E I D E N M O L L E R

PHIU which is an insertion in the main part; and by repeating the argument once more we obtain the following result. The linked-cluster expansion of the ground state expectation value of an operator different from the energy consists of all diagrams which cannot be divided into a PHI and a part containing the operator. The general structure of these diagrams is indicated in fig. 2.

a)

D

D

(b)

(c)

Fig. 1. Structure o f a general term in the linked-cluster e x p a n s i o n o f Eo if m a x i m u m overlap orbitals are used as single-particle basis. (a) does contribute whereas (b) a n d (c) do not. T h e b r o k e n line indicates a possible cut into a P H I D and a rest.

(a)

(b)

(c)

Fig. 2. Structure o f a general term in the linked-cluster e x p a n s i o n o f (~olO:;~Uo~(~P'oi~o'- ~, 0 ~ H'. (a) a n d (b) do contribute whereas (c) does not. T h e crossed part represents the part containing the operator O. M a x i m u m overlap orbitals are used.

(a)

(b)

- V_E (e) (h)

(c)

(d)

+ Q::V--x + x---c::V (f) (i)

(g) (k)

Fig. 3. First t e r m s in the hole-line e x p a n s i o n o f Up,. ~t Exchange diagrams are not shown. In the second order in the n u m b e r o f hole-lines there are further contributions from Bethe-Faddeev terms. F o r details on the s y m b o l s used here a n d in the following d i a g r a m s see appendix B.

DOUBLY CLOSED-SHELL NUCLEI

15

In order to calculate Uph in terms of diagrams it is advantageous to start from eq. (l. 1) which states that the sum of all PHID vanishes. With the help of the arguments developed above it is easy to show that this statement implies the following one: The sum ~ ' of all PHID which themselves do not contain PHID vanishes. The sum ~ ' contains only one diagram (fig. 3a) containing Uph but an infinite number of diagrams containing U~p. The relation X' = 0,

(3.3)

together with the requirement that U be hermitian gives an implicit equation for U~ [the index M stands for maximum overlap orbitals] which can be solved in perturbation theory up to any desired order. In fig. 3 we give the first few terms of U~, grouped in terms of the hole-line expansion. The symbols used in fig. 3 are in detail defined in appendix B. We return to fig. 3 in subsect. 3.3 where we discuss the mutual relationship of various choices of U. 3.2. GENERALIZED NATURAL ORBITALS AND THEIR IMPLICATIONS We recall from subsect. 1.3 that the single-particle functions t#~ which solve the variational problem A

N< = ~. (~Pola+ai[~o) = global maximum

(3.4)

i=l

are termed generalized natural orbitals. We also recall that condition (3.4) does not lead to a complete determination of the states tp~, but only to a decomposition of L 2 into two orthogonal subspaces, that of the hole states and of the particle states, respectively. The Euler-Lagrange equations of this variational problem read

(~ola+a~l~o) = 0,

i < A,

a > A,

(3.5a)

(~Pola~+aal~Po) = 0,

i < A,

a > a.

(3.5b)

Eqs. (3.5) are not equivalent to eq. (3.4) since they only state that N< is stationary. From the definition of the natural orbitals, it is easy to see that there are infinitely many choices of the set {tp~, i = 1. . . . . A} of hole states for which N< is stationary. We may supplement eq. (3.5) by the requirement that the second variation of N< is positive semidefinite. It is possible to show that the resulting conditions are equivalent to the conditions (3.4). In the following, we proceed in the same manner as in the case of maximum overlap orbitals, i.e. we use only the necessary (but not sufficient) conditions (3.5) for the generalized natural orbitals. We now investigate the consequences of eqs. (3.5) for the linked-cluster expansions. We first give some definitions. We call a dangling insertion (DI) that part of a diagram which can be separated from the rest by cutting one particle-line and one hole-line at the same horizontal level a). If the insertion dangles downwards (DID), the cut is to be performed right above the highest interaction vertex (belonging to the DID) in the particle-line, or the hole-line, which is to be cut. If the insertion dangles upwards (DIU),

16

L. SCH,~FER AND H. WEIDENMOLLER

one has to proceed analogously. According to this definition, all DI have the dimension of an energy. The D1 are different from the PHI defined in subsect. 3.1. The linked-cluster expansion of <~ola+ad~o>/<~ol~o> is given by the sum of all linked diagrams which contain the operator a~aa just once 4. 13). This set of diagrams can also be defined as the set of all diagrams which can be cut into a DID and a part which contains the operator a~a~ as the only vertex. According to eq. (3.5a) the sum of all these diagrams vanishes. We can separate this set into classes of diagrams, thc diagrams in a given class differing only in the relative position of the vertex a~aa and the DID forming the rest of the diagram. The sum of all diagrams belonging to one class is identical to the contribution of the DID (taken on shell) specifying that class, multiplied by ( e t - e o ) - ' . Here, e i and e~ denote the single-particle energies in the states tp~ and tp~, respectively. An example is given in fig. 4. We thus conclude that eq. (3.5a) implies that the sum of all on-shell DID vanishes, and by the same argument we can show that by virtue of eq. (3.5b) the sum of all on-shell DIU vanishes. It is straightforward to show that these statements are equivalent to the following one. The sum of all DID or of all DIU which themselves do not contain DI t,anishes. With an appropriate change of wording the proof is identical to the one given above showing the cancellation of all PHI in the linked-cluster expansion of the expectation value of any operator different from the energy (see subsect. 3.1). This proof also shows that DI do not occur in the linked-cluster expansion of the ground state expectation value of any operator different from the energy.

o-X

-X

;' Fig. 4. Example of the relation between <~ola,+a°l~/'o><~ol~Uo>-' and the dangling insertions. We now turn to the expansion of Eo. We consider first those diagrams which contain at least one DID which itself does not contain any DI. These diagrams cancel mutually. This can be shown in complete analogy to the reasoning used in the previous paragraph and with the help of the statement printed in italics. We now consider those members of the remaining set of diagrams which contain at least one DIU which itself does not contain any DI. These diagrams cancel mutually, too. As a result, all diagrams cancel which contain at least one DI which does not itself contain any DI. Obviously, these are all diagrams containing a DI. We conclude that the

DOUBLY CLOSED-SHELLNUCLEI

17

linked-cluster expansion of Eo is given by the sum of all diagrams which do not contain any DI. In the linked-cluster expansion of the ground state wave function ~Po, on the other hand, not all DI are cancelled since it is not possible to put all DI on the energy shell. In particular, the expansion of ~o contains lp-lh states as intermediate and final states. We now turn to the diagram expansion of the particle-hole matrix elements U~h of the single-particle potential for the generalized natural orbitals. We have seen that eqs. (3.5) imply that the sum of all DID without DI vanishes. Among these there is only the trivial diagram of fig. 5a which contains U~. We thus have the result that U~ is given by the sum of all DID different from fig. 5a which do not contain any DI. In fig. 5 we give the first terms of the hole-line expansion of U~. The situation is here different from the one encountered for the maximum overlap orbitals where we could not give an explicit diagrammatic representation for U~ in such a simple way. - 'z---x

:

'z..--.o

(aj

+

(,b)

(g) +

+

(c)

+

(f)

(h)

~

+*

(i)

* * "

(k) Fig. 5. First terms in the hole-lineexpansion of U~. We now turn to the natural orb~tals themselves. They are defined by the relations ( ~oia+ap[ ~o) -- 6~p(~ola+a~l ~o>-

(3.6)

This set of relations fully specifies the single-particle functions tp~, up to a possible degeneracy. However, the eqs. (3.6) do not determine which states are occupied or unoccupied, respectively, in ~o. This can only be decided by ordering the terms on the right-hand side of eqs. (3.6) according to their magnitudes. Except for the ordering of the single-particle states tp~, we thus see that the relations <~PoJa+apl~Po> = 0

for

~t ~ fl

(3.7)

fully determine the non-diagonal matrix dements of the single-particle potential U s . We investigate whether it is possible to obtain an explicit representation of all these matrix dements in terms of diagrams. In fig. 6 we give the first few terms (in the number of hole lines) of the linked-cluster expansion of the relation

<~Pola~'akl~Po) =

0;

i # k;

i, k < A.

(3.8)

18

L. SCHAFER ANt) H. WEIDENMOLLER

We use the fact that all DI cancel. In this expansion, the matrix elements UN of U N between hole states do not explicitly contribute in lowest order (see the first diagram in fig. 6, referred to as fig. 6a in the following). Thus we see that it is not possible to solve eqs. (3.8) for U~ using a perturbation expansion in the number of hole lines. We essentially reach the same conclusion if we try to represent U~ in terms of a perturbation series in the interaction W. The same results hold for the matrix elements U~p of U ~ between particle states. Hence, it is not possible to calculate the natural orbitals from a single-particle potential which itself is defined in terms of a linked-cluster expansion. If we want to use the natural orbitals in the framework of a calculation neglecting all but the lowest-order terms in the number of hole lines we have to diagonalize the contributions of the diagrams shown in figs. 6a and 7 selfconsistently, taking into account the matrix elements U~h as given in fig. 5b. (a)

Fig. 6. First terms in the hole-line expansion of the relation 0 - (Hola~a~jHo); k # i, k, i ~ A.

o Fig. 7. First term in the hole-line expansion of the relation 0 -: (Hola[a,ltFo) a ~ b; a, b :. A.

_ #--x

-- v----..-@

+

Fig. 8. Choice of Uph in Brueckner-Hartree-Fock calculations. 3.3. COMPARISON OF VARIOUS CHOICES OF U A comparison between fig. 3 and fig. 5 shows that in lowest non-vanishing order the diagrams contributing to the linked-cluster expansion of Upn are the same for maximum overlap orbitals, and for generalized natural orbitals (provided we use the same definition of Upp and Uhh in both cases). These diagrams are identical to the ones (fig. 8) usually employed to determine Uph in self-consistent Brueckner-Hartree-Fock calculations of finite nuclei 14). The latter choice can, therefore, be considered as a f i r s t step either towards the use o f m a x i m u m overlap orbitais, or o f generalized natural orbitals [see eq. (3.5)]. We have seen in subsect. 3.2 that there are infinitely many ways

of satisfying the conditions (3.5). Hence, the choice of Upn implied by fig. 8 is far from unique and depends strongly on the initial choice of hole states in terms of which the diagrams are calculated.

DOUBLY CLOSED-SHELLNUCLEI

19

In higher orders in the number of bole lines, the matrix elements of U;~ and U~ differ from each otber. Tbis is seen f.i. by looking at the second-order terms (figs. 3 and 5). The diagrams d and e occurring in the expansion of U;~ (fig. 3) are absent in the expansion of U~ (fig. 5). Generally speaking, the diagram expansion for U~h is simpler than that for U~. We have sbown above that also the linked-diagram expansions of the ground state expectation values of all operators are much simpler for generalized natural orbitals than for maximum overlap orbitals. Moreover, the choice of generalized natural orbitals leads to the cancellation of all dangling insertions and thus corresponds to the principle of cancellation of all factorizable insertions advocated by Brandow 4) and Kirson a). The particle-particle and bole-hole matrix elements of U are not defined for maximum overlap orbitals. Furthermore, various choices have been proposed for Upp in Brueckner-Hartree-Fock calculations. We, therefore, confine ourselves to a comparison of Uhh as determined from natural orbitals (U~), and from the Brueckner-HartreeFock conditions (U~). We show that these two conditions for Uhb are in general not compatible. The Brueckner-Hartree-Fock condition for U~h reads (il Ualk) = ½ ~, (ijlG(e, + ej) + G(e K + ej)]kj)~, js_A

(3.9)

where the index a denotes antisymmetrization of the matrix element. Details on the definition of the G-matrix may be found in appendix B. Eq. (3.9) implies the following self-consistency conditions for hole states. 0 = ( i i t l k ) + ½ ~_, (ij[G(e,+ej)+G(ek+ey)lkj),;

i ~ k;

i, k ~_ A;

(3.10)

j~_A

e, = ( i l t l i ) + Y~ (ijiG(e,+ej)lij),;

j~A

i < A;

(3.11)

where t denotes the operator of the kinetic energy. The condition for natural orbitals, on the other hand, can be read off from the diagram in fig. 6a, which gives the lowest-order term for hole states. This yields 0 = ~

(illG(et+ez)lab),(ek+el--e,,--eb)-l(e,+el--e,--eb) - l

a,b>A

× (abl6(~,+~k)Ikl);

i ~ k;

i,k < A.

(3.12)

Let us assume that we have found an interaction W such that eqs. (3.10) to (3.12) are all satisfied by the same set of hole states. We alter W in such a way that the matrix elements (ijlGIkl) and (ajlGIkl) ; a > A; i,j, k, 1 < A remain unchanged whereas the matrix elements (ablGlij); a, b > A; i , j < A are changed. One can convince oneself that such a change is always possible by a suitable modification of W. In lowest order in the number of hole lines, this change will not alter the separation of the space of single-particle functions into two orthogonal subspaces. It will also leave invariant the hole states and single-particle energies as determined from the Brueckner-Hartree-

20

L. SCH,AFER A N D H. W E I D E N M O L L E R

Fock condition. However, this change will modify eq. (3.12) and thus alter the natural orbitals. Hence, eqs. (3.10) and (3.12) are in general incompatible. In summary, we have seen that the physical content of the Brueckner-Hartree-Fock conditions for the choice of Upp, Upb and Uhh is quite different, although the prescriptions are similar. The prescription given for Uph can be interpreted as a first step towards the use of maximum overlap orbitals, or of generalized natural orbitais, both of which seem to be very reasonable sets of wave functions. The distinction between these two sets can only be made by the inclusion of higher-order terms. In neither case can we be sure that we do determine the chosen set by the conditions used for the diagrams. This ambiguity is caused by the fact that the conditions employed are not sufficient. Within this ambiguity, the cancellation of all dangling insertions leads to generalized natural orbitals. The self-consistent choice of U~h according to conditions (3.10) and (3.1 l) determines a particular set of maximum overlap orbitals, or of generalized natural orbitals. The physical content of this choice is not clear. It determines a set of orbitals which are, in general, different from the natural orbitals. This, however, is no cause for alarm since the important variational principle (1.2) is already satisfied by the generalized natural orbitals and there is no definite advantage in using the natural orbitals themselves. Probably the importance of the conditions (3.10) and (3.11) lies mainly in the determination of the single-particle energies ei, i =< A and is thus analogous to the case of nuclear matter. This situation resembles closely that encountered in Hartree-Fock theory. There, too, only the matrix elements Uph of the single-particle potential are determined by the variational principle, the choice of Upp and Uh~ being a matter of convenience.

Appendix A EXISTENCE OF MAXIMUM

OVERLAP ORBITALS

Let ~vo be a square-integrable function of the arguments xl, i = 1. . . . . ,4. Here, x~ stands for the space and (if needed) spin and isospin variables of the ith particle. An integration over x~ implies (if needed) a summation over spin and isospin indices. We define the functional z{goi} by the relation z{go,} = z(go, . . . . . g0A) =

f

~o(X 1 . . . . . XA) I-[ go~(x,)dxi" i=1

(A.1)

Throughout the appendix, all single-particle wave functions go~(x) are taken to be normalized to unity, I[golll = f Igo,(x)[2dx = 1.

(A.2)

We prove the following statement: There exists a set of,4 functions go~°)(x) with the property that for go~ = go,o) the functional [r{go~}[2 assumes its maximum value.

DOUBLY CLOSED-SHELLNUCLEI

21

Proof: According to the Schwartz inequality, Iz{q~,}l2 is bounded,

f

0 ~ Iz{~o,}l2 __< IWo(Xl. . . . . XA)Iz I-I dxj,

(m.3)

j=l

hence the supremum a of Iz{q~}l 2 exists. The supremum vanishes only in the trivial case ~o - 0. The norm of the A-tuple of functions (fl . . . . . fA) is defined by A

II(fl(xl) . . . . . fA(XA))Ila = E IIf,(X)II"

(A.4)

i=l

The functional z{~pi} is continuous with respect to {~p,} = (~Pl. . . . . ~PA)if the definition (A. 4) for the norm is used. According to the definition of the supremum, a sequence {~")} exists with the property that lim Ix(~")}l 2 = a. (A.5) N-'* ao

From {~")}, we will construct another sequence {~")} which also has the property (A.5), lim Ix{O[")}l2 = a, (A.6) II--* gO

and which in addition converges with respect to the norm defined in (A.4). Given this sequence, the completeness of the space of square-integrable functions guarantees the existence of the limiting functions ~p~O)defined by lim 0~") = ~p]O),

I1~0~°)11= 1,

i = 1. . . . . A.

(A.7)

n~oo

The continuity of z{cpi} gives the desired result, z{tp~°)} = a.

(A.8)

Hence, the proof is completed if we find a convergent series {~")} with the property (A.6). For this purpose, we define the A functionals M (°, i = 1 , . . . , A by the relations M o ) = M(O{~Pk}~,t =

f ~Po(Xt . . . . . XA) I-I q~(xj)dxj , d

.

(A.9)

j~i

It follows from the definitions (A.I) and (A.9) and from the Schwartz inequality that for any A-tuple {~p~} we have

IT{%}I < M 0)

for all i.

(A.10)

We now remove from the sequence { ~ ) } all those members for which I~{¢~")}1 is smaller than a fixed positive number ct, 0 < ~t < a. The new sequence {tp~")} still has the property (A.5), and by virtue of (A.10) we have ~t __
for all i.

(A.l 1)

22

L. SCHAFER AND H. WEIDENMIDLLER

We now define the sets C °) of ( A - l ) - t u p l e s {~oi}j, ~ to contain all ( A - l ) - t u p l e s with the property M (i) >= ~.

By (A. l l) all {rp(k~)}k,~ are contained in C (°. We define the operators D (i) operating on C (0 by '

O"){tpk}~¢, = (M(i)) -1

A

~o(Xl . . . . . x,~) 1-I c#*(xl)dxs = ~,(x,).

(A.12)

The operators D (° are compact. The proof of this statement can be given in full analogy to the proof of compactness of Hilbert-Schmidt operators as contained, for example, in the book by Riesz and Nagy *57. It rests upon the fact that kvo is squareintegrable. Furthermore, it is easy to see that I~{~o,}l _-< I~(~0~. . . . . ~0,_t, ¢~,, ~,+~ . . . . . ~0A)I.

(A.I3)

In the sequence {~0~')}, we now replace the functions tp~n) by the functions ~n) defined in (A.12). According to eq. (A.5), the definition of a, and the inequality (A.13), the new sequence fulfills the relation lim Iz(~0~n). . . . . ~o(~)_1,t~)12 = a. n.-.PO0

Because of the compactness of D tA}, we can select a subsequence {qo(~'). . . . . ~o~'_),, ~(x~'>} with the property that [l~n')-~("')li < e

if

m', n' > hA(e).

Repeating this procedure (A - I) times, we arrive at the desired result, i.e. a sequence {~I n)} that fulfills eq. (A.6) and has the property that II{~o~n)}-{~Im)}lla < e

if m, , > no(e).

(A.14)

The proof just given does not make use of any symmetry properties of kVo and applies therefore equally to fermions and bosons. In the case of fermions, antisymmetry of ~o implies the orthogonality of the different functions ~o~°), as shown in sect. 2. For bosons, the proof given implies the existence of a function ~o(°)(x) with the properties t~

i2

A

~'~(xl . . . . . XA) I-I ~°°(xi)dxj

= maximum,

j=l

f ~'g(x, .....

XA)p=l~O(°)(xj)dxjh(XA)dX,~

= 0

if

f

(A.15a)

q~(°)(x)h~(x)dx = O. (a.15b)

As a simple example we now consider the case of two fermions. The Euler-Lagrange equations (see sect. 2, eq. (2.777, of the variational problem ~e,[~{~o~}] = 0 read in this case

q~(x 7 = N-1 (q~*(Y)~o(x, y)dy, d

(A.16)

DOUBLY CLOSED-SHELL NUCLEI

23

/, ¢2(Y) =

N-IJ¢~(X)~o(X,

y)dx,

(A.17)

N = f dx dy ¢*(x)¢~(y)~Vo(X, y).

(A. 18)

We define the antilinear operator T by the relation

TIe) = f ¢*(X)~o(X, y)dx, and rewrite eqs. (A.16), (A.17) in the form 1¢, > = - 2T1¢2>,

(A.19)

1¢2) = +2T1¢,>.

(A.20)

Because of the antisymmetry of ~o, the expression (91T]¢) vanishes for every I¢). The integral operator B = T" T is of Hilbert-Schmidt type. This can be seen by using the explicit representation

B]¢) = f k(x, y)q~(x)dx, k(x, y)

= f d z IP~(x, z)~Vo(Z, y).

(A.21) (A.22)

There exists therefore a complete system of eigenvectors of B. Moreover, at least one eigenvalue # of B differs from zero, unless ~vo vanishes identically. Let Ig) be an eigenvector corresponding to eigenvalue/~. It is easy to see that Tlg), which is orthogonal to Ig), is also an eigenvector corresponding to the eigenvalue .u. We define Ih) by IITlv>ll-17'lv>. The pair Ig), Ih) satisfies the relations (A.19), (A.20) with ,:, = (h]Tlg). It follows that/~ = (gIT2[g) = -1212. We have thus shown that eigenvectors of B the eigenvalues of which differ from zero occur pairwise. Each pair is a solution of the eqs. (A.16), (A.17). There will be infinitely many such solutions, unless tP o can be represented as a finite sum of productsfi (x)gl(y). Any of such pair of solutions satisfies the condition (iii) defined in subsect. 1.2. The maximum overlap condition is satisfied by exactly one pair of solutions if and only if the eigenvalue of B with largest absolute value is not more than two-fold degenerate.

Appendix B SOME REMARKS ON DIAGRAM EXPANSIONS The diagram expansion can either be obtained by using 13) perturbation theory in IF, or by employing a hole-line expansion ~6). In either case, special care is required because of the finite size of the system which invalidates momentum conservation at each vertex. In carrying out the hole-line expansion, one may introduce the G-matrix,

24

L. SCH.~FER AND H. WEIDENMCrLLER

defined by

(co) = ( w - upp)-(w- u p s ) - Q - c(co), H o -co

(B I)

where we have defined Q =

E

Icd>
(B.2)

c,d>A

H o = ~, 8~,a+aa~,.

(a.3)

~t

Alternatively, one may work with the quantity (~(co) = W - I4

Q 6(09). Ho - Upp- co

(a.4)

The operators G and (~ are different, although they give the same lowest-order contribution to E0. From eq. (B.I) it is easy to see that G(a)) represents the sum of all ladder graphs containing an indefinite number of U-insertions into the particle lines. If we use eq. (B. I ), the energy denominators are to be calculated according to the usual rules [see ref. ' 7)]. This would not be true if we were to use eq. (B.4). It is clear that in the diagrams used in this paper U-insertions into particle lines will never appear since their effect is already contained in the definition (B.1) of G(co). The sign of an insertion has to be defined carefully. We recall that both particlehole insertions and dangling insertions are defined in such a way that we cut a particleline and a hole-line right above (below) an interaction vertex which belongs to the insertion in such a way that the dissected lines are replaced by short directed legs. In calculating the contribution of an insertion, we do not attach any energy denominators to these legs. However, we include a minus sign for each hole line, even in the case that the hole line manifests itself only in the form of a leg. According to the usual rules ,7), each closed loop and each hole line in a diagram contribute a minus sign. The rule for calculating an insertion given above guarantees that a diagram conraining an on-shell insertion can be calculated by taking the product of the main part and the insertion. By dissecting the diagram we have destroyed one closed loop. According to the definition given above, we count one hole-line twice and thus the overall sign does not change. References 1) 2) 3) 4) 5) 6) 7) 8)

D. H. Kobe, Phys. Rev. C3 (1971) 417 D. H. Kobe, J. Chem. Phys. 50 (1969) 5183 M. W. Kirson, Nucl. Phys. A139 (1969) 57 B. H. Brandow, Rev. Mod. Phys. 39 (1967) 771 R. K. Nesbet, Phys. Rev. 109 (1958) 1632 P. O. L6wdin, J. Math. Phys. 3 (1962) 1171 V. H. Smith and W. Kutzelnigg, Ark. Fys. 38 (1968) 309 W. Brenig, Nucl. Phys. 4 (1957) 363

DOUBLY CLOSED-SHELL NUCLEI 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

25

H. Lipkin, Phys. Rev. 109 (1958) 2071 H.J. Mang and W. Wild, Z. Phys. 154 (1959) 182 A. J. Col©man, Rev. Mod. Phys. 35 (1963) 668 H. A. Bethe, B. H. Brandow and A. G. Petchek, Phys. Rev. 129 (1963) 225 D. J. Thouless, The quantum mechanics of many-body systems (Academic Press, New York, 1961) M. Barang©r, in International School of Physics E. Fermi, Varenna, 1967, ed. M. Jean (Academic Press, New York, 1968) Riesz and Nagy, Vorlesungen fiber Funktionsanalysis (VEB Deutscher Verlag der Wissenschaften, Berlin, 1956) p. 165 T. H. Schucan and H. A. Weidenmfiller, Phys. Rev. C3 (1971) 1856 B. Day, Rev. Mod. Phys. 39 (1967) 719 B. H. Brandow, Ann. of Phys. 57 (1970) 214