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On the derivation of effective interactions by matrix diagonalization methods
Volume 56B, number 3
PHYSICS LETTERS
ON THE DERIVATION
OF EFFECTIVE
BY MATRIX DIAGONALIZATION
28 April 1975
INTERACTIONS METHODS ~
P.J. ELLIS School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Received 3 February 1975 A technique is developed which in certain cases allows the derivation of a completely linked two body effective interaction for two particles beyond a closed shell using matrix diagonalization methods. In particular this applies to a space consisting of 0 and 2;~to excitations. In order to understand the apparent lack of convergence o f the perturbation expansion for the effective interaction between two particles beyond a closed shell [1 ] and to attempt to provide more accurate results various matrix diagonalization methods have been used. We focus here on the diagonalization of 0fi~o configurations together with all or some of the excited 2h~o configurations (in an oscillator notation) in closed shell (non-degenerate system), closed shell plus one and closed shell plus two particle systems [2, 3]. These results are then used to obtain effective interactions. However a difficulty arises since unlinked terms are introduced, as has been pointed out by Mavromatis [4] and by Goode [3]. This is unfortunate since the perturbation series for the effective interaction, which formally contains no space truncation, does not contain any unlinked diagrams [5]. The purpose o f the present work is to show how these unlinked terms may be eliminated, thus yielding a more reliable comparison of the matrix results with other calculations and with experiment. Our results are only applicable to the case of a "linked Q space" which we now define. Neglecting energy denominators the perturbation expansion takes the form (~c]VQVQVQVIdPc) ,
(1)
where the operator Q forbids the non-degenerate unperturbed state qbc (or set of states in the degenerate case) from appearing in intermediate states. We are interested in the case where Q refers to a truncated space and we define a linked Q space by the requirement that eq. (1) can be put in the unlinked form, say (dPcl VQVI q,c ) (q~c I VI q~c) (cbc I VIq~c ) ,
(2a)
but it cannot be put in the form (q~cl VQVlePc) ( q'c I VQVl ~c> .
(2b)
Diagrammatically this corresponds to the possibility of unlinked diagrams only in the case wherethe piece(s) not linked to the main skeleton correspond to (4~cl Vlcb c) i.e., fig. IA is allowed but not fig. lB. We shall not explicitly refer to one body pieces of V, but they are to be understood. The unlinked diagrams o f fig. 1A are easy to deal with so we still refer to the space as a linked space, the important point being the exclusion of type lB. It is noted that in passing from top to bottom of fig. 1B the unlinked nature of the diagram requires that the unperturbed excitation energy change and that the number of particle-hole ( p - h ) excitations of the core charge. Thus any space consisting o f a fixed number of p - h pairs or consisting of all harmonic oscillator excitations of a given energy satisfies our definition of a linked Q space. In the degenerate case we shall have a set o f unperturbed states and the definition o f a linked Q space here is analogous. We break up the Hamiltonian into unperturbed and perturbing parts in the usual way Work supported in part by the U.S. Atomic Energy Commission Contract No. AT (11 -1 ) 1764. 232
Volume 56B, number 3 H=H 0+V
with
PHYSICS LETTERS
28 April 1975
HoePi=ei,l, i.
(3)
We choose a set of states i = 1 ..... n to span the degenerate model P space (the operator P projects onto this space) and a set of states i = n + 1..... n + m to span the Q space which we take to be linked. The (n + m) dimensional Hamiltonian matrix is then set up but rather than diagonalize the whole matrix, we diagonalize the Q and P space parts separately. H~Q = ~ Q ~ Q ;
n+m ~Q=i=~n+14cbi,
n
H~p = ~OpM ;
~bMp=i~=laMdp i.
(4)
The notation here is used to refer to the case of two particles beyond a closed shell. The closed shell and closed shell plus one systems will be denoted by using the subscripts c and ] respectively; no diagonalization of the P space is necessary here (assuming only one single particle state of a given j). Now the Schrodinger equation can be rewritten [1] as ~ e f f P ~ = EPkb,
(5)
with 1
~eff= PHP +PVQ E ~ -
Q QVP = PHP + h ( E ) ,
and using the basis (4) we obtain (~kM'lh(~)lCr~tp) = ~ ( ~ p 'IVI~d~Q)
1
(~k~lVl ~r~p) ,
(6)
where ~ is as yet unspecified. At this stage all diagrams consistent with our linked Q space are present and we indicate the four possible classes in fig. 2. Bloch and Horowitz have shown (see ref. [5]) that only the valence diagrams of type A are required in obtaing the valence effective interaction and that one obtains the analogue of eq. (5) with energies E replaced by valence energies measured with respect to the nondegenerate system. Here we are using a matrix method in (6) so that we need to remove explicitly the contribution of figs. 2B, C and D. The diagrams of fig. 2D are most easily removed by subtracting the term (OclHlOc) from the diagonal elements ofaU the matrices which we set up, eg. in (4); this also has the effect of measuring the unperturbed energies relative to that of the closed shel. We do not indicate this subtraction explicitly. The diagrams of fig. 2B can be explicitly calculated provided that the Q space of the closed shell corresponding to the valence Q space is wel defined. (The Q space of the closed shell plus one system should also be well defined.) Explicitly one obtains
<*clho( -,v)l,o> =
l
<
QIvI,o>,
(7)
d-O~cQ-e v where we assume without loss of generality [5] that the unperturbed two particle states are degenerate with energy e V. It is worth noting that if e e is the unperturbed energy of the closed shell the total energy in RayleighSchr~Sdinger perturbation theory is given by E = <~elHl~l,c> + (~clhc(ec)lobe),
(8)
a result which corresponds to summing all linked diagrams, since figure 1B does not arise in our linked Q space and (Cl,clHIcl,e) has been removed. This result has previously been given by Padjen and Ripka [6]. The remaining terms we wish to remove are those indicated in fig. 2C where the two valence particles must always remain within the P space since our Q space is linked. Writing
233
Volume 56B, number 3
PItYSICS LETTERS
28 April 1975
@.___@
[]
0----(3 A
Jl
B
Jz
dz
Ji
Fig. 1. Two types of unlinked diagrams.
[]
8
dj
d2 O
0---'~
0----0 di
A
B
C
d2
D
Fig. 2. Four classes of diagrams which occur in eq. (6).
Fig. 4. One body and unlinked diagrams which must be removed to obtain a pure linked two-body effective interaction. The diagrams are drawn in unfolded form without explicit indication of folds.
Fig. 3. Lowest order contribution of the type indicated in fig. 2C.
¢¢clhc(
- J)
l¢c 1
1+ d_~cQ~_ e V
('-~--~Q ~eg'/
. . . .
we see that the first term in parenthese yields eq. (7), the second term yields the contribution of fig. 3 and subsequent terms yields higher order contributions (the two particle states are labelled M and the basis is chosen such that H is diagonal). The geometric series is easily summed and simply replaces (~ - o~eO - e v) by (~ - wNQ w M) a result which is clear from eq. (6) since we have considered the case where the H-fimiltonians and wave functions factor into core and valence pieces, ignoring the exclusion principle, so that WNQQbecomes (wcNQ + wM). We are thus able to isolate the pure valence contribution to h (fig. 2A) as (~pM' IhVAL(e )1 ~bp M) =
(10)
Of course the second term here is not needed if there are no core correlations, for instance if one takes a l p - l h Q space and assumes Hartree-Fock selfconsistency so that the coupling to the core vanishes (in this case one could alternatively diagonalize in the complete P + Q space). In the first term the exclusion principle is obeyed in intermediate states but we may regard pairs of cancelling exclusion violating diagrams as also being present. In the second term the exclusion principle is ignored between the valence particles and the core diagrams. Then the difference between the two terms, apart from removing core contributions, will yield those valence diagrams which correct for exclusion violations in the core calculation. At this stage it might appear tempting to separate out the one and two-body terms from h VAL, however since the cross terms are needed we prefer to solve the BIoch-Horowitz equation 234
Volume 56B, number 3
PHYSICS LETTERS
28 April 1975
[H+ hVAL(6) -- 6 ]P~ = 0 ,
(11)
to yield the eigenvalues 6~ and eigenvectors, la) say, expressed in a / - / c o u p l e d basis via eq. (4). We may then solve for the effective Hamiltonian Herr = ~
Io~>6~(81,
(12)
o¢
where IF) is the biorthogonal completement. If only single particle energies are required eq. (11) is sufficient and corresponds to the linked, folded valence expansion. However we are interested in the two particle case so we now need to disentable the one and two-body contributions to eq. (12). Apart from purely two-body contributions, where both particles interact and the diagram is linked, we shall have included the classes of diagrams indicated in fig. 4 and these we require to remove. In fig. 4 Rayleigh-Schr6dinger energy denominators are now to be understood. Graphs A and B schematically give the nonfolded and once-folded contributions to the energy of single particle/1. (Note diagrams B - D are drawn in unfolded form without explicit indications of folds to avoid cluttering the figure). Summing the completely linked, folded expansion to all orders for a single particle is equivalent to solving the analogue of the Bioch-Horowitz eq. (11) for a single particle, i.e., Iv+
hW'L(6/,) -
= 0.
(13)
The graphs of fig. 4D can easily be summed as in eq. (9) by replacing h by hVAL(6 .. - Vh ) where we use the shorthand notation V: for the bubble insertion in single particle state/~. Only tl~ diag(~al term arises for a linked Q space. In like fa/hion we can sum in addition the diagrams of fig. 4ZCby using h V ~ ' L ( ~ / , - Vh + 6i2 - eh) if 6/2 is inserted (or folded) into 6/1 (The unperturbed energy of single particle state 1'2 is eh and (~h carries an extra minus sign associated with the fold). Now in the general case fig. 4C allows 6 h and 6/2 to be inserted back into each other so that a self-consistency problem arises and we obtain /, " /l + 6/2 - e/'2 - Vh) - (~/,]%, = 0 ' [H+'hVAL(6
[H+ hVAL(6 . . . Vh) . h " /2 + (~/, . eh
~/2ldPh = 0 .
(14)
The contributions of figs. 4 A - D is diagonal in j - j coupling and is given by the sum (~h + 6 h of the self-consistent solutions of eq. (14) (Note that unperturbed single particle energies are included). This result can then be subtracted from eq. (12) to yield the purely two-body effective interaction. The result (14) is really obvious; if we add the equations (14) together (using a wave function ~/,h ) we have the Bloch-Horowitz equation but with all interactions connecting the particles Jl and ]2 together deleted. All such terms belong to the two-body effective interaction and thus should not be considered here. This deletion implies that, apart from one qualification, the Bloch-Horowitz effective Hamiltonian breaks into the sum of a part referring to the particle labelled I1 and a part referring to 12" The qualification is that the Hartree-Fock insertions I'll and Vh are allowed so that the cross terms are present which involve the effective interaction for particle/1 producted with Vh insertions (and vice versa). The result is to replace 6 h - Vh thus demonstrating the cancellation of these particular unlinked diagrams against the corresponding unlinked folded diagrams. The cancellation of all the unlinked folded diagrams of fig. 4C does not take place because our linked Q space forbids the corresponding unlinked non-folded diagrams, i.e., it is a space truncation effect. It is perhaps worth pointing out two formal difficulties. Firstly, if we consider 2hw excitations by the present method we shall have included all possible folded diagrams. These diagrams formally have 2, 4, 6 .... hw energy denominators, although by taking sums of diagrams the factorization theorem [5] allows one to express everything in terms of 2h6o denominators. It is not clear whether it is best to compare such results with order-by-order calculations or whether one should strictly enforce 2h~o denominators (This could be carried out by evaluating eq. (10) using for t~ the appropriate unperturbed energy and then calculating the required folded diagrams explicitly.) A second complication is that some of the linked folded diagrams which we include would, in fact, cancel with nonfolded diagrams if the Q-space were extended (see the discussion of the reduced Bloch-Horowitz expansion in ref. [5]). 235
Volume 56B, number 3
PHYSICS LETTERS
28 April 1975
Summarizing, our prescription for obtaining the two-body effective interaction is to use eqs. (6), (7) and (10) to obtain a pure valence interaction which is then used to solve the Bloch-Horowitz equation. The eigenvalues and eigenvectors thus obtained allow the construction of an effective interaction which in addition to two body pieces contains unwanted one body, and products of one body, contributions. The latter are removed by explicitly calculating their contribution via eq. (14) - a coupled pair of Bloch-Horowitz equations for the two single particle energies. Thus finally we have obtained a completely linked two-body interaction using matrix techniques without the explicit calculation of diagrams. No difficulty arises if G-matrices are used provided some average starting energy is adopted, as is required for a matrix method. Probably the above method can be extended to nonLinked Q spaces by explicitly evaluating the contributions of the unlinked diagrams of fig. 1B, assuming that the space includes complete classes of such diagrams. This becomes complicated however and we shall not discuss it further. After this work was completed a preprint was received from Y. Starkand and M.W. Kirson concerning this same problem. These authors discuss the formalism in less generality than we have done, but give some numerical results. I would like to thank P. Goode for stimulating discussions.
References [11 B.R. Barrett and M.W. Kirson, Advances in Nuclear Physics, eds. E. Vogt and M. Baranger (Plenum, New York), Vol. 6, p. 219. 12] N. Lo ludice, D.J. Rowe and S.S.M. Wong, Phys. Lett. 37B (1971)44; Nucl. Phys. A219 (1974) 171; preprint; A. Watt, B.J. Cole and R.R. Whitehead, Phys. Lett. 51B (1974) 435. [31 P. Goode, preprint. [4] H.A. Mavromatis, Nucl. Phys. A206 (1973)477. [5] B.H. Brandow, Rev. Mode. Phys. 39 (1967) 771. [61 R. Padjen and G. Ripka, Nucl. Phys. A149 (1974) 273.
236
PHYSICS LETTERS
ON THE DERIVATION
OF EFFECTIVE
BY MATRIX DIAGONALIZATION
28 April 1975
INTERACTIONS METHODS ~
P.J. ELLIS School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Received 3 February 1975 A technique is developed which in certain cases allows the derivation of a completely linked two body effective interaction for two particles beyond a closed shell using matrix diagonalization methods. In particular this applies to a space consisting of 0 and 2;~to excitations. In order to understand the apparent lack of convergence o f the perturbation expansion for the effective interaction between two particles beyond a closed shell [1 ] and to attempt to provide more accurate results various matrix diagonalization methods have been used. We focus here on the diagonalization of 0fi~o configurations together with all or some of the excited 2h~o configurations (in an oscillator notation) in closed shell (non-degenerate system), closed shell plus one and closed shell plus two particle systems [2, 3]. These results are then used to obtain effective interactions. However a difficulty arises since unlinked terms are introduced, as has been pointed out by Mavromatis [4] and by Goode [3]. This is unfortunate since the perturbation series for the effective interaction, which formally contains no space truncation, does not contain any unlinked diagrams [5]. The purpose o f the present work is to show how these unlinked terms may be eliminated, thus yielding a more reliable comparison of the matrix results with other calculations and with experiment. Our results are only applicable to the case of a "linked Q space" which we now define. Neglecting energy denominators the perturbation expansion takes the form (~c]VQVQVQVIdPc) ,
(1)
where the operator Q forbids the non-degenerate unperturbed state qbc (or set of states in the degenerate case) from appearing in intermediate states. We are interested in the case where Q refers to a truncated space and we define a linked Q space by the requirement that eq. (1) can be put in the unlinked form, say (dPcl VQVI q,c ) (q~c I VI q~c) (cbc I VIq~c ) ,
(2a)
but it cannot be put in the form (q~cl VQVlePc) ( q'c I VQVl ~c> .
(2b)
Diagrammatically this corresponds to the possibility of unlinked diagrams only in the case wherethe piece(s) not linked to the main skeleton correspond to (4~cl Vlcb c) i.e., fig. IA is allowed but not fig. lB. We shall not explicitly refer to one body pieces of V, but they are to be understood. The unlinked diagrams o f fig. 1A are easy to deal with so we still refer to the space as a linked space, the important point being the exclusion of type lB. It is noted that in passing from top to bottom of fig. 1B the unlinked nature of the diagram requires that the unperturbed excitation energy change and that the number of particle-hole ( p - h ) excitations of the core charge. Thus any space consisting o f a fixed number of p - h pairs or consisting of all harmonic oscillator excitations of a given energy satisfies our definition of a linked Q space. In the degenerate case we shall have a set o f unperturbed states and the definition o f a linked Q space here is analogous. We break up the Hamiltonian into unperturbed and perturbing parts in the usual way Work supported in part by the U.S. Atomic Energy Commission Contract No. AT (11 -1 ) 1764. 232
Volume 56B, number 3 H=H 0+V
with
PHYSICS LETTERS
28 April 1975
HoePi=ei,l, i.
(3)
We choose a set of states i = 1 ..... n to span the degenerate model P space (the operator P projects onto this space) and a set of states i = n + 1..... n + m to span the Q space which we take to be linked. The (n + m) dimensional Hamiltonian matrix is then set up but rather than diagonalize the whole matrix, we diagonalize the Q and P space parts separately. H~Q = ~ Q ~ Q ;
n+m ~Q=i=~n+14cbi,
n
H~p = ~OpM ;
~bMp=i~=laMdp i.
(4)
The notation here is used to refer to the case of two particles beyond a closed shell. The closed shell and closed shell plus one systems will be denoted by using the subscripts c and ] respectively; no diagonalization of the P space is necessary here (assuming only one single particle state of a given j). Now the Schrodinger equation can be rewritten [1] as ~ e f f P ~ = EPkb,
(5)
with 1
~eff= PHP +PVQ E ~ -
Q QVP = PHP + h ( E ) ,
and using the basis (4) we obtain (~kM'lh(~)lCr~tp) = ~ ( ~ p 'IVI~d~Q)
1
(~k~lVl ~r~p) ,
(6)
where ~ is as yet unspecified. At this stage all diagrams consistent with our linked Q space are present and we indicate the four possible classes in fig. 2. Bloch and Horowitz have shown (see ref. [5]) that only the valence diagrams of type A are required in obtaing the valence effective interaction and that one obtains the analogue of eq. (5) with energies E replaced by valence energies measured with respect to the nondegenerate system. Here we are using a matrix method in (6) so that we need to remove explicitly the contribution of figs. 2B, C and D. The diagrams of fig. 2D are most easily removed by subtracting the term (OclHlOc) from the diagonal elements ofaU the matrices which we set up, eg. in (4); this also has the effect of measuring the unperturbed energies relative to that of the closed shel. We do not indicate this subtraction explicitly. The diagrams of fig. 2B can be explicitly calculated provided that the Q space of the closed shell corresponding to the valence Q space is wel defined. (The Q space of the closed shell plus one system should also be well defined.) Explicitly one obtains
<*clho( -,v)l,o> =
l
<
QIvI,o>,
(7)
d-O~cQ-e v where we assume without loss of generality [5] that the unperturbed two particle states are degenerate with energy e V. It is worth noting that if e e is the unperturbed energy of the closed shell the total energy in RayleighSchr~Sdinger perturbation theory is given by E = <~elHl~l,c> + (~clhc(ec)lobe),
(8)
a result which corresponds to summing all linked diagrams, since figure 1B does not arise in our linked Q space and (Cl,clHIcl,e) has been removed. This result has previously been given by Padjen and Ripka [6]. The remaining terms we wish to remove are those indicated in fig. 2C where the two valence particles must always remain within the P space since our Q space is linked. Writing
233
Volume 56B, number 3
PItYSICS LETTERS
28 April 1975
@.___@
[]
0----(3 A
Jl
B
Jz
dz
Ji
Fig. 1. Two types of unlinked diagrams.
[]
8
dj
d2 O
0---'~
0----0 di
A
B
C
d2
D
Fig. 2. Four classes of diagrams which occur in eq. (6).
Fig. 4. One body and unlinked diagrams which must be removed to obtain a pure linked two-body effective interaction. The diagrams are drawn in unfolded form without explicit indication of folds.
Fig. 3. Lowest order contribution of the type indicated in fig. 2C.
¢¢clhc(
- J)
l¢c 1
1+ d_~cQ~_ e V
('-~--~Q ~eg'/
. . . .
we see that the first term in parenthese yields eq. (7), the second term yields the contribution of fig. 3 and subsequent terms yields higher order contributions (the two particle states are labelled M and the basis is chosen such that H is diagonal). The geometric series is easily summed and simply replaces (~ - o~eO - e v) by (~ - wNQ w M) a result which is clear from eq. (6) since we have considered the case where the H-fimiltonians and wave functions factor into core and valence pieces, ignoring the exclusion principle, so that WNQQbecomes (wcNQ + wM). We are thus able to isolate the pure valence contribution to h (fig. 2A) as (~pM' IhVAL(e )1 ~bp M) =
(10)
Of course the second term here is not needed if there are no core correlations, for instance if one takes a l p - l h Q space and assumes Hartree-Fock selfconsistency so that the coupling to the core vanishes (in this case one could alternatively diagonalize in the complete P + Q space). In the first term the exclusion principle is obeyed in intermediate states but we may regard pairs of cancelling exclusion violating diagrams as also being present. In the second term the exclusion principle is ignored between the valence particles and the core diagrams. Then the difference between the two terms, apart from removing core contributions, will yield those valence diagrams which correct for exclusion violations in the core calculation. At this stage it might appear tempting to separate out the one and two-body terms from h VAL, however since the cross terms are needed we prefer to solve the BIoch-Horowitz equation 234
Volume 56B, number 3
PHYSICS LETTERS
28 April 1975
[H+ hVAL(6) -- 6 ]P~ = 0 ,
(11)
to yield the eigenvalues 6~ and eigenvectors, la) say, expressed in a / - / c o u p l e d basis via eq. (4). We may then solve for the effective Hamiltonian Herr = ~
Io~>6~(81,
(12)
o¢
where IF) is the biorthogonal completement. If only single particle energies are required eq. (11) is sufficient and corresponds to the linked, folded valence expansion. However we are interested in the two particle case so we now need to disentable the one and two-body contributions to eq. (12). Apart from purely two-body contributions, where both particles interact and the diagram is linked, we shall have included the classes of diagrams indicated in fig. 4 and these we require to remove. In fig. 4 Rayleigh-Schr6dinger energy denominators are now to be understood. Graphs A and B schematically give the nonfolded and once-folded contributions to the energy of single particle/1. (Note diagrams B - D are drawn in unfolded form without explicit indications of folds to avoid cluttering the figure). Summing the completely linked, folded expansion to all orders for a single particle is equivalent to solving the analogue of the Bioch-Horowitz eq. (11) for a single particle, i.e., Iv+
hW'L(6/,) -
= 0.
(13)
The graphs of fig. 4D can easily be summed as in eq. (9) by replacing h by hVAL(6 .. - Vh ) where we use the shorthand notation V: for the bubble insertion in single particle state/~. Only tl~ diag(~al term arises for a linked Q space. In like fa/hion we can sum in addition the diagrams of fig. 4ZCby using h V ~ ' L ( ~ / , - Vh + 6i2 - eh) if 6/2 is inserted (or folded) into 6/1 (The unperturbed energy of single particle state 1'2 is eh and (~h carries an extra minus sign associated with the fold). Now in the general case fig. 4C allows 6 h and 6/2 to be inserted back into each other so that a self-consistency problem arises and we obtain /, " /l + 6/2 - e/'2 - Vh) - (~/,]%, = 0 ' [H+'hVAL(6
[H+ hVAL(6 . . . Vh) . h " /2 + (~/, . eh
~/2ldPh = 0 .
(14)
The contributions of figs. 4 A - D is diagonal in j - j coupling and is given by the sum (~h + 6 h of the self-consistent solutions of eq. (14) (Note that unperturbed single particle energies are included). This result can then be subtracted from eq. (12) to yield the purely two-body effective interaction. The result (14) is really obvious; if we add the equations (14) together (using a wave function ~/,h ) we have the Bloch-Horowitz equation but with all interactions connecting the particles Jl and ]2 together deleted. All such terms belong to the two-body effective interaction and thus should not be considered here. This deletion implies that, apart from one qualification, the Bloch-Horowitz effective Hamiltonian breaks into the sum of a part referring to the particle labelled I1 and a part referring to 12" The qualification is that the Hartree-Fock insertions I'll and Vh are allowed so that the cross terms are present which involve the effective interaction for particle/1 producted with Vh insertions (and vice versa). The result is to replace 6 h - Vh thus demonstrating the cancellation of these particular unlinked diagrams against the corresponding unlinked folded diagrams. The cancellation of all the unlinked folded diagrams of fig. 4C does not take place because our linked Q space forbids the corresponding unlinked non-folded diagrams, i.e., it is a space truncation effect. It is perhaps worth pointing out two formal difficulties. Firstly, if we consider 2hw excitations by the present method we shall have included all possible folded diagrams. These diagrams formally have 2, 4, 6 .... hw energy denominators, although by taking sums of diagrams the factorization theorem [5] allows one to express everything in terms of 2h6o denominators. It is not clear whether it is best to compare such results with order-by-order calculations or whether one should strictly enforce 2h~o denominators (This could be carried out by evaluating eq. (10) using for t~ the appropriate unperturbed energy and then calculating the required folded diagrams explicitly.) A second complication is that some of the linked folded diagrams which we include would, in fact, cancel with nonfolded diagrams if the Q-space were extended (see the discussion of the reduced Bloch-Horowitz expansion in ref. [5]). 235
Volume 56B, number 3
PHYSICS LETTERS
28 April 1975
Summarizing, our prescription for obtaining the two-body effective interaction is to use eqs. (6), (7) and (10) to obtain a pure valence interaction which is then used to solve the Bloch-Horowitz equation. The eigenvalues and eigenvectors thus obtained allow the construction of an effective interaction which in addition to two body pieces contains unwanted one body, and products of one body, contributions. The latter are removed by explicitly calculating their contribution via eq. (14) - a coupled pair of Bloch-Horowitz equations for the two single particle energies. Thus finally we have obtained a completely linked two-body interaction using matrix techniques without the explicit calculation of diagrams. No difficulty arises if G-matrices are used provided some average starting energy is adopted, as is required for a matrix method. Probably the above method can be extended to nonLinked Q spaces by explicitly evaluating the contributions of the unlinked diagrams of fig. 1B, assuming that the space includes complete classes of such diagrams. This becomes complicated however and we shall not discuss it further. After this work was completed a preprint was received from Y. Starkand and M.W. Kirson concerning this same problem. These authors discuss the formalism in less generality than we have done, but give some numerical results. I would like to thank P. Goode for stimulating discussions.
References [11 B.R. Barrett and M.W. Kirson, Advances in Nuclear Physics, eds. E. Vogt and M. Baranger (Plenum, New York), Vol. 6, p. 219. 12] N. Lo ludice, D.J. Rowe and S.S.M. Wong, Phys. Lett. 37B (1971)44; Nucl. Phys. A219 (1974) 171; preprint; A. Watt, B.J. Cole and R.R. Whitehead, Phys. Lett. 51B (1974) 435. [31 P. Goode, preprint. [4] H.A. Mavromatis, Nucl. Phys. A206 (1973)477. [5] B.H. Brandow, Rev. Mode. Phys. 39 (1967) 771. [61 R. Padjen and G. Ripka, Nucl. Phys. A149 (1974) 273.
236