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Is Hartree-Fock self-consistency important in particle-hole calculations?
Volume 44B, number 2
PHYSICS LETTERS
16 April 1973
IS HARTREE-FOCK SELF-CONSISTENCY IMPORTANT IN PARTICLE-HOLE CALCULATIONS? * D.J. ROWE Department of Physics, Universtty of Toronto, Toronto, Ontario, Canada Received 12 January 1973 The systematic predictable effects of self-consxstency m particle-hole calculatmns are examined. It is shown that self-consistent radial wave functions are essentml for monopole excitanons and that self-consistency m deformanon is vital for the quadrupole and other vlbratmns of deformed nuclei. The TDA (Tamm-Dancoff Approximation) and RPA (Random-Phase Approximation) particle-hole theories are formally expressed in a HF (Hartree-Fock) basis, which automatically ensures that the Hamiltoman matrix elements connecting the 0 p - 0 h HF ground state to l p - l h excited states vanish. However, pracncal calculations have almost invariably employed some other basis, notably harmonic oscillator, for which the coupling does not vanish. Nevertheless the coupling has traditionally been ignored in the hope that the single-particle states are sufficiently close to Hf selfconsistency that the errors are small. The ~mportance of self-consistency corrections has been emphasized in several recent papers [1 4 ] . Our objective is to examine the systematic, predictable effects of self-consistency in particle-hole calculations; to find out when it is or is not necessary to take it into account; to see how the effects depend on the interaction strength and what fraction of the corrections is due to changes m the single-particle energies and what to wave function changes The latter is particularly important because, in practice, single-parncle energies are customarily taken from experiment, rather than from calculation. Thus a substantial part of the renormahzation of the particle-hole equations is automatically included. However, the same experimental energies are used regardless of the wave functions, so that it is the wave function effect of self-consistency that is of primary concern Interest m the problem arises largely because of the importance of core polarization in the renormahzatlon of the effective interacnon and transition operators * Work supported in part by the National Research Council of Canada
for valence shell nucleons. When the core excitations are treated in RPA there is a tendency for the J = T = 0 'breathing mode' to collapse, as first noted by Blomqvist [5]. Consequently, perturbation expansions for effective operators can diverge. The collapse can be prevented by the inclusion of propagator screening corrections and vertex renormahzations [ 5 - 8 ] . However, as Zamick [7] points out, this is not really a solution to the problem. For if the propagator renormalizatmn is iterated to all orders, so that the screening is effected by an RPA phonon rather than a particle-hole bubble, the effectwe parncle-hole mteracnon becomes very large and repulsive and, in particular, the monopole vibration rises to absurdly high energy Thus Zamick calls into question the vahdity of the RPA for monopole vibranon~ Now according to a theorem due to Thouless [9], if some solution of the unrenormalized RPA (aside from the spurious c. m. solunon) collapses, it is symptomatic of an unsatisfactory HF ground state. For example, the collapse of a JTr =2 +, T=0 excitation, in a spherical HF basis, implies the existence of a quadrupole deformed HF solution of lower energy than the spherical solunon employed. Similarly, the collapse of a monopole excitation implies a lower energy spherical HF solunon. Therefore, if one wishes to calculate meaningful monopole excitation energies m the RPA, it is essential that one start with correct radial wave functions. Similarly, for the quadrupole/3- or 3'- vibrations of a deformed nucleus, one must employ single-parncle wave functions corresponding to the corect equihbrium quadrupole deformation. In the following we show that numerical calculanons confirm the above predictions and that having correct 155
V o l u m e 44B, number 2
PHYSICS LETTERS
50
16 Aprd 1973 30
0 +, T = 0
~
( H 0
7t ÷ d =0,
s p e.'s
T=O
( E,'.PT( s p e ' s )
A >
"~
~.
~
"
20
>.- 20 (.9 n.." hi Z bJ
)lO Z
"~'~'..,.
HFTDA
~ , ~
~_ 0 ~-< IO I-G X W
~--~ _
",.
.....
HFTDA
H F R P A~ " IO
HORPA \
NX X HORR~\ \
.HOTDA
HOTDA
\ I
0
09
014
P
0.6
OP8
Ib'TER 'CTION
I
I0
I
12
I
1.4
1.6
0
0'2_
STRENGTH
I
I 0 4
l ~,i 018 1.0 12 INTER,o-.CTION STRENGTH 0 6
i14
16
Fig l Excitation energies for J~r=0+, T = 0 particle-hole excitations of 160 m T D A and RPA with harmonic oscillator ( H e ) and Hartree-Fock (HF) wave functions. In both cases the single-particle energies are fixed (1) at H e values (hto= 14MeV) and (b) at experimental values. Where a T D A result IS not shown It Is because it is not distinguishable from the corresponding RPA result and where a HORPA result is not shown It IS indistinguishable from HFRPA.
radial wave functions is much more important for particle-hole calculations of monopole excitations than for any other excitation. If an RPA solutions continues to fall very low in energy, even m the lowest energy HF representation, it implies that the nucleus is 'soft' with respect to vibrations of the low energy mode and that the HF ground state is approaching a phase transition. The ground state correlations are then very large and the RPA breaks down in a well-understood manner [10]. If this were the situation for the monopole vibrations of closedshell nuclei, then, as Zamick suggests [7], the RPA would not be valid. However, we know that monopole vibrations are not seen at low energies due to the incompressibility of nuclear matter. Thus one does not believe that nuclei are 'soft' against monopole polarization, as a genuine low-energy RPA solution would imply. In calculating the particle-hole excitations of 160 we followed the usual practxce of restricting the space to the lowest four major oscillator shells with an oscillator constant of h w = 14 MeV Instead of using several interactions we thought it mere instructive to employ just the bare G-matrix elements of Kuo [11] but to mulUply them by a constant, which varied from 0.2 to 1.6 Thus all our results are plotted as a function of lnteracUon strength in umts of the unmodified G-matrix strength. Wlthm the four major shell space we performed HF calculations for 16 O and subsequently used the wave 156
functions for TDA and RPA calculations of all the particle-hole excitations. The results are compared with corresponding calculations in the H e (harmonic oscillator) basis for the lowest j~r =0 +, T=0 excitations, in figs. 1 and 2. In fig. 1 (a) the H e and HF calculaUons were both with H e single-particle energies. In fig. 1 (b) expenmental energies were used and in fig 2 the singleparticle energies were calculated according to the HF expression
Jr~T
50
'
\
,
,
.-.....
/
I I
,I s ~,', \
~- 20
/,
,
I
I/
\ (,/
I-',FTDA \ \ \
\,
Ld Z I,l
"-L/
/,
HOTDA
Z 0 i-F- iO
(cclculofed s p e's)
U xi,i
J~= 0+, T =0
I
02
0'4
I
06
0'8
I ,".ITER.'~CTION
IlO
l
12
Ii4
I6
STRENGTH
Fig 2. Excitation energy of the lowest J~r=0+, T = 0 state o f 160 in HOTDA, HFTDA and according to the shell-model (SM) procedure outlined m the text Calculated single-particle energaes are used.
Volume 44B, number 2
50 J
~
PHYSICS LETTERS
T
=0
( EXPT'¢ s p e ' s )
'~.~.
I
{
16 April 1973
"~'~-~.
JT=5-, T=O ( EXPTa s p e ' s } >
26
(56) 3o (74)
"-"~..
691 (393)
I0 >-
,oi
I
(99) I . I z~s(zzs_! l
I
875 ( 7 3 9 ) " ~ ' ~ - - -
n~ w z w
-.
578(10,9)~, z o F-
223(192) I
450(1576) ~t
24 0 (64)
2 6 5 (100)
I
I
I
02
04
I
06
I
I
1
I
08
I0
12
14
16
INTERACTION STRENGTH
\
% I
x w
0
~.
0
02
04 0;6 O;B lu.O In2 14 INTERACTIOi~] STRENGTH
Fig. 3. Excitation energies for (a)Jn=2+, T=0 and (b)J~r=3-, T=0 particle-hole excitations of 160 in TDA and RPA with harmomc oscillator (HO) and Hartree-Fock (HF) wave functions. The same convention is used as in fig. 1. The numbers on the curves are reduced E~, ground state transition probabilities in units of f4 and f6 respectively. The upper numbers refer to HFTDA (HOTDA) resuits and the lower to HFRPA (HORPA).
OCC l
where T is the kinetic energy and V the two-body interaction. Thus only in fig. 2 is the effect of self-consistency on the single-particle energies included. Also shown in fig. 2 are results of a shell-model calculation in the combined 0 p - 0 h and I p - l h space. Comparing these figures, one notes that the largest effect of self-consistency is through the changed singleparticle energies. However, as fig. 1 demonstrated, the wave function effect alone is able to prevent the monopole excitation from collapse. Fig. 2 shows that while the shell-model procedure gives some of the trends of HFTDA, it is unreliable. This is because, while it allows the repulsion of the ground and excited states, due to coupling of the 0 p - 0 h and l p - l h configurations, it ignores the depression of the excited states due to coupling to 2 p - 2 h configurations. Clearly this neglected coupling is of the same order of magnitude as the coupling considered. The shell-model procedure is at a further disadvantage because it is restricted to the TDA and, more important, because one would in general prefer to use experimental singie-particle energies and thus not admit changes in them due to stir-consistency, as one automatically must in a simple calculation of this type. Fig. 3 shows results for T = O , J ~ = 2 + and 3 - excita-
tions, for experimental single-particle energies. It is seen that, for these excitations, the use Of self-consistent wave functions does not substantially change the excitation energies. The general trends of the above results were found for all other particle-hole excitations of 160. Reduced electric transition probabilities from each of the J :/: 0 excitations to the ground state are indicated by numbers on th~ figures. It is seen that here are very substantial differences between the HO and HF results. This is because, for strong interactions, the effect of self-consistency is to pull m the occupied orbitals at the expense of the unoccupied which are correspondingly pushed out to maintain orthogonahty. This results m reduced overlaps of the particle and hole wave functions for strong interactions and vice versa for weak interactions. The effect is certainly exaggerated in the present calculations due to the small space used for the HF calculations. Given a limited set of two-body matrix elements, we adopted the point of view that the wave functions should be at least selfconsistent within the space used. Our numbers give some indication of the confidence we can put in excitation energies and transition probabilities calculated with HO wave functions. But calculations in a much larger space are clearly desirable. Alternatively a small space would probably be acceptable provided Woods-Saxon rather than HO wave func157
Volume 44B, number 2
PHYSICS LETTERS
tions were used, as in Pradhan and Shakin's calculatlons [12]. The point remains that to describe monopole excitations at all reliably in a particle-hole model, considerable attention must be given to the self-consistency of the radial wave functions. Similarly, attention must be given to self-consistency in deformation for J = 2 and other excitations. The authors wishes to thank Mr. G. Rosensteel for extensive computational assistance and Dr. P. Goode for &scusslons which stimulated this investigation.
References [1] H.A. Mavromatis, Phys. Lett 32B (1970) 256.
158
16 April 1973
[2] P.J. Elhs and H.A. Mavromatls, Nucl. Phys A175 (1971) 309. [3] J.M Irvme and V F.E Pucknell, Nucl. Phys. A174 (1971) 634. [4] P.J. Elhs and E. Osnes, Phys. Lett. 41B (1972) 97. [5] J Blomqvist, Nucl. Phys. A103 (1967) 644 J Damgaard et al., Nucl. Phys. A121 (1968) 625 [6] J. Blomqvist and TT.S. Kuo, Phys. Lett 29B (1969) 544 [71 L. Zamlck, Phys. Lett. 31B (1970) 160. [8] M W. Kirson and L. Zamick, Ann. Phys. (N.Y.) 60 (1970) 188. [9] D.J Thouless, Nucl. Phys. 21 (1960) 225, 22 (1961) 78. [10] D.J. Rowe, Phys. Rev 175 (1968) 1283. [11] T.T.S Kuo, Nucl. Phys A103(1967) 71. [12] H.C. Pradhan and C.M. Shalon, Phys. Lett 37B (1971) 151.
PHYSICS LETTERS
16 April 1973
IS HARTREE-FOCK SELF-CONSISTENCY IMPORTANT IN PARTICLE-HOLE CALCULATIONS? * D.J. ROWE Department of Physics, Universtty of Toronto, Toronto, Ontario, Canada Received 12 January 1973 The systematic predictable effects of self-consxstency m particle-hole calculatmns are examined. It is shown that self-consistent radial wave functions are essentml for monopole excitanons and that self-consistency m deformanon is vital for the quadrupole and other vlbratmns of deformed nuclei. The TDA (Tamm-Dancoff Approximation) and RPA (Random-Phase Approximation) particle-hole theories are formally expressed in a HF (Hartree-Fock) basis, which automatically ensures that the Hamiltoman matrix elements connecting the 0 p - 0 h HF ground state to l p - l h excited states vanish. However, pracncal calculations have almost invariably employed some other basis, notably harmonic oscillator, for which the coupling does not vanish. Nevertheless the coupling has traditionally been ignored in the hope that the single-particle states are sufficiently close to Hf selfconsistency that the errors are small. The ~mportance of self-consistency corrections has been emphasized in several recent papers [1 4 ] . Our objective is to examine the systematic, predictable effects of self-consistency in particle-hole calculations; to find out when it is or is not necessary to take it into account; to see how the effects depend on the interaction strength and what fraction of the corrections is due to changes m the single-particle energies and what to wave function changes The latter is particularly important because, in practice, single-parncle energies are customarily taken from experiment, rather than from calculation. Thus a substantial part of the renormahzation of the particle-hole equations is automatically included. However, the same experimental energies are used regardless of the wave functions, so that it is the wave function effect of self-consistency that is of primary concern Interest m the problem arises largely because of the importance of core polarization in the renormahzatlon of the effective interacnon and transition operators * Work supported in part by the National Research Council of Canada
for valence shell nucleons. When the core excitations are treated in RPA there is a tendency for the J = T = 0 'breathing mode' to collapse, as first noted by Blomqvist [5]. Consequently, perturbation expansions for effective operators can diverge. The collapse can be prevented by the inclusion of propagator screening corrections and vertex renormahzations [ 5 - 8 ] . However, as Zamick [7] points out, this is not really a solution to the problem. For if the propagator renormalizatmn is iterated to all orders, so that the screening is effected by an RPA phonon rather than a particle-hole bubble, the effectwe parncle-hole mteracnon becomes very large and repulsive and, in particular, the monopole vibration rises to absurdly high energy Thus Zamick calls into question the vahdity of the RPA for monopole vibranon~ Now according to a theorem due to Thouless [9], if some solution of the unrenormalized RPA (aside from the spurious c. m. solunon) collapses, it is symptomatic of an unsatisfactory HF ground state. For example, the collapse of a JTr =2 +, T=0 excitation, in a spherical HF basis, implies the existence of a quadrupole deformed HF solution of lower energy than the spherical solunon employed. Similarly, the collapse of a monopole excitation implies a lower energy spherical HF solunon. Therefore, if one wishes to calculate meaningful monopole excitation energies m the RPA, it is essential that one start with correct radial wave functions. Similarly, for the quadrupole/3- or 3'- vibrations of a deformed nucleus, one must employ single-parncle wave functions corresponding to the corect equihbrium quadrupole deformation. In the following we show that numerical calculanons confirm the above predictions and that having correct 155
V o l u m e 44B, number 2
PHYSICS LETTERS
50
16 Aprd 1973 30
0 +, T = 0
~
( H 0
7t ÷ d =0,
s p e.'s
T=O
( E,'.PT( s p e ' s )
A >
"~
~.
~
"
20
>.- 20 (.9 n.." hi Z bJ
)lO Z
"~'~'..,.
HFTDA
~ , ~
~_ 0 ~-< IO I-G X W
~--~ _
",.
.....
HFTDA
H F R P A~ " IO
HORPA \
NX X HORR~\ \
.HOTDA
HOTDA
\ I
0
09
014
P
0.6
OP8
Ib'TER 'CTION
I
I0
I
12
I
1.4
1.6
0
0'2_
STRENGTH
I
I 0 4
l ~,i 018 1.0 12 INTER,o-.CTION STRENGTH 0 6
i14
16
Fig l Excitation energies for J~r=0+, T = 0 particle-hole excitations of 160 m T D A and RPA with harmonic oscillator ( H e ) and Hartree-Fock (HF) wave functions. In both cases the single-particle energies are fixed (1) at H e values (hto= 14MeV) and (b) at experimental values. Where a T D A result IS not shown It Is because it is not distinguishable from the corresponding RPA result and where a HORPA result is not shown It IS indistinguishable from HFRPA.
radial wave functions is much more important for particle-hole calculations of monopole excitations than for any other excitation. If an RPA solutions continues to fall very low in energy, even m the lowest energy HF representation, it implies that the nucleus is 'soft' with respect to vibrations of the low energy mode and that the HF ground state is approaching a phase transition. The ground state correlations are then very large and the RPA breaks down in a well-understood manner [10]. If this were the situation for the monopole vibrations of closedshell nuclei, then, as Zamick suggests [7], the RPA would not be valid. However, we know that monopole vibrations are not seen at low energies due to the incompressibility of nuclear matter. Thus one does not believe that nuclei are 'soft' against monopole polarization, as a genuine low-energy RPA solution would imply. In calculating the particle-hole excitations of 160 we followed the usual practxce of restricting the space to the lowest four major oscillator shells with an oscillator constant of h w = 14 MeV Instead of using several interactions we thought it mere instructive to employ just the bare G-matrix elements of Kuo [11] but to mulUply them by a constant, which varied from 0.2 to 1.6 Thus all our results are plotted as a function of lnteracUon strength in umts of the unmodified G-matrix strength. Wlthm the four major shell space we performed HF calculations for 16 O and subsequently used the wave 156
functions for TDA and RPA calculations of all the particle-hole excitations. The results are compared with corresponding calculations in the H e (harmonic oscillator) basis for the lowest j~r =0 +, T=0 excitations, in figs. 1 and 2. In fig. 1 (a) the H e and HF calculaUons were both with H e single-particle energies. In fig. 1 (b) expenmental energies were used and in fig 2 the singleparticle energies were calculated according to the HF expression
Jr~T
50
'
\
,
,
.-.....
/
I I
,I s ~,', \
~- 20
/,
,
I
I/
\ (,/
I-',FTDA \ \ \
\,
Ld Z I,l
"-L/
/,
HOTDA
Z 0 i-F- iO
(cclculofed s p e's)
U xi,i
J~= 0+, T =0
I
02
0'4
I
06
0'8
I ,".ITER.'~CTION
IlO
l
12
Ii4
I6
STRENGTH
Fig 2. Excitation energy of the lowest J~r=0+, T = 0 state o f 160 in HOTDA, HFTDA and according to the shell-model (SM) procedure outlined m the text Calculated single-particle energaes are used.
Volume 44B, number 2
50 J
~
PHYSICS LETTERS
T
=0
( EXPT'¢ s p e ' s )
'~.~.
I
{
16 April 1973
"~'~-~.
JT=5-, T=O ( EXPTa s p e ' s } >
26
(56) 3o (74)
"-"~..
691 (393)
I0 >-
,oi
I
(99) I . I z~s(zzs_! l
I
875 ( 7 3 9 ) " ~ ' ~ - - -
n~ w z w
-.
578(10,9)~, z o F-
223(192) I
450(1576) ~t
24 0 (64)
2 6 5 (100)
I
I
I
02
04
I
06
I
I
1
I
08
I0
12
14
16
INTERACTION STRENGTH
\
% I
x w
0
~.
0
02
04 0;6 O;B lu.O In2 14 INTERACTIOi~] STRENGTH
Fig. 3. Excitation energies for (a)Jn=2+, T=0 and (b)J~r=3-, T=0 particle-hole excitations of 160 in TDA and RPA with harmomc oscillator (HO) and Hartree-Fock (HF) wave functions. The same convention is used as in fig. 1. The numbers on the curves are reduced E~, ground state transition probabilities in units of f4 and f6 respectively. The upper numbers refer to HFTDA (HOTDA) resuits and the lower to HFRPA (HORPA).
OCC l
where T is the kinetic energy and V the two-body interaction. Thus only in fig. 2 is the effect of self-consistency on the single-particle energies included. Also shown in fig. 2 are results of a shell-model calculation in the combined 0 p - 0 h and I p - l h space. Comparing these figures, one notes that the largest effect of self-consistency is through the changed singleparticle energies. However, as fig. 1 demonstrated, the wave function effect alone is able to prevent the monopole excitation from collapse. Fig. 2 shows that while the shell-model procedure gives some of the trends of HFTDA, it is unreliable. This is because, while it allows the repulsion of the ground and excited states, due to coupling of the 0 p - 0 h and l p - l h configurations, it ignores the depression of the excited states due to coupling to 2 p - 2 h configurations. Clearly this neglected coupling is of the same order of magnitude as the coupling considered. The shell-model procedure is at a further disadvantage because it is restricted to the TDA and, more important, because one would in general prefer to use experimental singie-particle energies and thus not admit changes in them due to stir-consistency, as one automatically must in a simple calculation of this type. Fig. 3 shows results for T = O , J ~ = 2 + and 3 - excita-
tions, for experimental single-particle energies. It is seen that, for these excitations, the use Of self-consistent wave functions does not substantially change the excitation energies. The general trends of the above results were found for all other particle-hole excitations of 160. Reduced electric transition probabilities from each of the J :/: 0 excitations to the ground state are indicated by numbers on th~ figures. It is seen that here are very substantial differences between the HO and HF results. This is because, for strong interactions, the effect of self-consistency is to pull m the occupied orbitals at the expense of the unoccupied which are correspondingly pushed out to maintain orthogonahty. This results m reduced overlaps of the particle and hole wave functions for strong interactions and vice versa for weak interactions. The effect is certainly exaggerated in the present calculations due to the small space used for the HF calculations. Given a limited set of two-body matrix elements, we adopted the point of view that the wave functions should be at least selfconsistent within the space used. Our numbers give some indication of the confidence we can put in excitation energies and transition probabilities calculated with HO wave functions. But calculations in a much larger space are clearly desirable. Alternatively a small space would probably be acceptable provided Woods-Saxon rather than HO wave func157
Volume 44B, number 2
PHYSICS LETTERS
tions were used, as in Pradhan and Shakin's calculatlons [12]. The point remains that to describe monopole excitations at all reliably in a particle-hole model, considerable attention must be given to the self-consistency of the radial wave functions. Similarly, attention must be given to self-consistency in deformation for J = 2 and other excitations. The authors wishes to thank Mr. G. Rosensteel for extensive computational assistance and Dr. P. Goode for &scusslons which stimulated this investigation.
References [1] H.A. Mavromatis, Phys. Lett 32B (1970) 256.
158
16 April 1973
[2] P.J. Elhs and H.A. Mavromatls, Nucl. Phys A175 (1971) 309. [3] J.M Irvme and V F.E Pucknell, Nucl. Phys. A174 (1971) 634. [4] P.J. Elhs and E. Osnes, Phys. Lett. 41B (1972) 97. [5] J Blomqvist, Nucl. Phys. A103 (1967) 644 J Damgaard et al., Nucl. Phys. A121 (1968) 625 [6] J. Blomqvist and TT.S. Kuo, Phys. Lett 29B (1969) 544 [71 L. Zamlck, Phys. Lett. 31B (1970) 160. [8] M W. Kirson and L. Zamick, Ann. Phys. (N.Y.) 60 (1970) 188. [9] D.J Thouless, Nucl. Phys. 21 (1960) 225, 22 (1961) 78. [10] D.J. Rowe, Phys. Rev 175 (1968) 1283. [11] T.T.S Kuo, Nucl. Phys A103(1967) 71. [12] H.C. Pradhan and C.M. Shalon, Phys. Lett 37B (1971) 151.