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Two particle — two hole mixing in Hartree-Fock calculations
Volume
TWO
35B,
number
PARTICLE
6
PHYSICS
-
TWO
Laboratoive
HOLE
MIXING
LETTERS
IN
HARTREE-FOCK
J. A. RABBAT * and G. DO DANG de Physique ThGorique et Hades Energies, Received
5 July 1971
CALCULATIONS
Orsay **
10 May 1971
The constrained Hartree-Fock theory is used for studying the stability of the solutions against two particle - two hole excitations. Quadrupole and hexadecapole deformations are imposed on the HartreeFock equations to get the true energy minimum and the equilibrium shape. Energy corrections are calculated using second-order perturbations and complete diagonalization. Applications are made to the 20Ne and 2% nuclei.
A number of works have recently been done in a common effort to study the two particle two hole (2p-2h) and many particle - many hole correlations in the Hartree-Fock ground state. The interest is twofold. First, from the experimental point of view, these works are aimed at explaining the large correlation effects observed, even for doubly closed shell nuclei, in transfer reactions [l]. From the theoretical point of view, they are aimed at solving the ambiguity caused by the existence of near degenerate solutions of the HF equations [2]. The HF solution by its own definition is (and only is) stable against lp-lh excitation and thus from the start, it may be unstable against 2p-2h (or Np-Nh) excitations. One well-known example is the instability caused by the pairing force. Two questions may then be asked: (i) how are these excited states mixed in the ground state? and (ii) does this mixing affects its intrinsic structure ? A first attempt to answer these questions has been made by Faessler et al. [3] in the so-called “multiconfiguration Hartree-Fock theory”. The idea is simple and as a matter of fact had been used in atomic and molecular physics [4]. It consists in taking as the ground-state trial function a linear combination of the Hartree-Fock Op-Oh state and the set of 2p-2h (and Np-Hh) states. These states are defined in the usual way in * On leave from:
Departamento de Fisica, Facultad de Ciencias e Ingenieria, Univ. Nat. de Rosario, Rosario, Argentina. ** Laboratoire associe au Centre National de la Recherche Scientifique, Postal address: Laboratoire de Physique Theorique et Hautes Energies, Batiment 211, Universitd de Paris-Sud, Centre d’orsay, 91 - Orsay (France).
480
terms of a common set of orbital wave functions taken as variational parameters and with respect to which the minimization process is carried out. The mixing is defined by the condition that the total hamiltonian is diagonal in the space defined. Other works [5,6] follow the same line, in particular that of ref. [6] where a careful treatment of spurious components is done. There appear, however, many difficulties as recognized by the authors of ref. [3] which are both mathematical as for example the non-orthog. onality of the orbital wave functions and numerical because of the slow convergence of the iteration process. The diagonalization process, or more precisely the calculation of the matrix of the hamiltonian to be diagonalized, is a heavy task and for this reason applications to realistic calculations are very much hindered. In this note, we propose an alternative way to attack the problem which, though not as accurate, at least has the merit of being simple. From the point of view of the HF theory, the ground state energy can be viewed as the lowest point on an energy surface defined in terms of some observables as the quadrupole moment. Because of the mixing with other excited states, or of other effects, this surface is lowered and it is possible that the true minimum does not coincide with the one given by the HF equations. One way of looking for this minimum is to add constraints on the HF equations, an artifice which allows one to move around on the energy surface and at each point to make the desired corrections. The most common constraints are the quadrupole and also the hexadecapole deformation [7]. To the hamiltonian
Volume 35B, number 6
PHYSICS
LETTERS
5 July 1971
Table 1. Ground state energies of 20Ne in various approximations and quadrupole moments for different values of we now add a term
172.
H'=CqQ+ 1
l 1
where Q i is the Z1-pole operator. For a given set of parameters qz, the HF procedur? is carried out for the modified hamiltonian H = H - H’. It consists in the definition of a set of orbital wave functions, that is, a set of numbers & of the expansion
bx+= C ci ai CY
The Op-Oh state 10) is the one where the A nucleans occupy the lowest orbitals. Excited states can be defined in terms of the operators bi. The number of these states henceforth denotzd ) I) is limited both by axial symmetry which we assume and by time reversal invariance [6]. We now calculate the energy corrections. The simplest approximation (A) is given by the Brillouin-Wigner second-order perturbation, namely: (4) where EI is the excitation energy of the state 111, namely El = (l/H]Z) -E . A better approximation (By for the ground state energy (not for the excited states) can be obtained by the diagonalization of the total hamiltonian H in the space of Op-Oh and Zp-2h states. This requires the calculation of all the matrix elements of H, a task which, though straightforward, is somewhat time-consuming. We also note that in both approximations (A) and (B), care should be taken to eliminate the spurious component of the Zp-2h states coming from angular momentum projection. The problem now consists in looking for the minimum of the total energy E = E, + AE by varying the parameters vl. This, of course, is not a self-consistent procedure. The deformation of the nucleus may be different from what we may want it to be. However, nothing should prevent us from taking as many components 171as we wish and by varying 771one at a time we shall be able to study the stability of the HF solution SeParately against each kind of deformation. This is in contrast to the multiconfiguration or self-consistent HF theory where all the modes come into play simultaneously. We have carried
~..~
772 ___.__
- 0.30
- 0.15
0
E f$”
-34.60
-34.87
EMeV A
-34.99
E?!eV Q fm2
0.30
1.00
-34.88
-34.85
-34.80
-35.21
-35.29
-35.27
-35.22
-35.30
-35.60
-35.63
-35.62
-35, GO
23.75
24.96
25.14
25.37
25.59
out the calculations for the 20Ne and 28Si nuclei. The single particle energies are taken from ref. [7] and the force is a Rosenfeld mixture with a gaussian radial dependence. No effort has been made to fit these parameters to the experimental binding energies. Our aim is to show that the constrained HF theory can be a simple alternative to the more complicated multi-configuration HF theory and which is also capable of giving good ground state energy and better shape for the nucleus. We assume both for 20Ne and 2% that the most important components in the sum (2) are the quadrupole and hexadecapole deformations. The latter has been found [7] to be important for 28Si. The results for 20Ne are given in table 1 for 174 = 0 and various values of 172. It is found that the hexadecapole constraint plays only a negligible role. As can be seen from the table. the minima of the HF energies with and without corrections occur at the same place corresponding to the same value of the quadrupole moment. This means that the HF solution is stable against quadrupole deformations due to Zp-2h excitations. This was to be expected because of the sharp minimum of the HF energy curve. The energy gains due to Zp-2h excitations are found to be 0.41 MeV and 0.75 for approximations (A) and (B) respectively. The answer to the second question, namely that concerning the capability of the procedure to take account of the modification of the intrinsic structure of the HF solution, is given in the c;lst’ of the 2%i nucleus. The results are given in fig. 1 where the energy curves are drawn It11 various values of q2 and q4. It is seen 11~~11 bcsides the energy gain which is now 1.27 IkIck 1c,i approximation (A), the true minimum corresponds to a shape different from that of the sinrple HF theory. The corresponding deformntiol: parameters are now ~4 = -0.044 and 172 = 0.25. The quadrupole component is important, in con481
Volume 35B, number
PHYSICS
6
5 July 1971
of the pairing component of the interaction. In conclusion, the constrained HF theory may be viewed as a simple alternative to the more complicate multiconfiguration HF theory in the treatment of the effects of 2p-2h mixing in the ground state. Calculations using this procedure together with angular momentum projection are under way and will be reported separately.
- 122
One of us (J. A. R.) is grateful to Professor Jancovici for his kind hospitality at the Laboratoire de Physique Theorique et Hautes Energies. His stay would not have been possible without the financial support from the Centre International de Stages and the Universidad National de Rosario.
- 123
)
References
-124
- 0.2
0
0.2
0.4
0.6
Fig. .. Energies and quadrupole moment function of v2 for various of 34.
OB
n2
of 28Si as a
trast to the conclusion of ref. [7]. It is also to be noted that the 2p-2h mixing reduces the deformation as expected because it takes account in part
482
LETTERS
[1] I. Rapaport, T. A. Belote and W. E. Dorenbusch, Phys. Rev. 156 (1967) 1255. [2] G.Do Dang, Nucl. Phys. Al57 (1970) 231. [3] A. Faessler and A. Plastino, 2. Physik, 220 (1969) 88; A. Faessler, A. Plastino and K. W. Schmidt, Phys. Letters 34B (1971) 31. [4] J. Hinze and C. C. J. Roothan, Suppl. Progr. Theoret. Phys. 40 (1967) 37. [5] L. Sapathy and Q. Ho-Kim, Phys. Rev. Letters 25 (1970) 123. [6) R. Padjen and G. Ripka, Nucl. Phys. Al49 (1970) 273. [7] B. Caste1 and J. C. Parikh, Phys. Rev. Cl (1970) 990.
TWO
35B,
number
PARTICLE
6
PHYSICS
-
TWO
Laboratoive
HOLE
MIXING
LETTERS
IN
HARTREE-FOCK
J. A. RABBAT * and G. DO DANG de Physique ThGorique et Hades Energies, Received
5 July 1971
CALCULATIONS
Orsay **
10 May 1971
The constrained Hartree-Fock theory is used for studying the stability of the solutions against two particle - two hole excitations. Quadrupole and hexadecapole deformations are imposed on the HartreeFock equations to get the true energy minimum and the equilibrium shape. Energy corrections are calculated using second-order perturbations and complete diagonalization. Applications are made to the 20Ne and 2% nuclei.
A number of works have recently been done in a common effort to study the two particle two hole (2p-2h) and many particle - many hole correlations in the Hartree-Fock ground state. The interest is twofold. First, from the experimental point of view, these works are aimed at explaining the large correlation effects observed, even for doubly closed shell nuclei, in transfer reactions [l]. From the theoretical point of view, they are aimed at solving the ambiguity caused by the existence of near degenerate solutions of the HF equations [2]. The HF solution by its own definition is (and only is) stable against lp-lh excitation and thus from the start, it may be unstable against 2p-2h (or Np-Nh) excitations. One well-known example is the instability caused by the pairing force. Two questions may then be asked: (i) how are these excited states mixed in the ground state? and (ii) does this mixing affects its intrinsic structure ? A first attempt to answer these questions has been made by Faessler et al. [3] in the so-called “multiconfiguration Hartree-Fock theory”. The idea is simple and as a matter of fact had been used in atomic and molecular physics [4]. It consists in taking as the ground-state trial function a linear combination of the Hartree-Fock Op-Oh state and the set of 2p-2h (and Np-Hh) states. These states are defined in the usual way in * On leave from:
Departamento de Fisica, Facultad de Ciencias e Ingenieria, Univ. Nat. de Rosario, Rosario, Argentina. ** Laboratoire associe au Centre National de la Recherche Scientifique, Postal address: Laboratoire de Physique Theorique et Hautes Energies, Batiment 211, Universitd de Paris-Sud, Centre d’orsay, 91 - Orsay (France).
480
terms of a common set of orbital wave functions taken as variational parameters and with respect to which the minimization process is carried out. The mixing is defined by the condition that the total hamiltonian is diagonal in the space defined. Other works [5,6] follow the same line, in particular that of ref. [6] where a careful treatment of spurious components is done. There appear, however, many difficulties as recognized by the authors of ref. [3] which are both mathematical as for example the non-orthog. onality of the orbital wave functions and numerical because of the slow convergence of the iteration process. The diagonalization process, or more precisely the calculation of the matrix of the hamiltonian to be diagonalized, is a heavy task and for this reason applications to realistic calculations are very much hindered. In this note, we propose an alternative way to attack the problem which, though not as accurate, at least has the merit of being simple. From the point of view of the HF theory, the ground state energy can be viewed as the lowest point on an energy surface defined in terms of some observables as the quadrupole moment. Because of the mixing with other excited states, or of other effects, this surface is lowered and it is possible that the true minimum does not coincide with the one given by the HF equations. One way of looking for this minimum is to add constraints on the HF equations, an artifice which allows one to move around on the energy surface and at each point to make the desired corrections. The most common constraints are the quadrupole and also the hexadecapole deformation [7]. To the hamiltonian
Volume 35B, number 6
PHYSICS
LETTERS
5 July 1971
Table 1. Ground state energies of 20Ne in various approximations and quadrupole moments for different values of we now add a term
172.
H'=CqQ+ 1
l 1
where Q i is the Z1-pole operator. For a given set of parameters qz, the HF procedur? is carried out for the modified hamiltonian H = H - H’. It consists in the definition of a set of orbital wave functions, that is, a set of numbers & of the expansion
bx+= C ci ai CY
The Op-Oh state 10) is the one where the A nucleans occupy the lowest orbitals. Excited states can be defined in terms of the operators bi. The number of these states henceforth denotzd ) I) is limited both by axial symmetry which we assume and by time reversal invariance [6]. We now calculate the energy corrections. The simplest approximation (A) is given by the Brillouin-Wigner second-order perturbation, namely: (4) where EI is the excitation energy of the state 111, namely El = (l/H]Z) -E . A better approximation (By for the ground state energy (not for the excited states) can be obtained by the diagonalization of the total hamiltonian H in the space of Op-Oh and Zp-2h states. This requires the calculation of all the matrix elements of H, a task which, though straightforward, is somewhat time-consuming. We also note that in both approximations (A) and (B), care should be taken to eliminate the spurious component of the Zp-2h states coming from angular momentum projection. The problem now consists in looking for the minimum of the total energy E = E, + AE by varying the parameters vl. This, of course, is not a self-consistent procedure. The deformation of the nucleus may be different from what we may want it to be. However, nothing should prevent us from taking as many components 171as we wish and by varying 771one at a time we shall be able to study the stability of the HF solution SeParately against each kind of deformation. This is in contrast to the multiconfiguration or self-consistent HF theory where all the modes come into play simultaneously. We have carried
~..~
772 ___.__
- 0.30
- 0.15
0
E f$”
-34.60
-34.87
EMeV A
-34.99
E?!eV Q fm2
0.30
1.00
-34.88
-34.85
-34.80
-35.21
-35.29
-35.27
-35.22
-35.30
-35.60
-35.63
-35.62
-35, GO
23.75
24.96
25.14
25.37
25.59
out the calculations for the 20Ne and 28Si nuclei. The single particle energies are taken from ref. [7] and the force is a Rosenfeld mixture with a gaussian radial dependence. No effort has been made to fit these parameters to the experimental binding energies. Our aim is to show that the constrained HF theory can be a simple alternative to the more complicated multi-configuration HF theory and which is also capable of giving good ground state energy and better shape for the nucleus. We assume both for 20Ne and 2% that the most important components in the sum (2) are the quadrupole and hexadecapole deformations. The latter has been found [7] to be important for 28Si. The results for 20Ne are given in table 1 for 174 = 0 and various values of 172. It is found that the hexadecapole constraint plays only a negligible role. As can be seen from the table. the minima of the HF energies with and without corrections occur at the same place corresponding to the same value of the quadrupole moment. This means that the HF solution is stable against quadrupole deformations due to Zp-2h excitations. This was to be expected because of the sharp minimum of the HF energy curve. The energy gains due to Zp-2h excitations are found to be 0.41 MeV and 0.75 for approximations (A) and (B) respectively. The answer to the second question, namely that concerning the capability of the procedure to take account of the modification of the intrinsic structure of the HF solution, is given in the c;lst’ of the 2%i nucleus. The results are given in fig. 1 where the energy curves are drawn It11 various values of q2 and q4. It is seen 11~~11 bcsides the energy gain which is now 1.27 IkIck 1c,i approximation (A), the true minimum corresponds to a shape different from that of the sinrple HF theory. The corresponding deformntiol: parameters are now ~4 = -0.044 and 172 = 0.25. The quadrupole component is important, in con481
Volume 35B, number
PHYSICS
6
5 July 1971
of the pairing component of the interaction. In conclusion, the constrained HF theory may be viewed as a simple alternative to the more complicate multiconfiguration HF theory in the treatment of the effects of 2p-2h mixing in the ground state. Calculations using this procedure together with angular momentum projection are under way and will be reported separately.
- 122
One of us (J. A. R.) is grateful to Professor Jancovici for his kind hospitality at the Laboratoire de Physique Theorique et Hautes Energies. His stay would not have been possible without the financial support from the Centre International de Stages and the Universidad National de Rosario.
- 123
)
References
-124
- 0.2
0
0.2
0.4
0.6
Fig. .. Energies and quadrupole moment function of v2 for various of 34.
OB
n2
of 28Si as a
trast to the conclusion of ref. [7]. It is also to be noted that the 2p-2h mixing reduces the deformation as expected because it takes account in part
482
LETTERS
[1] I. Rapaport, T. A. Belote and W. E. Dorenbusch, Phys. Rev. 156 (1967) 1255. [2] G.Do Dang, Nucl. Phys. Al57 (1970) 231. [3] A. Faessler and A. Plastino, 2. Physik, 220 (1969) 88; A. Faessler, A. Plastino and K. W. Schmidt, Phys. Letters 34B (1971) 31. [4] J. Hinze and C. C. J. Roothan, Suppl. Progr. Theoret. Phys. 40 (1967) 37. [5] L. Sapathy and Q. Ho-Kim, Phys. Rev. Letters 25 (1970) 123. [6) R. Padjen and G. Ripka, Nucl. Phys. Al49 (1970) 273. [7] B. Caste1 and J. C. Parikh, Phys. Rev. Cl (1970) 990.