Variation-after-projection Research Articles

A novel algorithm to the shell model study of heavy deformed nuclei using the variation after projection approach

We introduce an extension to the conventional variation after projection (VAP) method, in which wave functions of different spin states projected from a shared intrinsic wave function are first varied simultaneously. Then a second variation is carried out for each individual state starting from the obtained intrinsic wave function. This novel algorithm enables the efficient extraction of a rotational band from chaotic non-rotational states by adopting only one Slater determinant, which can not be reached by the conventional VAP method. Some examples including the heavy deformed nuclei in the jj56 model space have been calculated. The calculated results show that this new method could be a useful tool to systematically study the collective rotational states in heavy deformed nuclei. Furthermore, the present method holds potential for probing significant correlations, such as octupole correlations, within nuclear structures.

Mixing of one-particle-one-hole projected states with the variation after projection wave functions

In this paper, we study the mixing of one-particle-one-hole projected states with the variation after projection (VAP) wave functions in an attempt to improve the approximation of this method. It turns out that, when minimizing only the lowest (yrast) energy with given spin and parity, the one-particle-one-hole projected states cannot be mixed into the converged VAP wave function, which is very similar to the situation of the Hartree-Fock method. However, if one minimizes the sum of several lowest energies with the same spin and parity, the one-particle-one-hole mixing can make some improvements to the VAP wave functions. We expect such one-particle-one-hole mixing may be useful in the calculations of the low-lying excited states in heavy nuclei with a large model space.

Finding optimal basis states for variation after projection nuclear wave functions* *Supported by the National Natural Science Foundation of China (11975314), the Key Laboratory of Nuclear Data foundation (JCKY2022201C158), and the Continuous Basic Scientific Research Project (WDJC-2019-13, BJ2000250)

The variation after projection (VAP) method is expected to be an efficient way of obtaining the optimized nuclear wave functions, which can be as close as possible to the exact shell model ones. However, we found that there are two additional problems that may seriously affect the convergence of the VAP iteration. The first problem is the existence of irrelevant projected basis states. At a VAP iteration, the Hill-Wheeler (HW) equation is composed of all updated projected basis states. If one of these projected basis states does not mix with a calculated wave function of interest, which is obtained by solving this HW equation, it is likely that this basis state will never mix with this wave function even after the VAP iteration converges. The other problem is the poor orthonormality among the projected basis states, which seriously affects the accuracy of the calculated VAP wave function. In the present work, solutions for these two problems are proposed, and examples are presented to test the validity. With the present solutions, the most important projected basis states can be reliably obtained, and the fully optimized VAP wave functions can be accurately and efficiently calculated.

Weighted variation after projection method for low-lying nonyrast states* *Supported by the National Natural Science Foundation of China (11975314), the Key Laboratory of Nuclear Data foundation (JCKY2022201C158), and the Continuous Basic Scientific Research Project (WDJC-2019-13, BJ20002501)

We propose a simple algorithm to further improve the previous variation after projection (VAP) wave functions for low-lying nonyrast states. We attach a weight factor to each calculated energy; then, the sum of these weighted energies is minimized. It turns out that a low-lying nonyrast VAP wave function can be further optimized when the weight factor for the corresponding energy is far larger than the other ones. Based on the improved WVAP wave functions, the energy-variance extrapolation method is applied to estimate the exact shell model energies. The calculated results for nuclei in the and model spaces clearly show that the extrapolated energies for all the calculated states are very close to the exact shell model ones within keV.

Energy-variance extrapolation for high-spin states with fully optimized variation after projection wave functions

We have implemented an energy-variance extrapolation method with the variation after projection (VAP) wave functions which are fully optimized. Thanks to the adopted VAP wave functions which are simplified by taking only one projected state for each selected reference state, the present energy variance extrapolation calculation can be easily applied to high-spin states, even over the backbending. Moreover, the energy variance can be calculated very quickly because the required matrix elements are already available in our previous VAP calculations. The calculated results in $pf$ model space have clearly shown that the extrapolated energies for all calculated spin states are very close to the exacted shell model ones within 20 keV.

Variation after projection calculations for high-spin states

In this paper, an improvement has been made on the variation after projection (VAP) method, which is crucial in the calculations of high-spin states. It turns out that, the form of the trial VAP wave function with spin J can be simplified by adopting just one projected state rather than previously adopting all (2J+1) angular momentum projected states for each selected reference state, |Φ〉. The present calculations show that such simplification still minimizes the angular momentum projected energy up to a very good approximation. In this simplified VAP scheme, one can obtain almost equivalent VAP wave functions starting from different sets of the projected basis states. This clearly shows that a nuclear state can not be identified with a single intrinsic state, while in the traditional nuclear collective models, an intrinsic state is usually assigned to a well deformed rotational band.

Axial shape asymmetry and high-spin states in nuclei with Z=100 suggested by the projected total energy surface approach

The axial-shape asymmetry of yrast states in $^{246--256}\mathrm{Fm}$ is studied by performing the projected total-energy surface (PTES) calculations, which consider the beyond-mean-field effects associated with the restoration of rotational symmetry and shape variation at the same time. The results show a large elongation deformation but also a considerable large triaxiality for their ground and high spin states, the triaxial deformation $\ensuremath{\gamma}\ensuremath{\approx}{11}^{\ensuremath{\circ}}$ in average. In comparison, the TRS calculations have also been performed for these nuclei, and the results show a well-established axial quadrupole shape in their ground states. The presence of the significant triaxial deformation can be attributed to the beyond-mean-field effects generated by the angular-momentum projection. The axial asymmetric shape for the yrast states of nuclei with $Z=100$, suggested by the present variation after projection (VAP) calculations, indicates that the triaxial degree of freedom may also play a significant role in other transfermium and even superheavy nuclei. The present PTES calculations have well reproduced the available experimental energies of the ground-band states and predict the rest yrast states up to spin 30 in each nucleus. The calculated yrast bands of $^{246--256}\mathrm{Fm}$ present the back bending phenomenon at about the state ${18}^{+}$, caused by the alignment excitations of the two quasiparticle neutrons of $\ensuremath{\nu}{j}_{15/2}[743]7/2$ or of $\ensuremath{\nu}{h}_{11/2}[761]1/2$. It is worth confirming the predicted band structures by the future spectroscopic experiments in the transfermium nuclei for the study of the single-particle structure in the superheavy mass region.

Generator coordinate method with a conjugate momentum: Application to particle number projection

We discuss an extension of the generator coordinate method (GCM) by taking simultaneously a collective coordinate and its conjugate momentum as generator coordinates. To this end, we follow the idea of the dynamical GCM (DGCM) proposed by Goeke and Reinhard. We first show that the DGCM method can be regarded as an extension of the double projection method for the center of mass motion. As an application of DGCM, we then investigate the particle number projection, for which we not only carry out an integral over the gauge angle as in the usual particle number projection but also take a linear superposition of BCS states which have different mean particle numbers. We show that the ground state energy is significantly lowered by such effect, especially for magic nuclei for which the pairing gap is zero in the BCS approximation. This suggests that the present method makes a good alternative to the variation after projection (VAP) method, as the method is much simpler than the VAP.

Half-Projected σ Self-Consistent Field For Electronic Excited States.

Fully self-consistent mean-field solutions of electronic excited states have been much less accessible compared to ground state solutions (e.g., Hartree-Fock). The main reason for this is that most excited states are energy saddle points, and hence energy-based optimization methods such as Δ-SCF often collapse to the ground state. Recently, our research group has developed a new method, σ-SCF [ J. Chem. Phys. 2017 , 147 , 214104 ], that successfully solves the "variational collapse" problem of energy-based methods. Despite the success, σ-SCF solutions are often spin-contaminated for open-shell states due to the single-determinant nature; unphysical behaviors such as disappearing solutions and discontinuous first-order energy derivatives are also observed along with the spontaneous breaking of spin or spatial symmetries. In this work, we tackle these problems by partially restoring the broken spin-symmetry of a σ-SCF solution through an approximate spin-projection scheme called half-projection. Orbitals of the projected wave function are optimized in a variation-after-projection (VAP) manner. The resulting theory, which we term half-projected (HP) σ-SCF, brings substantial improvement to the description of singlet and triplet excitations of the original σ-SCF method. Numerical simulations on small molecules suggest that HP σ-SCF delivers high-quality excited-state solutions that exist in a wide range of geometries with smooth potential energy surfaces. We also show that local excitations in HP σ-SCF can be size-intensive.

New algorithm in the variation after projection calculations for non-yrast nuclear states

We present a novel and simple algorithm in the variation after projection (VAP) approach for the non-yrast nuclear states. It is for the first time that the yrast state and non-yrast states can be varied on the same footing. The orthogonality among the calculated states is automatically fulfilled by solving the Hill-Wheeler equation. This avoids the complexity of the frequently used Gram-Schmidt orthogonalization, as adopted by the excited VAMPIR method. Thanks to the Cauchy's interlacing theorem in the matrix theory, the sum of the calculated lowest projected energies with the same quantum numbers can be safely minimized. Once such minimization is converged, all the calculated energies and the corresponding states can be obtained, simultaneously. The present VAP calculations are performed with time-odd Hartree-Fock Slater determinants. It is shown that the calculated VAP energies (both yrast and non-yrast) are very close to the corresponding ones from the full shell model calculations. It looks the present algorithm is not limited to the VAP, but should be universal, i.e., one can do the variation with different forms of the many-body wavefunctions to calculate the excited states in different quantum many-body systems.

Efficient Implementation of Variation after Projection Generalized Hartree-Fock.

Projected Hartree-Fock (PHF) theory can restore important symmetries to broken symmetry wave functions. Variation after projection (VAP) implementations make it possible to deliberately break and then restore a given symmetry by directly minimizing the projected energy expression. This technique can be applied to any symmetry that can be broken from relaxing constraints on single Slater determinant wave functions. For instance, generalized Hartree-Fock (GHF) wave functions are eigenfunctions of neither Ŝz nor S2. By relaxing these constraints, the wave function can explore a larger variational space and can reach lower energies than more constrained HF solutions. We have implemented spin-projected GHF (SGHF), which retains many of the advantages of breaking symmetry while also being a spin eigenfunction, with some notable improvements over previous implementations. Our new algorithm involves the formation of new intermediate matrices not previously discussed in the literature. Discretization of the necessary integration over the rotation group SO(3) is also accomplished much more efficiently using Lebedev grids. A novel scheme to incrementally build rotated Fock matrices is also introduced and compared with more standard approaches.

Implementation of the variation-after-projection approach in calculations with a time-odd Hartree-Fock mean field

We have implemented a new variation-after-projection (VAP) calculation based on a time-odd Hartree-Fock (HF) mean field. The exact Hessian matrix of the projected energy has been successively evaluated for the first time, which makes the VAP calculation much more stable. With the time-odd mean field, the present VAP can be applied to the yrast states not only in the even-even nuclei, but also in the odd-$A$ and odd-odd ones. The VAP energies throughout all spins for some calculated $sd$-shell nuclei are very close to the corresponding shell model ones. Our calculations clearly show that the spin projection is very important in achieving a good approximation to the full shell model.

Spin-Projected Matrix Product States: Versatile Tool for Strongly Correlated Systems.

We present a new wave function ansatz that combines the strengths of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as [Formula: see text], where [Formula: see text] is the spin projector for total spin S and |ΨMPS(N,M)⟩ is an MPS wave function with a given particle number N and spin projection M. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin adaptation (i.e., to create eigenfunctions of Ŝ2) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state |ΨMPS(N,M)⟩ in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for nonsinglet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) Due to the use of |ΨMPS(N,M)⟩ as its underlying state, the SP-MPS can be closely connected to broken-symmetry mean-field states. This allows one to straightforwardly generate the large number of broken-symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector [Formula: see text] naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin-projected theories. Unlike usual spin-projected theories based on determinants, the SP-MPS ansatz can be made essentially exact simply by increasing the bond dimensions in |ΨMPS(N,M)⟩. Computing excited states is also simple by imposing orthogonality constraints, which are simple to implement with MPS. To illustrate the versatility of SP-MPS, we formulate algorithms for the optimization of ground and excited states, develop perturbation theory based on SP-MPS, and describe how to evaluate spin-independent and spin-dependent properties such as the reduced density matrices. We demonstrate the numerical performance of SP-MPS with applications to several models typical of strong correlation, including the Hubbard model, and [2Fe-2S] and [4Fe-4S] model complexes.

Analytic energy gradient of projected Hartree-Fock within projection after variation.

We develop a geometrical optimization technique for the projection-after-variation (PAV) scheme of the recently refined projected Hartree-Fock (PHF) as a fast alternative to the variation-after-projection (VAP) approach for optimizing the structures of molecules/clusters in symmetry-adapted electronic states at the mean-field computational cost. PHF handles the nondynamic correlation effects by restoring the symmetry of a broken-symmetry single reference wavefunction and moreover enables a black-box treatment of orbital selections. Using HF orbitals instead of PHF orbitals, our approach saves the computational cost for the orbital optimization, avoiding the convergence problem that sometimes emerges in the VAP scheme. We show that PAV-PHF provides geometries comparable to those of the complete active space self-consistent field and VAP-PHF for the tested systems, namely, CH2, O3, and the [Cu2O2]2+ core, where nondynamic correlation is abundant. The proposed approach is useful for large systems mainly dominated by nondynamic correlation to find stable structures in many symmetry-adapted states.

Variation after projection with a triaxially deformed nuclear mean field

We implemented a variation after projection (VAP) algorithm based on a triaxially deformed Hartree-Fock-Bogoliubov vacuum state. This is the first projected mean field study that includes all the quantum numbers (except parity), i.e., spin ($J$), isospin ($T$) and mass number ($A$). Systematic VAP calculations with $JTA$-projection have been performed for the even-even $sd$-shell nuclei with the USDB Hamiltonian. All the VAP ground state energies are within 500 keV above the exact shell model values. Our VAP calculations show that the spin projection has two important effects: (1) the spin projection is crucial in achieving good approximation of the full shell model calculation. (2) the intrinsic shapes of the VAP wavefunctions with spin projection are always triaxial, while the Hartree-Fock-Bogoliubov methods likely provide axial intrinsic shapes. Finally, our analysis suggests that one may not be possible to associate an intrinsic shape to an exact shell model wave function.

Lipkin method of particle-number restoration to higher orders

Background: On the mean-field level, pairing correlations are incorporated through the Bogoliubov-Valatin transformation, whereupon the particle degrees of freedom are replaced by quasiparticles. This approach leads to a spontaneous breaking of the particle-number symmetry and mixing of states with different particle numbers. In order to restore the particle number, various methods have been employed, which are based on projection approaches before or after variation. Approximate variation-after-projection (VAP) schemes, utilizing the Lipkin method, have mostly been used within the Lipkin-Nogami prescription. Purpose: Without recurring to the Lipkin-Nogami prescription, and using instead states rotated in the gauge space, we derive the Lipkin method of particle-number restoration up to sixth order and we test the convergence and accuracy of the obtained expansion. Methods: We perform self-consistent calculations using the higher-order Lipkin method to restore the particle-number symmetry in the framework of superfluid nuclear energy-density functional theory. We also apply the Lipkin method to a schematic exactly solvable two-level pairing model. Results: Calculations performed in open-shell tin and lead isotopes show that the Lipkin method converges at fourth order and satisfactorily reproduces the VAP ground-state energies and energy kernels. Near closed shells, the higher-order Lipkin method cannot be applied because of a non-analytic kink in the ground-state energies in function of the particle number. Conclusions: In open-shell nuclei, the higher-order Lipkin method provides a good approximation to the exact VAP energies. The method is computationally inexpensive, making it particularly suitable, for example, for future optimizations of the nuclear energy-density functionals and simultaneous restoration of different symmetries.

Microscopic study of deformation systematics in some isotones in the A ≈ 100 mass region

Variation after projection (VAP) calculations in conjunction with Hartree-Bogoliubov (HB) ansatz have been carried out for N=60, 62 isotones in the mass region A=100. In this framework, the yrast spectra with JΠ ≥ 10+ B(E2) transition probabilities, quadrupole deformation parameter and occupation numbers for various shell model orbits have been obtained. The results of calculations indicate that the simultaneous increase in polarization of p1/2, p3/2 and f5/2 proton sub-shells is a significant factor into the development of the deformation in neutron rich isotones in the mass region A=100.

Microscopic study of low-lying yrast spectra and deformation systematics in neutron-rich 98–106Sr isotopes

Variation-after-projection (VAP) calculations in conjunction with Hartree-Bogoliubov (HB) ansatz have been carried out for A = 98–106 strontium isotopes. In this framework, the yrast spectra with J Π ≥ 10+, B(E2) transition probabilities, quadrupole deformation parameter and occupation numbers for various shell model orbits have been obtained. The results of the calculation for yrast spectra give an indication that it is important to include the hexadecapole-hexadecapole component of the two-body interaction for obtaining various nuclear structure quantities in Sr isotopes. Besides this, it is also found that the simultaneous polarization of p 3/2 and f 5/2 proton subshells is a significant factor in making a sizeable contribution to the deformation in neutron-rich Sr isotopes.

Study of N = Z nuclei in variation-after-projection framework

Variation-after-projection (VAP) calculations in conjunction with the Hartree- Bogoliubov (HB) ansatz have been carried out for A = 68-88, N = Z nuclei. In this framework, the yrast spectra with \(J^{\pi} \leqslant 10^{ + } \), B(E2) transition probabilities and deformation parameter (\(\beta_{2}\)) have been obtained. A pairing interaction for like particles as well as protons and neutrons has been included in the model for a two-body interaction.

MICROSCOPIC STUDY OF BACKBENDING PHENOMENA IN 126-132Ce

126-132 Ce isotopes, which exhibit enhanced collective behavior and occurrence of first backbend around 10+, are studied in the Variation after Projection (VAP) and Cranked Hartree–Bogoliubov (CHB) frameworks. The pairing plus-quadrupole-quadrupole (PQQ) interaction is employed as an effective interaction. The yrast spectra, B(E2) transition probabilities and variation of moment of inertia versus cranking frequency are studied. Our results show that the backbending arises as a result of the sudden alignment of a pair of protons in the h11/2 subshell.