L. Y. Jia recently proposed an algorithm that applies the variational method directly to coherent-pair condensates (VDPC) [L. Jia, Phys. Rev. C (2019).] Now I’ll make changes to the VDPC algorithm that applies it to the Constraint Hatree–Fock theory. I derived expressions of the mean energy extremum with linear constraints and the analytic form of the Lagrangian multiplier. In addition, I acquire an angle through the inverse tangent of the energy minimum to the electric quadrupole moment in Fig. 3, which describes the unconstrained energy under an equivalent rotating system. I demonstrate the new algorithm in a semirealistic example using the realistic [Formula: see text] interaction in the large model spaces (up to 15 harmonic-oscillator major shells). In this work, I am interested in a wave function [Formula: see text], which minimizes the total energy under the constraint that has a fixed expectation [Formula: see text]. I graphed the results as just an energy extreme versus electric quadrupole moment graph. The result shows that Fig. 3 is similar to Fig. 7.2 in [P. Ring and P. Schuck, The Nuclear Many-body Problem (Springer Science & Business Media, 2004)].
A rotating bosonic many-body system in a harmonic trap is studied with the three-dimensional cranked Hartree-Fock-Bogoliubov method at zero temperature, which has been applied to nuclear many-body systems at high spin. This method is a variational method extended from Hartree-Fock theory, which can treat the pairing correlations in a self-consistent manner. An advantage of this method is that a finite-range interaction between constituent particles can be used in the calculation, unlike the original Gross-Pitaevskii approach. To demonstrate the validity of our method, we present a calculation for a toy model---that is, a rotating system of ten bosonic particles interacting through the repulsive quadrupole-quadrupole interaction in a harmonic trap. It is found that the yrast states, the lowest-energy states for the given total angular momentum, do not correspond to the Bose-Einstein condensate, except for a few special cases. One such case is a vortex state, which appears when the total angular momentum
L is twice the particle number
N (i.e.,
L=2N).
The energy and properties of a many-electron atom or molecule may be directly computed from a variational optimization of a two-electron reduced density matrix (2RDM) that is constrained to represent many-electron quantum systems. In this paper we implement a variational 2RDM method with a representability constraint, known as the
T2 condition. The optimization of the 2RDM is performed with a first-order algorithm for semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] which, because of its lower computational cost in comparison to second-order methods, allows the treatment of larger basis sets. We also derive and implement a spin- and symmetry-adapted formulation of the
T2 condition that significantly decreases the size of the largest block in the
T2 matrix. The
T2 condition, originally derived by Erdahl [Int. J. Quantum Chem. 13, 697 (1978)], was recently applied via a second-order algorithm to atoms and molecules [Z. Zhao et al., J. Chem. Phys. 120, 2095 (2004)]. While these calculations were restricted to molecules at equilibrium geometries in minimal basis sets, we apply the 2RDM method with the
T2 condition to compute the electronic energies of molecules in both minimal and nonminimal basis sets at equilibrium as well as nonequilibrium geometries. Accurate potential energies curves are produced for BH, HF, and
N2. Results are compared with the 2RDM method without the
T2 condition as well as several wave-function methods.
We consider a large number of Bosons with interaction potential . In our earlier papers (Grillakis et al. in Comm. Math. Phys. (2010) and in Adv. Math. (2011), as well as Grillakis and Machedon in Comm. Math. Phys., (2013)) we considered a set of equations for the condensate ϕ and pair excitation function k and proved that they provide a Fock space approximation to the exact evolution of a coherent state for . In Grillakis and Machedon, J. Fixed Point Theory Appl., (2013), in the hope of treating higher values of β<1, we introduced a coupled refinement of our original equations. In that paper, we showed the coupled equations conserve the number of particles and energy. In the current paper, we prove that the coupled equations do indeed provide a Fock space approximation for , at least locally in time. In order to do that, we reformulate the coupled equations in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of those used by Chen and Holmer in J. Euro. Math. Soc. (2016) to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation function k. While our earlier papers provide background material, the methods of this paper paper are mostly new, and the presentation is self-contained.
The quantum-phase-transitional behavior of the spherical shell-model mean field plus the geometric quadrupole-quadrupole and standard pairing model within a single-
j shell is analyzed in detail. Various quantities, such as low-lying energy levels, some typical energy ratios, the overlaps of the excited states with those of the corresponding limiting cases,
B(E2) values and electric quadrupole moments of some low-lying states and their ratios, as functions of the control parameter of the model in a
j=15/2 shell are calculated as an example, in which only a crossover occurs due to the Pauli exclusion. The results show that there are noticeable changes within the crossover region of the rotation-like to the pair-excitation (superconducting) phase transition, especially in the half-filling case. As an application to realistic nuclear systems, a chain of isotones $^{212}\mathrm{Rn}\text{\ensuremath{-}}^{213}\mathrm{Fr}\text{\ensuremath{-}}^{214}\mathrm{Ra}\text{\ensuremath{-}}^{215}\mathrm{Ac}$ is chosen to be described by the model with valence protons in the
1h9/2 shell. As far as the low-lying energy levels, the experimentally observed
B(E2) values, and the electric quadrupole moment within the yrast band are concerned, these nuclei seem fitted reasonably well. The results indicate that these nuclei are all within the rotation-like to the pair-excitation phase transition near the crossover point.
Electronic structure calculations, in particular the computation of the ground state energy, lead to challenging problems in optimization. These problems are of enormous importance in quantum chemistry for calculations of properties of solids and molecules. Minimization methods for computing the ground state energy can be developed by employing a variational approach, where the second-order reduced density matrix defines the variable. This concept leads to large-scale semidefinite programming problems that provide a lower bound for the ground state energy. Upper bounds of the ground state energy can be calculated for example with the Hartree–Fock method or numerically more exact for a given basis set by full CI. However, Nakata et al. ( J. Chem. Phys. 2001 114 82828292) observed that due to numerical errors the semidefinite solver produced erroneous results with a lower bound significantly larger than the full CI energy. For the LiH, CH –, NH –, OH, OH –, and HF molecules violations within one mhartree were observed. We applied the software VSDP which takes all numerical errors due to floating-point arithmetic operations into consideration. For two test libraries VSDP provides tight rigorous error bounds lower than full CI energies reported with an accuracy of 0.1 to 0.01 mhartree. Only little computation work must be spent in order to compute close rigorous error bounds for the ground state energy.