Orbital angular momentum constraints in the variational optimization of the two-electron reduced-density matrix

Abstract

The direct variational determination of the two-electron reduced-density matrix (2-RDM) is usually carried out under the assumption that the 2-RDM is a real-valued quantity. However, in systems that possess orbital angular momentum symmetry, the description of states with a well-defined, nonzero z-projection of the orbital angular momentum requires a complex-valued 2-RDM. We consider a semidefinite program suitable for the direct optimization of a complex-valued 2-RDM and explore the role of orbital angular momentum constraints in systems that possess the relevant symmetries. For atomic systems, constraints on the expectation values of the square and z-projection of the orbital angular momentum operator allow one to optimize 2-RDMs for multiple orbital angular momentum states. Similarly, in linear molecules, orbital angular momentum projection constraints enable the description of multiple electronic states, and, moreover, for states with a non-zero zprojection of the orbital angular momentum, the use of complex-valued quantities is essential for a qualitatively correct description of the electronic structure. For example, in the case of molecular oxygen, we demonstrate that orbital angular momentum constraints are necessary to recover the correct energy ordering of the lowest-energy singlet and triplet states near the equilibrium geometry. However, care must still be taken in the description of the dissociation limit, as the 2-RDM-based approach is not size consistent, and the size-consistency error varies dramatically, depending on the z-projections of the spin and orbital angular momenta.

Revisiting the variational two-particle reduced density matrix for nuclear systems

“…However, the discrepancies between variational 2RDM using P QGT 1 T 2 and exact results using SM are still large. The largest energy difference occurs in 28 Si, where it is about 6.5 MeV. Conversely, the energy discrepancy is small in 20 Ne, as it is only about 150 keV.…”

“…Consequently, it is necessary in practice to couple angular momenta in Eqs. ( 14), ( 15), ( 16), (23), and (28). Atomic and molecular systems are considered in the LS scheme as spin S and orbital L angular momenta can be considered as good quantum numbers [51].…”

“…Isospin-breaking effects are not considered in the USDB interaction [64]. In the present work, we will first consider the ground-state energies of even-even oxygen isotopes, in which the nuclei only have valence neutrons; after that, we will focus on the N = Z eveneven nuclei, thus bearing both valence protons and neutrons, which consist of 20 Ne, 24 Mg, 28 Si, and 32 S. Due to the isospin conserving character of the USDB interaction, the configurations of valence neutrons and protons are the same in N = Z nuclei. In the latter results, we then just show the normalized occupations of valence neutrons.…”

“…In order to ponder out the influence of the isospin T = 0 part of the nucleon-nucleon interaction, we now consider nuclei in which protons and neutrons are both active. As testing grounds for our benchmarking purpose, we calculate the ground states of the N = Z eveneven sd-nuclei, 20 Ne, 24 Mg, 28 Si, and 32 S. The results are shown in Tab. III and Fig.…”

“…However, limitations in optimization software and computer resources prevented the practical development of the 2RDM method for several years. Significant progress has been made over the last decade in 2RDM methods, as the possible use of powerful computers allowed to devise numerical schemes otherwise impossible to apply in practice [8,10,[14][15][16][17][18][25][26][27][28][29]. Two approaches have been developed to calculate the ground state 2RDM : (i) variational 2RDM with energy minimization [10,17,18] and (ii) non-variational approaches based on solutions of the contracted Schrödinger equation [8,26,27].…”