This study presents two new relativistic Gaussian basis sets without variational prolapse of double- and triple-ζ quality, RPF-2Z and RPF-3Z, along with augmented versions including additional diffuse functions, aug-RPF-2Z and aug-RPF-3Z, which are available for all s and p block elements from Hydrogen to Oganesson. The exponents of the Correlation/Polarization (C/P) functions are obtained from a polynomial version of the generator coordinate Dirac-Fock method (p-GCDF). The choice of C/P functions was guided by multireference configuration interaction calculations with single and double excitations (MR-CISD) based on a valence active space. Finally, calculations of fundamental properties done for atomic and molecular systems (bond lengths, vibrational frequencies, dipole moments, and electron affinities) ensure the expected quality of these new basis sets, which may also exhibit some computational efficiency advantages. Additionally, the prolapse-free feature of these sets must provide a reliable description of properties more dependent on core electron distributions, as well.
The N-representability problem consists in determining whether, for a given p-body matrix, there exists at least one N-body density matrix from which the p-body matrix can be obtained by contraction, that is, if the given matrix is a p-body reduced density matrix (p-RDM). The knowledge of all necessary and sufficient conditions for a p-body matrix to be N-representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the p-body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the N-representability conditions grows exponentially with system size, and hence, the procedure quickly becomes intractable for practical applications. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N-representability conditions. The algorithm consists of applying to an initial N-body density matrix a sequence of unitary evolution operators constructed from a stochastic process that successively approaches the reduced state of the density matrix on a p-body subsystem, represented by a p-RDM, to a target p-body matrix, potentially a p-RDM. The generators of the evolution operators follow the well-known adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented by using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide whether a given p-body matrix is N-representable, establishing a criterion to determine its quality and correcting it. We apply the proposed hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, from the reduced Bardeen-Cooper-Schrieffer model with constant pairing, and from the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices toward different targets.
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