Frozen-Density Embedding for Including Environmental Effects in the Dirac-Kohn–Sham Theory: An Implementation Based on Density Fitting and Prototyping Techniques

Abstract

Frozen density embedding (FDE) represents an embedding scheme in which environmental effects are included from first-principles calculations by considering the surrounding system explicitly by means of its electron density. In the present paper, we extend the full four-component relativistic Dirac–Kohn–Sham (DKS) method, as implemented in the BERTHA code, to include environmental and confinement effects with the FDE scheme (DKS-in-DFT FDE). The implementation, based on the auxiliary density fitting techniques, has been enormously facilitated by BERTHA’s python API (PyBERTHA), which facilitates the interoperability with other FDE implementations available through the PyADF framework. The accuracy and numerical stability of this new implementation, also using different auxiliary fitting basis sets, has been demonstrated on the simple NH3–H2O system, in comparison with a reference nonrelativistic implementation. The computational performance has been evaluated on a series of gold clusters (Au n , with n = 2, 4, 8) embedded into an increasing number of water molecules (5, 10, 20, 40, and 80 water molecules). We found that the procedure scales approximately linearly both with the size of the frozen surrounding environment (consistent with the underpinnings of the FDE approach) and with the size of the active system (in line with the use of density fitting). Finally, we applied the code to a series of heavy (Rn) and super-heavy elements (Cn, Fl, Og) embedded in a C60 cage to explore the confinement effect induced by C60 on their electronic structure. We compare the results from our simulations, with respect to more-approximate models employed in the atomic physics literature. Our results indicate that the specific interactions described by FDE are able to improve upon the cruder approximations currently employed, and, thus, they provide a basis from which to generate more-realistic radial potentials for confined atoms.

Subsystem density‐functional theory (update)

“…While most of the work reviewed in this Update addresses non‐relativistic theory, we note that an FDE implementation within the framework of the full four‐component relativistic Dirac–Kohn–Sham Theory was recently presented in Ref. 41.…”

Development of analytic gradients for the Huzinaga quantum embedding method and its applications to large-scale hybrid and double hybrid DFT forces

“…The next step is to determine the Lagrange multipliers in Eq. (15). They can be calculated by using the fact that the Lagrangian has to be stationary with respect to the high-and low-level orbitals.…”

“…Although the division of the modeled system is a common idea of all embedding methods, there are many different realizations. The most notable ones are QM/MM 4,10 , density matrix embedding 11 , embedded mean-field theory 12 , frozen density embedding [13][14][15] , ONIOM [16][17][18] , subsystem DFT 19 , fragmentation methods [20][21][22][23][24][25] , projection based embedding (PbE) 6,[26][27][28][29][30] , and its Huzinaga-equation-based variant [31][32][33][34] .…”

Molecular properties in the linear response regime and beyond with relativistic coupled-cluster

“…Naturally, in the near future, we should extend all the code including response properties, and EOM energy to use a library suited for distributed memory computing architectures, such as the ExaTENSOR. Once that is carried out, we can combine the code with other developed methods such as the quantum embedding approach [113] to investigate the large molecules we could not compute before for both response properties and energy. We are working towards this goal.…”

“…In our calculations, we have profited from the components of an ongoing implementation in ExaCorr of the Cholesky-decomposition [112][113][114] approach to reduce the memory footprint of our calculations in the step to transform two-electron integrals from AO to MO basis, by avoiding the storage in memory of the whole AO basis two-electron integral tensor. The Cholesky vectors (generated with a conservative threshold of 10 −9 , as to retain most of them) are then used to explicitly form all six two-electron integral classes needed by the coupled cluster method.…”