In this paper, we present and review the most recent computational advances in the BERTHA code. BERTHA can be regarded as the state of the art in fully relativistic four-component Dirac–Kohn–Sham (DKS) software. Thanks to the implementation of various parallelization and memory open-ended distribution schemes in combination with efficient “density fitting” algorithms, it greatly reduces the computational burden of four-component DKS calculations. We also report the newly developed OpenMP version of the code, that, together with the berthmod Python module, provides a significant leap forward in terms of usability and applicability of the BERTHA software. Some applications of the recently developed natural orbitals for chemical valence/charge displacement bonding analysis and the real-time time dependent DKS implementation are also reported.

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We present a real-time time-dependent four-component Dirac–Kohn–Sham (RT-TDDKS) implementation based on the BERTHA code. This new implementation takes advantage of modern software engineering, including the prototyping techniques. The software design follows a three step approach: (i) the prototype implementation of a time-propagation algorithm in nonrelativistic real-time TDDFT within the Psi4NumPy framework, which provides a suitable environment for the creation of a clear, readable, and easy to test reference code in Python, (ii) the design of an original Python application programming interface for the relativistic four-component code BERTHA (PyBERTHA), which has an efficient computational kernel for relativistic integrals written in FORTRAN, and (iii) the porting of the time-propagation scheme enveloped within the Psi4NumPy framework to PyBERTHA. The propagation scheme consequently resides in a single readable Python computer code that is easy to maintain and in which the key quantities, such as the Dirac–Kohn–Sham and dipole matrices, can be accessed directly from the PyBERTHA module. For linear algebra operations (matrix–matrix multiplications and diagonalization) we use the highly optimized procedures implemented in the popular NumPy library. The overhead introduced by the Python interface to BERTHA is almost negligible (less than 1% evaluated on the SCF procedure), and the interoperability between different programming languages (FORTRAN, C, and Python) does not affect the numerical stability of the time-propagation scheme. Our new RT-TDDKS implementation has been employed to investigate the stability of the time-propagation procedure in combination with a density-fitting algorithm (both for the Coulomb and for the exchange-correlation matrix construction), which are employed in BERTHA to speed up the Dirac–Kohn–Sham matrix evaluation. On the basis of systematic calculations, employing several density-fitting basis sets of increasing accuracy, we showed that quantitative agreement can be achieved in combination with extended-fitting basis sets, with an error in the Coulomb energy below 1 μ-hartree. Convergence of the transition energies increasing of quality of the fitting basis sets has been also observed. Our data suggest that the error in the Coulomb energy may also represent a good estimate of the fitting basis set quality for real-time electron dynamics simulations. Further, we study the applicability of the RT-TDDKS method in combination with both weak- and extreme strong-field regime. Numerical results of excited-state transitions for the Group 12 atoms are reported and compared with a previous real-time Dirac–Kohn–Sham implementation (Repisky et al. J. Chem. Theory Comput.201511980991). Finally, calculations of high harmonic generation in the hydrogen molecule and Au dimer have been also carried out. We were able to generate high harmonics with relatively well-defined peaks up to the 21st and 13th order in the case of H2 and Au2, respectively. Our findings show that the four-component structure o...

Conversely to the H2O–CF4 adduct, an appreciable intermolecular bond stabilization by charge transfer is operative in the H2O–CCl4 system.

We present a four-component relativistic density functional theory study of the chemical bond and s-d hybridization in the group-11 cyanides MCN (M = Cu, Ag, Au). The analysis is carried out within the charge-displacement/natural orbital for chemical valence (CD-NOCV) scheme, which allows us to single out meaningful contributions to the total charge rearrangement that arises upon bond formation and to quantify the components of the Dewar-Chatt-Duncanson bonding model (the ligand-tometal donation and metal-to-ligand back-donation). The M-CN bond is characterized by a large donation from the cyanide ion to the metal cation and by two small backdonation components from the metal towards the cyanide anion. The case of gold cyanide elucidates the peculiar role of the relativistic eects in determining the characteristics of the AuC bond with respect to the copper and silver homologues. In AuCN, the donation and back-donation components are signicantly enhanced and the spinorbit coupling, removing the degeneracy of the 5d atomic orbitals, induces a substantial split in the back-donation components. A simple spatial analysis of the NOCV-pair density, related to the ligand-to-metal donation component, allows us to quantify, with unprecedented accuracy, the charge rearrangement due to the s-d hybridization occurring at the metal site. The s-d hybridization plays a key role in determining the shape and size of the metal: it removes electron density from the bond axis and induces a signicant attening at the metal site in trans position to the ligand. The s-d hybridization is present in all noble metal complexes, inuencing the bond distances, and its eect is enhanced for Au, which is consistent with the preference of gold to form linear complexes. A comparative investigation of simple complexes [AuL] +/0 of Au + with dierent ligands (L=F − , N-Heterocyclic Carbene, CO, PH 3) shows that the s-d hybridization mechanism is also inuenced by the nature of the ligand.