Spin-orbit coupling from a two-component self-consistent approach. I. Generalized Hartree-Fock theory

Abstract

Formal and computational aspects are discussed of a self-consistent treatment of spin-orbit coupling within the two-component generalisation of the Hartree-Fock theory. A molecular implementation into the Crystal program is illustrated, where the standard one-component code (typical of Hartree-Fock and Kohn-Sham spin-unrestricted methodologies) is extended to work in terms of two-component spinors. When passing from a one-to a two-component description, the Fock and density matrices become complex. Furthermore, apart from the αα and ββ diagonal spin blocks, one has also to deal with the αβ and βα off-diagonal spin blocks. These latter blocks require special care as, for open-shell electronic configurations, certain constraints of the one-component code have to be relaxed. This formalism intrinsically allows to treat local magnetic torque as well as non-collinear magnetization and orbital current-density. An original scheme to impose a specified noncollinear magnetization on each atomic center as a starting guess to the self-consistent procedure is presented. This approach turns out to be essential to surpass local minima in the rugged energy landscape and allow possible convergence to the ground-state solution in all of the discussed test cases.

Charge Density Analysis of Actinide Compounds from the Quantum Theory of Atoms in Molecules and Crystals

“…In our methodology, both molecular and crystalline orbitals are expressed as linear combinations of atomic orbitals (LCAO), which is a suitable representation when chemical features of bonding are to be analyzed. Quantum-mechanical calculations are performed with a developmental version of the Crystal program, , where the LCAO approach has recently been extended to g -type basis functions. , Scalar relativistic effects must be accounted for − , and here are described by use of small-core effective pseudopotentials (with 60 electrons in the core for U). , While the program has recently been extended to the treatment of spin–orbit coupling, − this relativistic effect is disregarded here. This is because, while making the calculations significantly more demanding, it has been previously shown to induce very minor changes to chemical bonding .…”

Fundamental Role of Fock Exchange in Relativistic Density Functional Theory

“…A fully relativistic description of the electronic structure of a quantum-mechanical system requires the Dirac equation to be solved (instead of the Schrödinger equation), and this leads to a four-component wave function (instead of a one-component wave function). Alternatively, the α and β small components of the wave function can be passed onto the Hamiltonian using appropriate decoupling transformations, resulting in the well-known two-component relativistic theories. − …”

“…We now discuss the effect of Fock exchange within a relativistic DFT approach on the description of the noncollinear magnetization and orbital current density, through test calculations on a representative example: the I 2 + open-shell molecule. All relativistic DFT calculations, with inclusion of SOC, are performed with a developmental version of the public Crystal program, , where we have recently implemented a self-consistent treatment of SOC in a two-component formalism. , We also perform spin–orbit configuration–interaction (SO-CI) calculations with the Epciso and Cipsi programs , to have a reference description from a correlated wave function approach. Computational details on these calculations are provided in the Supporting Information.…”

Topology of the Electron Density and of Its Laplacian from Periodic LCAO Calculations on f-Electron Materials: The Case of Cesium Uranyl Chloride

“…With respect to the original basis set optimized for molecular calculations, some very diffuse exponents have been removed (crucially the most diffuse p -type exponent) that were causing linear dependencies in the periodic calculations. While the program has recently been extended to the treatment of spin-orbit coupling [ 44 , 45 , 46 , 47 ], this relativistic effect is disregarded here. This is because, while making the calculations significantly more demanding, it has been previously shown to induce minor changes to chemical bonding for such systems [ 48 ].…”