Spin contamination analogy, Kramers pairs symmetry and spin density representations at the 2-component unrestricted Hartree–Fock level of theory

Abstract

Abstract"Kramers pairs symmetry breaking" is evaluated at the 2-component (2c) Kramers unrestricted and/or general complex Hartree-Fock (GCHF) level of theory, and its analogy with "spin contamination" at the 1-component (1c) unrestricted Hartree-Fock (UHF) level of theory is emphasized. The GCHF "Kramers pairs symmetry breaking" evaluation is using the square of overlaps between the set of occupied spinorbitals with the projected set of Kramers pairs. In the same fashion, overlaps between α and β orbitals are used in the evaluation of "spin contamination" at the UHF level of theory. In this manner, UHFŜ 2 expectation value is made formally extended to the GCHF case. The directly evaluated GCHF expectation value of theŜ 2 operator is considered for completeness. It is found that the 2c GCHF Kramers pairs symmetry breaking has a very similar extent in comparison to the 1c UHF spin contamination. Thus higher excited states contributions to the 1c so 2c unrestricted wave functions of open shell systems have almost the same extent and physical consequences. Moreover, it is formally shown that a single determinant wave function in the restricted open shell Kramers case has the expectation value ofK 2 operator equal to the negative number of open shell electrons, while the eigenvalue ofK 2 for the series of simple systems (H, He, He*-triplet, Li and Li*-quartet) are found to be equal to minus the square of the number of open shell electrons. The concept of unpaired electron density is extended to the GCHF regime and compared to UHF and restricted open shell Hartree-Fock spin density. The "collinear" and "noncollinear" analogs of spin density at the GCHF level of theory are considered as well. Spin contamination and/or Kramers pairs symmetry breaking, spin

Spin contamination and noncollinearity in general complex Hartree–Fock wave functions

“…This demonstrates that, in Table 4 of Ref. [34], the equality of the entries in column 2 (reference ROHF value plus spin contamination term) and column 4 (our Φ GCHF |S 2 |Φ GCHF value) does not imply no noncollinearity. In contrast, if for a given line of the table, these two quantities differ, then necessarily there will be some noncollinearity in the corresponding GCHF wave function.…”

“…This hints that the following systems Cl, HCl + , F e, Cu, Cu 2+ and [OsCl 5 (Hpz)] − reported in Table 4 of Ref. [34] would present stronger non collinearity than H 2 O + .…”

Calculations of the EPR g-tensor using unrestricted two- and four-component relativistic approaches within the HF and DFT frameworks

“…al. [34] discussed possible definitions of 'relativistic' spin-contamination. Without further discussing definitions of spin-contamination in the relativistic case the following will demonstrate that the use of unrestricted one-determinant wave functions in the context of the present approach leads to new problems in calculations of the g-tensor.…”

Dirac Material Graphene

“…Concerning graphene, recent estimation of [35] allows placing graphene far below the border with the AFM phases and attributing it to the SM phase if . However, the doubtless presence of the correlation of graphene p z electrons causes breaking of Kramers pairs of spinors [36], which violates the time-reversal symmetry, on the one hand, and stimulates the origin of dynamic SOC, on the other [32]. The findings shift graphene along the axis in the depth of the QSHI phase thus providing a vivid topological non-triviality of graphene that might be revealed by not only the SOC-stimulated energy gap splitting.…”