Accurate and general formalism for spin-mixing parameter calculations

The spin-admixture parameter $\gamma$ is key to understanding spin dynamics in molecules. It measures the influence of spin-orbit coupling on spin dynamics in virtually all models for molecular materials. However, the predictive quality of its current first-principles electronic structure theory formulation [Z. G. Yu, Phys. Rev. B 85, 115201 (2012)] is limited by multiple approximations. Consequently, current literature is lacking further first-principles predictions of $\gamma$ beyond those of the original paper, and the application of such in computational studies of the role of spin mixing in organic and molecular spintronics. Here, we generalize the previous formulation of $\gamma$ to remove all approximations limiting its predictive accuracy. The result is dramatic qualitative improvements in several model systems for organic spintronics. The significantly increased transferability of our method also allows its application to a range of heavier, metal-center “molecular qubit” molecules without modification. For these molecules, we identify molecular spin-orbit coupling as the key to their spin relaxation, with unprecedented accuracy. The accuracy and computational robustness of our reformulated method makes it ideal for the kind of large-scale, high-throughput computational exploration of chemical spaces used in several other areas of molecular nanotechnology.

Figure 1

(a) Benzene (${\mathrm{C}}_6{\mathrm{H}}_6$) and thiophene (${\mathrm{C}}_4{\mathrm{H}}_4\mathrm{S}$) molecules, with an illustration of the rotation of their conjugation plane with fixed spin quantization axis. (b) Biphenyl (${\mathrm{C}}_{12}{\mathrm{H}}_{10}$) with the definition of the phenyl ring dihedral angle. (c) Metal-phthalocyanine (MPc, ${\mathrm{C}}_{32}{\mathrm{H}}_{16}\mathbf{M}{\mathrm{N}}_8$).

Figure 2

Variation of ${\gamma }^2$ with rotation of conjugation plane relative to the spin quantization axis (see Fig. 1). Left $y$ axis: ROKS (blue) and UKS (red) results for benzene. Solid, dotted, and dashed lines represent ${\gamma }^2$, ${\gamma }_{\uparrow \uparrow }^2$, and ${\gamma }_{\uparrow \downarrow }^2$, respectively. Note the UKS (ROKS) asymmetry (symmetry) of ${\gamma }_{\uparrow \uparrow }^2$ and ${\gamma }_{\uparrow \downarrow }^2$. Right $y$ axis: UKS curve for thiophene (magenta). The thiophene ${\gamma }_{\uparrow \downarrow }^2$ and ${\gamma }_{\uparrow \uparrow }^2$ curves differ from benzene in magnitude only, and have been left out for clarity.

Figure 3

The UKS ${\gamma }^2$ curve of Fig. 2 repeated for fully local (PW92), hybrid (PBE0), and fully nonlocal (HF) xc approximations for cationic (solid line) and anionic (dotted-dashed line) benzene molecules.

Figure 4

${\gamma }^2$ as a function of dihedral torsion in a biphenyl molecule at ZORA/UKS/PBE0/TZVP level of theory (red) compared with Eq. (9) (green). Solid (dashed) lines indicate positive (negative) biphenyl ions.

Figure 5

Correlation plot of ${\gamma }^2$ vs aggregate spin relaxation rates ${\left(T_1\right)}^{-1}$ for the four MPc molecules, and a linear fit on the form ${\left(T_1\right)}^{-1}=\frac{{\gamma }^2}{\kappa }$ (see text).