Systematic study of relativistic and chemical enhancements of $\mathcal{P},\mathcal{T}$-odd effects in polar diatomic radicals

Polar diatomic molecules that have, or are expected to have, a ${}^2\mathrm{\Sigma }_{1/2}$-ground state are studied systematically with respect to simultaneous violation of parity $\mathcal{P}$ and time-reversal $\mathcal{T}$ with numerical methods and analytical models. Enhancements of $\mathcal{P},\mathcal{T}$-violating effects due to an electric dipole moment of the electron (eEDM) and $\mathcal{P},\mathcal{T}$-odd scalar-pseudoscalar nucleon-electron current interactions are analyzed by comparing trends within columns and rows of the periodic table of the elements. For this purpose electronic structure parameters are calculated numerically within a quasirelativistic zeroth order regular approximation (ZORA) approach in the framework of complex generalized Hartree-Fock (cGHF) or Kohn-Sham (cGKS). Scaling relations known from analytic relativistic atomic structure theory are compared to these numerical results. Based on this analysis, problems of commonly used relativistic enhancement factors are discussed. Furthermore, the ratio between both $\mathcal{P},\mathcal{T}$-odd electronic structure parameters mentioned above is analyzed for various groups of the periodic table. From this analysis an analytic measure for the disentanglement of the two $\mathcal{P},\mathcal{T}$-odd electronic structure parameters with multiple experiments in dependence of electronic structure enhancement factors is derived.

Figure 1

Comparison of relativistic enhancement factors for eEDM induced permanent EDMs of atoms. Factor by Sandars derived analytically with Casimirs method (CS) and empirical factor for hyperfine interaction found by Fermi and Segrè (FS). Plots are shown for the case of $j=\frac{1}{2}$ as in ${}^2\mathrm{\Sigma }_{1/2}$ states.

Figure 2

Comparison of combined relativistic enhancement factors and conversion factors for the ratio between $\mathcal{P},\mathcal{T}$-odd eEDM and nucleon-electron current interactions $W_{\text{d}}/W_{\text{s}}$. The relativistic factors $\tilde{R}$ derived from the analytically derived factor (CS) and the empirical factor (FS) are shown, as well as their analogs derived from an old relativistic enhancement factor for $W_{\text{s}} \widetilde{\stackrel{̃}{R}}$. Plots are shown for the case of $j=\frac{1}{2}$ as in ${}^2\mathrm{\Sigma }_{1/2}$ states. Mass numbers $A$ were assumed as the natural mass number corresponding to the next integer value of $Z$. Numerical values shown are from cGKS calculations.

Figure 3

Fit of the $Z$ dependence of the ratio between $\mathcal{P},\mathcal{T}$-odd eEDM and scalar-pseudoscalar nucleon-electron current interactions $W_{\text{d}}/W_{\text{s}}$. The values of $W_{\text{d}}/W_{\text{s}}$ for ${\mathrm{HfF}}^{+}$ and ThO were calculated by Fleig in a four-component configuration interaction framework in Ref. [57] and are shown for comparison but are not included in the fit. All other numerical values correspond to results from cGKS calculations.

Figure 4

Scaling of ${log}_{10}\left\{\frac{|W_{\text{s}}|}{R(Z,A)f\left(Z\right)\frac{\gamma +1}{2}}\times \frac{1}{h\mathrm{Hz}}\right\}$ with ${log}_{10}\left\{Z\right\}$ for group 2 fluorides (Mg-E120)F, group 3 oxides (Sc-E121)O, group 4 nitrides (Ti-Hf)N, group 12 hydrides (Zn-Cn)H, and group 13 oxides (B-Tl)O at the level of GKS-ZORA/B3LYP (top) and GHF-ZORA (bottom). Corresponding functional expressions of the fits are plotted in each panel as a solid line, long-dashed line, short-dashed line, dotted line, and dash-dotted line, respectively. Plot of the $f$-block groups (Ce-Th)N, (Yb-No)F, and (Lu-Lr)O without fit. Boron was not included in the fit of group 13 oxides (see text).

Figure 5

Scaling of ${log}_{10}\left\{|W_{\text{d}}|\gamma \left(4{\gamma }^2-1\right)\times {10}^{-24}\frac{e\mathrm{cm}}{h\mathrm{Hz}}\right\}$ with ${log}_{10}\left\{Z\right\}$ for group 2 fluorides (Mg-Ra)F, group 3 oxides (Sc-Ac)O, group 4 nitrides (Ti-Hf)N, group 12 hydrides (Zn-Cn)H, and group 13 oxides (B-Tl)O at the level of GKS-ZORA/B3LYP (top) and GHF-ZORA (bottom). Corresponding functional expressions of the fits are plotted in each panel as a solid line, long-dashed line, short-dashed line, dotted line, and dash-dotted line, respectively. Plot of the $f$-block groups (Ce-Th)N, (Yb-No)F, and (Lu-Lr)O without fit. Boron was not included in the fit of group 13 oxides (see text).

Figure 6

Scaling of ${log}_{10}\left\{|W_{\text{d}}|{\gamma }^4\times {10}^{-24}\frac{e\mathrm{cm}}{h\mathrm{Hz}}\right\}$ with ${log}_{10}\left\{Z\right\}$ for group 2 fluorides (Mg-E120)F, group 3 oxides (Sc-E121)O, group 4 nitrides (Ti-Hf)N, group 12 hydrides (Zn-Cn)H, and group 13 oxides (B-Tl)O at the level of GKS-ZORA/B3LYP (top) and GHF-ZORA (bottom). Corresponding functional expressions of the fits are plotted in each panel as a solid line, long-dashed line, short-dashed line, dotted line, and dash-dotted line, respectively. Plot of the $f$-block groups (Ce-Th)N, (Yb-No)F, and (Lu-Lr)O without fit. Boron was not included in the fit of group 13 oxides (see text).

Figure 7

Scaling of ${log}_{10}\left\{\frac{|W_{\text{s}}|}{R(Z,A)f\left(Z\right)\frac{\gamma +1}{2}}\times \frac{1}{h\mathrm{Hz}}\right\}$ with ${log}_{10}\left\{Z\right\}$ for row 4 (Ca-Ti; solid line), row 5 (Sr-Zr; dash-dotted line), row 6 (Ba-Ce; long-dashed line, Yb-Hf; dotted line), and row 7 (Ra-Th; short-dashed line) at the level of GKS-ZORA/B3LYP (top) and GHF-ZORA (bottom).

Figure 8

Scaling of ${log}_{10}\left\{|W_{\text{d}}|{\gamma }^4\times {10}^{-24}\frac{e\mathrm{cm}}{h\mathrm{Hz}}\right\}$ with ${log}_{10}\left\{Z\right\}$ for row 4 (Ca-Ti; solid line), row 5 (Sr-Zr; dash-dotted line), row 6 (Ba-Ce; long-dashed line, Yb-Hf; dotted line), and row 7 (Ra-Th; short-dashed line) at the level of GKS-ZORA/B3LYP (top) and GHF-ZORA (bottom).