Triple excitations in perturbed relativistic coupled-cluster theory and electric dipole polarizability of group-IIB elements

We use the perturbed relativistic coupled-cluster (PRCC) theory to compute the electric dipole polarizabilities $\alpha$ of Zn, Cd, and Hg. The computations are done using the Dirac-Coulomb-Breit Hamiltonian with the Uehling potential to incorporate vacuum polarization corrections. To assimilate the self-energy corrections we use the model self-energy operator of Shabaev et al. [Phys. Rev. A 88, 012513 (2013)]. The triple excitations are included perturbatively in the PRCC theory and nonperturbatively in the unperturbed sector. Our results for $\alpha$ for all three elements are in excellent agreement with the experimental data. The other highlight of the results is the orbital energy corrections from Breit interactions. In the literature we could only get the data of Hg [E. Lindroth et al., J. Phys. B 22, 2447 (1989)], which are a near perfect match with our results. We also present the linearized equations of the cluster amplitudes, including the triple excitations, with the angular factors.

Figure 1

Goldstone diagrams that contribute to the ${\mathbf{T}}_1^{\left(1\right)}$ equation in the LPRCC approximation. Diagrams (a)–(c) and (j) arise from $H_{\mathrm{N}}{\mathbf{T}}_1^{\left(1\right)}$, (d)–(i) arise from $H_{\mathrm{N}}{\mathbf{T}}_2^{\left(1\right)}$, and (k)–(o) arise from $\mathbf{d}{\mathbf{T}}^{\left(1\right)}$. The dashed lines ending with an open circle $(\circ )$ and those ending with a closed circle $(•)$ correspond to interactions associated with the single-body part of $H_N$ and $H_{\mathrm{int}}$, respectively. The vertices with an undulating line and those with a short vertical stump represent ${\mathbf{T}}_1^{\left(1\right)}$ and ${\mathbf{T}}_2^{\left(1\right)}$, respectively.

Figure 2

Goldstone diagrams that contribute to the ${\mathbf{T}}_2^{\left(1\right)}$ in the LPRCC approximation. Diagrams (a) and (b), (c)–(j), and (k) and (l) arise from $H_{\mathrm{N}}{\mathbf{T}}_1^{\left(1\right)},H_{\mathrm{N}}{\mathbf{T}}_2^{\left(1\right)}$, and $\mathbf{d}{\mathbf{T}}_2^{\left(1\right)}$, respectively. The dashed lines ending with an open circle $(\circ )$ and those ending with a closed circle $(•)$ correspond to interactions associated with the single-body part of $H_N$ and $H_{\mathrm{int}}$, respectively. The vertices with an undulating line and those with a short vertical stump represent ${\mathbf{T}}_1^{\left(1\right)}$ and ${\mathbf{T}}_2^{\left(1\right)}$, respectively.

Figure 3

Goldstone diagrams of approximate ${\mathbf{T}}_3^{\left(1\right)}$ obtained from perturbing ${\mathbf{T}}_2^{\left(1\right)}$ with one order of the electron-electron interaction $g=1/r_{12}+g_{12}^{\mathrm{B}}$, represented by dashed lines.

Figure 4

Diagrams of $\alpha$ that arise from $T_2^{\left(0\right)†}\mathbf{D}{\mathbf{T}}_3^{\left(1\right)}$, which represents the perturbative ${\mathbf{T}}_3^{\left(1\right)}$ contracted with $T_2^{\left(0\right)†}$ and $\mathbf{D}$. The ${\mathbf{T}}_3^{\left(1\right)}$ considered in the diagrams is obtained from particle-particle contraction of ${\mathbf{T}}_2^{\left(1\right)}$ and the electron-electron interaction Hamiltonian $g=1/r_{12}+g_{12}^{\mathrm{B}}$, represented by dashed lines.