Compound-tunable embedding potential method to model local electronic excitations on $f$-element ions in solids: Pilot relativistic coupled cluster study of Ce and Th impurities in yttrium orthophosphate, ${\mathrm{YPO}}_4$

A method to simulate local properties and processes in crystals with impurities via constructing cluster models within the frame of the compound-tunable embedding potential (CTEP) and highly accurate ab initio relativistic molecular-type electronic structure calculations is developed and applied to the Ce- and Th-doped yttrium orthophosphate crystals, ${\mathrm{YPO}}_4$, having xenotime structure. Two embedded cluster models are considered, the “minimal” one, ${\mathrm{YO}}_8$@${\mathrm{CTEP}}_{\min }$, consisting of the central ${\mathrm{Y}}^{3+}$ cation and its first coordination sphere of eight ${\mathrm{O}}^{2-}$ anions (i.e., with broken P–O bonds), and its extended counterpart, $\mathrm{Y}{\left({\mathrm{PO}}_4\right)}_6$@${\mathrm{CTEP}}_{\mathrm{ext}}$, implying the full treatment of all atoms of the ${\mathrm{PO}}_4^{3-}$ anions nearest to the central ${\mathrm{Y}}^{3+}$ cation. ${\mathrm{CTEP}}_{\min ,\mathrm{ext}}$ denote here the corresponding cluster environment described within the CTEP method. The Fock space relativistic coupled cluster (FS RCC) theory is applied to the minimal cluster model to study electronic excitations localized on ${\mathrm{Ce}}^{3+}$ and ${\mathrm{Th}}^{3+}$ impurity ions. Calculated transition energies for the cerium-doped xenotime are in a good agreement with the available experimental data (mean absolute deviation of ca. 0.3 eV for $4f\to 5d$ type transitions). For the thorium-doped crystal the picture of electronic states is predicted to be quite complicated, and the ground state is expected to be of the $6d$ character. The uncertainty for the excitation energies of thorium-doped xenotime is estimated to be within 0.35 eV. Radiative lifetimes of excited states are calculated at the FS RCC level for both doped crystals. The calculated lifetime of the lowest $5d$ state of ${\mathrm{Ce}}^{3+}$ differs from the experimentally measured one by no more than twice.

Figure 1

General scheme of construction of a compound-tunable embedding potential for a minimal cluster model exemplified by the Ce-doped xenotime. The oxygen pseudoatoms of the anionic layer in the extended cluster model and CTEP partial charges are not shown for clarity. In both cluster models chemical bonds are shown for the main cluster areas. Cerium, yttrium, phosphorus, and oxygen atoms are shown yellow, green, light-purple, and red, respectively; for the minimal cluster oxygen pseudoatoms are shown in faded red

Figure 2

Minimal xenotime cluster ${\mathrm{YO}}_8$@${\mathrm{CTEP}}_{\min }$. Chemical bonds are depicted between atoms of the main cluster $\left[{\mathrm{YO}}_8\right]$. The NCE consists of yttrium and phosphorus pseudoatoms, the NAE consists of oxygen pseudoatoms (shown in the faded red color).

Figure 3

The radial dependence of electronic density differences for the studied extended and minimal clusters [see Eq. (6) and text below the formula for details]. Black lines represent total densities. Due to the using of pseudopotentials for the central yttrium, cerium, and thorium ions, the radial total density maximums, associated with this ions are shifted from the point $r=0.0$. The height of the maximums for Y, Ce, and Th ions correspond to the number of their electrons treated explicitly. Colored solid lines correspond to the integral of absolute difference $\rho \left(r\right)$ multiplied by factor of 100 (30 in the case of cerium ion). Filled peaks in the background qualitatively represent positions of the neighbor atoms (color denotes an atom type, width at the bottom is equal to crystal radius, and the peak height is proportional to the number of atoms (right vertical axis) at the same distance from the center).

Figure 4

Energy levels of impurity ions in xenotime matrix. For ${\mathrm{Ce}}^{3+}$, the values estimated from experimental data [39] are plotted. The levels for ${\mathrm{Th}}^{3+}$ were estimated by the FS RCCSD calculation (see Table 7).