Towards photoassociation processes of ultracold rubidium trimers

We theoretically investigate the prospects for photoassociation (PA) of ${\mathrm{Rb}}_3$, in particular at close range. We provide an overview of accessible states and possible transitions. The major focus is placed on the calculation of equilibrium structures, the survey of spin-orbit effects, and the investigation of transition dipole moments. Furthermore we discuss Franck-Condon overlaps and special aspects of trimers including the (pseudo) Jahn-Teller effect and the resulting topology of adiabatic potential-energy surfaces. With this we identify concrete and suitable PA transitions to potentially produce long-lived trimer bound states. Calculations are performed using the multireference configuration-interaction method together with a large-core effective core potential and a core-polarization potential with a large uncontracted even-tempered basis set.

Figure 1

Strongly simplified illustration of the two different photoassociation (PA) schemes for the production of ${\mathrm{Rb}}_3$ species. PA 1 photoassociates a ${\mathrm{Rb}}_2$ dimer with a free Rb atom starting from the asymptote $|g_1\rangle$. PA 2 photoassociates three Rb atoms starting from the asymptote $|g_2\rangle$. Both PA processes can be also realized (in principle) starting from an inner turning point (ITP), as discussed in Sec. 3a and indicated on the very left. In both cases the excited trimer (more weakly bound for PA starting from $|g_1\rangle$ or $|g_2\rangle$) can radiatively decay to the ground state.

Figure 2

(a) Sketch of the ${\mathrm{Rb}}_3$ system in the $xz$ plane with internuclear distances $R_{12},R_{23},R_{13}$. (b) Illustration of the geometric interpretation of the perimetric coordinates for triatomic molecules. They represent the radii of three mutual tangent circles centered on the nuclei. (c) Division of the positive octant in perimetric coordinates to show special configuration subspaces of triatomic systems (i.e., $D_{\infty h}, D_{3h}$, and $C_{2v}$).

Figure 3

Two-dimensional contour plots of the quartet ground state $1{}^{4}B{}_1$ (a) and the first quartet excited state $1{}^{4}A{}_2$ (b) PESs in the subspace $R_1$ and $R_2=R_3$ of perimetric coordinates, i.e., along one space diagonal surface in Fig. 2. The space diagonal (i.e., $D_{3h}$ configurations) corresponds to the diagonal line shown in white, while the horizontal line [both in (a)] represents the special one-dimensional $C_{2v}$ cut for $R_{23}=6.094\AA$. The wavy character in (b) is due to the underlying spline interpolation of the corresponding ab initio data.

Figure 4

Upper panel: One-dimensional cut through the potential-energy surfaces (PESs) along the space diagonal in the perimetric coordinate space shown in Fig. 2 and in terms of the diagonal shown in Fig. 3 (i.e., $D_{3h}$ scan maintaining the equilateral triangular configuration). Doublet states are shown in (a) and quartet states in (b), respectively. The low-lying states discussed in the text are highlighted; the presence of further states is indicated by the gray lines. Lower panel: One-dimensional cut through the PESs along one special direction on one space diagonal surface shown in Fig. 2. This $C_{2v}$ scan corresponds to a fixed distance $R_{23}=R_e\left(a{}^3{\mathrm{\Sigma }}_u\right)=6.094\AA$. Doublets are shown in (c) and quartets in (d), respectively.

Figure 5

Energy-level diagram of the extremal points of doublet (right) and quartet (left) states of ${\mathrm{Rb}}_3$ optimized at MRCI($\mathrm{ECP}+\mathrm{CPP}$)/UET15 level of theory. The dissociation asymptotes into ${\mathrm{Rb}}_2+\mathrm{Rb}$ (with the corresponding equilibrium energies of the ${\mathrm{Rb}}_2$ states) or $\mathrm{Rb}+\mathrm{Rb}+\mathrm{Rb}$ are shown in the area highlighted in blue. Levels given in black belong to triangular equilibrium configurations (i.e., $D_{3h}$ or $C_{2v}$ symmetry) while levels given in orange represent linear equilibrium configurations (all of them $D_{\infty h}$ symmetry). Yellow boxes mark Jahn-Teller states, where the respective dashed lines correspond to saddle points showing isosceles triangular geometry. The ionized ${\mathrm{Rb}}_3^{+}$ states are shown at the top in terms of blue energy levels. All energies are given relative to the free atom-atom-atom limit (i.e., $3\times \mathrm{Rb}\left[5s\right]$).

Figure 6

Heat-map representation of the absolute values of the spin-orbit matrix $\sqrt{|{\widehat{H}}_{ij}^{\text{SO}}{|}^2}$ (without diagonal elements). The ${\mathrm{Rb}}_3$ geometry was fixed to the equilibrium configuration of the first excited quartet state $1{}^{4}A{}_2$ (see Table 2). The partitioning separates doublet-doublet ($\mathrm{D}\leftrightarrow \mathrm{D}$), quartet-quartet ($\mathrm{Q}\leftrightarrow \mathrm{Q}$), and quartet-doublet ($\mathrm{Q}\leftrightarrow \mathrm{D}$) couplings. The dashed blue lines mark the rows where the four components corresponding to the $1{}^{4}A{}_2$ state are found in the SO matrix. The SO matrix is sorted according to IRREPs ($C_{2v}$) in the sequence ($A_1/B_1/B_2/A_2$) where each of them (in zeroth-order basis) are ordered with respect to increasing energy and are accordingly combined with the $m_s$ spin function starting from $m_s=+1/2$ to $-1/2$ for doublets and $m_s=+3/2$ to $-3/2$ for quartets. This is a complete representation with respect to the energetically lowest 12 doublet (5/4/2/1) and 15 quartet states (4/5/3/3) leading to the $84\times 84$ SO matrix.

Figure 7

Inner turning point (ITP) locations and locations of equilibrium geometries in the configuration space of perimetric coordinates. (a) Since all equilibrium geometries show at least $C_{2v}$ symmetry the configuration space survey can be restricted to one of the space diagonal surfaces shown in Fig. 2. ITPs on the quartet ground-state PES with respect to either the ${\mathrm{Rb}}_2+\mathrm{Rb}$ or $3\times \mathrm{Rb}$ dissociation scenarios are given in light blue for the first one and in light red for the latter. States lying close to these lines are promising candidates for showing good Franck-Condon factors. Here we focus on the $1{}^{4}A{}_2$ state highlighted in purple. The numbers given in (b) along both ITP lines represent the electronic dipole transition strengths (in units of $\left[{\mathrm{D}}^2\right]$) between the quartet ground state and this $1{}^{4}A{}_2$ state at the corresponding ITP locations. Note that the numbers given in (b) do not correlate with the equilibrium geometries depicted in (a). The transition dipole strength at the equilibrium geometry of the first excited quartet state amounts to $0.018{\mathrm{D}}^2$. The ellipses shown in (a) and (b) give an estimate of the size of the vibrational ground-state wave function for the $1{}^{4}A{}_2$ state.

Figure 8

The quartet states (according to $C_{2v}$ nomenclature) $1{}^4{B}_2$ and $1{}^4{A}_2$ forming the Jahn-Teller pair $1{}^4{E}^{\prime \prime }$ in the higher-symmetry $D_{3h}$ subspace (equilateral triangle). For each equilateral triangular configuration the two states show a conical intersection (COIN) leading to a COIN seam (one-dimensional curve) in the full 3D configuration space. Lowering the symmetry, e.g., by scanning along the $C_{2v}$ preserving coordinate $Q_3$, leads to a splitting of both states [see (a) for $Q_1=8.335\AA$ and $Q_2=0.0\AA$ fixed]. Due to the Jahn-Teller character these states cannot be viewed separately. The interactions lead to the formation of a lower PES sheet $E_{-}$ showing tricorn topology (three equivalent wells alternating regularly with three saddle points, separated by the localization energy $E_{\text{loc}}=225{\mathrm{cm}}^{-1}$) and a parabolic shaped upper surface $E_{+}$. The $1{}^{4}A{}_2$ state represents a global minimum on $E_{-}$ while the $1{}^{4}B{}_2$ state is a saddle point on this surface. This behavior is illustrated by the lower inset in (a) for $Q_1=8.335\AA$. Including spin-orbit coupling (SOC) leads to the annihilation of the COIN and to an energy splitting $\mathrm{\Delta }$ as shown by the inset on the top left in (a). The topology of the PESs in the two-dimensional subspace of $Q_1$ and $Q_3$ is shown in (b). The degenerate line at $Q_3=0.0\AA$, where we have $D_{3h}$ symmetry, corresponds to the one-dimensional COIN seam. The white line at the bottom represents the one-dimensional cut shown in (a).