Fourier-transform spectroscopy, direct potential fit, and electronic structure calculations on the entirely perturbed (4)${}^1\mathrm{\Pi }$ state of RbCs

We perform a high-resolution Fourier-transform spectroscopic study of the $\left(4\right){}^1\mathrm{\Pi }$ state of the RbCs molecule by applying two-step $\left(4\right){}^1\mathrm{\Pi }\leftarrow A{}^1{\mathrm{\Sigma }}^{+}\sim b{}^3\mathrm{\Pi }\leftarrow X{}^1{\mathrm{\Sigma }}^{+}$ optical excitation followed by observation of the $\left(4\right){}^1\mathrm{\Pi }\to X{}^1{\mathrm{\Sigma }}^{+}$ laser-induced fluorescence (LIF) spectra. In many LIF progressions the collision-induced satellite rotational lines are observed, thus increasing the amount of term values and allowing us to estimate the $\mathrm{\Lambda }$-doubling effect in the $\left(4\right){}^1\mathrm{\Pi }$ state. The direct potential fit (DPF) of experimental term values of 777 rovibronic levels of both ${}^{85}\mathrm{RbCs}$ and ${}^{87}\mathrm{RbCs}$ isotopologues is performed by means of the robust weighted nonlinear least-squares method. The DPF analysis based on the adiabatic approximation and analytical expanded Morse oscillator potential reveals numerous regular shifts in the measured level positions. The spectroscopic studies of the $\left(4\right){}^1\mathrm{\Pi }$ state are supported by the electronic structure calculations including the potential energy curves of the singlet- and triplet-state manifold and spin-allowed transition dipole moments. The subsequent estimates of radiative lifetimes and corresponding vibronic branching ratios elucidate a dominant contribution of the $\left(4\right){}^1\mathrm{\Pi }\to A\sim b$ channel into the total radiative decay of the $\left(4\right){}^1\mathrm{\Pi }$ state. The relative intensity distributions simulated for $\left(4\right){}^1\mathrm{\Pi }\to X{}^1{\mathrm{\Sigma }}^{+}$ LIF progressions agree well with their observed counterparts even for the profoundly shifted levels of the entirely perturbed $\left(4\right){}^1\mathrm{\Pi }$ state. To get insight into the origin of the intramolecular perturbations, the relevant spin-orbit- and $L$-uncoupling electronic matrix elements are evaluated.

Figure 1

Based on Ref. [6] schema of the singlet-state manifold of the RbCs molecule involved in the consideration. The solid vertical arrows show the two-step laser excitation of the $\left(4\right){}^1\mathrm{\Pi }$ state through an intermediate level of the singlet-triplet $A{}^1{\mathrm{\Sigma }}^{+}\sim b{}^3\mathrm{\Pi }$ complex, as well as the observed LIF transitions. The dashed arrows denote the dominant (according to calculations) radiative decay channels of the $\left(4\right){}^1\mathrm{\Pi }$ state.

Figure 2

Shown on the right is an example of a LIF spectrum ($Q$ progression) representing two-step excitation of the $\left(4\right){}^1\mathrm{\Pi }$ state in transitions $\left(4\right){}^1\mathrm{\Pi }(v^{\prime }=0,J^{\prime }=10f,E^{\prime }=20913.462{\mathrm{cm}}^{-1})\leftarrow A{}^1{\mathrm{\Sigma }}^{+}\sim b{}^3\mathrm{\Pi }(E_{A\sim b}=10160.215{\mathrm{cm}}^{-1},J=10)\leftarrow X{}^1{\mathrm{\Sigma }}^{+}(v^{\prime \prime }=4,J^{\prime \prime }=9)$. The laser frequencies are ${\nu }_{L1}=9935.917{\mathrm{cm}}^{-1}$ and ${\nu }_{L2}=10753.243{\mathrm{cm}}^{-1}$. On the left is a fragment of the respective $A\sim b\to X$ LIF spectrum.

Figure 3

Example of a LIF spectrum ($P$ and $R$ progression) representing two-step one-color excitation with laser frequency ${\nu }_{L1}\equiv {\nu }_{L2}=10683.988$ ${\mathrm{cm}}^{-1}$ in transitions $\left(4\right){}^1\mathrm{\Pi }(v^{\prime }=19,J^{\prime }=115e,E^{\prime }=21620.507{\mathrm{cm}}^{-1})\leftarrow A{}^1{\mathrm{\Sigma }}^{+}\sim b{}^3\mathrm{\Pi }(E_{A\sim b}=10936.519{\mathrm{cm}}^{-1},J=116)\leftarrow X{}^1{\mathrm{\Sigma }}^{+}(v^{\prime \prime }=0,J^{\prime \prime }=117)$: (a) the entire spectrum and (b) close-up of a fragment with transitions to $v^{\prime \prime }=0$ with satellite lines due to collisional population of neighboring rotational levels of the upper state of both $e$ and $f$ parities. The red (solid), blue (dashed), and green (short) bars below the spectrum mark $P$, $R$, and $Q$ lines, respectively. Indices denote the $J^{\prime \prime }$ value of the $X$ state.

Figure 4

Data field of experimentally observed $\left(4\right){}^1\mathrm{\Pi }$ state rovibrational levels as dependent on $J^{\prime }$. Red open circles denote the $e$ component of the ${}^{85}\mathrm{RbCs}$ isotopologue, black closed circles denote the $f$ component of the ${}^{85}\mathrm{RbCs}$, larger green closed circles denote the $f$ component of the ${}^{87}\mathrm{RbCs}$ isotopologue, blue open squares denote $e$ component of the ${}^{87}\mathrm{RbCs}$ isotopologue, and the column with stars denotes term values of the ${}^{85}\mathrm{RbCs}$ isotopologue predicted with the present mass-invariant EMO potential for $J^{\prime }=5$.

Figure 5

Present calculated difference-based potentials of the $\left(4\right){}^1\mathrm{\Pi }$ state and of the states which are presumably responsible for both local and regular perturbations observed in the $\left(4\right){}^1\mathrm{\Pi }$ state. The present EMO PEC of the $\left(4\right){}^1\mathrm{\Pi }$ state is marked by a solid red (bold) line and the RKR potential from Ref. [10] is marked by open squares; the red solid horizontal line denotes the experiment-based region of the present EMO PEC, while the black dashed horizontal line refers to the RKR potential [10].

Figure 6

The ab initio spin-orbit-coupling matrix element calculated between the singlet $\left(4\right){}^1\mathrm{\Pi }$ and triplet $\left(2\right){}^3\mathrm{\Delta }$ and (5,6)${}^3{\mathrm{\Sigma }}^{+}$ states of the RbCs molecule.

Figure 7

(a) The ab initio $L$-uncoupling electronic matrix elements of the RbCs molecule evaluated between the $\left(4\right){}^1\mathrm{\Pi }$ state and the (1–7)${}^1{\mathrm{\Sigma }}^{+}$ states. The dashed lines denote the previous results obtained in the framework of the multipartition perturbation theory calculation in Ref. [34]. (b) The partial contribution of the (4–7)${}^1{\mathrm{\Sigma }}^{+}$ states in the sum-over-states $Q\left(R\right)$ function as defined by Eq. (7).

Figure 8

The ab initio electronic transition dipole moments obtained between the $\left(4\right){}^1\mathrm{\Pi }$ state and the (1–6)${}^1{\mathrm{\Sigma }}^{+}$, (1–3)${}^1\mathrm{\Pi }$, and (1)${}^1\mathrm{\Delta }$ states of the RbCs molecule.

Figure 9

(a) Differences between experimental and calculated by the present EMO PEC rovibronic term values of the RbCs $\left(4\right){}^1\mathrm{\Pi }$ state as dependent on the vibrational quantum number $v^{\prime }$. (b) Open circles represent the empirical $q_{v^{\prime }}$ factors obtained during the DPF analysis based on the EMO potential (5), the fixed ab initio $Q\left(R\right)$ function (7), and the fitted parameter $s=0.908$; the solid line represents the theoretical $q_{v^{\prime }}$ values predicted due to the approximate sum rule (8) for $J^{\prime }=1$ levels of the ${}^{85}\mathrm{RbCs}$ isotopologue using the constrained $s\equiv 1$ parameter.

Figure 10

Comparison of experimental and calculated relative intensity distributions in the full $\left(4\right){}^1\mathrm{\Pi }(v^{\prime },J^{\prime })\to X{}^1{\mathrm{\Sigma }}^{+}\left(v^{\prime \prime }\right)$ LIF progressions starting from the weakly perturbed vibrational levels $v^{\prime }=2$, 10, and 42 of the $\left(4\right){}^1\mathrm{\Pi }$ state of the ${}^{85}\mathrm{RbCs}$ isotopologue; the respective residuals between the experimental and calculated term values are $-0.009$, 0.01, and 0.332 ${\mathrm{cm}}^{-1}$.

Figure 11

Comparison of experimental and calculated relative intensity distributions in full $\left(4\right){}^1\mathrm{\Pi }(v^{\prime }=25;J^{\prime })\to X{}^1{\mathrm{\Sigma }}^{+}\left(v^{\prime \prime }\right)$ LIF progressions starting from locally perturbed $J^{\prime }=$ 41 and 114 rovibronic levels of the ${}^{85}\mathrm{RbCs}$ isotopologue; the respective residuals between the experimental and calculated term values are 4.385 and $-2.532$ ${\mathrm{cm}}^{-1}$.

Figure 12

(a) Radiative lifetimes and (b) vibronic branching ratios of vibrational levels $v^{\prime }$ of the ${}^{85}\mathrm{RbCs}$ $\left(4\right){}^1\mathrm{\Pi }$ state evaluated using the present EMO PEC for the upper state as well as the corresponding ab initio ETDM functions and adiabatic PECs evaluated for the lower-lying singlet-state manifold by Eq. (12). The lifetimes of related atomic levels are presented in (a) by short solid lines.