Rydberg-State Double-Well Potentials of Van der Waals Molecules

Abstract

Recent progress in studies of Rydberg double-well electronic energy states of MeNg (Me = 12-group atom, Ng = noble gas atom) van der Waals (vdW) molecules is presented and analysed. The presentation covers approaches in experimental studies as well as ab initio-calculations of potential energy curves (PECs). The analysis is shown in a broader context of Rydberg states of hetero- and homo-diatomic molecules with PECs possessing complex 'exotic' structure. Laser induced fluorescence (LIF) excitation spectra and dispersed emission spectra employed in the spectroscopical characterization of Rydberg states are presented on the background of the diverse spectroscopic methods for their investigations such as laser vaporization-optical resonance (LV-OR), pump-and-probe methods, and polarization labelling spectroscopy. Important and current state-of-the-art applications of Rydberg states with irregular potentials in photoassociation (PA), vibrational and rotational cooling, molecular clocks, frequency standards, and molecular wave-packet interferometry are highlighted.

Keywords: Rydberg electronic state; ab initio calculations; double-well potential; rotational energy structure; van der Waals molecule; vibrational energy structure.

Figure 26

Experimental LIF excitation spectra terminating at the vibrational energy structure of the $E^3{\mathsf{\Sigma }}_1^{+}$-state inner potential well in CdKr, recorded using (a) $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }⟵A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=9$ and (c) $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }⟵B^{3}1,{\upsilon }_B^{″}=1$ transitions, with intensities of the spectrum in (a) normalized because of the large spectral region in which the total spectrum for ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }$ was recorded (different dye lasers and carrier gas pressures were applied). (b) Simulation of positions only and (d) simulation of positions and intensities performed assuming the $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+}$ potential representation as the result of the IPA method using the (b) LEVEL [205] and (d) LEVEL and PGOPHER [206] programs, assuming (d) $T_{\mathrm{r}\mathrm{o}\mathrm{t}}=3 \mathrm{K}$ and ${\mathsf{\Delta }}_{\mathrm{L}}={\mathsf{\Delta }}_{\mathrm{G}}=0.15 \mathrm{c}{\mathrm{m}}^{-1}$. (e) B-S plot for the ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=0–21⟵{\upsilon }_A^{″}=6$ (full squares) and ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=22–28⟵{\upsilon }_B^{″}=1$ (empty squares) progressions, showing distinct nonlinearity for ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }\ge 24$. The size of the points in the plot equals the error bars for $\mathsf{\Delta }G$.

Figure 1

Illustration of the anti-crossing (avoided crossing) of the U₁ and U₂ potential energy curves that causes the formation of a potential energy barrier in the U₂ separating two, inner and outer, potential wells.

Figure 2

(a) ${\mathrm{C}\mathrm{d}}^{+}$–$e^{-}$ interaction potential and square of the module of ${\mathsf{\Psi }}_{6s}$ atomic orbital of Cd based on the results of ab initio calculations taken from Ref. [4], where values of the orbital are in arbitrary units (they do not correspond to the values on vertical axis); (b) energy shift $E_s\left(R\right)$ due to the Rydberg electron $e^{-}$–Ng interaction calculated using Equation (1) for the $E^3{\mathsf{\Sigma }}_1^{+}$ Rydberg state of CdNg molecules; (c) charge ${\mathrm{M}\mathrm{e}}^{+}$–induced-dipole (Ng) interaction energy representing the dominating contribution to the long-range ${\mathrm{M}\mathrm{e}}^{+}$–Ng interaction; and (d) ab initio-calculated PECs of the $E^3{\mathsf{\Sigma }}_1^{+}$ state of CdNg taken from Refs. [39,40]. The dashed vertical line in (a) and (b) indicates the classical range of the Rydberg 6s electron. Note: 1 a.u.(1 bohr) = 0.5292 $Å$. For details, see text.

Figure 3

Photoassociation from the rubidium 5s ²S_1/2 ground state asymptote [reaction (1) and (1’)] to (a) a typical long-range attractive potential; (b) a double-well potential, attractive at a long distance [e.g., $0_{\mathrm{g}}^{-}$(5p ²P_3/2)]; and (c) a double-well potential, repulsive at a long distance [e.g., $0_{\mathrm{g}}^{+}$(5p ²P_3/2)]. The system decays by spontaneous emission either back to the continuum [reaction (2)] or to a bound level of a lower electronic state [reaction (3)] [e.g., the $a^3{\mathsf{\Sigma }}_u^{+}$(5s ²S_1/2) state]. For case (a), reaction (3) is usually unlikely. Note: 1 a.u.(1 bohr) = 0.5292 $Å$. (from Ref. [89], under permission of EDP Sciences, Springer-Verlag).

Figure 4

Illustration of optical–optical double resonance (OODR) method applied in molecular excitation. Potential energy curves (PECs) (solid lines) of the $X^1{\mathsf{\Sigma }}_{0^{+}}$(5s^{2 1}S₀), $A^3{\Pi }_{0^{+}}\left(5s{5p }^3{\mathrm{P}}_1\right)$, $B^{3}1\left(5s{5p }^3{\mathrm{P}}_1\right),$ and $E^3{\mathsf{\Sigma }}_1^{+}\left(5s{6s }^3{\mathrm{S}}_1\right)$ states in CdAr and electronic transition dipole moments squared ${\left|\mathrm{T}\mathrm{D}\mathrm{M}\right|}^2\left(R\right)$ (dashed lines) are plotted according to the recent result published by Krośnicki et al. [4]. ${\left|\mathrm{T}\mathrm{D}\mathrm{M}\right|}^2$ are plotted for (a) the first step of OODR: the $A^3{\Pi }_{0^{+}}\leftarrow X^1{\mathsf{\Sigma }}_{0^{+}}$ and $B^{3}1\leftarrow X^1{\mathsf{\Sigma }}_{0^{+}}$ transitions, and (b) the second step of OODR: the $E^3{\mathsf{\Sigma }}_1^{+}\leftarrow A^3{\Pi }_{0^{+}}$ and $E^3{\mathsf{\Sigma }}_1^{+}\leftarrow B^{3}1$ transitions. Examples of vibrational transitions (a) ${\upsilon }_{A,B}^{″}\leftarrow {\upsilon }_X=0$ and (b) ${\upsilon }_E^{\prime }\leftarrow {\upsilon }_{A,B}^{″}$ used in OODR are shown with vertical lines. The intensity of the vibrational transition depends on ${\left|\mathrm{T}\mathrm{D}\mathrm{M}\right|}^2\sim {\left|\int {\psi }_{\mathrm{e}\mathrm{l}}^{\prime \ast }{\mu }_{\mathrm{e}\mathrm{l}}{\psi }_{\mathrm{e}\mathrm{l}}^{″}d{\tau }_{\mathrm{e}\mathrm{l}}\right|}^2$ along with so-called overlap integrals (a) $\int {\psi }_{\upsilon }{\psi }_{{\upsilon }^{″}}dR$ and (b) $\int {\psi }_{{\upsilon }^{″}}{\psi }_{{\upsilon }^{\prime }}dR$ for the first and second transition, respectively, where ${\mu }_{\mathrm{e}\mathrm{l}}$ is an electric dipole operator, ${\psi }_{\mathrm{e}\mathrm{l}}$ are electronic eigenfunctions, and ${\psi }_{\upsilon }$ are $\upsilon$-level vibrational eigenfunctions (shown for each vibrational level). (c) Details of the second excitation. The ab initio-calculated height of the potential barrier is somewhat larger than that obtained from an experiment. Experimental positions of the ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }$ and ${\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }$ and ${\upsilon }_{A,B}^{″}$ levels are also depicted.

Figure 5

Diagram of interatomic potentials for (a) HgZn (from Ref. [153], under permission of Elsevier Science B.V.) and (b) Hg₂ (from Ref. [72], under permission of Elsevier Science B.V.) showing the relevant pump (blue arrows), probe (red arrows), and LIF (black arrows) processes. Arrows are added to the original figure. Note: 1 a.u. = 0.5292 $Å$.

Figure 6

The ‘V-scheme’ of polarization labelling spectroscopy (see the text for details).

Figure 7

LIF excitation spectrum recorded using the $A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}⟵X^1{\mathsf{\Sigma }}_1^{+},{\upsilon }_X=0$ and $B^{3}1,{\upsilon }_B^{″}⟵X^1{\mathsf{\Sigma }}_1^{+},{\upsilon }_X=0$ transitions in CdNe, as reported in Ref. [203], the former being first step of the excitation in the OODR process $E^3{\mathsf{\Sigma }}_1^{+},{\upsilon }^{\prime }⟵A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}⟵X^1{\mathsf{\Sigma }}_1^{+},{\upsilon }_X=0,$ which allowed for investigating the $E^3{\mathsf{\Sigma }}_1^{+}$-state potential and, partly, potential barrier. The arrow shows the ${\upsilon }_A^{″}=0,1$ that was used as the origin for the second transition in OODR. The asterisk depicts the vibrational band recorded in higher resolution and shown in Figure 8.

Figure 8

(a) LIF excitation spectrum showing the profile of the vibrational component recorded using the $A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=0,J_R^{″}⟵X^1{\mathsf{\Sigma }}_{0^{+}},{\upsilon }_X=0,J$ first OODR transition in CdNe. (b) Simulation performed using LEVEL [205] and PGOPHER [206] programs allowed for determining the $J_R^{″}$ assignment shown above the spectrum, which reveals the partly resolved structure of the R-branch (P-branch is also shown). In the simulation, $T_{\mathrm{r}\mathrm{o}\mathrm{t}}=5 \mathrm{K}$ (rotational temperature) and ${\mathsf{\Delta }}_{\mathrm{L}}={\mathsf{\Delta }}_{\mathrm{G}}=0.1 \mathrm{c}{\mathrm{m}}^{-1}$ were assumed (Lorentzian and Gaussian broadenings responsible for laser bandwidth and transversal divergence of molecular beam, respectively) as well as Morse representations of the $A^3{\Pi }_{0^{+}}$- and $X^1{\mathsf{\Sigma }}_{0^{+}}$-state potentials from Ref. [203]. The positions of the $J_R^{″}$ levels used as the intermediates in the OODR process (red ticks, compare with Figure 9) and a $\pm 10 \mathrm{c}{\mathrm{m}}^{-1}$ vertical bar representing the laser bandwidth are depicted.

Figure 9

Branches of rotational transitions involved in the realization of the selective $J^{\prime }$ excitation in the OODR experiment performed in CdNe using the $E^3{\mathsf{\Sigma }}_1^{+},{\upsilon }_E^{\prime }=0,J\prime ⟵A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=0,J_R^{″}⟵X^1{\mathsf{\Sigma }}_{0^{+}},{\upsilon }_X=0,J$ transition paths. Details are provided in the text.

Figure 10

(a) LIF excitation spectra recorded using the $E^3{\mathsf{\Sigma }}_1^{+},{\upsilon }_E^{\prime }=0,J\prime ⟵A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=0,J_R^{″}$ second OODR transition in CdNe, for $J_R^{″}=5,\dots ,12$ selected in the $A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=0,J_R^{″}⟵X^1{\mathsf{\Sigma }}_{0^{+}},{\upsilon }_X=0,J$ first OODR transition (see also Figure 9). (b) Separations $\mathsf{\Delta }\left(J_R^{″}\right)$ between energies of rotational transition recorded for the $\mathrm{P}(J_R^{″}-1)$ and $\mathrm{R}(J_R^{″}+1)$ branches. Linear regression allowed for determining the $B_{{\upsilon }^{\prime }=0}$ rotational constant.

Figure 11

(a) Partly rotationally resolved profiles of vibrational bands recorded in the LIF excitation spectra of the $E^3{\mathsf{\Sigma }}_1^{+},{\upsilon }_E^{\prime }⟵A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}$ second-step OODR transitions in CdNe. (b) Simulations performed using the LEVEL [205] and PGOPHER [206] programs: P-, Q-, and R-branches are shown. The intensity of Q-branch is damped, as concluded in Ref. [10]. All simulations are performed assuming $T_{\mathrm{r}\mathrm{o}\mathrm{t}}=5 \mathrm{K}$, ${\mathsf{\Delta }}_{\mathrm{L}}={\mathsf{\Delta }}_{\mathrm{G}}=0.15 \mathrm{c}{\mathrm{m}}^{-1}$, $J_{\mathrm{m}\mathrm{a}\mathrm{x}}=6–9$ depending on the transition and the isotopic shift between CdNe isotopologues (abundances > 3%) included in the simulation as approximately one order of magnitude smaller than their rotational structure.

Figure 12

The $E^3{\mathsf{\Sigma }}_1^{+}\left({5s6s }^3{\mathrm{S}}_1\right)$ Rydberg state potential well of CdNe determined experimentally [10,11] (black solid line) represented with a Morse function compared with the results of ab initio calculations from Czuchaj and Stoll [6] (blue: empty triangles and line), Czuchaj et al. [7] (green: empty squares and line), and Krośnicki and collaborators [39,40] (red: empty circles and line). Positions of ${\upsilon }^{\prime }$ levels supported by the potential well are depicted: $E\left({\upsilon }^{\prime }=0\right)=\mathrm{51,542} {\mathrm{c}\mathrm{m}}^{-1}$, $E\left({\upsilon }^{\prime }=1\right)=\mathrm{51,581} {\mathrm{c}\mathrm{m}}^{-1},$ $E\left({\upsilon }^{\prime }=2\right)=\mathrm{51,602.5} {\mathrm{c}\mathrm{m}}^{-1}$ as observed in the experiment. The energy limit beyond which no bound$⟵$bound transitions were observed is depicted.

Figure 13

(a) LIF excitation spectrum recorded using the $A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}⟵X^1{\mathsf{\Sigma }}_1^{+},\upsilon =0$ transition in CdAr, being first step of excitation in the OODR process $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }⟵A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}⟵X^1{\mathsf{\Sigma }}_1^{+},\upsilon =0$, which allowed for investigating the $E^3{\mathsf{\Sigma }}_{1\mathrm{i}\mathrm{n}}^{+}$-state inner well and, partly, potential barrier. (b) Simulation performed using the LEVEL [205] and PGOPHER [206] programs and data derived from the analysis of the experimental spectrum [13]. The arrow shows the ${\upsilon }_A^{″}=6$ that was used as the origin for the second transition in OODR. Also, the ${\upsilon }_A^{″}=6⟵\upsilon =0$ band recorded with higher resolution was used, among others, in the isotopologue selection experiment (see Section 5.4). The position of the atomic transition in Cd is depicted.

Figure 14

(a) LIF excitation spectrum of the ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=0–19⟵{\upsilon }_A^{″}=6$ progression recorded using $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+}⟵A^3{\Pi }_{0^{+}}$ in CdAr. (b) Simulations performed using a representation of the $E^3{\mathsf{\Sigma }}_1^{+}$-state inner well (b) obtained from the IPA method and (c) using a Morse function. As the figure compares only the positions of the vibrational components, their intensities in (a–c) were normalized. To make the comparison easier, several last ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }⟵{\upsilon }_A^{″}=6$ components are shown on different horizontal scales. Details of the ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=19⟵{\upsilon }_A^{″}=6$ component, shown in the rectangular frame. (d) B-S plot for the ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=0–19⟵{\upsilon }_A^{″}=6$ progression presenting a distinct nonlinearity for approx. ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }>12$.

Figure 15

Comparison of the $E^3{\mathsf{\Sigma }}_1^{+}\left({5s6s }^3{\mathrm{S}}_1\right)$-state inner well IPA representation in CdAr (black: full circles and line) with the results of the most recent ab initio calculations by Krośnicki et al. [4] (red: full circles and line) and ab initio calculations by Czuchaj and Stoll [6] (blue: empty triangles and line), Czuchaj et al. [7] (green: empty squares and line), and Krośnicki and collaborators [39,40] (red: empty circles and line). The inset shows vicinity of the potential barrier that separates the inner and the outer wells. The position of three ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }$ levels closest to the dissociation energy is shown. The height of the potential barrier $E_b^{\prime }$ is as estimated in Ref. [13] (grey rectangle) and as determined in Ref. [12] (horizontal black line) (also see Table 1).

Figure 16

(a) LIF excitation spectrum recorded using the $E^3{\mathsf{\Sigma }}_1^{+}⟵A^3{\Pi }_{0^{+}}$ transition in CdAr [13], showing the last ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=19⟵{\upsilon }_A^{″}=6$ quasi-bound$⟵$bound transition and profile of free$⟵$bound transitions starting from ${\upsilon }_A^{″}=6$. (b) and (c) Simulations of the bound$⟵$bound and free$⟵$bound transitions performed using the PGOPHER [206] and BCONT [209] programs, respectively.

Figure 17

(A) Partly rotationally resolved profiles of vibrational bands recorded in the LIF excitation spectrum of the $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }⟵A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=6$ second-step OODR transition in ¹¹⁶${\mathrm{C}\mathrm{d}}^{40}\mathrm{A}\mathrm{r}$ (with a small admixture of ¹¹⁴${\mathrm{C}\mathrm{d}}^{40}\mathrm{A}\mathrm{r}$). (a) Experimental spectra. (b) Simulations performed using the LEVEL [205] and PGOPHER [206] programs assuming $T_{\mathrm{r}\mathrm{o}\mathrm{t}}=2.5 \mathrm{K}\left({\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=2 \mathrm{a}\mathrm{n}\mathrm{d} 5\right),T_{\mathrm{r}\mathrm{o}\mathrm{t}}=3.5 \mathrm{K}\left({\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=11\right),$ $T_{\mathrm{r}\mathrm{o}\mathrm{t}}=$5 K (${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=17$), and ${\mathsf{\Delta }}_{\mathrm{L}}={\mathsf{\Delta }}_{\mathrm{G}}=0.1 \mathrm{c}{\mathrm{m}}^{-1}$. (c) Intensities of rotational P-, Q- and R-branches contributing to (b), depicted with a colour code in part for ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=17$. The intensity of the Q-branch is damped, as concluded in Ref. [10]. (B) The profile of the $A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=6⟵X^1{\mathsf{\Sigma }}_1^{+},\upsilon =0$ first-step OODR transition showing the possibility of selective excitation of ¹¹⁶${\mathrm{C}\mathrm{d}}^{40}\mathrm{A}\mathrm{r}$ with a small admixture of ¹¹⁴${\mathrm{C}\mathrm{d}}^{40}\mathrm{A}\mathrm{r}$ isotopologue (see also Section 5.4). (a) Experimental LIF excitation spectrum, (b) total simulation of the profile, and (c) contributions to the total simulated profile corresponding to different CdAr isotopologues. Vertical blue dashed line and wide bar depict a laser wavenumber of the first-step OODR transition and the laser bandwidth, respectively.

Figure 18

The $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+},{\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }⟵B^{3}1,{\upsilon }_B^{″}=2$ vibrational band of CdAr. (a) Experimental spectrum. Simulations performed using the PGOPHER program [206] assuming that (b) $R_{e \mathrm{o}\mathrm{u}\mathrm{t}}^{\prime }$ = 7.63 $Å$ [16] and (c) $R_{e \mathrm{o}\mathrm{u}\mathrm{t}}^{\prime }$ = 6.90 $Å$ were obtained as a result of the new method of bond length adjustment with the help of the $C\left(R_{e \mathrm{o}\mathrm{u}\mathrm{t}}^{\prime }\right)$ agreement parameter [15]. In both simulations, $T_{\mathrm{r}\mathrm{o}\mathrm{t}}=5 \mathrm{K}$ and ${\mathsf{\Delta }}_{\mathrm{L}}={\mathsf{\Delta }}_{\mathrm{G}}=0.1 \mathrm{c}{\mathrm{m}}^{-1}$ were used. Note: according to later studies, the figure shows the wrong ${\upsilon }_{E\mathrm{o}\mathrm{u}\mathrm{t}}^{\prime }$ assignment as it lacks the low-intensity ${\upsilon }_{E\mathrm{o}\mathrm{u}\mathrm{t}}^{\prime }=0⟵{\upsilon }_B^{″}=2$ component that was recorded during the investigation of the $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+},{\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }⟵B^{3}1,{\upsilon }_B^{″}$ transitions [12], causing correction of the spectroscopical characterization of the $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+}$ well. This is also illustrated in Figure 19 and Figure 20.

Figure 19

The $C\left(R_{e \mathrm{o}\mathrm{u}\mathrm{t}}^{\prime }\right)$ agreement coefficient describing the agreement between $I_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{t}}^{\left(i\right)}$ experimental and $I_{\mathrm{s}\mathrm{i}\mathrm{m}}^{\left(i\right)}\left(R_{e \mathrm{o}\mathrm{u}\mathrm{t}}^{\prime }\right)$ simulated intensities of the vibrational components in the LIF excitation spectrum recorded using the $E^3{\mathsf{\Sigma }}_{1\mathrm{o}\mathrm{u}\mathrm{t}}^{+},{\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }⟵B^{3}1,{\upsilon }_B^{″}$ transition (see Equation (2)) for ${\upsilon }_B^{″}=1$ (black full circles and line) [15] and ${\upsilon }_B^{″}=2$ (black empty circles and line) [15]. The plot is supplemented with the recent results of Ref. [12], which concluded in the correction of the ${\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }$ assignment (see Section 5.2.3) for ${\upsilon }_B^{″}=1$ (grey squares and line), ${\upsilon }_B^{″}=2$ (green squares and line), ${\upsilon }_B^{″}=3$ (blue squares and line), and ${\upsilon }_B^{″}=4$ (red squares and line). Symbols with error bars and corresponding rectangles depict the $R_{e \mathrm{o}\mathrm{u}\mathrm{t}}^{\prime }$ results of Refs. [12,15] and, for comparison, Ref. [16].

Figure 20

The $({\omega }_e^{\prime }-{\omega }_e^{\prime }{x}_e^{\prime })$ agreement plot drawn according to Equation (3), expressing simulation-to-experiment agreement with respect to the $E^3{\mathsf{\Sigma }}_{1\mathrm{o}\mathrm{u}\mathrm{t}}^{+}$ state of CdAr using LIF excitation spectra of the $E^3{\mathsf{\Sigma }}_{1\mathrm{o}\mathrm{u}\mathrm{t}}^{+},{\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }⟵B^{3}1,{\upsilon }_B^{″}$ transitions for (a) ${\upsilon }_B^{″}=1,$ as in Ref. [15], and (b) ${\upsilon }_B^{″}=3,$ as in Ref. [12], after recording an additional component in the spectrum and correction of the ${\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }$ assignment (see Section 5.2.3). The plot shows the dependency between ${\omega }_e^{\prime }$ and ${\omega }_e^{\prime }{x}_e^{\prime }$ vibrational constants. Error bars: uncertainties in ${\omega }_e^{\prime }$ and ${\omega }_e^{\prime }{x}_e^{\prime },$ as in Ref. [16] (red), Ref. [15] (yellow), and, finally, Ref. [12] (white).

Figure 21

LIF excitation spectrum recorded using the $B^{3}1,{\upsilon }_B^{″}⟵X^1{\mathsf{\Sigma }}_1^{+},{\upsilon }_X=0$ transition in CdAr, as reported in Ref. [210], which is the first step of excitation in the OODR process that allowed for investigating the $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+}$-state outer well [12,15] and potential barrier [12] using the $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+},{\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }⟵B^{3}1,{\upsilon }_B^{″}$ second-step OODR transition. Arrows show ${\upsilon }_B^{″}$ values that were used as origins in the second-step transition.

Figure 22

LIF excitation spectra (red lines) recorded using (a) the $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+},{\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }⟵B^{3}1,{\upsilon }_B^{″}=0–4$ bound$⟵$bound and (b) the $E^3{\mathsf{\Sigma }}_1^{+}⟵B^{3}1,{\upsilon }_B^{″}=0–4$ free$⟵$bound transitions in CdAr, and their respective simulations [205,206,209] (black lines). The position ${\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }=0⟵{\upsilon }_B^{″}$ component that was not previously recorded [15] is depicted (one asterisk). The position of the $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=17–19⟵B^{3}1,{\upsilon }_B^{″}$ vibrational components is shown, proving that the spectrum also contains transitions to the $E^3{\mathsf{\Sigma }}_1^{+}$-state inner well. Unlike in other spectra recorded using second-step OODR transition and presented in this review, here, laser wavenumbers are given with respect to the $X^1{\mathsf{\Sigma }}_1^{+}$-state asymptote. The energy corresponding to the $\left({5s6s }^3{\mathrm{S}}_1\right)$ Cd asymptote is depicted (vertical dashed line). The inset to (a) shows details of the very weak ${{\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }⟵\upsilon }_B^{″}=0$ bound$⟵$bound transitions with clearly visible ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=18⟵{\upsilon }_B^{″}=0$. (c) B-S plot for ${\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }⟵{\upsilon }_B^{″}=4$ progression and with corrected ${\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }$ assignment, providing final values for the $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+}$ outer well vibrational constants ${\omega }_e^{\prime }=4.36 {\mathrm{c}\mathrm{m}}^{-1}$ and ${\omega }_e^{\prime }{x}_e^{\prime }=0.207$ ${\mathrm{c}\mathrm{m}}^{-1}$.

Figure 23

Comparison of the CdAr interatomic potential of the $E^3{\mathsf{\Sigma }}_1^{+}\left({5s6s }^3{\mathrm{S}}_1\right)$ Rydberg-state outer well representations close to the dissociation limit, being the result of series of experiments using the OODR excitation method as follows: very first investigation by Koperski and Czajkowski [17] (grey dashed-dotted line), result of Urbańczyk and Koperski [15] (grey dotted line), and the most recent result of Sobczuk et al. [12] (black solid line)—also see Table 1. Note: for the inner potential well, the representations refer to Figure 15. Part of the whole $E^3{\mathsf{\Sigma }}_1^{+}$-state potential representation in the vicinity of the potential barrier as the result of Ref. [12] is also shown (blue line and points)—see the text for details. The ab initio-calculated potential by Krośnicki et al. [4] is shown for comparison (red line and points). The potential of the $B^{3}1$ intermediate state [16] used in the OODR process is also drawn (green solid line). Three ways to execute the excitation using the $E^3{\mathsf{\Sigma }}_1^{+}{⟵B}^{3}1,{\upsilon }_B^{″}=4$ transitions are depicted with vertical arrows as follows: (a) quasi-bound$⟵$bound to the $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+}$, (b,c) free$⟵$bound terminating at the inner and outer repulsive walls of the potential barrier, respectively, and (d) bound$⟵$bound to the $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+}$.

Figure 24

(a) Ab initio-calculated interatomic potentials of CdAr electronic energy states [4] used to illustrate the OODR process of the excitation of $E^3{\mathsf{\Sigma }}_1^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{″}=2$ from $X^1{\mathsf{\Sigma }}_1^{+},{\upsilon }_X=0$ via $A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=5$ (vertical solid arrows) followed by emission to several lower-lying molecular states, allowing for the characterization of their potentials using bound→free and bound→bound transitions. (b) Examples of simulated dispersed emission spectra [205,206,209] with bound→free (b→f) and bound→bound (b→b) transitions from $E^3{\mathsf{\Sigma }}_1^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{″}=2$ to $c^3{\mathsf{\Sigma }}_1^{+}\left({5s5p }^3{\mathrm{P}}_2\right)$, $b^3{\Pi }_2\left({5s5p }^3{\mathrm{P}}_2\right)$ and $a^3{\Pi }_{0^{-}}\left({5s5p }^3{\mathrm{P}}_0\right)$ electronic ‘dark’ states, which would serve as data to determine their PECs [202].

Figure 25

LIF excitation spectra recorded using the (a) $A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}⟵X^1{\mathsf{\Sigma }}_1^{+},{\upsilon }_X=0$ [215] and (b) $B^{3}1,{\upsilon }_B^{″}⟵X^1{\mathsf{\Sigma }}_1^{+},{\upsilon }_X=0$ [106] transitions in CdKr, both being the first steps of the excitation in OODR processes, i.e., $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }⟵A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}⟵X^1{\mathsf{\Sigma }}_1^{+},{\upsilon }_X=0$ and $E^3{\mathsf{\Sigma }}_1^{+},{\upsilon }_E^{\prime }⟵B^{3}1,{\upsilon }_B^{″}⟵X^1{\mathsf{\Sigma }}_1^{+},{\upsilon }_X=0,$ that allowed for investigating the inner $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+}$ and outer $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+}$ potential wells. Arrows show the ${\upsilon }_A^{″}=9$ and ${\upsilon }_B^{″}=0, 1, 4, 6$ that were used as origins for the second-step OODR transition. The asterisk depicts the ${\upsilon }_A^{″}=9⟵{\upsilon }_X=0$ vibrational band recorded with higher resolution and used, among others, in the isotopologue selection experiment (see Section 5.4). Note: irregular, i.e., non-Morse, behaviour of ${\upsilon }_B^{″}$ levels is due to the double-well character of the $B^{3}1$-state potential [106].

Figure 27

LIF excitation spectra (red traces) terminating at the ${\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }$ vibrational energy structure of the $E^3{\mathsf{\Sigma }}_1^{+}$-state outer well and the most upper-lying ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }$ levels of the $E^3{\mathsf{\Sigma }}_1^{+}$-state inner well in CdKr, recorded using $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+},{\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }⟵B^{3}1,{\upsilon }_B^{″}$ and $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }⟵B^{3}1,{\upsilon }_B^{″}$, respectively, starting at (a) ${\upsilon }_B^{″}=0$, (b) ${\upsilon }_B^{″}=1$, (c) ${\upsilon }_B^{″}=4$, and (d) ${\upsilon }_B^{″}=6$. Simulations of the $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+}⟵B^{3}1$ and $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+}⟵B^{3}1$ spectra (blue and grey traces, respectively) performed using the LEVEL [205] and PGOPHER [206] programs in which $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+}$, $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+}$ and $B^{3}1$-state representations were taken from Refs. [19,106], and $T_{\mathrm{r}\mathrm{o}\mathrm{t}}=3 \mathrm{K}$ and ${\mathsf{\Delta }}_{\mathrm{L}}={\mathsf{\Delta }}_{\mathrm{G}}=0.15 \mathrm{c}{\mathrm{m}}^{-1}$ were assumed. (e) B-S plot drawn for the $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+}$ potential well in CdKr based on the $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+}{,\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }⟵B^{3}1,{\upsilon }_B^{″}=6$ transition shown in (d), in which strong nonlinearity is present, and two regions, A (${\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }=0–8$) and B (${\upsilon }_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }=9–16$), each of linear behaviour, can be extracted.

Figure 28

Representations of the $E^3{\mathsf{\Sigma }}_1^{+}\left({5s6s }^3{\mathrm{S}}_1\right)$ Rydberg state potential of CdKr with an inset showing details of the potential barrier and outer potential well close to the dissociation limit. $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+}$ (red line and circles) and $E^3{\mathsf{\Sigma }}_{1 \mathrm{o}\mathrm{u}\mathrm{t}}^{+}$ (red line and squares) potential well representations as results of the IPA methodology and analysis of the spectra in Figure 26a,c and Figure 27d, respectively. A Morse representation of $E^3{\mathsf{\Sigma }}_{1 \mathrm{i}\mathrm{n}}^{+},$ being the result of the B-S plot analysis in Figure 26e, is also presented (blue solid line). The experimentally determined representations are compared with two results of ab initio calculations [216], showing their considerable difference, including those from 2008 performed by Krośnicki and collaborators [39,40] (black line and empty circles) and those most recently performed by Krośnicki et al. [208] (black line and full circles). The estimated height of the potential barrier $E_b^{\prime }$ is shown as a red horizontal line. For comparison, the result of the first experimental study [20] (grey solid lines) is included. The ranges of ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }$ and ${\mathsf{\upsilon }}_{E_{\mathrm{o}\mathrm{u}\mathrm{t}}}^{\prime }$ excited from $B^{3}1,{\upsilon }_B^{″}$ and analysed in [19] are shown (grey rectangles).

Figure 29

(A): (a) Profile of the LIF excitation spectrum recorded using the $A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=9⟵X^1{\mathsf{\Sigma }}_1^{+},\upsilon =0$ transition in CdKr. (b) Total simulation of the profile performed using the PGOPHER [206] program, assuming $T_{\mathrm{r}\mathrm{o}\mathrm{t}}=4 \mathrm{K}$ and ${\mathsf{\Delta }}_{\mathrm{L}}={\mathsf{\Delta }}_{\mathrm{G}}=0.1 \mathrm{c}{\mathrm{m}}^{-1}$ as well as $B_{\upsilon }$ and $D_{\upsilon }$ rotational constants, and transition energies calculated using the LEVEL [205] program for the $A^3{\Pi }_{0^{+}}$ [16] and $X^1{\mathsf{\Sigma }}_1^{+}$-state [215] potential characteristics. (c) Individual contributions to (b) that originate from different ${}^{A_{\mathrm{Cd}}}\mathrm{C}\mathrm{d}^{A_{\mathrm{Kr}}}\mathrm{K}\mathrm{r}$ isotopologues with abundances larger than 1%. (d) Positions of the ${\nu }_{\mathrm{l}\mathrm{a}\mathrm{s} 1}$ laser wavenumber of OODR first-step excitation used in (B); examples of ${\nu }_{\mathrm{l}\mathrm{a}\mathrm{s} 1}\pm 0.1 \mathrm{c}{\mathrm{m}}^{-1}$ are represented with blue vertical dashed lines and rectangles. (B): Profiles of LIF excitation spectra recorded using the $E^3{\mathsf{\Sigma }}_{1\mathrm{i}\mathrm{n}}^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=18⟵A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=9$ OODR second-step transition for different ${\nu }_{\mathrm{l}\mathrm{a}\mathrm{s} 1}\pm 0.1 \mathrm{c}{\mathrm{m}}^{-1}$ depicted in (A)-(d), i.e., for different combinations of excited ${}^{A_{\mathrm{C}\mathrm{d}}}\mathrm{C}\mathrm{d}^{A_{\mathrm{K}\mathrm{r}}}\mathrm{K}\mathrm{r}$ isotopologues. For the simulation shown with the black line (in position-2), refer to (C). (C): (a) Profiles of LIF excitation spectra recorded using the $E^3{\mathsf{\Sigma }}_{1\mathrm{i}\mathrm{n}}^{+},{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=0,3,18⟵A^3{\Pi }_{0^{+}},{\upsilon }_A^{″}=9$ transitions in ¹¹⁴${\mathrm{C}\mathrm{d}}^{86}\mathrm{K}\mathrm{r}$ (with small admixture of ¹¹⁶${\mathrm{C}\mathrm{d}}^{40}\mathrm{K}\mathrm{r}$)—excitation in position-2: ${\nu }_{\mathrm{l}\mathrm{a}\mathrm{s} 1}=\mathrm{30,557.58}\pm 0.1 \mathrm{c}{\mathrm{m}}^{-1}$ shown in (A). (b) Simulations performed using the PGOPHER [206] program, assuming $J_{A \mathrm{m}\mathrm{i}\mathrm{n}}^{″}=0, J_{A \mathrm{m}\mathrm{a}\mathrm{x}}^{″}=15$, $T_{\mathrm{r}\mathrm{o}\mathrm{t}}=4 \mathrm{K}$ (for ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=0 \mathrm{a}\mathrm{n}\mathrm{d} 3$), $T_{\mathrm{r}\mathrm{o}\mathrm{t}}=15 \mathrm{K} (\mathrm{f}\mathrm{o}\mathrm{r}{\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=18$), and ${\mathsf{\Delta }}_{\mathrm{L}}={\mathsf{\Delta }}_{\mathrm{G}}=0.1 \mathrm{c}{\mathrm{m}}^{-1}$, as well as $B_{\upsilon }$ and $D_{\upsilon }$ rotational constants, and transition energies calculated using the LEVEL [205] program for the $E^3{\mathsf{\Sigma }}_{1\mathrm{i}\mathrm{n}}^{+}$ [16] and $A^3{\Pi }_{0^{+}}$-state [16] potential characteristics. (c) Simulated distributions of P-, Q-, and R-branch components depicted with a colour code as in part for ${\upsilon }_{E_{\mathrm{i}\mathrm{n}}}^{\prime }=0$. The intensity of the Q-branch is damped, as concluded in Ref. [10].

Figure 30

(a) Scheme of the double-resonance technique for the detection of the ‘dark’ intermediate ${\upsilon }^{\prime }$ vibrational levels of the $c^{3}1$ state in HgAr using probe laser excitation to the potential barrier in the $E^{3}1$ Rydberg state. (b) $c^{3}1\left({\upsilon }^{\prime }\right)- X^1{0}^{+}({\upsilon }^{″}=0)$ excitation spectrum of the HgAr vdW complex plotted against the pump wavenumber (from Ref. [25], under the permission of the American Institute of Physics).

Figure 31

Pump control probe scheme for detecting the populations of the $A^3{0}^{+}$, ${\upsilon }_A^{″}$ vibrational levels created by a double-laser pulse from the $X^1{0}^{+}$, ${\upsilon }_X=0$ vibrational level (from Ref. [217], under the permission of the American Physical Society).

Figure 32

Dispersed emission spectrum corresponding to the $B^{3}1\left(5s{5p }^3{\mathrm{P}}_1\right),{\upsilon }^{″}=4\to X^1{\mathsf{\Sigma }}_{0^{+}}\left({5s^2 }^1{\mathrm{S}}_0\right),\upsilon$ transitions in the CdAr molecule (a) recorded using a SpectraPro HRS 750 spectrometer (Teledyne Princeton Instruments) equipped with a CCD camera with an image intensifier (PIMAX 4) and diffraction grating with 1200 grooves/mm. (b) Simulation of the dispersed emission spectrum from (a) performed using the LEVEL [205] and PGOPHER [206] programs and assuming parameters of the $B^{3}1$ and $X^1{\mathsf{\Sigma }}_{0^{+}}$-state potentials from Refs. [14,218], respectively, as well as Gaussian broadening responsible for spectrometer spectral throughput ${\mathsf{\Delta }}_{\mathrm{G}}=38 \mathrm{c}{\mathrm{m}}^{-1}$. (c) Simulation of the recorded background signal associated with the excitation laser beam at the $B^{3}1,{\upsilon }^{″}=4⟵X^1{\mathsf{\Sigma }}_{0^{+}},\upsilon =0$ transition. (d) Simulated [205,206] distribution of transitions to different $\upsilon$ values in the $X^1{\mathsf{\Sigma }}_{0^{+}}$ground state originating from ${\upsilon }^{″}=4$ in the $B^{3}1$ state performed for ${\mathsf{\Delta }}_{\mathrm{G}}=1.9 \mathrm{c}{\mathrm{m}}^{-1}$.