REActions in Ultracold Alkali Metal Systems

• REActions are processes where reactants transform into products through atomic rearrangements governed by kinetic and thermodynamic factors.
• Studies reveal that reaction enthalpies in alkali metal dimers dictate atom exchange, with exothermic transitions leading to reactive loss in ultracold traps.
• Advanced electronic structure methods, such as AQCC, enable mapping of potential energy surfaces, highlighting features like conical intersections and Jahn-Teller distortions.

A reaction is a process in which one or more initial species (reactants) undergo transformation into different species (products) via specific rearrangements of atomic constituents, often governed by both kinetic and thermodynamic factors. In theoretical and applied contexts, reactions are modeled at multiple levels, from the quantum description of electron transfer and bond rearrangement to stochastic and deterministic representations in chemical, biological, and physical systems. This entry provides a comprehensive treatment of reactions across physical chemistry, statistical mechanics, and quantum systems, integrating key results from the paper of ultracold alkali metal dimers (Zuchowski et al., 2010).

1. Energetics of Reactions in Alkali Metal Dimers

The energetics of reactions involving alkali metal dimers are determined by the comparative dissociation energies of heteronuclear and homonuclear species. A prototypical atom exchange process is

$$\text{KRb} + \text{KRb} \rightarrow \text{K}_2 + \text{Rb}_2,$$

where two diatomic heteronuclear molecules (KRb) are converted into two homonuclear dimers (K$_2$, Rb$_2$). The critical parameter governing reactivity is the reaction enthalpy, denoted $\Delta E_2$, calculated using accurate experimental dissociation energies (with uncertainties as small as $\pm$1 cm$^{-1}$).

Analysis indicates:

• Heteronuclear lithium dimers and KRb in their ground states have negative $\Delta E_2$ values, evidencing exothermic, thus energetically allowed, atom exchange. These dimers are therefore unstable with respect to loss via reactive collisions.
• Other heteronuclear alkali dimers have positive or near-zero $\Delta E_2$, indicating endoergic atom exchange, energetically forbidden under ultracold conditions.

These results underscore the species-dependent stability of quantum gases composed of alkali metal dimers. Only molecules protected by energetically disallowed atom exchange reactions are expected to be long-lived in ultracold traps.

2. High-Level Electronic Structure Calculations and Potential Energy Surfaces

For reactions where direct experimental energetics are inaccessible, particularly in trimer-forming reactions, high-level theoretical methods are required. Calculations were performed using the multireference average-quadratic coupled-cluster (AQCC) method within the MOLPRO suite, with the following key technical aspects:

• Effective core potentials (ECP) and core polarization potentials (CPP) were employed to model core-valence correlation.
• Cutoff parameters in core-valence potentials were adjusted to match experimental bond lengths and binding energies of homonuclear dimers at the same theoretical level, and the basis was augmented (with s, p, d, and f functions).
• Potential energy surfaces (PES) were computed as functions of molecular geometry for all heteronuclear trimers, considering both C$_{2v}$ and C$_s$ symmetries.

Results show that trimer formation reactions, e.g.,

$$\text{KRb} + \text{KRb} \rightarrow \text{K} + \text{KRb}_2 \text{ or } \text{K}_2\text{Rb} + \text{Rb},$$

are always endoergic for low-lying singlet dimers. This conclusion is supported by both AQCC computations and simple orbital/Hückel theory arguments. For instance, the Hückel model predicts the energy difference for atom transfer is about $|\beta|$, where $2|\beta|$ is the singlet dimer binding energy, and $3|\beta|$ the trimer's, hence such reactions are endoergic by roughly $|\beta|$.

Illustrative PES cuts (e.g., for Rb$_2$Cs) reveal conical intersections and Jahn–Teller distortions, highlighting the complex multi-sheeted nature of the reaction landscape.

3. Implications for the Stability of Quantum Gases

The interplay of energetic thresholds for different reaction channels dictates the experimental feasibility of stable quantum gases consisting of ultracold alkali metal dimers:

• For species with exothermic atom exchange, trap loss is inevitable due to kinetic release from the conversion to homonuclear dimers.
• For those where both atom exchange and trimer formation reactions are endoergic (as is generically the case for ground-state singlet dimers except for the unfavorable lithium and KRb systems), the molecular quantum gas is protected against two-body reactive loss.

The robustness of these findings is critical for the design and interpretation of ultracold molecular experiments. Notably, if the dimers are prepared in highly excited or triplet states, energetic allowances can change, potentially opening additional reactive channels.

4. Reaction Pathway Classification and Model Equations

The atomic transformations can be represented by specific equations, including:

Atom exchange:

$$\text{KRb} + \text{KRb} \rightarrow \text{K}_2 + \text{Rb}_2$$

Atom transfer/trimer formation:

$$\text{KRb} + \text{KRb} \rightarrow \text{K} + \text{KRb}_2 \quad \text{or} \quad \text{K}_2\text{Rb} + \text{Rb}$$

In Hückel theory, the relative energetics are summarized by: $\Delta E_{\text{atom transfer}} \sim |\beta|$ where $|\beta|$ is half the dimer's binding energy.

The distinction between exothermic/endothermic channels and the application of these model equations, in conjunction with AQCC-calculated energy surfaces, enables the systematic classification of allowable and forbidden reaction pathways under ultracold conditions.

5. Broader Context and Future Research Directions

The in-depth analysis of REActions in ultracold dimer systems informs several broader research trajectories:

• Preparation of highly stable, long-lived molecular quantum gases is crucial for exploring strongly correlated and dipolar many-body physics.
• Energetic suppression of reactive losses in specific species opens new avenues for the paper of controlled ultracold chemistry, non-adiabatic effects near conical intersections, and quantum-state-resolved reaction dynamics.
• There is particular interest in extending these computational strategies to molecules in triplet or vibrationally excited states, which may support alternative reaction pathways with different energy landscapes.
• High-level ab initio methods remain indispensable for accurate mapping of multi-dimensional PESs, especially for systems with significant Jahn–Teller coupling or multiple conical intersections.

6. Summary Table: Energetic Accessibility of Two-Body Reaction Channels

Dimer Species	Atom Exchange ΔE₂	Atom Exchange Allowed?	Trimer Formation (Singlet)	Trimer Formation Allowed?
Li-containing heteronuclear	Negative	Yes (Exothermic)	Substantially endoergic	No
KRb	Negative	Yes (Exothermic)	Substantially endoergic	No
Other heteronuclear dimers	Positive/≈0	No (Endoergic/Suppressed)	Substantially endoergic	No

For each entry: "Atom Exchange Allowed?" or "Trimer Formation Allowed?" is affirmative only if the corresponding energetic threshold is negative (exothermic).

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These results collectively provide a quantitative and theoretical framework for understanding and predicting the reactivity and stability of ultracold alkali metal dimers. The combination of precise dissociation energy measurements, advanced electronic structure computations, and quantum statistical considerations enables predictive control over reaction channels, supporting both fundamental chemical physics and the engineering of novel quantum matter (Zuchowski et al., 2010).