Calculation of the local environment of a barium monofluoride molecule in an argon matrix: A step towards using matrix-isolated BaF for determining the electron electric dipole moment

Barium monofluoride embedded in a solid argon matrix is being used [1] by the EDM³ collaboration for pursuing a measurement of the electron electric dipole moment (eEDM). Current measurements [2] of the eEDM already put strong limits on possible beyond-the-standard-model physics (for energy scales of up to 100 TeV) that would lead to the level of time-reversal (T) violation required to understand the asymmetry between matter and antimatter in the universe. Measurements of the eEDM at higher levels of accuracy will test T-violating physics at even higher energy scales.

In this work, we calculate the local environment of a BaF molecule embedded in an Ar matrix. A substitution of a BaF molecule for four Ar atoms is found to be strongly favoured energetically compared to substitutions for other numbers of Ar atoms. The equilibrium positions of the BaF molecule and of the neighbouring Ar atoms are calculated. Most importantly for the work of the EDM³ collaboration, the potential barriers that prevent the BaF molecule from migrating through the solid and from rotating within the solid are determined. These barriers will allow for a large sample of stationary, non-rotating BaF molecules (without the need for an external electric field to inhibit their rotations [1]).

In a previous work [3], we calculated the ground-state energies of the BaF-Ar triatomic system using high-precision all-electron relativistic quantum-mechanical calculations (using a scalar-relativistic approach within the framework of a second-order Douglas-Kroll-Hess approach) that include correlation and that are extrapolated to the complete basis set limit. These energies were calculated for a large range of positions (angles $\theta$ and distances $r$) of the Ar atom relative to the BaF molecule using the CCSD(T) method. The calculations provide a smoothly varying potential energy versus $r$ and $\theta$ for the interaction between BaF and Ar. This potential is used here, along with the well-known Ar-Ar interatomic potential to calculate the local environment near the embedded BaF molecule.

Recently, BaF molecules have been embedded in both Ne [4] and Ar [5] cryogenic solids by the EDM³ collaboration. In both cases, laser-induced fluorescence is observed. Earlier work [6] also studied matrix-isolated BaF molecules, including optical absorption and electron spin resonance studies. Knowledge of the position of the BaF molecule within the fcc Ar crystal is needed to calculate shifts of the BaF energy levels due to the Ar matrix and interpret the observed spectra. The present work will also allow for future calculations of the oscillatory modes (e.g., librational motion and centre-of-mass oscillatory motion) for the matrix-isolated BaF molecule. An understanding of the local environment of the BaF molecule within an Ar solid will help to guide continuing work of the EDM³ collaboration.

In this work, the geometry and energy of a BaF molecule in an Ar crystal is determined. An ideal Ar crystal forms a face-centred-cubic (fcc) structure with a cube of size $a=5.3118$ Å [7]. Relative to one Ar atom, there are $n_{1}=12$ nearest neighbours at $b_{1}=a/\sqrt{2}$, with subsequent sets (of size $n_{k}$) of nearest neighbours at $b_{k}=\sqrt{k}b_{1}$ (for subsequent integer values $k$). Our simulations start with a cluster of Ar atoms within a sphere of radius $b_{n}$. In the same manner as was done in Ref. [8], the outer part of this sphere (those farther than $b_{m}$ from the centre) have their positions fixed at the ideal Ar fcc crystal positions. Inside this shell, a single BaF molecule is situated near the centre, surrounded by $M$ Ar atoms. These $M$ Ar atoms and one BaF molecule are allowed to move to minimize the overall interaction energy of the system.

In particular, the interaction energy is minimized while varying $3M+5$ parameters: the positions of the $M$ Ar atoms and of the centre of mass of the BaF molecule, as well as the two angles that define the orientation of the BaF molecule. As in Ref. [3], the separation between the Ba and F nuclei is fixed at 2.16 Å, the separation determined from rotational spectroscopy [9]. The large BaF binding energy (6 eV [10, 11]) compared to the BaF-Ar binding energy (23 meV [3]), along with the larger equilibrium separations for the BaF-Ar system, leads to a much stronger restoring force for stretching this 2.16 Å separation compared to typical BaF-Ar forces, justifying a fixed BaF internuclear separation.

Two independent calculations using different methods are employed for this minimization to verify that the global minimum is found. One hundred independent simulated annealing runs with different random initial configuration and different cooling schedules from a temperature $T_{\rm high}$ (of between 50 and 100 K) to $T_{\rm low}<0.1$ K are carried out for each energy minimization performed. Averaging the low-$T$ configurations within a simulation yielded the lowest-energy configuration for that run. The five lowest energies found among the 100 runs are typically within a few meV. The lowest of these is further refined by local minimization to an accuracy of better than 0.1 meV. An independent program for minimization uses $10^{5}$ trials with randomly chosen initial positions and an adaptive gradient search. The lowest twenty energies obtained from these trials agree to better than 0.1 meV, and these results (their energies and positions) agree with the annealing results.

The number $M$ is chosen to be $S$ fewer than the number of atoms that would fully occupy the sphere ($M=1+\sum_{i=1}^{m}n_{i}-S)$, which allows the BaF molecule to substitute for $S$ Ar atoms. The total number of Ar atoms (including the fixed outer shell) in the simulations is $N=1+\sum_{i=1}^{n}n_{i}-S$.

The energy being minimized is the sum of pairwise interactions:

Here, $\vec{r}_{0}$ is the position of the BaF molecule, defined as the geometric average of $\vec{r}_{\rm Ba}$ and $\vec{r}_{\rm F}$ (i.e., the centre point between the Ba and F nuclei), $\vec{r}_{i}$ is the position of the $i^{\rm th}$ Ar nucleus, and $\theta_{i}$ is the angle between the internuclear axis ($\vec{r}_{\rm F}-\vec{r}_{0}$) and the $i^{\rm th}$ Ar atom ($\vec{r}_{i}-\vec{r}_{0}$).

We have recently calculated [3] the BaF-Ar interaction energy, $V_{\rm BaF-Ar}(r,\theta)$, by calculating the ground-state energies of the BaF-Ar triatomic system for 1386 values of $r$ and $\theta$ using high-precision all-electron relativistic quantum-mechanical calculations that include correlation and that are extrapolated to the complete basis set limit. These calculations used a scalar-relativistic approach within the framework of a second-order Douglas-Kroll-Hess approach. A fit provided in that work or, alternatively, interpolations and extrapolations provide $V_{\rm BaF-Ar}(r,\theta)$ for intermediate values of $r$ and $\theta$. Uncertainties from this calculation of $V_{\rm BaF-Ar}(r,\theta)$ are also provided in Ref. [3].

The interaction energy between two Ar atoms, $V_{\rm Ar-Ar}^{\rm pairwise}(r)$ is precisely known [12]. To correctly describe an Ar crystal, however, corrections must be included [13] to these pairwise Ar-Ar interactions. The dominant correction is due to the zero-point energy of the Ar atoms, which requires an averaging of $V_{\rm Ar-Ar}^{\rm pairwise}(r)$ over the positional probability distributions that results from the zero-point motion of the two atoms. A second, slightly-smaller correction is due to three-body Ar-Ar-Ar interactions. Without these corrections, a calculation using only the two-body potential predicts an fcc crystal with the incorrect lattice constant and cohesive energy. Following the example of Ref. [14], we compensate for these effects by using a modified potential

Coefficients $\alpha=0.8395$ and $\beta=0.9815$ are chosen to match the experimental Ar fcc cube dimension of $a=5.3118$ Å [7] and cohesion energy per atom of $E_{\rm coh}=80.05$ meV [15].

Four uncertainties in these simulations are investigated. The first is due to the finite size of the cluster used for the calculation. For all simulations, the calculations are repeated with increasing numbers $N$ and $M$ of Ar atoms and the convergence of our results with increasing cluster size provides an estimate of the resulting uncertainty.

A second comes from the uncertainty in our calculated BaF-Ar potentials in Ref. [3]. In that work, we repeat our calculations with increasing basis set sizes: $n\zeta$, with $n=2$ through $5$. We make two extrapolations of our results, one from $n=2$, $3$ and $4$, and the other (more precise one) from $n=3$, $4$ and $5$. Based on comparisons to measured quantities in BaF, Ba, Ba⁺ and Ar, we estimate the uncertainty in our 345 extrapolated results to be one quarter of the difference between these two extrapolations. To determine the effect of these uncertainties on our present cluster simulations, we repeat the simulations with both the 234 and 345 results from Ref. [3]. The uncertainty in our simulations is expected to be one quarter of the difference between the two results.

Thirdly, and most importantly, we investigate the approximation inherent in using the scaled Ar-Ar potential of Eq. (2). To do this, we repeated cluster calculations using $V_{\rm Ar-Ar}^{\rm pairwise}(r)$ in place of Eq. (2). These calculations led to a cohesive energy $E_{\rm coh}$ and Ar-Ar separation $b_{1}$ that differ from measured values by a factor of $\alpha$ and $\beta$, respectively. The difference between the calculations with and without scaling of the Ar-Ar potential, should give the scale for the effect of the approximation used here.

Finally, the fourth uncertainty involves four-atom BaF-Ar-Ar interactions. To estimate this effect, we calculate energies for the BaF-Ar-Ar four-atom system for twenty geometries with separations of between 3 and 7 Å, which covers the most important range of separations for our simulated solids. We compare the 234 extrapolation of this binding energy to the sum of the Ar-Ar binding energy plus the two BaF-Ar contributions (also calculated using a 234 extrapolation). The difference between the full BaF-Ar-Ar calculation and the sum of the two-body contributions is typically less than one percent. It is sometimes positive and sometimes negative, and gets smaller quickly with increasing distances. Therefore, we estimate the net effect due to this four-atom effect to be less than one percent.

Calculations of the local environment of a BaF molecule within an argon matrix are reported. It is found that the molecule strongly prefers to replace four Ar atoms and is aligned along the 111 axis of the argon crystal. The remaining Ar atoms are found to be only slightly displaced from their original fcc crystal positions. The single type of strongly-preferred site, the inability of the molecule to reorient or migrate at a temperature of 4 kelvin and the small perturbation of the rest of the Ar crystal are important features for the planned eEDM measurement by the EDM³ collaboration using matrix-isolated BaF molecules.

This work is supported by the Alfred P. Sloan Foundation, the Gordon and Betty Moore Foundation, the Templeton Foundation in conjunction with the Northwestern Center for Fundamental Physics, the Natural Sciences and Engineering Research Council of Canada and York University. Computations for this work required more than 15 core-years of CPU time and were enabled by support provided by Compute Canada.

The authors report there are no competing interests to declare.