We present a theoretical study of the Zeeman relaxation of the magnetically trappable lowest field seeking state of CrH(X^{6}Σ^{+}) in collisions with ^{3}He. A two dimensional potential energy surface (PES) was calculated with the partially spin-restricted coupled cluster singles, doubles, and non-iterative triples [RCCSD(T)] method. The global minimum was found for the collinear He$\cdots $Cr–H geometry with the well depth of 1143.84 cm^{−1} at *R*_{e} = 4.15 a_{0}. Since the RCCSD(T) calculations revealed a multireference character in the region of the global minimum, we performed additional calculations with the internally contracted multireference configuration interaction with the Davidson correction (*ic*-MRCISD+Q) method. The resulting PES is similar to the RCCSD(T) PES except for the region of the global minimum, where the well depth is 3032 cm^{−1} at *R*_{e} = 3.8 a_{0}. An insight into the character of the complex was gained by means of symmetry-adapted perturbation theory based on unrestricted Kohn-Sham description of the monomers. Close coupling calculations of the Zeeman relaxation show that although the $\Delta MJ=MJ\u2032\u2212MJ$ = −1 and −2 transitions are the dominant contributions to the collisional Zeeman relaxation, $\Delta MJ<\u22122$ transitions cannot be neglected due to the large value of CrH spin-spin constant. The calculated elastic to inelastic cross section ratio is 1600 for the RCCSD(T) PES and 500 for the MRCISD+Q PES, while the estimate from the buffer-gas cooling and magnetic trapping experiment is 9000.

## I. INTRODUCTION

Since its identification in 1937 by Gaydon and Pearse,^{1} the chromium monohydride molecule has become a focus of interest in astrophysics and ultracold chemistry. In the astronomy and astrophysics, CrH was identified in the spectra of S stars^{2} and sunspots.^{3} More importantly, the 1-0 and 0-0 bands of the CrH $A6\Sigma +\u2190X6\Sigma +$ transition serve as markers for L dwarf spectral class and allow for the determination of their effective temperature.^{4,5} The need for modeling of the spectral energy distribution of brown dwarfs drove several theoretical^{6–9} and experimental^{10–12} studies focused on the spectrum and low-lying electronic states of CrH complementing earlier findings.^{13–17} Most recently, Kuzmychov and Berdyugina^{18} investigated the Paschen-Back effect in the CrH molecule and concluded that the CrH lines can serve as a useful signature when detecting stellar magnetic fields on cool brown dwarfs observed with low spectral resolution.

Due to its unique properties, chromium monohydride drew attention also of the ultra-cold molecules research community. Bakker *et al.*^{19} proposed CrH together with MnH as promising candidates for direct cooling via thermalization with a cold inert buffer gas and subsequent trapping in a magnetic field, so called buffer-gas cooling. In order for buffer-gas cooling to be efficient, a favorably high ratio of elastic-to-inelastic collision rates must be achieved, as the inelastic processes lead to trap loss.^{20} Paramagnetic CrH and MnH were believed to meet this criterion because of their large rotational constants and relatively small spin-spin and spin-rotation coupling constants. Indeed, in 2008 both molecules were successfully cooled in ^{3}He buffer-gas and magnetically trapped at the temperature of 0.65 K.^{21}

Modeling of potential energy surfaces (PESs) for interaction of these diatoms with helium must be the first step towards understanding the mechanism of the collisional Zeeman relaxation in buffer-gas cooling and theoretical prediction of the elastic to inelastic cross section ratio. Up until now only the MnH(X^{7}Σ^{+})–He complex has been modeled. Starting from the analytical fit of the PES for the MnH(X ^{7}Σ^{+})–He dimer, Turpin *et al.*^{22} confirmed that the MnH Zeeman relaxation should be dominated by the $\Delta $M_{J} = −1,−2 transitions.^{22,23} Moreover, these authors recognized that the calculated ratio of the elastic to inelastic cross section strongly depends on the accuracy of the PES—a result reasserted by taking into account the hyperfine structure of MnH in the work of Stoecklin and Halvick.^{24}

The main aim of this work is the study of the second molecule buffer-gas-cooled by Stoll *et al.*^{21}—CrH(X^{6}Σ^{+}), its interaction potential with the He atom, and the mechanism of Zeeman relaxation in its collisions with helium buffer gas. We start with the analysis of the potential energy curves of the ground X^{6}Σ^{+} and spin-polarized ^{8}Σ^{+} states of the CrH molecule in Sec. II. Additionally, the dipole moment curve together with vibrational bound states, average diatomic distances, and rotational constants of the ground electronic X^{6}Σ^{+} state are presented. In Sec. III we describe the PES of the CrH(X^{6}Σ^{+})–He dimer and in Sec. IV its bound states for the total angular momentum quantum number *J* = 0 are given. Sec. V is devoted to the scattering calculations involving a diatomic molecule in the ^{6}Σ state: a brief introduction to the underlying close-coupling equations is given in Sec. V A, followed by the analysis of the vital parameters for the buffer gas cooling and magnetic trapping of a X^{6}Σ diatom in Sec. V B and comparison with the experimental results of Stoll *et al.*^{21} in Sec. V C. Section VI concludes the paper.

## II. MRCI POTENTIALS OF CrH AND ITS PROPERTIES

We focus on the ground electronic X^{6}Σ^{+} state of the CrH which originates from the 3*d*^{5}4*s*^{1} Cr valence shell and 1*s* of H atom giving rise to $4s\sigma 23d\sigma 13d\delta 23d\pi 2$ electronic diatom configuration. In our *ab initio* approach to calculate the potential and dipole moment, we used multiconfigurational self-consistent field (MCSCF) to obtain reference orbitals for subsequent internally contracted multireference configuration interaction calculations including explicitly single and double excitations (*ic*-MRCISD). Davidson correction was applied to account for the effects of higher excitations in an approximate manner. The Cr and H atoms were described by all-electron correlation consistent quadruple-zeta basis sets designed for Douglass-Kroll relativistic calculations (aug-cc-pVQZ-DK).^{25,26} A better description of the partial anionic character of the CrH dipole was achieved by adding *spdf* diffuse functions placed on H atom with the following exponents: *s*, 0.007 88, *p*, 0.0283, *d*, 0.063 and *f*, 0.13.

The reference wave function for the MCSCF calculations was obtained from the restricted Hartree-Fock (RHF) calculations for the high-spin case. The first step in the MCSCF calculations was to perform state-averaged calculations for the X^{6}Σ and ^{8}Σ states. This formed a starting point for subsequent single-state MCSCF calculations of the ground electronic states along the dissociation variable *r*, except for the results presented in Figure 1 when we obtained both states. The active space in the MCSCF calculation was composed of 11 orbitals in symmetry *A*_{1}, 3 orbitals of *B*_{1} and *B*_{2} symmetry each, and 1 orbital of *A*_{2} symmetry. We kept 3*s* and 3*p* orbitals correlated and always doubly occupied. All the electronic structure calculations were performed with the MOLPRO suite of programs.^{27}

In Figure 1 we show the diatomic potential for the ground X^{6}Σ^{+} state and for the spin-polarized octet ^{8}Σ^{+} as well. The high-spin state is practically repulsive in this plot in comparison to the ground sextet state. In our calculations the well depth, *D*_{e}, of the X^{6}Σ^{+} state is 18 485.21 cm^{−1} (2.292 eV) at *r*_{e} = 3.111 a_{0}. Our minimum is slightly deeper than the one of 2.11 eV obtained by Dai and Balasubramanian^{17} and close to the one of 2.34 eV obtained by Ghigo *et al.*^{9} The equilibrium distance agrees with the experimental one of 3.1275 a_{0} measured by Bauschlicher *et al.* in 2001.^{6} Using the discrete variable representation (DVR) approach, we calculated vibrational bound states for the CrH(X^{6}Σ^{+}) molecule which are presented in Table I along with average position $\u27e8r\u27e9$ and rotational constants *B*_{v}. The zero-point corrected dissociation energy of CrH(X) is *D*_{0} = 17 626.88 cm^{−1} which is 2.186 eV. Our *D*_{0} value agrees reasonably well with the experimental $D0exp=1.93\xb10.07$ eV of Bauschlicher^{6} and, even slightly better, with the estimation of the dissociation energy of 2.3 eV from the experimental CrH potential curve modeled by Rajamanickam.^{28} The rotational constant for the ground vibrational state of CrH is *B*_{0} = 6.203 cm^{−1}, which compares well to the experimental one of 6.13 cm^{−1} given by Huber and Herzberg.^{29}

v
. | E_{v}/eV
. | $\u27e8r\u27e9$/a_{0}
. | B_{v}/cm^{−1}
. |
---|---|---|---|

0 | −2.1855 | 3.150 | 6.203 |

1 | −1.9803 | 3.235 | 6.014 |

2 | −1.7858 | 3.325 | 5.821 |

3 | −1.6005 | 3.419 | 5.628 |

4 | −1.4240 | 3.517 | 5.436 |

5 | −1.2557 | 3.620 | 5.246 |

6 | −1.0955 | 3.728 | 5.054 |

7 | −0.9433 | 3.843 | 4.859 |

8 | −0.7992 | 3.967 | 4.660 |

9 | −0.6636 | 4.102 | 4.451 |

10 | −0.5368 | 4.253 | 4.230 |

11 | −0.4195 | 4.426 | 3.990 |

12 | −0.3127 | 4.632 | 3.721 |

13 | −0.2180 | 4.901 | 3.392 |

14 | −0.1387 | 5.271 | 2.982 |

15 | −0.0754 | 5.745 | 2.550 |

16 | −0.0316 | 6.581 | 1.925 |

17 | −0.0069 | 8.000 | 1.264 |

v
. | E_{v}/eV
. | $\u27e8r\u27e9$/a_{0}
. | B_{v}/cm^{−1}
. |
---|---|---|---|

0 | −2.1855 | 3.150 | 6.203 |

1 | −1.9803 | 3.235 | 6.014 |

2 | −1.7858 | 3.325 | 5.821 |

3 | −1.6005 | 3.419 | 5.628 |

4 | −1.4240 | 3.517 | 5.436 |

5 | −1.2557 | 3.620 | 5.246 |

6 | −1.0955 | 3.728 | 5.054 |

7 | −0.9433 | 3.843 | 4.859 |

8 | −0.7992 | 3.967 | 4.660 |

9 | −0.6636 | 4.102 | 4.451 |

10 | −0.5368 | 4.253 | 4.230 |

11 | −0.4195 | 4.426 | 3.990 |

12 | −0.3127 | 4.632 | 3.721 |

13 | −0.2180 | 4.901 | 3.392 |

14 | −0.1387 | 5.271 | 2.982 |

15 | −0.0754 | 5.745 | 2.550 |

16 | −0.0316 | 6.581 | 1.925 |

17 | −0.0069 | 8.000 | 1.264 |

The dipole moment of the CrH(X^{6}Σ^{+}) state is large, 1.382 a.u. (3.512 D), at the equilibrium position. We calculated the dipole moment function as the expectation value of the dipole moment operator using the *ic*-MRCISD wave function in the default procedure implemented in MOLPRO program.^{27} Figure 2 shows the dipole moment curve compared to results of Ghigo *et al.*^{9} who applied the multistate CASPT2 method based on reference wave functions from the state-averaged CASSCF with 16 molecular orbitals in the active space. We observe a good agreement between our *ic*-MRCISD and CASPT2 results for the dipole moment function.

## III. SPIN-RESTRICTED COUPLED CLUSTER SINGLES, DOUBLES, AND NON-ITERATIVE TRIPLES (RCCSD(T)) POTENTIAL ENERGY SURFACE FOR THE He–CrH(X^{6}Σ^{+}) COMPLEX

The potential energy surface for the He–CrH(X) complex was calculated at the coupled cluster singles, doubles, and non-iterative triples, RCCSD(T), level of theory. The reference wave function for the RCCSD(T) calculations was obtained by first performing two-state averaged CASSCF calculations, saving pseudo-canonical orbitals for the ground state, and then starting RHF calculations. We used aug-cc-pwCVQZ-DK basis set for Cr atom and aug-cc-pVQZ-DK basis sets for H and He. In RCCSD(T) calculations 1*s*, 2*s*, 2*p*_{y}, 2*p*_{z}, and 2*p*_{x} core orbitals of Cr were frozen. The interaction energies were obtained with the reference to the asymptotic total energy at *R* = 100 a_{0} and were not corrected for the basis set superposition error (BSSE).

The contour plot of the He–CrH(X) potential is shown in Figure 3. The global minimum with a well depth of *D*_{e} = 1143.84 cm^{−1} is located at *R*_{e} = 4.15 a_{0} for $\theta e=180\xb0$ which corresponds to the collinear He$\u22c5\u2063\u22c5\u2063\u22c5$Cr–H geometry. There is an additional local minimum at He$\u22c5\u2063\u22c5\u2063\u22c5$H–Cr linear geometry ($\theta =0\xb0$) located at *R* = 9.6 a_{0} with a well depth of 12.75 cm^{−1}. A T-shaped saddle point occurs at *R* = 9.05 a_{0} and $\theta =94\xb0$ with a barrier height of −6.6 cm.^{1} The T_{1} diagnostic was on the order of 0.11 and D_{1} diagnostic 0.44.

Because of the alarming D_{1} and T_{1} diagnostics, the accuracy of the RCCSD(T) PES demanded a more careful examination. Therefore, we also performed MRCISD+Q calculations of the whole surface (see supplementary material for details). The multireference approach confirmed the shape and main features of the PES. However, whereas the far-off shallow minimum has a similar depth of 12 cm^{−1}, the close-in deep minimum is almost three times as deep as the CCSD(T) one: ca. 3032 cm^{−1} before and 3002 cm^{−1} after counterpoise correction.^{30} Apparently, the multireference character of the complex manifests itself at the Cr end, for the collinear He–Cr–H geometry, and much less for the Cr–H–He form. Further confirmation was provided by the sensitivity of the global minimum to the selection of the active space in the case of MRCI, and to the choice of the frozen core orbitals in the case of RCCSD(T). Additional evidence of problems with single-reference approach is a steep, almost discontinuous drop within the global minimum well of the RCCSD(T) potential.

Due to the well-known size inconsistency problems of the MRCI method as well as difficulties in selecting the active space, we opt for the single reference RCCSD(T) approach. Both the MRCISD+Q PES and results of the subsequent scattering calculations are given in the supplementary material. One should recognize the challenge in the proper description of the global minimum region and need for further investigation.

The anisotropy of the He–CrH(*X*) PES is unusual for a van der Waals complex. The striking difference between the well depth of the global and local minima amounts to more than 1144 cm^{−1}. The reason behind such behaviour lies in the polar Cr^{+}H^{−} charge separation, as indicated by the pronounced dipole moment of the molecule. In order to show this, we calculated MCSCF/*ic*-MRCISD+Q(Davidson) and RCCSD(T) potential curves for He–H^{−}(^{1}S) presented in Figure 4 and He–Cr^{+}(^{6}S) presented on Figure 5. We expect those model systems to qualitatively reflect the contrasting character of the minima present in the He–CrH complex.

The minimum of the He–H^{−} system is located at *R* = 13 a_{0} and approximately 6 cm^{−1} deep in *ic*-MRCISD+Q calculations, and 4 cm^{−1} in RCCSD(T). The T_{1} and D_{1} diagnostics were approximately 0.01 indicating a single-reference character of this system. The minimum for He–Cr^{+} is present at *R* = 4.34 a_{0} with a well depth of 368.2 cm^{−1} at the *ic*-MRCISD+Q level. Our RCCSD(T) result, *R* = 4.33 a_{0}, *D*_{e} = 364.6 cm^{−1}, is in good agreement with the *ic*-MRCI+Q results of Partridge and Bauschlicher, *R* = 4.44 a_{0}, *D*_{e} = 364 cm^{−1}.^{31} The T_{1} diagnostic for this system was in the range from 0.015 to 0.019 indicating that the multireference character is slightly increased, but an order of magnitude less than for He–CrH system.

Both global and local minima of the He–CrH complex occur at a shorter distance and are two times (for local minimum) or three times (for global minimum) deeper than the minima of their model counterparts, He–Cr^{+} and He–H^{−}, respectively. This shows the substantial stabilizing role of the dispersion interaction in the He–CrH system. Nevertheless, the relative difference between the potentials of charge-separated systems, He–Cr^{+} and He–H^{−}, resembles the one observed for collinear ($\theta =0\xb0$ and 180°) cuts of the He–CrH PES.

Better insight can be derived from a detailed analysis of the SAPT(UKS) energy components, where SAPT(UKS) stands for symmetry-adapted perturbation theory based on unrestriced Kohn-Sham description of the monomers.^{32} The SAPT(UKS) calculations have been done in the Douglas-Kroll Hamiltonian (DKH) framework with the PBE0 xc functional.^{33,34} The aug-cc-pVQZ basis set with additional diffuse functions on the H atom has been chosen, as described in Sec. II.

The SAPT energy components reveal the so called “exchange cavity” on the chromium side—the reduced Pauli exchange repulsion accompanied by considerable induction due to the dipole moment of CrH. A similar effect has recently been observed for He–BeO($\Sigma +1$).^{35} We visualized the “exchange cavity” by plotting the *E*_{int} = 0 isosurface of the RCCSD(T) interaction potential in Fig. 6. The same effect can be illustrated by plotting the first order SAPT energy components, i.e., electrostatic and exchange energies, $Eelst(1)$ and $Eexch(1)$, respectively. In contrast, the MnH molecule has much lesser dipole moment and does not exhibit such a peculiar anisotropy in the He–MnH($\Sigma +7$) complex—see Figure S5 in the supplementary material^{23} for comparison of the first-order SAPT energy components for He–MnH and He–CrH.

Finally, we compare the RCCSD(T) results with the hybrid RHF+disp hybrid approach. The latter combines HF supermolecular energies with SAPT(UKS) dispersion contribution. In Figure 7 we show radial cuts that compare RCCSD(T) and RHF+disp long-range interaction energies for $\theta =0\xb0$, 90°, 140°, and 180°. The agreement is good, especially for $\theta =\u20090\xb0$ and 90°. For $\theta =\u2009140\xb0$ and 180° the RHF+disp method predicts only shallow minima a few wavenumbers deep, in contrast to both MRCI and RCCSD(T) results. We attribute this discrepancy to the multireference character of the complex in the vicinity of the Cr in CrH.

To perform bound states and scattering calculations, we constructed an analytical representation of potential energy surfaces. The PES model is a sum of a short-range function and a long-range function seamlessly connected by a switching function around $R=11a0$. The short-range function was determined by fitting the angular dependence of the *ab initio* interaction energies for each *R* distance to Legendre polynomial expansion retaining 17 expansion terms (including the isotropic one). The resulting radial expansion coefficients are then interpolated with the Reproducing Kernel Hilbert Space (RKHS) method of Ho and Rabitz.^{36} The long-range function is a sum of *R*^{−n} terms (*n* = 6,7,8,9), each multiplied by the appropriate coefficient and angular function, representing the long-range dispersion and induction interaction energies.

The long-range part of the potentials is especially important in the ultracold collisional processes. The first two leading van der Waals dispersion coefficients determined from the long-range fits are *C*_{60} = −3.0934 × 10^{7} and *C*_{62} = −3.1833 × 10^{6} cm$a06\u22121$ for the MRCISD PES and *C*_{60} = −3.0143 × 10^{7} and *C*_{62} = −6.0287 × 10^{5} cm$a06\u22121$ for the MRCISD PES. One can see that for the two PESs the isotropic *C*_{60} dispersion coefficients are similar, but the first anisotropic one, *C*_{62}, is significantly larger for the MRCISD potential. The discrepancy in *C*_{62} is not surprising, as the MRCI PES is much more attractive near 180° than the RCCSD(T) one. The difference of the long-range anisotropy will affect the scattering resonances in the cross sections. Fortran routines for RCCSD(T) and MRCISD+Q He–CrH PESs are available for download from Zenodo repository.^{37}

## IV. BOUND STATES OF THE He–CrH(X$\Sigma 6$^{+}) VAN DER WALLS COMPLEX

We calculated bound states supported by the potentials obtained in this work for total angular momentum quantum number *J* = 0. The approximation of CrH as a closed shell molecule was used (no spin-splitting). We applied the collocation method on a grid composed of 30 angular points corresponding to Gauss-Legendre weights and 200 radial points spread between 2.5 and 30 a_{0}. The reduced mass of the complex $\mu A\u2212BC=3.721\u200929$ a.m.u. was obtained taking atomic masses of the most abundant isotopes. The rotational constant of CrH(X) was taken as reported in Table III.

There are nine bound states supported by the RCCSD(T) potential and 14 states by the MRCISD one. The wave functions are mostly located in the collinear He$\cdots $Cr–H minimum and are shown in Fig. S6 in the supplementary material. The bound state energies, their rovibrational assignments, average distance, angle values, and rotational constants are shown in Table II. For the RCCSD(T) PES the *D*_{0} dissociation energy for the He–CrH system is 798 cm^{−1} and the rotational constant of the complex is 0.87 cm^{−1}, whereas for MRCISD PES they are 2146 cm^{−1} and 1.09 cm^{−1}, respectively. The RCCSD(T) PES supports up to six of pure vibrational quanta ($\nu s$), while MRCISD PES up to eight. These two sets of bound states obtained on two different PESs will provide a comparison for future experimental efforts.

RCCSD(T) PES . | MRCISD PES . | ||||||||
---|---|---|---|---|---|---|---|---|---|

$(J,\nu s,\nu b)$ . | E_{v}
. | $\u27e8R\u27e9$/a_{0}
. | $\u27e8\theta \u27e9$ . | B_{v}
. | $(J,\nu s,\nu b)$ . | E_{v}
. | $\u27e8R\u27e9$/a_{0}
. | $\u27e8\theta \u27e9$ . | B_{v}
. |

(0, 1, 0) | −586.02 | 4.394 | 165 | 0.867 | (0, 1, 0) | −1722.15 | 3.973 | 170 | 1.048 |

(0, 2, 0) | −361.43 | 4.601 | 164 | 0.807 | (0, 2, 0) | −1325.13 | 4.097 | 169 | 1.001 |

(0, 0, 2) | −337.33 | 4.451 | 154 | 0.827 | (0, 0, 2) | −1056.00 | 3.902 | 161 | 1.071 |

(0, 3, 0) | −186.32 | 5.017 | 164 | 0.698 | (0, 3, 0) | −961.16 | 4.249 | 169 | 0.946 |

(0, 1, 2) | −102.12 | 4.489 | 155 | 0.834 | (0, 1, 2) | −671.66 | 4.026 | 161 | 1.023 |

(0, 4, 0) | −66.42 | 5.710 | 160 | 0.554 | (0, 4, 0) | −638.51 | 4.448 | 169 | 0.877 |

(0, 5, 0) | −10.63 | 7.452 | 142 | 0.333 | (0, 5, 0) | −366.90 | 4.725 | 168 | 0.791 |

(0, 6, 0) | −2.03 | 10.915 | 89 | 0.162 | (0, 2, 2) | −325.04 | 4.185 | 161 | 0.964 |

(0, 6, 0) | −159.95 | 5.173 | 166 | 0.670 | |||||

(0, 0, 4) | −83.54 | 3.934 | 156 | 1.055 | |||||

(0, 7, 0) | −37.35 | 5.757 | 161 | 0.568 | |||||

(0, 3, 2) | −33.59 | 4.834 | 160 | 0.782 | |||||

(0, 8, 0) | −3.26 | 9.872 | 106 | 0.185 |

RCCSD(T) PES . | MRCISD PES . | ||||||||
---|---|---|---|---|---|---|---|---|---|

$(J,\nu s,\nu b)$ . | E_{v}
. | $\u27e8R\u27e9$/a_{0}
. | $\u27e8\theta \u27e9$ . | B_{v}
. | $(J,\nu s,\nu b)$ . | E_{v}
. | $\u27e8R\u27e9$/a_{0}
. | $\u27e8\theta \u27e9$ . | B_{v}
. |

(0, 1, 0) | −586.02 | 4.394 | 165 | 0.867 | (0, 1, 0) | −1722.15 | 3.973 | 170 | 1.048 |

(0, 2, 0) | −361.43 | 4.601 | 164 | 0.807 | (0, 2, 0) | −1325.13 | 4.097 | 169 | 1.001 |

(0, 0, 2) | −337.33 | 4.451 | 154 | 0.827 | (0, 0, 2) | −1056.00 | 3.902 | 161 | 1.071 |

(0, 3, 0) | −186.32 | 5.017 | 164 | 0.698 | (0, 3, 0) | −961.16 | 4.249 | 169 | 0.946 |

(0, 1, 2) | −102.12 | 4.489 | 155 | 0.834 | (0, 1, 2) | −671.66 | 4.026 | 161 | 1.023 |

(0, 4, 0) | −66.42 | 5.710 | 160 | 0.554 | (0, 4, 0) | −638.51 | 4.448 | 169 | 0.877 |

(0, 5, 0) | −10.63 | 7.452 | 142 | 0.333 | (0, 5, 0) | −366.90 | 4.725 | 168 | 0.791 |

(0, 6, 0) | −2.03 | 10.915 | 89 | 0.162 | (0, 2, 2) | −325.04 | 4.185 | 161 | 0.964 |

(0, 6, 0) | −159.95 | 5.173 | 166 | 0.670 | |||||

(0, 0, 4) | −83.54 | 3.934 | 156 | 1.055 | |||||

(0, 7, 0) | −37.35 | 5.757 | 161 | 0.568 | |||||

(0, 3, 2) | −33.59 | 4.834 | 160 | 0.782 | |||||

(0, 8, 0) | −3.26 | 9.872 | 106 | 0.185 |

## V. SCATTERING CALCULATIONS

### A. Preliminaries

In this part of the manuscript we model the buffer-gas cooling experiment of Meijer following our previous study of the ^{3}He–MnH(X$\Sigma +7$) collisions.^{22,23} We also briefly investigate the details of the Zeeman relaxation mechanism for collisions involving a diatomic molecule in a $\Sigma 6$ state and determine which diatomic constant is critical in the cooling and trapping process. Finally, we calculate the collisional Zeeman relaxation cross sections of the magnetically trappable lowest field seeking state *M*_{J} = 5/2 of CrH(X^{6}Σ^{+}) belonging to the sextet associated with *N* = 0, *J* = *S* = 5/2, where *N*, *S*, and *J* designate the quantum numbers associated with the rotational, the electronic spin, and the total angular momenta of CrH, without nuclear spin, while *M*_{N}, *M*_{S}, and *M*_{J} are the quantum numbers associated with their projections along the *z* space-fixed axis.

The close Coupling equations which we solve are derived as a simple extension of those developed by Krems and Dalgarno for collisions between a structureless atom and the $\Sigma 3$ molecules.^{38} They were presented in our previous work; therefore, we will only emphasize the differences as well as similarities between MnH(X^{7}Σ^{+}) and CrH(X^{6}Σ^{+}).

The rigid rotor Hamiltonian of the colliding system takes the form

where $L\u21922$ is the angular momentum associated with the intermolecular coordinate $R\u2192$ and $H^diatom$ is the Hamiltonian for the CrH(X^{6}Σ^{+}) monomer. In order to compare the two systems, we reported in Table III the parameters of the diatomic Hamiltonians of CrH and MnH as well as their dipole moments, and reduced masses in collisions with ^{3}He. As one can infer from Table III, the hyperfine structure of CrH is negligible as the rotational constant of CrH is large (B_{0} = 6.131 cm^{−1}) whereas the spin-rotation constant ($\gamma $ = 5.033 × 10^{−2} cm^{−1}), which is small, is fifty times larger than the largest hyperfine constant (b_{F}(Cr) = −1.16 × 10^{−3} cm^{−1}). Moreover, for the similar system ^{3}He–MnH, we showed^{24} that the hyperfine structure does not affect the ratio of elastic to inelastic cross section in the energy domain of the experiment of Meijer *et al.*^{21} Hence, we retain only the following terms of the Hamiltonian of Corkery *et al.*:^{39,41}

where *B*, *D*, $\gamma $, $\lambda $, $\gamma D$, and $\lambda D$ are the rotational constant, the first centrifugal distortion constant, the spin-rotation and spin-spin interaction constants, and the centrifugal distortion corrections to spin-rotation and spin-spin constants, respectively. The remaining constants are *θ*, which is the fourth order spin-spin splitting constant and the *g*_{S} factor of the diatomic, the values of which are given in Ref. 39.

. | CrH(^{6}Σ^{+})
. | MnH(^{7}Σ^{+})
. |
---|---|---|

B | 6.131 | 5.605 |

D | 3.945 × 10^{−4} | 3.0585 × 10^{−4} |

$\gamma $ | 5.033 × 10^{−2} | 3.073 × 10^{−2} |

$\gamma D$ | 3.451 × 10^{−6} | 0.0 |

$\lambda $ | 2.328 × 10^{−1} | −4.053 × 10^{−3} |

$\lambda D$ | 9.831 × 10^{−6} | 6.0 × 10^{−6} |

$\theta $ | −7.73 × 10^{−1} | 0.0 |

g_{S} | 6.677 × 10^{−2} | $\u2026$ |

b_{F} | −1.16 × 10^{−3} | 9.28 × 10^{−3} |

$\mu $ | 2.8534 | 2.8617 |

. | CrH(^{6}Σ^{+})
. | MnH(^{7}Σ^{+})
. |
---|---|---|

B | 6.131 | 5.605 |

D | 3.945 × 10^{−4} | 3.0585 × 10^{−4} |

$\gamma $ | 5.033 × 10^{−2} | 3.073 × 10^{−2} |

$\gamma D$ | 3.451 × 10^{−6} | 0.0 |

$\lambda $ | 2.328 × 10^{−1} | −4.053 × 10^{−3} |

$\lambda D$ | 9.831 × 10^{−6} | 6.0 × 10^{−6} |

$\theta $ | −7.73 × 10^{−1} | 0.0 |

g_{S} | 6.677 × 10^{−2} | $\u2026$ |

b_{F} | −1.16 × 10^{−3} | 9.28 × 10^{−3} |

$\mu $ | 2.8534 | 2.8617 |

The matrix elements of this Hamiltonian in the uncoupled basis set $\phi i=|NMN\u27e9|SMS\u27e9$ are given in Ref. 22. Below we present the expressions for the matrix elements of the centrifugal distortion corrections to the spin-rotation and spin-spin operators and those for the fourth order spin-spin operator which were not used in our previous work,

with $\alpha \xb1(a,b)=a(a+1)\u2212b(b\xb11)$, and

where *q* = *M*_{N′} −*M*_{N} = *M*_{S} −*M*_{S′}, while

The diagonalization of the diatomic Hamiltonian (2) $[CH^DiatomC\u22121]\alpha \beta =\xi \alpha \delta \alpha \beta $ in an uncoupled basis set $\varphi i=\chi \nu ,N(r)|NMN\u27e9|SMS\u27e9$ gives the diatomic energies.

Each energy level $\xi \alpha $ of the diatomic molecule is associated with a single value of *M*_{J} = *M*_{N} + *M*_{S} denoted $MJ(\alpha )$. For a given value of the projection of the total angular momentum along the direction of the *z* space-fixed axis, denoted *M*_{T}, and for a given *M*_{J}($\alpha $), the projection of the relative angular momentum *L* along the *z* space-fixed axis is then simply $ML=MT\u2212MJ(\alpha )$. The basis set describing the collision process is then obtained by adding the possible values of the quantum number *L* for each value of $\alpha $. This basis set is denoted by the quantum numbers $\alpha $, *M*_{L}, and *L*. The close coupling equations which have to be solved take the form

such as demonstrated in Ref. 38. We used the following parameters to perform these calculations. First, as we are interested in the low collision energy range, the Zeeman relaxation cross sections are calculated for the basis set associated with the single value *M*_{T} = *M*_{J} = 5/2 of the *z* space-fixed projection of the total angular momentum and for the positive parity of the system which includes the *s* wave. For the sake of numerical simplicity the calculations were performed for a field value of 1 G which is large enough to remove the degeneracy of the *N* = 0 multiplet, yet small enough not to change the results. Because of the large well depth of the RCCSD(T) PES and the relatively small reduced mass of the system, the scattering basis set had to include 9 values of the rotational quantum number *N* and 12 for the maximum value of the relative angular momentum quantum number *L*. The size of the matrices, which we had to propagate, was unusually large for such low collision energy calculations. Additionally, we verified the convergence of the Zeeman cross sections as a function of the radial propagation parameters.

### B. Zeeman relaxation mechanism

In our paper dedicated to He–MnH, we studied in detail the Zeeman transition mechanism in the case of a $\Sigma 7$ molecule. Our findings should also hold for collisions involving a $\Sigma 6$ molecule, as the main terms of the diatomic Hamiltonian are the same. For $\Sigma 7$ molecules we found that the Zeeman transitions follow an indirect mechanism due to the spin-spin interaction as they are mediated by the rotational level *N*^{′} = 2 and that the corresponding Zeeman transitions cross sections decrease in magnitude when $|\Delta MJ|$ increases. We showed that it is due to the spin-spin operator which couples *N* with $N\xb12$ and that collisional Zeeman relaxation for a $\Sigma 7$ molecule is dominated by the transitions $\Delta MJ$ = $MJ\u2032\u2212MJ$ = −1 and −2 in agreement with the numerical simulation of the MnH buffer gas cooling experiment by Stoll.^{42} The Zeeman transition cross sections are split into pairs with decreasing magnitude as $|\Delta MJ|$ increases. This is indeed what is also observed in Fig. 8. However, the splitting of the pairs of transitions cross sections is a lot smaller than for He–MnH. This is attributed to the large value of the spin-spin constant which is 2 orders of magnitude larger for CrH than for MnH. The Zeeman transitions cross sections were also checked to follow the expected $E\Delta MJ$ and $E[\Delta MJ+1]$ threshold law as demonstrated by Krems and Dalgarno^{43} for the even and odd $\Delta MJ$ transitions, respectively.

In Fig. 9 the two parity components (the parity is equal to (−1)^{N+l}) of the Zeeman relaxation and elastic cross sections for the *M*_{J} = 5/2 state of CrH are presented for a field of 1 G. This figure shows several resonances all identified as shape or orbiting resonances.^{44} This is in contrast with the He–MnH system which exhibited only one resonance in the same energy domain. This is due to the fact that the He–CrH potential is more attractive than He–MnH both in the short and long-range regions. The ratio of the elastic to the inelastic cross section which is the limiting factor for the cooling and trapping process is presented in Fig. 10. It clearly reaches several minima associated with each of these resonances. This aspect will be discussed in the next paragraph dedicated to the comparison with the experimental data.

### C. Comparison with experiment

An important point to stress before performing this comparison is related to the accuracy of the PES. As discussed in Section III, there is a significant difference in the well depths and bond energies predicted by MRCI and CCSD(T) calculations. At ultracold temperatures only the long-range region of the PES is involved in the collision process. Although the isotropic part of the long-range regions of both PESs is similar, they differ in the description of the long-range anisotropy (see Section III for the comparison of the *C*_{6l} coefficients). One should note that the CCSD(T) PES is expected to be more accurate in that region than the MRCISD PES. When one moves to the cold and thermal regime, the region of the potential well should also have a strong influence on the cross sections, thus a significant discrepancy between the results obtained with the two PESs is expected.

For a collision energy of *E* = 0.45 cm^{−1} we find a value of $\sigma \u2248340$ Å^{2} for the elastic cross section at zero field in the middle of the trap. This compares reasonably well with the experimental evaluation: 90 $\u2264$ $\sigma exp\u2264$ 190 Å^{2}. For the same values of the collision energy, we also calculate the ratio of the elastic to inelastic cross section which has to be large in order to allow the cooling and trapping processes. The calculated ratio for this energy is 1600 while its experimental estimate is 9000. The agreement between theory and experiment should be regarded as satisfactory considering the accuracy of both the theoretical and experimental results. The experimental ratio was deduced by Stoll^{42} from trajectory simulations of CrH in the buffer gas experiment. They compared the measured lifetime to the time dependence of the density of molecules in the center of the trapping region obtained from a simulation. The possible source of an error in their model was the approximation of the inelastic collision cross section simply with a step function. The use of a more realistic representation could certainly change the estimated experimental ratio.

On the theoretical side, the accuracy of scattering calculations strongly depends on the underlying PES. Therefore, it is interesting to compare the RCCSD(T)- and MRCI-based results (see supplementary material). The value of the elastic cross section obtained using the MRCI PES at the collision energy *E* = 0.45 cm^{−1} is 388 Å^{2} which is close to the RCCSD(T) value of 340 Å^{2}. In general, the profile of elastic cross sections as a function of collision energy is similar for both PESs, as shown in Fig. 8 and Fig. S2 of the supplementary material. The elastic to inelastic cross section ratio also shows analogous behavior for both MRCI and RCCSD(T): a large value at the lowest energy and a decrease until a plateau, interspersed by minima corresponding to resonances, is reached. Yet, the magnitudes of the ratios as well as the location of resonances are different. This shows that the relaxation cross sections in the energy range of the plateau are affected by differences in the long-range anisotropy and in potential well region: the more attractive character of the MRCI PES makes the resonances occur at lower collision energies. The calculations based on the MRCISD+Q PES give a ratio of 500 for a collision energy of *E* = 0.45 cm^{−1} which deviates by a factor of three from the RCCSD(T)-based result and by a factor of twenty from the experimental values.

## VI. CONCLUSIONS

We presented a complete theoretical study of the buffer gas cooling experiment of CrH(X^{6}Σ^{+}) by Meijer and compared our results to those obtained previously for MnH(X$\Sigma +7$).^{22–24} We examined the properties of the CrH molecule in the ground X^{6}Σ state and its interaction potential with He. Our *ic*-MRCISD+Q calculations of the CrH equilibrium distance, dissociation energy corrected for the zero-point energy, rotational constant, and dipole moment remain in good agreement with both the experimental^{6,28,29} and previous theoretical full second-order CI (SOCI)^{17} and CASPT2^{9} results.

The PES for the CrH–He complex features strong anisotropy with two minima for the two collinear forms: the global one He–Cr–H on the order of thousands of wavenumbers deep, and the local one He–H–Cr, ca. 12 cm^{−1} deep. The latter is a typical van der Waals well, and is accurately reproduced at the CCSD(T) level of theory. The former, however, exhibits binding on a par with a chemical bond. Combination of CCSD(T) and MRCI treatments exposed the multireference character: significant T_{1} diagnostics at the CC level of theory and large discrepancy of the values of the well depth, from ca. 1000 cm^{−1} for CCSD(T) to ca. 3000 cm^{−1} for MRCI. In addition, the scattering calculations carried out on the MRCISD+Q PES underestimated the ratio of elastic to inelastic cross sections. It is clear that a more elaborate MRCI treatment is warranted.

The origin of the PES anisotropy is traced to the dipole-induced-dipole attraction of polar Cr^{+}H^{−} and helium, and the concurrent “exchange cavity” effect. The latter means an electronic deficiency along the Cr–H molecular axis at the Cr end leading to dramatic reduction of Pauli repulsion, so that He may be pulled into close proximity of Cr. A similar effect was observed in the case of the He–BeO dimer.^{35} It should be stressed that the magnitude of the CrH–He interaction qualifies the complex as one of the neutral He-containing molecules (see, e.g., Ref. 46 and additional references therein). For a more complete picture we report the bound states supported by RCCSD(T) PES for *J* = 0, but they provide a qualitative picture only, as the CCSD(T) is only qualitatively reliable at the Cr end (see above).

In contrast to CrH(X^{6}Σ^{+})–He, the MnH(X$\Sigma +7$)–He complex lacks the “exchange cavity” effect and has a typical van der Waals character with a shallow well of ca. −13 cm^{−1}.^{23} Surprisingly, we have found that the collisional Zeeman relaxation mechanism is essentially the same for both molecules, and that the critical diatomic parameters for the buffer gas cooling and magnetic trapping in both cases are the spin-spin and rotational constants, as collisional Zeeman relaxation is controlled by the spin-spin operator. The collisional Zeeman relaxation is dominated by the $\Delta MJ=MJ\u2032\u2212MJ$ = −1 and −2 transitions, but the large value of spin-spin constant of CrH (which is two orders of magnitude bigger than that of MnH) gives also appreciable weight to larger $|\Delta MJ|$ transitions. Both elastic cross section and ratio of the elastic to inelastic cross section calculated with the RCCSD(T) PES are on the same order of magnitude as those experimentally measured. The MRCI approach shows larger discrepancy with respect to the experiment.

This qualitative agreement between experiment and theory indicates that the efficiency of the buffer gas cooling and trapping experiment will be theoretically predictable for similar paramagnetic molecules. As discussed in Sec. V C, the differences between theory and experiment could be attributed to the simplified model of inelastic cross section used in the experiment. Another plausible explanation is the formation of a metastable He–CrH complex which can survive between collisions. Indeed, the density of 10^{6} cm^{−3} reported in the experiment^{21} yields a mean free path which is long enough for that to occur. As shown by Brahms *et al.* in the study of AgHe,^{47} formation and dynamics of van der Waals molecules in buffer-gas traps is a complex event which entails collisions forming larger clusters, with two or more He atoms—the latter possibility may be particularly relevant here as He–CrH makes a strongly bound species. *Ab initio* modeling of the structure and scattering of the many-body system makes another theoretical challenge for future work.

To summarize, our study confirms that the limiting factor for the cooling and trapping of molecules like MnH or CrH in this temperature domain is the presence of low energy resonances for which the elastic to inelastic cross sections ratio is the lowest. Possible improvement in buffer gas cooling of these molecules will be pursued in future calculations by considering a combined action of electric and magnetic fields on these resonances. We hope that the present work will stimulate further studies, both experimentally and theoretically.

## SUPPLEMENTARY MATERIAL

See supplementary material for the details of calculations of MRCISD+Q PES and corresponding scattering calculations, comparison of anisotropy of the He–CrH and He–MnH PESs, and contour plots of selected bound state wavefunctions supported by RCCSD(T) PES. The Fortran routines for RCCSD(T) and MRCISD PESs are included as a supplementary PES.tar file and in Zenodo repository.^{37}

## ACKNOWLEDGMENTS

We acknowledge computational resources of Deepthought Supercomputer at the University of Maryland. M.H. was supported by the Polish Ministry of Science and Higher Education (Grant No. 2014/15/N/ST4/02179). G.C. was supported by the National Science Foundation (US), Grant No. CHE-1152474. G.C. is also a beneficiary of the MISTRZ Academic Grant for Professors. J. K. acknowledges financial support from U. S. National Science Foundation Grant No. CHE-1213332 to M. H. Alexander.

## REFERENCES

*ab initio*programs written by

^{6}Σ

^{+}) by

^{3}He atoms