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Efficient method for quantum calculations of molecule - molecule scattering properties in a magnetic field

We show that the cross sections for molecule - molecule collisions in the presence of an external field can be computed efficiently using a total angular momentum basis, defined either in the body-fixed frame or in the space-fixed coordinate system. This method allows for computations with much larger basis sets than previously possible. We present calculations for 15NH - 15NH collisions in a magnetic field. Our results support the conclusion of the previous study that the evaporative cooling of rotationally ground 15NH molecules in a magnetic trap has a prospect of success.

The evaporative cooling experiments rely on elastic collisions, which return the gas to thermal equilibrium after the most energetic particles are removed from the ensemble [30]. If evaporative cooling is performed in a magnetic trap, collisions between molecules in low-fieldseeking Zeeman states can lead to electron spin depolarization, causing trap loss [31][32][33][34].
An empirical rule indicates that evaporative cooling works if elastic collisions occur γ ∼ 100 times more frequently than inelastic collisions [3]. While evaporative cooling is widely used to create ultracold atomic gases [30], the applicability of this technique to molecules in lowfield-seeking states remains an open question. Molecular collisions at low temperatures are determined by strongly anisotropic interactions, which may lead to large inelastic collision rates. In order to predict whether or not evaporative cooling of specific molecules would be possible, it is necessary to determine the cross sections for elastic and inelastic scattering in molecule -molecule collisions in the presence of the trapping fields. Unfortunately, the fully quantum calculation of molecule -molecule scattering properties in external fields remains a daunting task. Selected calculations have been reported for OH( 2 Π) -OH( 2 Π) collisions in an electric field [35], collisions in a magnetic field [36] and NH( 3 Σ) -NH( 3 Σ) collisions in a magnetic field [37]. These calculations are based on the fully uncoupled space-fixed (SF) representation of the scattering wave functions [38,39], which results in prohibitively large basis sets and precludes fully converged computations with realistic intermolecular potentials. The basis set truncation error of such computations is often difficult to estimate.
Recently, Tscherbul and Dalgarno [40] showed that quantum calculations of atommolecule scattering properties in an external field can be performed using an expansion of the scattering wavefunction in eigenfunctions of the total angular momentum J of the collision complex in the body-fixed (BF) coordinate frame [40]. While J is not conserved in the presence of an external field, the total angular momentum representation allows for a much more efficient truncation of the basis set than the uncoupled representation [41]. In the present work, we use the total angular momentum representation to extend the 15 NH - 15 NH scattering calculations of Janssen et al [37] in order to compute the scattering cross sections with a much larger basis set including up to 7 rotational states of 15 NH. The imidogen radical (NH) is currently studied in several experiments aiming at the production of ultracold molecules [17,34,[42][43][44][45]. Our results demonstrate that the method developed in Ref. [40] can be applied to molecule -molecule scattering after a suitable modification to allow for the exchange symmetry of the two-molecule wave function. Our calculations corroborate the conclusions of Janssen et al [37] and remove the uncertainties due to the basis set truncation errors. Our results demonstrate the effect of increasing the basis set on molecule -molecule scattering in a magnetic field at low temperatures and can be used as benchmark for future calculations.

II. THEORY
There are two equivalent formulations of the scattering theory that can be used for the calculations presented here. The body-fixed formulation introduced in Ref. [40] is based on representing the scattering wave function (ψ) of two molecules by an expansion in terms of BF basis functions. This basis set is convenient for evaluating the matrix elements of the intermolecular interaction potential, but leads to complicated expressions for the matrix elements of the angular momentum operator that describes the rotation of the collision complex in the laboratory frame and the molecule -field interaction operators. expansion in terms of eigenfunctions of the total angular momentum operator defined in the space-fixed frame and evaluating the matrix elements of the Hamiltonian directly in the SF basis. The SF and BF basis sets are related by a unitary transformation. The numerical efficacy of these two basis sets is the same and the results obtained with these two basis sets must be identical. We present below both formulations. Calculating the scattering observables with these two different basis sets is usually a good test of the accuracy of the computations.

A. Body-fixed formulation
The Hamiltonian for a non-reactive collision of two 3 Σ molecules A and B may be written in the following form [40] where µ is the reduced mass of the collision complex, R is the separation between the centers of mass of the molecules,l = (Ĵ −N A −Ŝ A −N B −Ŝ B ) is the orbital angular momentum describing the rotational motion of the collision complex,Ĵ is the total angular momentum, N i andŜ i are the rotational angular momentum and electronic spin, respectively, of the molecule i, and the sets of angles (θ i , φ i ) describe the orientation of the molecules with respect to the BF quantization axis z directed alongR. The hat over the symbol is used to denote angular momentum operators and unit vectors.
The Hamiltonian operators describing the separated molecules can be written as followŝ where B e is the rotational constant,Ŝ Z is the Z-component ofŜ i , µ 0 is the Bohr magneton, γ SR is the spin-rotation interaction constant [46], λ SS is the spin-spin interaction constant [46] and [Ŝ ⊗Ŝ] (2) q is a spherical tensor product ofŜ with itself [47]. The SF quantization axis Z is chosen to point along the direction of the magnetic field.
The operator describing the magnetic dipole-dipole interaction between the molecules is given by [39] where g s is the electron g-factor, α is the fine-structure constant and [Ŝ A ⊗Ŝ B ] (2) q is the spherical tensor product ofŜ A andŜ B [47]. Note that g 2 s µ 2 0 ≈ 1.0023 2 e 2 /m 2 e [48]. This factor, approximately equal to one when expressed in atomic units, was omitted from the corresponding expressions in Ref. [39].
The interaction of two molecules with electronic spins S A = S B = 1 gives rise to three potential energy surfaces labeled by S = 0, 1 and 2, whereŜ =Ŝ A +Ŝ B . As in the previous studies of 15 NH-15 NH collisions [37,48,49], we assume that all three spin states of the [50]. The intermolecular interaction potential in Eq. (1) can then be expanded in angular basis functions [40] where Y λ i m (θ i , φ i ) are the spherical harmonics and the parentheses denote the 3j-symbols [47]. This expansion is particularly useful for evaluating the matrix elements of the interaction potential in the BF basis: where Ω, K N i , and Σ i are the projections ofĴ,N i , andŜ i on the BF quantization axis z, In Eq. (5), where D J M Ω is a Wigner D-function, describing the rotation of the collision complex with angular momentum J in the SF frame [47].
Ref. [40] presents the matrix elements of the interaction potential (4) and thel 2 operator in the BF basis (5). The matrix elements of the asymptotic Hamiltonians (2) can be obtained in the same basis by recoupling the angular momenta as was done for the atom -molecule collision system in Ref. [40]. The expression on the right-hand side of Eq. (3) is a contraction of spherical tensor operators of the same rank and therefore it is independent of the choice of the coordinate frame. SettingR = 0 and using the Wigner-Eckart theorem [47], we obtain the matrix elements of the dipole -dipole interaction operator in the BF basis (5) For collisions of identical molecules, the basis set in Eq. (5) should be modified to account for the effects of the permutation symmetry. This has been done by a number of authors in the SF frame (see, e.g. Refs. [36,37,39,51,52]). In the case of the BF total angular momentum representation given by Eq. (5), the symmetrization can be accomplished by performing the transformation to the SF frame [47] and applying the symmetrization operator 1 +P AB to the right-hand side of Eq. (5) [51,52]. As a result, we obtain the symmetrized orthonormal BF basis states for two identical molecules where η = +1(−1) for identical bosons (fermions), Ω ≥ 0, and we assume that the states are well-ordered.
Eqs. (7) and (8) and the expressions for the matrix elements in Ref. [40] can be combined to obtain the elements of the Hamiltonian matrix describing the interaction of indistinguishable molecules.

B. Space-fixed formulation
The matrix elements of the operatorl 2 in the BF basis (5) can be lengthy and cumbersome to program. An alternative formulation of the collision problem can be derived using the total angular momentum representation directly in the SF coordinate frame. To do this, we re-write the expansion (5) as whereĵ A is defined as the vector sum ofN A andŜ A ,ĵ B as the vector sum ofN B andŜ B ,ĵ as the vector sum ofĵ A andĵ B , andĴ as the vector sum ofĵ andl. The basis set representation (9) is most suitable for molecule -molecule scattering problems in the absence of external fields [48] but it can also be used for computations of cross sections for molecular collisions in external fields. The matrix elements of the operatorl 2 in the SF representation (9) are particularly simple: The basis states (9) with a given value of J form J-manifolds that lead to J-blocks of the Hamiltonian matrix as described in Ref. [40]. The blocks with different J values are coupled by the interaction of the molecules with the external magnetic field given by the last term in Eq. (2). To evaluate the matrix elements of this operator in the basis (9), we note that the operatorsŜ The operatorsŜ can be obtained using Eq. (5.72) of Ref. [47] to yield where the factors in the curly braces are 6j symbols.
The matrix elements of the molecule -molecule interaction potential operator in the SF basis (9) can be obtained using the theory described in Ref. [54]. In particular, if the molecule -molecule interaction potential for each spin multiplicity is represented by the expansion given in Eq. (3) of Ref. [54], then the matrix elements of the total interaction potential (including the Heisenberg exchange interaction [36]) in the total angular momentum basis (9) are given by Eq. (33) of Ref. [54] after substituting L A and L B with N A and N B . If the potentials of different spin multiplicity of the molecule -molecule complex are assumed to be identical, the matrix elements of the interaction potential can be evaluated using a simpler expression given in Eq. (A1) of Ref. [48]. Ref. [48] also gives the expressions for the matrix elements of the operators (3) for zero magnetic field and the matrix elements of the magnetic dipole -dipole interaction (3) in a SF total angular momentum basis. Note that the total angular momentum basis set used in Ref. [48] is based on a different coupling scheme. However, additional recoupling transformation required to bring the Hamiltonian matrix defined in Ref. [48] to the basis set representation (9) can be readily obtained by a simple modification of Eq. (30) in Ref. [54]. We recommend the use of the representation (9) over the total angular momentum basis used in Ref. [48] because it allows one to construct the basis by concatenation of single-molecule basis sets.
As mentioned in the previous section, the symmetrization procedure for collisions of identical molecules can be applied directly in the SF coordinate frame. Using the SF total angular momentum representation (9) as the basis, we can write the symmetrized orthonormal SF basis states for two identical molecules as follows [52]:

III. NUMERICAL RESULTS
For numerical computations, the basis sets (1) and (9) Figure 5 presents the magnetic field dependence of γ for 15 NH -15 NH collisions computed for several collision energies. In the limit of low collision energies, the elastic cross sections are energy independent and the inelastic cross sections are inversely proportional to the collision velocity [55]. Therefore, γ must decrease when the collision energy decreases.
At the same time, the rate of spin relaxation tends to zero when the magnetic field lifting the degeneracy of the Zeeman levels vanishes [38,56]. The results of Figure 5 show that γ for 15 NH -15 NH collisions remains ≥ 100 for all magnetic fields at collision energies above 10 −3 cm −1 . These results indicate that the evaporative cooling of rotationally ground 15 NH molecules in a magnetic trap is likely to be feasible even in the presence of a strong magnetic field. At very low collision energies, γ increases dramatically with the field strength, so evaporative cooling to ultracold temperatures would only be possible at B ≤ 10 −3 T. Overall, the present results support the conclusion of Janssen et al. [37] that the evaporative cooling of rotationally ground 15 NH molecules in a magnetic trap has a prospect of success.

IV. SUMMARY
The present work considers the problem of Zeeman relaxation in molecule -molecule collisions in a magnetic field. Accurate calculations of cross sections for elastic scattering and Zeeman relaxation in molecule -molecule collisions are much needed for understanding the prospects of evaporative cooling of molecules in a magnetic trap to ultracold temperatures.
Although the theory of molecular scattering in external fields was previously developed [38,39], the uncoupled space-fixed basis used in all previous studies leads to a prohibitively large number of coupled differential equations. In this work, we show that the cross sections for molecule -molecule collisions can be computed more efficiently using a total angular momentum basis, defined either in the body-fixed frame or in the space-fixed coordinate system. The total angular momentum representation allows for a more physical truncation of the basis set than the fully uncoupled representation, permitting more relevant basis functions to be included in the basis. The fully uncoupled representation [39] became popular for the theoretical description of molecular collisions in external fields because it leads to simple expressions for the Hamiltonian matrix elements that are easy to evaluate. In the present work, we show that compact expressions for the Hamiltonian matrix elements can also be derived in the space-fixed total angular momentum representation.
We repeated the previous calculations of cross sections for elastic scattering and Zeeman relaxation in 15 NH -15 NH collisions using the total angular momentum representation leading to a larger basis set. We have obtained converged results demonstrating that the probability of elastic 15 NH -15 NH scattering remains > 100 times greater than the probability of Zeeman relaxation at magnetic fields between 10 −4 and 0.1 T and collision energies above 10 −3 cm −1 . For collision energies below 10 −3 cm −1 , γ displays a strong magnetic field dependence and evaporative cooling appears to be feasible only at B ≤ 10 −3 T. These results support the conclusions previously reached by Janssen et al. [37] and remove the uncertainty of the basis set truncation error that constrained the previous results.
It is well established that low-temperature molecule-molecule scattering properties are extremely sensitive to small variations in the interaction PES. Therefore, while the results presented in Figs. 3 and 5 can be considered converged for a given interaction potential, they are likely far from being quantitatively accurate. A more careful theoretical analysis is in order to assess the prospects of evaporative cooling of 15 NH in a magnetic trap. Such an analysis can be performed using the methodology outlined in this work and would require extensive computations of thermally averaged ratios of elastic to inelastic collision rates over a wide range of magnetic fields and potential scaling parameters. Because of the large uncertainties in the ab initio PES, the calculations may quickly become computationally intensive, so it would be best to constrain the 15 NH-15 NH interaction potential based on future experimental measurements of 15 NH-15 NH scattering cross sections, as often done for ultracold atoms [57]. This will be a challenging task requiring efficient iterative calculations of molecule-molecule scattering cross sections in a magnetic field.
Molecule-molecule scattering properties determine the possibility of the creation and manipulation of cold molecular gases. Recent theoretical work on dipolar collisions and chemical reactions of ultracold molecules [58][59][60] began to uncover the universal properties of molecule-molecule scattering in the quantum regime. However, not all regimes of molecular collisions are universal, and details of collision dynamics often depend on the intricate interplay between intermolecular and intramolecular interactions at short range. Our work complements recent theoretical efforts [58][59][60] by providing an efficient numerical technique for computing the collision properties of polar molecules for arbitrary collision energies, external field strengths, and interaction potentials. Our proposed methodology can be extended to include the effects of external electric fields, multiple potential energy surfaces and non-adiabatic couplings, and hyperfine interactions, providing a valuable numerical tool for future exploration of molecular collision dynamics in the quantum regime.