Collisions of paramagnetic molecules in magnetic fields: an analytic model based on Fraunhofer diffraction of matter waves

# Collisions of paramagnetic molecules in magnetic fields: an analytic model based on Fraunhofer diffraction of matter waves

Mikhail Lemeshko    Bretislav Friedrich Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany
July 18, 2019
###### Abstract

We investigate the effects of a magnetic field on the dynamics of rotationally inelastic collisions of open-shell molecules (, , and ) with closed-shell atoms. Our treatment makes use of the Fraunhofer model of matter wave scattering and its recent extension to collisions in electric [M. Lemeshko and B. Friedrich, J. Chem. Phys. 129, 024301 (2008)] and radiative fields [M. Lemeshko and B. Friedrich, Int. J. Mass. Spec. in press (2008)]. A magnetic field aligns the molecule in the space-fixed frame and thereby alters the effective shape of the diffraction target. This significantly affects the differential and integral scattering cross sections. We exemplify our treatment by evaluating the magnetic-field-dependent scattering characteristics of the He – CaH (), He – O () and He – OH () systems at thermal collision energies. Since the cross sections can be obtained for different orientations of the magnetic field with respect to the relative velocity vector, the model also offers predictions about the frontal-versus-lateral steric asymmetry of the collisions. The steric asymmetry is found to be almost negligible for the He – OH system, weak for the He – CaH collisions, and strong for the He – O. While odd transitions dominate the He – OH integral cross sections in a magnetic field parallel to the relative velocity vector, even transitions prevail in the case of the He – CaH () and He – O () collision systems. For the latter system, the magnetic field opens inelastic channels that are closed in the absence of the field. These involve the transitions with .

Rotationally inelastic scattering, paramagnetic molecules, alignment and orientation, Zeeman effect, models of molecular collisions.
###### pacs:
34.10.+x, 34.50.-s, 34.50.Ez

## I Introduction

All terrestrial processes, including collisions, take place in magnetic fields. And yet, quantitative studies of the effects that magnetic fields may exert on collision dynamics are mostly of a recent date, having been prompted by the newfashioned techniques to magnetically manipulate, control and confine paramagnetic atoms and molecules. Theoretical accounts of molecular collisions in magnetic fields are usually based on rigorous close-coupling treatments Krems04 (). Analytic models of such collisions are scarce, and limited to the Wigner regime, see, e.g., ref. sadeghpour2000 (). Here we present an analytic model of state-to-state rotationally inelastic collisions of closed-shell atoms with open-shell molecules in magnetic fields. The model, applicable to collisions at thermal and hyperthermal collision energies, is based on the Fraunhofer scattering of matter waves Drozdov ()Faubel () and its recent extension to include collisions in electrostatic LemFri1 () and radiative LemFri2 () fields. The magnetic field affects the collision dynamics by aligning the molecular axis with respect to the relative velocity vector, thereby changing the effective shape of the diffraction target. We consider open-shell molecules in the , , and electronic states, whose body-fixed magnetic dipole moments are on the order of a Bohr magneton  FriHer2sig ()WeinsteinCaH (). These states coincide with the most frequently occurring ground states of linear radicals, which are exemplified in our study by the CaH(), O), and OH() species. We take, as the closed-shell collision partner, a He atom. Helium is a favorite buffer gas, used to thermalize molecules and radicals produced by laser ablation and other entrainment techniques Doyle ().

The paper is organized as follows: in Section II, we briefly describe the field-free Fraunhofer model of matter-wave scattering. In Sections IIIIV, and V, we extend the Fraunhofer model to account for scattering of open-shell molecules with closed-shell atoms in magnetic fields: in Section III, we work out closed-form expressions for the partial and total differential and integral cross sections and the steric asymmetry of collisions between closed-shell atoms and paramagnetic molecules, and apply them to the collision system; in Sec. IV we present the analytic theory for molecules and apply it to the scattering; in Section V we develop the theory for collisions of molecules and exemplify the results by treating the inelastic scattering. Finally, in Section VI, we compare the results obtained for the collisions of the different molecules with helium and draw conclusions from our study.

## Ii The Fraunhofer model of field-free scattering

The Fraunhofer model of matter-wave scattering was recently described in Refs. LemFri1 () and LemFri2 (). Here we briefly summarize its main features.

The model is based on two approximations. The first one replaces the amplitude

 fi→f(ϑ)=⟨f|f(ϑ)|i⟩ (1)

for scattering into an angle from an initial, , to a final, , state by the elastic scattering amplitude, . This is tantamount to the energy sudden approximation, which is valid when the collision time is much smaller than the rotational period, as dictated by the inequality , where

 ξ=ΔErotkR02Ecoll≈BkR0Ecoll, (2)

is the Massey parameter, see e.g. Refs. Nikitin96 (),NikitinGasesBook (). Here is the rotational level spacing, the rotational constant, the collision energy, the wavenumber, the reduced mass of the collision system, and the radius of the scatterer.

The second approximation replaces the elastic scattering amplitude in Eq. (1) by the amplitude for Fraunhofer diffraction by a sharp-edged, impenetrable obstacle as observed at a point of radiusvector r from the scatterer, see Fig. 1. This amplitude is given by the integral

 f(ϑ)≈∫e−ikRϑcosφdR (3)

Here is the asimuthal angle of the radius vector R which traces the shape of the scatterer, , and with the initial wave vector. Relevant is the shape of the obstacle in the space-fixed plane, perpendicular to , itself directed along the space-fixed -axis, cf. Fig. 1.

We note that the notion of a sharp-edged scatterer comes close to the rigid-shell approximation, widely used in classical Beck79 ()Marks_ellips (), quantum Bosanac (), and quasi-quantum Stolte () treatments of field-free molecular collisions, where the collision energy by far exceeds the depth of any potential energy well.

In optics, Fraunhofer (i.e., far-field) diffraction BornWolf () occurs when the Fresnel number is small,

 F≡a2rλ≪1 (4)

Here is the dimension of the obstacle, is the distance from the obstacle to the observer, and is the wavelength, cf. Fig. 1. Condition (4) is well satisfied for nuclear scattering at MeV collision energies as well as for molecular collisions at thermal and hyperthermal energies. In the latter case, inequality (4) is fulfilled due to the compensation of the larger molecular size by a larger de Broglie wavelength pertaining to thermal molecular velocities.

For a nearly-circular scatterer, with a boundary in the plane, the Fraunhofer integral of Eq. (3) can be evaluated and expanded in a power series in the deformation ,

 f(ϑ)=f0(ϑ)+f1(ϑ,δ)+f2(ϑ,δ2)+⋯ (5)

with the amplitude for scattering by a disk of radius

 f0(ϑ)=i(kR20)J1(kR0ϑ)(kR0ϑ) (6)

and the lowest-order anisotropic amplitude,

 f1(ϑ)=ik2π∫2π0δ(φ)e−i(kR0ϑ)cosφdφ (7)

where is a Bessel function of the first kind. Both Eqs. (6) and (7) are applicable at small values of , i.e., within the validity of the approximation .

The scatterer’s shape in the space fixed frame, see Fig. 1, is given by

 R(α,β,γ;θ,φ)=∑κνρΞκνDκρν(αβγ)Yκρ(θ,φ) (8)

where are the Euler angles through which the body-fixed frame is rotated relative to the space-fixed frame, are the polar and azimuthal angles in the space-fixed frame, are the Wigner rotation matrices, and are the Legendre moments describing the scatterer’s shape in the body-fixed frame. Clearly, the term with corresponds to a disk of radius ,

 R0≈Ξ00√4π (9)

Since of relevance is the shape of the target in the plane, we set in Eq. (8). As a result,

 δ(φ)=R(α,β,γ;π2,φ)−R0=R(φ)−R0=∑κνρκ≠0ΞκνDκρν(αβγ)Yκρ(π2,φ) (10)

By combining Eqs. (1), (7), and (10) we finally obtain

 fi→f(ϑ)≈⟨f|f0+f1|i⟩=⟨f|f1|i⟩=ikR02π∑κνρκ≠0κ+ρ evenΞκν⟨f|Dκρν|i⟩FκρJ|ρ|(kR0ϑ) (11)

where

 Fκρ=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩(−1)ρ2π(2κ+14π)12(−i)κ√(κ+ρ)!(κ−ρ)!(κ+ρ)!!(κ−ρ)!! for κ+ρ ~{}even~{% }and~{} κ≥ρ0 elsewhere (12)

For negative values of , the factor is to be replaced by .

## Iii Scattering of 2Σ molecules by closed-shell atoms in a magnetic field

### iii.1 A 2Σ molecule in a magnetic field

The field-free Hamiltonian of a rigid molecule

 H0=BN2+γN⋅S (13)

is represented by a matrix, diagonal in the Hund’s case (b) basis, . Here and are the rotational and (electronic) spin angular momenta, is the rotational constant and the spin-rotation constant. Its eigenfunctions

 Ψ±(J,M)=1√2[∣∣S,12⟩∣∣J,Ω,M⟩±∣∣S,−12⟩∣∣J,−Ω,M⟩], (14)

are combinations of (electronic) spin functions with Hund’s case (a) (i.e., symmetric top) functions pertaining to the total angular momentum , whose projections on the space- and body-fixed axes are and , respectively. The Hund’s case (a) wavefunctions are given by:

 |J,M,Ω⟩=√2J+14πDJ∗MΩ(φ,θ,γ=0) (15)

The and states are conventionally designated as and states, for which the rotational quantum number and , respectively. Equation (14) can be recast in terms of instead of :

 |Ψϵ(N,M)⟩=1√2[∣∣S,12⟩∣∣N+ϵ2,Ω,M⟩+ϵ∣∣S,−12⟩∣∣N+ϵ2,−Ω,M⟩], (16)

with .

The eigenvalues corresponding to states and are given by

 E+(N+12,M;F1)=BN(N+1)+γ2N (17)
 E−(N−12,M;F2)=BN(N+1)−γ2(N+1), (18)

whence we see that the spin-rotation interaction splits each rotational level into a doublet separated by .

In a static magnetic field, , directed along the space-fixed axis, the Hamiltonian acquires a magnetic dipole potential which is proportional to the projection, , of on the axis

 Vm=SZωmB, (19)

with

 ωm≡gSμBH2B (20)

a dimensionless interaction parameter involving the electron gyromagnetic ratio , the Bohr magneton , and the rotational constant .

The Zeeman eigenproperties of a molecule can be readily obtained in closed form, since the operator couples states that differ in by or and, therefore, the Hamiltonian matrix, , factors into blocks for each :

 H=−ωmB⎛⎜ ⎜ ⎜ ⎜ ⎜⎝−M2N+1+E−ωmB12[1−M2(N+1/2)2]1212[1−M2(N+1/2)2]12M2N+1+E+ωmB⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ (21)

As a result, the Zeeman eigenfunctions of a molecule are given by a linear combination of the field-free wavefunctions (16),

 ψ(~N,~J,M;ωm)=a(ωm)∣∣Ψ−(N,M)⟩+b(ωm)∣∣Ψ+(N,M)⟩, (22)

with the hybridization coefficients and obtained by diagonalizing Hamiltonian (21). Although and are no longer good quantum numbers in the magnetic field, they can be employed as adiabatic labels of the states: we use and to denote the angular momentum quantum numbers of the field-free state that adiabatically correlates with the given state in the field. Since the Zeeman eigenfunction comprises rotational states with either even or odd, the parity of the eigenstates remains definite even in the presence of the magnetic field; it is given by .

The degree of mixing of the Hund’s case (b) states that make up a Zeeman eigenfunction is determined by the splitting of the rotational levels measured in terms of the rotational constant, : for the mixing (hybridization) is incomplete, while it is perfect in the high-field limit, . We note that in the high-field limit, the eignevectors can be found from matrix (21) with . As an example, Table 1 lists the values of the hybridization coefficients and for the states of the CaH molecule in the high-field limit, which is attained at .

The degree of molecular axis alignment is given by the alignment cosine, , which, in the case, can be obtained in closed form. To the best of our knowledge, this result has not been presented in the literature before; therefore, we give it in Appendix C. The dependence of the alignment cosine on the magnetic field strength parameter is shown in Fig. 2 for the two lowest states of the CaH molecule. One can see that for , the alignment cosine smoothly approaches a constant value, corresponding to as good an alignment as the uncertainty principle allows.

### iii.2 The field-dependent scattering amplitude

In what follows, we consider scattering from the state to some state in a magnetic field. Since the state of a molecule is not aligned, the effects of the magnetic field on the scattering arise solely from the alignment of the final state.

In order to account for an arbitrary direction of the electric field with respect to the initial wave vector , we introduce a field-fixed coordinate system , whose -axis is defined by the direction of the electric field vector . By making use of the relation

 DJ∗MΩ(φ♯,θ♯,0)=∑ξDJξM(φε,θε,0)DJ∗ξΩ(φ,θ,0) (23)

we transform the wavefunctions (22) to the space-fixed frame. For the initial and the final states we have:

 (24)
 ⟨f(N′,M′)|=1√4π∑ξ′{a′(ωm)√N′DN′−12ξ′M′(φε,θε,0)[DN′−12∗ξ′Ω(φ,θ,0)−DN′−12∗ξ′−Ω(φ,θ,0)]+b′(ωm)√N′+1DN′+12ξ′M′(φε,θε,0)[DN′+12∗ξ′Ω(φ,θ,0)−DN′+12∗ξ′−Ω(φ,θ,0)]} (25)

where for a molecule.

By substituting from Eqs. (24) and (25) into Eq. (11), we finally obtain the scattering amplitude for inelastic collisions of molecules with closed-shell atoms in a magnetic field:

 fωmi→f(ϑ)=ikR04π∑κρκ≠0κ+ρ evenΞκ0Dκ∗−ρ,ΔM(φε,θε,0)FκρJ|ρ|(kR0ϑ)[(−1)κ+(−1)ΔN]×{a(ωm)a′(ωm)√NN′C(N−12,κ,N′−12;Ω0Ω)C(N−12,κ,N′−12;MΔMM′)+a(ωm)b′(ωm)√NN′+1C(N−12,κ,N′+12;Ω0Ω)C(N−12,κ,N′+12;MΔMM′)+a′(ωm)b(ωm)√N+1N′C(N+12,κ,N′−12;Ω0Ω)C(N+12,κ,N′−12;MΔMM′)+b(ωm)b′(ωm)√N+1N′+1C(N+12,κ,N′+12;Ω0Ω)C(N+12,κ,N′+12;MΔMM′)} (26)

As noted above, there is no hybridization of the initial state for the collisions, i.e., , in Eq. (26). By making use of the properties of the Clebsch-Gordan coefficients Zare (),Varshalovich (), the expression for the scattering amplitude from the state to an state simplifies to

 fωm0,12,±12→N′,J′,M′(ϑ)=ikR02πΞN′02N′+1⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩∑ρρ+N′evendN′−ρ,ΔM(θε)FN′ρJ|ρ|(kR0ϑ)⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭×[±a′(ωm)√N′∓M′+12+b′(ωm)√N′±M′+12] (27)

The amplitude is seen to be directly proportional to the Legendre moment. We note that the cross section for the transition differs from that for the scattering. This is because the magnetic field completely lifts the degeneracy of the states, in contrast to the electric field case LemFri1 ().

### iii.3 Results for He -- CaH(X2Σ,J=1/2→J′) scattering in a magnetic field

Here we apply the analytic model scattering to the He – CaH( collision system. The CaH molecule, employed previously in thermalization experiments with a He buffer gas WeinsteinCaH (), FriedrichCaH (), has a rotational constant cm and a spin-rotational interaction parameter cm MartinCaH (). Such values of molecular constants result in an essentially perfect mixing (and alignment) of the molecular states for field strengths Tesla, see Sec. III.1.

According to Ref. PESHe-CaH (), the ground-state He–CaH potential energy surface has a global minimum of cm. Such a weak attractive well can be neglected at a collision energy as low as cm (which corresponds to a wave number Å). The corresponding value of the Massey parameter, , warrants the validity of the sudden approximation to the He – CaH collision system from this collision energy on. The “hard shell” of the potential energy surface was found by a fit to Eq. (8) for , and is shown in Fig. 3. The coefficients obtained from the fit are listed in Table 2. According to Eq. (9), the coefficient determines the hard-sphere radius , which is responsible for elastic scattering.

#### iii.3.1 Differential cross sections

The state-to-state differential cross sections for scattering in a field parallel () and perpendicular () to are given by

 Iωm,(∥,⊥)0→J′(ϑ)=∑M′Iωm,(∥,⊥)0,0→J′,M′(ϑ) (28)

with

 Iωm,(∥,⊥)0,0→J′,M′(ϑ)=∣∣∣fωm,(∥,⊥)0,0→~J′,M′(ϑ)∣∣∣2 (29)

They are presented in Figs. 45 for He–CaH collisions at zero field, , as well as at high field, (corresponding to =2.75 T for CaH), where the hybridization and alignment are as complete as they can get.

From Eq. (27) for the scattering amplitude, we see that the differential cross section for the transitions is proportional to the Legendre moment. According to Table 2, the Legendre expansion of the He–CaH potential energy surface is dominated by . Therefore, the transition provides the largest contribution to the cross section.

The field dependence of the scattering amplitude, Eq. (27), is encapsulated in the coefficients and , whose values cannot affect the angular dependence, as this is determined solely by the Bessel functions, . Furthermore, the summation in Eq. (27) includes only even for even , and odd for odd . From the asymptotic properties of Bessel functions Watson (), we have for large angles such that :

 (30)

For the He – CaH system, the phase shift between the and Bessel functions, which contribute to the transitions, is negligibly small at angles up to about 30. Therefore there is no field-induced phase shift, neither in the parallel nor in the perpendicular case, as illustrated by Figs. 45.

Figs. 4 and 5 show that the magnetic field induces only small changes in the amplitudes of the cross sections, without shifting their oscillations. The amplitude variation is so small because the magnetic field fails to mix contributions from the different Legendre moments, in contrast to scattering in electrostatic LemFri1 () and radiative LemFri2 () fields. The changes in the amplitudes of the differential cross sections are closely related to the field dependence of the partial integral cross sections, which are analyzed next.

#### iii.3.2 Integral cross sections

The angular range, , where the Fraunhofer approximation applies the best, comprises the largest-impact-parameter collisions that contribute to the scattering the most, see Figs. 4 and 5. Therefore, the integral cross section can be obtained, to a good approximation, by integrating the Fraunhofer differential cross sections, Eq. (28) and (29), over the solid angle , with and .

The integral cross-sections thus obtained for the magnetic field oriented parallel and perpendicular to the initial wave vector are presented in Figs. 6 and 7. A prominent feature of the cross sections for the transitions is that, in the parallel field geometry, they increase for the final states and decrease for the states, while it is the other way around for the perpendicular geometry.

In order to make sense of these trends in the field dependence of the -averaged cross sections, let us take a closer look at the partial, -resolved cross sections for the channels and the two field geometries, also shown in Figs. 6 and 7.

(i) Magnetic field parallel to the initial wave vector, . In this case, the real Wigner matrices reduce to the Kronecker delta functions, , and the scattering amplitude (27) becomes:

 fωm,∥0,12,±12→N′,J′,M′(ϑ)=ikR02πΞN′02N′+1FN′,−ΔMJ|ΔM|(kR0ϑ)×[±a′(ωm)√N′∓M′+12+b′(ωm)√N′±M′+12] (31)

Eq. (31) allows to readily interpret the dependences presented in Fig. 6. First, we see that the coefficients, defined by eq. (12), lead to a selection rule, namely that the cross sections vanish for odd. Therefore, the partial cross sections for such combinations of and do not contribute anything to the trends seen in Fig. 6 that we wish to explain. Equally absent are contributions from the transitions leading to the states with , since these states exhibit no alignment, see Fig. 2 (a), (c).

As we can see from Fig. 6, the field-dependence of the cross section for the transitions is a result of a competition among the partial -resolved cross sections. Therefore we need to account for the relative magnitudes of the non-vanishing -dependent cross sections. Let us do it for the scattering channel . Substituting the coefficients from Table 1 into Eq. (31), we see that the term in the square brackets vanishes for and for , but equals for and . In addition, taking into account that , we see that the -averaged cross section must go up with increasing field strength.

More generally, the dependence of the cross sections on the magnetic field is contained in the two hybridization coefficients and . In the field-free case, and for collisions leading to states, whereas , for collisions that produce states. In a magnetic field, the and coefficients assume values ranging between and . As a consequence, the and coefficients have the same signs for the states and opposite signs for the states. Clearly, then, for an state, increases with the field strength, while decreases. Hence the factor in the square brackets of Eq. (31) increases with for and decreases for , because of the opposite sign of the coefficient.This is reversed for the final states, i.e., the cross sections increase for , and decrease for .

(ii) Magnetic field perpendicular to the initial wave vector, . In this case, Eq. (27) takes the form:

 fωm,⊥0,12,±12→N′,J′,M′(ϑ)=ikR02πΞN′02N′+1⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩∑ρρ+N′evendN′−ρ,ΔM(π2)FN′ρJ|ρ|(kR0ϑ)⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭×[±a′(ωm)√N′∓M′+12+b′(ωm)√N′±M′+12] (32)

which is more involved than for the parallel case, although the field-dependent coefficients and remain outside of the summation. The difference between the parallel and perpendicular cases is related to the values of the real Wigner -matrices mixed by eq. (32). For instance, an inspection of the coefficients from Table 1, along with the matrices, reveals that the -averaged cross sections for the transition (i.e., transition to an final state) will decrease with , in contrast to the parallel case.

#### iii.3.3 Frontal-versus-lateral steric asymmetry

As in our previous work LemFri1 ()LemFri2 (), we define a frontal-versus-lateral steric asymmetry by the expression

 Si→f=σ∥−σ⊥σ∥+σ⊥ (33)

where the integral cross sections correspond, respectively, to and . The field dependence of the steric asymmetry for the He – CaH collisions is presented in Fig. 8. One can see that a particularly pronounced asymmetry obtains for the scattering into the final states, while it is smaller for the channels. Moreover, the steric asymmetry exhibits a sign alternation: it is positive for final states and negative for final states. This behavior is a reflection of the alternation in the trends of the integral cross sections for the and final states, discussed in the previous subsection, cf. the corresponding -dependent integral cross sections, Figs. 6 and 7.

We note that within the Fraunhofer model, elastic collisions do not exhibit any steric asymmetry. This follows from the isotropy of the elastic scattering amplitude, Eq. (6), which depends on the radius only: a sphere looks the same from any direction.

## Iv Scattering of 3Σ molecules by closed-shell atoms in a magnetic field

### iv.1 A 3Σ molecule in a magnetic field

The field-free Hamiltonian of a electronic state consists of rotational, spin-rotation, and spin-spin terms

 H0=BN2+γN⋅S+23λ(3S2z−S2) (34)

where and are the spin-rotation and spin-spin constants, respectively. In the Hund’s case (b) basis, the field-free Hamiltonian (34) consists of matrices pertaining to different values (except for =0). The matrix elements of Hamiltonian (34) can be found, e.g., in ref. amiot () (see also BocaFri () and Lefebvre-Brion ()). The eigenenergies of are (in units of the rotational constant ):

 E1(J)/B=J(J+1)+1−3γ′2−λ′3−XE2(J)/B=J(J+1)−γ′+2λ′3E3(J)/B=J(J+1)+1−3γ′2−λ′3+X (35)

with

 X≡[J(J+1)(γ′−2)2+(γ′+2λ′−22)2]1/2γ′≡γ/Bλ′≡λ/B

The eigenenergies (35) correspond to the three ways of combining rotational and electronic spin angular momenta and for into a total angular momentum ; the total angular momentum quantum number takes values , , and for states which are conventionally designated as , , and , respectively. For the case when , , the sign of the term should be reversed Herzberg (). The parity of the states is .

The interaction of a molecule with a magnetic field is given by:

 Vm=SZωmB, (36)

where

 ωm≡gSμBHB (37)

is a dimensionless parameter characterizing the strength of the Zeeman interaction, cf. Eq. (20).

We evaluated the Zeeman effect in Hund’s case (b) basis

 |N,J,M⟩=c1NJ|J,1,M⟩+c0NJ|J,0,M⟩+c−1NJ|J,−1,M⟩ (38)

using the matrix elements of the operator given in Appendix A.

The Zeeman eigenfunctions are hybrids of the Hund’s case (b) basis functions (38):

 ∣∣~N,~J,M;ωm⟩=∑NJa~N~JNJ(ωm)|N,J,M⟩ (39)

and are labeled by and , which are the angular momentum quantum numbers of the field-free state that adiabatically correlates with a given state in the field. Since couples Hund’s case (b) states that differ in by or , the parity remains definite in the presence of a magnetic field, and is given by . However, the Zeeman matrix for a molecule is no longer finite, unlike the Zeeman matrix for a state.

Using the Hund’s case (b) rather than Hund’s case (a) basis set makes it possible to directly relate the field-free states and the Zeeman states, via the hybridization coefficients .

The alignment cosine, , of the Zeeman states can be evaluated from the matrix elements of Appendix B. The dependence of on the magnetic field strength parameter is exemplified in Fig. 9 for Zeeman states of the O molecule.

### iv.2 The field-dependent scattering amplitude

We consider scattering from an initial state to a final state. We transform the wavefunctions (39) to the space-fixed frame by making use of Eq. (23) – cf. Section III.2. As a result, the initial and final states become:

 |i⟩≡∣∣~N,~J,M,ωm⟩=∑NJ√2J+14πa~N~JNJ(ωm)∑ΩcΩNJ∑ξDJξM(φε,θε,0)DJ∗ξΩ(φ,θ,0) (40)
 ⟨f|≡⟨~N′,~J′,M′,ωm|=∑N′J′√2J′+14πb~N′~J′N′J′(ωm)∑Ω′cΩ′N′J′∑ξ′DJ′ξ′M′(φε,θε,0)DJ′∗ξ′Ω′(φ,θ,0) (41)

On substituting from Eqs. (40) and (41) into Eq. (11) and some angular momentum algebra, we obtain a general expression for the scattering amplitude:

 fωmi→f(ϑ)=ikR04π∑κρκ≠0κ+ρ evenΞκ0Dκ∗−ρ,ΔM(φε,θε,0)FκρJ|ρ|(kR0ϑ)×∑NJN′J′√2J+12J′+1a~N~JNJ(ωm)b~N′~J′N′J′(ωm)C(JκJ′;MΔMM′)∑ΩcΩNJcΩN′J′C(JκJ′;Ω0Ω) (42)

### iv.3 Results for He--O2(X3Σ,N=0,J=1→N′,J′) scattering in a magnetic field

The O molecule has a rotational constant cm, a spin-rotation constant cm, and a spin-spin constant cm TomutaO2 (). According to Ref. PESHe-O2 (), the ground state He – O potential energy surface has a global minimum of cm, which can be neglected at a collision energy cm (corresponding to a wave number Å). A small value of the Massey parameter, , ensures the validity of the sudden approximation. The “hard shell” of the potential energy surface at this collision energy is shown in Fig. 3, and the Legendre moments , obtained from a fit to the potential energy surface of Ref. PESHe-O2 (), are listed in Table 2. Since the He – O potential is of D symmetry, only even Legendre moments are nonzero.

Furthermore, since the nuclear spin of O is zero and the electronic ground state antisymmetric (a state), only rotational states with an odd rotational quantum number are allowed. We will assume that the O molecule is initially in its rotational ground state, .

Expression (42) for the scattering amplitude further simplifies for particular geometries. In what follows, we will consider two such geometries.

(i) Magnetic field parallel to the initial wave vector, , in which case the scattering amplitude becomes:

 fωm,∥1,0,0→N′,J′,M′(ϑ)=ikR04πJ|M′|(kR0ϑ)∑κ≠0κ evenΞκ0FκM′×∑NJN′J′√2J+12J′+1a10NJ(ωm)b~N′~J′N′J′(ωm)C(JκJ′;0M′M′)∑ΩcΩNJcΩN′J′