Para-ortho hydrogen conversion: Solving a 90-year oldmystery

Funding information ChinaScholarshipCouncil,NationalNatural ScienceFoundationofChina,Grant/Award Number: 11604118;ChinaPostdoctoral ScienceFoundation,Grant/AwardNumber: 2015M581390 Abstract Almost ninety years have passed since the experiments of Farkas and Sachsse [Z. Phys. Chem. B 1933; 23:1] on para-ortho hydrogen conversion catalyzed by paramagnetic species such as O2, but a detailed and quantitative understanding of the conversion process and its temperature dependencewas still lacking. Here, we present a complete and quantitative theoretical treatment of this catalytic process. Both interactions causing the conversion are included: the magnetic dipole-dipole coupling between the electron spin of O2 and the nuclear spins in H2 and the Fermi contact coupling from spin densities at the H-nuclei induced by O2. The latter were extracted from ab initio electronic structure calculations. State-to-state conversion cross sections and rate coefficients are obtained from quantummechanical coupled-channel calculations including the full anisotropic O2-H2 interaction potential and by treating both the spin-dependent couplings perturbatively. The total rate coefficient agrees with the experimental value recently measured by Wagner [Magn. Reson. Mater. Phys., Biol. Med. 2014; 27:195] in O2-H2 gas mixtures and explains the temperature dependence observed in the 1933measurements mentioned above.


INTRODUCTION
F I G U R E 1 Setup used in 1933 by L. Farkas and Sachsse 6 to measure pH 2 -oH 2 conversion in gas mixtures with O 2 , NO, and NO 2 . The technique to determine the oH 2 /pH 2 concentration ratio was developed by A. Farkas. 11 Picture copied from Ref. [ 6 ] the wave functions of the two protons in H 2 must be antisymmetric. In pH 2 , the two proton spins are coupled to total spin I = 0. The I = 0 nuclear spin wave function is antisymmetric, which implies that the spatial wave functions of pH 2 must be symmetric and that the molecule can only possess rotational states with even values of the angular momentum quantum number j. By contrast, the three nuclear spin wave functions of oH 2 with the proton spins coupled to I ′ = 1 and M ′ I = −1, 0, 1 are symmetric and so its spatial wave functions must be antisymmetric and its rotational quantum numbers j are odd. The equilibrium ratio of about 1 to 3 of pH 2 and oH 2 at room temperature (293 K) is determined by the multiplicity of their nuclear spin wave functions with I = 0 and I ′ = 1, respectively. The j = 1 ground state of oH 2 is higher in energy than the j = 0 ground state of pH 2 by about 120 cm −1 (173 K). By passing the mixture of pH 2 and oH 2 over a suitable catalyst at liquid hydrogen temperature (20 K), one can convert it completely into the more stable pH 2 in the rotational ground state with j = 0. Although strongly out of equilibrium at room temperature, this pure pH 2 can be kept for several weeks in special gas cylinders and used in NMR or MRI to bring also the nuclear spins of other nuclei in a sample out of equilibrium and thereby strongly enhance the measured signals.
Para-ortho conversion of hydrogen by interaction with gas phase paramagnetic species was reported in 1933 in three papers in the same issue of the Zeitschrift für Physikalische Chemie. Two of them, by Farkas and Sachsse, 6,7 described measurements of the para-ortho H 2 conversion rate in gas mixtures with paramagnetic molecules such as O 2 , NO, and NO 2 , see Figure 1. The third was a theoretical paper contributed by Eugene Wigner. 8 He argued that the magnetic dipole of a paramagnetic molecule produces an inhomogeneous magnetic field that in a nearby H 2 molecule may be different at the positions of its two H nuclei. The nuclear spins of these two nuclei interact with this field, which makes them inequivalent.
This breaks the exchange symmetry and causes para-ortho H 2 conversion. From an estimate of the frequency of collisions of H 2 with the paramagnetic molecule, the time they stay in each other's vicinity, the strength of the different nuclear spin interactions at an estimated distance of closest approach, Wigner obtained a realistic estimate of the para-ortho H 2 conversion rate. A more qualitative discussion of para-ortho conversion in H 2 and D 2 by paramagnetic species and, in particular, of how to determine the ratio of the proton and deuteron magnetic moments from this process 9 was presented in 1935 by Kalckar and Teller. 10 The subject of para-ortho H 2 conversion attracted more attention again when it was discovered in 1997 1-5 that para-H 2 is an important species in NMR and MRI. Since O 2 , which is one of the few stable small molecules with an open-shell ground state and a nonvanishing electron spin (S = 1), is abundant in air and under pressure dissolves in water and other liquids, this also caused renewed interest in para-ortho H 2 conversion by interactions with O 2 . A recent remeasurement of the conversion rate coefficient in O 2 -H 2 gas mixtures by Wagner 12 was triggered by the need to understand the detrimental effect of oxygen when applying para-hydrogen induced enhancement methods in NMR and MRI.
Over the years the magnetic dipole-dipole coupling mechanism proposed by Wigner was invoked in several qualitative theoretical model studies of para-ortho H 2 conversion on various catalyst surfaces and in solid or liquid hydrogen with impurities, [13][14][15][16] which accompanied experimental work on this subject. Ilisca and Sugano 17 and Minaev and Ågren 18 suggested in 1986 and 1995, respectively, that para-ortho H 2 conversion is also caused by another mechanism that they believed to be much more effective than the one proposed by Wigner. Their idea was that the exchange interaction between a spin triplet (S = 1) molecule such as O 2 and a nearby H 2 molecule induces spin density in the latter by mixing a small amount of its lowest triplet (S = 1) excited state into its closed-shell (S = 0) ground state. In asymmetric collisions this causes a different electron spin density at the two H nuclei which, via the Fermi contact interaction, leads to para-ortho H 2 conversion. Both mechanisms were included in subsequent model studies 19 of H 2 adsorbed on clean Ag surfaces and on Ag surfaces with co-adsorbed O 2 , and it was concluded that the Fermi contact interaction is nearly as effective as the magnetic dipole-dipole coupling when O 2 and H 2 are close to each other on the surface, while it is less effective when the adsorbed molecules are further apart. In our description of the couplings, we explain this distance effect. With the aid of density-functional theory (DFT) electronic structure calculations, order of magnitude estimates of the conversion time were given 19 and it was found that the presence of O 2 on the Ag surface accelerates the conversion. The only more quantitative theoretical study of the catalyzing effect of gas phase O 2 on para-ortho H 2 conversion is the 1967 paper by Nielsen and Dahler. 20 These authors employed scattering theory based on a hard-sphere isotropic intermolecular potential to model the O 2 -H 2 interactions, which allowed them to write the scattering wave functions analytically, but neglects the scrambling of the rotational states of O 2 and H 2 that also takes place during the collision. Only the rotational ground states of O 2 and H 2 were included and only the leading term of the magnetic dipole-dipole coupling was taken into account. These rather crude approximations and the absence of the Fermi contact coupling mechanism caused their rate coefficients calculated at 77.3 and 86 K to be smaller than the experimental values of Farkas 21 by a factor of 3 to 4. So, in spite of all the work carried out since 1933 a complete and quantitative explanation of the para-ortho hydrogen conversion by paramagnetic molecules was still lacking. Also the gas phase measurements 6, 12 did not provide insight into the processes occurring during O 2 -H 2 collisions or on the conversion mechanisms.
The present paper describes the first complete and quantitative theoretical calculation of the para-ortho H 2 conversion rate due to collisions with O 2 from first principles. Both the Fermi contact coupling and the magnetic dipole coupling between the electron spin of O 2 and the nuclear spins in H 2 are evaluated as functions of the geometry of the O 2 -H 2 collision complex. The spin density at the two H-nuclei induced by the exchange interaction with O 2 that appears in the formula for the Fermi contact coupling is obtained from ab initio electronic structure calculations. Both couplings are then included in nearly exact quantum mechanical coupled-channel (CC) scattering calculations 22,23 for the collisions between O 2 and H 2 , which also include the full anisotropic O 2 -H 2 potential surface 24 and yield the para-ortho H 2 conversion cross sections and rate coefficients for temperatures up to 400 K. The latter are compared with the rate recently measured in H 2 -O 2 gas mixtures at room temperature 12 and with the older experimental data 6 measured for a range of temperatures.
Also para-ortho H 2 conversion by an exchange reaction with H atoms or H + and H + 3 ions has been studied theoretically and experimentally, see Ref. [ 25 ] and references therein. This is of interest especially in astrophysical environments where free H atoms and H + or H + 3 ions are abundant, but it is not the subject of the present paper.

PARA-ORTHO H 2 COUPLINGS
The anisotropic O 2 -H 2 interaction potential leads to rotationally inelastic collisions of the pH 2 and oH 2 species, but since it depends only on the spatial coordinates of the molecules and pH 2 and oH 2 have different nuclear spin functions, it does not produce any pH 2 -oH 2 conversion. Interactions that do lead to conversion must also act on the nuclear spin functions. Since the exchange (P 12 ) parities of the pH 2 and oH 2 nuclear spin functions are opposite to each other, they must be antisymmetric under exchange of the spin coordinates of the nuclei H 1 and H 2 . And since the rotational wave functions of pH 2 and oH 2 have opposite parities as well, these interactions must also be antisymmetric under exchange of the H 1 and H 2 spatial coordinates.
The magnetic dipole-dipole coupling between the electron spin of a paramagnetic molecule (O 2 in our case) and the proton spins in H 2 proposed as a conversion mechanism by Wigner 8 can be written as followŝ where g e = 2.0023193 and g p = 5.5856947 are the gyromagnetic factors of the electron and the proton, respectively, B is the bohr magneton, N the nuclear magneton, and 0 the vacuum permeability. The vector operatorŜ with components (Ŝ x ,Ŝ y ,Ŝ z ) is the electron spin operator of O 2 , and the vector operatorsÎ(H i ) are the nuclear spin operators of the nuclei H i (i = 1, 2) in H 2 . The symbol T (2) (R i ) denotes the second rank dipole-dipole interaction tensor (in Cartesian components a 3 × 3 matrix), which depends on the length R i and the direction of the vector R i pointing from the center of O 2 to the nuclei H i . The Fermi contact term proposed as a coupling mechanism by Ilisca and Sugano 17 and by Minaev and Ågren 18 readŝ with (H i ) being the spin density at nucleus H i (i = 1, 2). (1) and (2)  at the complete-active-space self-consistent field (CASSCF) and multi-reference configuration-interaction (MRCI) levels with the program package

The operators in Equations
Molpro. 26 Details are given in the Supporting Information. Dipole-dipole coupling is a long-range interaction and the interaction tensors T (2) The H-nuclei are relatively close to one another compared to the O 2 -H 2 distance R, and the difference tensor [T (2) The induced spin density and, therefore, also the Fermi contact interaction is an overlap effect which decays exponentially with R. Hence, the magnetic dipole-dipole interaction dominates at long range, the Fermi contact interaction at short range.

PARA-ORTHO H 2 CONVERSION BY COLLISIONS WITH O 2
Collisions between molecules that lead to (de-)excitation of the rotational states of one or both of the molecules can be studied in detail experimentally in a crossed molecular beam setup. By preparing the molecules before the collision in specific states and state-specifically detecting them after the collision, one can measure state-to-state cross sections, which are proportional to the probabilities that specific (de-)excitation processes occur.
Also O 2 -H 2 collisions have been studied in this way 24,[27][28][29] and these measurements were accompanied by quantum mechanical coupled-channels (CC) calculations with the use of an O 2 -H 2 interaction potential computed in Ref. [ 24 ]. Good agreement was found between the calculated and measured collision cross sections, which shows that the interaction potential is accurate. The pH 2 -oH 2 conversion cross sections that correspond to transitions between rotational states of pH 2 with even j and of oH 2 with odd j, or vice-versa, are about twelve orders of magnitude smaller (see below) than those of the observed rotational transitions within the even and odd j manifolds. 27 Hence, no pH 2 -oH 2 conversion can be detected in a crossed-beam setup. Neither was it predicted in the CC calculations, 24, 27-29 because the latter did not include the required coupling terms given above. To our knowledge, the only experimental data available on pH 2 -oH 2 conversion by collisions with O 2 is the rate coefficient recently measured at room temperature in O 2 -H 2 gas mixtures 12 and the older data 6 measured for a range of temperatures. These measurements do not provide any insight into the processes occurring during O 2 -H 2 collisions or on the conversion mechanisms. Therefore, such information has to be provided by theory, which is the aim of the present paper.
Coupled-channel calculations as carried out for rotationally inelastic O 2 -H 2 collisions 24, 27-29 are not feasible for pH 2 -oH 2 conversion. In addition to the anisotropic O 2 -H 2 potential one has to include the hyperfine interaction operators in Equations (1) and (2), which act both on the spatial and spin coordinates. The channel bases containing the rotational states of O 2 -pH 2 with even j H 2 and those of O 2 -oH 2 with odd j H 2 would have to be combined and extended with the three electron spin functions of O 2 (S = 1) and the nuclear spin functions of pH 2 (I = 0) and oH 2 (I ′ = 1). This would increase the size of the channel basis by more than a factor of 10, the required computer memory by more than a factor of 100, and the cpu time by more than a factor of 1000, which implies that CC calculations become impractical.
In our approach, we take advantage of the large difference in coupling strength between the anisotropic terms in the O 2 -H 2 interaction potential that couple the rotational states of O 2 and H 2 (but not those with even and odd j H 2 ), and the much weaker hyperfine terms in Equations (1) and (2). We use the CC method to compute the wave functions for the rotationally inelastic collisions exactly, but separately for pH 2 and oH 2 , without including the hyperfine coupling terms. Next, we compute the cross sections for pH 2 -oH 2 transitions from the matrix elements of the hyperfine coupling terms over these scattering wave functions. This method is known as the distorted-wave Born approximation (DWBA), 30 and since both the Fermi contact coupling and the magnetic dipole-dipole coupling are about a million times smaller than the anisotropy in the O 2 -H 2 interaction potential, it is extremely accurate in this case. Calculating matrix elements involving scattering wave functions is also demanding, but we made use of an efficient algorithm developed in our group. 31 One problem remains, however. The spatial part of the magnetic dipole-dipole coupling in Equation (1)

RESULTS AND DISCUSSION
The O 2 molecule with total spin S = 1 in its electronic ground state has two unpaired electrons. They are coupled by a spin-spin coupling term of about 4 cm −1 and the total spin is coupled to the rotational angular momentum n O 2 by a much smaller spin-rotation coupling term to produce the total angular momentum j O 2 . These internal O 2 spin couplings are very small in comparison with the energy gaps of at least 120 cm −1 between the states of pH 2 and oH 2 , so we expect that their effect on the pH 2 -oH 2 conversion rate will be negligible. They might have some effect at low  Figure S3.
Although the j H 2 = 2 state is higher in energy than the j H 2 = 0 ground state by about 360 cm −1 (518 K), its population at room temperature is not much smaller than that of the ground state, because of the multiplicity factor 2j H 2 + 1 = 5. The initial states that we take into account are those with j H 2 = 0, 2 and n O 2 = 1, 3, … , n max , with n max = 15 for j H 2 = 0 and n max = 13 for j H 2 = 2. When j H 2 = 0 and O 2 is in its ground state with n O 2 = 1 there is an energy threshold of about 120 cm −1 to get to the lowest state of oH 2 which gradually decreases for increasing n O 2 . For j H 2 = 2 there is no threshold to get into the lowest j ′

State-to-state cross sections
Some typical cross sections for pH 2 -oH 2 conversion from the j = 0 ground state and the j = 2 excited state of pH 2 to the j ′ = 1 ground state of oH 2 are depicted in Figure 2, a more complete set is shown in Figures S1 and S2 of the Supporting Information. They are about twelve orders of magnitude smaller than the cross sections for rotationally inelastic collisions within pH 2 or oH 2 , cf. Figure 10 of Ref. [ 24 ]. These figures also illustrate the energy thresholds for pH 2 (j = 0)-oH 2 conversion: 120 cm −1 when the O 2 collision partner stays in its ground state with n = 1, and higher when O 2 is simultaneously excited. The threshold vanishes when O 2 is initially excited to n = 9 and gets de-excited to n ′ = 1, because of energy exchange between the collision partners. The cross sections for transitions from the excited j = 2 state of pH 2 to the j = 1 ′ state of oH 2 look similar to those for the j H 2 = 0 → 1 transitions, but they are smaller. They mostly have no energy threshold, so they yield a substantial contribution to the overall pH 2 -oH 2 conversion rate coefficient.
The cross sections for j H 2 = 0 → 3 transitions are not shown; they are smaller than those in Figure 2 by more than three orders of magnitude, because of the much larger energy gap of nearly 720 cm −1 . Moreover, such transitions occur only for collision energies above this gap, which makes their contribution negligible. The cross sections for j H 2 = 2 → 3 transitions are smaller than those for j H 2 = 2 → 1 transitions by about one order of magnitude and they have an energy threshold of about 360 cm −1 , so they yield a small but nonnegligible contribution to the conversion rate coefficient.
The two panels in Figure 2 also illustrate that the cross sections are largest for Δn = n ′ are nearly the same for all initial n O 2 . We return to this point in our discussion of the rate coefficients.
Another feature that can be seen in Figure 2 are the narrow peaks in the cross sections for collision energies just above the threshold. They correspond to scattering resonances, but have only a small effect on the rate coefficients except at temperatures below 50 K. F I G U R E 4 T-dependent pH 2 -oH 2 conversion rate coefficients for initial j H 2 = 0 and 2 (blue and red dashed lines), and the total rate (closed black line) obtained by Boltzmann averaging over these initial states. The value measured by Wagner 12 is indicated by a circle with error bars, the older experimental values of Farkas and Sachsse 6 by diamonds.

Rate coefficients
State-to-state rate coefficients for conversion of pH 2 into oH 2 for temperatures up to 400 K were obtained from the ab initio calculated state-tostate cross sections by numerical integration over the collision energy, cf. Equation (S63) in the Supporting Information. By exponential extrapolation of the integrand in Equation (S63) they can even be extended to higher temperatures.
We do not explicitly show all the state-to-state rate coefficients k n ′  Figure 4 not only shows the total rates, but also the contributions from the initial states with j H 2 = 0 and 2. One may observe that the rates for initial j H 2 = 0 increase with the temperature, while those for initial j H 2 = 2 are nearly temperature-independent. This is related to the energy threshold that the pH 2 (j = 0) states must overcome to convert into oH 2 (j = 1), while the pH 2 (j = 2) states mostly have no threshold. Wigner 8 already discussed this T-dependence. He only considered the initial pH 2 state with j = 0 and the final oH 2 state with j = 1 and he suggested an increase of the TA B L E 1 Contributions from magnetic dipole-dipole coupling and Fermi contact interactions to the j H 2 = 0 → 1 conversion rate coefficients (in L mol −1 min −1 ) at different temperatures conversion rate with temperature because of this energy threshold, which is more easily overcome when the temperature is higher and the colliding molecules have more kinetic energy. On the other hand, he also noted that there may be an opposite effect, because the collisions are shorter and probably less effective when the molecules move faster. It is clear now from Figure 4 that the j H 2 = 0 → 1, 3 rates are dominated by the threshold effect, while those for j H 2 = 2 → 1, 3 rates are nearly T-independent because the most dominant contributions do not experience any threshold.
Also the overall pH 2 -oH 2 conversion rate increases with T, but this is moderated at higher temperature by the contributions from the pH 2 (j = 2) state. The role of this pH 2 (j = 2) state was not previously considered, although its population at room temperature is nearly as high as that of the j H 2 = 0 ground state. Neither were the excited O 2 states that are substantially populated up to n O 2 = 15 at room temperature, while even higher O 2 states become excited during the collision. Both the magnitude of the conversion rate coefficient and its temperature dependence are significantly affected by the contributions from these excited states.
The contributions from the magnetic dipole-dipole coupling in Equation (1) and the Fermi contact interaction in Equation (2) to the pH 2 -oH 2 conversion rates are listed in Table 1. It turns out that these contributions are of comparable importance, so the suggestion of Ilisca and Sugano 17 and of Minaev and Ågren 18 that the Fermi contact interaction is much more important than the magnetic dipole-dipole coupling is not correct.
As mentioned in Section Para-ortho H 2 couplings, the Fermi contact interaction is a short range effect, while the magnetic dipole-dipole coupling acts also at larger O 2 -H 2 distances. Figure S4 in the Supporting Information illustrates that this is reflected by the calculated cross sections: the magnetic dipole-dipole coupling contributes to the conversion for higher values of the total angular momentum J than the Fermi contact coupling. Table 1 shows, however, that this difference is not important for the temperature dependence of the conversion rate.
Also the rate coefficients measured at room temperature by Wagner 12 in 2014, with error bars, and by Farkas and Sachsse 6 for several temperatures are shown in Figure 4. Our ab initio calculated value at room temperature lies just at the lower error bar of Wagner's result. The much older and, in view of the applied methods, probably less accurate measurements of Farkas and Sachsse, yielded somewhat higher values and no error bars were given. Their value at room temperature is higher than Wagner's more recent value and at all temperatures higher than our calculated values.
It is satisfactory, though, that their measured temperature dependence of the rate is nicely reproduced by our calculated data. We cannot present error bars for our calculated rate coefficients, but we may quote some values at room temperature from different calculations. In the calculations by which all the results discussed are obtained, we assumed that the electron spin S = 1 of O 2 is localized at the center of the molecule. In alternative calculations for some initial and final states we placed two spins s = 1∕2 at the O-nuclei. The rate coefficients from the latter calculations only differ from those presented here by about 1%. Another alternative considered was to use spin densities in the Fermi contact interaction in Equation (2) from ab initio electronic structure calculations at the CASSCF level instead of the MRCI level. The room temperature rate coefficient with the MRCI calculated Fermi contact term is 6.94 L mol −1 min −1 , with the CASSCF calculated term we estimate it to be 7.9 L mol −1 min −1 . The latter value agrees even better with the value of 8.27 ± 1.30 L mol −1 min −1 measured by Wagner, but we emphasize that it is less accurate according to theory because the CASSCF method takes less electron-electron correlation into account than the MRCI method.
Only the rate coefficients for the conversion of pH 2 to oH 2 are presented in this paper. The rates for the reverse process can easily be obtained by the detailed balance relation, Equation (S65) in the Supporting Information. We also computed some of the state-to-state oH 2 -pH 2 conversion rate coefficients by using the oH 2 states with odd j H 2 as the initial states and those of pH 2 with even j H 2 as the final states, and we found 32 that our results accurately obey this relation.

CONCLUSION
Finally, almost 90 years after the first measurement 6 of para-ortho H 2 conversion in collisions with paramagnetic molecules such as O 2 , a complete and detailed picture of this process has been obtained. Ab initio computations of the nuclear-spin-dependent interactions that cause the conversion, the magnetic dipole-dipole coupling proposed by Wigner 8 in 1933 and the Fermi contact coupling proposed later, 17,18 and the inclusion of both these interactions in nearly exact quantum mechanical coupled-channel calculations yield state-to-state collision cross sections and rate coefficients that agree with the most recent measurements within the error bars. A complete characterization of the conversion process is thus obtained and the observed temperature dependence of the rate coefficients is explained. It is established that the Fermi contact mechanism yields a substantial contribution but is not dominant, as suggested in Refs. [ 17,18 ]. With the efficient algorithm -three orders of magnitude faster than direct inclusion of the relevant spin-dependent coupling terms in CC calculations-and the software developed in this study, it will be also possible to study the effects of paramagnetic molecules and ions other than O 2 that may hinder the application of para-hydrogen induced polarization in NMR and MRI. The early measurements 6 have shown already that interesting and unexpected differences may occur, such as the observation that NO with electron spin S = 1∕2 is four times more effective in converting pH 2 into oH 2 than O 2 with spin S = 1.