## Introduction

The behavior/dynamics of H2 on surfaces strongly depend on the H2 molecular orientation/rotational states (rotational quantum number J, magnetic quantum number m) (cf., e.g., ref. 1 and references therein). On transition metal surfaces such as copper (Cu) and palladium (Pd), H2 exhibits rotational alignment (cf., e.g., refs. 1,2 and references therein). One could then tune or design the structure of the metal (alloy) surfaces to dynamically filter the quantum rotational states of desorbing or scattered H2 (dynamical quantum filtering), and control the H2 dynamics (cf., e.g., refs. 1,2,3,4,5,6,7 and references therein). The resulting H2 dynamics could, in turn, be used to probe local surface reactivity8,9 (e.g., via the H2(D2) diffraction spectra10,11,12). On ionic crystal surfaces, the H2 quadrupole moment interacts with the surface local electric field gradient to couple the translational and rotational degrees-of-freedom13. On SrTiO3(001) (STO(001)), an ionic crystal material14,15,16,17,18,19,20,21 with tunable surface terminations (cf., e.g., refs. 22,23), H2 adsorbs with the H-H bond oriented parallel (polar coordinate θ = π/2) to the TiO2-terminated surface at the Ti-site, and perpendicular (θ = 0) to the SrO-terminated surface on top of the O-site (cf., Fig. 1)24. The strong orientationally anisotropic potential ($$\Delta {E}_{{\rm{anisotropy}}}^{\theta }=| {E}_{\theta =0}-{E}_{\theta =\pi /2}|$$) results in adsorbed H2 with (hindered) rotational states (Jm) different from that of gas phase H2. These strongly hindered adsorption states lead to (Jm)-dependent thermal desorption energies24,25,26,27,28,29,30, suggesting the possibility of separating para-H2 [p-H2(J = 0, m = 0)] and ortho-H2 [o-H2(J = 1, m = ±1)] through an adsorption-desorption process. This could find significant applications not only in H2 storage and transport applications, but also in realizing materials with pre-determined characteristic properties.

As with metal (alloy) surfaces5,8,9, the H2 dynamics would be susceptible to the positive and negative charges that corrugate ionic crystal surfaces. In the following, we will show that on STO(001), under the influence of the orientationally anisotropic potential, on top of the surface lateral corrugation, p-H2 scatter strongly at specific angles from the TiO2-terminated and SrO-terminated STO(001). This dynamical filtering/scattering selectivity allows for more economical (less heat consumption) and more efficient means to rotationally separate o-H2 and p-H2, than the usual adsorption-desorption process1,2,3,4,5,24,25,26,27,28,29,30.

## Results and Discussion

### H2/SrTiO3(001) System

Figure shows a H2 interacting with STO(001). X gives the surface lateral coordinate of the H2 center-of-mass (CM) along the most corrugated directions on the two STO(001) terminations, viz., along [100] for TiO2-termination (Ti-O-Ti row) and along [110] for SrO-termination (Sr-O-Sr row), respectively (cf., Fig. 1(a,b)). Z gives the normal distance of the H2 CM from the surface. θ gives the polar angular orientation of the H-H bond with respect to the surface normal. ϕ (not shown) gives the azimuthal angular orientation of the H-H bond about the surface normal, with respect to the X-axis, at each site on STO(001). Θi(f) gives H2 incidence (scattering) angle measured with respect to the surface normal (cf., Fig. 1(c,d)). In the following, given H2(JimiEiΘi), we determine the probability of finding H2(JfmfEfΘf). H2(JimiEiΘi) indicates a H2 with initial rotational state (Jimi), impinging STO(001) with an initial incident translational energy Ei and at an incidence angle Θi with respect to the surface normal. H2(JfmfEfΘf) indicates a H2 with final rotational state (Jfmf), scattered from STO(001) with a final translational energy Ef and at a scattering angle Θf with respect to the surface normal.

### Orientationally anisotropic electrostatic potential

In Figs. 2 and 3, we plot the (electrostatic interaction energies) dot products of the (induced) dipole moment with the surface electric field Ui and the (induced) quadrupole moment with the gradient of the surface electric field Vi,j, for the TiO2-terminated and SrO-terminated surfaces, respectively. We can see that the orientational (θ) anisotropy $$\Delta {E}_{{\rm{anisotropy}}}^{\theta }$$ becomes important when the impinging H2 comes sufficiently near the surface, viz., at Z ≤ 2.4 Å above the Ti-site and Z ≤ 2.6 Å above the O-site on the TiO2- and SrO-terminated surfaces, respectively. On the other hand, far from the surface, viz., at Z ≥ 3.2 Å above the O-site and Z ≥ 3.0 Å above the Sr-site of the TiO2- and SrO-terminated surfaces, respectively, only a small $$\Delta {E}_{{\rm{anisotropy}}}^{\theta }$$ can be observed. This orientational anisotropy $$\Delta {E}_{{\rm{anisotropy}}}^{\theta }$$, on top of the surface lateral corrugation, would prove to be useful in our attempt to control the H2 scattering dynamics, as we will discuss in detail in the next sections.

### H2 Scattering along STO(001)[100] on TiO2-terminated STO(001)

In Fig. 4(a–c), we show the calculated (scattering) probabilities/spectra $${P}_{{J}_{{\rm{i}}}\to {J}_{{\rm{f}}}}^{{m}_{{\rm{i}}}\to {m}_{{\rm{f}}}}({E}_{{\rm{i}}}=80\,{\rm{meV}},{E}_{{\rm{f}}},{\varTheta }_{{\rm{i}}}={15.9}^{\circ },{\varTheta }_{{\rm{f}}})$$ of finding H2(JimiEi = 80 meV, Θi = 15.9°) scattered as H2(JfmfEfΘf) along the [100] direction of a TiO2-terminated STO(001). (The corresponding initial surface perpendicular translational energy $${E}_{{\rm{i}}}\times {\cos }^{2}{\varTheta }_{{\rm{i}}}=80\,{\rm{meV}}\times {\cos }^{2}({15.9}^{\circ }) \sim 74.0$$ meV and initial surface lateral translational energy $${E}_{{\rm{i}}}\times {\sin }^{2}{\varTheta }_{{\rm{i}}}=80\,{\rm{meV}}\times {\sin }^{2}({15.9}^{\circ }) \sim 6.00$$ meV). This corresponds to the maximum p-H2 to o-H2 ratio RJ=0/J=1(Ei = 80 meV, Θi = 15.9°, Θf) on a TiO2-terminated STO(001) (cf., Table 1). The resulting trends can be explained by inspecting the corresponding orientational anisotropy and surface lateral corrugation of the potential encountered by the impinging H2(JimiEi = 80 meV, Θi = 15.9°) (i.e., $$\langle {Y}_{J}^{m}| V(Z,\theta ,X)| {Y}_{J}^{m}\rangle$$, cf., Fig. 4(d)). Note that along STO(001)[100], H2(Ji = 0, mi = 0) encounters a strongly corrugated and orientationally anisotropic PES (cf., solid curve in Fig. 4(d)), that favors molecular adsorption atop the Ti-site, with H-H bond oriented parallel to the surface, and an adsorption energy Eads = − 191 meV24. Atop the O-site, H2 preferentially adsorbs with the H-H bond oriented perpendicular to the surface, with Eads = − 72.5 meV24. As a result, in Fig. 4(a), we find a corresponding spectra dominated by off-specularly scattered H2, with dominant components coming from backscattering. Most of the H2(Ji = 0, mi = 0) would be molecularly adsorbed (due to reorientation/steering31,32), and only a small fraction would be elastically scattered (due to shadow effect9). Considering the angle of incidence, the inelastically scattered H2 would come from those hitting the repulsive part of the potential well near the O-site (shadow effect, cf., solid curve in Fig. 4(d)), resulting in a change in surface lateral momentum and off-specular scattering angles. Energy transfer from the translational degree-of-freedom (DOF) to the rotational DOF allows the molecule more time to sample the anisotropic surface through reorientation/steering. Those that succeed would molecularly adsorb. Those that fail, would be rotationally de-excited on the way back to the gas phase, because of the smaller anisotropic potential further (out in the vacuum) from the surface. Thus, the scattering spectra shows negligible rotationally excited H2, i.e., H2(Jf = 2, mi = 0). H2(Ji = 1, mi = ±1) also encounters a strongly anisotropic PES (cf., dash-dot curve in Fig. 4(d)). Thus, we see the same trend in Fig. 4(c), i.e., a corresponding spectra dominated by off-specularly scattered H2, with dominant components coming from backscattering. On the other hand, H2(Ji = 1, mi = 0) encounters an almost flat PES (cf., dotted curve in Fig. 4(b)). As a result, we see strong (dominant) specular scattering of H2(Ji = 1, mi = 0) (Fig. 4(b)). Finally, we could observe a p-H2 to o-H2 ratio as large as ca. 4.96 (cf., Table 1) compared to that of normal-H2 (n-H2) (i.e., >1/3 at 300 K) at scattering angle Θf = 5.26° (Fig. 4).

### H2 Scattering along STO(001)[110] on SrO-terminated STO(001)

In Fig. 5(a–c), we show the calculated (scattering) probabilities/spectra corresponding to the maximum para-to-ortho ratio RJ=0/J=1(Ei = 60 meV, Θi = 76.5°, Θf), along the [110] direction of a SrO-terminated STO(001) (cf., Table 2). (The initial surface perpendicular translational energy $${E}_{{\rm{i}}}\times {\cos }^{2}{\varTheta }_{{\rm{i}}}=60\,{\rm{meV}}\times {\cos }^{2}({76.5}^{\circ }) \sim 3.27\,{\rm{meV}}$$ and the initial surface lateral translational energy $${E}_{{\rm{i}}}\times {\sin }^{2}{\varTheta }_{{\rm{i}}}=60\,{\rm{meV}}\times {\sin }^{2}({76.5}^{\circ }) \sim 56.7\,{\rm{meV}}$$.) Again, the resulting trends can be explained by inspecting the corresponding orientational anisotropy and surface lateral corrugation of the potential encountered by the impinging H2(JimiEi = 60 meV, Θi = 76.5°) (i.e., $$\langle {Y}_{J}^{m}| V(Z,\theta ,X)| {Y}_{J}^{m}\rangle$$, cf., Fig. 5(d)). Note that along STO(001)[110] on the SrO-terminated STO(001), H2(Ji = 0, mi = 0) encounters a strongly corrugated and orientationally anisotropic PES (cf., solid curve in Fig. 5(d)), that now favors molecular adsorption atop the O-site, with H-H bond oriented perpendicular to the surface, and an adsorption energy Eads = −151 meV24. Atop the Sr-site, H2 adsorbs with the H-H bond oriented parallel to the surface, and Eads = −111 meV24. As a result, in Fig. 5(a), we find a corresponding spectra dominated by off-specularly scattered H2, with dominant components coming from backscattering. Again, most of the H2(Ji = 0, mi = 0) would be molecularly adsorbed (due to reorientation/steering31,32), and only a small fraction would be elastically scattered (possibly due to shadow effect9). Considering the angle of incidence, the inelastically scattered H2 would come from those hitting the repulsive part of the potential well near the Sr-site (shadowing effect, cf., solid curve in Fig. 4(d)), resulting in a change in surface lateral momentum and off-specular scattering angles. Note that the impinging H2 has a larger surface lateral momentum as compared to the corresponding surface perpendicular component. This allows the impinging H2 more time to explore the anisotropic surface. Those that succeed would molecularly adsorb via reorientation/steering. Those that fail, would be rotationally de-excited on the way back to the gas phase, because of the smaller anisotropic potential further (out in the vacuum) from the surface. Thus, the scattering spectra shows negligible rotationally excited H2, i.e., H2(Jf = 2, mi = 0). H2(Ji = 1, mi = ± 1) also encounters a strongly anisotropic PES (cf., dash-dot curve in Fig. 5(d)). But now recall that the preferred adsorption site is at the O-site, with the H-H bond oriented perpendicular to the surface. Thus, we see strong specular scattering in Fig. 5(c). H2(Ji = 1, mi = 0) shows higher backscattering probabilities (cf., Fig. 5(b)) due to the larger surface anisotropy along X, making it more susceptible to reorientation/steering. Finally, we could observe a p-H2 to o-H2 ratio as large as ca. 16.1 (cf., Table 2) compared to that of n-H2 (i.e., >1/3 at 300 K) at scattering angle Θf = − 76.5° (Fig. 5).

### (Normalized) p-H2 to o-H2 Ratios

In Tables 1 and 2, we show the calculated p-H2 to o-H2 ratio RJ=0/J=1(EiΘiΘf) and the normalized p-H2 to o-H2 ratio $$\bar{{R}_{J=0/J=1}}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})$$ of H2 scattered from the TiO2-terminated and SrO-terminated STO(001), respectively. The incident energy ranges from 10 meV ≤ Ei ≤ 80 meV. Assuming Maxwell-Boltzmann distribution, at 300 K, 90% of the impinging H2 would have kinetic energies less than 80 meV. In general, the SrO-terminated STO(001) show a much higher p-H2 yield compared to the TiO2-terminated STO(001). On TiO2-terminated STO(001), we observe a maximum p-H2 to o-H2 ratio of $${R}_{J=0/J=1}^{\max }=4.96$$ for Ei = 80.0 meV, Θi = 15.9°, Θf = 5.26°. Whereas on SrO-terminated STO(001), we observe a maximum p-H2 to o-H2 ratio of $${R}_{J=0/J=1}^{\max }=16.1$$ for Ei = 60 meV, Θi = 76.5°, Θf = − 76.5°. The results suggest that, for all incident energies Ei considered, the p-H2 to o-H2 ratios at specific scattering angles Θf exceed that of n-H2, i.e., RJ=0/J=1(EiΘiΘf) > 1/3, at room temperature (T = 300 K).

## Conclusion

Here, we reported increased para-H2 (p-H2) to ortho-H2 (o-H2) ratio of H2 scattered from SrTiO3(001) (viz., ca. 4.96 and 16.1 along the SrTiO3(001)[100] and SrTiO3(001)[110] of the TiO2- and SrO-terminated SrTiO3(001), respectively). For reference, normal-H2 (n-H2) have a p-H2 to o-H2 ratio of 1/3, at room temperature (T = 300 K). We attribute this to the strongly orientation-dependent (electrostatic) interaction potential between the H2 (induced) quadrupole moment and the surface electric field of ionic SrTiO3(001). These results suggest that ionic surfaces (with tunable surface terminations) could function as a scattering/filtering media to realize rotationally state-resolved H2. This could find significant applications not only in H2 storage and transport, but also in realizing materials with pre-determined characteristic properties.

We can compare the present results to previous reports for the inelastic scattering of H2 on LiF(001), at normal incidence (Θi = 0°) and incident energy of Ei = 100 meV33. From the scattering probability data33, we estimate a maximum p-H2 to o-H2 ratio RJ=0/J=1 of ca. 2 (as compared to ca. 4.96 and 16.1 along SrTiO3(001)[100] and SrTiO3(001)[110] of the TiO2- and SrO-terminated SrTiO3(001), respectively). (Note that the results of ref. 33 have since been confirmed experimentally34).

Considering the Debye temperature of SrTiO3 (ca. 413.3 K35), we may expect some thermal modulation/attenuation. With Ti having a smaller mass than Sr, we would expect more pronounced effect (modulation/attenuation) on TiO2-terminated SrTiO3(001), as compared to SrO-terminated SrTiO3(001). Consider for example a surface temperature of 100 K and incidence angle Θi: [0°, 66°]. For TiO2-terminated SrTiO3(001), we estimated Debye-Waller factor values (cf., e.g., ref. 12 for more details) ranging from ca. 0.3 to 0.6, increasing with increasing incidence angle Θi. In comparison, for SrO-terminated SrTiO3(001), we estimated12 Debye-Waller factor values ranging from ca. 0.6 to 0.8, increasing in value with increasing incidence angle Θi. The modulation/attenuation would become more pronounced with increased temperature. However, regardless of the degree of modulation/attenuation, the maximum p-H2 to o-H2 ratio for each surface termination remained almost the same (viz., ca. 4.48 and 16.7 along the SrTiO3(001)[100] and SrTiO3(001)[110] of the TiO2- and SrO-terminated SrTiO3(001), respectively, at 100 K).

Finally, because of the wide spread of the scattering angle, we could expect small scattering probabilities at each particular scattering angle. Furthermore, the larger the (surface lateral) incident energy, the more number of (surface lateral) diffraction channels (would be) involved (excited). As a result, the normalized p-H2 to o-H2 ratio $$\bar{{R}_{J=0/J=1}}$$ shows small values (cf., Tables 1 and 2). Thus, so far, we can collect only a small amount of H2 through any given scattering event. (But with high purity!) For engineering applications, scattering at optimum surface angles should resolve this problem.

## Methods

### Model Hamiltonian

To study the dynamics of H2 scattering on STO(001) (Fig. 1), we performed quantum dynamical calculations (cf., e.g., refs. 1,2,3,4,5,8,9,24,31,32) by solving the corresponding time-independent Schrödinger equation for H2, in the vibrational ground state (ν = 0), under the influence of an orientationally anisotropic potential energy (hyper-) surface (PES), using the coupled-channel method36,37,38,39,40,41,42. The dynamical variables we considered include the H2 center-of-mass (CM) distance Z from the surface, the H2 bond-length r, the polar and azimuthal angular orientations of the H-H bond with respect to the surface, θ and ϕ, respectively, and the position of the H2 CM X, along STO(001)[100] and STO(001)[110] of the TiO2- and SrO-terminated STO(001), respectively. Considering that the energy scale of the H2 molecular vibration (ω = 516 meV) exceeds the energy range relevant to our current study (i.e., Ei: [10, 80] meV), we can neglect the molecular vibrational excitations, and fix the H2 interatomic distance at r = 0.74 Å. The small variation of the potential energy with respect to ϕ allows us to further neglect the ϕ-dependence24. Thus, we can reduce the original 6-Dimensional (6-D) Hamiltonian (for the diatomic molecule-surface system) to the following simplified 3-D form :

$$H=-\frac{{\hslash }^{2}}{2m}\left(\frac{{{\rm{\partial }}}^{2}}{{\rm{\partial }}{Z}^{2}},+,\frac{{{\rm{\partial }}}^{2}}{{\rm{\partial }}{X}^{2}}\right)-\frac{{\hslash }^{2}}{2I}\left[\frac{1}{\sin \theta },\frac{{\rm{\partial }}}{{\rm{\partial }}\theta },(\sin \theta \frac{{\rm{\partial }}}{{\rm{\partial }}\theta }),+,\frac{1}{{\sin }^{2}\theta },\frac{{{\rm{\partial }}}^{2}}{{\rm{\partial }}{\phi }^{2}}\right]+V(Z,\theta ,X).$$
(1)

m and I correspond to the H2 total mass and moment of inertia, respectively.

### Potential Energy (Hyper-) Surface: PES

The 3-D PES V(ZθX) in Eq. (1) comes from previously performed density functional theory (DFT)-based total energy calculations43,44,45 for H2 adsorption at the Ti and O surface sites on the TiO2-terminated STO(001), the Sr and O surface sites on the SrO-terminated STO(001), and H2 polar orientations θ = 0 and π/224. We fitted the potential energy curves for each configuration using Morse potentials:

$${V}_{{\rm{A}}({\rm{B}}),\theta }={D}_{{\rm{A}}({\rm{B}}),\theta }\times \{\exp [-\,2{\alpha }_{{\rm{A}}({\rm{B}}),\theta }(Z-{Z}_{{\rm{A}}({\rm{B}}),\theta }^{{\rm{e}}{\rm{q}}})]-2\exp [{\alpha }_{{\rm{A}}({\rm{B}}),\theta }(Z-{Z}_{{\rm{A}}({\rm{B}}),\theta }^{{\rm{e}}{\rm{q}}})]\},$$
(2)

and connected the Morse potentials for θ = 0 and π/2 at each surface site and for different surface sites to get

$$V(Z,\theta ,X)=[{V}_{{\rm{A}},\theta =\pi /2}{\sin }^{2}\theta +{V}_{{\rm{A}},\theta =0}{\cos }^{2}\theta ]{\cos }^{2}gX+[{V}_{{\rm{B}},\theta =\pi /2}{\sin }^{2}\theta +{V}_{{\rm{B}},\theta =0}{\cos }^{2}\theta ]{\sin }^{2}gX.$$
(3)

DA(B),θ, αA(B),θ, $${Z}_{{\rm{A}}({\rm{B}}),\theta }^{{\rm{e}}{\rm{q}}}$$ give the corresponding potential depth, potential width, and equilibrium (normal/perpendicular) distance of H2 from the surface cation sites (A:[Ti, Sr]) and oxygen site (B), at H-H bond angle θ, respectively. The reciprocal lattice constant gX(= π/aX), with corresponding direct lattice constants $${a}_{X}^{[100]}=3.91$$ Å and $${a}_{X}^{[110]}=5.52$$ Å along [100] and [110] of the TiO2- and SrO-terminated STO(001), respectively46. Table 3 shows the fitted parameters for each configuration. We used spherical harmonics and plane waves as basis sets for the rotational motion and the translational motion (perpendicular to the surface and along the surface direction X), respectively.

To extract the contribution of the electrostatic interaction between the H2 quadrupole moment and the surface local electric field, as discussed above, we used the charge density distribution obtained from previous DFT-based total energy calculations24. We calculated the induced dipole ΔDi (i = xyz) and quadrupole moments ΔQi,j (ij = xyz) from the charge density difference ($$\Delta \rho (Z)={\rho }_{{{\rm{H}}}_{2}/STO}(Z)-{\rho }_{{{\rm{H}}}_{2}}-{\rho }_{{\rm{STO}}}$$) as a function of Z for the Ti and O sites (TiO2-termination) and Sr and O sites (SrO-termination) with θ = 0 and π/2. We also calculated the H2 quadrupole moment Qi,j from the charge density distribution of the isolated system. We used the pristine STO(001) to calculate the surface electric field and its gradient.

### Scattering probability

Consider a H2 impinging with an initial rotational state (Jimi), incident energy Ei, and angle of incidence with respect to the surface normal Θi (cf., e.g., Fig. 1(c,d), and Table 4). Using the coupled-channel method36,37,38,39,40,41,42, we calculated the probability $${P}_{{J}_{{\rm{i}}}\to {J}_{{\rm{f}}}}^{{m}_{{\rm{i}}}\to {m}_{{\rm{f}}}}({E}_{{\rm{i}}},{E}_{{\rm{f}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})$$ of finding the H2 scattered with a final rotational state (Jfmf), final kinetic energy Ef, scattering at an angle of Θf with respect to the surface normal. (We carefully checked the convergence for calculations with maximum quantum numbers $${J}_{\max }=10$$ and $$| {G}_{{X}_{\max }}| =30$$). From the calculated scattering probability $${P}_{{J}_{{\rm{i}}}\to {J}_{{\rm{f}}}}^{{m}_{{\rm{i}}}\to {m}_{{\rm{f}}}}({E}_{{\rm{i}}},{E}_{{\rm{f}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})$$ of H2 on STO(001), we evaluated the corresponding p-H2 to o-H2 ratio $${R}_{{p-{\rm{H}}}_{2}(J=0)/{o-{\rm{H}}}_{2}(J=1)}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})={R}_{J=0/J=1}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})$$, given the incident energy Ei, incident angle Θi, and scattering angle Θf, i.e.,

$${R}_{J=0/J=1}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})=\frac{\int d{E}_{{\rm{f}}}\,\,{P}_{{J}_{{\rm{i}}}=0\to {J}_{{\rm{f}}}=0}^{{m}_{{\rm{i}}}=0\to {m}_{{\rm{f}}}=0}({E}_{{\rm{i}}},{E}_{{\rm{f}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})}{\int d{E}_{{\rm{f}}}{\sum }_{m=-1}^{1}{P}_{{J}_{{\rm{i}}}=1\to {J}_{{\rm{f}}}=1}^{{m}_{{\rm{i}}}\to {m}_{{\rm{f}}}={m}_{{\rm{i}}}}({E}_{{\rm{i}}},{E}_{{\rm{f}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})}=\frac{{P}_{0}^{0}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})}{{\sum }_{m=-1}^{1}{P}_{1}^{m}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})}.$$
(4)

Note that

$$\int \ d{E}_{{\rm{f}}}\,\sum _{{J}_{{\rm{f}}},{m}_{{\rm{f}}},{\varTheta }_{{\rm{f}}}}{P}_{{J}_{{\rm{i}}}\to {J}_{{\rm{f}}}}^{{m}_{{\rm{i}}}\to {m}_{{\rm{f}}}}({E}_{{\rm{i}}},{E}_{{\rm{f}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})=1.$$
(5)

And since we are considering the case when Ji = Jf and mi = mf, in Eq. (4) we have

$${P}_{{J}_{{\rm{i}}}\to {J}_{{\rm{f}}}}^{{m}_{{\rm{i}}}\to {m}_{{\rm{f}}}}({E}_{{\rm{i}}},{E}_{{\rm{f}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})={P}_{{J}_{{\rm{i}}}\to {J}_{{\rm{f}}={\rm{i}}}}^{{m}_{{\rm{i}}}\to {m}_{{\rm{f}}={\rm{i}}}}({E}_{{\rm{i}}},{E}_{{\rm{f}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})={P}_{J}^{m}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}}).$$
(6)

We also calculated the normalized p-H2 to o-H2 ratio $$\bar{{R}_{J=0/J=1}}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})$$, i.e.,

$$\bar{{R}_{J=0/J=1}}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})={R}_{J=0/J=1}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})\times \frac{{P}_{0}^{0}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})+{\sum }_{m=-1}^{1}{P}_{1}^{m}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})}{{\sum }_{{\varTheta }_{{\rm{f}}}=-9{0}^{\circ }}^{9{0}^{\circ }}[{P}_{0}^{0}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})+{\sum }_{m=-1}^{1}{P}_{1}^{m}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})]}.$$
(7)

In Eq. (7), we multiplied the angle specific p-H2 to o-H2 ratio by a normalization factor, so as to evaluate the efficiency of rotational state separation with respect to the incident n-H2. Note that the p-H2 to o-H2 ratio of n-H2 corresponds to 0.333 (1/3) at room temperature (T = 300 K)