## Abstract

Quantum dynamics calculation, performed on top of density functional theory (DFT)-based total energy calculations, show dynamical quantum filtering via enhanced scattering of *para*-H_{2} on SrTiO_{3}(001). We attribute this to the strongly orientation-dependent (electrostatic) interaction potential between the H_{2} (induced) quadrupole moment and the surface electric field gradient of ionic SrTiO_{3}(001). These results suggest that ionic surfaces could function as a scattering/filtering media to realize rotationally state-resolved H_{2}. This could find significant applications not only in H_{2} storage and transport, but also in realizing materials with pre-determined characteristic properties.

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## Introduction

The behavior/dynamics of H_{2} on surfaces strongly depend on the H_{2} 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), H_{2} 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 H_{2} (dynamical quantum filtering), and control the H_{2} dynamics (cf., e.g., refs. ^{1,2,3,4,5,6,7} and references therein). The resulting H_{2} dynamics could, in turn, be used to probe local surface reactivity^{8,9} (e.g., via the H_{2}(D_{2}) diffraction spectra^{10,11,12}). On ionic crystal surfaces, the H_{2} quadrupole moment interacts with the surface local electric field gradient to couple the translational and rotational degrees-of-freedom^{13}. On SrTiO_{3}(001) (STO(001)), an ionic crystal material^{14,15,16,17,18,19,20,21} with tunable surface terminations (cf., e.g., refs. ^{22,23}), H_{2} adsorbs with the H-H bond oriented parallel (polar coordinate *θ* = *π*/2) to the TiO_{2}-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 H_{2} with (hindered) rotational states (*J*, *m*) different from that of gas phase H_{2}. These strongly hindered adsorption states lead to (*J*, *m*)-dependent thermal desorption energies^{24,25,26,27,28,29,30}, suggesting the possibility of separating para-H_{2} [*p*-H_{2}(*J* = 0, *m* = 0)] and ortho-H_{2} [*o*-H_{2}(*J* = 1, *m* = ±1)] through an adsorption-desorption process. This could find significant applications not only in H_{2} storage and transport applications, but also in realizing materials with pre-determined characteristic properties.

As with metal (alloy) surfaces^{5,8,9}, the H_{2} 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*-H_{2} scatter strongly at specific angles from the TiO_{2}-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*-H_{2} and *p*-H_{2}, than the usual adsorption-desorption process^{1,2,3,4,5,24,25,26,27,28,29,30}.

## Results and Discussion

### H_{2}/SrTiO_{3}(001) System

Figure 1 shows a H_{2} interacting with STO(001). *X* gives the surface lateral coordinate of the H_{2} center-of-mass (CM) along the most corrugated directions on the two STO(001) terminations, viz., along [100] for TiO_{2}-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 H_{2} 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). *Θ*_{i(f)} gives H_{2} incidence (scattering) angle measured with respect to the surface normal (cf., Fig. 1(c,d)). In the following, given H_{2}(*J*_{i}, *m*_{i}, *E*_{i}, *Θ*_{i}), we determine the probability of finding H_{2}(*J*_{f}, *m*_{f}, *E*_{f}, *Θ*_{f}). H_{2}(*J*_{i}, *m*_{i}, *E*_{i}, *Θ*_{i}) indicates a H_{2} with initial rotational state (*J*_{i}, *m*_{i}), impinging STO(001) with an initial incident translational energy *E*_{i} and at an incidence angle *Θ*_{i} with respect to the surface normal. H_{2}(*J*_{f}, *m*_{f}, *E*_{f}, *Θ*_{f}) indicates a H_{2} with final rotational state (*J*_{f}, *m*_{f}), scattered from STO(001) with a final translational energy *E*_{f} and at a scattering angle *Θ*_{f} 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 *U*_{i} and the (induced) quadrupole moment with the gradient of the surface electric field *V*_{i,j}, for the TiO_{2}-terminated and SrO-terminated surfaces, respectively. We can see that the orientational (*θ*) anisotropy \(\Delta {E}_{{\rm{anisotropy}}}^{\theta }\) becomes important when the impinging H_{2} 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 TiO_{2}- 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 TiO_{2}- 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 H_{2} scattering dynamics, as we will discuss in detail in the next sections.

### H_{2} Scattering along STO(001)[100] on TiO_{2}-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 H_{2}(*J*_{i}, *m*_{i}, *E*_{i} = 80 meV, *Θ*_{i} = 15.9°) scattered as H_{2}(*J*_{f}, *m*_{f}, *E*_{f}, *Θ*_{f}) along the [100] direction of a TiO_{2}-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*-H_{2} to *o*-H_{2} ratio *R*_{J=0/J=1}(*E*_{i} = 80 meV, *Θ*_{i} = 15.9°, *Θ*_{f}) on a TiO_{2}-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 H_{2}(*J*_{i}, *m*_{i}, *E*_{i} = 80 meV, *Θ*_{i} = 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], H_{2}(*J*_{i} = 0, *m*_{i} = 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 *E*_{ads} = − 191 meV^{24}. Atop the O-site, H_{2} preferentially adsorbs with the H-H bond oriented perpendicular to the surface, with *E*_{ads} = − 72.5 meV^{24}. As a result, in Fig. 4(a), we find a corresponding spectra dominated by off-specularly scattered H_{2}, with dominant components coming from backscattering. Most of the H_{2}(*J*_{i} = 0, *m*_{i} = 0) would be molecularly adsorbed (due to *reorientation/steering*^{31,32}), and only a small fraction would be elastically scattered (due to *shadow effect*^{9}). Considering the angle of incidence, the inelastically scattered H_{2} 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 H_{2}, i.e., H_{2}(*J*_{f} = 2, *m*_{i} = 0). H_{2}(*J*_{i} = 1, *m*_{i} = ±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 H_{2}, with dominant components coming from backscattering. On the other hand, H_{2}(*J*_{i} = 1, *m*_{i} = 0) encounters an almost flat PES (cf., dotted curve in Fig. 4(b)). As a result, we see strong (dominant) specular scattering of H_{2}(*J*_{i} = 1, *m*_{i} = 0) (Fig. 4(b)). Finally, we could observe a *p*-H_{2} to *o*-H_{2} ratio as large as ca. 4.96 (cf., Table 1) compared to that of *normal*-H_{2} (*n*-H_{2}) (i.e., >1/3 at 300 K) at scattering angle *Θ*_{f} = 5.26° (Fig. 4).

### H_{2} 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 *R*_{J=0/J=1}(*E*_{i} = 60 meV, *Θ*_{i} = 76.5°, *Θ*_{f}), 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 H_{2}(*J*_{i}, *m*_{i}, *E*_{i} = 60 meV, *Θ*_{i} = 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), H_{2}(*J*_{i} = 0, *m*_{i} = 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 *E*_{ads} = −151 meV^{24}. Atop the Sr-site, H_{2} adsorbs with the H-H bond oriented parallel to the surface, and *E*_{ads} = −111 meV^{24}. As a result, in Fig. 5(a), we find a corresponding spectra dominated by off-specularly scattered H_{2}, with dominant components coming from backscattering. Again, most of the H_{2}(*J*_{i} = 0, *m*_{i} = 0) would be molecularly adsorbed (due to *reorientation/steering*^{31,32}), and only a small fraction would be elastically scattered (possibly due to *shadow effect*^{9}). Considering the angle of incidence, the inelastically scattered H_{2} 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 H_{2} has a larger surface lateral momentum as compared to the corresponding surface perpendicular component. This allows the impinging H_{2} 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 H_{2}, i.e., H_{2}(*J*_{f} = 2, *m*_{i} = 0). H_{2}(*J*_{i} = 1, *m*_{i} = ± 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). H_{2}(*J*_{i} = 1, *m*_{i} = 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*-H_{2} to *o*-H_{2} ratio as large as ca. 16.1 (cf., Table 2) compared to that of *n*-H_{2} (i.e., >1/3 at 300 K) at scattering angle *Θ*_{f} = − 76.5° (Fig. 5).

### (Normalized) *p*-H_{2} to *o*-H_{2} Ratios

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

## Conclusion

Here, we reported increased *para*-H_{2} (*p*-H_{2}) to *ortho*-H_{2} (*o*-H_{2}) ratio of H_{2} scattered from SrTiO_{3}(001) (viz., ca. 4.96 and 16.1 along the SrTiO_{3}(001)[100] and SrTiO_{3}(001)[110] of the TiO_{2}- and SrO-terminated SrTiO_{3}(001), respectively). For reference, *normal*-H_{2} (*n*-H_{2}) have a *p*-H_{2} to *o*-H_{2} ratio of 1/3, at room temperature (*T* = 300 K). We attribute this to the strongly orientation-dependent (electrostatic) interaction potential between the H_{2} (induced) quadrupole moment and the surface electric field of ionic SrTiO_{3}(001). These results suggest that ionic surfaces (with tunable surface terminations) could function as a scattering/filtering media to realize rotationally state-resolved H_{2}. This could find significant applications not only in H_{2} 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 H_{2} on LiF(001), at normal incidence (*Θ*_{i} = 0°) and incident energy of *E*_{i} = 100 meV^{33}. From the scattering probability data^{33}, we estimate a maximum *p*-H_{2} to *o*-H_{2} ratio *R*_{J=0/J=1} of ca. 2 (as compared to ca. 4.96 and 16.1 along SrTiO_{3}(001)[100] and SrTiO_{3}(001)[110] of the TiO_{2}- and SrO-terminated SrTiO_{3}(001), respectively). (Note that the results of ref. ^{33} have since been confirmed experimentally^{34}).

Considering the Debye temperature of SrTiO_{3} (ca. 413.3 K^{35}), we may expect some thermal modulation/attenuation. With Ti having a smaller mass than Sr, we would expect more pronounced effect (modulation/attenuation) on TiO_{2}-terminated SrTiO_{3}(001), as compared to SrO-terminated SrTiO_{3}(001). Consider for example a surface temperature of 100 K and incidence angle *Θ*_{i}: [0°, 66°]. For TiO_{2}-terminated SrTiO_{3}(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 *Θ*_{i}. In comparison, for SrO-terminated SrTiO_{3}(001), we estimated^{12} Debye-Waller factor values ranging from ca. 0.6 to 0.8, increasing in value with increasing incidence angle *Θ*_{i}. The modulation/attenuation would become more pronounced with increased temperature. However, regardless of the degree of modulation/attenuation, the maximum *p*-H_{2} to *o*-H_{2} ratio for each surface termination remained almost the same (viz., ca. 4.48 and 16.7 along the SrTiO_{3}(001)[100] and SrTiO_{3}(001)[110] of the TiO_{2}- and SrO-terminated SrTiO_{3}(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*-H_{2} to *o*-H_{2} 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 H_{2} 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 H_{2} 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 H_{2}, in the vibrational ground state (*ν* = 0), under the influence of an orientationally anisotropic potential energy (hyper-) surface (PES), using the coupled-channel method^{36,37,38,39,40,41,42}. The dynamical variables we considered include the H_{2} center-of-mass (CM) distance *Z* from the surface, the H_{2} 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 H_{2} CM *X*, along STO(001)[100] and STO(001)[110] of the TiO_{2}- and SrO-terminated STO(001), respectively. Considering that the energy scale of the H_{2} molecular vibration (*ℏ**ω* = 516 meV) exceeds the energy range relevant to our current study (i.e., *E*_{i}: [10, 80] meV), we can neglect the molecular vibrational excitations, and fix the H_{2} 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*-dependence^{24}. Thus, we can reduce the original 6-Dimensional (6-D) Hamiltonian (for the diatomic molecule-surface system) to the following simplified 3-D form :

*m* and *I* correspond to the H_{2} total mass and moment of inertia, respectively.

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

The 3-D PES *V*(*Z*, *θ*, *X*) in Eq. (1) comes from previously performed density functional theory (DFT)-based total energy calculations^{43,44,45} for H_{2} adsorption at the Ti and O surface sites on the TiO_{2}-terminated STO(001), the Sr and O surface sites on the SrO-terminated STO(001), and H_{2} polar orientations *θ* = 0 and *π*/2^{24}. We fitted the potential energy curves for each configuration using Morse potentials:

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

*D*_{A(B),θ}, *α*_{A(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 H_{2} from the surface cation sites (A:[Ti, Sr]) and oxygen site (B), at H-H bond angle *θ*, respectively. The reciprocal lattice constant *g*_{X}(= *π*/*a*_{X}), with corresponding direct lattice constants \({a}_{X}^{[100]}=3.91\) Å and \({a}_{X}^{[110]}=5.52\) Å along [100] and [110] of the TiO_{2}- and SrO-terminated STO(001), respectively^{46}. 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 H_{2} quadrupole moment and the surface local electric field, as discussed above, we used the charge density distribution obtained from previous DFT-based total energy calculations^{24}. We calculated the induced dipole Δ*D*_{i} (*i* = *x*, *y*, *z*) and quadrupole moments Δ*Q*_{i,j} (*i*, *j* = *x*, *y*, *z*) 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 (TiO_{2}-termination) and Sr and O sites (SrO-termination) with *θ* = 0 and *π*/2. We also calculated the H_{2} quadrupole moment *Q*_{i,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 H_{2} impinging with an initial rotational state (*J*_{i}, *m*_{i}), incident energy *E*_{i}, and angle of incidence with respect to the surface normal *Θ*_{i} (cf., e.g., Fig. 1(c,d), and Table 4). Using the coupled-channel method^{36,37,38,39,40,41,42}, 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 H_{2} scattered with a final rotational state (*J*_{f}, *m*_{f}), final kinetic energy *E*_{f}, scattering at an angle of *Θ*_{f} 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 H_{2} on STO(001), we evaluated the corresponding *p*-H_{2} to *o*-H_{2} 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 *E*_{i}, incident angle *Θ*_{i}, and scattering angle *Θ*_{f}, i.e.,

Note that

And since we are considering the case when *J*_{i} = *J*_{f} and *m*_{i} = *m*_{f}, in Eq. (4) we have

We also calculated the normalized *p*-H_{2} to *o*-H_{2} ratio \(\bar{{R}_{J=0/J=1}}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})\), i.e.,

In Eq. (7), we multiplied the angle specific *p*-H_{2} to *o*-H_{2} ratio by a normalization factor, so as to evaluate the efficiency of rotational state separation with respect to the incident *n*-H_{2}. Note that the *p*-H_{2} to *o*-H_{2} ratio of *n*-H_{2} corresponds to 0.333 (1/3) at room temperature (*T* = 300 K)

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## Acknowledgements

This work is supported in part by MEXT Grants-in-Aid for Scientific Research (18H05518, 17K06818, 17H01057, 15H05736, 15KT0062, 15K14147, and 26248006); NEDO Project “R&D Towards Realizing an Innovative Energy Saving Hydrogen Society based on Quantum Dynamics Applications”, Grants-in-Aid for JSPS Fellows; and Kawasaki Heavy Industries, Ltd. Some of the numerical calculations presented here done using the computer facilities at the following institutes: CMC (Osaka University), ISSP, KEK, NIFS, and YITP. Structure figures were plotted using the VESTA package^{47}.

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K.S., W.A.D., H.N., K.F., and H.K. conceived the model and performed the calculation. All authors contributed to the discussion, analyses, and writing the manuscript.

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A.Y. is an employee of Kawasaki Heavy Industries, Ltd. The other authors declare no competing interests.

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Shimizu, K., Diño, W.A., Nakanishi, H. *et al.* Dynamical Quantum Filtering via Enhanced Scattering of *para*-H_{2} on the Orientationally Anisotropic Potential of SrTiO_{3}(001).
*Sci Rep* **10**, 5939 (2020). https://doi.org/10.1038/s41598-020-62605-8

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DOI: https://doi.org/10.1038/s41598-020-62605-8

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