Dynamical Quantum Filtering via Enhanced Scattering of para-H₂ on the Orientationally Anisotropic Potential of SrTiO₃(001)

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-H₂ on SrTiO₃(001). We attribute this to the strongly orientation-dependent (electrostatic) interaction potential between the H₂ (induced) quadrupole moment and the surface electric field gradient of ionic SrTiO₃(001). These results suggest that ionic surfaces 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.

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 reactivity^8,9 (e.g., via the H₂(D₂) diffraction spectra^10,11,12). 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 material^{14,15,16,17,18,19,20,21} 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 (J, m) different from that of gas phase H₂. 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-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.

Figure 1

A depiction of H₂ adsorption (a) atop the Ti-site and (b) atop the O-site on the TiO₂- and SrO-terminated SrTiO₃(001), with corresponding preferential orientations θ = π/2 and 0, respectively. The lower left panel shows which colored balls correspond to which element. (c) A depiction of H₂ with the H₂ center-of-mass (CM) at a distance Z from the surface, and the H-H bond at an orientation θ with respect to the surface normal. (d) A depiction of H₂ scattering on SrTiO₃(001) with angle of incidence Θ_i and scattering angle Θ_f. X corresponds to the surface lateral position of the H₂ CM along the (a) [100] and (b) [110] direction on the TiO₂- and SrO-terminated SrTiO₃(001), respectively.

As with metal (alloy) surfaces^5,8,9, 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 process^{1,2,3,4,5,24,25,26,27,28,29,30}.

Results and Discussion

H₂/SrTiO₃(001) System

Figure 1 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). Θ_i(f) gives H₂ incidence (scattering) angle measured with respect to the surface normal (cf., Fig. 1(c,d)). In the following, given H₂(J_i, m_i, E_i, Θ_i), we determine the probability of finding H₂(J_f, m_f, E_f, Θ_f). H₂(J_i, m_i, E_i, Θ_i) indicates a H₂ 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. H₂(J_f, m_f, E_f, Θ_f) indicates a H₂ 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 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.

Figure 2

Calculated electrostatic potential contributions from (a) the induced dipole moment ΔD_i with the surface electric field U_i, (b) the induced quadrupole moment ΔQ_i,j and (c) the H₂ quadrupole moment Q_i,j with the gradient of the surface electric field V_i,j, on the Ti-site (squares) and the O-site (circles) with θ = 0 (open symbols) and θ = π/2 (filled symbols) for the TiO₂-terminated STO(001). (d) The sum of the contributions (a–c). i, j = (x, y, z). The dotted vertical lines intercepting the abscissas indicate locations of the corresponding potential energy minima (H₂-surface equilibrium distance) at each respective surface site.

Figure 3

Calculated electrostatic potential contributions from (a) the induced dipole moment ΔD_i with the surface electric field U_i, (b) the induced quadrupole moment ΔQ_i,j and (c) the H₂ quadrupole moment Q_i,j with the gradient of the surface electric field V_i,j, on the Sr-site (squares) and the O-site (circles) with θ = 0 (open symbols) and θ = π/2 (filled symbols) for the SrO-terminated STO(001). (d) The sum of the contributions (a–c). i, j = (x, y, z). The dotted vertical lines intercepting the abscissas indicate the locations of the corresponding potential energy minima (H₂-surface equilibrium distance) at each respective surface site.

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(J_i, m_i, E_i = 80 meV, Θ_i = 15.9°) scattered as H₂(J_f, m_f, E_f, Θ_f) 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 R_J=0/J=1(E_i = 80 meV, Θ_i = 15.9°, Θ_f) 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(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], H₂(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 meV²⁴. Atop the O-site, H₂ preferentially adsorbs with the H-H bond oriented perpendicular to the surface, with E_ads = − 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(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 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(J_f = 2, m_i = 0). H₂(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 H₂, with dominant components coming from backscattering. On the other hand, H₂(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 H₂(J_i = 1, m_i = 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 Θ_f = 5.26° (Fig. 4).

Figure 4

(a–c) Calculated H₂ scattering probability (J_i, ∣m_i∣) → (J_f, ∣m_f∣) from initial rotational state (J_i, ∣m_i∣) to final rotational state (J_f, ∣m_f∣) on the TiO₂-terminated STO(001) as a function of the scattering angle Θ_f, for incident energy E_i = 80 meV and incident angle Θ_i = 15.9°. The dotted vertical lines intercepting the abscissas indicate specular scattering. (d) Solid, dotted, and dash-dot lines indicate constant energy surfaces of $\langle {Y}_{J}^{m}| V(Z,\theta ,X)| {Y}_{J}^{m}\rangle$ (with 80 [meV] $\times {\cos }^{2}{\varTheta }_{{\rm{i}}}$: corresponding to H₂ surface normal translational energy) encountered by a H₂ impinging with initial rotational states (J_i, m_i): [(0, 0), (1, 0), (1, 1)], respectively. Same scales used for X- and Z-coordinates. The arrows indicate the direction of incidence of the impinging H₂.

Table 1 Calculated R_J=0/J=1(E_i, Θ_i, Θ_f) and $\bar{{R}_{J=0/J=1}}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})$ on TiO₂-terminated STO(001) at the incident energy range 10 meV ≤ E_i ≤ 80 meV. For each incident energy, we list only the maximum R_J=0/J=1(E_i, Θ_i, Θ_f) with the corresponding incident and scattering angles.

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 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 H₂(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), H₂(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 meV²⁴. Atop the Sr-site, H₂ adsorbs with the H-H bond oriented parallel to the surface, and E_ads = −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(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 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(J_f = 2, m_i = 0). H₂(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). H₂(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-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 Θ_f = − 76.5° (Fig. 5).

Figure 5

(a–c) Calculated H₂ scattering probability (J_i, ∣m_i∣) → (J_f, ∣m_f∣) from initial rotational state (J_i, ∣m_i∣) to final rotational state (J_f, ∣m_f∣) on the SrO-terminated STO(001) as a function of the scattering angle Θ_f, for incident energy E_i = 60 meV and incident angle Θ_i = 76.5°. The dotted vertical lines intercepting the abscissas indicate specular scattering. (d) Solid, dotted, and dash-dot lines indicate constant energy surfaces of $\langle {Y}_{J}^{m}| V(Z,\theta ,X)| {Y}_{J}^{m}\rangle$ (with 60 [meV] $\times {\cos }^{2}{\varTheta }_{{\rm{i}}}$: corresponding to H₂ surface normal translational energy) encountered by a H₂ impinging with initial rotational states (J_i, m_i): [(0, 0), (1, 0), (1, 1)], respectively. Same scales used for X- and Z-coordinates. The arrows indicate the direction of incidence of the impinging H₂.

Table 2 Calculated R_J=0/J=1(E_i, Θ_i, Θ_f) and $\bar{{R}_{J=0/J=1}}({E}_{{\rm{i}}},{\varTheta }_{{\rm{i}}},{\varTheta }_{{\rm{f}}})$ on SrO-terminated STO(001) at the incident energy range 10 meV ≤ E_i ≤ 80 meV. For each incident energy, we list only the maximum R_J=0/J=1(E_i, Θ_i, Θ_f) with the corresponding incident and scattering angles.

(Normalized) p-H₂ to o-H₂ Ratios

In Tables 1 and 2, we show the calculated p-H₂ to o-H₂ ratio R_J=0/J=1(E_i, Θ_i, Θ_f) 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 ≤ E_i ≤ 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 E_i = 80.0 meV, Θ_i = 15.9°, Θ_f = 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 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-H₂ to o-H₂ ratios at specific scattering angles Θ_f exceed that of n-H₂, 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-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 (Θ_i = 0°) and incident energy of E_i = 100 meV³³. From the scattering probability data³³, we estimate a maximum p-H₂ to o-H₂ ratio R_J=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 Θ_i: [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 Θ_i. 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 Θ_i. 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 method^{36,37,38,39,40,41,42}. 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 (ℏω = 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 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(Z, θ, X) in Eq. (1) comes from previously performed density functional theory (DFT)-based total energy calculations^43,44,45 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)

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 H₂ 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 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.

Table 3 Fitted values of the potential depth D, potential width α, and equilibrium distance from the surface Z^eq, at each surface site, in Eq. (3).

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 ΔD_i (i = x, y, z) and quadrupole moments ΔQ_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 (TiO₂-termination) and Sr and O sites (SrO-termination) with θ = 0 and π/2. We also calculated the H₂ 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 H₂ 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 H₂ 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 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 E_i, incident angle Θ_i, and scattering angle Θ_f, 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 J_i = J_f and m_i = m_f, 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)

Table 4 Some relevant physical values corresponding to H₂(E_i, J_i, m_i, Θ_i) impinging a TiO₂- and SrO-terminated SrTiO₃(001). Incident angles presented in this work doubly underlined.