María Pilar
de Lara-Castells
*^{a} and
Alexander O.
Mitrushchenkov
*^{b}
^{a}Instituto de Física Fundamental (C.S.I.C.), Serrano 123, E-28006, Madrid, Spain. E-mail: Pilar.deLara.Castells@csic.es
^{b}Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, Université Paris-Est, 5 bd Descartes, 77454 Marne-la-Vallée, France. E-mail: Alexander.Mitrushchenkov@u-pem.fr
First published on 6th August 2018
A first-principles study of the spectroscopy of a single hydrogen molecule rotating inside and outside of carbon nanotubes is presented. Density functional theory (DFT)-based symmetry-adapted perturbation theory (SAPT) is applied to analyze the influence of the rotation in the dispersionless and dispersion energy contributions to the adsorbate–nanotube interaction. A potential model for the H_{2}–nanotube interaction is proposed and applied to derive the molecular energy levels of the rotating hydrogen molecule. The SAPT-based analysis shows that a subtle balance between the dispersionless and dispersion contributions is key in determining the angular dependence of the H_{2}–nanotube interaction, which is strongly influenced by the diameter of the carbon nanotubes. As a consequence, the structure of molecular energy levels is very different in wide and narrow nanotubes with the diameter above and below 1 nanometer, respectively. Strong anisotropy effects lead to a rather constrained rotation of molecular hydrogen inside narrow nanotubes.
The crucial role of quantum nuclear effects in the motion of light molecular species inside carbon nanotubes has been demonstrated in experimental observations at low temperatures. Thus, a recent work by Ohba^{6} revealed a quenched adsorption of helium atoms in carbon nanopores with a diameter below 0.7 nm at very low temperatures of 2–5 K. The same work also showed that more molecules of nitrogen than helium atoms adsorb in these narrow nanotubes despite the larger kinetic diameter of molecular nitrogen.^{6} A subsequent helium density-functional theory (DFT)-based study of carbon nanotubes immersed in helium nanodroplets pointed out that the quenched adsorption can be attributed to very large zero-point energies in the motion of ^{4}He atoms, including the formation of cavities with zero helium densities.^{7} A follow-up study revealed that the experimental observation can also be explained as a result of the extreme one-dimensional confinement existing along the axis of narrow nanotubes for ^{4}He atoms and N_{2} molecules.^{8} In the particular case of molecular hydrogen inside carbon nanotubes, a quantum-induced reverse trend in H_{2} and D_{2} diffusion rates upon lowering the temperature was experimentally detected^{9} and theoretically confirmed.^{10} The impact of quantum nuclear effects has also been highlighted when characterizing the (superfluid or crystalline) phase of parahydrogen molecules inside carbon nanotubes at zero^{11,12} and ultra-low temperatures.^{13}
Recent studies using high-resolution neutron spectroscopy have allowed accurate characterization of the motion of molecular hydrogen under confinement conditions provided by carbon-based nanoporous materials such as carbon nanohorns^{14} and aligned carbon nanotubes.^{15} A key ingredient when aiming at reproducing and predicting experimental results is a realistic model for the molecule–nanotube interaction. Ab initio intermolecular interaction theory allows an accurate description of van der Waals (vdW)-dominated interactions of molecules with carbon nanostructures.^{7,8,16–23} Recent options are mixed schemes combining second-order Möller–Plesset perturbation theory with the coupled-cluster approach,^{16,17} as well as non-periodic implementations of the incremental method^{24} applied at the coupled-cluster level^{25–29} with periodic calculations using dispersionless density functional theory.^{28,30–32} An alternative methodological protocol^{8} combines DFT-based symmetry-adapted perturbation theory (SAPT), which is used to derive the parameters of potential models describing gas adsorption to the carbon material, with an ad hoc-developed adsorbate wave function treatment. This method has very recently been applied in the precise characterization of the nature of bound states of atomic helium, molecular nitrogen, and molecular deuterium clusters inside carbon nanotubes.^{5,8}
In previous works^{5,8} molecules of nitrogen and deuterium were treated as structureless point-like particles using the full configuration interaction Nuclear Orbital (FCI-NO) approach (see, e.g., ref. 33–35 and references therein). As a step forward, we propose here an extension of the adsorbate wave function treatment allowing explicit inclusion of the rotational mode of molecular hydrogen. This extension is also motivated by very recent experimental evidence demonstrating the appearance of a high-density orientationally ordered phase in H_{2} adsorbed in subnanometer pores.^{4} This way, our focus will be on analyzing the effect of including the H_{2} orientational mode upon increasing the carbon tube diameter from the subnanometer scale. The next section is dedicated to a brief description of the computational methods. The results are reported in Section 3. Finally, the main conclusions are provided in Section 4.
V_{H2–CNT}(ρ_{1},ρ_{2}) = V_{H–CNT}(ρ_{1}) + V_{H–CNT}(ρ_{2}) |
(1) |
ϕ → = ϕ − Φ |
(2) |
(3) |
(4) |
(5) |
The integration over x and ϕ can then be performed analytically using spherical harmonics as a basis, Y_{jm}(θ,ϕ), with x = cos(θ) and j ≥ m. Using this basis, the Hamiltonian matrix can be explicitly written as
(6) |
Fig. 1 Radial scan of the total interaction energies between a single H_{2} molecule and the short carbon nanotube (sCNT) of helicity index (5,5) represented in the right-hand panel. Gray spheres represent carbon atoms while blue spheres stand for one H_{2} molecule inside. The sCNT was saturated with hydrogen atoms (not shown here). The interaction energies are represented as a function of the distance from the H_{2} center-of-mass to the nanotube axis, R. Full, dotted and dashed lines correspond to θ values of 0°, 45°, and 90°, respectively, with θ as the angle between the H–H internuclear axis and the nanotube axis. The different SAPT-based energy contributions to the total interaction energy are also shown. The H–H bond length was fixed to the vibrationally averaged rigid rotor value, r_{0} = 0.7508 Å, from ref. 18. |
As can be observed in Fig. 1 (left-hand panel) and also discussed in ref. 5, the H_{2}/CNT attractive interaction is dispersion-dominated, with the net repulsive dispersionless contribution mainly characterized by the exchange–repulsion term. The potential minimum is located at the center of the narrow nanotube because this symmetric position allows the adsorbate to benefit from the dispersion interaction with carbon atoms at both sides of the carbon cage. The exchange–repulsion contribution grows exponentially as the distance between the H_{2} and the carbon cage decreases although such behavior is somewhat smoothed out by the attractive electrostatic contribution. The induction term contributes very little in the relevant range of radial distances in this work.
Focusing on the influence of the particular H_{2} orientation on the total energy, it can be observed that the parallel orientation (θ = 0°, full lines) is clearly energetically preferred over the perpendicular configuration (θ = 90°, dashed lines). It can also be seen that this preference arises from a smaller exchange–repulsion contribution when the molecule is located along the tube axis. Contrarily, the dispersion contribution becomes more pronounced as the θ value increases. Therefore, the preferential orientation arises from the counterbalance between exchange–repulsion and dispersion terms.
The subtle balance between dispersion and dispersionless contributions is very much influenced by the diameter of the nanotube. To illustrate this, we apply the potential model to longer nanotubes with helicity indexes (5,5), (10,5) and (10,10), obtaining the θ-dependence of total, dispersionless and dispersion energies presented in Table 1 (see also Fig. 2). First, it can be observed in Table 1 that the potential model provides the total energies at the potential minimum (values in parentheses) agreeing with the SAPT(DFT) values to within 6%. Second, it can also be seen that the weight of the exchange–repulsion at the potential minimum decreases as the diameter of the tube increases and, then, the dispersion becomes more relevant in determining the θ-dependence of the total energies. This is the reason why the energy difference between parallel and perpendicular orientations becomes smaller when the tube diameter increases, as can be clearly seen in Fig. 2. From Table 1, we can also notice the very weak ϕ-dependence of the total energies, reflecting the small corrugation of the carbon material.
CNT | s(5,5)/SAPT(DFT) | (5,5)/model | (10,5)/model | (10,10)/model |
---|---|---|---|---|
d _{CNT} | 6.7 | 6.7 | 10.4 | 13.6 |
R _{0} | 0.0 | 0.0 | 2.0 | 3.6 |
(θ,ϕ) | E ^{disp-less}_{min} | E ^{disp}_{min} | E ^{tot}_{min} | E ^{disp-less}_{min} | E ^{disp}_{min} | E ^{tot}_{min} | E ^{disp-less}_{min} | E ^{disp}_{min} | E ^{tot}_{min} | E ^{disp-less}_{min} | E ^{disp}_{min} | E ^{tot}_{min} |
---|---|---|---|---|---|---|---|---|---|---|---|---|
(0°,0°) | 90 | −236 | −146 (−148) | 96 | −279 | −183 | 37 | −127 | −90 | 31 | −103 | −72 |
(45°,0°) | 108 | −247 | −139 (−137) | 118 | −289 | −171 | 51 | −133 | −82 | 44 | −109 | −65 |
(90°,0°) | 123 | −254 | −131 (−124) | 141 | −300 | −159 | 66 | −139 | −73 | 57 | −114 | −57 |
(45°,45°) | 108 | −247 | −139 (−137) | 118 | −289 | −171 | 46 | −132 | −86 | 32 | −100 | −68 |
Fig. 2 Scan of relative energies E − E_{min} as a function of θ for one H_{2} molecule inside carbon nanotubes (CNTs) of helicity indexes (5,5), (10,5), and (10,10). |
CNT(5,5) | CNT(10,5) | CNT(10,10) | CNT(10,10)-ext | |
---|---|---|---|---|
The energies at the potential minima (E_{min}, in cm^{−1}) and the zero-point energies (E_{zp}, in cm^{−1}) are also collected. The energy values are relative to that of the ground state (0 0 0). The energy of the ground state (0 0 0) is given in parentheses. For Λ > 0, ΔE_{Λ} = (E − E_{Λ=0})/Λ^{2}. The energies of the lowest bound states without including the rotation are presented in brackets. Additional energy levels are provided in Section S2 of the ESI. | ||||
d _{CNT}, Å | 6.7 | 10.5 | 13.6 | 13.6 |
E _{min}, cm^{−1} | −1474 | −725 | −575 | −296 |
E _{zp}, cm^{−1} | 474 | 99 | 99 | 81 |
n | j | m | E _{ Λ=0} | ΔE_{Λ=1} | E _{ Λ=0} | ΔE_{Λ=1} | E _{ Λ=0} | ΔE_{Λ=1} | E _{ Λ=0} | ΔE_{Λ=1} |
---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0.0 | 154.088 | 0.0 | 2.905 | 0.0 | 0.727 | 0.0 | 0.081 |
(−997.908) | (−625.627) | (−475.369) | (−215.515) | |||||||
0 | [−1147.07] | [−650.433] | [−502.037] | [−233.840] | ||||||
0 | 1 | 0 | 61.429 | 150.275 | 110.148 | 2.778 | 109.371 | 0.714 | 112.222 | 0.082 |
0 | 1 | 1s | 340.182 | 231.611 | 140.229 | 4.183 | 138.983 | 0.828 | 130.974 | 0.082 |
0 | 1 | 1a | 309.649 | −144.839 | 112.058 | 1.646 | 109.775 | 0.645 | 112.313 | 0.081 |
0 | 2 | 0 | 338.445 | 154.303 | 345.202 | 2.072 | 342.789 | 0.680 | 346.558 | 0.081 |
0 | 2 | 1s | 510.758 | 229.605 | 367.277 | 4.263 | 365.256 | 0.829 | 361.193 | 0.083 |
0 | 2 | 1a | 488.408 | −143.881 | 345.509 | 1.360 | 342.748 | 0.619 | 346.560 | 0.080 |
0 | 2 | 2s | 825.042 | 284.147 | 382.259 | 11.179 | 377.855 | −0.254 | 367.951 | 0.092 |
0 | 2 | 2a | 835.032 | −251.237 | 375.399 | −4.388 | 367.170 | 0.129 | 361.452 | 0.071 |
1 | 0 | 0 | 522.302 | 169.873 | 117.289 | 5.473 | 123.191 | 0.895 | 93.195 | 0.076 |
[486.978] | [126.178] | [130.899] | [100.079] | |||||||
1 | 1 | 0 | 571.230 | 165.199 | 230.937 | 5.068 | 235.674 | 0.876 | 208.018 | 0.076 |
1 | 1 | 1s | 912.150 | 233.919 | 256.851 | 8.366 | 258.877 | 1.070 | 220.864 | 0.076 |
1 | 1 | 1a | 857.645 | −142.973 | 234.331 | 0.745 | 236.068 | 0.748 | 208.069 | 0.075 |
1 | 2 | 0 | 877.251 | 151.240 | 467.850 | 3.594 | 470.371 | 0.765 | 443.506 | 0.074 |
1 | 2 | 1s | 1058.936 | 239.767 | 486.606 | 8.764 | 487.858 | 1.082 | 453.361 | 0.077 |
1 | 2 | 1a | 1019.278 | −148.993 | 469.366 | 0.100 | 470.381 | 0.698 | 443.503 | 0.074 |
1 | 2 | 2s | 1390.569 | 320.492 | 505.672 | 20.008 | 499.811 | 0.215 | 458.592 | 0.088 |
1 | 2 | 2a | 1414.431 | −267.805 | 500.797 | −10.555 | 490.208 | −0.112 | 453.576 | 0.064 |
2 | 0 | 0 | 1087.333 | 183.886 | 205.702 | 14.606 | 218.204 | 1.247 | 155.322 | 0.069 |
[1022.891] | [220.287] | [231.836] | [166.508] | |||||||
2 | 1 | 0 | 1123.970 | 177.651 | 321.341 | 13.555 | 332.906 | 1.211 | 271.706 | 0.069 |
2 | 1 | 1s | 1521.755 | 221.946 | 353.967 | 18.234 | 350.725 | 1.632 | 279.919 | 0.069 |
2 | 1 | 1a | 1435.031 | −137.1229 | 332.991 | −6.195 | 333.508 | 0.886 | 271.743 | 0.067 |
2 | 2 | 0 | 1465.285 | 141.410 | 560.816 | 9.167 | 569.158 | 0.498 | 507.901 | 0.067 |
2 | 2 | 1s | 1646.533 | 247.978 | 584.240 | 19.849 | 581.976 | 1.679 | 514.236 | 0.071 |
2 | 2 | 1a | 1592.477 | −152.7319 | 568.653 | −7.433 | 568.833 | 0.771 | 507.907 | 0.067 |
2 | 2 | 2s | 1992.091 | 351.905 | 614.377 | 33.948 | 592.094 | 4.737 | 518.548 | 0.078 |
2 | 2 | 2a | 2029.057 | −279.984 | 612.508 | −22.713 | 585.367 | −1.427 | 514.422 | 0.063 |
From the values of zero-point energies E_{zp} in Table 2, it can be observed that the strong confinement of H_{2} upon adsorption inside the (5,5) tube leads to a considerable zero-point energy due to the steepness of the interaction potential along the radial R distance (see Fig. 1). Similar to atomic helium as the adsorbate,^{7} it arises from a very pronounced exchange–repulsion contribution even at the center of the nanotube. This finding is consistent with the experimental observations of Ohba,^{6} showing the quenched adsorption of species as light as helium atoms in nanopores with diameters below 7 Å at low (2–5 K) temperatures.
It is also interesting to compare the lowest-energy states in Table 2 with those calculated by considering only the H_{2} center-of-mass motion R in ref. 5, with the H_{2} molecule lying parallel to the tube axis. The energies of these states are shown in brackets and they would be associated with the set (j m) = (0 0). Notice that the wave function for states with j = m = 0 is quite broad in θ, which results in penalty from the less attractive interaction potential for a perpendicular configuration and a higher energy when including the H_{2} rotation into our treatment. The energy differences with and without including the H_{2} rotation are more pronounced for H_{2} inside the narrow nanotube where, for (j m) = (0 0), the number of vibrational bound states is reduced from three to just two. Contrarily, the less steep H_{2}–CNT(10,10) interaction potential supports more than six vibrational states with (j m) = (0 0).
To better understand the dependence of the energy levels on the different quantum numbers, the total energies have been decomposed in the following contributions: (1) the average potential energy term 〈V_{H2–CNT}〉; (2) the kinetic R-dependent term, 〈K(R)〉, with (i.e., the first term from the kinetic energy operator in eqn (3)); (3) the “internal” kinetic term 〈K_{int}〉 (i.e., the terms multiplied by the H_{2} rotational constant in eqn (3)); and (4) the “external” kinetic term 〈K_{ext}〉 (i.e., the terms multiplied by the “external” rotational constant in eqn (3)). This decomposition is presented in Table 3 for the particular case of the (5,5) nanotube.
n | j | m | Λ | E _{tot} | ΔE_{tot} | 〈V_{H2–CNT}〉 | 〈K(R)〉 | 〈K_{int}〉 | 〈K_{ext}〉 |
---|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | −998 | 0.0 | −1206 | 188 | 19 | 0.1 |
0 | 1 | 0 | 0 | −936 | 61 | −1246 | 184 | 125 | 0.0 |
0 | 1 | 1s | 0 | −658 | 340 | −1073 | 168 | 128 | 120 |
0 | 1 | 1a | 0 | −688 | 310 | −1087 | 160 | 126 | 113 |
0 | 2 | 0 | 0 | −659 | 338 | −1191 | 191 | 340 | 0.2 |
1 | 0 | 0 | 0 | −476 | 522 | −986 | 475 | 34 | 0.9 |
2 | 0 | 0 | 0 | 89 | 1087 | −760 | 796 | 50 | 3.3 |
0 | 0 | 0 | 1 | −844 | 154 | −1138 | 159 | 23 | 112 |
0 | 1 | 0 | 1 | −786 | 212 | −1178 | 156 | 126 | 109 |
0 | 1 | 1s | 1 | −426 | 572 | −971 | 163 | 129 | 253 |
0 | 1 | 1a | 1 | −833 | 165 | −1153 | 195 | 125 | 0.3 |
Focusing first on the energy levels with (m Λ) = (0 0), we can notice from Tables 2 and 3 that their j-dependence deviates by just 10% from that of gas-phase H_{2} molecules and expressed as B_{H2}j(j + 1). For Λ > 0, it can be observed that the Λ-dependence of the energy levels is very weak for the wide CNT(10,10) tube, being simply accounted for with the energy contribution B_{ext}Λ^{2}. This can be explained by considering the large value of the equilibrium distance R_{0}, of 3.6 Å (see Table 1), leading to a very small value of the “external” rotational constant (about 1 cm^{−1}).
As opposed to the case of the wide CNT(10,10) tube, the potential minimum for the H_{2}–CNT interaction is located at the center of narrow nanotubes (i.e., R_{0} = 0) so that the value of the rotational constant B_{ext} is very large, differing by less than 13% from the value of the H_{2} rotational constant B_{H2}. The quantum numbers Λ and m are thus strongly correlated and the Λ-dependence of the energies can be approximated as B_{ext}(Λ − m)^{2}. Therefore, as can be observed Table 3, the “external” kinetic term 〈K_{ext}〉 is almost zero for levels with Λ = m.
Focusing now on the energy levels sharing the same (n j) quantum numbers and different m values, we can notice that these m-splittings decrease as the diameter of the nanotube increases. This feature is directly related to the anisotropy (i.e., the θ-dependence) of the H_{2}–CNT interaction. For instance, the θ-dependence of the wave-functions for (0 1 0) and (0 1 1s) levels is mainly described with the functions cos(θ) and sin(θ), favoring parallel and perpendicular configurations, respectively. As can be seen in Fig. 2, the parallel configuration is energetically favored for H_{2} molecules inside carbon nanotubes. Consequently, the (0 1 0) level is below the (0 1 1s) levels for any nanotube size.
It is important to note that the levels (n 1 1a) and (n 1 0) are nearly degenerate with the exception of the narrow (5,5) tube. This outcome can be explained as follows: the most important term in the potential expansion splitting m = 0 and m = 1 states corresponds to that with indexes I = K = 1, see eqn (5) above (the terms with K = 0 give no splitting), which is directly related to the V_{2} coefficient. However, it can be demonstrated that the I = K = 1 term contributes equally to (n 1 0) and (n 1 1a) Hamiltonian matrix elements, while it is three times larger for the (n 1 1s) counterparts. A detailed analysis of the degeneracy of (n 1 1a) and (n 1 0) levels is provided in Section S4 of the ESI.† For the narrow (5,5) tube, the anisotropy of the H_{2}–CNT interaction is much more marked. In this case, the B_{ext}(Λ − m)^{2} rotational term contributes to an increase in the energy of the (0 1 1) level by ca. 250 cm^{−1}. The energy pattern is then determined by the whole combination of different contributions. This way, the energy differences between (0 1 0) and (0 1 1) levels are much larger such as the levels (n 1 1a) and (n 1 0) are no more degenerate. In stark contrast, the m-splitting of (n j) = (0 1) levels inside and outside of the wider (10,10) tube is below 20 cm^{−1} (see Table 1).
Recent experimental measurements using high-resolution neutron spectroscopy have provided precise values of m-splittings for H_{2} adsorbed in the outside of carbon nanohorns^{14} and aligned carbon nanotubes.^{15} Specifically, values of ca. 4 and 8 cm^{−1} have been reported for the energy difference between (0 1 0) and (0 1 1) levels in carbon nanohorns^{14} and carbon nanotubes, respectively.^{15} By extending our potential model to the outside of the CNT(10,10) nanotube (see Table 1), the estimated value of the m-splitting is 18 cm^{−1}. This value is probably overestimated because our model potential is based on ab initio calculations with H_{2} placed inside the carbon cage. As shown in ref. 8, the repulsive core of adsorbate–CNT interaction is less marked for the adsorbate localized in the outside of the carbon cage and, then, the anisotropy of the interaction. Moreover, as discussed in ref. 49, it is well known that there is an electronic charge displacement directed toward the outside of the carbon cage in nanotubes. It is clear that this redistribution of the electronic density must affect the H_{2}–nanotube interaction, in particular the exchange–repulsion contribution. Hence, more precise estimations would require the refitting of dispersionless contributions using ab initio values for H_{2} placed outside the nanotube. Using preliminary calculations of SAPT(DFT) interaction energies with the H_{2} molecule located outside of the carbon cage, a refitting of the pairwise potential model indicates that the m-splitting is reduced from 18 to 11 cm^{−1}, with the global structuring of molecular energy levels being almost unmodified. Finally, a suitable modeling of assemblies of CNTs used in experimental measurements^{15} is probably needed.
Footnote |
† Electronic supplementary information (ESI) available: Details of the fittings to potential models, four tables with complete lists of calculated molecular energy levels, complementary expressions for the calculation of Hamiltonian matrix elements, and analysis of degenerate energy levels appearing in wide nanotubes. See DOI: 10.1039/c8cp04109a |
This journal is © the Owner Societies 2019 |