Quantum behavior of water nano-confined in Beryl

Transcription

1 Quantum behavior of water nano-confined in Beryl Y. Finkelstein, 1,a R. Moreh, 2 S. L. Shang, 3 Y. Wang, 3 and Z. K. Liu 3 1 Nuclear Research Center-Negev, Beer-Sheva 84190, Israel 2 Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 3 Department of Materials Science and Engineering, the Pennsylvania State University, University Park, PA 16802, USA ABSTRACT The proton mean kinetic energy, Ke(H), of water confined in nanocavities of Beryl (Be 3 Al 2 Si 6 O 18 ) at 5 K was obtained by simulating the partial vibrational density of states (VDOS) from density functional theory (DFT) based first-principles calculations. The result, Ke(H) = mev, is in remarkable agreement with the 5 K deep inelastic neutron scattering (DINS) measured value of 105 mev. This is in fact the first successful calculation that reproduces an anomalous DINS value regarding Ke(H) in nano-confined water. The calculation indicates that the vibrational states of the proton of the nano-confined water molecule distribute much differently than in ordinary H 2 O phases, most probably due to coupling with lattice modes of the hosting Beryl nano-cage. These findings may be viewed as a promising step towards the resolution of the DINS controversial measurements on other H 2 O nano-confining systems, e.g. H 2 O confined in single and double walled carbon nanotubes. Keywords: deep inelastic neutron scattering; density functional theory; proton kinetic energy, phonon density of states, Beryl, H 2 O a Finklfam@gmail.com 1

2 1. Introduction Over the last two and a half decades, a great deal of theoretical and experimental work has been devoted to study the proton mean kinetic energy, Ke(H), in quest of exploring the proton dynamics in hydrogen bonds (HBs) containing systems. 1,2,3,4,5,6,7 Particular progress was achieved owing to the advent of the DINS in which one probes a unique kinematic region in neutron scattering which provides a direct measurement of the proton zero point kinetic energy (ZPKE). 8,9 Currently, such capability is exclusively provided only by the Vesuvio spectrometer (a pulsed neutron source) at the ISIS facility, Rutherford Appleton Laboratory-UK. 9 The majority of the DINS measurements was focused on ordinary stable H 2 O phases, and was theoretically accompanied by state of the art simulation methods such as path integral Car- Parrinello molecular dynamics (PICPMD) 10,11,12,13 and path integral molecular dynamics (PIMD) 14,15,16,17 calculations. Another, much simpler and straight forward calculation used to understand the experimental results, is the semi-empirical (SE) approach which assumes the harmonic approximation (HA) and uses as input data the experimental fundamental vibrational modes of the system. 2,3,4,6,7,18 The SE method was found to yield in general very successful Ke(H) predictions of DINS results on pure H 2 O phases, but completely failed to reproduce anomalous DINS Ke(H) values measured for nano-confined water samples, e.g. in single (SWCNT) 19,20,21 and double (DWCNT) 22,23 wall carbon nanotubes, as well as in other H 2 O confining systems. 22,24,25,26,27 In fact, all other known theoretical methods were not able to predict any of those DINS findings despite all efforts taken to understand their origin, leaving this issue under scientific debate. 8,10 As the proton potential in nano-confined water is highly anharmonic, the HA underlying the SE approach, was pointed out as the primary argument against the validity of the SE method in calculating Ke(H) in nano-confined water. 4 Note that the DINS technique measures the momentum 2

3 distribution of the proton ground state and hence it's actual ZPKE, E 0. The SE approach uses as input data, fundamental frequencies taken from inelastic neutron scattering (INS), infrared (IR) or Raman methods where the first excited state E 1 is measured and the excitation energy is E = E 1 -E 0, with E 0 the ground state. Here, the harmonic assumption means E 0 = E/2 and was believed to set a serious drawback on its validity to predict kinetic energy values in systems involving strong anharmonicity. In fact, it was recently shown that by simulating the vibrational (phonon) density of states (VDOS) using density functional theory (DFT), and combining it with the SE formulation, an accurate prediction of DINS Ke(H) values is obtained also for strong anharmonic systems, such as the ferroelectric crystal, KH 2 PO 4 (KDP) and the super protonic conductor Rb 3 H(SO 4 ) 2. 4 Worth mentioning in that regard is the work of Bakker et. al. 28 who used ultrafast femtosecond IR pulses to excite and probe the O-H covalent bond vibration in liquid water, thereby capturing the delocalization of the proton between the oxygen atoms in the O-H O configuration. Using a quantum mechanical calculation of vibrational wave functions, they reproduced the experimental absorption spectrum. By examining the optimized highly anharmonic potential along the O-H O direction that reproduced the experiment, one may note that the harmonic relation E 0 / E ~ ½ is fulfilled within nearly 10 %. This study with the above SE findings considerably increased our confidence that the SE method which involves the HA can be used to study samples involving nano-confined water. Recently, in a unique THz-IR study, Gorshunov et. al. 29,30 showed that at 5 K, the H-atoms of water, residing as isolated molecules in the ~ 5Å cavities of Beryl, are delocalized and tunnel between the six symmetrically equivalent positions of the crystalline channel. This intriguing discovery was most recently confirmed by a combined INS/DINS/Ab-initio study by Kolesnikov et. al. 31 Interestingly, the DINS measurements, using the Vesuvio spectrometer, 9 revealed that the proton ZPKE of water nano-confined in Beryl, is Ke(H) = 105 mev, anomalously lower than that measured for ordinary ice and stable water (~ 152 mev). This result also accords 3

4 with previous reports of Ke(H) DINS anomalies, exclusively found in other H 2 O confining systems. 22 In view of the above it was tempting to carry out a DFT calculation on H 2 O@Beryl, in an attempt to test to what extent the simulated VDOS could reproduce the DINS measured Ke(H) result. Remarkably, the predicted DFT value revealed a perfect match with experiment in spite of the reliance on the HA. This result is very encouraging, hoping that it would help resolving the longstanding issue regarding the anomalous DINS Ke(H) values in water in strong confining environments. 2 Theoretical remarks 2.1 SE calculations Only few methods were used for measuring atomic kinetic energies. One of the early techniques utilized nuclear resonance photon scattering (NRPS) from the 15 N nucleus in the form of molecular gases and solids, 32,33,34 for measuring total and directional zero-point energies (related to nuclear quantum effects), molecular binding and molecular orientations. 35,36 These and other NRPS studies firmly established the validity of the SE approach. By now, its formulation may be found in a large variety of works published over the last 40 years in relation to NRPS studies, and more recently also in DINS 18 and electron Compton scattering (ECS) 37,38,39 studies. From a theoretical point of view, atomic kinetic energies are mostly calculated by the SE approach which utilizes experimental input data, and assumes the harmonic approximation and decoupling between the various modes of motion of the atom in the studied molecule. 32 We hereby explain how the SE approach is utilized for calculating the proton kinetic energy in H 2 O. 4

5 In order to calculate Ke(H) in an isolated water molecule, we first note that the kinetic energy of the nucleus is very nearly equal to that of the whole atom as it is assumed that the electrons are not disrupted from the nucleus and follow its motion. Two types of motion contribute to Ke(H): (1) the intra-atomic motions of the H 2 O atoms, i.e. the internal normal modes of the molecule, and (2) the external lattice modes caused by the motion of the H 2 O molecule as a whole. We hereby focus on a single isolated H 2 O molecule which represents the case of water in the vapor phase Ke(H) in H 2 O vapor Water is a symmetrically planar, tri-atomic molecule (N=3). It thus have nine degrees of freedom, (3 translational, 3 free rotations, and 3N-6 = 3 internal normal vibrations), all contribute to Ke(H). The three internal vibrations consist of symmetric and asymmetric stretching and HOH bending modes. Ke(H) of H 2 O in the vapor phase may thus be written as follows:, (1) where, 3kT/2, represent the classical kinetic energy of translation and of rotation of the entire molecule with T the thermodynamic temperature, and S Tr and S L the kinetic energy fractions shared by a single proton in each of the two motions. The third term is the kinetic energy of the three internal vibrational modes. Each mode of vibration is represented by a quantum harmonic oscillator of frequency j and total energy,, 5 / ; k is the Boltzmann constant and the factor ½ multiplying the third term in Eq. (1) is due to fact that the kinetic energy equals half the total energy of the harmonic oscillator. Each H atom shares only a fraction, S j, of the kinetic energy of each vibrational mode. S Tr, the kinetic energy fraction of translation is obtained from the mass ratio,. Similarly, S L, the kinetic energy fraction of rotation, is the sum over S x L, S L y and S L z, the shares of a single H atom in the rotation of H 2 O around each of its three principal axes; these are calculated by considering the moments of inertia along the three axes passing

6 through its center of mass; S L thus depends on the OH bond lengths and the OHO angle. In the vapor phase, 2 the OH bond lengths (R OH = Å), and the OHO angle (104.3 ) dictate that S L = To calculate S j (j=1,2,3), the energy fraction shared by the proton in the j th internal vibration of H 2 O, we note that it is equal to the energy ratio of the j th vibrational mode of one H- atom relative to that of the entire molecule, /, with M i the mass of the i th atom in the molecule, and A ij the amplitude of the i th atom in the j th vibrational mode of H 2 O. 32 It is deduced by solving the equations of motion of a molecular system containing N particles under simple decoupled harmonic motions. The method of calculation is treated in textbooks on infrared and Raman Spectroscopy, 40 and it contains the following key steps: mass weighing the Hessian matrix, determining the principal axes of inertia, generating coordinates in the rotating and translating frames, transforming the Hessian to internal coordinates, diagonalizing, calculating the frequencies, and finally deducing the reduced masses, force constants and Cartesian displacements. It is useful to check the correctness of the resulting energy fractions by using the sum rule which applies to each atom of the molecule (H 2 O); it states that the weighted sum of all energy fractions equals unity. Using eq. (1) this may be written as: 1 (2) Ke(H) in condensed H 2 O In the liquid and solid phases one has to account for the formation of hydrogen bonds (HBs) which exert external bonding forces on the H 2 O molecules. The net effect of the above has two major consequences: (1) deformation of the OHO angle and elongation of the covalent bond length, R OH, thus weakening its strength and hence decrease its vibrational frequency while leaving that of bending nearly intact, and (2) hindering the translational and rotational motions, causing the free translation and rotation to transform into vibration and libration of the entire molecule with 6

7 characteristic frequencies Tr and L respectively. These lattice modes have an inherently larger contribution to Ke(H) than that of the vapor phase due to the zero-point motion of vibration. All in all, with decreasing temperature, the average number of HBs per water molecule increases (reaching a maximum of 4 HBs in the solid phase); the geometrical parameters of the H 2 O molecules are also modified, the OH bond length, R OH, increases, while that of the HB, R O H, decreases. This causes a red-shift of the stretching modes, coupled by a blue shift of the lattice frequencies (libration and translational-vibration). In cases where the characteristic frequencies Tr and L of translation and libration are available from experiment or theory, Ke(H) in the condensed phase may be written as: 3 2, 1 2, (3), In cases where the frequency distributions g Tr ( ) and g L ( ) of translation and libration are available from inelastic neutron scattering (INS) measurements or from simulations, Ke(H) takes the following form:,,, (4) Tr, t, l and L are the limits of integration; g Tr ( ) and g L ( ) are normalized so that: = 1 (5) As pointed out in the introduction, the SE formulation which utilizes the five measured discrete characteristic fundamental frequencies to deduce Ke(H), was found to work flawlessly in predicting DINS Ke(H) values in pure H 2 O phases, but completely failed to do so in cases of nano-confined water (e.g. in single and double walled carbon nanotubes). 7 Based on scattering and spectroscopic measurements, it is imperative to mention in this regard, that the five fundamental frequencies of 7

8 H 2 O are not dramatically affected by confinement, 21,41 nor is the geometry of the molecule. 7 As a result, the SE formalism dictates close S l values in either ordinary H 2 O phases or nano-confined water. Adding to that, the fact that in the SE approach the value of S Tr is fixed at m H /m H2O = 1/18, regardless of the H 2 O phase (solid, liquid, vapor or confined), it is not surprising that overall, the SE value of Ke(H) in nano-confined H 2 O is close to that of ordinary ice. Yet, having realized the above, the exact origin of the DINS Ke(H) anomalies still remains unknown Deducing Ke(H) from VDOS Eqs. 1 and 3 utilize the measured discrete characteristic fundamental frequencies to deduce Ke(H) in vapor and condensed phases respectively. In practice however, the motion of the atom in condensed phases is better represented by the VDOS which has a continuous distribution. Hence, by accounting in particular for the partial VDOS of the H atom,, one can deduce Ke(H) directly from Eq. (6) as it becomes unnecessary to calculate the energy fractions being self-contained in the VDOS:, / (6) with 0 and f the frequency limits of. 2.2 DFT calculations All DFT based first-principles calculations were carried out by Vienna Ab initio Simulation Package (VASP 5.4.1). 42 The ion-electron interaction was accounted for by the projector augmented wave method. 43 The exchange-correlation (X-C) energy functional was described by the improved generalized gradient approximation (GGA) for densely packed solids and their surfaces, i.e., the PBEsol (hereby denoted by PS). 44 In addition, the extra contribution of van der Waals correction by the D3 method 45 was employed, i.e., the GGA of Perdew-Burke-Ernzerhof (PBE), 46 PBE+D3, and the 8

9 PS+D3 methods. For the elements used in VASP calculations, 3 electrons (3s 2 3p 1 ) were treated as valence electrons for Al, 2 (2s 2 ) for Be, 1 (1s 1 ) for H, 6 (2s 2 2p 4 ) for O, and 4 (3s 2 3p 2 ) for Si. Energy convergence criterion for electronic self-consistency was at least 10 6 ev per atom. Other details including the employed supercells, plane wave cutoff energies, k-points meshes, and the smearing methods are provided in Table I. For each structure of interest, particularly that of the lower energy structures, the equilibrium structural properties were determined by fitting the DFT energy versus volume data points according to a four-parameter (i.e., the third order) Birch-Murnaghan equation of state (EOS), 47 E ( V) k k V k V k V (7) 2/3 4/ where k 1, k 2, k 3, and k 4 are fitting parameters. Equilibrium properties determined from this EOS include energy (E 0 ), volume (V 0 ), bulk modulus (B 0 ), and the first derivative of bulk modulus with respect to pressure (B ). Six to eight data points within the volume range of -10% < (V-V 0 )/V 0 < 10% were typically used for the present EOS fittings. Phonon calculations were performed by the supercell method as implemented in the YPHON code. 48,49 Force constants, i.e., the Hessian matrix, were calculated directly using the VASP code in terms of the finite difference method. Correspondingly, phonon properties were calculated by a parameter-free, mixed-space approach as implemented in the YPHON code 48,49 with more details given in a review article 48 and references therein. It should be mentioned that the longitudinal optical and transverse optical (LO-TO) splitting should be excluded herein since the present H 2 O molecules of interest are isolated in the H 2 O@Beryl system without the long-range dipole-dipole interaction among H 2 O molecules; while the LO-TO splitting was considered in our previous phonon calculations in polar solids, such as hexagonal ice 50 and other polar materials. 49 9

10 TABLE I. Details of DFT calculations for each compound including the space group, total atoms used in the supercell, k-points mesh, cutoff energy (E cut in ev), and the smearing method. Compound Pure Beryl (Al 2 Be 3 Si 6 O 18 ) Proton Site None Space group Atoms k-mesh E cut Smear a H 2 O@Beryl 4e 24m P6/mcc b 500 b 0 a The same as the setting in VASP code, the number 0 represents the Gaussian smearing method. b k-mesh and cutoff energy of 500 ev are mainly used for phonon calculations. Both pure beryl (Al 2 Be 3 Si 6 O 18 ) and its monohydrate form, H 2 O@Beryl, have a hexagonal structure with space gorup P6/mcc. 51,52,53 However, atomic positions of hydrogen (H) in H 2 O@Beryl are undetermined. 52 Artioli et al. 51 proposed that H atoms occupy the site 4e or partly located in site 24m (based on X-ray and neutron data for different alkali and H 2 O contents, e.g., Al 2 Be 2.65 Li 0.32 Si 6 O 18 (Na 0.22 Cs H 2 O)). In the 64-atom supercell used here (i.e., Al 4 Be 6 Si 12 O 36 2H 2 O), four H atoms can fully occupy site 4e along the c-axis direction (structure-4e), or partly occupy site 24m, deviating from the c-axis direction (structure-24m). In the present work, all possible configurations in structure-24m were generated by ATAT code. 54 For the lowest energy configuration of structure-24m within the 64-atom supercell, the DFT result indicates that (i) each layer in site 24m along the c-axis direction (i.e., z, -z, z+0.5, and -z+0.5) possesses only one H atom; (ii) each H-O-H angle is roughly with the angle opened to approximately different a-axis (or the equivalent b-axis) directions for different H-O-H layers; and (iii) the O-H bond length and H-H distance are roughly Å and Å, respectively (Fig. 1). 10

11 FIG. 1 Calculated b-axis view of the low energy structure of the H 2 O@Beryl system. Note that the H 2 O molecules are oriented with their HH vector nearly parallel (~0.49 ) to the channel axis (c-axis), which accords with the free alkali, type I structure. 55 Table II summaries the DFT calculated structural properies of beryl and H 2 O@Beryl in comparsion with experimental data. 56,52,53 It is seen that the X-C of PS+D3 is best in predicting the V 0, B 0, and B properties of pure beryl, and this X-C functional is hence the key selection in the present work. In addition, structure-24m is more stable than that of structure-4e (~ 39 mev/atom), indicating that H atoms partially occupy site 24m of H 2 O@Beryl. In particular, phonon calculations of structure-4e yield various imaginary phonons (not shown), indicating its instability. 11

12 TABLE II. DFT calculated structural properties of pure Beryl and H 2 O@Beryl in comparison with experimental data, including equilibrium volume (V 0, Å 3 /atom), relative energy ( E 0, mev/atom), bulk modulus (B 0, GPa), and its pressure derivative (B ). Compound H Site Method E 0 V 0 B 0 B Pure beryl (Al 2 Be 3 Si 6 O 18 ) H 2 O@Beryl PS PBE+D PS+D Expt a 180±2 a, 4.2±0.5 a 178.5±2.5 c 3.90±0.05 c PS e PBE+D PS+D m PS+D Expt b a Based on data from single crystal synchrotron X-ray diffraction and a 4-parameter (3-order) Birch- Murnaghan EOS fitting. 53 b Measurement by X-ray diffraction from single crystal. 52 c Average over aquamarine and goshenite Beryls Results and discussion 3.1 Validation of the VDOS approach In this study, the H- and O- partial VDOS in H 2 O@Beryl were simulated by means of DFT and further utilized to deduce the S j values and hence Ke(H) and Ke(O) in the confined water molecule. As a first step, however, it was necessary to pre-validate the suggested approach in two test cases: in light (H 2 O) and heavy (D 2 O) ice XI, by completing the following steps (i) comparing the simulated phonon spectra with the experimental frequencies, (ii) checking to what extent Eq. (6) can predict the DINS measured Ke(H,D) values, and (iii) testing the procedure of deducing the S j values from VDOS. In the above, ice XI was chosen because it is the low-temperature equilibrium structure of ice Ih 58 whose Ke(H,D) DINS values represent benchmark reference experimental values for theoretical investigations. In the following the terms ice Ih and ordinary ice are used. 12

13 Fig. 2 depicts the DFT simulated H- and O-VDOS of light and heavy ice Ih based on previous DFT calculations. 50 FIG. 2 Upper panel: DFT calculated partial VDOS of H 2 O-ice Ih (H-solid blue, O-solid red). 50 Lower panel: same for D 2 O-ice Ih (D-dashed blue, O-dashed red). The five characteristic phonon bands are indicated. Fig. 2 reveals the full resolution, captured by the DFT simulation, between the five familiar phonon bands of the fundamental lattice and internal modes in condensed pure water phases. Also notable, is the expected redshift of the phonon energies in the deuterated ice compared to its protonated 13

14 form. The data of Fig. 2, was first utilized to account for the total H 2 O-VDOS = 2 H-VDOS+1 O- VDOS. The total VDOS was further utilized for deducing the five fundamental vibrational modes of ordinary ice, by taking the frequency weighted averaged along each of its five characteristic phonon bands, with the phonon amplitudes,, as weights: / ; j = T r, L, 1, 2, 3 (8) with j0 and jf the limits of each characteristic phonon band. The results are summarized in Table III, in which the literature experimental values ice Ih are also indicated. TABLE III. Experimental and DFT calculated vibrational frequencies of ice Ih (in cm -1 units). T r, L, 1, 2, 3 denote translation, libration, symmetric stretch, bending and asymmetric stretch respectively. Bracketed italic values are the percentage deviation of the calculated frequencies from experiment. The last row depicts the DFT frequency ratios, i.e. the isotopic effect. T r L Exp. 2, H 2 O (+1.5) (+6.7) (-1.2) (+0.3) (+1.6) DFT D 2 O H / D Table III illustrates the good accuracy of the DFT calculation in simulating the experimental frequencies. The somewhat larger deviation between the experimental and calculated librational mode in ice Ih may be understood by noting that the experimental value is by itself deduced from a weighted average over the librational band of the measured INS spectra of ice Ih. 2 Note also that apart for translation, the H 2 O/D 2 O frequency ratios of all remaining modes conform to the square 14

15 root of the reduced OD and OH mass ratios, while that of the translational mode is, as expected, close to the D 2 O/H 2 O molecular mass ratios. As a final validation step, we use the computed partial VDOS as input data to Eq. (6) to deduce Ke(X), X = H, D, O, and compare the results to published DINS values. The results are given in Table IV, together with those recently obtained for the HB containing crystals: 4 KH 2 PO 4 and Rb 3 H(SO 4 ) 2. TABLE IV. Calculated Ke(H,D) and Ke(O) values in mev units using SE (Eq. 3) and DFT (Eq. 6), versus DINS results of ice Ih and HBs containing crystals. SE DFT DINS T(K) H, D O H, D O H, D O 4 KH 2 PO 4 4 Rb 3 H(SO 4 ) ± ± 2 D 2 O ± ± ± ± 4 60 H 2 O ± ± ±2 5 The values of Table IV reveal a good agreement, 2%, between the SE and DFT calculations and experiment. Recalling that in the SE approach one solves the problem of an isolated water molecule, while "absorbing" the overall effect of the environment in the input experimental frequencies, the above result is quite impressive. It is interesting to examine the relative differences between the amplitudes distributions of the hydrogen (protium/deuterium) and oxygen atoms as emerge from Fig. 2. In each X-VDOS, individual states (phonons) represent in fact sub-modes of motion in which the X atom is participating. 15

16 Moreover, the amplitude of each discrete phonon may be related to the kinetic energy fraction shared by the X atom in that vibrational state. The relative kinetic energy shares of the 2 H and O in each phonon state are obtained by a normalization procedure and are depicted in Fig. 3 for ice Ih. FIG. 3 Relative partial VDOS of the two protons (, blue) and of the oxygen atom (, red) in ice Ih. These were obtained by normalizing the calculated partial VDOS, and to the total H 2 O VDOS ( + ). Clearly, due to the above normalization procedure, the relative 2H- and O-VDOS of Fig. 3 sum up to unity for each none zero phonon state, and follow the five (T r, L, 1, 2, 3) band structure of ice (see Fig. 2). Moreover, the above shows that the intensities along each of the five phonon bands accord with the nature of the assigned modes in the sense that they comply with the S j fractions (see theoretical section). In the translational region, for example (0 to ~ 350 cm -1 ) the fractions of oxygen 16

17 are seen in Fig. 3 to be ~ 8 fold larger than those of the two protons, as expected from the m O /m 2H mass ratio. To deduce S j (H,O), the intensities of the phonon states constituting each of the five characteristic bands in the partial H- and O-VDOS are integrated and normalized to the total H 2 O-VDOS: / ; X = H, O ; j = T r, L, 1, 2, 3 (9) The resulting values are given in Table V. TABLE V. DFT (Eq. 9) and SE kinetic energy fractions (S j ) shared by the X atom (X=H, D, O) in light and heavy ice Ih. The bottom row depicts the Ke(X) values (in mev units) at 5K, deduced by introducing the SE/DFT S j values and the experimental (SE)/calculated (DFT) frequencies, into Eqs. (3) and (6) respectively. DFT SE S j H 2 O D 2 O H 2 O H O D O H O S Tr S L S S S Ke(X) Worth noting in Table V is the fact that the calculated S j of all atoms in the unit cell (two protons and one oxygen atom) not only sum up to unity for any of the phonon branches (2S j (H) + S j (O) = 1 ; j = T r, L, 1, 2, 3), thus satisfying the orthonormality condition, 61 but at the same time they also fulfil the closure condition 61 (the sum rule of Eq. (2)). Note that the DFT simulated S j values of the H and O atoms in H 2 O ice Ih are nearly the same as that of the SE method (Table V) where only a single H 2 O molecule is considered. 2 In addition, the agreement for H is better than that for O. Note that by inserting the five discrete DFT values of S j and j into Eq. (3) (results not listed in table V), the same 17

18 Ke(H) and Ke(O) are obtained as those of the DFT continuous VDOS, as are, in Eq. (6). Moreover, these values are close to those of SE by ~ < 2 %. Overall, the above analysis may be viewed as a successful validation of the DFT calculation in simulating VDOS and its utilization for the prediction of DINS Ke(H) values in pure ice Ih. 3.2 H 2 O nano-confined in Beryl We now turn to the DFT results of the H 2 O@Beryl system. Figure 4 shows the simulated H- (blue) and O- (red) VDOS of pure ice Ih and of H 2 O@Beryl. For clarity, the two systems are presented in a mirror style. FIG. 4 Partial VDOS of the proton (g H ( ), blue) and of the oxygen (g O ( ), red) atoms in ice Ih (dotted curves 50 in the lower panel) and in Type-I H 2 O@Beryl (upper panel). The inset is a 300 fold 18

19 magnification of the lattice H 2 O@Beryl phonon band up to 1200 cm -1. The lattice bands of H 2 O@Beryl from 0 to 500 cm -1 and at ~ 1224 cm -1 are not indexed due to their uncertain assignment, as discussed in the text. 19

20 Several notable key differences between pure ice Ih and confined H 2 O@Beryl immediately emerge from Fig. 4. First, the stretch frequencies in H 2 O@Beryl occur at 3662 (symmetric) and 3752 (asymmetric) cm -1, in excellent agreement with IR 62,63 (298 K) and INS 41 (5 K) measurements. Being close to the frequencies of the pure vapor, they fully accord with the fact that H 2 O resides in the crystalline cavity of Beryl as a gas-like HBs free molecule. As for the bending frequency, it is redshifted by just ~ 3% relative to that of H 2 O in the vapor phase. Nonetheless, the DFT VDOS structure between ~ 600 and 1050 cm -1, characteristic of H 2 O librations, is markedly different from ordinary ice; 20

21 it is practically empty of any phonon states. In fact, it populates only extremely weak phonon states (see inset to Fig. 4), whose contribution to Ke(H,O) is clearly negligible. Instead, a new VDOS phonon band emerges at ~ 1224 cm -1. It is interesting to note that experimentally, several lines were observed in this energy region: of particular interest is the INS peak observed at ~152 mev (1226 cm -1 ), by Anovitz et. al. 64 together with other peaks; it was conjectured that these peaks may correspond to coupling of vibrations of H 2 O to those of the Beryl nano-cage. In an IR study by Kolesov 55,65 an absorption IR peak was observed at a nearby frequency of 1067 cm -1. In another IR study of hydrated 21

22 Beryl samples, 63 peaks were evident at 1020, 1060 and 1200 cm -1 and were assigned as absorption bands of Si-O-Al, Si-O-Be and Si-O-Si ring vibrations. In view of the above, it is tempting to suggest that the DFT feature at 1224 cm -1 may be due to a coupling of the H 2 O vibrations to those of the Beryl nano-cage. To test the above possibility, we carried a supplementary detailed analysis of all simulated partial VDOS, i.e. of all atoms of dry Beryl as well as those of H 2 O@Beryl as shown in Fig. 5. Note that the intensities in Fig. 5 are averaged per atom and must be multiplied according to their 22

23 stoichiometry in H 2 O@Beryl if the total VDOS is to be calculated. Note also that the O-VDOS (red profile in Fig. 5) corresponds only to the O-atoms of the Beryl skeleton. Fig. 5. Calculated partial VDOS (in arbitrary units and averaged per atom) of all atoms of Beryl skeleton in type-i H 2 O@Beryl, covering the lattice and bending frequencies of H 2 O (upper panel), and those of the H 2 O stretching modes (lower panel). Also presented, is the H-VDOS (blue profile) of the encaged H 2 O molecule. 23

24 It is clearly noted in Fig. 5 that all atoms of Beryl skeleton in H 2 O@Beryl contribute to different extents to all frequency bands that are associated with the confined H 2 O molecule. Explicitly, along the cm -1 regime, characteristic of the lattice and internal bending modes of pure H 2 O, significant shares of all Beryl atoms are noticed, while in the high energy stretching modes of pure H 2 O, only the Si and O-atoms of Beryl have significant intensities. It should importantly be noted in that regard, that the partial VDOS of dry Beryl, all terminate at ~ 1300 cm -1 (not shown). This behavior seems to support the occurrence of mutual vibrational coupling between the confined H 2 O and the host Beryl skeleton as is probably manifested in the appearance of the presently DFT calculated 1224 cm -1 peak in the H-VDOS. In view of the above considerations it will be interesting to carry out refined spectroscopic studies of this band and establish its precise nature. The third notable difference between the simulated VDOS of ice Ih and H 2 O@Beryl (Fig. 4), regards the seemingly distorted structure below ~ 500 cm -1. Note however that in the 0 to ~ 200 cm -1 range both the O- and H-atoms are in-phase following similar structure, while along 200 to 500 cm -1 the VDOS of the two atoms vary out of phase. This behavior could most probably be indicative of a large coupling between the H 2 O molecule and the Beryl lattice vibrations which manifest themselves differently for the O- and H-atoms above 200 cm -1. At this point, it would be tempting to try and deduce the S j values of the H and O atoms in H 2 O@Beryl by analogy to the case of pure ice Ih. However, in strong contrast to ice Ih, for which the simulated partial O and H VDOS of translation, libration, bend and stretch, are well resolved and accurately assigned, the situation in H 2 O@Beryl is by far more complicated. This is due to the large number of modes, 87 of beryl and 9 of H 2 O, and the very probable mixing and coupling, causing a serious difficulty in trying to assess the closure condition formulated by Eq. (2). Nonetheless, it is obvious from Fig. 4 that in H 2 O@Beryl, the share of the proton at low phonon frequencies, below ~ 500 cm -1, is much larger compared to its shares in the remaining phonon bands, than it is in ordinary ice. This fact is nicely captured in Fig. 6 that compares 24

25 the relative partial VDOS of the water proton in Beryl (Blue) and in ordinary ice Ih (red). Fig. 6 clearly shows that at low frequencies, from 0 to ~ 440 cm -1, the S j values of H-VDOS in H 2 O@Beryl differ by 2 to 10 fold compared to ice Ih. Overall, this implies that in H 2 O@Beryl, Ke(H) includes ~ 10 fold larger shares at low frequencies than in ice Ih, and accordingly ~ 2 fold smaller shares in the higher phonon states. Thus in H 2 O@Beryl, one may expect a lower Ke(H) value than in ice Ih. The Ke(H,O) in H 2 O@beryl were deduced by utilizing the simulated partial H and O VDOS of H 2 O in Beryl as input data into Eq. (6). The results are summarized in Table VI. Fig. 6 Relative partial VDOS of H in ordinary ice (red) and in H 2 O@Beryl (Blue). The H 2 O@Beryl H- VDOS has negligible intensities between ~ 450 and 1180 cm -1 (see inset to Fig. 4). The numbers denote kinetic energy fractions (Sj) shared by the proton in the five phonon branches of the confined H 2 O. Note that here, the three internal Sj values selfcontain the 1/3 coefficient of the sum rule (Eq. (2)). 25

26 TABLE VI. DFT calculated kinetic energy fractions shared by the H and O atoms in pure ice Ih and in H 2 O@Beryl at 5 K. Tr, L, B, SS and AS correspond to translation, libration, bending, symmetric stretch, asymmetric stretch respectively. Note that while the two low energy bands in the nanoconfined H 2 O are labeled by N/A (not assigned) due to their uncertain assignments, the high energy bands purely arise from the internal vibrations of H 2 O. The last row depicts the DFT calculated Ke(H,O) (Eq. (6), and the measured DINS values 1,31 at 5K (in mev units). DFT DINS Ice H 2 O@Beryl Ice H 2 O@Beryl Mode H O Mode H O H T r N/A L N/A B B SS SS AS AS Ke(H,O) The calculated values of Table VI show that at 5 K, Ke(H) in H 2 O@Beryl is mev, being lower by 32 %, than that of ice Ih (~ 155 mev), both in remarkable agreement with the DINS 1,31 measured values. Similarly, Ke(O) is lower by ~ 47 % (DINS value not available). It should be recalled, that the DFT Ke(H,O) values of Table VI were calculated directly (without the aid of kinetic energy fraction analysis) by introducing the complete, as are, DFT calculated H- and O- VDOS into Eq. (6). This is because by using Eq. (6) it becomes unnecessary to calculate the energy fractions being self-contained in the VDOS. 26

27 4 Conclusions This study was motivated by the large discrepancy between the SE calculated and the DINS measured Ke(H) value in H 2 O@Beryl. 7,31 We have shown that in the case of pure H 2 O, e.g. ice Ih, the calculated Ke(H), as obtained from DFT simulated VDOS, conforms to the SE and DINS values. However, for the H 2 O@Beryl system, the SE approach fails to account properly for the energy fractions shared by the constituent atoms of the confined H 2 O, leading to faulty Ke(H,O) values that resemble those of pure ice Ih. This was in fact indirectly noted by the authors of Ref. 31 from the experimental point of view, by commenting that the measured kinetic energy of the water protons in Beryl is in complete disagreement with accepted SE methods based on the energies of its vibrational modes. The current DFT key result is a predicted Ke(H) of mev at 5 K which is in excellent agreement with the 5 K DINS measured value of 105 mev. 31 This is in fact the first time that an anomalous DINS measured Ke(H) value is accurately predicted. The present result seems to have critical implications as it changes completely our view on the behavior of Ke(H) in nano-confined water and thus on the origin for the reported DINS anomalies. While these were presumed to stem from the anharmonicity of the potential sensed by the proton in its confining environment, the present findings hint that a correct treatment should rely on partial VDOS simulations, and that the assumption of the HA introduces only a small error in Ke(H) determinations. It is also interesting to note that in accurately deducing Ke(H), we did not account for the six fold proton tunneling, observed in Refs. 29, 30,31. Finally, it should be noted that anomalies in the shape of the proton momentum distribution, e.g. secondary features, as systematically observed by DINS measurements of nanoconfined H 2 O systems, are often of crucial importance to describe the anomalous behavior of the proton, but cannot be explained using the harmonic approximation. The present success in approaching the H 2 O@Beryl system by utilizing DFT simulations, illustrates the advantage of DFT calculated partial VDOS over the SE approach in properly accounting for the kinetic energies of the H- and O-atoms of H 2 O in H 2 O containing systems, and may thus be viewed as a promising tool for resolving the long standing debate regarding the Ke(H) 27

28 values measured by DINS in nano-confined H 2 O. Overall, these findings warrant more DFT studies to simulate the partial VDOS of other systems involving nano-confined water such as H 2 O@SWCNT and H 2 O@DWCNT, for which anomalous Ke(H) values were measured by DINS. 22 Another related system worth studying in that regard is the H 2 O@Bikitaite 7 in which the water molecules reside as a one dimensional ice chain along the crystallographic channel. Such studies are essentially required towards a comprehensive fundamental understanding of the problem. While the present result seems to provide a large step forward in trying to explain the huge difference between the measured and calculated values of Ke(H) in the H 2 O@Beryl system, it still remains to fully understand the exact nature of the DFT simulated line at ~ 1220 cm -1 and its experimental counterpart at nearby energies. Acknowledgments S.L.S, Y.W. and Z.K.L. would like to thank the financial support by U.S. National Science Foundation (NSF) with Grants no. DMR First-principles calculations were carried out partially on the LION clusters at the Pennsylvania State University, partially on the resources of NERSC supported by the Office of Science of the U.S. DOE under contract no. DE-AC02-05CH11231, and partially on the resources of XSEDE supported by NSF with Grant no. ACI We would also like to thank the anonymous reviewers for their efforts and helpful remarks. 28

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