Article pubs.acs.org/JPCC
Ab Initio, Embedded Local-Monomer Calculations of Methane Vibrational Energies in Clathrate Hydrates Chen Qu and Joel M. Bowman* Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, United States ABSTRACT: We present an anharmonic, coupled-mode vibrational analysis of CH4 in four clathrate cages, 512, 51262, 435663, and 51264, employing a general, fulldimensional, ab initio potential energy surface for CH4(H2O)n clusters. This potential is expressed in a many-body representation, truncated at the three-body level, for the water and the methane−water interactions. The embedded localmonomer model is used to determine the energies of the intramolecular vibrations of the confined methane. This model is validated by comparing the harmonic density of states using local-monomer and standard full normal-mode analyses for the 512 CH4@(H2O)20 clathrate. The agreement in the region of the methane intramolecular vibrations is excellent. Vibrational self-consistent field and virtual-state configuration interaction theory is employed to calculate the vibrational energies of methane in four cages using the code MULTIMODE, and comparisons with experiment are given. The zero-point energy and wave function of the enclathrated methane molecule are obtained in full dimensionality, but with a rigid cage, using diffusion Monte Carlo calculations. The results indicate substantial rotational delocalization. Dissociation energies are reported based on these calculations for methane in 512 and 51262 cages. the frequencies of methane in large cages (51262 and 51264) are lower than those in small cages (512). Therefore, Raman spectroscopy has been used to identify hydrate structures and cage occupancies. A simple “tight cage and loose cage” model,11 which was originally proposed to explain the frequency shifts of diatomic solutes due to interactions with liquid solvents, can be used to qualitatively explain the frequency shifts of enclathrated methane. The IR spectrum of the methane stretching mode exhibits substructure that resembles the gas-phase rovibrational lines, indicating that the trapped methane in the cage is a quasirotor displaying gaseous behavior.2 Later, the IR spectra of CH4/CF4 clathrate mixtures were recorded to decipher the assignment of the stretching modes in different cages.3 Because CF4 always occupies the large cage, in CH4/CF4 clathrate mixtures, the absorption of methane in the large cages became weaker, and thus, it could be distinguished from the absorption of methane in the small cages. It was also found that the asymmetric stretching frequencies in the large 51262 cages, as for the symmetric stretch, are also lower than those in the small 512 cages. In addition, the spectrum for the overtones and combination modes of the trapped methane was also recorded, and the transitions were assigned. The rotation−translation eigenstates of enclathrated guest methane were probed by inelastic neutron scattering.12−18 The transitions between the three lowest rotational states are between 8 and 27 cm−1,13 and translational excitations appear at higher energies;16,17 however,
1. INTRODUCTION Clathrate hydrates are crystalline inclusion compounds in which small molecules are trapped in hydrogen-bonded water cages. Among the clathrate hydrates, methane hydrate has attracted considerable interest, as it is a potential source of energy, as well as a potent source of a greenhouse gas. It might also be responsible for flow reductions in gas pipelines.1 Methane clathrate might also play an important role in many astrophysical bodies, and its infrared (IR) spectrum has been recorded to provide identification for its presence.2,3 Three typical crystal structures of clathrate hydrates have been observed: cubic structure I (sI), cubic structure II (sII), and hexagonal structure H (sH). In a unit cell of sI, 46 water molecules form two pentagonal dodecahedral (512) and six tetrakaidecahedral (51262) cages. An sII unit cell consists of 136 water molecules that form sixteen 512 and eight hexakaidecahedral (51264) cages. A unit cell of sH is made of 34 water molecules forming three 512 cages, two irregular dodecahedral (435663) cages, and one icosahedral (51268) cage. Simple methane hydrates crystallize in the sI structure, and the methane molecules occupy both the small (512) and large (51262) cages.1 Mixed hydrates with methane and other guest molecules can form other crystal structures. Vibrational spectroscopy is a valuable tool for investigating the properties of clathrate hydrates, such as structures, cage occupancies, and hydrate compositions. Raman spectroscopy has been applied to study the vibrational energies of enclathrated methane molecules.4−10 The symmetric stretch of methane is Raman-active, and its frequency red shifts compared to the corresponding gas-phase value. Furthermore, © XXXX American Chemical Society
Received: November 12, 2015 Revised: January 20, 2016
A
DOI: 10.1021/acs.jpcc.5b11117 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry C
frequencies of asymmetric stretches and a few overtones and combination bands are reported. Bending fundamentals are also provided, even though they have not been reported experimentally. The vibrational ground-state properties have been characterized by rigorous quantum diffusion Monte Carlo (DMC) calculations. The binding energies of methane with two cages are also presented. In section 2, we summarize the PES that we are using and present the geometries of the clusters of interest. In section 3, we describe the local-monomer model and compare the vibrational densities of states at the harmonic level determined using this model and normal-mode analysis in full dimensionality. The theory of VSCF and VCI calculations is summarized briefly in section 4, and the intramolecular frequencies of guest methane in four different cages (512, 435663, 51262, and 51264) are presented. In section 5, we present the DMC calculations and the zero-point properties of trapped methane. Finally, in section 6, a summary and concluding remarks are provided.
the assignment of the peaks is complicated because of lattice vibrations. Theoretically, the intramolecular vibrations of enclathrated methane have been calculated using ab initio molecular dynamics simulations and Fourier transformation of the autocorrelation functions.19−23 These simulations applied simple model potentials such as the Kuwagai−Kawamura− Yokokawa potential24 and the consistent valence force field (CVFF),25 or they used density functional theory approaches such as SIESTA26 and the Perdew−Burke−Ernzerhof (PBE) functional.27 These studies were able to reproduce the experimental trend of the frequency shift of the symmetric stretch but not to provide quantitative agreement. In addition to molecular dynamics, the vibrations of confined methane have been investigated using direct harmonic normal-mode analysis applying a model potential28 or using computationally efficient methods such as Hartree−Fock29 and density functional theory.29−31 As expected, the harmonic approximation can achieve only qualitative agreement with the experimental trend. It is also worth noting that the “independent molecule model” in ref 28 is very similar to our “local-monomer model” (which is discussed later in this article) at the harmonic level. The translation−rotation energy levels of enclathrated methane molecules have been investigated by fully coupled sixdimensional quantum calculations using a simple pairwiseadditive potential for the methane−water interaction.32 The calculated results for rotational and translational transitions agree reasonably well with the experiments. However, the angular anisotropy of the methane−cage interaction is exaggerated, and the translational fundamentals are underestimated, because of the deficiencies of the potential applied in the study. However, the early investigations of the guest vibrations did not apply an accurate potential energy surface (PES) for methane hydrate. The potential was evaluated either by using efficient low-level ab initio theories (for instance, density functional theory, Hartree−Fock theory, or second-order Møller−Plesset perturbation theory) or by assuming that the methane−water interaction is pairwise-additive. Furthermore, in calculations of the guest intramolecular vibrations, the methods applied were normal-mode analysis or molecular dynamics, which are not able to account for anharmonicity or quantum effects. Therefore, more reliable results would be expected if an accurate PES and a rigorous method for vibrations were applied. Recently, we presented accurate, fulldimensional, intrinsic PESs for CH4−H2O and CH4(H2O)2 and, thus, constructed a potential for the CH4(H2O)n system using the many-body expansion form.34,35 The methane−water two-body interaction was calculated using the high-level CCSD(T)-F12b method, and we went beyond the pairwiseadditive model by including the CH4−H2O−H2O three-body interaction at the MP2-F12 level. With respect to vibrational calculations, vibrational self-consistent field (VSCF) plus virtual-state configuration interaction (VCI)36 combined with the local-monomer model37 was used. This method has been successfully applied to molecular clusters characterized by hydrogen bonds.38−45 In this article, we apply the new many-body PES for CH4(H2O)n and perform local-monomer quantum anharmonic calculations for the intramolecular vibrations of guest methane molecules in different clathrate cages. Specifically, the frequency of the methane symmetric stretch has been calculated and compared with experimental Raman spectroscopy data, and the
2. POTENTIAL ENERGY SURFACE The potential of CH4(H2O)n is represented in a many-body expansion form as (1) VCH4(H 2O)n = V CH + 4
(2) ∑ V H(1)O(i) + ∑ VCH − H O(i) i
+
4
i
2
(3) ∑ V H(2)O(i) − H O(j) + ∑ VCH − H O(i) − H O(j) i
