Understanding Vibrational Anharmonicity and Phonon Dispersion in

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Understanding Vibrational Anharmonicity and Phonon Dispersion in Solid Ammonia Borane Shawn M. Kathmann,*,† Christopher J. Mundy,† Gregory K. Schenter,† Tom Autrey,† Philippe C. Aeberhard,‡ Bill David,‡,§ Martin O. Jones,‡ and Timmy Ramirez-Cuesta§ †

Chemical and Materials Sciences Division, Pacific Northwest National Laboratory, Richland, Washington 99352, United States Department of Chemistry, University of Oxford, Oxford, United Kingdom § ISIS, Rutherford Appleton Laboratory, Oxford, United Kingdom ‡

ABSTRACT: We compute the hydrogen vibrational spectra for bulk ammonia borane (NH3BH3), for both protonated and deuterated cases, using harmonic and anharmonic methodologies as well as low frequency coupling and quantifying dispersive effects to understand their influence on the resulting spectra. Even at 10 K the accounting of anharmonic effects on the incoherent inelastic neutron scattering (IINS) signal is more significant than approximations introduced by sampling and interpolating a finite simulation cell to capture dispersion effects. We compare our computed anharmonic spectrum with IINS measurements and find excellent agreement.

I. INTRODUCTION The use of increasingly sophisticated treatments of computational vibrational spectroscopy is important because it enhances the correspondence between the simulation and measurement of the dynamics of atoms and molecules in condensed matter. Inelastic incoherent neutron scattering (IINS) is an extremely powerful technique1−6 that can be directly related to the motions of the scattering atoms and thus allows one to employ modern computational approaches to better quantify molecular structure and dynamics.7−15 In IINS experiments, one observes how the strength of the neutron scattering varies with energy and momentum transfer via the system lattice atoms absorbing energy from the neutrons in particular directions. Neutrons are particularly useful in studying hydrogen dynamics because hydrogen’s scattering cross section is ten times higher than any other element. Furthermore, neutron spectroscopy is not subject to the selection rules of infrared and Raman spectroscopies, and hence provides a complementary measure of the vibrational dynamics. Here we wish to describe the vibrational dynamics in ammonia borane (NH3 BH3 or more simply AB) and deuterated AB (ND3BD3) employing ab initio molecular dynamics to compare directly with experimental IINS spectra. AB’s structure was determined via neutron diffraction by Klooster and co-workers16 and was chosen to investigate because of its capacity to store large amounts of hydrogen and the desire to better understand the unique dihydrogen bonding in this molecular crystal. Dihydrogen17 bonds in AB are a special type of intermolecular hydrogen bond where a hydrogen atom bonded to boron (referred to as a hydridic hydrogen and denoted by Hδ−) interacts favorably with a hydrogen atom © 2012 American Chemical Society

bonded to nitrogen (referred to as protonic hydrogen and denoted by Hδ+). The different charge character of these hydridic and protonic hydrogen atoms arises because of the difference in electronegativities between the boron (χB = 2) and nitrogen (χN = 3) atoms yielding a Hδ−···Hδ+ dihydrogen bond. In the IINS measurements the neutrons probe the energy and momentum transfer to the protonic and hydridic atoms in the 0−1000 cm−1 spectral range, which is where the dihydrogen bonding modes reside. Thus, our measurements and vibrational analysis provides both experimental and computational benchmarks of the hydrogen dynamics (including the dihydrogen bonds) in AB. We will show the benefits and limitations of the spectra computed within the harmonic approximation compared to the fully anharmonic dynamical treatment. Additionally, we address the vibrational dispersion in the spectra by using increasingly larger simulation cells to sample points in phonon k-space. The harmonic approximation assumes that small amplitude motions on the potential energy landscape exhibit the same period as large amplitude motions. We will show that this is not the case, even at 10 K for molecular crystals of AB. An anharmonic treatment considers how the potential energy landscape changes with increasing temperature, with increasing amplitude of motions. Furthermore, as the amplitude of the motion increases, the nature of the coupling of the characteristic motions can change. The small amplitude coupling corresponding to a harmonic approximation (i.e., normal Received: August 5, 2011 Revised: December 24, 2011 Published: January 9, 2012 5926

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the incoherent dynamical structure factor), Sinc(k,ω), is determined2 from the Fourier transform of the correlation function

modes) can differ significantly from the large amplitude coupled motion sampled in ab initio molecular dynamics. An anharmonic treatment of the motion explores the nature of the potential energy landscape. It allows one to compare the difference between small amplitude motions and large amplitude motions and their coupling. Inelastic incoherent neutron scattering measurements sample the hydrogen self-dynamics or, equivalently, the hydrogen velocity autocorrelation function of solid NH3BH3 and can be compared to the simulated spectrum computed from ab initio molecular dynamics. The harmonic approximation to the vibrational spectra can be found in many1,18 textbooks. We note that the vibrational spectra computed using the harmonic approximation, although computationally convenient, can potentially be misleading if the full anharmonic treatment is unavailable for comparison and validation. Typically, the harmonic approximation works quite well for high frequency vibrations (e.g., above 1000 cm−1), however, it does not describe the low frequency vibrations along the entrance channels to the dihydrogen coupling reaction pathways important in the release of hydrogen from AB. This paper provides essential benchmarks on how well the vibrational dynamics of AB and deuterated AB in the 0−1000 cm−1 range are described by the harmonic versus ab initio molecular dynamics phonon spectra when compared to IINS measurements. Vibrational dispersion arises from the variation in the relative phases of motion of atoms in the lattice. Here we are using the term dispersion only as a measure of the degree of instantaneous correlation between the vibrations of one atom with the vibrations of another and not in the context of dispersion interactions arising from instantaneous induced multipole moments (e.g., van der Waals/London dispersion interactions that vary as r−6). We can effectively sample more of the dispersive motion by using increasingly larger simulation to sample the longer wavelengths. This is because the size of the simulation cell determines the longest wavelengths that scale as λmax ≈ 2L, where L is the longest dimension of the simulation cell; see Figure 1. The IINS signal (or sometimes referred to as

Sinc(k, ω) =

∫

dt ei ωt ∑ ⟨ei k·rl(t )e−i k·rl(0)⟩ (1)

l

where k represents the neutron wave vector (or the neutron momentum transfer) and rl(t) represents the time dependent position of hydrogen atoms. The scattering intensity of an atom l (within the low temperature approximation2) is S(k, ων)n ∝ σl

(k·v ul)2n exp[−(k· ∑ v ul)2 ] n! ν

δ(Ei − Ef + nℏω)

(2)

where vul is the amplitude of the displacement of atom l in the vibrational mode ν, σl is the cross section of the atom l, and n is the order of the transition. This means that the contribution from overtones and combinations can have a great impact in the overall shape of the IINS spectrum. The term exp[− (k· ∑ v ul)2 ] = exp[− (k·Ul)2 ] = exp[− 2W (k)] (3)

ν

is the well-known Debye−Waller factor and manifests as an attenuation of the spectral intensity as the momentum transfer k increases. Ul, the total amplitude, is the sum of all of the amplitudes of the motion of atom l over all frequencies. The exact calculation of the IINS spectra, in the harmonic approximation, using the eigenvectors and eigenvalues of the static dynamical matrix calculation can be done using the aClimax software;19 see ref 19 for more details. In the case of using molecular dynamics methods to calculate the phonon density of states, there is no exact way of including multiphonon events and an approximation has to be made as shown in ref 2. Moreover, if the system is not harmonic, the multiphonon events do contribute to the spectral intensity as a general featureless background that increases with energy transfer, as it is in the case of the AB system under consideration in the paper. If we take the limit for low momentum transfer, the multiphonon contribution is small, and we obtain the relation I(ω) ∝ lim

ω2

1

Sinc(k, ω) 2 k → 0 |k| 1 + coth(ℏωβ/2)

(4)

where I(ω) is the vibrational spectrum. This formula approximates the relation between the density of vibrational states and the IINS signal, the coth (ℏωβ/2) term is negligible when ω > 50 cm−1 if the sample is cooled below 20 K. The limit |k| → 0 is not reached in general, and in particular for an indirect geometry instrument, with very low final energy transfer like TOSCA, the dynamical trajectory can be approximated by |k| ∝ √ω when ω > 50 cm−1; see ref 2. Therefore, for ω > 50 cm−1, the peak positions of the fundamental vibrations and relative intensities can be approximated by I(ω) ∝ ωSinc(k, ω)

(5)

The IINS measurements were performed with the TOSCA spectrometer at the ISIS neutron spallation source in the Rutherford Appleton Laboratory, in the United Kingdom. TOSCA, with a spectral resolution of ΔE/E ≈ 1% is the world’s

Figure 1. Showing at 3 × 3 × 3 simulation cell of AB molecules and the maximum wavelength λmax that can fit into the cell of dimension L. 5927

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the 23 spectra. We contrast these results with the broadband anharmonic vibrational spectra at 10 K for the 23, 33, and 43 cells. Again, the cell size alters the relative intensities with the anharmonic spectra being more similar for all cell sizes than that found with the harmonic spectra. In Figure 2 we compare

highest resolution indirect geometry IINS spectrometer. Samples were loaded in aluminum cans and cooled to a base temperature of 12 K. The raw data was converted from time-offlight using standard ISIS routines. In the analysis of both the harmonic system as well as the classical dynamics analysis of the anharmonic system motion, in this study we consider the intermolecular motion of a finite system consisting of a solid that is extended beyond the unit cell. We employ 2 × 2 × 2, 3 × 3 × 3, and 4 × 4 × 4 simulation cells to investigate the influence of system size on the vibrational spectra. We use these systems to extract correlation of motion across the Brillouin zone. Because of the finite size of the simulation cells, these calculations only account for correlated motions at the symmetry zone boundaries of the extended cells. The averaging of the motion effectively interpolates the k dependence of the intensity to represent correlated motion that corresponds to points within the Brillouin zone of the unit cell. With larger cells, the distance in k-space between sampled points becomes smaller and the interpolation of the full Sinc(k,ω) becomes more converged. Dispersion curves correspond to the location of the peaks in Sinc(k,ω) as the parametric curve ωi(k). Here i denotes different branches of the dispersion curves, corresponding the fundamental motions in the unit cell. The ab initio hydrogen velocity autocorrelation (VAC) can be used to obtain the anharmonic vibrational phonon spectrum, I(ω), by Fourier transformation I(ω) =

+∞

∫−∞

exp(i ωt )Cvv(t ) dt

Figure 2. Comparison of the broadband harmonic (green) and anharmonic (blue) vibrational spectra for the largest 4 × 4 × 4 AB simulation cell showing differences in the relative intensities of the vibrational peaks heights across the entire spectral range and that largest discrepancies in peak positions and intensities occur below about 1000 cm−1.

the broadband harmonic (green) and anharmonic (blue) vibrational spectra for the 43 cell showing differences in the relative intensities of the vibrational peaks heights across the entire spectral range with the largest discrepancies occurring for peak positions and intensities below about 1000 cm−1. We also compared the low frequency region (below 1000 cm−1) of the harmonic spectra for the 23, 33, and 43 cells showing similar peak positions, however, the relative peak heights are shifted with the larger cells (33 and 43) being more similar than the smallest cell. The low frequency region of the anharmonic vibrational spectra at 10 K for the 23, 33, and 43 cells shows changes in relative peak heights in the vibrational spectra. However, the spectra for all cell sizes are in better agreement than those found in the harmonic cases. To test the influence of the chosen atomic displacements used in the calculation of the harmonic frequencies, a 23 cell was used for computational convenience. Testing the influence of the atomic displacements used in the calculation of the harmonic spectra is important because of its use in the numerical evaluation of the Hessian matrix. The Hessian is a matrix of second order derivatives of the system energy with respect to geometry. The diagonalized Hessian matrix yields vibrational eigenvectors and eigenvalues, the roots of which are the fundamental frequencies of the normal modes. A harmonic potential energy landscape by definition rises parabolically around a minimum; however, the actual potential energy is not parabolic. The consequence of this nonparabolic topography is that the frequencies will become increasingly red-shifted which is a consequence of increasing the atomic displacement parameters chosen to calculate the curvature around the minima. Indeed, the resulting harmonic spectra in Figure 3 show that, as the step size gets larger, the harmonic frequencies are red-shifted due to the greater inclusion of the anharmonic (nonparabolic) part of the potential energy landscape. However, these results show that the full anharmonicity in

(6)

where Cvv(t) is the VAC function given by Cvv(t ) =

1 N

N

∑ ⟨vj(t )·vj(0)⟩ j=1

(7)

N is the number of hydrogen atoms in the system and vj(t) is the velocity of a hydrogen atom j at time t. This allows a direct comparison between the computed phonon spectra and the IINS measurements.

II. METHODS AND COMPUTATIONAL DETAILS The bulk AB16 was simulated using the simulation package CP2K with central simulation cells containing 2 × 2 × 2, 3 × 3 × 3, and 4 × 4 × 4 AB extended units cells at a temperature T = 10 K in the NVE ensemble. The lattice vectors (Lx, Ly, Lz) for these simulation cells are (in Angstroms): (11.09114, 9.23686, 9.98501), (16.63671, 13.85529, 14.97752), and (22.18228, 18.47372, 19.97002). A total of 5 ps, after equilibration, were used for the analysis of the ab initio anharmonic trajectory. For the harmonic modes, symmetry constrained optimizations were used (unless noted otherwise). The PBE density functional theory functional was used as in our previous studies8,10,20,21 as it provides a reasonably accurate description of the structural and vibrational observables for hydrogen-rich systems. We use TZVP-GTH basis sets with the GTH pseudopotentials. III. RESULTS AND DISCUSSION First we compare the broadband (0−3500 cm−1) harmonic vibrational spectra for the 2 × 2 × 2 (23), 3 × 3 × 3 (33), and 4 × 4 × 4 (43) AB simulation cells. The spectra (not shown) show that the cell size mainly influences the relative peak heights. The 33 and 43 cells are more consistent compared to 5928

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Figure 5. Comparison of the low frequency region of the measured IINS spectra (black with circles) with the harmonic (green) and anharmonic (blue) vibrational spectra for the largest 4 × 4 × 4 AB (NH3BH3) simulation cell. We note that both the harmonic and anharmonic vibrational spectra I(ω) have been divided by ω to make a direct comparison with what is measured experimentally.

Figure 3. Comparison of the low frequency harmonic vibrational spectra using a 2 × 2 × 2 simulation cell using increasingly larger displacement steps to calculate the harmonic frequencies. These results show that as the step size gets larger, the harmonic frequencies are red-shifted due to the greater inclusion of the anharmonic part of the potential energy.

circles) with the harmonic (green) and anharmonic (blue) vibrational spectra for the largest 43 AB simulation cell. We note that both the harmonic and anharmonic vibrational spectra I(ω) have been divided by ω to make a direct comparison with what is measured experimentally − as shown in eq 5. The results in Figure 5 show that the anharmonic vibrational spectrum agrees much better with the measured IINS spectra compared to the harmonic spectra. We note that the small peaks at 400 and 500 cm−1 are likely combination and overtone bands of the 100−200 cm−1 bands that we were unable to sample during our simulation. We also investigated the phonon spectra (not shown) from the ab initio molecular dynamics projected onto internal coordinates, via Fourier transformation of the velocity autocorrelation functions of internal coordinates, and found that all the resulting spectra provided no simplification of the modes into the clear spectral assignments found at higher frequencies as would be evident from single peaks in the coordinate systems. This means that the low frequency anharmonic modes of interest to ammonia borane (AB) are highly coupled with respect to the internal coordinates. In Figure 6 we compare the experimental IINS for deuterated AB (ND3BD3) with both harmonic and anharmonic vibrational spectra. It is important to highlight two essential differences in the harmonic versus anharmonic analysis: (1) in the 200−400 cm−1 region the harmonic peak does not recover the vibrational broadening displayed by both experiment and anharmonic phonon spectra and (2) the deuterated harmonic spectra in the 200−400 cm−1 contains only a single peak whereas the undeuterated (NH3BH3) harmonic spectra in the same region (as shown in Figure 5) show two distinct peaks compared to the anharmonic treatment which shows only a single broad peak in this region. We also note that the signal from the deuterated system and the hydrogenated system are related to each other by a scaling factor for both the experimental IINS and anharmonic phonon spectra in the 0−1000 cm−1 region. The experimental deuterated AB spectrum is well described by multiplying the undeuterated AB IINS spectra by 2−1/2 as shown in Figure 7. We clearly recognize that this scaling is only heuristic; since various effective masses associated to the system

the spectra cannot be recovered simply by using larger displacements in the estimation of harmonic frequencies as evidenced by the number, widths, and positions of the spectral peaks. In Figure 4 we compare the low frequency region of the harmonic (green), harmonic with no symmetry constraint

Figure 4. Comparison of the low frequency region of the harmonic (green), harmonic with no symmetry constraint (red), and anharmonic (blue) vibrational spectra for the largest 4 × 4 × 4 AB simulation cell. These results show that vibrational peak positions and intensities are quite different in the 0−500 and 650−900 cm−1 regions.

(red), and anharmonic (blue) vibrational spectra for the largest cell (43). It is important to note that the use of symmetryconstrained optimization for the harmonic modes is essential when computing the low frequency regions of the IINS spectra. This is due to the multiple minima problem associated with hydrogen disorder in the solid state, which is clearly evident by the two peaks that occur at 300 and 400 cm−1 when symmetry is not enforced in the optimization to the minimum energy configuration of the crystal. These results show that harmonic versus anharmonic vibrational peak positions and intensities are quite different in the 0−500 and 650−900 cm−1 regions. In Figure 5 we compare of the measured IINS spectra (black with 5929

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Figure 8. Plot of the calculated harmonic dispersion curves (left in black) compared to a plot of the measured (right black with circles) and computed harmonic (right in green) and anharmonic (right in blue) vibrational spectra.

Figure 6. Comparison of the low frequency region of the measured IINS spectra (black with circles) with the harmonic (green) and anharmonic (blue) vibrational spectra for the deuterated AB (ND3BD3) 2 × 2 × 2 simulation cell. Note that in the 200−400 cm−1 region the harmonic peak does not recover the vibrational broadening displayed by both experiment and anharmonic phonon spectra.

concise benchmarks for the hydrogen dynamics in the dihydrogen bonded AB system. We considered harmonic, anharmonic, isotopic, and dispersive contributions to the computed phonon spectrum. Even at 10 K the accounting of anharmonic effects on the IINS signal are more significant than approximations introduced by sampling and interpolating a finite simulation cell to capture dispersion effects. We find converged results using a 33 cell. We find that anharmonic, isotopic, coupling, and dispersive effects can be significant below 1000 cm−1. Above 1000 cm−1, a classical dynamics anharmonic analysis of the phonon spectrum is consistent with the harmonic analysis. This is because classical excursions at 10 K from a minimum energy structure, along the normal mode directions corresponding to these higher frequencies, are not large enough to sample the anharmonic regions of the potential energy surface. In order to achieve even better quantitative agreement with experiment, we expect that a quantum mechanical treatment of the vibrational problem may be required to account for the zero-point sampling of the anharmonic regions of the potential energy landscape. Future studies will focus on the role of higher temperatures in the computed anharmonic spectra and measured IINS as well as the role of quantum mechanical hydrogen tunneling.

Figure 7. Experimental deuterated AB spectra (black with circles) is very well described by multiplying the undeuterated AB IINS spectra (red with circles) by 2−1/2.

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ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy’s (DOE) Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences program and was performed in part using the Molecular Science Computing Facility (MSCF) in the William R. Wiley Environmental Molecular Sciences Laboratory, a DOE national scientific user facility located at the Pacific Northwest National Laboratory (PNNL). This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DEAC02-05CH11231. Pacific Northwest National Laboratory is operated by Battelle for the U.S. Department of Energy.

normal modes (although anharmonic) are not always simply given by the proton/deuteron mass ratio. To further address the role of dispersion in the AB system, Figure 8 shows the calculated harmonic dispersion curves as function of the reduced wave vector coordinate (left in black) using a 23 cell compared to a the measured IINS spectrum (right black with circles), the harmonic 43 spectrum (right in green) and the anharmonic 43 spectrum (right in blue). From the slopes of the dispersion curves on the left, the major dispersive contribution arises from the acoustic modes that occur below 100 cm−1. The optical B−N stretching modes between 700 and 900 cm−1 are rather flat across the entire Brillouin zone. However, between 100 and 400 cm−1 the optical modes do show some dispersive broadening.

■

REFERENCES

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IV. CONCLUSIONS We have presented calculations of the ab initio vibrational spectra of solid ammonia borane and compared them with the measured IINS spectrum. This analysis provides clear and 5930

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