SUPPLEMENTAL MATERIAL
Obtaining the Hessian from the force covariance matrix:
Application to crystalline explosives PETN and RDX
Andrey Pereverzev and Thomas D. Sewell
Department of Chemistry, University of Missouri-Columbia, Columbia, MO 65211-7600
[email protected] and [email protected]
S-I. Vibrational Density of States from the Velocity Autocorrelation Function
In Fig. S1 we present, for PETN at 298 K (panel (a)) and 20 K (panel (b)), the
vibrational density of states (VDOS) obtained from the Fourier transform of the massweighted velocity autocorrelation function (VACF).1 These results can be compared to
results for the VDOS in the main article, where the VDOS was obtained by binning with
1 cm-1 resolution the mode frequencies obtained from the the standard Hessian (0 K) and
effective Hessian (20 K and 298 K). To generate the results shown here, ten 100.0 ps
isochoric-isoergic (NVE) trajectories were calculated at “298 K” and “20 K”, from which
the ensemble-averaged VACF was calculated and subsequently Fourier transformed.
More details about these trajectories are given below in Sec. S-II. Atomic velocities were
recorded every 4.0 fs for construction of the VACF. Only the first 20.0 ps of the 100.0 ps
VACF were used in the Fourier transform because the precision of the VACF becomes
low for times comparable to the total simulation time. The resolution of the resulting
VDOS is approximately 1 cm-1, similar to what is used in the binning approach for the
VDOS shown in Figs. 2-5 of the main article. Although clearly the details differ,
especially at high frequencies where the significant frequency dispersion in the effective
1
Hessian calculations—due, we think, to incomplete sampling as discussed in the main
article—leads to considerable broadening in the high-frequency peaks, the VDOS peaks
in Fig. S1 are in general agreement with the results in the main article obtained from both
the effective and standard Hessian calculations.
Figure S1. Vibrational density of states for PETN crystal obtained from the Fourier
transform of the velocity autocorrelation function. (a) 298 K; (b) 20 K.
S-II. Comparison of Vibrational Frequencies for PETN Calculated Using the
Effective Hessian Based on NVE and NVT Trajectory Integration
In the main article we opted to construct the force covariance based on ensembles
of NVE trajectories rather than isochoric-isothermal (NVT) trajectories because in the
former case Newton’s equations of motion are solved directly with no need to introduce
the thermostat, and the associated empirical thermostat coupling constant, into the
equations of motion. While the systems studied were rather large, such that the
differences between the NVE and NVT ensembles should be small compared to the
precision of our simulations, the formal expressions from which the effective Hessian is
derived are based on an average over the canonical ensemble. For this reason, we attempt
here to assess possible effects of using the NVE equations of motion rather than the NVT
2
equations of motion by analyzing simulations performed for those two cases starting from
identical initial conditions corresponding to the desired temperatures (20 K or 298 K).
The initial conditions for this assessment were obtained as described in the main text and
the results here are based on ensembles of ten 100.0 ps trajectories. Figure S2 contains a
comparison of frequencies obtained from the effective Hessian for the two sets of
equations of motion, for 298 K (panel (a)) and 20 K (panel (b)). Results for the NVE and
NVT integrations are shown in blue and purple, respectively. The two sets of results
overlap closely and are nearly indistinguishable to the eye for both temperatures.
Therefore, to clarify the differences, the inset in each panel shows the relative percentage
difference (RPD), plotted versus mode number, between the NVE- and NVT-based
frequencies. For both temperatures the mean RDP is less than one percent. Thus, even for
ensembles of only ten trajectories, NVE and NVT simulations give very similar results
for both temperatures, implying practical equivalence of the two ensembles in the
effective Hessian calculations.
(
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)
(
4
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( )
12
3
10
RPD
RPD
)
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2
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0
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2000
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Mode
Mode
Figure S2. Frequencies, plotted versus mode number, of PETN crystal normal modes
obtained from the effective Hessian at 298 K (panel (a)) and 20 K (panel (b)). Blue and
purple symbols correspond to results obtained from NVE and NVT trajectory
integrations, respectively. Insets show relative percentage difference (RPD), plotted
versus mode number, between the NVE- and NVT-based frequencies.
3
S-III. Squared Overlap Matrices for PETN Effective Hessians at 298 K and 20 K
Compared to the Standard Hessian (T = 0 K)
Normal modes obtained both by standard Hessian analysis and from the effective
Hessian define orthonormal complete sets. Although a detailed mode-by-mode
comparison of the eigenvectors obtained from the two methods (or more generally for
two different non-zero temperatures when the effective Hessian is used) is difficult, a
simple way to compare the modes qualitatively is to expand the eigenvectors obtained
using one method in terms of the eigenvectors obtained using the other. We have done
this for PETN by expanding the 298 K and 20 K normal modes from the effective
Hessian in terms of the 0 K modes obtained from standard normal mode analysis. The
results are shown in Fig. S3 for 298 K (left-hand panel) and 20 K (right-hand panel). The
abscissa corresponds to the standard Hessian (0 K) and the ordinates correspond to the
effective Hessians at the respective temperatures. Prior to expanding one set of
eigenvectors in terms of the other, the modes in each set were ordered according to
frequency (lowest to highest). The results in Fig. S3 are for the k = 0 subspace (which
contains 171 modes, neglecting the three translational modes with zero frequency that are
not shown in the figure), but similar results were obtained for the other values of k
studied, including the largest value of k in the Brillouin zone consistent with the
simulation supercell used. For both temperatures the overlap matrices are approximately
diagonal. Stronger anharmonic effects are visible in the case of 298 K for low-frequency
modes (mode numbers below ≈50) for which the elements of the overlap matrix exhibit
significant amplitude further away from the diagonal.
4
50
100
150
50
100
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150
150
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100
50
1.0
Mode number
Mode number
1.0
0.0
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Mode number
0.0
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150
Mode number
Figure S3. Squared overlap matrices between the orthonormal complete sets of modes
for PETN obtained from the effective Hessian and the standard Hessian for k = 0. Left:
298 K; Right: 20 K. Lower mode number corresponds to lower vibrational frequency.
1 P. H. Berens, D. H. J. Mackay, G. M. White, and K. R. Wilson, J. Chem. Phys. 79, 2375
(1983). 5