Efficient spectra from atomistic simulation: a generalized master equation study of the air-water interface

TL;DR

This work addresses the high cost of deriving condensed-phase spectra from atomistic simulations by employing a data-driven generalized master equation (GME) with a memory kernel $\bm{\mathcal{K}}(t)$. The authors apply a minimal projection operator $\mathcal{P}=|\bm{A})(\bm{A}|\bm{A})^{-1}(\bm{A}|$ to predict VSFG spectra at the air–water interface from AIMD data, investigating whether the kernel lifetime $\tau_K$ can be made shorter than the correlation time $\tau_{eq}$ to gain efficiency; they extend the projector to include polarizability, quadrupoles, and layer-resolved observables to search for further improvements. The key finding is that, although GME can yield a modest ~2× (≈50%) reduction in data requirements, most projector augmentations do not shorten $\tau_K$ or dramatically improve spectral accuracy, leaving the search for a substantially more efficient projector an open challenge. This work highlights the subtle balance between observable choice and memory effects in atomistic spectroscopic predictions and suggests future work may need to operate at different modeling hierarchies or develop new collective variables to approach Markovian behavior.

Abstract

Computing condensed phase spectra from atomistic simulations requires calculating correlation functions from molecular dynamics and can be very expensive. A totally general, data-driven method to reduce cost is to employ an exact rewriting to a generalized master equation characterized by a memory kernel. The decay time of the kernel can be less than the original function, reducing the amount of data required. In this paper we construct the minimal projection operator to predict vibrational sum-frequency generation spectra and apply it to the air-water interface simulated using ab initio molecular dynamics. We are able to obtain a modest reduction in cost of just under 50\%. We explore various avenues to use more of the available data to expand the projector in an attempt to reduce the cost further. Interestingly, we are not able to effect any change by including quadrupoles, inter-molecular couplings, or a depth-dependence. How to strategically go about maximally reducing cost using projection operators remains an open question.

Figures (10)

• Figure 1: Correlation matrix appropriate to Eq. \ref{['eq:proj_vsfg']} and its memory kernel. Respectively dotted and dashed vertical lines show when all elements fall below $\epsilon$. Time series were normalized before construction, see Appendix \ref{['app:Units']}. Elements are arranged in the order $\langle \mathrm{Row}(0) \mathrm{Column}(t)\rangle$ as listed in Eq. \ref{['eq:proj_vsfg']}, however multiplication by $(\bm{A}\vert \bm{A})^{-1}$ makes it difficult to precisely identify the panels of the figure with each correlation function, visually. Numerically, the original correlation functions are easily retrieved by undoing the normalization steps. We note that elements including $\alpha_{zz}(t)$ have more noise.
• Figure 2: Error defined as the pointwise difference between the 'converged' spectrum with window-to-zero ending at $64$ steps (dotted line), and the predicted spectrum with an earlier/later window-to-zero (circles) or produced from GME with $\tau_ \mathcal{K} =$cutoff and then windowed (crosses). Circles are filled when they have lower error than the corresponding GME result. The error is normalized by the number of predicted points in time. Horizontal lines represent the average GQME error over the last 5 points, where there is a plateau. Dashed vertical line as in Fig. \ref{['fig:5by5CK']}; by error the cutoff is longer for the IR, but the two results are already in good agreement, see Fig. \ref{['fig:5by5_spectra']}.
• Figure 3: The four predicted spectra using the projector of Eq. \ref{['eq:proj_vsfg']} and lengths of correlation function as defined in the legend of 34 (color) and 64 (black) steps respectively. Grey line has no window applied. Dashed coloured line is simple windowing while solid coloured line uses the GME. Sampling steps are 4 fs in duration and the underlying timestep was 0.5 fs. In the IR spectrum the arrows mark the peak heights of the central band, with the dashed line underestimating by about 30%. For the three VSFG spectra, only the stretching region is shown for clarity. The highest frequencies are not well-captured with these cutoffs.
• Figure 4: Correlation matrix appropriate to Eq. \ref{['eq:proj_quad']}. The upper-left $5 \times 5$ is very close to Fig. \ref{['fig:5by5CK']}, but we emphasise they are only equal before (respective) multiplication by the normalizing matrix $(\bm{A}\vert \bm{A})^{-1}$. The lower-right $3 \times 3$ block comes from the new, quadrupole elements. Close inspection reveals a non-zero off-diagonal at $\mathcal{C} _{0,5}$, however the memory kernel here is effectively zero.
• Figure 5: Zoom of a $6 \times 6$ block on the diagonal of the matrix Eq. \ref{['eq:C_molecules']}. Color bar shows the average distance between molecules $i$ and $j$ of a particular element over the first frames of the averaging windows used to construct the correlation function (the same value used to determine if a molecule is beyond the cutoff). The full-element sum of this matrix gives the total dipole-dipole correlation function used in previous figures.