text:
\section{INTRODUCTION}

Molecular solids have been extensively used as key components in chemical industries such as medicine \cite{lee2011crystal}, fertilizers, dyes \cite{zhuo2022organic}, pesticides \cite{Yang-ANIE-2017}, and high energy explosives \cite{liu2018polymorphism}, as well as electronics industry since the discovery of bulk conductivity in polycyclic aromatic compounds in 1950s \cite{kallmann1960bulk}. While some molecules can aggregate with no particular order, such as amorphous solids, most organic solids are crystalline, and their physical properties largely depend on regularly repeating intermolecular packing. Nevertheless, understanding the packing of molecules is a rather elusive subject due to the variety of molecular shapes. Unlike the packing of spheres in the atomic crystals, geometry analysis on the irregularly shaped molecules appears to be more of an art than a science. It is visually difficult to capture the pattern even for the experienced researchers \cite{day2006experiment}.

To ease the visual challenge, most early works focused on the simple shapes. Among them, Robertson first proposed to divide planar aromatic hydrocarbons (PAH) into two categories based on the ratio of molecular area to the thickness \cite{Robertson-1951}. It was found that the disk-like molecules tend to stack together via rigid translation (\textit{stack-promoting}), while the molecules with smaller areas prefer the glide-like reflections (\textit{glide-promoting}). Using energetic as well as geometrical criteria, Desiraju and Gavezotti \cite{desiraju1989crystal} extended the categorization of PAHs into four groups and used them as a guide for prediction of crystal packing for new molecules. As shown in Fig. \ref{Fig1}, they include (i)\textit{herringbone}, in which each molecular column has interactions with the nonparallel neighbouring molecules; (ii) \textit{sandwich}, consisting of the herringbone motif with sandwich-type diads; (iii) $\gamma$, the flattened-out herringbone; and (iv) $\beta$, a layered structure made up of graphite-like planes.

\begin{figure}[ht]
\centering
\includegraphics[width=0.48 \textwidth]{Fig1-PAH.pdf}
\caption{\label{Fig1} Four packing motifs on the crystals made of planar aromatic hydrocarbons (PAHs).}
\end{figure}

The packing motif concept has been applied to establish the correlations between these materials' packing and observed physical properties (e.g., charge transport for organic semiconductors \cite{Campbell-JMC-2017, zhang2018theoretical} and insensitivity for molecular explosives \cite{mathieu2017sensitivity}). Despite its popularity in crystal analysis, the definition of four patterns, like many other chemical nomenclatures, lacks a mathematical rigor. From the definition, it is hard to define the boundary between $\gamma$ and $\beta$. Even within the herringbone group, some crystals clearly have different packing as compared to others. Although several tools have also been developed to automate the assignment of packing motif \cite{Campbell-JMC-2017, loveland2020automated, ito2020estimation}, they sometimes yield inconsistent results due to the subjective choice of projection plane or simply due to the ambiguity of the definition itself.

In addition, Hirshfield fingerprint has been commonly used to analyze the complex information contained in a molecular crystal structure into a single, unique full colour plot \cite{Mckinnon:2007}. Derived from the Hirshfeld surface, these 2D-fingerprint plots provide a visual summary of the frequency of each combination of internal and external distances across the surface of a molecule, so they do not only indicate which intermolecular interactions are present, but also the relative area of the surface corresponding to each kind of interaction. Other analysis techniques \cite{motherwell2010molecular, Spackman-2016, carugo2017packing}  were also proposed recently. But the interpretation is either too abstract or too cumbersome, and they also lack a simple criterion to judge if two crystal packings are truly similar or not. On the other hand, the COMPACK algorithm \cite{compack} is also commonly used to detect the similarity, but it is mostly designed to check if two structures are identical or not on when they consist of the similar molecules. Therefore, it is not suitable to compare the similarity between two packing with different constituents, which is important in many crystal engineering applications.

To address the aforementioned challenge, we introduce the image approach in combination with spherical harmonics expansion to quantify the packing similarity of organic crystals. This essentially involves a two-step process. First, we construct the spherical images to represent the molecular packing and energy distribution in a crystal. Second, the similarity of two spherical images (i.e., crystal packing) is computed by the maximum normalized cross correlation in the Fourier space. In the following, we present the methodology and its application to a set of hydrocarbon crystal data. Due to its mathematical rigor, this approach can be fully automated for a rapid classification of distinct crystal packing motifs in a large data set,  regardless of the molecular choices.

\section{Computational Methodology}

\subsection{Image Representation of Molecular Packing}
Although it is highly subjective to describe the crystal packing, there is a general consensus that at least two factors are of critical importance. First, the geometry has been widely used to understand the packing. The simplest analysis is to count the coordination number as a function of the cutoff distance. Moreover, the spatial distribution of neighboring molecules can also analyzed with more sophisticated tools \cite{BO-PRB-1983, CNA-JPC-1987}. However, crystal packing is not simply a geometry problem. The intermolecular energy should be taken into account as well, as described by Desiraju and Gavezotti \cite{desiraju1989crystal}, as well as the 2D-fingerprint of the Hirshfeld surface \cite{Mckinnon:2007}. Using the naphthalene crystal (referred as NAPHTA in the Cambridge Crystal Database) as an example, a molecule can be generally considered to have 16 neighbors (see Fig. \ref{Fig2}). Based on the intermolecular interactions, these molecules can divided to four tiers, colored in yellow (6), green (6), light blue (2) and deep blue (2). The last tier is often omitted because their interactions are much weaker. One can consider that the most important yellow tier molecules form the equator while the less important green and blue molecules are distributed on the Southern and Northern Hemispheres by following some pattern. Therefore, a good representation should be able to probe such characteristics of the spatial distribution of both molecules and energetics.

\begin{figure}[ht]
\centering
\includegraphics[width=0.48 \textwidth]{Fig2-NAPHTA-Real.pdf}
\caption{\label{Fig2} Different projections of NAPHTA crystal from the top (a) and side (b) views. The center molecule is shown as spheres while the surrounding molecules are shown by their van de Waals surface colored by the interaction energy.}
\end{figure}

Inspired by those previously developed approaches, we construct the spherical image to represent the packing as follows,

\begin{enumerate}
    \item Choose the center molecule ($M_0$) and find the neighboring molecules or intermolecular short contacts $\{M_i\}$.
    \item For each $M_i$, compute the bonding energies ($\{E_i\}$) with respect to $M_0$. If $M_i$ denotes a molecule, $E_i$ is the sum of all interactions between $M_i$ and $M_0$.
    \item Project each $M_i$ onto the unit sphere centered at $M_0$) with a spherical coordinates $d_i = (\theta_i, \phi_i)$. If $M_i$ denotes a short contact (e.g., C$\cdots$C or C$\cdots$H pairs), the center of the atomic pair is used as the reference point to determine $d_i$. For the case of molecule, the molecular center shall be used as the reference.
    \item Place a Gaussian based on $E_i\exp\frac{(d-d_i)^2}{2\sigma^2}$ in each $M_i$.
\end{enumerate}

Therefore, we come up with two definitions of spherical functions as follows:
\begin{equation}
    f = \begin{cases}
        \sum\limits_{i=1}^{\textrm{molecules}} E_i \exp \frac{(d-d_i)^2}{2\sigma^2} & \textrm{Coarse grained model}\\
        \sum\limits_{i=1}^{\textrm{contacts}}  E_i \exp \frac{(d-d_i)^2}{2\sigma^2} & \textrm{Fine resolution model}
    \end{cases}
\end{equation}

In general, $f_\textrm{molecule}$ can be considered as a coarse-grained descriptor for the molecular packing. Fig. \ref{Fig3}a shows the such representation for a naphthalene crystal (see Fig. \ref{Fig2}). Unlike other descriptors such as Hirshfield fingerprint plots \cite{Mckinnon:2007}, the spherical image can be easily interpreted since it retains the spatial information. For instance, the center of each hot spot denotes the position of the neighboring molecule. In NAPHTA, one can clearly find six bright spots at the equator plane, and four less bright spots at each of the hemisphere. These information is consistent with our analysis from Fig. \ref{Fig2}. It is essentially the arrangement of hot spots in the sphere. For such a spherical image, it is also common to plot its cylindrical projection as used in the map of the Earth. On the other hand, $f_\textrm{contact}$, shown in Fig. \ref{Fig3}b is a break-down version of $f_\textrm{molecule}$ or a projection of Hirshfield surface into the unit sphere \cite{Spackman-2016}. It can capture more details when describing the molecular packing. As one can see in the projection map in Fig. \ref{Fig3}b, some bright spots becomes rather diffusive, indicating that the intermolecular interaction are from multiple short contacts between the center and surrounding molecules, while the very hot spots suggest that there exist very strong interactions such as C$\cdots$H bonds for the PAH system, or hydrogen bonds for the general cases. Therefore, $f_\textrm{contact}$ may be more advantageous for a direct comparison of two different images.

\begin{figure}[ht]
\centering
\includegraphics[width=0.48 \textwidth]{Fig3-NAPHTA-proj.pdf}
\caption{\label{Fig3} Two sphere image representations for NAPHTA, (a) the coarse grained model and (b) the fine resolution model.}
\end{figure}

When computing the intermolecular energy, one can use different approaches from empirical force fields to more advanced electronic structure methods. In our case, we follow an empirical atom-atom potential \cite{gavezzotti1994crystal} due to its simplicity. In addition, it is important to note that the choice of Gaussian smearing is to ensure a smooth distribution when comparing two distinct local environments. As such, it can be also used to characterize the packing environments in an aggregate, amorphous solid, or even liquid. In the following, we will keep $\sigma$ = 0.1 \AA~  at the unit sphere for all the subsequent calculations. To construct the spherical image, we uniformly sampling 10,000 grids on the unit sphere according to the Fibonacci method, which is more efficient than the evenly spaced grid sampling on ($\theta, \phi$).

\subsection{Real Spherical Harmonic Expansion}
For a real valued spherical function $f(\theta, \phi)$, it can be expressed as a series of spherical harmonic functions,

\begin{equation}
    f(\theta, \phi) = \sum_{l=0}^\infty \sum_{m=-l}^l f_{lm}Y_{lm} (\theta, \phi),
\end{equation}

where $f_{lm}$ is the spherical harmonic coefficient, $Y_{lm}$ is the spherical harmonic function, and $l, m$ are the spherical harmonic degree and order. The details of $Y_{lm}$ are given in the appendix.

Similar to the Fourier transform of a one dimension spectrum, the purpose of spherical harmonic expansion is to efficiently extract the features from the frequency domain with only a few $f_{lm}$ coefficients. It is straightforward to show that $f_{lm}$ can be calculated by the integral
\begin{equation}
\label{isph}
    f_{lm} = \frac{1}{4\pi} \int _\Omega f(\theta, \phi) Y_{lm} (\theta, \phi) d\Omega
\end{equation}

In general, $f_{lm}$ consist of $2\times l\times l$ real valued numbers. Fig. \ref{Fig4}a shows the power spectrum of $f_{lm}$ obtained from the naphthalene crystal. Compared to the grids in the real space, these $f_{lm}$ coefficients provide an economical way to store all information to reconstruct the images. With the increase of spherical harmonic degree $l$, it is expected that more details of the function can be described. However, the $f_{lm}$ at very high order simply means the noise. Therefore, there should be a cutoff when it is found that further increase of $l$ no longer brings any notable improvement.

\begin{figure}[ht]
\centering
\includegraphics[width=0.50 \textwidth]{Fig4-NAPHTA-converge.pdf}
\caption{\label{Fig4} Spherical harmonic expansion. (a) shows the spectrum as a function of spherical harmonic degree ($l$) and order ($m$. (b) plots the reconstructed images from different cutoff values of $l$.}
\end{figure}

In practice, we construct the spherical image by sampling 10,000 grids on the unit sphere according to the Fibonacci method, and compute the expansion coefficients from eq. \ref{isph}. Fig. \ref{Fig4}b plots the recovered images from different $l_\textrm{max}$ values. In this work, we choose $l_\textrm{max}$ = 13 that should be sufficient to recover all spherical images in high accuracy.

\subsection{Cross-correlation on the Sphere}
Having established the compact description for one crystal packing ($f$), we can apply the same scheme to describe another crystal named $g$. Obviously, we arrive at the point to address the original question, namely, how to quantify the similarity between $f$ and $g$? It is clear that the use of expansion coefficients is advantageous to reconstruct the original function. Therefore, we wish to compute the similarity based on these coefficients.

One possible solution is to derive a set of rotation-invariant arrays. For instance, the power spectrum has been popularly used in distinguishing the local environments of an atomic crystal \cite{BO-PRB-1983, Bartok-PRB-2013}. However, it can lead to a substantial dimensionality reduction from the original coefficients ($2\times l\times l$) to a power spectrum ($l+1$), This indicates that a lot of information is lost during this process. Indeed, we found this is not sufficient since many different packing pattern may share a similar power spectrum.

To avoid unnecessary information loss, we aim to seek a better metric by considering all $f_{lm}$ coefficients when comparing the similarity. For the image similarity analysis, a common way is analyze the cross-correlation spectrum between two functions $f$ and $g$,
\begin{equation}
    C_{l} (f, g) = \int f(\Omega)g(\Omega)
\end{equation}

From the cross-correlation, we can further define a scalar metric at the Fourier space that is bounded between 0 and 1,
\begin{equation}
    S(f, g) = \frac{\sum\limits_{lm} f_{lm}g_{lm}}
    {\sqrt{\sum\limits_{lm} f^2_{lm} \sum\limits_{lm} g^2_{lm}}}
\end{equation}

According to the definition, $f$ and $g$ is identical if $S=1$. When $S$ has a strong deviation from 1, it means $f$ and $g$ are less similar. However, it is important to note that the coefficients are subject to change under a rotation ($R$, which can also be denoted by a set of Euler angles $\{\alpha, \beta, \gamma\}$). Therefore, we seek to maximize $S$ by sampling all possible rotational space of the SO(3). In practice, we perform deterministic quasi random sampling to generate a uniform grids of $\{\alpha, \beta, \gamma\}$ on the sphere. Then we rotate $g$ accordingly, and $S$ is further maximized based on the derivative-free optimization method as implemented in the Scipy.optimize tool box. Finally, the search returns the $S_{\textrm{max}}(f, g)$, a scalar number between 0 and 1, as well as the rotation ($R$) on $g$ that achieves the best match. Fig. \ref{Fig5} shows the initial and optimized rotation for PHENAN when it is compared with naphthalene by following this procedure. For the initial state in Fig. \ref{Fig5}a, it returns $S=0.253$. However, the true $S$ becomes 0.931 after the best rotation is found.

\begin{figure}[ht]
\centering
\includegraphics[width=0.50 \textwidth]{Fig5-correlation.pdf}
\caption{\label{Fig5} The spherical function's projection and power spectrum of PHENAN before (a) and after (b) rotation with respect to NAPTHA.}
\end{figure}

\section{Results}
With this new image approach, we proceed to build the first link between inorganic and organic crystallography. Considering hydrocarbon molecules as the simplest category, like elemental allotropes in the atomic crystals, can we find classify them by some packing motif? If so, what will be the most common motifs, like body-centered-cubic (bcc), face-center-cubic (fcc), hexagonal-close-packing (hcp) and diamond types? In the following, we will first validate the approach in a small set of previously investigated systems and then present our own categorization for a more extended data consisting of over 2000 crystals.

\subsection{Regroup of 30 PAH crystals}
In 1989, Desaraju and Gavezotti have defined four prototypical packing for about 30 crystals of disk-shaped PAH molecules \cite{desiraju1989crystal}. Their pioneering work has significantly influenced following studies in crystal engineering. With this new image approach, we revisited this data set. We calculated the similarity function for each structure pair. From the computed similarity matrix, we performed an unsupervised hierarchical agglomerative clustering, Fig. \ref{Fig6} displays the resulting dendrogram plot of the 30 PAH crystals. Clearly, our results largely agree with the previous assignment \cite{desiraju1989crystal}. Out of 30 samples, we found that 24 of them are closely clustered into three groups, corresponding the previously assigned herringbone, sandwich and $\gamma/\beta$ types. This is encouraging, given that our calculation are fully automated without any supervision. From these results, we can also analyze each cluster in detail.

\begin{figure*}[ht]
\centering
\includegraphics[width=0.9 \textwidth]{Fig6-linkage.pdf}
\caption{\label{Fig6} The hierarchical clustering of 30 PAH crystals. The structures belonging herringbone, sandwich are colored in blue and black, respectively. The $\gamma$ and $\beta$ types are indistinguishable, and hence they are all colored in red. In addition, several structures that cannot be matched to the previous assignments are colored in grey.}
\end{figure*}

\textbf{Herringbone} is the most common group. Using NAPHTA as the reference, we found that most crystals within this group (colored in red in Fig. \ref{Fig6}) have rather high $S$ values, including CRYSEN (0.975), ANTCEN (0.974), ZZZOYC01 (0.945), QUPHEN (0.941), BIPHEN (0.938), PHENAN (0.931), BEANTR (0.896).

\textbf{Sandwich} was the least common group with only five examples in its original proposal \cite{desiraju1989crystal}. Compared to the herringbone group, the $S$ values between each members are smaller. However, they can be clearly grouped to the same big branch (colored in black in Fig. \ref{Fig6}) with the unsupervised clustering method, indicating a strong correlation. According to our analysis, there actually exist two types of sandwich crystals. The details will be discussed in the following section.

$\bm{\gamma/\beta}$ were almost identical if one only focuses on the arrangement of molecular centers in Figs. \ref{Fig1}c-d. They can be only distinguished counting the ratio of C$\cdots$C and C$\cdots$H interactions \cite{desiraju1989crystal}. Indeed, the converted coarse grained tends to group them together (colored in red in Fig. \ref{Fig6}) with the high threshold value among three major clusters. However, one can always use the fine resolution model to search for really matched crystals.

Despite the overall agreement, some structures, colored in grey in Fig. \ref{Fig6} cannot be assigned to the same groups as suggested in the previous literature. For instance, the benzene crystal (BENZEN) was grouped to the herringbone type, but the authors also mentioned that it was more like an outlier \cite{desiraju1989crystal}. Similarly, several other crystals (including BZPHAN, TRIPHE, DBNTHR01, TBZPER, NAPANT01) clearly exhibit different patterns from any of four types as shown in Fig. \ref{Fig1}. The differences can be directly detected by a trivial visual analysis. Therefore, our image approach is more advantageous since it can provide a robust way to detect such outliers without ambiguity.

\subsection{Common prototypes on a larger data set}
Encouraged by the successful application on the 30 PAH crystals, we systematically inquired all crystals from the Cambridge Structural Database by searching for the systems containing only C and H elements and no more than one molecule in the asymmetric unit. After removing the duplicated entries, we obtain 2007 crystals (see more details in the supplementary materials). This data set serves as the test bed to further validate our image approach.

\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\textwidth]{Fig7-naphta-all.pdf}
\caption{\label{Fig7} The naphthalene crystal. (a) is the similarity ($S$) distribution of all PAH crystals with respect to NAPHTA. (b)-(e) show several representative crystals with different $S$ values.}
\end{figure*}

\subsubsection{The naphthalene family}
In the early day of small molecule crystallography, the naphthalene crystal is considered to be the common salt of organic crystal \cite{kitaigorodsky2012molecular}. This type of crystal packing was assigned to the herringbone group, which was found to occur more often than any other group. Fig. \ref{Fig7}a plots the histogram of the distances of between NAPHTA with all other hydrocarbon crystals. The entire distribution can be roughly described by some normal distribution with a mean around 0.6-0.7 (more details can be found in the supplementary materials). Here we focus on the region with high $S$ values. Among the entire data set, we found CEKWEU (Fig. \ref{Fig7}b) has the best match with NAPHTA. Although the molecule in CEKWEU is not flat, the mapping is obvious as they share the same space group ($P2_1/c$) and Wykcoff position, as well as similar molecular orientation. In addition, Figs. \ref{Fig7}c-e displays several structures with different $S$ values. By analyzing these pattern as well as the 3D structure, we decide to set a threshold of 0.892, leading to 86 crystals belonging to the naphthalene family.

It is important to note that our similarity calculation does impose any symmetry constraints. As shown in Table \ref{table1}, several space groups, e.g.,  $P2_1/c$, $Pbca$, $P\overline{1}$, $P2_1$, occur more often. Clearly, all of them follow group-subgroup relations. For instance, the molecules containing four or more aromatic rings crystallize in the $P\overline{1}$ with Z=2 (e.g., TENCEN, PENCEN), but the overall packing is still close to that of NAPHTA. It would be interesting to collect more data and use them to predict the possible phase transition.

\begin{table}[]
\caption{The space group distribution of 86 naphthalene crystals.}\label{table1}
\begin{tabular}{ccc}
\hline\hline
~~Space group~~     & ~~\# of molecules per cell ~~& ~~Occurrence~~ \\ \hline\hline
$P2_1/c$        & 2                        & 34        \\
$Pbca$          & 4                        & 14        \\
$P2_1/c$        & 4                        & 9         \\
$P\overline{1}$ & 2                        & 7         \\
$P2_1$          & 2                        & 5         \\
$Pnma$          & 4                        & 5         \\
$P2_12_12_1$    & 4                        & 3         \\
$Aea2$          & 4                        & 2         \\
$C2/c$          & 4                        & 2         \\
$Pbcn$          & 4                        & 2         \\ \hline\hline

\end{tabular}
\end{table}

Finally, we emphasize that the coarse-grained image model is still limited by its molecular approximation. Therefore, it is not able to detect the difference in terms of molecular shape. This can be corrected by the fine-resolution model. If we simulate the similarity based on $f_\textrm{contact}$, only three crystals ANTCEN (0.917), CEKWEU (0.834), DMANTR (0.833) can match NAPHTA well. Therefore, we suggest the use of $f_\textrm{contact}$ when it is necessary to find the really good match on a reference crystal.

\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{Fig8-others.pdf}
\caption{\label{Fig8} Other representative crystals, including (a) benzene type, (b) sandwich type 1, (c) sandwich type 2, (d) $\gamma$ and (e) $\beta$. The numbers in the parenthesis indicate the number of similar structures identified in the 2007 data set.}
\end{figure}

\subsubsection{Other common prototypes}
We also checked other common prototypes. The results are summarized in Fig. \ref{Fig8}.

\textbf{BENZEN}. The crystal structure of benzene, is an outlier of herringbone and was less investigated in terms of packing. It was considered to have a pseudo-fcc arrangement due to a strong X-ray reflections at the (111) plane \cite{benzene-rmp-1958}. However, from the projection image as shown in Fig. \ref{Fig8}a, such type can also be explained to have four strongly interacted molecules surrounding the center molecule forms the equator plane, while four secondary interactions are evenly distributed in each of the Northern and Southern Hemisphere. In our explanation, we take into account the fact that the 12 neighboring molecules do not have the same interactions with the centering molecules, which is different from a typical fcc atomic crystal. To our knowledge, there are no previous reports on the crystal sharing the same packing with BENZEN. Here our analysis suggests that at least 14 crystals belong to this family with the threshold distance value of 0.89, such as NUNCUW (0.966) and FAPZOL (0.962).

\textbf{Sandwich}. As shown in Figs. \ref{Fig8}b-c, both PYRENE and PERlEN01 has one very bright spot in the upper left of the projection image, however, other less  notably differ from each other. Such subtle difference can be easily ignored when one just focus on the common projection plane. The $S$ between PYRENE and PERLEN01 is only 0.753. Among the entire dataset, such packings do not appear often, with only 3 PYRENE and 4 PERLEN03 types.

$\bm{\gamma/\beta}$. Two representative crystals (CORONE and TBZPYR), as shown in Figs. \ref{Fig8}d-e, are nearly identical in terms of the strong characteristics. Both CORONE (210) and TBZPYR (186) can find a lot of structures sharing the same packing. The $S$ value between CORONE and TBZPYR is 0.975. Therefore, these two families are expected to have a lot of overlaps.

\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{Fig9-all.pdf}
\caption{\label{Fig9} The complete clustering results and several abundant packing motifs. (a) is the histogram of similar structures for each crystal with a threshold of 0.88. (b)-(h) show six representative structures' pattern featured by two, four, and six bright spots in the projected equatorial plane. The numbers in the parenthesis indicate the number of similar structures identified in the 2007 data set.}
\end{figure}

\subsubsection{A complete categorization}
We also perform a systematic pairwise distance analysis for the entire 2007 crystal data set. Using the threshold of 0.88, we count the number of similar structures for each crystal as shown in  Fig. \ref{Fig9}a. From the distribution, it is clear that many structures with the pattern of two strong bright spots (see Figs. \ref{Fig9}b-c) are most abundant. This pattern is also consistent with what we have observed in $\gamma/\beta$. However, we note that other neighbors in the crystal can adopt different packing sequences. Therefore, there exist many subtypes within this family. The second large group is featured by the pattern of six bright spots forming the equatorial plane (see Figs. \ref{Fig9}d-e)). Therefore, the most important feature of this group is that the molecules form a close packing layers with strong intermolecular interactions \cite{Kitaigorodskii-JAC-2003}. Within this group, ANTCEN belongs to the common naphthalene family. However, the projection map in Fig. \ref{Fig9}d is rotated to put the six bright spots in the equatorial plane. Another frequent-occurring subgroup is shown in Fig. \ref{Fig9}e of FLUREN. The structure differs from ANTCEN by the relative shifting of molecules in the adjacent layers. As shown in Fig. \ref{Fig10}, the strong similarity between ANTCEN and FLUREN can be easily detected in the real space. However, it can also be seen that ANTCEN follows the herringbone pattern and FLUREN adopts the sandwich type from other choice of projection. This drastic difference clearly suggests the limitation of interpreting structure by visual projection. On the other hand, our spherical image representation provides more rigorous comparison by capturing both similarity and dissimilarity at all directions. Finally, we also observe another group of structures with four bright spots in the equatorial plane (see Figs. \ref{Fig9}g-h). This is less common. Each structure of this group has only 30-50 similar structures within the 2007 data set.

\begin{figure}[ht]
\centering
\includegraphics[width=0.45\textwidth]{Fig10-6.pdf}
\caption{\label{Fig10} Crystal packing comparison between (a) ANTCEN and (b) FLUREN.}
\end{figure}

\section{Conclusions}
In sum, we present an image approach to characterize and classify molecular packing from the local neighboring environment. The similarity between two crystals (i.e., images) can be evaluated from the spherical harmonics expansion based on the maximum cross-correlation. We apply this approach to investigate the packing similarity analysis over 2000 C-H crystal data sets and categorize them by similarity. Compared to the previous approaches based on visual comparison in the real space, our spherical harmonics expansion provides a robust way to measure the packing similarity. It is suitable for a rapid search for the similar crystal structures by crystal packing from a large data set. In the future, a more extensive crystal packing motif library can be built by using this criterion. When such template becomes available, it can map the relation between molecule properties and crystal packing to guide the design of new materials. Alternatively, it can also be used to complement the crystal structure prediction by chemical substitution of the well defined structural prototypes \cite{Reilly-Acta-2016}.

\section*{Acknowledgments}
Q.Z. acknowledge the NSF (DMR-2142570) and Sony Group Corporation for their financial supports. The computing resources are provided by XSEDE (TG-DMR180040).

redpajama_set_name: RedPajamaArXiv


text:
By CHARLES MWANIKI
KRA to exceed tax target by record Sh140bn revenue
Wednesday June 15 2022
Kenya Revenue Authority (KRA) Commissioner-General Githii Mburu. PHOTO | JEFF ANGOTE | NMG
The Kenya Revenue Authority (KRA) expects to beat its initial revenue collection target for the current fiscal year by up to Sh140 billion, driven by an improved collection of consumption and income taxes as the economy continues its post-Covid recovery.
KRA Commissioner General Githii Mburu said his agency has exceeded two upward revenue target revisions brought in through supplementary budgets, pointing to the enhanced tax enforcement and recovery measures the taxman has been implementing in the period.
The 2021/2022 budget had set an ordinary revenue target of Sh1.77 trillion for the KRA, which was then revised to Sh1.8 trillion in the 2022 budget policy statement.
KRA has said in the past that it collects higher taxes in the last quarter of the fiscal year, as corporates including banks normally clear arrears from April.
"I believe we will be over the original target of Sh1.8 trillion by Sh140 billion. Right now we are above it by Sh126 billion, so we hope this month we can do another Sh15 billion, so it will be about Sh140 billion," said Mr Mburu.
ALSO READ: Treasury raises tax goal for KRA by Sh33 billion
The higher collection however comes in an environment of rising cost of goods and services largely because of global supply constraints which have been exacerbated by the Russia-Ukraine war.
In addition to VAT, income, excise, and custom taxes, the KRA also collects other revenue such as levies, rent of buildings, fines, forfeitures, and other smaller taxes on behalf of the government.
The KRA has also been the principal revenue collection agent for the Nairobi County government since March 2020.
In the year ending June 2021, the KRA also exceeded its revenue collection target, raising Sh1.669 trillion versus an initial target of Sh1.652 trillion.
This was primarily driven by higher excise and corporation taxes, which offset a decline in VAT and payroll taxes.
This year, Treasury budget outturn documents as of March 2022 showed that the revenue body was ahead of target in the VAT and income tax categories by Sh10.8 billion and Sh17.8 billion respectively, signalling a recovery in consumption and payroll taxes on the back of the continued recovery of the job market.
Hundreds of thousands of Kenyans had lost their jobs in 2020 when the government imposed restrictions to curb the spread of Covid-19.
ALSO READ: Seven-month tax receipts exceed Sh1trn for first time
Trade was also negatively affected by global restrictions which affected supply chains and production, hitting revenue collection targets in the period.
With the lifting of the restriction measures, trade has picked up significantly, resulting in imports in the 12-months to March 2022 rising by 30 percent to a record Sh2.22 trillion, thus boosting custom and VAT revenue.
Higher petroleum prices have also factored into improved excise collection for the government.
The taxman has also been ramping up compliance enforcement measures to cut tax losses.
The agency said earlier this year that it would make extensive use of data and intelligence to unearth unpaid taxes and use technology to simplify tax processes.
It has also raised its pursuit of tax cheats, resulting in several high-profile businesses being hauled before court for non-payment of taxes.
New and enhanced tax rates introduced in the 2021/2022 budget have also helped the KRA raise more revenue.
ALSO READ: Kenyans rush to comply as KRA targets cheats on social media
Some of the new taxation measures this financial year include 16 percent levy on cooking gas, a raising of excise duty on airtime and data to 20 percent from 15 percent, the introduction of a 20 percent duty on fees and commissions earned on loans as well as a 7.5 percent tax on gambling wins.
The International Monetary Fund in April backed Kenya's "important tax policy measures", arguing they have resulted in "strong" collections.

redpajama_set_name: RedPajamaCommonCrawl


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Issey Miyake rare and iconic Flying Saucer pleat dress.
Constructed in black solid and sheer pleated polyester with high cowl funnel neck and maxi length sleeve that allows this piece of wearable art to be manipulated to create different forms.
For the true Issey Miyake collector. This is a museum worthy piece.
In great condition for it's age.
Made in Japan. Size Medium.


Finally succeeded in grabbing a download of an episode of 80s kids show "The Adventure Game", which I haven't seen in 20 years or so. This leads me to want to write an RPG system that explores the ideas behind why the Rangdo of Arg felt the need to import hapless adventurers to torture every week. What sinister conspiracy lurks behind the genial public face of Arg's smiling yet strangely unhelpful eccentric citizenry? And what exactly is the real purpose of the Vortex - why does it matter whether or not these unfortunate visitors have to walk the billions of miles back to earth. I'm even imagining a freeform with real BBC Micros.
Made me smile to see the program again. It was every bit as lovely as I remember. Gronda gronda and so forth.
I particularly remember with fondness Chris Lever, who was played by Gandor. He was the butler.
Challenge TV showed about three episodes of TAG a while back, and we taped them (though goodness only knows where the tape is now). Rumour has it that only a few episodes still exist.
Season 4, Episode 5. With Joanno Munro, George Layton and Val Prince, whoever they might be. I have others that I'm waiting to download, but it's hit and miss whether they'll make it, I suspect.
YEEE Adventure Game! I loved that program!
Comment by The One known as D.T.
Ah, Gronda the Aspidistra. The only television programme that I have ever applied to be on.


Vancouver lies just across the Columbia River from Portland, Oregon. While most people would skip over it in favor of the more well known city, Vancouver is a lively town with an especially festive Fourth of July.
The premiere place to stay is the Hilton Vancouver, a hotel right in the center of downtown. The lobby is extensive with high ceilings and three elevators, and is connected to Gray's Restaurant right next door.
Our room was on the fourth floor and gave us a beautiful view of Esther Short Park across the street, which included the Salmon Run Bell and Clock Tower. The design evokes art styles of the local Chinook Native American tribes and chimes its bells at 12, 1, 2, and 4 pm.
The room included a king size bed and a full bathroom, along with a mini fridge and television. Additionally in our room we were set up with in-room delivery items which included a delicious Washington Hills Cabernet Sauvignon wine, and a plate of fruit and cheese and crackers.
Down on the third floor is both a fitness room for exercise, and a room adjacent featuring a full pool and small spa hot tub. It's the perfect place to wind down in the hot tub or cool down in the pool on a hot day.
We ate at Gray's restaurant, connected to the hotel lobby. Filled with tables and big booths there's plenty of room for either couples or big families. The menu was filled with great choices, but we were interested in the tater tots, topped with melted cheese and sprinkled sriracha ketchup. We also began with a cocktail, the seasonal sangria, which delivered both a nice sweet fruity taste and a cool refreshing drink for the summer evening. Paired with the caesar salad, it was a perfect opener for the meal.
For our main courses we tried the juicy Sirloin Steak, and Gray's Mac 'n Five Cheese. The steak came with a house made dipping sauce and creamy mashed potatoes, and had a large quantity of meat and very little fat. The Mac and Cheese was made from shell style pasta and baked in five cheeses, and covered in gorgonzola.
Unfortunately it was just too much food for us to include dessert, but the menu was very enticing. Fudge cake, salted caramel cookie, creme brulee, even drinks like the Irish Coffee, and the Warm Brandy Alexander. Should we return, it will absolutely be our next thing to try.
After eating we walked over to Fort Vancouver to watch the display of fireworks for the Fourth of July festivities. The Fort itself is on the list of national historic sites, and was originally an important stop for fur traders and travelers on the Oregon Trail. Today it has been restored to good condition and can be toured during the day, and typically has an all day celebration on Fourth of July. However, they have retooled the festivities this year to focus on an evening Firework show accompanied by music.
The entry is free, and aside from a brief security check to prevent alcohol from entering, you're given free reign to wander throughout the outdoor areas of the fort. The show is centered on a giant grass field, there is a radio station tent offering giveaways, and some food and drink stands to replenish yourself. There's nothing provided to sit with, so you'll have to bring your own towel, cushion, or seat to reside on.
When the sun has set and 10 pm hits, you get treated to a twenty minute deluge of fiery colorful explosions, and various tunes reflecting American styles like country, rock, and traditional marching band. Watching these fireworks under the stars in a warm summer night among hundreds of people is just how they were meant to be seen, and it was a second to none experience.
In addition to the fireworks, there are several shops and restaurants downtown to explore. We were very struck by the art demonstration that was the Veterans of Foreign Wars Memorial Wall. We had heard about it as a nice tribute to those who gave their lives for our country, but upon seeing it we were struck by its emotional impact and size. The wall extends around 250 feet and has separate murals denoting each major war in US history since the second World War. It's quite poetic in both its heroic and dour portrayal of war, something you don't typically see in a memorial.
Vancouver is a great spot to visit and check out on your way to Seattle or into Portland. It's right on the Columbia River and has an amazing historic vibe with lots of things to do. Visit Vancouver has a great guide of things to do and see to plan your next trip perfectly!


Old Abe would have driven the A.C.L.U. nuts with this proclamation.
Population has steadily increased, notwithstanding the waste that has been made in the camp, the siege and the battle-field; and the country, rejoicing in the consciousness of augmented strength and vigor, is permitted to expect continuance of years, with large increase of freedom.
In testimony whereof, I have hereunto set my hand, and caused the seal of the United States to be affixed Done at the city of Washington, this third day of October, in the year of our Lord one thousand eight hundred and sixty-three, and of the independence of the United States the eighty-eighth.


text:
10 US sailors missing after warship collides with tanker near Singapore
SINGAPORE Asia Pacific
Aug 21, 2017 12:00 am GMT+3
A general view shows the guided-missile destroyer USS John S. McCain with a hole on its left portside after a collision with oil tanker, outside Changi naval base in Singapore on August 21, 2017. (AFP Photo)
At least 10 U.S. sailors are missing after the USS John S. McCain collided with a tanker early Monday east of Singapore, the second accident involving a ship from the Navy's 7th Fleet in the Pacific in two months.
The Navy said five other sailors were hurt in the collision between the guided-missile destroyer and the 183-meter (600-foot) Alnic MC, an oil and chemical tanker. Four of them were evacuated by a Singaporean navy helicopter to a hospital Singapore for treatment of non-life threatening injuries and one did not require further medical attention.
The McCain had been heading to Singapore on a routine port visit after conducting a sensitive freedom of navigation operation last week by sailing near one of China's man-made islands in the South China Sea.
It wasn't immediately clear if the Liberian-flagged Alnic sustained damage or casualties.
The McCain was damaged on its port side aft, or left rear, from the collision that happened at 5:24 a.m., the Navy's 7th Fleet said, but was heading to port under its own power.
Malaysia's navy chief Ahmad Kamarulzaman Ahmad Badaruddin tweeted that two ships as well as aircraft from its navy and air force have been deployed to help look for the missing U.S. sailors.
The Navy said Osprey aircraft and Seahawk helicopters from the USS America were assisting in the search. It also said tugboats and Singaporean naval and coast guard vessels were in the area to render assistance.
There was no immediate explanation for the collision and the Navy said an investigation would be conducted. Singapore, a city-state at the southernmost tip of Malaysia, is one of the world's busiest ports and a U.S. ally, with its naval base regularly visited by American warships.
The collision is the second involving a ship from the Navy's 7th Fleet in the Pacific in two months. Seven sailors died in June when the USS Fitzgerald and a container ship hit each other in waters off Japan.
The Fitzgerald's captain was relieved of command and other sailors were being punished after the Navy found poor seamanship and flaws in keeping watch contributed to the collision, the Navy announced last week. An investigation into how and why the Fitzgerald collided with the other ship was not finished, but enough details were known to take those actions, the Navy said.
President Donald Trump has expressed concern for the McCain's crew.
Trump returned to Washington on Sunday night from his New Jersey golf club. When reporters shouted questions to him about the McCain, he responded, "That's too bad."
About two hours later, Trump tweeted that "thoughts and prayers" are with the McCain's sailors as search and rescue efforts continue.
The 154-meter (505 foot) McCain, named after U.S. Sen. John McCain's father and grandfather who were both U.S. admirals, is based at the 7th Fleet's homeport of Yokosuka, Japan. It was commissioned in 1994 and has a crew of 23 officers, 24 chief petty officers and 291 enlisted sailors, according the Navy's website.
Last Update: Aug 21, 2017 10:35 am


Hayaka Unta "into the Woods" Spring Camp held at Camp Bond in Reagan, Okla. on April 6, 2019.
Children's Fair held at the Agri Plex in Ada, Okla. on April 6, 2019.
Running Clinic held at Ada High School track in Ada, Okla. on March 19, 2019.
Veterans Billiards Tournament held at the Veterans Lodge in Ada, Okla. on February 23, 2019.
Elders Valentine Banquet and Coronation held at the Ardmore Community Center in Ardmore, Okla. on February 8, 2019.
Halloween Head Start and Child Development Center Trick-or-Treat event held at the Chickasaw Cultural Center in Sulphur, Okla. on October 30, 2018.
Harvest Fest held at Emet, Okla. on October 27, 2018.


Think of your cover letter like it's the first page of a book. Most hiring managers only scan a cover letter for a couple of seconds before skipping to the next one. You want your cover letter to grasp the reader's' attention right away in a way that makes them want to keep reading.
The most important key to an attention-grabbing cover letter is the very first sentence. You want your opening sentence to catch the reader's attention. So it needs to be interesting. One way to do this is by starting with a professional anecdote. It hooks the reader and gives your cover letter some personality.
A slogan should express the value you bring to an employer. It's a brief statement that can be placed under your name at the top of your letter in the far-left margin, or in italics at the bottom of your cover letter. This type of tagline helps to leave a lasting impression on the hiring manager.
Showing confidence in your cover letter shows an employer that you believe you have what it takes to do the job you are applying for. Don't end your cover letter with a phrase like, "I hope to hear from you soon". Instead, try something like, "I look forward to hearing back from you". This shows confidence. You want the job and you are qualified to do it, so act like it but don't be over-confident. Being overly confident can make you seem "overqualified".
You want your cover letter to show who you are. You want to give the reader an idea of how you will be a great culture fit for their company. Show a little bit of your personality in your cover letter, add some humor.
Would you want to read through over 100 cover letters that all essentially say the same thing? No, you aren't a robot, and luckily, robots don't read your cover letter, humans do. So, don't be afraid to show your humanity.
Do everything you can to find out who will be reading your cover letter so that you can address them directly. If your cover letter is addressed, "To Whom it May Concern", then you have already made your first mistake, and you haven't even gotten to the actual letter.
If the name isn't listed in the job posting or in an email that was sent to you, go to LinkedIn. Look up the companies name and try to find the person who is responsible for doing the hiring for the position you are applying for.
If you can't find it there, try calling the company and asking for the hiring manager's name. If all else fails and you can't find an actual name to address your cover letter to, default to "Dear Hiring Manager".
Researching the company you are applying for is vital. Personalizing your cover letter with information about the company shows that you want the job. You want to make it evident that your resume is tailored to the company you are applying for.
Hiring managers automatically assume that you are going to send off the same resume and cover letter to every job that you apply to. Don't just use the name of the company a couple of times. Use keywords from their mission statement and explain why you will be an excellent fit for the company, not just the position you are applying for.
You can also use information from the job posting to show how your skills and achievements align with what they are searching for.
5 Best Executive Career Coaches in Los Angeles, CA. Check out the five best Los Angeles career coaches for executives. This list serves surrounding areas including Pasadena, Beverly Hills, and Long Bank. Find the top rated executive career coaches with pricing and reviews.
An interview is all about giving employers the opportunity to evaluate you. And judge you they do! You're going to be held to a very high standard of performance by your interviewers. Here are some common interview mistakes and how to avoid them. Lateness. If you arrive after your appointed time, it's pretty likely that you won't get the job.


text:
Stuff I found on the Beach
Tear-sheets
Born and raised in Miami, Florida, Michael Prince graduated from the University of Florida with a degree in Communications in 1985. After enjoying a bohemian and somewhat itinerant lifestyle, he married and moved to Florence, Italy to study photography.
Michael left Italy in late 1989 to pursue a professional career in photography. First working as a photojournalist; he later turned to fine art photography, and had his first solo show at the Howard Greenberg Gallery in New York City. He has had solo exhibitions in Chicago, Seattle, San Francisco and Los Angeles galleries as well. His work is in the Library of Congress, several Museums and many private collections.
In 2000 he began shooting commercially, quickly gained a broad clientele, and has enjoyed numerous awards and high recognition.
Michael lives near Boston with his wife, Kristin; and Romi, the best dog in the world.
Bayer Pharmaceuticals
Cedars Sinai
Children's Healthcare of Atlanta
Goodby Silverstein
Grey Advertising
Hill Holliday
McCann Erikson
McGarryBowen
Milwaukee Community Health Centers
Pereira & O'Dell
Publicis Kaplan Thaler
Regence
TracyLocke
University of Phoenix
US Postal Services
Walton Isaacson
Institutional Collections
Communication Arts Photo Annual 2018
Santa Barbara Museum of Art
Houston Museum of Art
Marianne Campbell Associates
marianne@mariannecampbell.com
Michael Prince
michael@michaelprince.com
Instagram / © Michael Prince 2019. All rights reserved.


Publications by authors named "Thomas Glatzel"
Tantulocarida (Crustacea) from the Southern Ocean deep sea, and the description of three new species of Tantulacus Huys, Andersen & Kristensen, 1992.
Syst Parasitol 2010 Oct 18;77(2):131-51. Epub 2010 Sep 18.
Department Deutsches Zentrum für Marine Biodiversitätsforschung, Senckenberg am Meer, 26382 Wilhelmshaven, Germany.
Human metapneumovirus infections cause similar symptoms and clinical severity as respiratory syncytial virus infections.
Eur J Pediatr 2006 Jul 23;165(7):467-75. Epub 2006 Mar 23.
Children's Hospital, Medical Center, University of Bonn, Bonn, Germany.
Frequency of human metapneumovirus in the upper respiratory tract of children with symptoms of an acute otitis media.
Eur J Pediatr 2005 Jun 17;164(6):400-1. Epub 2005 Mar 17.
Institut für Medizinische Mikrobiologie und Immunologie, Universität Bonn, Sigmund-Freud-Strasse 25, 53105, Bonn, Germany.
Human metapneumovirus RNA in encephalitis patient.
University of Bonn, Institute for Medical Microbiology and Immunology, Sigmund-Freud-Strasse 25, 53105 Bonn, Germany.
New variant of the human metapneumovirus (HMPV) associated with an acute and severe exacerbation of asthma bronchiale.
Institute for Medical Microbiology and Immunology, University Hospital Bonn, Sigmund-Freud-Strasse 25, 53105 Bonn, Germany.


text:
Q: Create a Linked Server in On-Prem SQL Server to Azure SQL Database using Domain Service Account I am trying to connect from a On Prem SQL Server to an Azure SQL Database. We are able to telnet & connect directly but when trying to create a Linked Server it fails if we use Domain Authentication. This works great using SQL Login.
I scrambled across multiple documentation but couldn't find much for authentication using a Domain Service Account.
Would someone know if this works & any documentation on the setup will be helpful. Thanks

A: Thank you @Conor Cunningham MSFT for the comment, posting it as the answer.
SQL DB doesn't support domain authentication as it is not part of your domain and Linked servers don't directly take AAD auth, you will have to supply a password as well.
Refer to this document for more information.

redpajama_set_name: RedPajamaStackExchange


"A traditional novel will always be central to human civilization, but I get the impression a new form of storytelling is being called for. I call it dépêche fiction: fast-changing fiction," he writes.
Metro is pleased to announcean exclusive collaboration with celebrated author Douglas Coupland.
His novella, "Temp," will be running online and in the pages of your daily Metro each weekday from Nov. 4-29. Charlotte Empey, editor-in-chief of Metro Canada, arranged the collaboration, saying that the novella "explores what it means to be young, smart and only temporarily employed, through the eyes of a contemporary cast of characters just trying to carve a place for themselves in an uncertain world."
When we reached out to the famed Vancouver-based novelist about why he chose Metro to publish his work, he said, "Metro is the tom-toms of the global village. It's the new face of old media. It reaches millions of people who devour every word and it all ends up being discussed in lunchrooms, meeting rooms and dinner tables around the world."
Below, please read Coupland's letter to readers about "Temp" and his collaboration with your favorite daily newspaper.
In the next four weeks you'll be reading the daily story of a very likeable character named Shannon. You already know Shannon — everyone knows Shannon; you may well be Shannon. She's that slightly geeky gal who temps at every office you've ever worked at. She's funny, she's got her head screwed on right, and most of all, Shannon lives in our collective real world. She lives in the year 2013 and Shannon's world is changing and mutating as rapidly as yours.
Change: that dreaded word. If only we could have a one-year holiday from change — but that's not going to happen and we all know it. So then, how do we make the most of life as it's really lived? How do we cope and find the good stuff amidst the bad stuff? That's what Shannon's all about.
But wait. Why Metro? And why me writing about a character named Shannon in Metro? Two reasons. One, I always set my books in what I call 'the extreme present tense.' As living creatures, the present moment is all we have. It truly is… it's freaky and messy and weird, but it's ours and soon enough the present tense will become the Good Old Days — so enjoy it.
What to expect? Well, it's not 'The Office,' and it's not a comic strip. 'Temp'is something new — and you get to be a part of the experiment. And Metro is the perfect slot for 'Temp.' Metro is read by people who are out there in the world living a full-on modern experience. Fiction is about readers being able to say, 'Wait — that's how I see the world!' And I hope that's what you feel. And now let the eggs hatch.


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Comme Esteban Ocon, qui pour gagner le "Kartmag" en Cadet?


DENVER — Week after week, Von Miller has had the same message for his fellow pass-rusher, rookie Bradley Chubb.
It was "the week" indeed.
Chubb broke through for three sacks on Sunday, twice as many as he had registered this season coming into the afternoon. Chubb joins Elvis Dumervil (2006), Reggie Hayward (2001) and Rulon Jones (1980) as Denver rookies to record three sacks in a single game. He's the first rookie with a three-sack game since 2015.
Chubb showed his full skill set on Sunday and was used all over the field. He lined up at both outside linebacker positions and also in a three-point stance as a defensive end, the same position he played when he wreaked havoc on college offenses at North Carolina State.
That Chubb was able to play and create pressure from different positions shows that he's getting more and more comfortable at the NFL level. And as he becomes more acquainted with playing multiple positions, the Broncos' defense can bring more looks to its game plan.
It's no coincidence that the Broncos defense was at its best Sunday when it was creating pressure. After allowing 13 points and recording no sacks on the Rams' first three drives, the Denver defense allowed just 10 points and recorded five sacks over the final 40-plus minutes of the game.
In one of the key junctures of the game, Chubb showed his finest work. After Case Keenum was sacked on fourth and 3, the Rams got the ball at midfield with a 13-3 lead and looking to add on before halftime.
Instead, Chubb came through with two sacks in three plays.
On the first, he simply bull-rushed Rams All-Pro tackle Andrew Whitworth right into quarterback Jared Goff's lap, leaving the Rams quarterback with no escape.
Two plays later, on third down, Chubb backed Whitworth into Goff again. This time, Goff stepped up in the pocket, looking to extend the play. But Chubb disengaged from Whitworth and reached a powerful left arm out to grab Goff before pulling him to the ground. The Rams, given terrific field position just moments earlier, were forced to punt the ball away.
In the fourth quarter, Chubb completed his trio of sacks when he dipped inside of Whitworth and gobbled up Goff for a 10-yard loss. Like his first two sacks, Chubb's third takedown proved huge on the scoreboard. Instead of attempting a short kick, the Rams were forced to try a 47-yarder in the second-coldest October game in Broncos history. Cairo Santos pushed his try wide right.
Chubb credited his fellow pass-rushers for each of his sacks.
With Chubb winning his individual battles, the Broncos' pass rush as a whole was able to get on track, too. Miller, who hadn't registered a sack since Week 2, had 1.5. The team finished with five sacks on the day, its highest output since six in Week 1 against Seattle.
The team made a conscious effort to get the pass rushers on the field as much as possible, according to Head Coach Vance Joseph.
Going against a spread offense like the Rams', Denver's pass rush had more opportunities in one-on-one situations. It's a recipe for success for a team with a deep pass-rushing unit.
While the five-sack performance was a positive, all members of the effort acknowledged the numbers on the scoreboard at the end of the game need to change.
As Miller put simply, today was exactly what he imagined the Miller-Chubb duo could achieve, but it lacked the correct result.
With a short week upcoming, the Broncos need continued production out of their pass rush. Sunday's performance can become a trend after a three-week draught pressuring the passer. Chubb had his week this week. That doesn't mean it can't happen again.