Continuous Scatterplot and Image Moments for Time-Varying Bivariate Field Analysis of Electronic Structure Evolution

TL;DR

The paper presents a CSP-based framework for analyzing time-varying bivariate fields derived from hole and particle NTOs during photoinduced electronic transitions. It introduces a 4D image-moment descriptor for CSPs, projects these descriptors with PCA to produce tracks that summarize temporal evolution, and couples this with a CSP peel- and fiber-surface–driven pipeline for coarse-to-fine analysis. The approach is demonstrated on two excited-state dynamics case studies (MVK and cis-stilbene), showing how to identify active atoms, key time steps near conical intersections, and evolving donor/acceptor patterns. This method provides a compact, interpretable representation of complex electronic-structure dynamics, potentially aiding materials design and photochemical understanding, with future work focusing on scalability, automation, and broader applicability.

Abstract

Photoinduced electronic transitions are complex quantum-mechanical processes where electrons move between energy levels due to light absorption. This induces dynamics in electronic structure and nuclear geometry, driving important physical and chemical processes in fields like photobiology, materials design, and medicine. The evolving electronic structure can be characterized by two electron density fields: hole and particle natural transition orbitals (NTOs). Studying these density fields helps understand electronic charge movement between donor and acceptor regions within a molecule. Previous works rely on side-by-side visual comparisons of isosurfaces, statistical approaches, or bivariate field analysis with few instances. We propose a new method to analyze time-varying bivariate fields with many instances, which is relevant for understanding electronic structure changes during light-induced dynamics. Since NTO fields depend on nuclear geometry, the nuclear motion results in numerous time steps to analyze. This paper presents a structured approach to feature-directed visual exploration of time-varying bivariate fields using continuous scatterplots (CSPs) and image moment-based descriptors, tailored for studying evolving electronic structures post-photoexcitation. The CSP of the bivariate field at each time step is represented by a four-length image moment vector. The collection of all vector descriptors forms a point cloud in R^4, visualized using principal component analysis. Selecting appropriate principal components results in a representation of the point cloud as a curve on the plane, aiding tasks such as identifying key time steps, recognizing patterns within the bivariate field, and tracking the temporal evolution. We demonstrate this with two case studies on excited-state molecular dynamics, showing how bivariate field analysis provides application-specific insights.

Figures (10)

• Figure 1: Analyzing behavior of two carbon atoms across two electronic transition states. (Left) Individual electronic density fields ($\phi_h$, $\phi_p$) for both states and CSPs for C3 (orange) and C4 (green) for three time steps. Both CSPs align vertically at $t=30$ fs. (Right) Molecular structure highlighting the two carbons under consideration, C3 and C4, and tracks in the PC$_1$-PC$_3$ plane for State 1 C3 (S1_C3, orange) and State 2 C4 (S2_C4 green). The tracks are annotated at selected time steps with corresponding CSPs. The tracks appear to be similar in the section from $A$ to $E$ and $A'$ to $E'$, with a distinct pattern from $C$ to $D$ and $C'$ to $D'$. Points located nearby have similar CSPs, for example, ($C, D', E'$) or ($C', D, E$).
• Figure 2: CSPs and tracks in the PCA plot of synthetic time-varying bivariate fields. (Top) Two synthetic time-varying bivariate fields $(f_1,f_R)$ and $(f_1,f_S)$, where $f_1(x,y) = x$ and $f_R, f_S$ are rotated and scaled version of $f_1$. The angle of rotation and the scale factor increases with time (sampled at $0,12,24,39,48$). (Middle) CSPs of the two bivariate fields $(f_1, f_R)$ and $(f_1, f_S)$ at each time step. An increase in rotation angle causes the CSP area to increase, and a decrease in scale factor causes the slope to decrease over time. CSPs are annotated with corresponding normalized moment vectors. Values smaller than 0.01 are marked with *. All CSPs in this paper are shown using a yellows () color map in log scale. (Bottom) Tracks of the two time-varying bivariate fields begin close to each other in both PCA plots (PC$_1$-PC$_2$ and PC$_2$-PC$_3$ space), indicating similarity between the CSPs during the initial time steps. The larger range covered by the moments for $(f_1,f_R)$ is better depicted in the PC$_1$-PC$_2$ space. The plot in the PC$_2$-PC$_3$ space provides good insight into the similar evolution pattern of the two tracks.
• Figure 3: Fibers and fiber surface. (a) The CSP of a bivariate field ($f_1$: electron density, $f_2$: reduced gradient) defined on an ethane-diol molecule. The black vertical line represents an isovalue for $f_1$, and the green horizontal line represents an isovalue for $f_2$. They intersect at the pink point ($1.5, -0.5$). (b) Corresponding isosurfaces. The green isosurface encloses regions corresponding to both covalent and non-covalent bonds and some individual atoms. The two isosurfaces intersect along the pink fiber. (c) The fiber surface (blue), corresponding to the blue control polygon selected in the CSP, highlights the covalent bonds.
• Figure 4: Visual analysis pipeline. The input consists of a time-varying bivariate field together with domain-specific information regarding potential regions of interest. The first step computes a segmentation of the domain, either using a topological or geometric approach, using the domain-specific information provided with the input. Next, the CSP and moment vectors are computed for all the segments for all time steps. In the following step, the 4D moment vectors are projected to the plane using standard dimensionality reduction techniques. Different projections and the tracks within each PCA plot are analyzed visually to identify interesting tracks. The selected tracks can be further analyzed at a finer level of granularity by studying the individual CSPs and performing comparative analysis. The blue arrow in the PCA plot highlights the intersection point of the selected tracks. The fine-grained analysis (insets) shows a similarity in the CSPs of segments at the intersection point. The fiber surfaces (red and green) in the vicinity of the corresponding atoms (indicated by the blue arrow) further confirm the observation.
• Figure 5: Image moments to describe atom's CSP (blue oval). $M_{00}$: captures the area, larger area implies larger activity. $M_{20}$ captures variation along horizontal hole_NTO axis or donor character, $M_{02}$ captures acceptor behavior, and a diagonal CSP signifies local movement of charge.