Statistical Exploration of Electronic Structure of Molecules from Quantum Monte-Carlo Simulations
In this report, we present results from analysis of Quantum Monte Carlo (QMC) simulation data with the goal of determining internal structure of a 3N-dimensional phase space of an N-electron molecule. We are interested in mining the simulation data for patterns that might be indicative of the bond rearrangement as molecules change electronic states. We examined simulation output that tracks the positions of two coupled electrons in the singlet and triplet states of an H2 molecule. The electrons trace out a trajectory, which was analyzed with a number of statistical techniques. This project was intended to address the following scientific questions: (1) Do high-dimensional phase spaces characterizing electronic structure of molecules tend to cluster in any natural way? Do we see a change in clustering patterns as we explore different electronic states of the same molecule? (2) Since it is hard to understand the high-dimensional space of trajectories, can we project these trajectories to a lower dimensional subspace to gain a better understanding of patterns? (3) Do trajectories inherently lie in a lower-dimensional manifold? Can we recover that manifold? After extensive statistical analysis, we are now in a better position to respond to these questions. (1) We definitely see clustering patterns, and differences between the H2 and H2tri datasets. These are revealed by the pamk method in a fairly reliable manner and can potentially be used to distinguish bonded and non-bonded systems and get insight into the nature of bonding. (2) Projecting to a lower dimensional subspace ({approx}4-5) using PCA or Kernel PCA reveals interesting patterns in the distribution of scalar values, which can be related to the existing descriptors of electronic structure of molecules. Also, these results can be immediately used to develop robust tools for analysis of noisy data obtained during QMC simulations (3) All dimensionality reduction and estimation techniques that we tried seem to indicate that one needs 4 or 5 components to account for most of the variance in the data, hence this 5D dataset does not necessarily lie on a well-defined, low dimensional manifold. In terms of specific clustering techniques, K-means was generally useful in exploring the dataset. The partition around medoids (pam) technique produced the most definitive results for our data showing distinctive patterns for both a sample of the complete data and time-series. The gap statistic with tibshirani criteria did not provide any distinction across the 2 dataset. The gap statistic w/DandF criteria, Model based clustering and hierarchical modeling simply failed to run on our datasets. Thankfully, the vanilla PCA technique was successful in handling our entire dataset. PCA revealed some interesting patterns for the scalar value distribution. Kernel PCA techniques (vanilladot, RBF, Polynomial) and MDS failed to run on the entire dataset, or even a significant fraction of the dataset, and we resorted to creating an explicit feature map followed by conventional PCA. Clustering using K-means and PAM in the new basis set seems to produce promising results. Understanding the new basis set in the scientific context of the problem is challenging, and we are currently working to further examine and interpret the results.