[Paper Review] The particle-hole map: a computational tool to visualize electronic excitations

Yonghui Li, Carsten A. Ullrich|arXiv (Cornell University)|Oct 14, 2015

Photochemistry and Electron Transfer Studies38 references3 citations

TL;DR

This paper introduces the particle-hole map (PHM), a nonlocal visualization tool that maps the origins and destinations of electron and hole movements during electronic excitations in molecules, using time-dependent density-functional theory (TDDFT). The PHM provides a statistically interpretable, physically consistent representation of charge-transfer processes, outperforming traditional tools like transition density matrices in clarity and interpretability for complex excitonic dynamics.

ABSTRACT

We introduce the particle-hole map (PHM), a visualization tool to analyze electronic excitations in molecules in the time or frequency domain, to be used in conjunction with time-dependent density-functional theory (TDDFT) or other ab initio methods. The purpose of the PHM is to give detailed insight into electronic excitation processes which is not obtainable from local visualization methods such as transition densities, density differences, or natural transition orbitals. The PHM is defined as a nonlocal function of two spatial variables and provides information about the origins, destinations, and connections of charge fluctuations during an excitation process; it is particularly valuable to analyze charge-transfer excitonic processes. In contrast with the transition density matrix, the PHM has a statistical interpretation involving joint probabilities of individual states and their transitions, it satisfies several sum rules and exact conditions, and it is easier to read and interpret. We discuss and illustrate the properties of the PHM and give several examples and applications to excitations in one-dimensional model systems, in a hydrogen chain, and in a benzothiadiazole based molecule.

Motivation & Objective

• To address the limitations of local visualization tools—such as transition densities and density differences—in capturing nonlocal charge-transfer and excitonic behavior during electronic excitations.
• To develop a nonlocal, physically interpretable alternative to the transition density matrix (TDM) that explicitly tracks electron and hole movement during excitation.
• To provide a computational framework that enables intuitive interpretation of complex excitonic processes, especially charge-transfer excitations in donor-acceptor systems.
• To establish a consistent link between time- and frequency-domain representations of electronic excitations via the PHM, ensuring agreement with fundamental sum rules.
• To enable practical application of the PHM in real-world systems, including molecules and periodic materials, using standard TDDFT codes with minimal modification.

Proposed method

• The PHM is defined as a nonlocal function of two spatial coordinates, r and r′, representing the joint probability of an electron being excited from orbital i at r and a hole being left at orbital j at r′.
• The frequency-domain PHM (PHM_ω) is derived from the TDDFT response in the frequency domain, using Kohn-Sham orbital densities and transition amplitudes.
• The time-domain PHM (PHM_t) is constructed from the time evolution of the Kohn-Sham orbitals under a weak, monochromatic external field, capturing real-time charge redistribution.
• Spatial partitioning schemes (e.g., binning on a grid or using atom-centered basis sets) are applied to make the PHM computationally feasible in 3D molecular systems.
• The PHM satisfies key physical sum rules, including normalization and dipole selection rules, ensuring consistency with fundamental quantum mechanical constraints.
• The method is implemented in the octopus code using grid-based representations, with extensions to atom-centered basis sets for molecular applications.

Experimental results

Research questions

• RQ1 How can electronic excitations, especially charge-transfer processes, be visualized with greater physical clarity than current local methods?
• RQ2 Can a nonlocal visualization tool be constructed that explicitly maps the origin and destination of charge fluctuations during excitation?
• RQ3 How does the particle-hole map compare to the transition density matrix in interpretability and physical consistency?
• RQ4 To what extent can the PHM reveal intramolecular charge-transfer character and donor-acceptor connectivity in complex molecules?
• RQ5 Can the PHM be consistently defined in both time and frequency domains, and do they yield equivalent physical insights?

Key findings

• The PHM successfully visualizes the origin and destination of charge transfer in a 1D model system with triple potential wells, clearly identifying donor and acceptor regions.
• In a linear H8 chain, the PHM reveals delocalized charge-transfer character across the chain, consistent with the expected behavior of excitonic states.
• For a benzothiadiazole-based donor-acceptor molecule, the PHM identifies the BT unit as the primary site of charge redistribution, highlighting its role as an acceptor.
• The PHM ω and PHM_t are found to be consistent with each other across all test systems, validating the time-frequency correspondence of the method.
• The PHM exhibits strong agreement with sum rules, including normalization and dipole selection rules, confirming its physical consistency.
• The method outperforms the TDM in interpretability, as it directly maps joint probabilities of electron and hole positions, offering a clearer picture of excitonic coherence.